Post on 30-Sep-2020
Mathematical and StatisticalModelling
of Infectious Diseases in Hospitals
Emma McBrydeMBBS (Honours)
University of QueenslandFRACP
A thesis submitted in partial fulfilment of the requirements for the degree ofDoctor of Philosophy
November 2006
Principal Supervisor: Professor Sean McElwainAssociate Supervisors: Professor Tony Pettitt and Dr Mike Whitby
Queensland University of TechnologySchool of Mathematical Sciences
Faculty of ScienceBrisbane, Queensland, 4001, AUSTRALIA
c© Copyright by Emma McBryde 2006All Rights Reserved
ii
I dedicate this work to Liam Eisen McBryde.
iv
Acknowledgements
A number of people have assisted me in producing this work. My principal
supervisor, Professor Sean McElwain, has encouraged me from our first en-
counter, has been generous with his time and provided inspiration, technical
assistance and practical advice. Professor Tony Pettitt has been an exacting
and thorough supervisor imparting expertise in statistical methods.
A owe a number of people and groups thanks for assistance with data gath-
ering. The CHRISP group at the Princess Alexandra Hospital, for providing
data, my associate supervisor Dr Mike Whitby and Dr David Cook, Director
of Intensive Care at the Princess Alexandra Hospital. The Taiyuan Centre of
Disease prevention and control and coauthors Ms Zhang and Mr Zhao for the
Shanxi SARS dataset.
Thank you also to those who gave me practical assistance and support dur-
ing my time as a PhD candidate. In particular I would like to thank Jenny
Eisen my mother-in-law for devotedly looking after Liam and giving me an
opportunity to work.
Funding sources for this thesis include Australian Research Council Linkage
Scheme (LP347112), Australian Postgraduate Award and National Health
and Medical Research Council scholarship number 290541. Additionally the
Princess Alexandra Hospital provided me with a research scholarship for the
year 2003.
Thank you also to the Australian Mathematical Sciences Institute (AMSI)
for providing me with a room and equipment while I was collaborating
and working externally from Melbourne. The Royal Brisbane and Womens’
Hospital also assisted me by allowing dedicated research time within my
appointment as staff specialist from June 2004 to April 2005.
v
vi
Abstract
Antibiotic resistant pathogens, such as methicillin-resistant Staphylococcus
aureus (MRSA), and vancomycin-resistant enterococci (VRE), are an increas-
ing burden on healthcare systems. Hospital acquired infections with these
organisms leads to higher morbidity and mortality compared with the sensi-
tive strains of the same species and both VRE and MRSA are on the rise world-
wide including in Australian hospitals. Emerging community infectious dis-
eases are also having an impact on hospitals. The Severe Acute Respiratory
Syndrome virus (SARS Co-V) was noted for its propensity to spread through-
out hospitals, and was contained largely through social distancing interven-
tions including hospital isolation. A detailed understanding of the transmis-
sion of these and other emerging pathogens is crucial for their containment.
The statistical inference and mathematical models used in this thesis aim
to improve understanding of pathogen transmission by estimating the
transmission rates of contagions and predicting the impact of interventions.
Datasets used for these studies come from the Princess Alexandra Hospital
in Brisbane, Australia and Shanxi province, mainland China.
Epidemiological data on infection outbreaks are challenging to analyse
due to the censored nature of infection transmission events. Most datasets
record the time on symptom onset, but the transmission time is not ob-
servable. There are many ways of managing censored data, in this study
we use Bayesian inference, with transmission times incorporated into the
augmented dataset as latent variables. Hospital infection surveillance data
is often much less detailed that data collected for epidemiological studies,
often consisting of serial incidence or prevalence of patient colonisation
with a resistant pathogen without individual patient event histories. Despite
the lack of detailed data, transmission characteristics can be inferred from
such a dataset using structured Hidden Markov Models (HMMs).
Each new transmission in an epidemic increases the infection pressure on
vii
those remaining susceptible, hence infection outbreak data are serially de-
pendent. Statistical methods that assume independence of infection events
are misleading and prone to over-estimating the impact of infection control
interventions. Structured mathematical models that include transmission
pressure are essential. Mathematical models can also give insights into
the potential impact of interventions. The complex interaction of different
infection control strategies, and their likely impact on transmission can be
predicted using mathematical models.
This dissertation uses modified or novel mathematical models that are
specific to the pathogen and dataset being analysed. The first study es-
timates MRSA transmission in an Intensive Care Unit, using a structured
four compartment model, Bayesian inference and a piecewise hazard meth-
ods. The model predicts the impact of interventions, such as changes to
staff/patient ratios, ward size and decolonisation. A comparison of results of
the stochastic and deterministic model is made and reason for differences
given. The second study constructs a Hidden Markov Model to describe
longitudinal data on weekly VRE prevalence. Transmission is assumed to be
either from patient to patient cross-transmission or sporadic (independent
of cross-transmission) and parameters for each mode of acquisition are
estimated from the data. The third study develops a new model with a
compartment representing an environmental reservoir. Parameters for
the model are gathered from literature sources and the implications of the
environmental reservoir are explored. The fourth study uses a modified
Susceptible-Exposed-Infectious-Removed (SEIR) model to analyse data from
a SARS outbreak in Shanxi province, China. Infectivity is determined before
and after interventions as well as separately for hospitalised and community
symptomatic SARS cases. Model diagnostics including sensitivity analysis,
model comparison and bootstrapping are implemented.
viii
Keywords
Bayesian inference
epidemic modelling
environmental reservoir
hidden Markov models
infectious diseases
mathematical modelling
methicillin resistant Staphylococcus aureus (MRSA)
severe acute respiratory syndrome (SARS)
statistical modelling
stochastic processes
vancomycin resistant enterococci (VRE)
epidemiology
public health
infectious disease
ix
x
List of Publications and Manuscripts
Presented in this Thesis
McBryde, E. S., Pettitt, A.N., McElwain, D. L.S., 2006c. A stochastic
mathematical model of methicillin resistant Staphylococcus aureus trans-
mission in an intensive care unit: Predicting the impact of interven-
tions. The Journal of Theoretical Biology, e-published November 2006
http://dx.doi.org/10.1016/j.jtbi.2006.11.008. (Chapter 3)
McBryde, E. S., Bradley, L. C., Whitby, M., McElwain, D. L.S., 2004. An inves-
tigation of contact transmission of methicillin-resistant Staphylococcus au-
reus. The Journal of Hospital Infection 58 (2), 104-8. (Chapter 4)
McBryde, E. S., Pettitt, A.N., Cooper, B.S., McElwain, D. L.S., 2006b. Char-
acterising outbreaks of vancomycin-resistant enterococci using a statistical
method. Submitted to Public Library of Science-Medicine, September 2006.
(Chapter 5)
McBryde, E. S., McElwain, D. L.S., 2006. A mathematical model investigat-
ing the impact of an environmental reservoir on the prevalence and control
of vancomycin-resistant enterococci. The Journal of Infectious Diseases 193
(10), 1473-4. (Chapter 6)
McBryde, E., Gibson, G., Pettitt, A.N., Zang, Y., Zhao, B., McElwain, D. L.S.,
2006a. Bayesian Modelling of an epidemic of Severe Acute Respiratory
Syndrome. The Bulletin of Mathematical Biology 68(4), 889-917. DOI:
10.1007/s11538-005-9005-4. (Chapter 7)
xi
xii
Additional Publications & Manuscripts
involving Quantitative studies of
Infectious Diseases by the Candidate
during Ph.D. Candidature
McBryde, E.S., 2003. Severe Acute Respiratory Syndrome. Australian and
New Zealand Journal of Medicine 33, 10suppl(A70).
McBryde, E.S., 2004. Using mathematical models to investigate transmission
of methicillin-resistant Staphylococcus aureus (MRSA) in the healthcare set-
ting. Internal Medicine Journal 34, 9-10suppl (A68).
McBryde, E.S., 2004. An investigation of contact transmission of methicillin-
resistant Staphylococcus aureus (MRSA). Internal Medicine Journal 34,
9-10suppl (A71).
McBryde, E.S., 2004. Severe Acute Respiratory Syndrome: Predicting the epi-
demiology using mathematical modelling and data analysis techniques. In-
ternal Medicine Journal 34, 11suppl(A80).
McBryde, E.S., Tilse, M., McCormack, J., 2005. Comparison of contamination
rates of catheter-drawn and peripheral blood cultures. Journal of Hospital In-
fection 60, 118-121.
xiii
xiv
Contents
Acknowledgements v
Abstract vii
Keywords ix
Publications xi
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Burden of antibiotic resistant bacteria . . . . . . . . . . 2
1.1.2 The role of mathematical and statistical modelling in
infection control research . . . . . . . . . . . . . . . . . 3
1.2 Overall objectives of the thesis . . . . . . . . . . . . . . . . . . . 5
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Literature review and outline of thesis 13
2.1 Review of pathogens discussed in this thesis . . . . . . . . . . . 13
2.1.1 Methicillin resistant Staphylococcus aureus . . . . . . . 13
2.1.2 Vancomycin-resistant enterococci . . . . . . . . . . . . 16
2.1.3 Environmental pathogens . . . . . . . . . . . . . . . . . 18
2.1.4 Severe Acute Respiratory Syndrome Coronavirus . . . . 19
2.2 Mathematical models of human infectious diseases . . . . . . 20
2.2.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 The Susceptible-Infectious-Removed model . . . . . . 21
2.2.3 The Susceptible-Exposed-Infectious-Removed model . 23
2.2.4 The basic reproduction ratio, R0 . . . . . . . . . . . . . 24
2.2.5 Adaptation of the Ross-MacDonald model to the
healthcare setting . . . . . . . . . . . . . . . . . . . . . . 25
2.2.6 Single population models . . . . . . . . . . . . . . . . . 27
xv
2.2.7 Stochastic models . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Relationship of current literature to thesis . . . . . . . . . . . . 30
2.3.1 Studies based on the Ross-MacDonald model . . . . . . 30
2.3.2 Stochastic epidemic models based on the Susceptible-
Infectious model with migration . . . . . . . . . . . . . 35
2.3.3 Environmental models of transmission . . . . . . . . . 36
2.3.4 Epidemic models of Severe Acute Respiratory Syndrome 36
2.3.5 Other important models of transmission of nosocomial
pathogens . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Review of methodology used in stochastic epidemic modelling 42
2.4.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . 43
2.4.2 Methods used to manage serial dependence in infec-
tion data . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.3 Methods used to manage censored transmission data . 47
2.4.4 Hidden Markov models . . . . . . . . . . . . . . . . . . . 51
2.4.5 Assessing convergence of MCMC . . . . . . . . . . . . . 52
2.4.6 Model checking and improvement . . . . . . . . . . . . 53
2.4.7 Model selection and comparison . . . . . . . . . . . . . 54
2.5 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Mathematical model of MRSA 73
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.1 Patients and Setting . . . . . . . . . . . . . . . . . . . . . 79
3.3.2 Surveillance of colonisation . . . . . . . . . . . . . . . . 79
3.3.3 Parameter estimates . . . . . . . . . . . . . . . . . . . . 79
3.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Formula for daily hazard of MRSA cross-transmission . 80
3.4.2 Bayesian inference to estimate φ . . . . . . . . . . . . . 82
3.4.3 Estimates of the attack rate and the ward reproduction
ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.4.4 Model for the impact of interventions . . . . . . . . . . 83
3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.1 Estimate ward transmission: attack rate and the ward
reproduction ratio . . . . . . . . . . . . . . . . . . . . . 86
3.5.2 Predicted impact of interventions . . . . . . . . . . . . . 86
xvi
3.5.3 Model adequacy and sensitivity . . . . . . . . . . . . . . 87
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.A Bayesian estimation of the transmission parameter . . . . . . 94
3.A.1 Likelihood of the complete dataset . . . . . . . . . . . . 95
3.A.2 Gibbs update for the transmission parameter, φ . . . . 96
3.A.3 Latent variable imputation . . . . . . . . . . . . . . . . . 96
3.A.4 Incorporating uncertainty of model parameters . . . . 97
3.A.5 Markov chain Monte Carlo algorithm to estimate the
transmission parameter, φ . . . . . . . . . . . . . . . . . 98
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4 Contact transmission of MRSA 105
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.1 Hand sampling . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.2 Laboratory technique . . . . . . . . . . . . . . . . . . . . 108
4.2.3 Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.3.1 Detection of MRSA . . . . . . . . . . . . . . . . . . . . . 109
4.3.2 Pre-handwash sample . . . . . . . . . . . . . . . . . . . 110
4.3.3 Post-handwash sample . . . . . . . . . . . . . . . . . . . 110
4.3.4 Compliance with infection control procedures . . . . . 110
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
4.5 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5 Characterising outbreaks of VRE using statistical methods 117
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.1 Description of outbreak and infection control interven-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2.2 Serial surveillance data used for statistical analysis . . . 121
5.2.3 Data used for cluster analysis . . . . . . . . . . . . . . . 121
5.2.4 Model of transmission . . . . . . . . . . . . . . . . . . . 122
5.2.5 Hidden Markov model . . . . . . . . . . . . . . . . . . . 124
5.2.6 Constructing a transition probability matrix . . . . . . . 125
5.2.7 Observation Model . . . . . . . . . . . . . . . . . . . . . 126
xvii
5.2.8 Bayesian framework . . . . . . . . . . . . . . . . . . . . 127
5.2.9 Comparison of cluster analysis results using genotyping
with statistical analysis . . . . . . . . . . . . . . . . . . . 128
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
5.3.1 Parameter estimation . . . . . . . . . . . . . . . . . . . . 128
5.3.2 Comparison of statistical model and genotyping data . 129
5.3.3 Model selection and validation . . . . . . . . . . . . . . 129
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Appendix to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.A Likelihood computation . . . . . . . . . . . . . . . . . . . . . . 133
5.B Monte Carlo Markov chain algorithm . . . . . . . . . . . . . . . 134
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
6 Environmental reservoir model for VRE 143
6.A Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.B Elaboration of Environment Model . . . . . . . . . . . . . . . . 146
6.B.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.B.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
6.B.3 Further discussion . . . . . . . . . . . . . . . . . . . . . 150
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
7 Bayesian modelling of an epidemic of SARS 155
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model . . . . 159
7.3 SARS Data from Shanxi Province . . . . . . . . . . . . . . . . . 162
7.4 Challenges and specific aims of the study . . . . . . . . . . . . 166
7.5 Estimation of time to transmission and incubation period . . . 166
7.5.1 Bayesian approach to estimating incubation period . . 167
7.5.2 Results: Time to transmission and incubation period . 168
7.5.3 Discussion: Time to transmission and incubation period 170
7.6 Estimation of other transition periods . . . . . . . . . . . . . . 171
7.6.1 Results: Estimation of other transition periods . . . . . 172
7.7 Model for estimating coefficients of infectivity . . . . . . . . . 172
7.7.1 Bayesian approach to estimation of the transmission
coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7.7.2 Prior specification . . . . . . . . . . . . . . . . . . . . . . 174
7.7.3 Likelihood estimation . . . . . . . . . . . . . . . . . . . 175
7.7.4 Results: Change point Estimation . . . . . . . . . . . . . 176
7.7.5 Results: Coefficients of Infectivity . . . . . . . . . . . . . 176
xviii
7.7.6 Results: Reproduction ratio . . . . . . . . . . . . . . . . 178
7.8 Individual Infectivity profiles . . . . . . . . . . . . . . . . . . . 179
7.9 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . 180
Appendix to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.A Gantt chart of Shanxi epidemic . . . . . . . . . . . . . . . . . . 182
7.B Computations for time to transmission and incubation period 183
7.C Diagnostics: Convergence and Sensitivity analysis . . . . . . . 184
7.C.1 Sensitivity of estimate of incubation period to model
choice and hazard of transmission parameter . . . . . . 185
7.D Estimated values of shape and scale parameters for the
Gamma distributions . . . . . . . . . . . . . . . . . . . . . . . . 187
7.E Statistical inference used to estimate infectivity and change
points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.E.1 Augmented data . . . . . . . . . . . . . . . . . . . . . . . 187
7.E.2 Computations to determine posterior distributions of
the coefficients of infectivity and the change point . . . 188
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8 Conclusions and suggestions for future work 195
8.1 What has been achieved? . . . . . . . . . . . . . . . . . . . . . . 196
8.1.1 Estimation of basic reproduction ratio and cross-
transmission rates . . . . . . . . . . . . . . . . . . . . . . 196
8.1.2 Development of new models . . . . . . . . . . . . . . . 196
8.1.3 Using of models to inform health policy . . . . . . . . . 197
8.1.4 Methodological framework for future studies . . . . . . 197
8.1.5 Model comparison . . . . . . . . . . . . . . . . . . . . . 198
8.1.6 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . 198
8.2 Limitations and opportunities for extensions . . . . . . . . . . 198
8.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
xix
xx
List of Figures
2.1 The application of the Ross-MacDonald model to the trans-
mission of nosocomial pathogens . . . . . . . . . . . . . . . . . 26
2.2 Single population nosocomial transmission model . . . . . . . 28
2.3 Predicted linear relationship between number of patients
colonised and number of healthcare workers colonised. . . . . 28
3.1 Four compartment model of nosocomial pathogen transmission 78
3.2 Data collected over period of study . . . . . . . . . . . . . . . . 85
3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.4 Effect of changing parameters on attack rate . . . . . . . . . . . 90
3.5 Effect of cohorting on attack rate . . . . . . . . . . . . . . . . . 90
3.6 Effect of ward size on attack rate . . . . . . . . . . . . . . . . . . 91
4.1 Flow diagram of study participants and results of MRSA testing. 109
4.2 Boxplot of time taken to wash hands, based on type of health-
care worker. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 Prevalence data for VRE over 157 weeks . . . . . . . . . . . . . 122
5.2 The transmission of bacterial pathogens in the hospital ward. 123
5.3 Hidden Markov model . . . . . . . . . . . . . . . . . . . . . . . 124
5.4 Posterior distribution of proportion of VRE acquisitions that
are due to ward transmission . . . . . . . . . . . . . . . . . . . 130
6.1 Comparison of environmental and standard model predictions. 146
6.2 Environmental model of VRE transmission in the hospital setting147
7.1 The schematic of the SEIR model. . . . . . . . . . . . . . . . . . 159
7.2 The schematic of the extended SEIHRD model . . . . . . . . . 160
7.3 Histogram of daily admissions to hospital. . . . . . . . . . . . . 162
7.4 Histogram of time from first exposure to another SARS case to
symptom onset. . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7.5 Recorded time interval from symptom onset to hospitalisation. 164
xxi
7.6 Recorded time interval from symptom onset to recovery. . . . 164
7.7 Recorded time interval from symptom onset to death. . . . . . 165
7.8 Posterior distribution for the hazard of transmission, λ. . . . . 169
7.9 Estimated distribution of the incubation period . . . . . . . . . 169
7.10 Estimated best fit Gamma distribution for time from symptom
onset to hospitalisation . . . . . . . . . . . . . . . . . . . . . . . 172
7.11 Estimated best fit Gamma distribution for time from symptom
onset to recovery . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.12 Estimated best fit Gamma distribution for time from symptom
onset to death . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.13 Posterior distribution for change-point. . . . . . . . . . . . . . 176
7.14 Posterior distribution for the reproduction ratios prior to and
after March 29, 2003 . . . . . . . . . . . . . . . . . . . . . . . . . 179
7.15 Infectivity profile versus time since symptom onset. . . . . . . 180
7.16 Gantt chart of epidemic . . . . . . . . . . . . . . . . . . . . . . . 183
7.17 Convergence of a series of Markov chains . . . . . . . . . . . . 185
7.18 Sensitivity of the incubation period estimate to model assump-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
xxii
List of Tables
2.1 Comparison of studies that applied the Ross-MacDonald
model to nosocomial transmission of hospital pathogens . . . 32
2.2 Comparison of studies that estimated infectivity of Severe
Acute Respiratory Syndrome . . . . . . . . . . . . . . . . . . . . 38
3.1 Parameters used in the model for MRSA transmission. . . . . 88
4.1 Compliance with glove use amongst different healthcare
worker groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 Parameters used in the model . . . . . . . . . . . . . . . . . . . 125
5.2 Comparison of different models using the Deviance Informa-
tion Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.1 Model assumptions and their justifications. . . . . . . . . . . . 151
6.2 Table of parameters, their symbols and default values . . . . . 152
7.1 The posterior mean and standard deviation (in days) of the
times to hospitalisation, hospital discharge and death. . . . . . 174
7.2 Table of transmission coefficients . . . . . . . . . . . . . . . . . 177
7.3 The estimated mean and standard deviation (in days) of the
incubation period comparing the estimates using the assump-
tion of constant hazard, used by the current study, and the as-
sumptions of uniform probability and immediate transmission. 186
7.4 Estimated values for the parameters of the Gamma distribu-
tions for sojourn times . . . . . . . . . . . . . . . . . . . . . . . 187
xxiii
xxiv
Statement of Original Authorship
The work contained in this thesis has not been previously submitted
for a degree or diploma at any other higher education institution. To
the best of my knowledge and belief, the thesis contains no material
previously published or written by another person except where due
reference is made.
Signature:Emma McBryde
Date:
xxv
CHAPTER 1
Introduction
1.1 Motivation
Many infectious diseases owe their existence to the hospital environment.
Patients are vulnerable to infection for a number of reasons: intravenous
catheters and other devices breach fundamental barriers, illnesses and
medication can lead to impairment of both the innate and acquired immune
system and antibiotic exposure alters normal host flora. Healthcare workers
have been recognised as the agents of transmission of bacteria since the
treatise of Semmelweiss (1861).
Hospital acquired infections are serious, causing expense, morbidity
and mortality. The incidence of infections caused by antibiotic resistant
pathogens is increasing and often leads to greater morbidity and mortality
and hospital costs than their antibiotic-sensitive counterparts. Infection
control recommendations, aimed at reducing the burden of antibiotic resis-
tant pathogens, are rarely based on high level evidence such as randomised
controlled trials (Cooper et al., 2003).
In the current political environment, hospitals are strategic sites of contain-
ment of community epidemics. In planning policy for bed utilisation, iso-
lation facilities and quarantine, we can learn from past events including the
role of the hospital in the Severe Acute Respiratory Syndrome pandemic. This
viral agent (SARS Co-V) emerged in the community. Nevertheless, hospitals
became sites of intensified spreading in several countries and effective con-
tainment in others.
This thesis analyses healthcare-associated infections. Firstly, attention
is turned to antibiotic-resistant bacteria, arising from populations and
pressures that are peculiar to hospitals. The aim is to estimate the trans-
mission characteristics of methicillin-resistant Staphylococcus aureus and
2 Chapter 1. Introduction
vancomycin-resistant enterococci. Pathogen characteristics, available data
and questions being addressed determine the choice of model and method-
ology. Secondly, the thesis considers the role of the hospital in reflecting,
intensifying and potentially preventing community infection, using SARS as
an example.
1.1.1 Burden of antibiotic resistant bacteria
Staphylococcus aureus is a major cause of healthcare-associated bacteraemia
and post-operative wound infection (Mandell et al., 2005). This pathogen
has achieved resistance to successive antibiotics, such as aminoglycosides,
macrolides and tetracyclines following their introduction through the 1950s.
Methicillin-resistant S. aureus (MRSA) emerged in Europe in the early 1960s,
soon after methicillin was first used (Ericksen and Erichsen, 1963), has con-
tinued to increase in prevalence and is now found throughout the world. In-
fection with MRSA is associated with higher morbidity, mortality (Engemann
et al., 2003) and expense (Capitano et al., 2003) compared with methicillin-
sensitive S. aureus (MSSA). The rise of MRSA appears to be additive to the
underlying rate of MSSA (Cooper et al., 2004a).
Most hospital-acquired MRSA (HA-MRSA) isolates are also resistant to
other classes of antibiotics. During the 1990s, some MRSA isolates showed
reduced susceptibility to the “last line” antibiotics, linezolid (Tsiodras et al.,
2001; Meka et al., 2004) and vancomycin (Hiramatsu et al., 1997; MMWR,
2002; Chang et al., 2003). MRSA isolates with reduced susceptibility to
vancomycin have now been described in Australia (Gosbell et al., 2003).
This requires a rethink in the strategies for control of MRSA, with renewed
emphasis on prevention of cross-transmission of the pathogen whether
causing asymptomatic colonisation or infection.
Enterococci are part of the normal gastrointestinal flora; harmless colonis-
ers of healthy people. In contrast, hospitalised patients are at high risk of
invasive enterococcal infection, especially those with compromised immune
systems and breached integument. Enterococci are intrinsically resistant to a
number of antibiotics (Weinstein, 2005). Many strains are resistant to ampi-
cillin, leading to the reliance upon vancomycin as a commonly used effec-
tive bactericidal agent against enterococci for many decades. Vancomycin-
resistant enterococci (VRE) emerged in the late 1980s, long after the intro-
duction of vancomycin. Since its emergence, VRE has spread rapidly from
Europe to the United States. It was first identified in Australia in 1994 and
1.1 Motivation 3
its incidence is increasing (Nimmo et al., 2003). It is now regarded as one of
the most important nosocomial pathogens (Murray, 2005). Infection with en-
terococci carrying one of the vancomycin resistant genes is associated with
higher mortality (Lodise et al., 2002); in one study attributable mortality was
as high as 30% (Edmond et al., 1996). Strains of enterococci have been found
with resistance to all conventional antibiotics, including linezolid (Gonzales
et al., 2001). Of great concern is that the vanA resistance gene has been shown
to be able to cross genus into the methicillin-resistant S. aureus (Chang et al.,
2003; Tenover et al., 2004). In view of the failure of antibiotic development
to keep pace with VRE’s acquisition of resistance, VRE control strategies (as
with MRSA) must necessarily focus on containment and prevention.
Other antibiotic resistant pathogens are emerging and spreading within
and between hospitals. These include antibiotic resistant Gram-negative
pathogens such as Pseudomonas spp. and Acinetobacter spp. (Pimentel JD,
2005). These pathogens are not discussed further in this thesis; however
methods and models developed in this thesis could be applied to such
organisms.
1.1.2 The role of mathematical and statistical modelling in
infection control research
Hospital infection control is a relatively recent discipline. Recommended
infection control interventions include surveillance and patient isolation,
antibiotic restriction, ward cleaning, personal protective equipment for staff
and hand hygiene (CDC Guidelines, 1995). Infection control practices are
expensive and a burden for both patients and healthcare workers (Ridwan
et al., 2002). It is therefore essential that the efficacy of these be established.
Infection control strategies in place today frequently are not supported by
rigorous scientific evidence, with some exceptions (Cooper et al., 2004b).
A study by Pittet et al. (2000) showed improvements in hand hygiene was
accompanied by a reduction in MRSA “attack rate”1. Cepeda et al. (2005)
found that there was no reduction in MRSA transmission when patient
isolation was practised. It must be taken into account that in this study,
surveillance was conducted only weekly, a time interval larger than the
median length of stay in the unit and that laboratory turn around time was 4
days. Therefore, the proportion of time that each colonised patient spent in
1Attack rate was defined in this study as the number of transmissions of MRSA per 10 000uncolonised patient days.
4 Chapter 1. Introduction
isolation would be well under 50% which would dilute the effect of isolation
considerably. Nevertheless, the lag-time experienced in the study is likely
to be a realistic reflection of isolation practices outside of study conditions.
Grundmann et al. (2002) found that low staff/patient ratio was the only risk
factor for colonisation with MRSA, allowing for colonisation pressure in a
multivariate risk analysis. De la Cal et al. (2004) found that patients could
be decolonised through a combined approach of enteral vancomycin and
nasal mupiricin. This approach was criticised because of concerns about
increasing vancomycin resistance in enterococci and potentially MRSA as
well as methodological flaws (Daschner, 2005; Humphreys and Smyth, 2005).
The belief that antibiotic exposure predisposes patients to antibiotic-
resistant pathogen acquisition, while compelling, is not based on ran-
domised control trials. Antibiotic exposure has been found to be a risk
factor in a number of case control studies for VRE and MRSA (Rao, 1998;
Weinstein, 2005). Such studies are frequently subject to confounders and
rarely consider length of hospital stay, co-morbidities and colonisation
pressure. Antibiotics have been shown to increase the stool density and
environmental contamination of VRE (Donskey et al., 2000), suggesting that
antibiotic restriction aimed at colonised patients may be more effective than
those aimed at uncolonised patients. Prevalence of Healthcare Associated
(HA-) MRSA is lower in countries with strict antibiotic policies such as the
Netherlands and Scandinavian countries (Voss et al., 1994).
Many challenges face the infection control investigator. Logistical difficulties
and ethical considerations limit the application of randomised controlled
trials. The data available on hospital acquired infection or colonisation
are often time series data. Frequently this entails a retrospective, quasi-
experimental analysis of an outbreak, after multiple interventions have
taken place (Cooper et al., 2004b).
Interpretation of studies involving planned interrupted time series inter-
ventions requires caution. Data on any contagious disease by its nature is
not serially independent. Each infection or colonisation leads to greater
infection pressure on remaining susceptible individuals. This serial de-
pendence leads to both autocorrelation and overdispersion of colonisation
events. Analyses of serial colonisation data that use standard statistical
techniques, such as Cox proportional hazards survival analysis, applied
to serial infection data, may thus be misleading and are likely to result in
erroneous conclusions (Cooper et al., 2003).
1.2 Overall objectives of the thesis 5
Cluster-randomised control trials would be one way of achieving appropriate
comparison groups in the area of hospital infection control and some are cur-
rently underway or recently published (Cepeda et al., 2005). However, these
are expensive and lengthy and may not be sufficient to determine the opti-
mal infection control strategy. The number of events required to substantiate
an effect (such as a different isolation strategy) is larger in studies in which
the events are serially dependent and the outcomes are over-dispersed com-
pared with studies which can correctly assume serial independence. Such
studies also fail to capture dynamic interactions that contribute to the trans-
mission of bacteria in the healthcare setting (D’Agata et al., 2005), including
the likely impact of combined infection control strategies and the different
equilibria (steady state of proportion of patients colonised) that are possible
following infection control interventions, as found in a theoretical model by
Cooper et al. (2004a).
An additional challenge for the infection control investigator arises because
transmission events such as timing and chains of transmission, are not
wholly observable (Becker, 1989). Colonisation is asymptomatic and acts
as a carrier state for transmission antibiotic-resistant bacterial pathogens.
Serial infection data fail to capture the colonisation that underlies the small
number of patients who manifest disease. Statistical models ideally take
into account the interval-censored nature of hospital transmission data, the
potential for unobserved infectious cases.
Mathematical models can be useful in the area of hospital infection control
for two reasons. Firstly, they can be used to predict quantitatively the course
of an epidemic, predicting its total size, peak; and time to peak and the im-
pact of infection control interventions including nonlinear interactions that
occur when multiple interventions are undertaken. Secondly, they can in-
form the design of trials and structure statistical analyses to avoid assump-
tions of serial independence and difficulties with interval censoring and un-
known numbers of infectious cases.
1.2 Overall objectives of the thesis
The overall objectives of this thesis are to develop mathematical and statis-
tical models in order to improve the understanding of the transmission of
infectious agents in the hospital and to use these models to inform infection
control practitioners as to the likely impact of interventions.
6 Chapter 1. Introduction
The first part of the analyses in this thesis is the development of plausible
models based on biological and epidemiological knowledge of each or-
ganism with full recognition of model assumptions and limitations. The
second part is the analysis of datasets to determine parameters that govern
the underlying epidemic model. Statistical inference techniques used in
this thesis were mostly Bayesian, including the use of latent variables to
represent unobserved or missing data, with computation by Markov chain
Monte-Carlo techniques including the Metropolis-Hastings algorithm.
Expectation-Maximisation (EM) algorithms and piecewise constant hazard
models are also used throughout the thesis as are standard model selection
and model checking techniques. These are elaborated in Chapter 2. The
third part of the analysis is the prediction of the impact of infection control
procedures. By changing model parameters and assessing large scale behav-
iour of the model, we can make reasonable estimates of the likely outcome
of infection control interventions.
The strategy in each study is to a develop pathogen-specific mathematical
model and use this as the foundation of a statistical model to probe unpub-
lished datasets and quantify transmission. In two of the studies, described
in Chapters 5 and 7, infection control interventions occurred during the data
collection period, allowing estimates to be made of the impact of these in-
terventions on pathogen transmission. The methodology aims to account
for both serial dependence in infection data and the censored nature of the
transmission events. In Chapter 5, the thesis extends the notion of censored
data to develop a model and methodology that could make use of simple ser-
ial surveillance data in the absence of patient event data or perfect detection
of colonisation, using a hidden Markov model structure.
A major objective of this thesis, as in many studies in the area, is to estimate
the basic reproduction ratio (defined in Section 2.2.4). Estimates are not
confined to this measure however. In the study described in Chapter 3,
the transmission rate was low and the reproduction ratio was below unity.
Despite this, infection control interventions were shown to reduce trans-
mission, which could be quantified using the attack rate. Characterising
the source of infection (endemic or epidemic) using statistical methods was
an important outcome of the work described in Chapter 5. Estimation of
differences in infectivity following interventions (in Chapters 5 and 7) or
depending on the hospitalisation status of the individual (in Chapter 7) was
also undertaken.
1.2 Overall objectives of the thesis 7
One of the objectives of this thesis is to consider infection control interven-
tions that have not previously been considered by mathematical modellers.
Local policies and practices, for example local cohorting policy, HCW/patient
contact rates and ward size, were specifically considered in models devel-
oped. Recent research findings, for example the use of enteral vancomycin
to reduce MRSA colonisation as described by de la Cal et al. (2004), were in-
corporated into model predictions.
Both deterministic and stochastic models are considered in this thesis. The
results are compared and the reasons for any discrepancies are discussed.
Stochasticity is an important consideration given that the scale of the popu-
lations was small.
Where more than one model is plausible, several different models are com-
pared and the model with the best balance between parsimony and data fit,
as defined by model comparison criteria, is selected. Finally, this thesis aims
to validate, where possible, the models developed. This is achieved using ex-
ternal validation sources if available and internal validation to show that the
model techniques estimated the parameters without bias and with adequate
precision.
8 Chapter 1. Introduction
Bibliography
Becker, N., 1989. Analysis of Infectious Diseases Data. Chapman and Hall/CRC.
Capitano, B., Leshem, O. A., Nightingale, C. H., Nicolau, D. P., 2003. Cost effect ofmanaging methicillin-resistant Staphylococcus aureus in a long-term care facility.J Am Geriatr Soc 51 (1), 10–16.
CDC Guidelines, 1995. Recommendations for preventing the spread of vancomycinresistance. Hospital Infection Control Practices Advisory Committee (HICPAC).Infect Control Hosp Epidemiol 16 (2), 105–13.
Cepeda, J. A., Whitehouse, T., Cooper, B., Hails, J., Jones, K., Kwaku, F., Taylor, L., Hay-man, S., Cookson, B., Shaw, S., Kibbler, C., Singer, M., Bellingan, G., Wilson, A. P.,2005. Isolation of patients in single rooms or cohorts to reduce spread of MRSA inintensive-care units: prospective two-centre study. Lancet 365 (9456), 295–304.
Chang, S., Sievert, D. M., Hageman, J. C., Boulton, M. L., Tenover, F. C., Downes,F. P., Shah, S., Rudrik, J. T., Pupp, G. R., Brown, W. J., Cardo, D., Fridkin, S. K., theVancomycin-Resistant Staphylococcus aureus Investigative Team, 2003. Infectionwith vancomycin-resistant Staphylococcus aureus containing the vanA resistancegene. N Engl J Med 348 (14), 1342–1347.
Cooper, B. S., Medley, G. F., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A.,Duckworth, G., Lai, R., Ebrahim, S., 2004a. Methicillin-resistant Staphylococcusaureus in hospitals and the community: stealth dynamics and control catastro-phes. Proc Natl Acad Sci U S A 101 (27), 10223–8.
Cooper, B. S., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A., Medley, G. F.,Duckworth, G., Lai, R., Ebrahim, S., 2004b. Isolation measures in the hospital man-agement of methicillin resistant Staphylococcus aureus (MRSA): systematic reviewof the literature. Brit Med J 329 (7465), 533.
Cooper, B. S., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A., Medley, G. F.,Duckworth, G. J., Lai, R., Ebrahim, S., 2003. Systematic review of isolation poli-cies in the hospital management of methicillin-resistant Staphylococcus aureus:a review of the literature with epidemiological and economic modelling. HealthTechnol Assess 7 (39), 1–194.
D’Agata, E. M., Webb, G., Horn, M., 2005. A mathematical model quantifying the im-pact of antibiotic exposure and other interventions on the endemic prevalence ofvancomycin-resistant enterococci. J Infect Dis 192 (11), 2004–11.
Daschner, F., Mar 2005. MRSA-really time for a pragmatic approach. J Hosp Infect 59,259.
10 BIBLIOGRAPHY
de la Cal, M. A., Cerda, E., van Saene, H. K., Garcia-Hierro, P., Negro, E., Parra, M. L.,Arias, S., Ballesteros, D., 2004. Effectiveness and safety of enteral vancomycinto control endemicity of methicillin-resistant Staphylococcus aureus in a med-ical/surgical intensive care unit. J Hosp Infect 56 (3), 175–83.
Donskey, C., Chowdhry, T., Hecker, M., Hoyen, C., Hanrahan, J., Hujer, A., Hutton-Thomas, R., Whalen, C., Bonomo, R., Rice, L., 2000. Effect of antibiotic therapy onthe density of vancomycin-resistant enterococci in the stool of colonized patients.New Eng J Med 343 (I), 1925–1932.
Edmond, M. B., Ober, J. F., Dawson, J. D., Weinbaum, D. L., Wenzel, R. P., 1996.Vancomycin-resistant enterococcal bacteremia: natural history and attributablemortality. Clin Infect Dis 23 (6), 1234–9.
Engemann, J. J., Carmeli, Y., Cosgrove, S. E., Fowler, V. G., Bronstein, M. Z., Trivette,S. L., Briggs, J. P., Sexton, D. J., Kaye, K. S., 2003. Adverse clinical and economic out-comes attributable to methicillin resistance among patients with Staphylococcusaureus surgical site infection. Clin Infect Dis 36 (5), 592–8.
Ericksen, K., Erichsen, I., 1963. Clinical occurrence of methicillin-resistant strains ofStaphylococcus aureus. Ugeskr Laeger 125, 1234–40.
Gonzales, R. D., Schreckenberger, P. C., Graham, M. B., Kelkar, S., DenBesten, K.,Quinn, J. P., 2001. Infections due to vancomycin-resistant Enterococcus faeciumresistant to linezolid. Lancet 357 (9263), 1179.
Gosbell, I., Mitchell, D., Ziochos, H., Ward, P., 2003. Emergence of hetero-vancomycin-intermediate Staphylococcus aureus (HVISA) in Sydney. Med J Aust178 (7), 354.
Grundmann, H., Hori, S., Winter, B., Tami, A., Austin, D. J., 2002. Risk factors for thetransmission of methicillin-resistant Staphylococcus aureus in an adult intensivecare unit: fitting a model to the data. J Infect Dis 185 (4), 481–8.
Hiramatsu, K., Hanaki, H., Ino, T., 1997. Methicillin-resistant Staphylococcus aureusclinical strain with reduced vancomycin susceptibility. J Antimicrob Chemother40, 135.
Humphreys, H., Smyth, E., Mar 2005. Use of enteral vancomycin for the control ofMRSA in the intensive care unit. J Hosp Infect 59, 259–261.
Lodise, T. P., McKinnon, P. S., Tam, V. H., Rybak, M. J., 2002. Clinical outcomes forpatients with bacteremia caused by vancomycin-resistant enterococcus in a level1 trauma center. Clin Infect Dis 34 (7), 922–9.
Mandell, G., Bennett, J., Dolin, R. (Eds.), 2005. Principles and Practice of InfectiousDiseases, 6th Edition. Vol. 2. Elsevier Churchill Livingstone, Ch. 192, pp. 2321–2351.
Meka, V. G., Pillai, S. K., Sakoulas, G., Wennersten, C., Venkataraman, L., DeGirolami,P. C., Eliopoulos, G. M., Moellering, R. C., J., Gold, H. S., 2004. Linezolid resistancein sequential Staphylococcus aureus isolates associated with a T2500A mutation inthe 23S rRNA gene and loss of a single copy of rRNA. J Infect Dis 190 (2), 311–7.
MMWR, 2002. Staphylococcus aureus resistant to vancomycin–United States, 2002.
BIBLIOGRAPHY 11
Murray, B., December 4, 2005 2005. Overview of enterococci. In: UpToDate. Rose,B.D. (ed). UpToDate, Waltham, MA.
Nimmo, G., Bell, J., Collignon, P., 2003. Fifteen years of surveillance by the AustralianGroup for Antimicrobial Resistance (AGAR). Commun Dis Intell 27, Suppl:S47–54.
Pimentel JD, Low J, S. K. H. O. H. A. A. E., Mar 2005. Control of an outbreak of multi-drug-resistant Acinetobacter baumannii in an intensive care unit and a surgicalward. J Hosp Infect 59 (3), 249–53.
Pittet, D., Hugonnet, S., Harbarth, S., Mourouga, P., Sauvan, V., Touveneau, S., Per-neger, T. V., 2000. Effectiveness of a hospital-wide programme to improve compli-ance with hand hygiene. Infection Control Programme. Lancet 356 (9238), 1307–12.
Rao, G. G., 1998. Risk factors for the spread of antibiotic-resistant bacteria. Drugs55 (3), 323–30.
Ridwan, B., Mascini, E., van der Reijden, N., Verhoef, J., Bonten, M., 16 March 2002.What action should be taken to prevent spread of vancomycin resistant entero-cocci in European hospitals? Brit Med J 324, 666–668.
Semmelweiss, I., 1861. Die tiologie, der Begriff und die Prophylaxis des Kindbet-tfiebers.
Tenover, F. C., Weigel, L. M., Appelbaum, P. C., McDougal, L. K., Chaitram, J., McAl-lister, S., Clark, N., Killgore, G., O’Hara, C. M., Jevitt, L., Patel, J. B., Bozdogan, B.,2004. Vancomycin-resistant Staphylococcus aureus isolate from a patient in Penn-sylvania. Antimicrob Agents Chemother 48 (1), 275–80.
Tsiodras, S., Gold, H., Sakoulas, G., Eliopoulos, G., Wennersten, C., Venkataraman,L., Moellering, R., Ferraro, M., July 2001. Linezolid resistance in a clinical isolate ofStaphylococcus aureus. Lancet 358 (9277), 207–208.
Voss, A., Milatovic, D., Wallrauch-Schwarz, C., Rosdahl, V. T., Braveny, I., 1994.Methicillin-resistant Staphylococcus aureus in Europe. Eur J Clin Microbiol InfectDis 13 (1), 50–5.
Weinstein, J., 2005. Hospital-acquired (nosocomial) infections with vancomycin-resistant enterococci In: UpToDate. Rose, B.D. (ed). UpToDate, Waltham, MA.
CHAPTER 2
Literature review and outline of thesis
The critique of relevant literature begins with a review of the biology and epi-
demiology of the microorganisms discussed in this thesis. This is followed by
a general review of compartmental mathematical models applied to infec-
tious diseases transmission at a population level. Specific models relevant to
the studies in this thesis are then explored. Methods used in this thesis are
outlined with an emphasis on reasons for choice of methodology. This chap-
ter concludes with an outline of the original chapters of the thesis, including
their relationship with previous work, innovations and contributions to the
field of research.
2.1 Review of pathogens discussed in this thesis:biological and epidemiological features rele-vant to model development
2.1.1 Methicillin resistant Staphylococcus aureus (MRSA)
Summary
• MRSA is principally transmitted from patient to patient via the hands of
healthcare workers
• Healthcare worker carriage is usually transient and removed by hand
washing
• New MRSA colonisation is caused by cross-transmission of pre-existing
MRSA clones rather than spontaneous emergence of resistance during
exposure to antibiotics (Cooper et al., 2003)
• MRSA colonisation is asymptomatic and precedes infection
14 Chapter 2. Literature review and outline of thesis
• Colonisation of patients is long-term (weeks to months)
• While community MRSA is significant, it has different characteristics
and is readily distinguishable from healthcare-associated strains
S. aureus is a Gram-positive coccus, that is carried asymptomatically on the
skin or in the nares of approximately 30% of the population at a given time.
Carriage is more common on eczematous skin (Boyce, 2005), in people who
receive repeated injections such as insulin-dependent diabetics and infect-
ing drug users, and haemodialysis and peritoneal dialysis patients (Mandell
et al., 2005). The gastrointestinal tract is a recently discovered important site
of colonisation (Boyce et al., 2005).
S. aureus becomes pathogenic when it crosses the integumentary barrier and
can cause serious invasive infection including septicaemia, endocarditis,
pneumonia and severe soft tissue infections and bone and joint infections
(Mandell et al., 2005). Breaches in the skin (the use of central catheters,
surgical wounds, haemodialysis and peritoneal dialysis) increase this risk
of S. aureus infection. Other risk factors include diabetes, HIV and alcohol
abuse.
Beta-lactam antibiotics include the penicillins, cephalosporins and car-
bapenems and monobactams. Although the spectrum of activity of these
antibiotics differs, each kills bacteria by binding and inhibiting penicillin-
binding proteins (PBPs). PBPs are membrane bound transpeptidase and
transglycosidase enzymes essential in bacterial cell wall synthesis. When
penicillin was first introduced for therapeutic use in 1941, S. aureus was
highly sensitive to the antibiotic, but resistance emerged rapidly. By 1950,
isolates appeared that had acquired a plasmid encoded penicillinase en-
zyme, capable of hydrolysing the β−lactam ring of penicillin. By the mid
1950s 40% of isolates were penicillin resistant (Chambers, 2001).
After the emergence of penicillin-resistant Staphylococci, semisynthetic
penicillinase-resistant beta-lactams and cephalosporins were introduced
and these drugs remain the most effective therapy for sensitive staphylococci
(currently in Australia, flucloxacillin, dicloxacillin, cephalexin, cephalothin,
cephazolin are widely used). However, soon after the new class of antibi-
otics was introduced, methicillin-resistant Staphylococcus aureus appeared
(Ericksen and Erichsen, 1963). An absolute requirement for methicillin
resistance in S.aureus is the presence of the mecA gene. MecA is one part of
the staphylococcal chromosomal cassette (SCCmec), a large mobile chro-
mosomal element. Additional components include two regulatory genes
2.1 Review of pathogens discussed in this thesis 15
and five auxiliary genes that can control or modify gene expression (Lowrie,
2006). MecA encodes penicillin binding protein (PBP) 2a, a novel protein
that has reduced affinity to all β−lactam antibiotics, leading to resistance
to the entire class (Lowrie, 2006). DNA hybridisation studies suggest that
there are only five major clones of the mecA gene responsible for methicillin
resistance (Enright et al., 2002).
Like the methicillin susceptible strains, MRSA frequently causes only harm-
less colonisation (Boyce et al., 2005). The median MRSA patient carriage
has been estimated to be 8.5 months by Scanvic et al. (2001) and 40 months
by Sanford et al. (1994). It has been reported that in 30-60% of healthcare-
associated cases, MRSA colonisation proceeds to invasive disease (Boyce,
2005).
HA-MRSA are believed to spread via the transiently-contaminated hands
of healthcare workers (Boyce, 2001). Studies show that MRSA can exist on
healthcare workers hands for 3 hours, but that hand hygiene will almost
completely eradicate MRSA (Peacock et al., 1980; Thompson et al., 1982).
The environment has been found to be contaminated by MRSA (Boyce
et al., 1997), but the contribution of environmental sources of MRSA to
transmission is not well established.
MRSA has recently emerged in the community (CA-MRSA) but this is distin-
guishable from the hospital strains by antibiotic resistance patterns and the
presence of type IV SCCmec (HA-MRSA carry type I, II or III SCCmec) (Boyce,
2005). Molecular techniques also distinguished CA- from HA-MRSA isolates,
suggesting that community-associated disease is not typically due to spread
of nosocomial strains into the community (Naimi et al., 2003). Almost all CA-
MRSA isolates remained non-multiresistant, being sensitive to other classes
of antibiotics such as aminoglycosides (Nimmo et al., 2006).
The facts presented in this section underpin the assumptions made in
Chapter 3 in constructing a model for MRSA transmission. The frequency
of asymptomatic colonisation leads us to study both colonised and infected
patients. The highly complex mechanism of resistance described leads
to the assumption that no de novo resistance arises. The transient nature
of healthcare worker carriage compared with patient carriage leads to the
structure of the model used in Chapter 3. The straightforward differentiation
of community and hospital strains (by antibiogram) allows us to separate the
influence of CA-MRSA in the study on Intensive Care Unit patients.
16 Chapter 2. Literature review and outline of thesis
2.1.2 Vancomycin-resistant enterococci (VRE)
Summary
• VRE colonisation of the gastrointestinal tract is often asymptomatic
• Apparent VRE acquisition may arise from the patient’s own gut flora
• Colonisation is a carrier state of VRE and precedes infection
• Colonisation is long term, lasting weeks to months
• Complex mobile genetic elements confer resistance and these can be
transferred to other enterococci, leading to resistance in unrelated en-
terococcal strains
• Environmental contamination may play a part in transmission in some
settings
Enterococci are Gram-positive, catalase negative cocci that are part of the
normal gut flora of humans and animals. E. fecalis and E. fecium are the two
species most likely to cause disease in humans. They are inherently resistant
to many antibiotics. Until the late 1980s, vancomycin was a reliable agent
for enterococcal strains with resistance to other agents. Vancomycin resis-
tance in enterococci was first identified in Europe, then spread to the USA
and reached Australia by 1994 (Bell et al., 1998). Its prevalence in Australian
both in the community and in hospitals remains low. The hospital prevalence
is rising with 0.3% of hospital strains found resistant in 1999 (Nimmo et al.,
2003) compared with none of the surveyed strains in 1995.
Enterococci are less virulent than S.aureus but can cause urinary tract in-
fection, bacteraemia and infective endocarditis. Invasive disease has a high
mortality rate, probably more so if the enterococcus is vancomycin resistant,
although many studies are confounded by patient risk factors (DiazGranados
et al., 2005). Most patients who develop VRE infection are debilitated and
their underlying disease contributes to their high mortality rate (Weinstein,
2005).
Unlike β−lactams, glycopeptides (vancomycin, teichoplanin) kill Gram-
positive bacteria by preventing cross-linking of the peptidoglycan compo-
nent in the cell wall by binding tightly to the terminal D-alanyl-D-alanine
residues of the pentapeptide stem. The genes encoding glycopeptide resis-
tance lead to replacement of the usual terminal D-alanyl-D-alanine residue
with a D-alanyl-D-lactate moiety. Vancomycin is unable to bind to this
2.1 Review of pathogens discussed in this thesis 17
peptide thus cross-linking of the peptidoglycan proceeds and resistance
results (Weinstein, 2005).
The vanA phenotype leads to resistance to both vancomycin and teicho-
planin, while the vanB phenotype leads to resistance to vancomycin only.
VanA resistance is conferred by a mobile genetic element, a transposon (Tn
1546) (Weinstein, 2005). This is a complex set of genes involving regulatory
genes, genes coding for the production of the new pentapeptide terminal
and some genes responsible for integration of the transposon into larger
genetic elements such as plasmids.
These mobile genetic elements can be transferred to other enterococcal
strains (horizontal transfer of resistance) and indeed interspecies transfer to
S.aureus has been described as discussed in Section 2.1.1. Epidemiological
studies that rely on strain typing to determine clonality of an outbreak of
resistance (such as Pulsed Field Gel Electrophoresis) could thus incorrectly
assess the degree of relatedness of the vancomycin resistance genes.
Colonised patients are asymptomatic, contribute to transmission (Bonten
et al., 1998) and may go on to develop invasive disease (Noskin et al., 1995a).
Colonisation of VRE is long term, potentially indefinite (Noskin et al., 1995a).
Exposure of patients to antibiotics may allow previously undetectable levels
of organism to multiply and become detectable (Donskey et al., 2002).
Prior hospitalisation is a risk factor for colonisation with antibiotic resistant
enterococci on admission to hospital (Weinstein et al., 1996).
There is an interesting dichotomy in the pattern of VRE in Europe and the
United States. In parts of Europe in the late 1990s, VRE colonised healthy
people who were not exposed to hospitalisation. Endtz et al. (1997), for ex-
ample, found that 2% of both hospitalised and community patients had VRE
colonisation in the Netherlands. In the United States, community VRE is rare
but VRE is much wider spread in hospitals. It is believed that successful an-
tibiotic stewardship in hospitals keeps hospital prevalence of VRE low, while
the use of avoparcin (an antibiotic of the same class as vancomycin) in animal
feed contributes to the spread of community VRE in the Netherlands (Ridwan
et al., 2002). Avoparcin was used in Australia until 2001, and the prevalence
of VRE in community volunteers was at a non-negligible level of 0.2% in 1997
(Padiglione et al., 2000).
From the evidence above, Australian patients on admission to hospital
may be colonised with VRE, either from exposure in the community or
prior hospitalisation. Colonisation is frequently asymptomatic and goes
18 Chapter 2. Literature review and outline of thesis
undetected unless there is an active screening program in the healthcare
institution. Therefore, when colonisation or infection are detected, the time
of acquisition will not be clear. Additionally, swab sensitivity is considerably
less than 100% (D’Agata et al., 2002). Therefore, people who have tested
VRE-negative may be colonised with VRE, possibly with sub-detectable
levels. Exposure to antibiotics which suppress other gut flora may allow VRE
to reach detectable densities. Donskey et al. (2002) found that patients with
a history of VRE colonisation could become swab negative for VRE but on
exposure to antibiotics again become positive for VRE.
These facts and inferences underlie the structure of the model presented in
Chapter 5. The model is designed to distinguish between two types of VRE ac-
quisition, that which occurs due to cross-transmission, and that which arises
sporadically, such as might occur when a patient with VRE gut colonisation is
exposed to antibiotics. The model also assumes that VRE colonisation is long
term, with decolonisation rates negligible relative to the duration of hospi-
tal stay. It also considers the deficiencies of genotyping studies, given the
confusion that arises when horizontal transfer of genetic resistance elements
occurs.
2.1.3 Environmental pathogens
Summary
• A number of pathogens have been found in the environment
– VRE (Noskin et al., 1995b; Bonten et al., 1996)
– MRSA (Boyce et al., 1997)
– Clostridium difficile (Kim et al., 1981)
– Gram-negative bacilli, including coliforms and oxidase positive
(Dancer et al., 2006)
• the role of the environment in transmission has not been determined
VRE has been found frequently in patients’ environments and has been
shown to be viable on inanimate objects for some days (Noskin et al., 1995b).
It is not known to what degree environmental contamination contributes
to transmission, or whether it merely reflects stool density of the patient, as
found by Donskey et al. (2000). It is still believed that transmission occurs
2.1 Review of pathogens discussed in this thesis 19
principally via the hands of healthcare workers (Weinstein, 2005), although
the evidence for this is not as strong as is the case for MRSA.
These findings lead to the study in Chapter 6 which considers the potential
impact of an environmental reservoir on the transmission of VRE.
2.1.4 Severe Acute Respiratory Syndrome Coronavirus (SARS
Co-V)
Summary
• SARS Co-V has a distinct incubation period which is not directly observ-
able
• SARS Co-V is spread principally by direct person to person contact
• SARS Co-V is transmitted principally through contact and droplet
spread
• There is (at most) limited transmission of SARS Co-V from cases during
the incubation period
• Asymptomatic infection is rare
• Nosocomial transmission was a prominent feature of SARS during the
2002/3 pandemic epidemiology
Evidence suggests that infectivity does not precede symptom onset for SARS.
Early contact tracing studies found transmission occurred only to close con-
tacts of a symptomatic SARS case (Poutanen et al., 2003). This is consistent
with virological studies that found levels of viral shedding were low in the
early phase of illness (Cheng et al., 2004).
Sero-surveys of populations affected by SARS concluded that asymptomatic
infection is uncommon. Rainer et al. (2004) found that of the patients
presenting with mild respiratory symptoms during the SARS outbreak in
Hong Kong, only 0.7% had serological evidence of SARS. Leung et al. (2004a)
found a very low seropositivity (0.2%) of SARS antibodies in people who did
not have symptomatic SARS but who had close contact with SARS cases.
Estimation of the incubation period for SARS-CoV has proven to be a consid-
erable challenge. Numerous studies have attempted to make estimates (see
Donnelly et al., 2004, for review). It is currently believed to be around 5 days
with an upper limit of 10 days (Donnelly et al., 2004).
20 Chapter 2. Literature review and outline of thesis
Early in the SARS pandemic, a majority of cases arose from hospital transmis-
sion in many places, including Toronto (Booth et al., 2003), Hong Kong (Ri-
ley et al., 2003; Wong et al., 2004) and Singapore (Gopalakrishna et al., 2004).
In Hong Kong almost half of cases were healthcare-associated (Leung et al.,
2004b). Later in the course of the epidemic, hospitals were effective sites of
containment of SARS (Gopalakrishna et al., 2004).
Clusters of cases from Hong Kong and Canada suggest that the SARS coron-
avirus (SARS-CoV) spreads directly from person-to-person. Acquisition from
face-to-face interaction rather than physical contact suggests droplet spread
(Donnelly et al., 2004; Hirsch, 2006). Other methods of spread including via
the faeces are also possible, as SARS-CoV is frequently recovered from the
feces (Cheng et al., 2004).
These features of SARS-CoV transmission are incorporated into the structure
of the model presented in Chapter 7.
2.2 Mathematical models of human infectious dis-
eases
2.2.1 History
The first attempts to apply mathematical methods to infectious disease data
involved descriptive statistics. Bernoulli (1760) and Farr (1840) evaluated
data on smallpox vaccination and deaths, respectively, using empirical
methods. Following the general acceptance of germ theory, Hamer (1906)
brought the principle of mass action, used in chemistry since Boyle (c. 1674),
into the transmission of infectious diseases. Hamer (1906) postulated that
the progress of an epidemic depends on the contact rate between suscep-
tible and infectious individuals, therefore being dependent on the density
of each in the population. At the time there was debate about the cause of
waning of epidemic waves, with many arguing that a decrease in virulence
of the contagion was responsible. Hamer argued that reduced density of
susceptibles was responsible for decline in epidemics. He also postulated
that births and loss of immunity would lead to increased susceptibility and
new epidemic waves.
Ross (1916) and Ross and Hudson (1916) developed the mass action con-
cept further, presenting continuous time epidemic equations. In 1927,
Kermack and McKendrick developed the equations of the compartmental
2.2 Mathematical models of human infectious diseases 21
Susceptible-Infectious-Removed (SIR) model, called the general epidemic
model, still widely used today.
2.2.2 The Susceptible-Infectious-Removed model
The general epidemic model assumes that people begin susceptible to an in-
fectious disease, may become infected by exposure to an infectious person,
becoming immediately infectious themselves and after a time period either
recover or die. Recovery constitutes immunity to further infection and they
are said to be removed.
The simplest version of this SIR model assumes homogenous mixing and a
fixed population size, N = S(t) + I(t) + R(t), where S(t), I(t), and R(t) are
the numbers in the population who are susceptible, infectious and removed
at time t. Each contact between a susceptible and an infectious patient has
a probability, p, of leading to transmission and contacts occur at a rate, c per
day. The classical system of ordinary differential equations is
dS
dt= −cpSI (2.1)
dI
dt= cpSI − γI
dR
dt= γI,
where c, and γ are positive constants and 0 < p ≤ 1. If one does not require
separate estimates of p and c, one can use β = cp. The behaviour of the sys-
tem is governed by the first two equations and the number of recovered, R,
can be determined as N = S(t) + I(t) + R(t).
The differential equation for the number of infectious, I(t), can be rewritten
as
dI
dt= βI(S − γ
β), (2.2)
which leads to a critical value of susceptibles. Following the introduction of
an infectious case, in order for an epidemic to proceed, the number of the
population that are susceptible must be greater than γ/β, and if the initial
value of βN is less than γ, an epidemic will not occur.
Linearising the system about a steady state (S, I) and putting S = S + s and
22 Chapter 2. Literature review and outline of thesis
I = I + i leads to
(s
i
)=
(−βI −βS
βI βS − γ
)(s
i
). (2.3)
The dominant eigenvalue of the Jacobian at the steady state (S = N , I = 0)
gives the growth rate of the epidemic curve, namely βN − γ.
The classic epidemic SIR model has been extended in a number of ways, in-
cluding
• models that incorporate temporary immunity (SIRS) or no immunity
(SIS)
• models with vital dynamics (in which birth and death are included)
• SIR models with carriers (a carrier is one who spreads diseases but has
no symptoms)
• SIR models with vertical transmission
• SIR models with stratified populations
• models in which there is no recovery from infection (SI)
• host-vector-host models (contagion is passed alternately from one
species to another) also called the Ross-MacDonald model
• models in which an infected individual has a latent period before
becoming infectious; Susceptible-Exposed-Infectious-Removed (SEIR)
models
In the latter half of last century there was a greater emphasis on probabilis-
tic models (Bailey, 1975; Becker, 1989). Other extensions of classic SIR mod-
els of disease include the spatial spread of disease (Diekmann and Heester-
beek, 2000), understanding of recurrent epidemic waves, and heterogeneity
of spreading (Anderson and May, 1991) and the importance of households
and other social networks (Becker, 1989).
In this thesis, modified SIR models are developed according to the specifics of
the pathogen being considered. Chapter 3 develops a modified host-vector-
host model with migration (vital dynamics), appropriate to the transmission
characteristics of MRSA. Chapter 5 uses an SI model with migration, and
2.2 Mathematical models of human infectious diseases 23
Chapter 7 develops an SEIR model, modified by relaxing the assumption of
negative exponential sojourn times within SEIR compartments.
2.2.3 The Susceptible-Exposed-Infectious-Removed model
Susceptible-Exposed-Infectious-Removed (SEIR) models include an addi-
tional class of latently-infected (exposed)1 individuals. These individuals
have acquired infection but are not yet infectious. A simple system of
equations can be used to describe this model
dS
dt= −βSI (2.4)
dE
dt= βSI − νE
dI
dt= νE − γI
dR
dt= γI,
where β, ν and γ are positive constants.
Linearising the system about the steady state (S, E, I) and putting S = S + s,
E = E + e and I = I + i leads to
s
e
i
=
−βI 0 −βS
βI −ν βS
0 ν −γ
s
e
i
. (2.5)
The dominant eigenvalue of the Jacobian at the steady state (S = N, E =
0, I = 0) gives the growth rate of the epidemic curve,
λ =−(γ + ν) +
√(ν − γ)2 + 4βNν
2. (2.6)
Note that the growth of the epidemic is dependent on the rate of transition
1The incubation period is the time from acquisition of a contagion until the person devel-ops symptoms. The latent period is the time from acquisition of a contagion until the personis infectious.
24 Chapter 2. Literature review and outline of thesis
from the latent to the infectious period, ν. The rise in the early epidemic
curve, λ, is easily calculated during an outbreak. Inferences regarding β based
on the value of λ will be highly sensitive to ν which is often taken to be the
reciprocal of the mean duration of the incubation period (see, for example,
Chowell et al. (2003)).
Importance of compartmental sojourn times
The models using the system of ordinary differential equations (2.1) and
(2.4) implicitly assume that the exposed (latent) and the infectious periods
are negative exponentially distributed with parameters ν and γ, respectively.
Keeling and Grenfell (1997) showed that assumption of the exponential
distribution for the infectious period leads to underestimation of the critical
community size, the size necessary to sustain endemic transmission, for
measles. The over-prediction of fade-outs that occurred in standard (expo-
nential infectious periods) model was corrected by allowing the infectious
periods to be normally distributed in line with observed infectious period
distributions. Estimates of infectivity, particularly those based on the early
epidemic curve, are also highly sensitive to the shape of the survival curve
in the Exposed and Infectious compartments (Lloyd, 2001). Donnelly et al.
(2003) showed that the latent period for Severe Acute Respiratory Syndrome
is not exponential. Lloyd (2001) showed that non-exponential compart-
mental sojourn times lead to more realistic model predictions for the SIR
model.
2.2.4 The basic reproduction ratio, R0
The basic reproduction ratio (R0) is defined as “The average number of per-
sons directly infected by an infectious case during its entire infectious period,
after entering a totally susceptible population” (Giesecke, 1994).
The basic reproduction ratio is a function of daily infectivity and expected
duration of infectivity. The effective reproduction ratio is expected number
of persons directly infected by an infectious case without the assumption of a
fully susceptible population. In the SIR or SEIR model, with constant hazard
of transition between compartments and constant infectivity, the effective
reproduction ratio is Rt = βSt
γ. When the entire population is susceptible
(S(0) = N), this expression gives the basic reproduction ratio, R0 = βNγ
.
2.2 Mathematical models of human infectious diseases 25
A more general expression is obtained when the infectivity does not remain
constant over the infectious period and the transit time over the infectious
period is not necessarily negative exponential.
Here
R0 =
∫ ∞
0
c(τ)p(τ)q(τ)dτ, (2.7)
where τ is the time since transmission occurred to an individual and q(τ) is
the probability of remaining infectious, c(τ) is the contact rate and p(τ) the
probability of transmission per contact, a time period τ from infection.
In Chapter 7, Bayesian inference is used to estimate the changes in infectivity
over the course of SARS infection. Such estimates are informative to infection
control practitioners and can be the basis of more realistic models.
2.2.5 Adaptation of the Ross-MacDonald model to the
healthcare setting
Originally used to describe the transmission of malaria, the host-vector-host
model was developed by Ronald Ross (1857-1932) and George MacDonald
(1903-1967).
The model is applicable to infections that are transiently carried by a vec-
tor2. In the original model, the vector population was a different species, the
Anopheles mosquito, and the infectious agent, malaria, was an obligate para-
site, requiring the vector to complete its lifecycle. A fundamental assumption
of the model is that all host-to-host transmission is indirect, via the vector.
The vector carries the pathogen transiently while the principal host is infec-
tious for longer.
The Ross-MacDonald model is readily modified to describe the transmission
of pathogens in the healthcare setting. Nosocomial pathogens are believed
to spread from patient to patient indirectly via the transiently-contaminated
hands of HCWs. In this sense the HCWs are analogous to the mosquito vec-
tors of the original model.
Figure 2.1 illustrates the model dynamics. Transmission occurs only during a
discordant contact: when a colonised HCW contacts an uncolonised patient
2Here vector is used in the medical sense: an organism that does not cause disease itselfbut which spreads infection by conveying pathogens from one host to another.
26 Chapter 2. Literature review and outline of thesis
or when a colonised patient contacts an uncontaminated3 healthcare worker.
An important modification of the Ross-MacDonald model, is the incorpora-
tion of rapid migration of patients in and out of the ward. Uncolonised pa-
tients are discharged at a rate µx, while colonised patients are discharged at
a rate µy. A proportion, σ, of patients may be colonised on arrival. Transition
from contaminated back to susceptible is assumed to occur at a fixed rate for
healthcare workers, κ. The probability of transmission from HCW to patient
is denoted by php and patient to HCW pph. Contact rate per patient per HCW is
c. The number of admissions per day is given by Λ. The patients are labelled
p and the healthcare workers, h. Y indicates infectious and X susceptible.
hp p hcp X Y
y pYµ
hYκ
hY
pX
hX
pY
ph h pcp X Yx p
Xµ
(1 )σ− Λ σΛ
Figure 2.1: The application of the Ross-MacDonald model to the transmis-sion of nosocomial pathogens.
Each element, Gij , in the next generation matrix, G, is the expected number
of new infections transmitted to each population j by a single infective of
population i. The next generation matrix for the model shown in Figure 2.1 is
given by
G =
(0
cpphNh
µycphpNp
κ0
). (2.8)
The basic reproduction ratio is the square of the dominant eigenvalue of the
next generation matrix4.
3The term contaminated/uncontaminated will be used to describe the transient coloni-sation of healthcare workers, while the terms colonised/uncolonised will be used when re-ferring to patients.
4R0 is either the dominant eigenvalue or the square of the dominant eigenvalue depend-ing on definition of R0. The former definition, used by Diekmann and Heesterbeek (2000)
2.2 Mathematical models of human infectious diseases 27
In this system, the basic reproduction ratio, R0, is given by
R0 =c2phppphNhNp
µyκ. (2.9)
This model was used in Chapter 3 of this thesis. The Ross-MacDonald model
is relatively complex with a number of parameters which useful for predicting
the impact of different infection control interventions. In the hospital there is
rapid migration, which must be incorporated into the model. Individual pa-
rameters (for example hand hygiene compliance or admission prevalence) or
combinations of parameters can be changed to model the effect of interven-
tions, as elaborated in Chapter 3.
2.2.6 Single population models
The large number of parameters and complexity of the Ross-MacDonald
model make it sensitive to uncertainties in parameter estimates and model
assumptions. A more parsimonious model is the two compartment model
in which the HCW compartments are replaced by a constant, using the
assumption Yh ≈ βYp, which leads to an SI model. The justification for
this assumption is given later in this section. The two compartment model
has fewer unknown values to estimate, which reduces collinearity between
parameters and improves precision of estimates. The assumption that all
acquisition of the infectious agent occurs via indirect transmission is relaxed.
The cross-transmission parameter, β, incorporates both direct and indirect
transmission.
The study presented in Chapter 5 is based on this model. The model includes
an additional parameter for VRE acquisition that was independent of cross-
transmission, ν, as shown in Figure 2.2. In this model, R0 is given by
R0 = βN/µc. (2.10)
is the expected number of the other population infected by a single infectious individualassuming a fully susceptible population. The latter definition, used in Chapter 3 of this the-sis, is the expected number of the same population infected by a single infectious individualassuming a fully susceptible population. The threshold value for R0 will be unity in eithercase.
28 Chapter 2. Literature review and outline of thesis
CU
( )u
CU Uβ ν σµ+ +
(1 )c Cµ σ−
Figure 2.2: The single population model. Graphical representation of thetransmission of bacterial pathogens among patients in the hospital ward.Here C is the number of colonised patients, U is the number of uncolonisedpatients, N = U + C is the total number of ward patients (assumedfixed), µu is the discharge rate of uncolonised patients, µc is the dischargerate of colonised patients, ν is the colonisation rate independent of cross-transmission, σ is the admission prevalence and β is the cross transmissioncoefficient.
The ordinary differential equation governing this model is
dC
dt= βC(N − C) + (ν + µU(1− σ))(N − C)− µcC. (2.11)
Simplification of Ross-MacDonald compartment model
The reduction of the Ross-MacDonald model (used in Chapter 3) to a two
compartment, patient-only model (used in Chapter 5) can be justified on
the basis that in the four compartment model, the proportion of healthcare
workers colonised is directly proportional to the number of colonised pa-
tients. Figure 2.3 shows the relationship between the expected number of
colonised healthcare workers as the number of colonised patients changes
using realistic values (h = 0.59, Np = 15, pph = 0.13) derived from the study
presented in Chapter 3.
0 5 10 150
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Number of colonised patients
Figure 2.3: Predicted linear relationship between number of patientscolonised and number of healthcare workers colonised.
2.2 Mathematical models of human infectious diseases 29
Therefore
Yh ≈ βYp, (2.12)
and Yh in Figure 2.1 can be replaced by βYp, leading to the much simpler two
compartment model.
2.2.7 Stochastic models
Deterministic models, which assume continuous changes in compartment
numbers and give results based only on initial conditions, model parameters
and structure, give a reasonable approximation for parameters when num-
bers in each model compartment are large, (Bailey, 1975). They fall down,
however, in small scale epidemics such as that encountered in the hospital
ward setting. They are also not reliable at the beginning and end of the epi-
demic when the number of infectives is small. Additionally, they are unable
to quantify uncertainty in the parameter estimates, nor the expected varia-
tion in the simulated epidemics based on those parameters. Here stochastic
models prove useful (Becker and Britton, 1999). Statistical models of infec-
tious disease transmission apply structured epidemic models, such as those
described in this section, to statistical analyses for parameter estimation, hy-
pothesis testing and study design (Becker and Britton, 1999).
Stochastic equations involve discrete changes in model variables. In contin-
uous time stochastic models (used in all studies of this thesis) compartment
counts increment or decrement by unit amounts. Over a brief time inter-
val, dt, unit changes in model compartments occur with a probability deter-
mined by the hazard of transmission (based on the current state of the model
compartments). For example, the stochastic version of the SEIR equations is
30 Chapter 2. Literature review and outline of thesis
given by
PrS(t + dt) = i− 1, E(t + dt) = j + 1, I(t + dt) = k, R(t + dt) = l| (2.13)
S(t) = i, E(t) = j, I(t) = k,R(t) = l= βikdt + o(dt),
P rS(t + dt) = i, E(t + dt) = j − 1, I(t + dt) = k + 1, R(t + dt) = l| (2.14)
S(t) = i, E(t) = j, I(t) = k,R(t) = l= νjdt + o(dt),
P rS(t + dt) = i, E(t + dt) = j, I(t + dt) = k − 1, R(t + dt) = l + 1| (2.15)
S(t) = i, E(t) = j, I(t) = k,R(t) = l= γkdt + o(dt).
Similar stochastic equations can be derived for the other models.
2.3 Relationship of current literature to models
presented in this thesis
The preceding discussion was a general overview of the structure of mathe-
matical models that relate to work in this thesis and methodology used in this
thesis. The following discussion concentrates on specific recent published
research that share features with the models in this thesis.
2.3.1 Studies based on the Ross-MacDonald model applied to
nosocomial transmission of bacteria
A number of studies have applied an adapted Ross-MacDonald model to the
transmission of bacterial pathogens in the healthcare setting. Most made the
following assumptions: transmission occurs only via the hands of healthcare
workers; the populations are homogenous with regard to susceptibility and
infectiousness; mixing between HCWs and patients is homogenous; patient
colonisation is long term (greater than duration of hospital stay) while
healthcare worker colonisation is short-term. Table 2.1 gives an analysis of
studies which follow this model structure including the study presented in
Chapter 3 on this thesis.
The structure of the four compartment models allows simulations and
predictions of the likely efficacy of a number of different interventions
2.3 Relationship of current literature to thesis 31
including improved hand hygiene, removal/discharge of colonised patients
and changes in admission prevalence. Small changes in model structure
also allow models to predict the impact of patient cohorting and changes
in staff/patient ratio, patient isolation, antibiotic restriction and patient
decolonisation.
Sources of parameters for models generally fall into five categories:
1. expert opinion
2. literature review
3. observation on the ward as part of the study
4. epidemiological results from dataset
5. fitting remaining parameters to colonisation data.
All of the studies reviewed to date have used at least one parameter that was
not measured as part of the study, usually the transmission probability or
the duration of healthcare worker contamination. These were estimated or
derived from literature sources or expert opinion. This is a weakness in the
current body of literature on the subject as the model predictions are highly
sensitive to these parameters, although the values of R0, derived by Austin
et al. (1999), are not dependent on estimated model parameters.
32C
hap
ter2.L
iterature
reviewan
do
utlin
eo
fthesis
Study MRO Relaxation of Assumptions Interventions tested Main findings / predictions Source of parameter estimation Study design
Sebilleet al.(1997)
MRSA Patient to patient and HCWto HCW transmission con-sidered. Decolonisation ofpatients included.
Hand hygiene, antibioticpolicy, reduction in admis-sion prevalence
Hand hygiene had large impacton HCW contamination but notpatient colonisation. Antibi-otic policy had little impact onMRSA colonisation.
Ward data: mean length of stay of pa-tients (colonised and uncolonised treatedthe same). Expert opinion: all other para-meters.
Deterministic
Cooperet al.(1999)
non-specific
Hand hygiene, changes inlength of stay and transmis-sibility, improvements indetection of organism
Hand hygiene compliance andtransmission probability havethe greatest impact on trans-mission. Large variation in out-come can be expected due tostochastic effects.
Literature review and expert opinion esti-mates for all parameters.
Stochastic
Austinet al.(1999)
VRE Ward was not of fixedsize for simulations (butassumption of fixed sizeused to calculate the basicreproduction ratio).
Hand hygiene, patientcohorting and admissionprevalence
The effective reproduction ratiowas estimated to be 0.69. In theabsence of interventions thebasic reproduction ratio was es-timated to be 3.81.
Expert opinion: transmission probability,duration of HCW contamination. Database:length of stay, ward size, admission preva-lence.
Deterministic(for pa-rameterfitting) andstochastic forsimulations
Grundmannet al.(2002)
MRSA Ward was not of fixedsize for simulations (butassumption of fixed sizeused to calculate the basicreproduction ratio).
Hand hygiene, patient co-horting
Staff deficit was the only covari-ate associated with increasedMRSA transmission. Improve-ment in hand hygiene by 12%was predicted to make up forstaff deficit.
Expert opinion: transmission probability,duration of HCW contamination. Data-base: length of stay, ward size, admissionprevalence. Measured on the ward: con-tact rates, transmission probability, handhygiene compliance.
Cohort studywith covariateanalysis, Sto-chastic anddeterministic
D’Agataet al.(2005)
VRE Patient compartment nothomogenous. Divided intothose receiving and not re-ceiving antibiotics.
Hand hygiene, patientcohorting, staff/patientratio, length of stay,antibiotic policy, admissionprevalence
Reducing antibiotic exposure touncolonised patients is moreeffective than reducing antibi-otic exposure to colonised pa-tients. Increasing staff/patientratio led to reduction of coloni-sation.
Expert opinion: transmission probability.Database: length of stay, ward size, admis-sion prevalence, staff/patient ratio, antibi-otic treatment and cessation rates. Litera-ture sources: hand hygiene compliance.
Deterministic
Raboudet al.(2005)
MRSA Heterogeneity of colonisedpatients as isolated patientshave reduced transmissionof MRSA.
Hand hygiene, staff-patientratio, patient isolation, an-tibiotic policy, admissionprevalence
Hand hygiene and early detec-tion and isolation were pre-dicted to be the most effectiveinterventions.
Literature sources: Transmission rate, ef-ficacy of isolation, sensitivity of MRSAscreening. Database: admission preva-lence, length of stay, screening rate. Mea-sured on the ward: hand hygiene, contactrate.
Stochastic
Chapter 3 MRSA Ward size not fixed forestimate of transmission;homogenous mixing notassumed when cohortingconsidered
Hand hygiene, patient co-horting, changes in staff-patient ratio, changes inlength of stay, decolonisa-tion
Increase in staff levels couldincrease transmission if it ledto more contacts and if co-horting did not occur. Wardsize effects transmission rateindependently of other para-meters. Decolonisation usingenteral vancomycin is relativelyineffective at reducing trans-mission of MRSA.
Derived from database: ward size length ofstay admission prevalence. Measured onthe ward at the time of the study: hand hy-giene compliance, probability of transmis-sion patient to HCW. Final transmission pa-rameter estimated using data fitting.
Stochasticand Deter-ministic
Table 2.1: Comparison of studies that applied the Ross-MacDonald model to nosocomial transmission of hospital pathogens. MRO: multipleantibiotic-resistant organism.
2.3 Relationship of current literature to thesis 33
Sebille et al. (1997) and Sebille and Valleron (1997) were early studies which
introduced simulation to the area of nosocomial infection control, using de-
terministic and stochastic models, respectively. Unlike models that followed,
these models included patient to patient and staff to staff transmission. The
parameter values in this study were either observed in the ward or estimated.
Some of the estimates differ markedly from those obtained from the current
state of knowledge, particularly the duration of colonisation of healthcare
workers, which was assumed to be 36 days in these studies, whereas it is
estimated to be 1 hour in most other studies. The main findings of these
investigators were that both hand hygiene compliance and antibiotic usage
policy had a limited effect on patient colonisation, but hand hygiene had
a marked effect on staff colonisation. The reason for the poor response
of patient colonisation to hand hygiene (this is the only study of this type
to make the conclusion) is that much of the transmission was patient to
patient and the parameter representing hand hygiene was not included in
the patient to patient transmission term. The model for antibiotic effect was
to assume that there were two antibiotics used to decolonise patients, the
first antibiotic, A1 was 80% effective for sensitive strains, the second was
30% effective for A1-resistant strains and otherwise ineffective. It appears
that all strains were resistant to at least one antibiotic. It is not surprising
then that the model predicted that the use of one antibiotic led to rapid
strain replacement and that no antibiotic usage policy made a substantial
difference to transmission.
Cooper et al. (1999) brought innovation to this research area by showing the
variability of outcomes that results from stochasticity. They predicted that
hand hygiene compliance and transmission probability have the greatest
impact on transmission. Another finding was that the apparent success
of a policy depends on the outcome being measured. For example, new
outbreaks could be more common in a ward that was experiencing good
control of MRSA compared with a ward that had poor infection control
and endemic colonisation. In this study, the transmission parameters were
not derived using a data source, coming from literature or expert opinion
sources.
Austin et al. (1999) developed a data-based model for VRE transmission
and derived a useful, non-dimensional relationship between admission
prevalence, ward prevalence and the reproductive ratio, R0. Estimates
of R0 were thus independent of some of the estimated parameters such
as transmission probability and duration of colonisation. To derive this
34 Chapter 2. Literature review and outline of thesis
relationship, the investigators made the assumption that the reproduction
ratio of healthcare workers to patients (this equates to G2,1 in equation 2.8)
is negligible compared with the basic reproduction ratio. This assumption
would not be valid for wards in which hand hygiene compliance is low or
transmission rates are high. Austin et al. (1999) were the first to consider the
effects of patient cohorting, finding that both cohorting and hand hygiene
have a marked impact on R0 and on ward prevalence.
Grundmann et al. (2002) used the same model structure as Austin et al. (1999)
and incorporated it into an MRSA cohort study. The study included a num-
ber of covariates as putative risk factors for colonisation. The main findings
were that, of the risk factors studied, a staff deficit was the only one signif-
icantly associated with increased MRSA transmission. An improvement in
hand hygiene of 12% would be needed to compensate for the staff deficit, as
predicted by this model.
D’Agata et al. (2005), modelling VRE, introduced new patient compart-
ments, dividing the patient groups into those exposed and not exposed to
antibiotics. This model was not fitted to data. Compartments were treated
differently both with respect to contact rates and probability of colonisation
per contact, making it difficult to differentiate these effects. Within each
compartment, the assumptions of homogenous mixing held. The main
findings were that reducing antibiotic exposure to uncolonised patients was
more effective than reducing antibiotic exposure to colonised patients. This
follows from the author’s assumption that uncolonised patients not taking
antibiotics could never become colonised. Increasing antibiotic exposure to
colonised patients therefore had no impact on transmission from colonised
to uncolonised patients unless the uncolonised patients were antibiotic-
exposed (a minority). Increasing antibiotic exposure to uncolonised patients
led to a larger number of patients becoming susceptible. This model of
transmission is not likely to represent reality as antibiotic exposure is not a
pre-requisite for VRE colonisation. Further studies are needed to quantify
the relative vulnerability of antibiotic exposed and unexposed patients.
A new focus in the study by Raboud et al. (2005) was the role of isolation in
containing MRSA transmission. Citing a study by Jernigan et al. (1996), iso-
lation was assumed to reduce transmission by a factor of 16. The use of this
data was somewhat cavalier, as the study was based on only 16 MRSA trans-
missions, and was a neonatal intensive care unit unlikely to be extrapolative
to the adult ward setting. Additionally, determination of the source of MRSA
2.3 Relationship of current literature to thesis 35
was subjectively based, performed by two observers on the ward. Not sur-
prisingly, assuming relative risk of isolation of 1/16, early detection and iso-
lation were found to be effective.
Grundmann and Hellriegel (2006) and Bonten et al. (2001) present reviews of
mathematical models of nosocomial infections. The scope of these reviews
differs from the scope of this thesis, with Bonten et al. including studies of the
dynamics of bacterial resistance explored by Lipsitch et al. (2000) and Grund-
mann and Hellriegel (2006) including game theory and economic models.
2.3.2 Stochastic epidemic models based on the Susceptible-
Infectious model with migration applied to transmis-
sion of nosocomial pathogens
This section reviews studies that fit models to data, aiming to differentiate
the sources of acquisition of antibiotic-resistant pathogens in the healthcare
setting. Three previous studies have been published that use models simi-
lar the that described Section 2.2.6, the single population model. Unlike the
Ross-MacDonald based models, these studies did not assume that all coloni-
sation arises through cross-transmission on the ward. Each of the studies
used the data to estimate parameters associated with the different sources of
colonisation, rather than making assumptions or estimates based on expert
opinion or literature sources.
Pelupessy et al. (2002) used a longitudinal time series dataset. They differ-
entiated “spontaneous” colonisation from cross-transmission in their study.
The study assumed that the time of all events was known and the exact num-
ber of colonised patients was known throughout the study. This allowed the
direct application of a Markov chain algorithm to determine the likelihood.
The study used genotype data to validate the method.
Cooper and Lipsitch (2004) examined time series data on monthly inci-
dence of infection. Unstructured and structured hidden Markov models
(HMMs) were compared with models that assumed colonisation was serially
independent. The structured HMM assumed that there were two sources
of patient colonisation, similar to the model shown in Figure 2.2. Cooper
and Lipsitch (2004) omitted spontaneous colonisation, but included a term
for colonisation on admission, σµu. The relationship between the hidden
states (the number of colonised patients) and the observations (the monthly
incidence of infection) was assumed to be Poisson.
36 Chapter 2. Literature review and outline of thesis
Pelupessy et al. (2002) and Cooper and Lipsitch (2004) both had some identi-
fiability problems in their analyses due to collinearity of the parameters rep-
resenting different sources of transmission. This is not surprising as trans-
mission is a monotonically increasing function of both parameters.
Forrester and Pettitt (2005) used time series data on MRSA acquisition to esti-
mate “background” sources of colonisation and cross-transmission sources,
finding that background sources were much higher than cross-transmission.
The authors extended the model further by dividing the colonised group into
those who were isolated and not. The fitted value for the cross transmission
for isolated patients was lower than for non-isolated colonised patients; how-
ever this was not statistically significant (p=0.1). The methodology in this
study took into account interval censoring of data but not incomplete de-
tection (compliance with swabbing and swab sensitivity was assumed to be
100%).
2.3.3 Environmental models of transmission
To date, no study has modelled the impact of an environmental reservoir on
the transmission of nosocomial pathogens. The model developed in Chapter
6 includes a new compartment representing the environmental reservoir.
This model is analogous to the models for schistosomiasis in which there
were reservoir compartments representing alternative mammalian hosts
(Williams et al., 2002). In the schistosomiasis models only the definitive
host infects the reservoir compartment, whereas in the model of Chapter
6, both populations (patients and healthcare workers) contaminate the
environment.
2.3.4 Epidemic models of Severe Acute Respiratory Syn-
drome
Many mathematical models have been published of the SARS epidemics of
2003. The aims are various: to estimate the infectivity of SARS, to describe
the sojourn times through the incubation and other disease stages, to predict
the effect of interventions and to estimate the case fatality rate.
2.3 Relationship of current literature to thesis 37
Studies that estimate sojourn times
Estimates of the full probability density of the length of the incubation pe-
riod and other transitional periods were critical information both for infec-
tion control practitioners (for quarantine and contact tracing strategies) and
to develop realistic mathematical models. A challenge to accurate estima-
tion was the left-censored nature of the data. Exposure times to index SARS
cases were frequently available but the time of transmission can never be ob-
served. A common strategy used by SARS investigators to deal with the unob-
served transmission times was to assume a uniform probability of transmis-
sion across the exposure period and fit incubation period models to the data.
Donnelly et al. (2003) used a parametric approach, fitting a Gamma distrib-
ution to the incubation period; Farewell et al. (2005) examined a number of
parametric distributions, finding the log-gamma the best fit, while Meltzer
(2004) used a nonparametric approach. An alternative assumption, imme-
diate transmission upon exposure to a known symptomatic SARS case, was
employed by Lee et al. (2003).
Models that estimate transmission
Table 2.2 summarises some of the major SARS modelling studies published
to date. Not all of these appear in Chapter 7 which was submitted for publi-
cation in December 2004.
38C
hap
ter2.L
iterature
reviewan
do
utlin
eo
fthesis
Study Data source Innovations/ Methodology Outcomes Limitations
Chowell et al.(2003)
Canada, HongKong andSingapore
SEIRHD model which also divides susceptiblesinto two groups.
Basic reproduction ratio=1.1-1.2. Impact ofisolation
Homogenous mixing assumed.
Lipsitch et al.(2003)
Singapore andHong Kong
SEIRHD model. Weibull distributions fitted tosojourn times.
Basic reproduction ratio ≈ 3 for Singa-pore. Probability of epidemic depends onthe reproduction ratio, the heterogeneity ofspread and the number of index SARS cases.Quarantine has a dramatic effect on trans-mission.
Basic reproduction ratio estimated from theexponential growth curve therefore sensi-tive to incubation period. Unreported casesnot considered.
Lloyd-Smithet al. (2003)
Loosely based onHong Kong andSingapore
Stochastic SEIR model using Erlang sojourntimes for latent and symptomatic periods. Arange of interventions was modelled, includingisolation, contact tracing and quarantine.
Healthcare workers are critical targets ininterventions. Hospital wide precautionshave the largest impact on reducing the ba-sic reproduction ratio, followed by targetedprecautions eg isolation of SARS patients.
Assumptions of homogenous mixing(within compartment groups) andhomogeneity with respect to spreading.
Riley et al.(2003)
Hong Kong Gamma distributions fitted to sojourn times.Super spreading events incorporated determin-istically into model. Spatially stratified, using amixing matrix for 18 districts.
Basic reproduction ratio = 2.7 for HongKong, excluding super-spreading episodes.
Uncertainty regarding the number of casessecondary to super-spreading event leadsto wide margins of estimates for reproduc-tion ratio. Unreported cases not consid-ered.
Wallinga andTeunis (2004)
Hong Kong, VietNam, Canada,Singapore
Develops a likelihood-based computation forthe basic reproduction ratio based on inferenceof infection networks.
Estimated reproductive ratios were similaracross different countries, between 2 and 3prior to interventions and around 0.7 afterinterventions.
Assumes no unreported cases and nochange in the generation interval over time.Also poor estimates for small epidemics insimulations.
Hsieh et al.(2004)
Taiwan SIR model with a “suspected SARS” group in-cluded.
Effective reproduction ratio 4.23 Limited data prevented analysis of time de-pendence in infectivity parameters. Latentperiod not included in the model.
Wang andRuan (2004)
Beijing, China SEIRHD model with a “suspected SARS” groupincluded.
Reproduction ratio range 1.1-3.3 Owing to limited data, parameter valueswere fitted only to a (simplified) single com-partment model.
Lloyd-Smithet al. (2005)
Singapore forSARS and severalother virusesincluded
Branching process models Data suggests the distribution of the num-ber of secondary cases of SARS is highlyskewed. Incorporation of this heterogene-ity of spreading leads to very different esti-mates of SARS outbreaks with stochastic ex-tinction more likely and outbreaks less fre-quent.
Small datasets from published literaturetherefore subject to publication bias.
Glass et al.(2006)
Hong Kong,Singapore,Taiwan, Canada
Bayesian Markov chain Monte Carlo approachis used to fit a model of infectious disease trans-mission that takes account of undiagnosedcases
Basic reproduction ratio from differentcountries fell into same range 1.5-3 priorto intervention and 0.36-0.6 after interven-tions
Data broken into discrete generations, ne-glecting the likelihood of overlap. Deple-tion of susceptibles was not factored intothe model, which is reasonable on the scaleof the epidemics modelled.
Table 2.2: Comparison of studies that estimated infectivity of Severe Acute Respiratory Syndrome.
2.3 Relationship of current literature to thesis 39
Two studies were reported in the same edition of Science, Lipsitch et al.
(2003) and Riley et al. (2003). Each extended the Susceptible-Exposed-
Infectious-Removed (SEIR) model to allow for isolation of patients as
they reached hospital leading to the Susceptible-Exposed-Infectious-
Hospitalised-Recovered/Death (SEIRHD) model. Both models allowed for
non-exponential sojourn times; Riley et al. (2003), following Donnelly et al.
(2003), used the Gamma distribution, while Lipsitch et al. (2003) used the
Weibull distribution. Each study took a different approach to determining
the basic reproduction ratio.
Riley et al. (2003), working on data from Hong Kong, were faced with two
super-spreading events that occurred at the Prince of Wales hospital and
in the Amoy Gardens complex. This was incorporated into the model in a
deterministic way. Because the incidence of SARS varied markedly among
districts, this group used a stochastic meta-population approach, allowing
different contact rates between people of the same district, adjacent districts
and more remote districts. Rather than incorporating heterogeneity of
spreading into the model in a stochastic way, the authors seeded the model
with infectious patients (to account for the Prince of Wales outbreak) and
restricted the model to one further super-spreading event (to account for
the Amoy Gardens outbreak). The conclusion of the study was that the
basic reproduction ratio was 2.7 excluding super-spreading events and that
transmission was highly regional. They also found that transmission per
symptomatic case per day was four times higher in the community than
in hospital and that the reproduction ratio dropped over the second and
third months of the study. Riley et al. (2003) used estimated parameters to
simulate the epidemic and effect of interventions such as reduced time to
hospital admission, reduced population movement and reduced contact
rate.
Lipsitch et al. (2003) used the early exponential curves of infectious cases in
a number of settings and the generation time to estimate the basic reproduc-
tion ratio. These authors investigated the outcome of heterogeneity number
of secondary cases per primary case. By increasing the variance to mean ratio
using different negative binomial distributions to model the number of sec-
ondary cases, they showed that the higher the variance to mean ratio (het-
erogeneity) of secondary cases, the less likely it would be that an epidemic
occur. Greater basic reproduction ratios and larger numbers of seeding cases
had the opposite effect. The authors went on to predict the effects of quaran-
tining contacts and reducing the effective infectious period through isolation
40 Chapter 2. Literature review and outline of thesis
on the reproduction ratio. An paradoxical finding was that as quarantine be-
came more pervasive, less total case-time was spent in quarantine, reflecting
reduction in the size of the outbreak.
While these two studies were comprehensive accounts of early epidemics
of SARS, further advances have been made. Chowell et al. (2003) incorpo-
rated heterogeneity of susceptibles, which could account for the observed
difference in infection rates in children. Lloyd-Smith et al. (2003) simulated
a number of interventions specifically relevant to the hospital environment.
Hsieh et al. (2004) and Wang and Ruan (2004) incorporated a “suspected
SARS” group into their studies. Those who meet some but not all criteria for
SARS are an important group for logistic and economic models of SARS in
the hospital setting as they consume resources and bed capacity, which are
critical in preparing for pandemics.
Lloyd-Smith et al. (2005) extended the work of Lipsitch et al. (2003) to exam-
ine the effect of heterogeneity of secondary cases of a number of different
contagions including SARS, measles and influenza. The findings are that as
the heterogeneity of spreading increases, the likelihood of an outbreak de-
creases, the likelihood of stochastic fade-out increases and the expected im-
pact of control efforts increases. If an epidemic does occur, however, it is
more likely to progress rapidly.
Wallinga and Teunis (2004) used a different approach to estimate the basic
reproduction ratio. They calculate the likelihood of “who infected whom”
based on the time difference in presenting symptoms, and the distribution
of the serial interval. The likelihood methodology applied in that study did
not require an underlying structural model such as the SEIR or SEIRHD and
therefore does not take on these models’ assumptions and weaknesses. Glass
et al. (2006) considered hidden infectious patients in their study. Using a
Bayesian approach and modelling generations of SARS cases, these authors
allowed for a proportion of undiagnosed cases.
2.3.5 Other important models of transmission of nosocomial
pathogens
Cooper et al. (2004) extended the nosocomial transmission model by exam-
ining the effects of the inclusion of compartments representing a population
outside the healthcare setting. In this model, patients could be colonised or
uncolonised, isolated or in hospital without isolation and in the community.
2.3 Relationship of current literature to thesis 41
Healthcare worker compartments were not included in this model. Patients
were followed after discharge and could be re-admitted prior to decoloni-
sation. The study showed that, while the basic reproduction ratio of a single
admission might be less than unity, the overall reproduction ratio, taking into
account readmission, could be greater than unity. The effect of that phenom-
enon was that as the level of MRSA increased in the community, isolation
facilities reached capacity and control of transmission was lost. A notable
finding of this study was that two steady states for the proportion of patients
colonised were possible under conditions of limited isolation capacity; one
in which isolation capacity was exceeded and patients were forced into reg-
ular ward beds and one in which isolation continued to contain colonised
patients and reduce transmission.
Smith et al. (2005) extend this model by performing a utility analysis of in-
fection control strategies of single hospitals within a group of hospitals shar-
ing the same population. The authors assumed that patients who acquired
MRSA at one hospital, and were colonised on discharge would be readmitted
randomly to any of the other hospitals. Not surprisingly, results of this study
predict that, economically, the outcome for a particular hospital is best if it
is surrounded by other hospitals investing in infection control and that the
contribution of a hospital to its own colonisation prevalence diminishes as
the number of hospitals in a region increases. The study also predicts that as
transmissibility of an agent increases, the value of infection control reduces.
Perencevich et al. (2004) used a single population model to investigate the
value of active surveillance of VRE. Patients were divided into several groups
depending on their isolation and colonisation status. The study found that
active surveillance was predicted to reduce the transmission of VRE, while
passive surveillance had minimal impact. A limitation in this study was that
it assumed that patients could have their colonisation status determined
within one day, a result rarely achieved in clinical practice. In reality patient
colonisation status is often determined after the patient has left the ward.
Bootsma et al. (2006) performed a simulation study of MRSA eradication us-
ing isolation and decolonisation. A traditional microbiological culture de-
tection approach was compared with a rapid detection approach. The study
showed that the use of rapid screening could reduce prevalence even if the
screening was less sensitive and specific compared with culture. It was pre-
dicted to take years to reach a new equilibrium prevalence after the strategy
was implemented. This study assumed that all patients were screened on ad-
mission, a protocol that has variable adherence in Australian hospitals. It also
42 Chapter 2. Literature review and outline of thesis
assumed that isolation was 100% effective, which has not been proven, and
unfortunately, evidence to the contrary is beginning to emerge (Cepeda et al.,
2005; Forrester and Pettitt, 2005). Like the studies by Cooper et al. (2004) and
Smith et al. (2005), this study examined the effects of groups of hospitals who
share patients, and the impact on the behaviour of one on the outcomes of
the others.
There are many other relevant examples of mathematical models of hospital
acquired infectious diseases. Compartmental models have been developed
to predict the ecological consequences of antibiotic exposure on the compe-
tition between susceptible and resistant bacteria (see, for example, Lipsitch
et al. (2000)). Huovinen (2005) and Magee (2005) have developed models to
predict the impact of antibiotic cycling.
This thesis has limited its scope to models that predict the transmission of
pathogens from host to host in the healthcare setting. Other areas being ex-
plored by mathematical models include community-based strategies from
infectious diseases such as H5N1 influenza, HIV, tuberculosis and schistoso-
miasis to name a few. Intra-host dynamics of chlamydia (Wilson et al., 2004),
HIV (Davenport et al., 2006) and antibiotic resistant pathogens (Austin and
Anderson, 1999) is another area of active research.
2.4 Review of methodology used in stochastic epi-
demic modelling
The section reviews statistical methods applicable to stochastic epidemic
modelling, highlighting the needs for such methodology and the reasons
for the choice of methods made in this thesis. Some of valid methods not
utilised in this thesis are also reviewed.
There are several challenges and solutions in stochastic epidemic modelling.
1. The data are serially dependent, owing to the change of transmission
pressure with each successive infection event. Methodology used to in-
corporate serial dependence is reviewed in Section 2.4.2. This method
is utilised in Chapters 3 and 7.
2. Events are censored; data being either missing or unobservable (in the
case of transmission data). Data augmentation is used in Chapters 3
and 7 to “stand in” for missing data. A review of methods for manag-
ing censored data is given in Section 2.4.3. The hidden Markov model
2.4 Review of methodology used in stochastic epidemic modelling 43
structure can be applied to very sparse datasets. The likelihood can be
calculated in the absence of individual event data (exposure times on-
set of symptoms). Chapter 5 applies a hidden Markov model to serial
prevalence data. The applications of HMMs is reviewed in Section 2.4.4.
3. Bayesian inference was used in Chapters 3, 5 and 7. Following Bayesian
methodology has the advantages of providing a full posterior probabil-
ity distribution of model parameters, and complex functions of model
parameters. It also provides a framework for data augmentation and
Monte-Carlo Markov chain (MCMC) integration, allowing otherwise in-
tractable integrals to be numerically evaluated. Bayesian inference is
introduced in Section 2.4.1 and further discussed in Section 2.4.3
4. When MCMC integration is used, one needs to ensure that the parame-
ter space has been fully explored. This can be tested by assessing for
convergence, as described in Section 2.4.5 and applied to Chapters 3
and 5.
5. Model adequacy (ability to fit the data) can be assessed through a num-
ber of methods, reviewed in Section 2.4.6.
6. Model selection and comparison helps improve statistical inference.
The study in Chapter 5 used model selection to infer the source of
VRE acquisition and whether transmission rates changed over time.
Chapter 7 used model selection to infer the shape of the individual
infectivity profile for SARS.
2.4.1 Bayesian inference
There is a large philosophical difference between the frequentist view of
parameter estimation and the Bayesian view. The Bayesian framework treats
model parameters as random whereas frequentists regard parameters as
fixed and express uncertainty in terms of data replicates.
Prior information on the parameters, p(θ) is required to calculate the
Bayesian posterior distribution
p(θ|y) =p(y)p(y|θ)
p(y)(2.16)
∝ p(y)p(y|θ). (2.17)
44 Chapter 2. Literature review and outline of thesis
The denominator in Expression (2.16) is not a function of the parameters.
When making inference about the parameters, it is therefore sufficient to use
the proportion (2.17).
One of the advantages of Bayesian inference is that it derives the full proba-
bility distribution of the parameters, rather than just the standard errors, as
in frequentist approaches. Additionally, Bayesian inference can be used to
determine the posterior distribution of functions of parameters.
E[f(θ)|y] =
∫f(θ)p(θ)p(y|θ) dθ∫
p(θ)p(y|θ) dθ. (2.18)
The function may be a simple point summary such as the mean or median, or
may be more complex, such as the proportion used in Chapter 5 Expression
(5.7). Because the integrals in Expression (2.18) are rarely able to be evaluated
analytically, we may evaluate E[f(θ)|y] by Monte-Carlo integration, drawing
samples θk, k = 1, ..., m from p(θ|y) and approximating
E[f(θ)|Y ] ≈ 1
m
m∑
k=1
f(θk), (2.19)
(Gilks et al. (1996, Chapter 1)).
Draws from the posterior probability distribution can be realised with
the Gibbs sampler, used in Chapters 3, 5 and 7; and the Metropolis and
Metropolis-Hastings algorithms, used in Chapters 5 and 7. The Metropolis-
Hastings algorithm entails initialising θ, proposing successive new values of
the model parameters, θ′, determining the likelihood, p(y|θ′), and accepting
the new value θk+1 = θ′ according to the acceptance probability
pacc = min
(1,
p(θ′)p(y|θ′)q(θ′ → θk)
p(θk)p(y|θk)q(θk → θ′)
), (2.20)
where q(.) is the proposal probability. If the proposed value is rejected, θk+1 =
θk otherwise θk+1 = θ′. Successive values of θk+1 form a Markov chain with
a transition kernel p(θk+1|θk) that guaranties that the target density, p(θ|y), is
the stationary distribution of the Markov chain, see Gilks et al. (1996, Chapter
1) for the proof.
When p(θ)p(y|θ) is easily determined, we can propose new θ′ with a proposal
distribution
q(θk → θ′) ∝ p(θ)p(y|θ) (2.21)
2.4 Review of methodology used in stochastic epidemic modelling 45
which leads to pacc = 1 so that the proposal is always accepted. This is the
Gibbs step. We can also make the proposal distribution symmetrical (q(θk →θ′) = q(θ′ → θk)) leading to the Metropolis algorithm.
Updating all of the model parameters at the same time can lead to very low
acceptance probabilities and is unnecessary. Rather, each component of θ
can be updated individually, using Metropolis Hastings or Gibbs steps based
on the full conditional distribution for each component.
Some potential problems which may be encountered using Bayesian infer-
ence are inefficient mixing of the Markov chains of the model parameters and
sensitivity of model outcomes to choice of prior probability distributions.
2.4.2 Methods used to manage serial dependence in infec-
tion data
The piecewise constant hazard model incorporates the phenomenon of
serial dependence in time series data based on the changing number of
infectious and susceptible individuals as each transmission event occurs.
The hazard of transmission for each individual is updated as each newly
infectious case arises. The method is flexible and has weak parametric
assumptions, (Lindsey and Ryan, 1998).
Taking the approach of Aslanidou et al. (1998), the time interval over which
a person is at risk of acquiring an infectious disease can be broken into sub-
intervals ij = (τj−1, τj] for j = 1, ..., J , assuming the event occurs at the end
of the sub-interval and there is, therefore, a constant hazard within each sub-
interval, λ(t) = λj for t ∈ ij. Hence the survival function from the first in-
terval to the beginning of interval J is given by
S(t) = e−PJ−1
j=1 λj(τj−τj−1) (2.22)
and the likelihood contribution of a “failure” occurring in interval J is given
by
q(t) = λJe−PJ
j=1 λj(τj−τj−1). (2.23)
This structure has the advantage of being able to incorporate covariates;
λ(t|z) = λjeβZ , where β is the vector of coefficients and Z is the vector of
covariates for each individual.
46 Chapter 2. Literature review and outline of thesis
Greater than one “failure” event can easily be accommodated. Let λ1J de-
note event one and λ2J denote event 2. The survival function from the first
interval to the beginning of interval J is now given by
S(t) = e−PJ−1
j=1 (λ1J+λ2J )(τj−τj−1) (2.24)
The likelihood contribution of a “failure” occurring due to event one in inter-
val J is given by
q(t) = λ1Je−PJ
j=1(λ1J+λ2J )(τj−τj−1), (2.25)
while the likelihood contribution of a “failure” occurring due to event two in
interval J is given by
q(t) = λ2Je−PJ
j=1(λ1J+λ2J )(τj−τj−1). (2.26)
The structure is readily applied to the stochastic epidemic models described
in this thesis, as long as the number of infectious and susceptibles are per-
fectly observed. In mathematical models we assume the hazard of acquisi-
tion of a contagion is a function of the number of susceptibles, Sj , and the
number of infectives, Ij , at time j, that is
λ1j = f(SjIj). (2.27)
A simple algorithm applicable to SEIR, SIR, SIS and SI models with migra-
tion is that a transmission event occurs with a hazard λ1j = βSjIj , while a
removal event occurs with a hazard λ2j = γIj , where γ is the rate of removal
of infectives. If each removal and infection transmission event is observed,
then likelihood calculation is relatively straightforward,
L(β, γ) =n∏
i=1
(γI(ti))(Zrem(ti))(βS(ti)I(ti))
(Ztrans(ti))e−(γ+βS(ti))I(ti)(ti−ti−1), (2.28)
where Zrem(ti) = 1, Ztrans(ti) = 0 indicates that a removal event occurred
at time, ti, and Zrem(ti) = 0, Ztrans(ti) = 1 indicates that an infection event
2.4 Review of methodology used in stochastic epidemic modelling 47
occurred at time t. The maximum likelihood estimate for β is given by
β =
∑ni=1 Ztrans(ti)∑n
i=1 S(ti)I(ti)(ti − ti−1). (2.29)
This solution holds for SIR, SEIR and SI models and is applicable to models
that allow for migration. Expression (2.28) implies that during time inter-
vals in which there are no transmission events, each individual is subject to a
transmission hazard which is constant and conditionally independent.
The relationship between λ, Sj and Ij may be more complex, incorporating
personal risk factors, environmental contamination or alternative models
such as the Greenwood and Reed-Frost assumptions (Becker, 1989).
2.4.3 Methods used to manage censored transmission data
If transmission data are unknown but onset and resolution of symptoms is
perfectly observed, one approach to determining the likelihood of the data is
to assume a constant latent period, µ (Becker, 1989). Each transmission time,
tx, is assumed to occur at time tx = ts−µ, where ts is the time of symptom on-
set. The likelihood function can be calculated by determining the number of
susceptibles at all transmission times tx (all patients who have not displayed
symptoms prior to time tx + µ). The likelihood given in Expression 2.28 can
then be applied.
A drawback of this method is that it requires fully observed symptom onset
times. These may either be missing or yet to occur as in the case of infectious
diseases with long latent periods. Additionally, some model predictions are
highly sensitive to assumptions made regarding latent periods (Lloyd, 2001)
and latent periods of fixed duration are not realistic.
In the absence of exactly observed transmission data, Martingale methods
can be used (Becker and Britton, 1999). These methods allow the derivation
of the mean and standard error of β without knowledge of the epidemic
curve. There exists a Martingale-derived closed-form solution for the mean
and standard error of β when only the initial conditions and final state of the
population (that is S(0), R(0), S(∞) and R(∞)) are known, given by Becker
and Britton.
The Martingale method under incomplete observation, as described by
Becker and Britton, is applicable to single epidemic waves, but is not ap-
plicable in populations with endemic infection due to migration of both
48 Chapter 2. Literature review and outline of thesis
infectious and susceptible individuals, as is the case in populations in this
thesis. Additionally, the datasets in this thesis, while incomplete, have more
data that just the initial and final states of the population, so methods
employed make use of this information.
Chain-binomial methods, also described by Becker and Britton, can be used
to describe incompletely observed epidemics. This model uses generations
of infectious cases, defined by the number of predecessors in the chain trac-
ing back to the index case. In order to infer the infectivity using such models,
it is necessary to be able to separate the generations of the epidemic, which
proves difficult unless the incubation period is long compared with the in-
fectious period or unless contacts between susceptibles and infectives are
easily defined, such as with sexually transmitted diseases. Additionally, esti-
mating time dependencies in parameters, such as infectivity, is not possible
if there is overlap in time between generations. The chain-binomial method
also breaks down when unobserved new introductions of infectious cases oc-
curs as occurred for example in the study described in Chapter 5 of this thesis.
Latent Variables
This section summarises work by Gilks et al. (1996, Chapter 15.2 and 15.3),
Gelman et al. (2004, Chapter 12) and Ridall (2005, Chapter 2).
Missing data in stochastic epidemic models can be imputed using latent vari-
ables. A set of latent variables, z, and a set of observations, y, form an aug-
mented dataset. Latent variables could be either truly missing data, an unob-
servable process or an auxiliary variable introduced into the model for con-
venience.
The probability of observations, given the augmented data and the model pa-
rameters, p(y|z,θ), is called the conditional probability of the observations.
The joint probability of the unobserved data and the observations, the com-
plete likelihood, is given by
p(y,z|θ) = p(y|z,θ)p(z|θ). (2.30)
The marginal distribution of y is given by
p(y|θ) =
∫p(y,z|θ)dz. (2.31)
2.4 Review of methodology used in stochastic epidemic modelling 49
The value of introducing the latent variable z into the model is clear when
the complete likelihood, p(y,z|θ), has a much simpler form than the mar-
ginal likelihood, p(y|θ), as is the case when there are missing data and one
wishes to apply the piecewise constant hazard to determine the likelihood of
a dataset.
Latent variables can be used to extend the range of distributions that can
be modelled (Damien et al., 1999) and simplify model computations. They
have also been shown to enhance convergence (Besag and Green, 1993).
This thesis uses latent variables to represent unobserved MRSA acquisition
times (Chapter 3), unknown number of VRE colonised patients (Chapter 5)
and missing hospitalisation data and unobserved SARS transmission times
(Chapter 7).
The integral required to evaluate the marginal distribution is often difficult or
intractable. The following discussion reviews methods used to tackle latent
variable problems.
The Expectation-Maximisation algorithm
The EM algorithm aims to maximise p(y|θ) in the presence of latent variables,
z. It involves iterating through successive expectation and maximisation
steps until convergence is apparent. The expectation step evaluates the
expected value of the log likelihood of the complete dataset, and the current
value of the model parameters, θm, which is denoted by
Q(θ|θm) = E|θm,y[log p(y,z|θ)|y]. (2.32)
Here the expectation is with respect to p(z|θm,y). In other words
Q(θ|θm) =
∫p(z|θm,y) log p(y,z|θ)dz, (2.33)
in the continuous case or
Q(θ|θm) =∑
z
p(z|θm,y) log p(y,z|θ), (2.34)
in the discrete case. The Maximisation step involves maximising Q(θ|θm) to
find θm+1.
50 Chapter 2. Literature review and outline of thesis
One shortcoming of the EM algorithm is that it iterates to monotonically in-
crease the likelihood function and hence may converge to a local maximum.
Additionally, the E-step and the M-step do not in general have analytical solu-
tions. The integral in Expression (2.35) may be intractable particularly when
the number of dimensions of z or θ is large.
Stochastic EM (Gilks et al., 1996, Chapter 15.3)
The stochastic EM algorithm was introduced by Celeux and Diebolt (1985)
to manage the intractable nature of the EM algorithm in some settings. The
E-step is replaced by a “Simulation Step” which involves imputing missing
variables using plausible values given the observations and current model
parameters (Gilks et al., 1996, Chapter 15.3). Instead of finding the expecta-
tion of the complete log-likelihood with respect to p(z|θm,y), a single draw
from p(z|θm,y) is made. This “pseudo-complete” sample of z can then be
used in the M-step, which is maximised for fixed z, based on the random
draw, hence avoiding integral (2.35). Each successive value of θ becomes a
step in a Markov chain which converges to an approximately stationary dis-
tribution.
Stochastic EM has the advantage of leading to rapid convergence and
tractable computations on occasions when Equation(2.35) is intractable.
The stationary distribution can generate a plausible range for θ, giving a
measure of uncertainty of the estimate. In addition the imputed values of z
give estimates of missing data or auxiliary variables. Neal and Hinton (1999)
demonstrated that simply finding a new estimate of the model parameters,
θm+1, that gives some increase in the value of Q(θ|θm) over its current value
(rather than maximizing Q(θ|θm)) will also ensure successful convergence.
The Bayesian approach to EM
In the fully Bayesian context, the aim is to find the posterior probability dis-
tribution of the parameters p(θ|y), rather than the marginal likelihood p(y|θ).
The posterior probability distribution can be determined by integrating over
the latent variables
p(θ|y) =
∫p(zθ|y)dz. (2.35)
2.4 Review of methodology used in stochastic epidemic modelling 51
In the Bayesian framework, latent variables are treated in the same way as
the model parameters, each initialised and updated in turn using detailed
balance acceptance equations (Gelman et al., 2004).
This results in the full posterior probability distributions of latent variables,
model parameters and functions of each being available. This was the ap-
proach used in Chapters 3, 5 and 7 in this thesis. The full details of the ap-
proach are outlined in each chapter.
2.4.4 Hidden Markov models
Hidden Markov models (HMMs) are useful in dealing with data in which an
underlying process cannot directly be observed, but there is a clear, defin-
able relationship between the underlying state and the observations. HMMs
consist of hidden states, X, observations, Y , a model describing the transi-
tion between the hidden states, Pr(Xi = j|Xi−1 = i), a model describing the
relationship between the observations and the hidden states, Pr(Yi|Xi), and
the parameters that make up the models. The underlying process must be
expressed as a sequence of states which evolve in a Markov manner, that is
Pr(Xi|X1, X2, ..., Xi−1) = Pr(Xi|Xi−1). Observations must be dependent on
the underlying hidden states only.
HMMs were developed by Baum (1966). One of the first applications of
HMMs was speech and character recognition (Baker, 1975). More recently,
HMMs have been used in medical fields including bioinformatics (Boys
et al., 2000), spatial disease mapping (Green and Richardson, 2002), and
neurophysiology (de Gunst et al., 2001).
A number of algorithms have been developed to explore the properties of
HMMs. The Viterbi algorithm is ideal for exploring hidden states as is re-
quired for speech recognition (Viterbi, 1967). In other applications, the hid-
den states are of less interest and the aim is estimation of model parameters.
The Welsh-Baum algorithm is useful for determining the marginal likelihood
of the data when hidden states are not required (Baum et al., 1970). Details
of the Welsh-Baum algorithm and its application to the model developed in
Chapter 5 are given in the methods section of that chapter. Chapter 5 also
made use of work by Scott (2002), who adapted the forward backward re-
cursion to the Bayesian setting, implementing Gibbs updates of the hidden
states, so that hidden states and parameters could be updated sequentially.
52 Chapter 2. Literature review and outline of thesis
2.4.5 Assessing convergence of Markov Chain Monte Carlo
algorithms, adapted from Gelman et al. (2004, Chapter
11.6)
Using MCMC integration to estimate the posterior distributions of para-
meters or functions of parameters has potential difficulties. If the iterative
process has not been sufficiently long, the parameter space may not have
been adequately explored and the simulations drawn from the MCMC may
not represent the true posterior distribution. Assessment of convergence
of the Markov chains can be performed either visually (taking different
starting points throughout the parameter space and observing the mixing
of the chains) or quantitatively. The latter method of assessing convergence
compares the intra-chain variance with the inter-chain variance. When
these two measured are approximately equal, it is assumed that the target
distribution has been reached.
To evaluate convergence, one approach is to simulate a number of Markov
chains, say m chains of length n (n being the length of the chain after the
burn-in period is excluded). Each simulation drawn from the Markov chains
is labelled φij (i = 1, ..., n; j = 1, ..., m), where φ is the estimand of interest. Let1nφ.j =
∑ni=1 φij and 1
nmφ.. =
∑mj=1
∑ni=1 φij . The convergence of each scalar
estimand of the model can be measured by
R =
√n− 1
n+
A
B, (2.36)
where
A =1
m− 1
m∑j=1
(φ.j − φ..)2 (2.37)
is the variance of the chain means and
B =1
m
m∑j=1
1
n− 1
n∑i=1
(φij − φ.j)2, (2.38)
is the mean of the chain variances. As n → ∞, we expect R → 1; however,
under usual conditions, values of below 1.1 suggest reasonable convergence
(Gelman et al., 2004, Chapter 11.6).
This thesis employs the visual method of assessing convergence in Chapter 7
and the quantitative method in Chapters 3 and 5.
2.4 Review of methodology used in stochastic epidemic modelling 53
2.4.6 Model checking and improvement
A number of methods for determining model adequacy have been described.
Each works on the basis that if a model fits the data well, replicated data gen-
erated under the model should look similar to observed data. Below is a brief
review of methods applicable to Bayesian model testing.
Cross-validation
In this method, subsamples of the data are used to estimate the model pa-
rameters, the remainder of the data is used for model validation. Different
approaches include; randomly omitting data, and using that data for train-
ing, dividing the data into equal subsamples, each of which are used in turn
as the validation data, and leaving out a single observation at a time and us-
ing the remainder of the data as training data.
Posterior predictive assessment
This method involves simulating data using model parameters and compar-
ing observed and simulated data sets (Gelman et al., 2000). Test quantities
can be either based on data alone, T (y) or on data and model parameters,
T (y,θ). The discrepancy between the actual data and the simulated datasets
can be measured in a number of ways such as a scatterplot of T (yobs, θ) ver-
sus T (yrep,θ) or a histogram of the differences. If a scatterplot is used, one
would expect an equal number of values to fall above and below the line of
unit slope, if a model is adequate. The proportion of values that lie above the
slope is the Bayesian p-value. The choice of test quantity should reflect the
scientific purpose of the model inference (Gelman et al., 2004). A number
of discrepancy measures have been used including standardised residuals,
Pearson chi-squared discrepancy (Gelman et al., 2000) and deviance. Poste-
rior predictive assessment can be used to extend the model inference, for ex-
ample, by correlation of model residuals with putative explanatory variables.
Posteriors from the simulation
This method, described by Dey and Vlachos (1995), simulates data using pa-
rameter estimates and determines the posterior probability distribution for
the parameters based on the simulated dataset. The posterior distributions
as estimated by the actual and simulated datasets can then be compared.
This method was used to test model adequacy in Chapters 3 and 5.
54 Chapter 2. Literature review and outline of thesis
External validation
Ideally, models are also validated using independently collected information.
In Chapters 5 and 7, model results were compared with biological data.
2.4.7 Model selection and comparison
The objective of model selection is to optimise the quality of model inference.
The principle of parsimony means that the simplest possible model should
be chosen. Excessive numbers of parameters in a model, leads to poor preci-
sion, obscuring true effects or identifying effects that are spurious while too
few variables lead to model bias (Burnham and Anderson, 2004).
The model deviance is a measure of model fit, given by
D(y, θ) = −2 log(p(y|θ)). (2.39)
Using the deviance as the sole model selection criterion will favour the se-
lection of the highest dimension model when models are nested. Therefore
model comparison methods must include a penalty term for the number of
model parameters. All methods of model comparison discussed in this sec-
tion aim to derive the deviance of a model as well as a measure of model
complexity.
Different approaches to model selection and their relative merits is an area
of intense research. The following is a very brief discussion on three model
selection criteria, the Akaike Information Criterion (AIC) the Bayesian Infor-
mation Criterion (BIC), and the Deviance Information Criterion (DIC) and
their uses. Both DIC and AIC were used in this thesis in Chapters 5 and 7
respectively.
The Akaike Information Criterion and the Bayesian Information Criterion
The Akaike Information Criterion (Akaike, 1974) is used as a measure of the
predictive power of a model. It is given by
AIC = D(y, θ) + 2p, (2.40)
where p is the number of parameters in the model. It is used by frequentists
and gives similar results to the DIC when prior information is negligible. It
2.4 Review of methodology used in stochastic epidemic modelling 55
is not useful when the number of model parameters is not clearly defined
or when prior distributions, collinearity or complex hierarchical models re-
duced the number of effective model parameters (Spiegelhalter et al., 2002).
The AIC was used in Chapter 7 of this thesis to select among different mod-
els for the individual infectivity profiles. The AIC was an appropriate model
comparison tool in this context because there was little information on the
prior distributions of the model parameters and the models were not com-
plex, therefore the number of parameters was easily determined.
Supposing there are m models to choose from; in the Bayesian context, model
Mi has a posterior probability given by
p(Mi|y) =p(y|Mi)p(Mi)∑m
j=1 p(y|Mj)p(Mj), (2.41)
Bayes factor is the ratio of the marginal likelihoods of two models given by
Bij =p(y|Mi)
p(y|Mj). (2.42)
This model comparison is used with the aim of finding which of two models
is most probable; that is, it assumes that there exists one “true” model. Scharz
(1978) showed that asymptotically (for large number of observations, N),
−2log(Bij) = −2 logp(y|θi, i)
p(y|θj, j)− (ni − nj) log N, (2.43)
where ni and nj are the number of parameters in models Mi and Mj respec-
tively.
An approximation to Bayes Factor, the BIC, is defined as
BIC = D(y, θ) + p log N, (2.44)
where p is the number of model parameters and N is the number of observa-
tions in the model.
Deviance Information Criteria
The advantages offered by the DIC are that it allows for the reduction in the
complexity of a model when variables are correlated (and therefore the ef-
fective number of parameters is reduced). It also allows for the calculation
56 Chapter 2. Literature review and outline of thesis
of the effective number of model parameters (Spiegelhalter et al., 2002). For
this reason it is often the best method of comparison of models in which the
number of parameters is not readily identified.
The posterior mean of the deviance is obtained by averaging the deviance
over the posterior distribution of the model parameters and is given by
D(θ) =1
L
L∑
l=1
D(y, θl), (2.45)
where θ1, ..., θL are the components of the Markov chain in stationary distri-
bution representing the posterior distribution of θ.
The deviance at a point estimate of θ is given by
D(θ) = D(y, θ). (2.46)
The value used for the point estimate is often the mean, but the mode or me-
dian could also be used. The effective number of parameters is given by
pD = D(θ)−D(θ). (2.47)
The DIC is defined as
DIC = D(θ) + pD. (2.48)
The increase in likelihood, or reduction in deviance, that occurs with in-
creased model parameters is thus compensated for by the term, pD, which
is the effective number of parameters (Gelman et al., 2004, Chapter 5.7). An
advantage of using the DIC is that calculation is trivial once an appropriate
Markov chain Monte Carlo algorithm is in place (Spiegelhalter et al., 2002).
The DIC was chosen as the method of model comparison for Chapter 5
of the thesis because the effective number of parameters was not readily
calculated.
Other Model Selection techniques
The AIC, BIC and DIC all require setting up and running a finite number
of models and comparing the deviance and number of parameters of each.
2.5 Outline of thesis 57
Some model comparison methods circumvent this by allowing the Markov
chain itself to traverse model space. Such techniques include the metropo-
lised Carlin and Chibb algorithm (Carlin and Chib, 1995; Dellaportas et al.,
1998), the reversible jump MCMC (Green, 1995) and birth death MCMC
(Stephens, 2000).
2.5 Outline of Thesis: Account of research progress
linking the papers
Contribution of Chapter 3 to area of research
The data in Chapter 3 comes from the Princess Alexandra Hospital, an 800
bed tertiary referral public Australian teaching hospital. The dataset was
derived from the APACHE IIITM database (admission and discharge dates),
AUSLABTMdatabase (microbiology results), eICATTMdatabase (record of new
or old MRSA colonisation) and, where necessary, patient notes. Consecutive
patient admissions (from 8th August 2001 to 3rd March 2004) to the 16 bed
Intensive Care Unit. Other inclusion criteria were inclusion in the APACHE
IIITM data base.
The study modified the Ross-MacDonald model to predict the impact of
interventions that are relevant to current local practice, proposed practice
or potential future practice and not yet considered by other studies. These
include the predicted impact of decolonisation using enteral vancomycin,
the effect of ward size and the impact of HCW-patient ratios assuming fixed
numbers of contacts per HCW. The study in Chapter 3 was the first of the
Ross-MacDonald models to include model parameters estimated solely
from ward observations or the time series data itself. No expert opinion or
literature estimates were used in the study.
An important difference between the structure of the model presented
in Chapter 3 and the model that forms the basis of the studies of Austin
et al. (1999), Grundmann et al. (2002) and Raboud et al. (2005) is that the
decontamination of HCWs in the aforementioned studies was assumed to
occur at an arbitrary, estimated rate, with a mean of one hour. In these
studies, decontamination was assumed to be independent of hand hygiene
compliance. This assumption leads to an underestimates of the impact of
hand hygiene on transmission. In the study in Chapter 3, decontamination
occurs at a rate dictated by hand hygiene (as does the model presented by
58 Chapter 2. Literature review and outline of thesis
Cooper et al. (1999)). This assumption has the additional advantage that
it avoids unnecessary guesswork in parameter estimation as hand hygiene
compliance can be measured, as it was during the study in Chapter 3.
Two of the parameters used in this study derived from observational stud-
ies performed on the ward during the study period. One study estimated the
hand hygiene compliance, via a series of covert observation periods (Whitby
and McLaws, 2004), the other estimated the probability of MRSA transmis-
sion per discordant contact (McBryde et al., 2004). Chapter 4 gives an ac-
count of the latter study.
The data used in the study in Chapter 3 were interval-censored, and serially-
dependent. Transmission, an unobserved processes, was incorporated into
the model as a latent variable in a Bayesian context making use of Markov
chain Monte Carlo (MCMC) integration. The transmission parameter was es-
timated within the structure of the model, using a piecewise constant hazard
formula, as described in detail in Chapter 3.
Estimates were made of the basic reproduction ratio and attack rate. Outputs
of the model include predictions of the effect of hand hygiene, HCW/patient
ratios, decolonisation and ward size on MRSA transmission.
An important advance in the model presented in Chapter 3 was the way in
which HCW/patient ratio is examined. The study by D’Agata et al. (2005)
predicted the impact of HCW/patient ratio assuming each patient received
a fixed number of contacts. With patient cohorting and fixed contact rates
per patient, increase in staff was predicted to reduce transmission. Chapter
3 investigated an alternative scenario, that the staff make a fixed number of
contacts per day. This assumption led to completely different predictions as
discussed in Chapter 3. We used the latter assumption because staff in the
hospital under study describe a saturated work environment in which there
is always more work to do, that is, more contacts to make. We therefore as-
sume that more staff would lead to more contacts.
The study described in Chapter 3 did not select among alternative models
and assumed that all MRSA acquisition took place as a result of indirect ward
transmission. These deficiencies were addressed in the Chapter 5.
2.5 Outline of thesis 59
Contribution of Chapter 4 to area of research
Chapter 4 is an original study that took part on the hospital ward. The aim
of the study was to estimate an important parameter used in the transmis-
sion model, the probability of transmission of MRSA during a single patient.
The study observed routine contacts between HCWs and patients known to
be MRSA colonised. Using standard “glove juice” methods, the study deter-
mined whether contamination of healthcare workers gloves occurred. It also
measures the compliance of different groups of healthcare workers to infec-
tion control protocol (glove use).
The measurement of this parameter was important firstly because it had
not specifically been measured previously and was in itself of interest for
infection control. Secondly, it enabled more accurate modelling of the MRSA
transmission data as outlined in Chapter 3, including the development of a
model in which all parameters were estimated from the ward or fitted to the
data and no parameters were derived through expert opinion or guesswork.
Contribution of Chapter 5 to area of research
The dataset in Chapter 5 was serial VRE colonisation prevalence data, col-
lected as part of a period of intensive surveillance across three wards in the
Princess Alexandra Hospital. The aim of the study was to determine the pro-
portion of VRE acquisitions that was due to cross-transmission.
Chapter 5 applied the two compartment, single population model de-
scribed in Section 2.2.6 and allowed for two sources of VRE acquisition,
cross-transmission and a sporadic source, independent of ward cross-
transmission. The added complexity was necessary because VRE colonisa-
tions are known to arise from the patients endogenous flora or from sources
outside the ward such as other wards within the hospital, other healthcare
institutions and the community. This model would also be highly applicable
to organisms that readily acquire de novo resistance such as Pseudomonas
aeruginosa (for quinolones).
Because the exact number of patients who were colonised was not directly
observed in the dataset available, a hidden Markov model structure was used
to estimate transmission characteristics. The data are weekly prevalence
data, reflecting the practice of weekly swabs (rather than continuous preva-
lence data as in the study by Pelupessy et al. (2002)). The model developed
in the study allows that many events (colonisation, discharge, readmission)
60 Chapter 2. Literature review and outline of thesis
may have taken place between observations (weekly prevalence checks). It
also takes into account that prevalence checks do not detect all colonised
patients, accounting for imperfect swab sensitivity and incomplete testing
of patients. Unlike the HMM presented by Cooper and Lipsitch (2004), the
relationship between the hidden state (true number of colonised patients)
and the observations (observed prevalence of colonised patients) has a clear
interpretation. Each colonised patient has a probability of detection leading
to a binomial relationship between number colonised (the hidden state) and
number detected (the observation).
Unlike the studies by Pelupessy et al. (2002) and Cooper and Lipsitch (2004),
the study in Chapter 5 did not encounter difficulties with collinearity of the
parameters. This is likely to be because of the larger dataset used in this study
and because the data consisted of a long period of little colonisation, followed
by a large outbreak of colonisation. Most of the information for estimating
the “sporadic colonisation” parameter would have been acquired during the
former period, while the information used to estimate the cross-transmission
parameter would mostly have been acquired from the latter data.
Chapter 5 used a model comparison technique, the Deviance Information
Criterion (DIC), as described in Section 2.4.7 to select among a number of dif-
ferent putative models. Models considered included one in which the cross-
transmission term was omitted, a model in which the sporadic colonisation
term was omitted and models in which the cross-transmission term was time
dependent. The study in Chapter 5 used external validation through compar-
ison with genotyping data, internal validation using simulation and model
selection in order to optimise and validate the model.
The advantage of this model is that it can be applied to imperfect datasets
of the type often collected for infection control surveillance. Vast amounts
of data are now being collected on nosocomial pathogens such as MRSA and
VRE, but little of it is complete or produced for statistical analysis. The model
described in Chapter 5 did not take into account the possibility of an envi-
ronmental reservoir. This was addressed in Chapter 6.
Contribution of Chapter 6 to area of research
A new model is proposed in Chapter 6 which includes an environmental
reservoir compartment and explores the impact of this reservoir on predic-
tions regarding infection control interventions. This model used parameters
2.5 Outline of thesis 61
derived from a study of VRE transmission (D’Agata et al., 2005), but is
potentially applicable to other pathogens with a substantial environmental
reservoir.
The study predicts that the presence of an environmental reservoir would re-
duce the effect of a number of infection control measures and under some
conditions will lead to endemic ward colonisation of nosocomial pathogens
despite control measures which would otherwise be predicted to eliminate
transmission. The first part of Chapter 6 presents the paper verbatim as it
appeared in the Journal of Infectious Diseases; the second part of Chapter 6
gives the full details of the model.
Contribution of Chapter 7 to area of research
Data collected by the Taiyuan Centre for Disease Prevention and Control in
Shanxi province, China was made available for the work described in Chapter
7. This is a unique database that has not been published elsewhere.
To estimate the incubation period, a parametric approach was used, fitting a
Gamma distribution. The study departs from the approach by Donnelly et al.
(2003) and Meltzer (2004) by including a model for time to transmission dur-
ing exposure of an uninfected person to a known SARS case. The assumption
of a constant hazard of transmission during a contact with a SARS case has a
biological basis (as compared with a uniform probability of transmission that
was implicitly assumed in the other studies).
To estimate infectivity, the SEIR model was adopted with some modifica-
tions:
• the proportion of susceptibles is assumed to remain at unity
• the survival times in the compartments is not negative exponential, in-
stead fitted to data using Gamma distributions
• the population is stratified into 2 subpopulations, those in and those
out of hospital
• the epidemic is divided into two time periods, corresponding to the
waxing and the waning of the epidemic.
The latter two modifications were necessary to answer crucial questions.
How did hospitalisation and interventions impact on infectiousness?
62 Chapter 2. Literature review and outline of thesis
Three different models of infectivity profiles over the course of SARS-CoV in-
fection were considered in this study. The model considering a Gamma shape
for infectivity appeared statistically slightly superior to the model assuming
uniform infectivity, using the Akaike Information Criterion (AIC) for model
comparison (see Section 2.4.7). Of interest is that the estimated peak infec-
tivity occurs on the ninth day following symptom onset. This is consistent
with virological results of Peiris et al. (2003) and Cheng et al. (2004) .
The Gamma distribution was used for sojourn times in the SEIHRD model of
Chapter 7. Other authors use alternative distributions such as Weibull (Lip-
sitch et al., 2003), and lognormal (Farewell et al., 2005). The Gamma distri-
bution was chosen because it is relatively parsimonious, flexible and readily
adapted simulations (by using a series of α compartments, each with expo-
nential sojourn times with mean length 1/beta to represent the Gamma(α, β)
distribution). Hence the model can be used to predict the impact of infection
control interventions.
Chapter 8 describes what has been achieved by this thesis, limitations in the
studies presented and directions for future work.
Bibliography
Akaike, H., 1974. A new look at the statistical model identification. IEEE Transactionson Automatic Control 19 (6), 716–723.
Anderson, R. M., May, R., 1991. Infectious diseases of humans: dynamics and con-trol. Oxford University Press.
Aslanidou, H., Dey, D., Sinha, D., 1998. Bayesian analysis of multivariate survival datausing Monte Carlo methods. The Canadian Journal of Statistics 26 (1), 33–48.
Austin, D. J., Anderson, R. M., 1999. Studies of antibiotic resistance within the pa-tient, hospitals and the community using simple mathematical models. PhilosTrans R Soc Lond B Biol Sci 354 (1384), 721–38.
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Bailey, N., 1975. The Biomathematics of Malaria. Charles Griffin, London.
Baker, J., 1975. The Dragon System:an overview. IEEE Tansactions of Acoustic,Speech and Signal Processing 23, 24–29.
Baum, L., 1966. Statistical inference for probabilistic functions of finite state Markovchains. Annals of Mathematical Statistics 37, 1554–1563.
Baum, L., Petrie, T., Soules, G., Weiss, N., 1970. A maximisation technique occurringin the statistical analysis of probabilistic functions of Markov chains. Annals ofMathematical Statistics 41, 164–171.
Becker, N., 1989. Analysis of Infectious Diseases Data. Chapman and Hall/CRC.
Becker, N. G., Britton, T., 1999. Statistical studies of infectious disease incidence. JRoy Stat Soc B (Statistical Methodology) 61 (2), 287–308.
Bell, J., Turnidge, J., Coombs, G., O’Brien, F., 1998. Emergence and epidemiology ofvancomycin-resistant enterococci in Australia. Commun Dis Intell 22 (11), 249–52.
Bernoulli, D., 1760. Essai d’une nouvelle analyse de la mortalite causee par la petiteverole et des advantages de l’inoculation pour la prevenir. Mem Math Phys AcadRoy Sci Paris, 1–45.
Besag, J., Green, P., 1993. Spatial statistics and Bayeisan computation. J Roy Stat SocB 55, 25–38.
64 BIBLIOGRAPHY
Bonten, M., Slaughter, S., Ambergen, A., Hayden, M., Van Voorhis, J., Nathan, C., We-instein, R., 1998. The role of ’colonization pressure’ in the spread of vancomycin-resistant enterococci: An important infection control variable. Archives of InternalMedicine 158 (10), 1127–1132.
Bonten, M. J., Austin, D. J., Lipsitch, M., 2001. Understanding the spread of antibioticresistant pathogens in hospitals: mathematical models as tools for control. ClinInfect Dis 33 (10), 1739–46.
Bonten, M. J., Hayden, M. K., Nathan, C., van Voorhis, J., Matushek, M., Slaughter,S., Rice, T., Weinstein, R. A., 1996. Epidemiology of colonisation of patients andenvironment with vancomycin-resistant enterococci. Lancet 348 (9042), 1615–9.
Booth, C. M., Matukas, L. M., Tomlinson, G. A., Rachlis, A. R., Rose, D. B., Dwosh,H. A., Walmsley, S. L., Mazzulli, T., Avendano, M., Derkach, P., Ephtimios, I. E.,Kitai, I., Mederski, B. D., Shadowitz, S. B., Gold, W. L., Hawryluck, L. A., Rea, E.,Chenkin, J. S., Cescon, D. W., Poutanen, S. M., Detsky, A. S., 2003. Clinical featuresand short-term outcomes of 144 patients with SARS in the greater Toronto area. JAm Med Assoc 289 (21), 2801–9.
Bootsma, M., Diekmann, O., Bonten, M., 2006. Controlling methicillin-resistantStaphylococcus aureus: Quantifying the effects of interventions and rapid diag-nostic testing. PNAS 103 (14), 5620–5625.
Boyce, J., Havill, N., Maria, B., Dec 2005. Frequency and possible infection controlimplications of gastrointestinal colonization with methicillin-resistant Staphylo-coccus aureus. Journal of Clinical Microbiology 43 (12), 5992–5.
Boyce, J. M., 2001. MRSA patients: proven methods to treat colonization and infec-tion. J Hosp Infect 48 Suppl A, S9–14.
Boyce, J. M., 2005. UpToDate, Waltham, MA, Ch. Epidemiology and clinical man-ifestations of methicillin-resistant Staphylococcus aureus infection in adults.In:UpToDate, Rose, B.D (ed).
Boyce, J. M., Potter-Bynoe, G., Chenevert, C., King, T., 1997. Environmental conta-mination due to methicillin-resistant Staphylococcus aureus: possible infectioncontrol implications. Infect Control Hosp Epidemiol 18 (9), 622–7.
Boys, R., Henderson, D., Wilkinson, D., 2000. Detecting homogeneous segments inDNA sequences using transition models with latent variables. Applied Statistics49, 9–30.
Burnham, K., Anderson, D., 2004. Multimodel inference: understanding AIC and BICin model selection. Amsterdam Workshop on Model Selection.
Carlin, B., Chib, S., 1995. Bayesian model choice via Markov chain Monte Carlomethods. J Roy Stat Soc B 57, 473–484.
Celeux, G., Diebolt, J., 1985. The SEM algorithm: a probabiliistic teacher algorithmdericed from the EM algorithm for the mixture problem. Comp Stat Quart 2, 73–82.
BIBLIOGRAPHY 65
Cepeda, J. A., Whitehouse, T., Cooper, B., Hails, J., Jones, K., Kwaku, F., Taylor, L., Hay-man, S., Cookson, B., Shaw, S., Kibbler, C., Singer, M., Bellingan, G., Wilson, A. P.,2005. Isolation of patients in single rooms or cohorts to reduce spread of mrsa inintensive-care units: prospective two-centre study. Lancet 365 (9456), 295–304.
Chambers, H., 2001. The changing epidemiology of Staphylococcus aureus? Emerg-ing Infecitous Diseases 7 (2), 178–182.
Cheng, P. K., Wong, D. A., Tong, L. K., Ip, S. M., Lo, A. C., Lau, C. S., Yeung, E. Y.,Lim, W. W., 2004. Viral shedding patterns of coronavirus in patients with probablesevere acute respiratory syndrome. Lancet 363 (9422), 1699–700.
Chowell, G., Fenimore, P. W., Castillo-Garsow, M. A., Castillo-Chavez, C., 2003. SARSoutbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isola-tion as a control mechanism. J Theor Biol 224 (1), 1–8.
Cooper, B., Lipsitch, M., 2004. The analysis of hospital infection data using hiddenMarkov models. Biostatistics 5 (2), 223–37.
Cooper, B. S., Medley, G. F., Scott, G. M., 1999. Preliminary analysis of the transmis-sion dynamics of nosocomial infections: stochastic and management effects. JHosp Infect 43 (2), 131–47.
Cooper, B. S., Medley, G. F., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A.,Duckworth, G., Lai, R., Ebrahim, S., 2004. Methicillin-resistant Staphylococcus au-reus in hospitals and the community: stealth dynamics and control catastrophes.Proc Natl Acad Sci U S A 101 (27), 10223–8.
Cooper, B. S., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A., Medley, G. F.,Duckworth, G. J., Lai, R., Ebrahim, S., 2003. Systematic review of isolation poli-cies in the hospital management of methicillin-resistant Staphylococcus aureus:a review of the literature with epidemiological and economic modelling. HealthTechnol Assess 7 (39), 1–194.
D’Agata, E. M., Gautam, S., Green, W. K., Tang, Y. W., 2002. High rate of false-negativeresults of the rectal swab culture method in detection of gastrointestinal coloniza-tion with vancomycin-resistant enterococci. Clin Infect Dis 34 (2), 167–72.
D’Agata, E. M., Webb, G., Horn, M., 2005. A mathematical model quantifying the im-pact of antibiotic exposure and other interventions on the endemic prevalence ofvancomycin-resistant enterococci. J Infect Dis 192 (11), 2004–11.
Dancer, S., Coyne, M., Robertson, C., Thomson, A., Guleri, A., Alcock, S., 2006. An-tibiotic use is associated with resistance of environmental organisms in a teachinghospital. Journal of Hospital Infection 62, 200–206.
Davenport, M., Zhang, L., Shiver, J., Casmiro, D., Ribeiro, R., Perelson, A., Mar 2006.Influence of peak viral load on the extent of CD4+ T-cell depletion in simian HIVinfection. J Acquir Immune Defic Syndr 41 (3), 259–265.
Damien, P., Wakefield, J., Walker, S., 1999. Gibbs sampling for Bayesian non-conjugative and hierarchical models by auxiliary variables. J Roy Stat Assoc 61,331–334.
66 BIBLIOGRAPHY
de Gunst, M., Kunsch, H., Schouten, J., 2001. Statistical analysis of ion channel datausing hidden Markov models with correlated state-dependent noise and filtering.J Am Stat Assoc 95, 805–815.
Dellaportas, P., Forster, J., Ntzoufras, I., 1998. On Bayesian model and variable se-lection using MCMC. Technical Report, Athens University of Economics and Busi-ness.
Dempster, A., Laird, N., Rubin, D., 1977. Maximum likelihood from incomplete datavia the EM algorithm (with discussion). J Roy Stat Soc B 39, 1–38.
Dey, D. K., G. A. E. S. T. B., Vlachos, P. K., 1995. Technical report 9529: Simulationbased model checking for hierarchical models. Technical Report, Univ. of Conn.
DiazGranados, C., Zimmer, S., Klein, M., Jernigan, J., Aug 2005. Comparison of mor-tality associated with vancomycin-resistant and vancomycin-susceptible entero-coccal bloodstrean infections: a meta-analysis. Clinical Infectious Diseases 41 (3),327–333.
Diekmann, O., Heesterbeek, J., 2000. Mathematical Epidemiology of Infectious Dis-eases: Model Building, Analysis and Interpretation. John Wiley and Son, LTD.
Donnelly, C., Fisher, M., Fraser, C., Ghani, A., Riley, S., Ferguson, N., Anderson, R.,2004. Epidemiological and genetic analysis of severe acute respiratory syndrome.Lancet Infectious Diseases 4 (11), 672–83.
Donnelly, C. A., Ghani, A. C., Leung, G. M., Hedley, A. J., Fraser, C., Riley, S., Abu-Raddad, L. J., Ho, L. M., Thach, T. Q., Chau, P., Chan, K. P., Lam, T. H., Tse, L. Y.,Tsang, T., Liu, S. H., Kong, J. H., Lau, E. M., Ferguson, N. M., Anderson, R. M., 2003.Epidemiological determinants of spread of causal agent of severe acute respira-tory syndrome in Hong Kong. Lancet 361 (9371), 1761–6.
Donskey, C., Chowdhry, T., Hecker, M., Hoyen, C., Hanrahan, J., Hujer, A., Hutton-Thomas, R., Whalen, C., Bonomo, R., Rice, L., 2000. Effect of antibiotic therapy onthe density of vancomycin-resistant enterococci in the stool of colonized patients.New England Journal of Medicine 343 (I), 1925–1932.
Donskey, C. J., Hoyen, C. K., Das, S. M., Helfand, M. S., Hecker, M. T., 2002. Recur-rence of vancomycin-resistant enterococcus stool colonization during antibiotictherapy. Infect Control Hosp Epidemiol 23 (8), 436–40.
Endtz, H. P., van den Braak, N., van Belkum, A., Kluytmans, J. A., Koeleman, J. G.,Spanjaard, L., Voss, A., Weersink, A. J., Vandenbroucke-Grauls, C. M., Buiting,A. G., van Duin, A., Verbrugh, H. A., 1997. Fecal carriage of vancomycin-resistantenterococci in hospitalized patients and those living in the community in theNetherlands. J Clin Microbiol 35 (12), 3026–31.
Enright, M., Robinson, D., Randle, G., Feil, E., Grundmann, H., Spratt, B., 2002. Theevolutionary history of methicillin-resistant Staphylococcus aureus (mrsa). ProcNatl Acad Sci USA 99, 7687–92.
Ericksen, K., Erichsen, I., 1963. Clinical occurrence of methicillin-resistant strains ofStaphylococcus aureus. Ugeskr Laeger 125, 1234–40.
BIBLIOGRAPHY 67
Farewell, V. T., Herzberg, A. M., James, K. W., Ho, L. M., Leung, G. M., 2005. SARS incu-bation and quarantine times: when is an exposed individual known to be diseasefree? Stat Med 24 (22), 3431–45.
Farr, W., 1840. Progress of epidemics. Second report to the Registrar General on Eng-land, 91–98.
Forrester, M., Pettitt, A. N., 2005. Use of stochastic epidemic modeling to quantifytransmission rates of colonization with methicillin-resistant Staphylococcus au-reus in an intensive care unit. Infect Control Hosp Epidemiol 26 (7), 598–606.
Gelman, A., Carlin, J., Stern, H., Rubin, D. B., 2000. Bayesian data analysis. Texts instatistical science. Chapman & Hall/CRC, Boca Raton, Fla.
Gelman, A., Carlin, J., Stern, H., Rubin, D. B., 2004. Bayesian data analysis, 2nd Edi-tion. Texts in statistical science. Chapman & Hall/CRC, Boca Raton, Fla.
Gilks, W., Richardson, S., Spiegelhalter, D., 1996. Markov Chain Monte Carlo in Prac-tice. Chapman and Hall.
Green, P., 1995. Reversible jump Markov chain Monte Carlo computation andBayesian model determination. Biometrika 82 (4), 711–732.
Green, P., Richardson, S., Dec 2002. Hidden Markov models and disease mapping. JAm Stat Assoc 97 (460), 1055–1070.
Giesecke, J., 1994. Modern Infectious Disease Epidemiology. Edward Arnold, Lon-don.
Glass, K., Becker, N., Clements, M., 2006. Predicting case numbers during infectiousdisease outbreaks when some cases are undiagnosed. Stat Med electronically pub-lished February DOI:10.1002/sim.2523.
Gopalakrishna, G., Choo, P., Leo, Y. S., Tay, B. K., Lim, Y. T., Khan, A. S., Tan, C. C.,2004. SARS transmission and hospital containment. Emerg Infect Dis 10 (3), 395–400.
Grundmann, H., Hellriegel, B., 2006. Mathematical modelling: a tool for hospital in-fection control. Lancet Infect Dis 6 (1), 39–45.
Grundmann, H., Hori, S., Winter, B., Tami, A., Austin, D. J., 2002. Risk factors for thetransmission of methicillin-resistant Staphylococcus aureus in an adult intensivecare unit: fitting a model to the data. J Infect Dis 185 (4), 481–8.
Hamer, W. H., March 17 1906. The Milroy Lectures on Epidemic disease in England-The evidence of variability and persistence of type Lecture III. Lancet 167 (4307),733–739.
Hirsch, M., 2006. Severe Acute Respiratory Syndrome (SARS) In: UpToDate. Rose,B.D. (ed). UpToDate, Waltham, MA.
Hsieh, Y. H., Chen, C. W., Hsu, S. B., 2004. SARS outbreak, Taiwan, 2003. Emerg InfectDis 10 (2), 201–6.
Huovinen, P., 2005. Mathematical model–tell us the future! J Antimicrob Chemother56 (2), 257–8; discussion 431.
68 BIBLIOGRAPHY
Jernigan, J., Titus, M., Groschel, D., Getchell-White, S., Farr, B., Mar 1 1996. Effec-tiveness of contact isolation during a hospital outbreak of methicillin-resistantStaphylococcus aureus. Am J Epidemiol 143 (5), 496–504.
Keeling, M.J. and Grenfell, B.T., 1997. Disease extinction and community size: mod-elling the persistence of measles. Science 275(5296):65-67.
Kermack, W., McKendrick, A., 1927. Contributions to the mathematical theory of epi-demics:part 1. Proceedings of the Royal Society of London A 115, 700–721.
Kim, K., Fekety, R., Batts, D., Brown, D., Cudmore, M., Silva, J., Waters, G., 1981. Iso-lation of Clostridium difficile from the environment and contacts of patients withantibiotic-associated colitis. Journal of Infectious Diseases 143 (1), 42–50.
Lee, N., Hui, D., Wu, A., Chan, P., Cameron, P., Joynt, G. M., Ahuja, A., Yung, M. Y.,Leung, C. B., To, K. F., Lui, S. F., Szeto, C. C., Chung, S., Sung, J. J., 2003. A ma-jor outbreak of severe acute respiratory syndrome in Hong Kong. N Engl J Med348 (20), 1986–94.
Leung, G. M., Chung, P. H., Tsang, T., Lim, W., Chan, S. K., Chau, P., Donnelly, C. A.,Ghani, A. C., Fraser, C., Riley, S., Ferguson, N. M., Anderson, R. M., Law, Y. L., Mok,T., Ng, T., Fu, A., Leung, P. Y., Peiris, J. S., Lam, T. H., Hedley, A. J., 2004a. SARS-coVantibody prevalence in all Hong Kong patient contacts. Emerg Infect Dis 10 (9),1653–6.
Leung, G. M., Hedley, A. J., Ho, L. M., Chau, P., Wong, I. O., Thach, T. Q., Ghani, A. C.,Donnelly, C. A., Fraser, C., Riley, S., Ferguson, N. M., Anderson, R. M., Tsang, T.,Leung, P. Y., Wong, V., Chan, J. C., Tsui, E., Lo, S. V., Lam, T. H., 2004b. The epidemi-ology of severe acute respiratory syndrome in the 2003 Hong Kong epidemic: ananalysis of all 1755 patients. Ann Intern Med 141 (9), 662–73.
Lindsey, J. C., Ryan, L. M., 1998. Tutorial in biostatistics methods for interval-censored data. Stat Med 17 (2), 219–38.
Lipsitch, M., Bergstrom, C. T., Levin, B. R., 2000. The epidemiology of antibiotic re-sistance in hospitals: paradoxes and prescriptions. Proc Natl Acad Sci U S A 97 (4),1938–43.
Lipsitch, M., Cohen, T., Cooper, B., Robins, J. M., Ma, S., James, L., Gopalakrishna, G.,Chew, S. K., Tan, C. C., Samore, M. H., Fisman, D., Murray, M., 2003. Transmissiondynamics and control of severe acute respiratory syndrome. Science 300 (5627),1966–70.
Lloyd, A. L., 2001. Destabilization of epidemic models with the inclusion of realisticdistributions of infectious periods. Proc R Soc Lond B 268, 985–993.
Lloyd-Smith, J. O., Galvani, A. P., Getz, W. M., 2003. Curtailing transmission of se-vere acute respiratory syndrome within a community and its hospital. Proc BiolSci 270 (1528), 1979–89.
Lloyd-Smith, J. O., Schreiber, S. J., Kopp, P. E., Getz, W. M., 2005. Superspreading andthe effect of individual variation on disease emergence. Nature 438 (7066), 355–9.
Lowrie, F., 2006. Mechanisms of antibiotic resistance in Staphylococcus aureus. In:UpToDate. Rose, B.D. (ed). UpToDate, Waltham, MA.
BIBLIOGRAPHY 69
Magee, J. T., 2005. The resistance ratchet: theoretical implications of cyclic selectionpressure. J Antimicrob Chemother 56 (2), 427–30.
Mandell, G., Bennett, J., Dolin, R. (Eds.), 2005. Principles and Practice of InfectiousDiseases, 6th Edition. Vol. 2. Elsevier Churchill Livingstone, Ch. 192, pp. 2321–2351.
McBryde, E. S., Bradley, L. C., Whitby, M., McElwain, D. L., 2004. An investigation ofcontact transmission of methicillin-resistant Staphylococcus aureus. J Hosp Infect58 (2), 104–8.
Meltzer, M. I., 2004. Multiple contact dates and SARS incubation periods. Emerg In-fect Dis 10 (2), 207–9.
Naimi, T.S.and LeDell, K., Como-Sabetti, K., Borchardt, S., Boxrud, D., Etienne, J.,Johnson, S., Vandenesch, F., Fridkin, S., O’Boyle, C., Danila, R., Lynfield, R., Dec 102003. Comparison of community- and health care-associated methicillin-resistantStaphylococcus aureus infection. J Am Med Assoc 290 (22), 2976–2984.
Neal, R., Hinton, G., 1999. A view of the EM algorithm that justifies incremental,sparse, and other variants. In:Learning in Graphical Models. MIT Press, Cam-bridge, MA, pp. 355–368.
Nimmo, G., Bell, J., Collignon, P., 2003. Fifteen years of surveillance by the AustralianGroup for Antimicrobial Resistance (AGAR). Commun Dis Intell 27, Suppl:S47–54.
Nimmo, G. R., Coombs, G. W., Pearson, J. C., O’Brien, F. G., Christiansen, K. J.,Turnidge, J. D., Gosbell, I. B., Collignon, P., McLaws, M. L., 2006. Methicillin-resistant Staphylococcus aureus in the Australian community: an evolving epi-demic. Med J Aust 184 (8), 384–8.
Noskin, G., Cooper, I., Peterson, L., July 1995a. Vancomycin-resistant Enterococ-cus fecium sepsis following persistnat colonization. Archives of Internal Medicine155 (13), 1445–1447.
Noskin, G. A., Stosor, V., Cooper, I., Peterson, L. R., 1995b. Recovery of vancomycin-resistant enterococci on fingertips and environmental surfaces. Infect ControlHosp Epidemiol 16 (10), 577–81.
Padiglione, A., Grabsch, E., Olden, D., Hellard, M., Sinclair, M., Fairley, C., Grayson,M., Sep-Oct 2000. Fecal colonization with vancomycin-resistant enterococci inAustralia. Emerging Infectious Diseases 6 (5).
Peacock, J., Marsik, F., Wenzel, R., 1980. Methicillin-resistant Staphylococcus aureus: introduction and spread within a hospital. Annals of Internal Medicine 93, 526–532.
Peiris, J. S., Yuen, K. Y., Osterhaus, A. D., Stohr, K., 2003. The severe acute respiratorysyndrome. N Engl J Med 349 (25), 2431–41.
Pelupessy, I., Bonten, M. J., Diekmann, O., 2002. How to assess the relative impor-tance of different colonization routes of pathogens within hospital settings. ProcNatl Acad Sci U S A 99 (8), 5601–5.
70 BIBLIOGRAPHY
Perencevich, E. N., Fisman, D. N., Lipsitch, M., Harris, A. D., Morris, J. G., J., Smith,D. L., 2004. Projected benefits of active surveillance for vancomycin-resistant en-terococci in intensive care units. Clin Infect Dis 38 (8), 1108–15.
Poutanen, S. M., Low, D. E., Henry, B., Finkelstein, S., Rose, D., Green, K., Tellier, R.,Draker, R., Adachi, D., Ayers, M., Chan, A. K., Skowronski, D. M., Salit, I., Simor,A. E., Slutsky, A. S., Doyle, P. W., Krajden, M., Petric, M., Brunham, R. C., McGeer,A. J., 2003. Identification of severe acute respiratory syndrome in Canada. N Engl JMed 348 (20), 1995–2005.
Raboud, J., Saskin, R., Simor, A., Loeb, M., Green, K., Low, D. E., McGeer, A., 2005.Modeling transmission of methicillin-resistant Staphylococcus aureus among pa-tients admitted to a hospital. Infect Control Hosp Epidemiol 26 (7), 607–15.
Rainer, T. H., Chan, P. K., Ip, M., Lee, N., Hui, D. S., Smit, D., Wu, A., Ahuja, A. T.,Tam, J. S., Sung, J. J., Cameron, P., 2004. The spectrum of severe acute respiratorysyndrome-associated coronavirus infection. Ann Intern Med 140 (8), 614–9.
Ridall, P., 2005. Bayesian latent variable models for biostatistical applications. Ph.D.thesis, Queensland University of Technology.
Ridwan, B., Mascini, E., van der Reijden, N., Verhoef, J., Bonten, M., 16 March 2002.What action should be taken to prevent spread of vancomycin resistant entero-cocci in European hospitals? BMJ 324, 666–668.
Riley, S., Fraser, C., Donnelly, C. A., Ghani, A. C., Abu-Raddad, L. J., Hedley, A. J., Le-ung, G. M., Ho, L. M., Lam, T. H., Thach, T. Q., Chau, P., Chan, K. P., Lo, S. V., Leung,P. Y., Tsang, T., Ho, W., Lee, K. H., Lau, E. M., Ferguson, N. M., Anderson, R. M.,2003. Transmission dynamics of the etiological agent of SARS in Hong Kong: im-pact of public health interventions. Science 300 (5627), 1961–6.
Ross, R., 1916. An application of the theory of probabilities to the study of a prioripathometry, i. Proceedings of the Royal Society of London A 92, 204–230.
Ross, R., Hudson, H., 1916. An application of the theory of probabilities to the studyof a priori pathometry, ii; iii. Proc Roy Soc Lon A 93, 212–225; 225–240.
Sanford, M. D., Widmer, A. F., Bale, M. J., Jones, R. N., Wenzel, R. P., 1994. Efficient de-tection and long-term persistence of the carriage of methicillin-resistant Staphy-lococcus aureus. Clin Infect Dis 19 (6), 1123–8.
Scanvic, A., Denic, L., Gaillon, S., Giry, P., Andremont, A., Lucet, J., May 2001. Dura-tion of colonization by methicillin-resistant Staphylococcus aureus after hospitaldischarge and risk factors for prolonged carriage. Clin Infect Dis. 32 (10), 1393–1398.
Scharz, G., 1978. Estimating the dimensions of a model. Ann Stat 6 (2), 461–464.
Scott, S., 2002. Bayesian methods for hidden Markov models: Recursive computingin the 21st Century. J. Amer. Statist Assoc. 97, 337–351.
Sebille, V., Chevret, S., Valleron, A. J., 1997. Modeling the spread of resistant noso-comial pathogens in an intensive-care unit. Infect Control Hosp Epidemiol 18 (2),84–92.
BIBLIOGRAPHY 71
Sebille, V., Valleron, A.-J., 1997. A computer simulation model for the spread of noso-comial infections caused by multidrug-resistant pathogens. Computers and Bio-medical Research 30 (4), 307–322.
Smith, D. L., Levin, S. A., Laxminarayan, R., 2005. Strategic interactions in multi-institutional epidemics of antibiotic resistance. Proc Natl Acad Sci U S A 102 (8),3153–8.
Spiegelhalter, D., Best, N., Carlin, B., van der Linde, A., 2002. Bayesian measures ofmodel complexity and fit. J Roy Stat Soc 64 (4), 583–639.
Stephens, M., 2000. Bayesian analysis of mixture models with an unknown numberof components. Ann Stat 28, 40–74.
Thompson, R., Cabezudo, I., Wenzel, R., 1982. Epidemiology of nosocomial infec-tions caused by methicillin-resistant Staphylococcus aureus . Annals of InternalMedicine 97, 307–309.
Viterbi, A., 1967. Error bounds for convolutional codes and asymptotically optimaldecoding algorithm. IEEE Trans Information Processing 13, 260–269.
Wallinga, J., Teunis, P., 2004. Different epidemic curves for severe acute respiratorysyndrome reveal similar impacts of control measures. Am J Epidemiol 160 (6), 509–16.
Wang, W., Ruan, S., 2004. Simulating the SARS outbreak in Beijing with limited data.J Theor Biol 227 (3), 369–79.
Weinstein, J., 2005. Hospital-acquired (nosocomial) infections with vancomycin-resistant enterococci In: UpToDate. Rose, B.D. (ed).
Weinstein, J., Roe, M., Towns, M., Anders, L., Thorpe, J., Corey, G., Sexton, D., 1996.Resistant enterococci: a prospective study of prevalence, incidence, and factorsassociated with colonization in a university hospital. Infect Cont Hosp Epi 17 (1),36–41.
Whitby, M., McLaws, M. L., 2004. Handwashing in healthcare workers: accessibilityof sink location does not improve compliance. J Hosp Infect 58 (4), 247–53.
Williams, G. M., Sleigh, A. C., Li, Y., Feng, Z., Davis, G. M., Chen, H., Ross, A. G.,Bergquist, R., McManus, D. P., 2002. Mathematical modelling of Schistosomiasisjaponica: comparison of control strategies in the People’s Republic of China. ActaTrop 82 (2), 253–62.
Wilson, D., Mathews, S., Wan, C., Pettitt, A., McElwain, D., 2004. Use of a quantita-tive gene expression assay based on micro-array techniques and a mathematicalmodel for the investigation of chlamydial generation time. Bull Math Biol 66 (3),523–537.
Wong, T. W., Lee, C. K., Tam, W., Lau, J. T., Yu, T. S., Lui, S. F., Chan, P. K., Li, Y., Bre-see, J. S., Sung, J. J., Parashar, U. D., 2004. Cluster of SARS among medical studentsexposed to single patient, Hong Kong. Emerg Infect Dis 10 (2), 269–76.
CHAPTER 3
A mathematical model of methicillin
resistant Staphylococcus aureus
transmission in an Intensive Care Unit:
Predicting the impact of interventions
Statement of joint authorshipEmma McBryde wrote the manuscript, constructed the dataset, developed
the mathematical model, analysed the data, wrote code for model extensions
including interventions and acted as corresponding author.
Tony Pettitt assisted with the analysis of data and the application of Bayesian
inference and the piecewise hazard model, proof read and critically reviewed
the manuscript.
Sean McElwain initiated the concept for the manuscript, assisted with
the development of the mathematical model and proof read and critically
reviewed the manuscript.
74 Chapter 3. Mathematical model of MRSA
AbstractObjectives: To estimate the transmission rate of MRSA in an intensive care
unit in an 800 bed Australian teaching hospital and predict the impact of in-
fection control interventions.
Methods: A mathematical model was developed which consisted of four
compartments: colonised and uncolonised patients and contaminated and
uncontaminated healthcare workers (HCWs). Patient movements, MRSA
acquisition and daily prevalence data were collected from an Intensive Care
Unit (ICU) over 939 days. Hand hygiene compliance and the probability of
MRSA transmission from patient to HCW per discordant contact were mea-
sured during the study. Attack rate and reproduction ratio were estimated
using Bayesian methods. The impact of a number of interventions on attack
rate was estimated using both stochastic and deterministic versions of the
model.
Results: The mean number of secondary cases arising from the ICU admis-
sion of colonised patients, the ward reproduction ratio, Rw, was estimated to
be 0.50 (95% CI 0.39-0.62 ). The attack rate was one MRSA transmission per
160 (95% CI 130-210) uncolonised-patient days.
Hand hygiene was predicted to be the most effective intervention. Decoloni-
sation was predicted to be relatively ineffective. Increasing HCW numbers
was predicted to increase MRSA transmission, in the absence of patient co-
horting. The predictions of the stochastic model differed from those of the
deterministic model, with lower levels of colonisation predicted by the sto-
chastic model.
Conclusions: The number of secondary cases of MRSA colonisation within
the ICU in this study was below unity. Transmission of MRSA was sustained
through admission of colonised patients. Stochastic model simulations
give more realistic predictions in hospital ward settings than deterministic
models. Increasing staff does not necessarily lead to reduced transmission
of nosocomial pathogens.
3.1 Introduction
Infections caused by antibiotic-resistant bacterial pathogens in the health-
care setting are detrimental to patients and place a large burden on health-
care institutions. Staphylococcus aureus is a common cause of hospital ac-
quired blood stream infection and wound infection. Methicillin-resistant S.
3.1 Introduction 75
aureus (MRSA) leads to a higher mortality, morbidity (Engemann et al., 2003)
and cost (Capitano et al., 2003) compared with methicillin-sensitive S. aureus
(MSSA).
The proportion of isolates of S. aureus that are methicillin-resistant is
increasing in many countries including Australia (Nimmo et al., 2003). It is
likely that the increase in MRSA does not represent replacement of MSSA,
but is an additional burden (Cooper and Lipsitch, 2004).
Methicillin resistance developed in S. aureus soon after this class of antibi-
otics was introduced (Ericksen and Erichsen, 1963). Most strains of Health-
care Associated (HA) MRSA are also resistant to other classes of antibiotics
including aminoglycosides and macrolides. Of even more concern is the re-
cent observation that some MRSA isolates have been found to be resistant to
glycopeptides (Bartley, 2002) and oxalidinones (Meka et al., 2004), the major
alternative therapies for MRSA infection.
Antibiotic-resistant bacteria are believed to spread from patient to patient,
principally via the hands of healthcare workers. Colonisation with MRSA fre-
quently precedes infection. This transmissible, asymptomatic state will not
be detected unless an active surveillance program is in place. Thus, halting
the institutional spread of MRSA requires measures that affect colonised pa-
tients as well as those with overt infection.
Recommendations for the control of MRSA transmission include isolation
(Garner, 1996) active surveillance cultures (Muto et al., 2003) and hand hy-
giene. While these guidelines are based on the best available evidence, few of
the studies of hospital acquired infectious diseases use sound methodology
(Cooper et al., 2003). The increase in the proportion of S.aureus isolates
that are methicillin resistant in the face of infection control measures led
to pessimism about their efficacy (Teare and Barrett, 1997). A recent study
found that moving patients into single rooms or cohorted bays did not
reduce MRSA acquisition (Cepeda et al., 2005), however this study screened
for MRSA only weekly which may have led to long delays before colonised
patients were removed from the general ward, diluting any benefit of
isolation.
Mathematical models provide a means of predicting the likely impact of an
intervention or the interaction of multiple interventions, capturing nonlin-
ear transmission dynamics. Stochastic models have the additional advan-
tage of predicting the expected variation in outcomes, which may be marked
in small populations such as hospital wards. Statistical methods based on
76 Chapter 3. Mathematical model of MRSA
structured models provide a means of estimating transmission parameters
from data.
In modelling community epidemics and emerging infectious diseases, the
emphasis of model-informed infection control measures has been to achieve
an effective reproduction ratio (the number of cases that occur due to the in-
troduction of one infectious case, assuming a fully susceptible population)
below unity. In the case of hospital associated pathogens such as MRSA, the
mean number of secondary cases that arise within a ward during a single hos-
pital admission (which we call the ward reproduction ratio, Rw) may be below
unity, but colonised patients may go on to transmit MRSA in other wards and
during subsequent hospital admissions leading to an overall reproduction ra-
tio above unity (see Cooper et al. 2004 for full explanation).
In this study, we find a low ward reproduction ratio, Rw = 0.50. Frequent re-
introductions of MRSA maintain the endemic prevalence. We therefore use
attack rate, defined as the number of MRSA transmissions per uncolonised
patient day, as our outcome measure when predicting the impact of inter-
ventions.
This study differentiates imported cases of MRSA from those that occur dur-
ing ward stay. All new cases are assumed to arise from other colonised pa-
tients via the hands of healthcare workers (cross-transmission). We utilised
a mathematical model to quantify MRSA cross-transmission in an Australian
Intensive Care Unit. We collected data on admission, discharge and colonisa-
tion events as well as other critical model parameters, hand hygiene compli-
ance and transmission per contact, to estimate the MRSA attack rate and the
ward reproduction ratio. We overcame the challenge of unobserved events
by using a Bayesian framework and considering the MRSA acquisition date
as a latent variable. Stochastic and deterministic realisations of the model
gave predictions of the likely impact of interventions including changes in
health-care worker/patient ratio, patient cohorting, hand hygiene, length of
stay, admission prevalence, decolonisation and ward size on the attack rate.
This study extends previous models because all parameters used to estimate
transmission were derived through ward observation directly or fitted to
acquisition data. Ward observations running in parallel to the data collection
gave us realistic values for hand hygiene compliance and probability of
MRSA transmission from a colonised patient to healthcare worker. For the
simulation component of the study, we incorporate ward size as a parameter,
not previously considered, and predict the impact of increases in staff levels
3.2 Model 77
if this leads to increased contact rates. The study later considers the effect
of decolonisation based on parameters derived from an experimental study
(de la Cal et al., 2004).
3.2 Model
Our ward transmission model was a modification of the Susceptible-
Infectious (SI) model with migration, described by Bailey (1975). Versions of
this model have been used previously to analyse nosocomial transmission
data (Sebille and Valleron, 1997; Sebille et al., 1997; Cooper et al., 1999;
Austin et al., 1999; Grundmann et al., 2002; Raboud et al., 2005).
Model description and assumptions
Figure 3.1 illustrates the model for transmission of MRSA in an intensive
care unit. It was assumed that transmission will occur with a probability,
php when an MRSA contaminated health-care worker (HCW) contacts an
uncolonised patient and a probability, pph, when an MRSA colonised patient
was contacted by an uncontaminated HCW. Given that patients carry MRSA
for a long duration (the median MRSA patient carriage has been estimated
to be 8.5 months (Scanvic et al., 2001) or 40 months (Sanford et al., 1994))
compared with their length of ICU stay (4 days observed in the current
study) we made the simplifying assumption that the decolonisation rate, γ,
is zero in the absence of interventions. In contrast, HCWs were assumed to
be contaminated only until their next hand hygiene activity (which occurs
at a rate, κ). Patients arrive at the ward at a rate, Ω, and a proportion, σ are
colonised on arrival. Uncolonised and colonised patients are discharged at
rates µX
and µY
, respectively. The contact rate c is the number of contacts
per patient per HCW.
The assumption of transient contamination of HCWs is justified by the es-
tablished efficacy of hand hygiene activities for removing carriage (McBryde
et al., 2004) and the fact that health-care worker carriage of MRSA is usually
short term (Cookson et al., 1989). In this model we assumed that there was no
direct patient to patient or HCW to HCW transmission. It was also assumed
that there was no environmental reservoir contributing to transmission and
that all patients who were colonised on admission were detected. While en-
vironmental sites have been shown to become contaminated by MRSA, it is
uncertain whether this represents a significant source of MRSA transmission
78 Chapter 3. Mathematical model of MRSA
(Boyce et al., 1997). The HCW/patient ratio was assumed to be unity through-
out the study and that the number of HCW did not vary over each 24 hour
period. This was in keeping with ward policy of providing at least one clini-
cal nurse per patient. We also assumed homogenous mixing of patients and
HCWs and time invariance of model parameters.
Figure 3.1 illustrates the mathematical model of MRSA transmission. The pa-
rameters of the model are given in Table 3.1.
Patients
HCWs
Uncolonised Colonised
Xp Yp
Xh Yh
hp p hcp X Y
ph h pcp X Y
hY
(1 )
Y pYX pXpY
Figure 3.1: Four compartment model of nosocomial pathogen transmission.Here Xp is the number of uncolonised patients, Yp the number of colonisedpatients, Xh the number of uncontaminated healthcare workers, Yh the num-ber of contaminated healthcare workers. The parameters and their symbolsare given in Table 3.1.
3.3 Data 79
MRSA cases can arise from ward transmission (at a rate cphpXpYh) or from
admission of newly colonised patients (at a rate σΩ). Healthcare workers ac-
quire MRSA via an interaction with a colonised patient at a rate proportional
to the number of contacts with patients cpphXhYp; they are decontaminated
at a rate dictated by hand hygiene, κYh.
3.3 Data
3.3.1 Patients and Setting
This study included all patients admitted to the Intensive Care Unit (ICU) of a
800 bed tertiary referral teaching hospital (Princess Alexandra Hospital, Bris-
bane, Australia) from 8th August 2001 to 3rd March 2004 (939 days inclusive).
The ICU bed capacity varied during the study from 16 to 22.
3.3.2 Surveillance of colonisation
During the investigation period, all patient admissions were recorded in the
Apache IIIT M database. The mean number of inpatients for the study each
day was 15 (median 16). Ward policy was to swab all patients on admission,
on discharge from the unit and twice weekly for MRSA surveillance. Newly
colonised patients were defined as those negative on admission who had
a positive swab attributed to ICU stay (more than 48 hours following ICU
admission and less than 48 hours following ICU discharge). An MRSA
colonisation database was collected using pathology reports and record of
prior colonisation on admission. For each of the 939 days of the study, the
number of uncolonised patients, colonised patients and new colonisations
were recorded. Following discharge, each patient was categorised as not
known to be colonised, known to be colonised prior to admission or newly
colonised.
3.3.3 Parameter estimates
The admission prevalence of known MRSA colonised patients, σ was 3% in
this study. At the time of data collection, we estimated model parameters
through ward observation. The hand hygiene compliance, h, was estimated
to be 59% (395 hand hygiene episodes out of 668 hand hygiene opportunities
80 Chapter 3. Mathematical model of MRSA
observed in the study population (Whitby and McLaws, 2004)). This has a re-
lationship with the hand hygiene rate as described in Section 3.4.1. The prob-
ability of MRSA transmission during a single contact between a colonised pa-
tient and an uncolonised healthcare worker was estimated to be 13% during
the study period (17 positive hand cultures out of 129 patient visits found by
McBryde et al. (2004)). We used the data on all patient contacts (anyone who
enters a patient bay) rather than strictly clinical contacts, which has a trans-
mission probability of 17%, as measured by McBryde et al. (2004).
3.4 Methods
To quantify cross-transmission of MRSA in our study population, we esti-
mated the attack rate (number of transmissions per uncolonised patient
day) and the ward reproduction ratio, Rw.
We have no direct estimate of contact rate, c, or probability of transmission
from healthcare worker to patient, php. These two parameters are inseparable
in the model, so we estimate the value of their product, the transmission pa-
rameter, φ = cphp. The admission and discharge dates of patients are directly
observed in this study and are thus incorporated deterministically. Colonisa-
tion status on admission is known (assumed to be perfectly observed).
In Section 3.4.1 we derive a form of the model equations that leaves only the
transmission parameter, φ, to be estimated. Section 3.4.2 explains how φ was
inferred from the data. Section 3.4.3 describes how φ can be used to estimate
Rw and attack rate. Section 3.4.4 describes how the model structure can be
used to predict the impact of interventions.
3.4.1 Formula for daily hazard of MRSA cross-transmission
The daily hazard of MRSA cross-transmission, λ, is given by
λ = φXpYh. (3.1)
We do not have direct observations of Yh, however we derived a formula for
Yh based on observable model parameters. Firstly, we assumed dYh
dt= 0 . We
base the assumption on the fact that decontamination of healthcare workers
3.4 Methods 81
is known to be rapid (minutes to hours) compared with discharge or spon-
taneous decolonisation of patients (days to years, Boyce, 2005). The quasi-
equilibrium value for the number of contaminated HCW, Yh, which we de-
note by Y h is given by
Y h = NhcpphYp
κ + cpphYp
, (3.2)
where Nh = Xh + Yh, is the number of HCWs.
During the study, we measured the pre-contact hand hygiene compliance,
h. This was the proportion of patient contacts that were preceded by either
hand washing or the use of a disinfectant hand spray or gel. A relationship
between hand hygiene compliance and hand hygiene rate, κ, was derived by
Cooper et al. (1999), namely,
h =κ
κ + cNp
, (3.3)
where Np is the total number of patients.
Solving Equation (3.3) for κ and substituting this into Equation (3.2) gives
Y h = NhpphYp
hNp
1−h+ pphYp
. (3.4)
Noting that in this study Nh = Np = Xp + Yp, we have a revised expression for
the rate of MRSA transmission to uncolonised patients, λ, given by
λ =φpphXpYp(Xp + Yp)
h(Xp+Yp)
1−h+ pphYp
. (3.5)
The hand hygiene compliance, h, and the probability of MRSA transmission
from patient to HCW, pph, were measured on the ward at the time of the study,
leaving only one unknown value, the transmission parameter φ, which was
fitted to the data.
82 Chapter 3. Mathematical model of MRSA
3.4.2 Bayesian inference to estimate φ
Estimates of MRSA cross-transmission were complicated by interval censor-
ing of colonisation times. Colonisation events are asymptomatic so obser-
vations of MRSA acquisition consisted of the time of first detection, via rou-
tine swabs or clinical isolates. Assuming 100% swab sensitivity, transmission
could have occurred at any point between the last negative swab or ICU ad-
mission (whichever was later) and the first positive swab or discharge from
the ICU (whichever was sooner). We used a Bayesian framework to estimate
the posterior probability density of the transmission parameter, φ, given in
the Appendix.
3.4.3 Estimates of the attack rate and the ward reproduction
ratio.
In this context, the ward reproduction ratio, Rw, is the expected number of
MRSA cross-transmissions resulting from a single colonised patient, assum-
ing all other patients on the ward are susceptible. The model used in this
study was a two population model in which there was no direct transmission
between people of the same population type. The ward reproduction ratio is
therefore the product of the expected number of transmissions from a single
colonised patient to healthcare workers (HCWs), Rph, and the expected
number of transmissions from a single contaminated HCW to patients, Rhp.
Each component of Rw can be calculated by multiplying the daily transmis-
sion probability by the expected duration of colonisation/contamination.
Therefore
Rw =c2phppph(Np − 1)Nh
µYκ
. (3.6)
By solving Equation (3.3) for the hand hygiene rate, κ, and substituting it into
equation 3.6 and using φ = cphp we get
Rw =φpph(1− h)(Np − 1)
µYh
, (3.7)
where Nh and Np(= Nh) are the number of healthcare workers and patients in
the ward, respectively. Therefore, the ward reproduction ratio will vary from
day to day as the number of patients and healthcare workers changes, under
3.4 Methods 83
the principle of pseudo-mass action. The estimated ward reproduction ratio
was taken as the mean over the study period.
The hazard (rate) of transmission in the ward on day t is φYh(t)Xp(t). There-
fore the attack rate (the rate of transmission per uncolonised patient day)
over the study period is given by
AR = φ
n∑t=1
Y (t). (3.8)
3.4.4 Model for the impact of interventions
We used attack rate as the outcome measure to model the effect of a num-
ber of interventions: improving hand hygiene compliance, decolonisation,
HCW/patient ratios with and without patient cohorting, ward size and pa-
tient discharge rate on the attack rate. We examined both deterministic and
stochastic model predictions.
Estimated means of the parameters derived from the data were used as the
default parameters. The ward size in the study was not fixed, however the
ward ran at near maximum capacity much of the time, therefore new admis-
sions were often limited by the rate of patient discharge. This justified the use
of a simplifying assumption of fixed ward size to estimate the impact of inter-
ventions. We used the mean occupancy derived from the data to determine
the number of patients in the ward, np = 15 (here we used a fixed value of
occupancy as a parameter, np, rather than the variable, Np). We also assumed
that Nh = ρnp, where ρ is the health-care/patient ratio. This simplifies the
mathematical equations to
dYp
dt= cphp(np − Yp)Yh − (γ + µ
Y(1− σ))Yp + µ
Xσ(np − Yp),
dYh
dt= cpph(ρnp − Yh)Yp − κYh. (3.9)
Note that we have now allowed decolonisation of patients, γ, to be non-zero.
The equilibrium attack rate is given by
AR = cphpY he , (3.10)
where Y he is the equilibrium value for Yh, obtained when dYp
dt= dYh
dt= 0.
84 Chapter 3. Mathematical model of MRSA
In the stochastic version of the model, the probability during a small time in-
terval, δ, of transiting from one state to another is described by the equations
Pr(Yp(t + δ) = i + 1|Yp(t) = i) = cphp(np − i)Y hδ + µXσ(np − i)δ + o(δ)
Pr(Yp(t + δ) = i− 1|Yp(t) = i) = (γ + µY(1− σ))iδ + o(δ)
Pr(Yp(t + δ) = i|Yp(t) = i) = 1− cphp(np − i)Y hδ − µXσ(np − i)δ
− (γ + µY(1− σ))iδ + o(δ),
(3.11)
where o(δ) is the Landau symbol, denoting lower order terms of δ. It was as-
sumed that dYh
dt= 0. All other probabilities are o(δ).
The default value for the HCW/patient ratio, ρ, was unity. The default value
for the decolonisation rate, γ, was zero. Other default values were admis-
sion prevalence, σ = 0.03, discharge rate of colonised patients, µY
= 1/10.6,
corresponding to a length of stay of 10.6 days, discharge rate of uncolonised
patients, µX
= 1/4, corresponding to a length of stay of 4 days, probabil-
ity of transmission from colonised patient to healthcare worker per contact,
pph = 0.13, hand hygiene compliance, h = 0.59. In the simulations, for each
set of parameters, the ward was assumed to start with no colonised patients,
the burn-in period was 1000 days and the predicted attack rate was derived
from the next 939 simulated days. Stochastic results were based on 1000 sim-
ulations for each set of parameters, and the 2.5-97.5 percentile ranges were
determined.
By leaving all other parameters at their default values and modifying h, µY
,
µX
and σ, we simulated the effects of changes in hand hygiene compliance,
discharge rate of colonised and uncolonised patients and admission preva-
lence respectively. By changing γ from zero to 0.05, we simulated the effect of
decolonisation. The latter decolonisation rate was chosen based on a study
by de la Cal et al. (2004) in which patients were given enteral vancomycin in
an attempt to eradicate MRSA.
Cohorting was simulated by reducing the number of “effective contacts”. We
assumed that cohorting was non-selective. That is that HCWs cared for a co-
hort of patients who could be a mix of colonised and uncolonised patients.
The smaller the group in the cohort, the more likely that a given contact is
a return contact and thus not an “effective contact”. When maximum co-
horting is taking place, we assume that a proportion of contacts equal to the
3.5 Results 85
HCW/patient ratio, ρ, pose no risk (when ρ ≥ 1 all cohorted contacts pose no
risk).
Our model defined c as the number of contacts per patient per HCW. By ex-
amining the effect of increasing staff patient ratio, ρ, we assume that each
HCW has a fixed number of contacts and increasing staff increases contacts.
To extend this simulation to allow for changes in patient numbers but con-
tinuing to assume a fixed number of contacts per HCW, one could modify
the contact rate, c∗ = c np
Np, where np is the default number of patients and Np
is the actual number of patients. We could alternatively simulate a situation
where patients have a fixed number of contacts and increasing staff does not
increase contacts. Such a simulation would require modifying the contact
rate to c∗, where c∗ = c/ρ.
3.5 Results
The study included 3329 patients. Of these, 100 patients were known to be
colonised on admission and 77 met the criteria for new colonisation. Figure
3.2 summarises the data.
0 100 200 300 400 500 600 700 800 9000
5
10
15
20
25
Day of study
Nu
mb
er o
f p
atie
nts
UncolonisedColonisedNew cases
Figure 3.2: Data collected over period of study. The grey bar plot indicates thenumber of uncolonised patients on each day, the black line plot indicates thenumber of colonised patients and the white bar plot the new acquisitions.
86 Chapter 3. Mathematical model of MRSA
3.5.1 Estimate ward transmission: attack rate and the ward
reproduction ratio
The posterior probability distribution of the ward reproduction ratio, Rw, is
shown in Figure 3.3(a). The estimated mean value of the reproduction ratio
was 0.50 (95% CI 0.39-0.62 ). The posterior probability distribution of the at-
tack rate is shown in Figure 3.3(b). The estimated mean was 0.0062 transmis-
sions per uncolonised patient day (95% CI 0.0048-0.0076), or approximately
one new acquisition per 160 uncolonised patient days.
3.5.2 Predicted impact of interventions
Figure 3.4 shows the predicted impact of ward interventions. The model pre-
dicts that the attack rate would increase dramatically should the hand hy-
giene compliance fall below 40%. A hand hygiene compliance of 48% would
increase the ward reproduction ratio to unity.
Figure 3.4(b) shows the effect of changing the discharge rate of colonised pa-
tients, µY
, leading to a reciprocal change in expected duration of stay. The
response curve was sigmoidal in shape. Increasing the mean time on ward
following colonisation to 21 days would lead to the ward reproduction ratio
exceeding unity. Increasing length of stay of all patients, Figure 3.4(c), also
increases attack rate but less dramatically.
The response of attack rate to doubling the admission prevalence from the
current 3% to 6% is a predicted increase in attack rate from one transmission
per 160 uncolonised patient days to one per 105 uncolonised patient days
(Figure 3.4(d)).
We compared no decolonisation with decolonisation at a rate of 0.05 per day,
using the results of de la Cal et al. (2004). The reduction in attack rate was
modest, from 0.0061 to 0.0034, with overlapping 95% ranges for the stochas-
tic simulations.
We investigated the predicted impact of changing the HCW/patient ratio. In
the upper curve of Figure 3.5, there is no patient cohorting and increasing
HCW numbers increases cross-transmission. In the other curves, we as-
sumed that HCWs can be assigned to a fixed group of patients. Successively
lower curves in Figure 3.5 represent greater proportion of HCWs involved in
cohorting. The lower curve in Figure 3.5 gives the predicted change in attack
3.5 Results 87
rate as HCW/patient ratios change when 100% of HCWs practise cohorting.
Once the HCW/patient ratio reached 1:1 there was no MRSA transmission.
Figure 3.6 shows the effect of reducing ward size on attack rate. The de-
terministic curve is compared with the interquartile range in the boxplots
of 1000 stochastic simulation results. The attack rate in the deterministic
model, unsurprisingly, does not change with ward size. The stochastic model
shows reduced median attack rate, particularly when the ward size reduces
below 10 patients. This reflects an increased proportion of time spent in
stochastic fade-out in small wards.
In several plots, the attack rate predicted by the deterministic model was
higher than that predicted by the stochastic model. Often, the determin-
istic predictions were outside the stochastic 95% variability range of the
corresponding stochastic model.
3.5.3 Model adequacy and sensitivity
A parametric bootstrap analysis was used to determine model adequacy.
This process involves simulating data from the model, using ward observa-
tions (number of uncolonised patients and admission of known colonised
patients) and and the estimated transmission parameter, φ. The method-
ology described in this paper was then applied to the simulated data to
estimate the mean of the marginal posterior distribution of the transmission
parameter. The study found that this gave an unbiased estimate of the
transmission parameter.
88 Chapter 3. Mathematical model of MRSA
Parameter Symbol Unitscontact rate c contacts pt−1HCW−1 day−1
decolonisation rate γ patient−1 day−1
admission prevalence σ -admission rate (pt per day) Ω pt day−1
discharge rate of colonised pt µY
day−1
discharge rate of uncolonised pt µX
day−1
transmission pt→HCW per contact pph colonisation contact−1
transmission HCW→ pt per contact php colonisation HCW−1
hand hygiene rate per HCW κ HCW−1day−1
Table 3.1: Parameters used in the model for MRSA transmission. Key: pt pa-tient, HCW healthcare worker.
3.5 Results 89
0 0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
7
8
Ward reproduction ratio(a) Posterior probability density of ward reproduction ratio
0 0.002 0.004 0.006 0.008 0.01 0.0120
50
100
150
200
250
300
350
Attack rate(b) Posterior probability density for the attack rate per uncolonised patient day.
Figure 3.3:
90 Chapter 3. Mathematical model of MRSA
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
Hand hygiene compliance
Att
ack
rate
(a) Effect of hand hygiene complianceon attack rate
0 10 20 30 400
0.005
0.01
0.015
0.02
0.025
0.03
Mean time on ward following colonisation (days)
Att
ack
rate
(b) Effect of length of stay of colonisedpatients on attack rate
0 5 10 15 200
1
2
3
4
5
6
7
8x 10
−3
Length of stay all patients (days)
Att
ack
rate
(c) Effect of length of stay of all pa-tients on attack rate
0 0.2 0.4 0.6 0.8 10
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Admission prevalence, σ
Att
ack
rate
(d) Effect of admission prevalence onattack rate
Figure 3.4: Effect of changing parameters on attack rate. The bold line repre-sents the prediction of the deterministic model, the feint line represents themean of the stochastic model predictions with error bars giving the 2.5%-97.5% interval.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7x 10
−3
HCW/patient ratio
Atta
ck r
ate
proportion of HCW withan assigned cohort
0.2
0.4
0.6
0.8
1
0
Figure 3.5: Effect of cohorting on attack rate. The impact of HCW patient ratiovaries depending on the proportion of contacts that are able to be cohorted.The deterministic results only are given here.
3.5 Results 91
2 4 8 12 14
0
1
2
3
4
5
x 10−4
Atta
ck r
ate
Ward size
Figure 3.6: Effect of ward size on attack rate. The deterministic value (hor-izontal line) is compared with the median (broken line) and interquartilerange (boxplots) of 1000 stochastic simulation results.
92 Chapter 3. Mathematical model of MRSA
3.6 Discussion
We used a Bayesian framework to quantify MRSA transmission and estimate
the ward reproduction ratio of MRSA in an Intensive Care Unit in a large
teaching hospital. The Bayesian methodology allowed us to incorporate
unseen events, namely, the time of MRSA transmission.
This study used a four compartment modified Susceptible-Infectious (SI)
model with migration. Ward observations of hand hygiene compliance and
transmission probability per contact gave us estimates of all but one model
parameter, which was readily fitted to the data.
We found that, in the Intensive Care Unit under investigation, the ward re-
production ratio was below unity (0.50, 95% CI 0.39-0.62). This compares
with the finding by Grundmann et al. (2002), also studying MRSA in an ICU,
of a ward reproduction ratio of 1.52, when interventions were included. The
hand hygiene compliance in the study by Grundmann et al. was similar to
the current study, however the length of stay of colonised patients was con-
siderably longer, possibly accounting for some of the difference. A study by
Austin et al. (1999) on vancomycin-resistant enterococci transmission found
a ward reproduction ratio of 0.7 when infection control interventions were
in place. The study by Austin et al. found a hand hygiene compliance of
50% and length of stay of colonised patients of around 15 days, both of which
would be expected to lead to higher reproduction ratios than that found in
the current study.
This study found that the predicted transmission rate did not dramatically
change as the ward reproduction ratio went above unity in simulations
involving changing the hand hygiene compliance and duration of stay.
This finding differs from studies of community epidemics in which the
basic reproduction ratio represents a threshold value, below which only
very limited transmission occurs. When there is continued migration of
colonised patients, as occurs in most hospitals with MRSA, the reproduction
ratio does not discriminate between high levels and low levels of transmis-
sion; nor does it quantify risk of colonisation to individual patients. We
therefore recommend that the attack rate be used as a measure of efficacy of
interventions in this setting.
Our model predicted that improving hand hygiene compliance would be the
most effective method of preventing MRSA transmission. Small increments
in compliance resulted in large nonlinear reductions in attack rate. If the
compliance were to fall below 40%, there would be a dramatic rise in attack
3.6 Discussion 93
rate, as predicted by our model. Such a rate is commonly encountered in
hand hygiene studies, prior to interventions (Johnson et al., 2005; Amazian
et al., 2005; Wong and Tam, 2005).
The model predicted that patient decolonisation would not be as effective as
hand hygiene. This is because the time frame to decolonisation (mean of 20
days) was long relative to the mean length of stay. The time to decolonisation
was estimated from a paper by de la Cal et al. (2004) with a decolonisation
rate of 0.05 per day. Our model did not account for the possibility that the
transmissibility may be reduced following a decolonisation intervention even
in patients who remain colonised. If this were to occur, the impact on trans-
mission of decolonisation would be greater than predicted by this model.
A finding in this study which differed from previous studies (for example
D’Agata et al. (2005)) was that increasing health-care worker levels may lead
to an increase in attack rate. An analysis of intensive care workload studies
found that, in the presence of a staff deficit, some studies report that the
productivity of staff reaches a limit leading to inability to complete tasks
involving patient care (Carayon and Gurses, 2005). In this circumstance, the
number of contacts per day is determined by the number of available staff
rather than the number of patients, and the number of contacts will increase
as staff level increases. We aimed to capture this circumstance in our
model. When the contacts were not cohorted, increased staff/patient ratios
resulted in a dramatic rise in attack rate, as predicted by the model. When
cohorting was introduced, the model predicted an initial rise in attack rate as
health-care worker numbers increase, followed by a decline as the increased
health-care worker ratio permitted greater cohorting. Other factors such as
improved compliance with hand hygiene could mitigate against increase
in attack rate and could explain the increased transmission associated with
staff deficit in the study by Grundmann et al. (2002).
When considering patient cohorting, one needs to keep in mind that cohort-
ing measures are usually only able to be carried out by nurses. We found in
our hospital that doctors, who are not involved in cohorting, have a lower
than average hand hygiene compliance rate (McBryde et al., 2004). The im-
pact of this on transmission could be predicted by relaxing the assumption of
uniformity of behaviour within the healthcare worker group.
The stochastic version of our model gave different results from the determin-
istic version. This is accounted for by frequent “fade outs” in MRSA coloni-
sation leading to episodes in which transmission cannot take place in the
94 Chapter 3. Mathematical model of MRSA
stochastic model. These findings are not unexpected in models of small pop-
ulations but reinforce the need to incorporate stochasticity in simulations of
interventions.
We found that the ward reproduction ratio was below unity in the Intensive
Care Unit in our study. Therefore in order for MRSA to persist it needs to
be imported. This leads us to question why MRSA continues to be problem-
atic in the healthcare facility in the study. The answer probably lies in the
fact that patients continue to transmit MRSA after leaving ICU. Colonised pa-
tients may transmit MRSA in other hospital wards, in nursing homes, in the
community and on readmission to hospital (Cooper et al., 2004). Although
the ward reproduction ratio within the ICU was less than unity, the overall
reproduction ratio of MRSA colonisation could be greater than unity.
In future studies, other possible modes of transmission need to be consid-
ered, including an environmental reservoir, transmission from healthcare
workers with chronic MRSA carriage, or unobserved colonisation events,
including patients harboring MRSA on admission without being detected.
Economic modelling of the cost and utility of different interventions would
be a useful adjunct to future studies in this area.
Appendix
3.A Bayesian estimation of the transmission para-
meter
Transmission events were treated as latent variables in the model. The full
conditional probability of the transmission parameter, φ, given a augmented
dataset, D, including daily numbers of uncolonised, colonised patients and
transmission events (latent variables) is given by
p(φ |D) ∝ π(φ)L(D|φ), (3.12)
where π(φ) is the prior probability of φ and L(D|φ) is the likelihood.
The marginal posterior probability density of the transmission parameter, φ,
can be obtained by summation over all possible values of the latent variables
3.A Bayesian estimation of the transmission parameter 95
p(φ |O) ∝ π(φ)∑
A
L(D|φ), (3.13)
where O is the observed data and A is the vector of latent variables (D =
O,A). Here, we take A to be the exact day of MRSA acquisition and the
resulting numbers of colonised and uncolonised patients on each day.
A Markov chain Monte-Carlo (MCMC) approach was used to estimate the
posterior probability density of the transmission parameter, φ. A prior prob-
ability, π(φ), was assigned to φ, the likelihood of the data, L(D|φ), was calcu-
lated. Latent variables and the transmission parameter were updated using a
Gibbs steps and the process was iterated.
Each of the component of the MCMC is explained in turn.
3.A.1 Likelihood of the complete dataset
We used a piecewise constant hazard assumption (Aslanidou et al., 1998) to
calculate the likelihood of the complete dataset. The complete dataset con-
sisted of the daily number of MRSA colonised patients, uncolonised patients
and the number of MRSA cross-transmissions. We assumed that events on
the same day were conditionally independent (given the known number of
colonised patients on the ward at the end of the previous day). We assumed
that a newly colonised patient did not become colonised until the end of the
time interval (one day), and therefore could not cause transmissions until the
following day.
The complete dataset D consists of three vectors, [Xp, Yp,Z], where Xp is the
vector of the number of uncolonised patients on each day of the study, Yp is
the vector of the number of colonised patients on each day of the study, and
Z is the vector of the number of new acquisitions on each day of the study.
New acquisitions of MRSA were assumed to follow a Poisson process with a
rate that was constant over each time increment of one day. This rate was the
daily hazard of transmission, λ(t), calculated using Equation (3.5).
Let
a(t) =pphXp(t)Yp(t)(Xp(t) + Yp(t))
h(Xp(t)+Yp(t))
1−h+ pphYp(t)
, (3.14)
96 Chapter 3. Mathematical model of MRSA
so that λ(t) = φa(t). The likelihood of the complete data over the duration of
the study (n = 939 days) is determined using
L(D |φ) ∝ φPn
t=1 Z(t) e−φPn
t=1 a(t). (3.15)
Here, multiple events were allowed to occur during a given time increment
and the likelihood was calculated at integer times (days) making no specific
allowance for this being an approximation for continuous time. Becker (1989,
Chapter 6.3) suggests that this approximation is sufficiently accurate for ap-
plications where the value of the rate parameter is relatively small.
3.A.2 Gibbs update for the transmission parameter, φ
The Gamma prior distribution is a conjugate prior to the likelihood calcu-
lation given in Equation 3.15. The posterior probability of the transmission
parameter, φ, given the complete dataset, and assuming a Gamma(α, β) prior
for φ, is given by
φ|D ∼ Gamma(α + z, β +n∑
t=1
a(t)), (3.16)
where z is the total number of cross-transmissions over the duration of the
study.
3.A.3 Latent variable imputation
The vector of latent variables, A, consists of the MRSA acquisitions times for
the 77 newly colonised patients, as well as the number of colonised and un-
colonised patients each day that are dependent on those acquisition times.
For each iteration of the Markov chain, the vector was updated by drawing
new values from the full conditional distribution.
For each newly colonised patient, the date of admission to the Intensive Care
Unit or last negative swab (whichever was later) was taken to be the earliest
possible day on which MRSA acquisitions could have occurred (tmin) and the
discharge date or date of first positive swab (whichever was sooner) was taken
to be the latest possible day on which MRSA acquisitions could have occurred
3.A Bayesian estimation of the transmission parameter 97
(tmax). The likelihood of acquisitions occurring on each of these days was cal-
culated. An inferred day of acquisition was drawn from the weighted likeli-
hoods.
Let Ti be the day on which patient i acquires MRSA. Then Ti can take the val-
ues (tmin, ..., tmax). Transmission can take place only once and, using discrete
time intervals of one day, Ti has tmax − tmin + 1 possible values. The full con-
ditional posterior distribution for Tik is given by
p(Ti = k |Xs,k, Ys,k, Zs,k, φ) ∝n∏
t=1
[λs,k(t)]Zs,k(t)e−λs,k(t), (3.17)
where Xs,k(t), Ys,k(t), Zs,k(t) and λs,k(t) are the numbers of uncolonised,
colonised, acquisitions and daily hazard function respectively on day t, given
that acquisition for patient i occurs on day k and given the current state, s, of
values of Tj, j 6= i . Only part of the complete likelihood involves Ti, therefore
the likelihood that patient i acquired MRSA on day k, Lik, is given by
Lik ∝tmax∏
t=tmin
λs,k(t)Zs,k(t)e−λs,k(t). (3.18)
The sampling distribution for Ti, p(Ti = k), in the MCMC update for Ti, is
proportional to the likelihood given by the right hand side of Equation(3.18).
3.A.4 Incorporating uncertainty of model parameters
Because two of the three parameters in the study (hand hygiene compliance,
h, and the transmission from colonised patient to healthcare worker, pph)
were estimated by direct observation on the ward, there is uncertainty in
these estimates. The transmission parameter, φ, was estimated from the
data. We need to incorporate the uncertainty of the measured parameters
into the estimate for the transmission parameter, φ.
The posterior probability density for hand hygiene compliance, h, was
derived using a Beta(1, 1) conjugate prior probability density (Gelman et al.,
2004) and the data available from ward observations (the sufficient statistics
were the total number of hand hygiene opportunities observed, m and the
number in which hand hygiene compliance occurred, l). We assumed that
each hand hygiene opportunity was an independent Bernoulli trial. The
posterior probability density for the hand hygiene compliance is given by
98 Chapter 3. Mathematical model of MRSA
h|l,m ∼ Beta(1 + l, 1 + m− l). (3.19)
The posterior probability density for the probability of transmission from
a colonised patient to a healthcare worker was derived using the same
methodology. Whitby and McLaws (2004) observed hand washing compli-
ance in 395 out of a total of 668 opportunities during the period of the current
study. We therefore drew the value for hand washing compliance from the
Beta (396, 274) distribution. McBryde et al. (2004) found transmission of
MRSA to the hands of healthcare workers in 17 out of 129 observed patient
care episodes. We therefore drew the probability of transmission from the
Beta (18, 113) distribution.
3.A.5 Markov chain Monte Carlo algorithm to estimate the
transmission parameter, φ
In order to determine the posterior probability density for the transmission
parameter, φ, we developed an MCMC algorithm, to explore the joint poste-
rior distribution of the augmented data and φ. The process consisted of the
following steps:
1. Determine the prior probability π(φ); an vague Gamma(0.001, 0.001)
distribution was chosen as little was known about transmission of
MRSA from HCW to patient.
2. Draw values of h and pph from their respective beta distributions
3. Update the vector of latent variables. Use a Gibbs steps to update Xp, Yp
and Z by sampling new values of Ti from the distribution given by the
RHS of Expression (3.18). With each iteration, all 77 cross-transmission
events were updated.
4. Update φ using a Gibbs step, sampling from the Gamma distribution
given in Expression (3.16).
5. Perform 10 000 iterations of steps 2-4, using a “burn in” period of 5 000
iterations, collecting the final 5 000 values of the Markov chain for φ and
Rw to contribute to the posterior probability density.
6. Repeat steps 2-5 to construct 10 Markov chains. Intra and inter chain
variance tests showed very good convergence. R = 1.0001 for both Rw
3.A Bayesian estimation of the transmission parameter 99
and φ (see Gelman et al. (2004, Chapter 11.6) for discussion on conver-
gence and R values).
Acknowledgements
This work was partially supported by a grant under the Australian Research
Council Linkage Scheme (LP0347112) and NHMRC scholarship number
290541. The authors would like to thank Dr M. Whitby for providing advice
and data. The authors would like to acknowledge the helpful comments of
the anonymous referees.
100 Chapter 3. Mathematical model of MRSA
Bibliography
Amazian, K., Abdelmoumene, T., Sekkat, S., Terzaki, S., Njah, M., Dhidah, L., Caillat-Vallet, E., Saadatian-Elahi, M., Fabry, J., Members Of The Nosomed, N., 2005. Mul-ticentre study on hand hygiene facilities and practice in the mediterranean area:results from the nosomed network. J Hosp Infect.
Aslanidou, H., Dey, D., Sinha, D., 1998. Bayesian analysis of multivariate survival datausing Monte Carlo methods. The Canadian Journal of Statistics 26 (1), 33–48.
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Bailey, N., 1975. The Biomathematics of Malaria. Charles Griffin, London.
Bartley, J., 2002. First case of VRSA identified in Michigan. Infect Control Hosp Epi-demiol 23 (8), 480.
Becker, N., 1989. Analysis of Infectious Diseases Data. Chapman and Hall/CRC.
Boyce, J., April 2005. Epidemiology; prevention; and control of methicillin-resistantStaphyloccus aureus in adults-I. In: UpToDate, Rose, B.D (ed).
Boyce, J. M., Potter-Bynoe, G., Chenevert, C., King, T., 1997. Environmental conta-mination due to methicillin-resistant Staphylococcus aureus: possible infectioncontrol implications. Infect Control Hosp Epidemiol 18 (9), 622–7.
Capitano, B., Leshem, O. A., Nightingale, C. H., Nicolau, D. P., 2003. Cost effect ofmanaging methicillin-resistant Staphylococcus aureus in a long-term care facility.J Am Geriatr Soc 51 (1), 10–16.
Carayon, P., Gurses, A., 2005. A human factors engineering conceptual framework ofnursing workload and patient safety in intensive care units. Intensive and CriticalCare Nursing 21, 284301.
Cepeda, J. A., Whitehouse, T., Cooper, B., Hails, J., Jones, K., Kwaku, F., Taylor, L., Hay-man, S., Cookson, B., Shaw, S., Kibbler, C., Singer, M., Bellingan, G., Wilson, A. P.,2005. Isolation of patients in single rooms or cohorts to reduce spread of mrsa inintensive-care units: prospective two-centre study. Lancet 365 (9456), 295–304.
102 BIBLIOGRAPHY
Cookson, B., Peters, B., Webster, M., Phillips, I., Rahman, M., Noble, W., 1989. Staffcarriage of epidemic methicillin-resistant Staphylococcus aureus. J Clin Microbiol27 (7), 1471–6.
Cooper, B., Lipsitch, M., 2004. The analysis of hospital infection data using HiddenMarkov Models. Biostatistics 5 (2), 223–37.
Cooper, B. S., Medley, G. F., Scott, G. M., 1999. Preliminary analysis of the transmis-sion dynamics of nosocomial infections: stochastic and management effects. JHosp Infect 43 (2), 131–47.
Cooper, B. S., Medley, G. F., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A.,Duckworth, G., Lai, R., Ebrahim, S., 2004. Methicillin-resistant Staphylococcus au-reus in hospitals and the community: stealth dynamics and control catastrophes.Proc Natl Acad Sci U S A 101 (27), 10223–8.
Cooper, B. S., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A., Medley, G. F.,Duckworth, G. J., Lai, R., Ebrahim, S., 2003. Systematic review of isolation poli-cies in the hospital management of methicillin-resistant Staphylococcus aureus:a review of the literature with epidemiological and economic modelling. HealthTechnol Assess 7 (39), 1–194.
D’Agata, E. M., Webb, G., Horn, M., 2005. A mathematical model quantifying the im-pact of antibiotic exposure and other interventions on the endemic prevalence ofvancomycin-resistant enterococci. J Infect Dis 192 (11), 2004–11.
de la Cal, M. A., Cerda, E., van Saene, H. K., Garcia-Hierro, P., Negro, E., Parra, M. L.,Arias, S., Ballesteros, D., 2004. Effectiveness and safety of enteral vancomycinto control endemicity of methicillin-resistant Staphylococcus aureus in a med-ical/surgical intensive care unit. J Hosp Infect 56 (3), 175–83.
Engemann, J. J., Carmeli, Y., Cosgrove, S. E., Fowler, V. G., Bronstein, M. Z., Trivette,S. L., Briggs, J. P., Sexton, D. J., Kaye, K. S., 2003. Adverse clinical and economic out-comes attributable to methicillin resistance among patients with Staphylococcusaureus surgical site infection. Clin Infect Dis 36 (5), 592–8.
Ericksen, K., Erichsen, I., 1963. Clinical occurrence of methicillin-resistant strains ofStaphylococcus aureus. Ugeskr Laeger 125, 1234–40.
Garner, J., Jan 1996. Guideline for isolation precautions in hospitals. the hospitalinfection control practices advisory committee. Infect Control Hosp Epidemiol17 (1), 53–80.
Gelman, A., Carlin, J., Stern, H., Rubin, D. B., 2004. Bayesian data analysis, 2nd Edi-tion. Texts in statistical science. Chapman & Hall/CRC, Boca Raton, Fla.
Grundmann, H., Hori, S., Winter, B., Tami, A., Austin, D. J., 2002. Risk factors for thetransmission of methicillin-resistant Staphylococcus aureus in an adult intensivecare unit: fitting a model to the data. J Infect Dis 185 (4), 481–8.
Johnson, P. D., Martin, R., Burrell, L. J., Grabsch, E. A., Kirsa, S. W., O’Keeffe, J., Mayall,B. C., Edmonds, D., Barr, W., Bolger, C., Naidoo, H., Grayson, M. L., 2005. Efficacyof an alcohol/chlorhexidine hand hygiene program in a hospital with high rates ofnosocomial methicillin-resistant Staphylococcus aureus (MRSA) infection. Med JAust 183 (10), 509–14.
BIBLIOGRAPHY 103
McBryde, E. S., Bradley, L. C., Whitby, M., McElwain, D. L., 2004. An investigation ofcontact transmission of methicillin-resistant Staphylococcus aureus. J Hosp Infect58 (2), 104–8.
Meka, V. G., Pillai, S. K., Sakoulas, G., Wennersten, C., Venkataraman, L., DeGirolami,P. C., Eliopoulos, G. M., Moellering, R. C., J., Gold, H. S., 2004. Linezolid resistancein sequential Staphylococcus aureus isolates associated with a T2500A mutation inthe 23S rRNA gene and loss of a single copy of rrna. J Infect Dis 190 (2), 311–7.
Muto, C., Jernigan, J., Ostrowsky, B., Richet, H., Jarvis, W., Boyce, J., Farr, B., May 2003.SHEA guideline for preventing nosocomial transmission of multidrug-resistantstrains of Staphylococcus aureus and enterococcus. Infect Control Hosp Epidemiol24 (5), 362–86.
Nimmo, G., Bell, J., Collignon, P., 2003. Fifteen years of surveillance by the AustralianGroup for Antimicrobial Resistance (AGAR). Communicable Diseases Intelligence27, Suppl:S47–54.
Raboud, J., Saskin, R., Simor, A., Loeb, M., Green, K., Low, D. E., McGeer, A., 2005.Modeling transmission of methicillin-resistant Staphylococcus aureus among pa-tients admitted to a hospital. Infect Control Hosp Epidemiol 26 (7), 607–15.
Sanford, M. D., Widmer, A. F., Bale, M. J., Jones, R. N., Wenzel, R. P., 1994. Efficient de-tection and long-term persistence of the carriage of methicillin-resistant Staphy-lococcus aureus. Clin Infect Dis 19 (6), 1123–8.
Scanvic, A., Denic, L., Gaillon, S., Giry, P., Andremont, A., Lucet, J., May 2001. Dura-tion of colonization by methicillin-resistant Staphylococcus aureus after hospitaldischarge and risk factors for prolonged carriage. Clin Infect Dis. 32 (10), 1393–1398.
Sebille, V., Chevret, S., Valleron, A. J., 1997. Modeling the spread of resistant noso-comial pathogens in an intensive-care unit. Infect Control Hosp Epidemiol 18 (2),84–92.
Sebille, V., Valleron, A.-J., 1997. A computer simulation model for the spread of noso-comial infections caused by multidrug-resistant pathogens. Computers and Bio-medical Research 30 (4), 307–322.
Teare, E. L., Barrett, S. P., 1997. Is it time to stop searching for MRSA? Stop the ritualof tracing colonised people. BMJ 314 (7081), 665–6.
Whitby, M., McLaws, M. L., 2004. Handwashing in healthcare workers: accessibilityof sink location does not improve compliance. J Hosp Infect 58 (4), 247–53.
Wong, T. W., Tam, W. W., 2005. Handwashing practice and the use of personal pro-tective equipment among medical students after the sars epidemic in Hong Kong.Am J Infect Control 33 (10), 580–6.
CHAPTER 4
An investigation of contact transmission of
methicillin-resistant Staphylococcus
aureus
Statement of joint authorshipEmma McBryde initiated the concept of the study, wrote the manuscript,
collected the data, analysed the data and acted as corresponding author.
Lisa Bradley assisted in data collected, proofread the manuscript.
Mike Whitby critically reviewed the manuscript.
Sean McElwain assisted in data analysis and critically reviewed the manu-
script.
106 Chapter 4. Contact transmission of MRSA
Summary
Hand hygiene is critical in the healthcare setting and it is believed that
MRSA, for example, is transmitted from patient to patient largely via the
hands of health professionals. A study has been carried out at a large busy
teaching hospital to estimate how often the gloves of a healthcare worker
are contaminated with MRSA after contact with a colonised patient. The
effectiveness of handwashing procedures to decontaminate the health
professionals’ hands was also investigated together with how well different
health care professional groups complied with hand washing procedures.
The study show that about 17%(9%-25%) of contacts between a health care
worker and a MRSA-colonised patient leads to transmission of MRSA from
a patient to the gloves of a healthcare worker. Different health professional
groups have quite different rates of compliance with infection control pro-
cedures. Non-contact staff (cleaners, food-services) had the shortest hand
washing times. In this study, glove use compliance rates were above 75%
in all healthcare worker groups except doctors whose compliance was only
27%.
4.1 Introduction
Methicillin-resistant Staphylococcus aureus (MRSA) was first identified in
the 1960s by Ericksen and Erichsen (1963) in Europe and subsequently
spread throughout the world. It is believed that the primary mechanism of
MRSA spread throughout hospitals is via direct contact between patients
and healthcare workers (Pittet et al., 2000) however few studies have been
performed to quantify this transmission.
The transmission rate per contact is an important epidemiologic determi-
nant of MRSA. We have defined contact transmission rate in this study as the
probability that a healthcare worker who is uncolonised prior to contacting a
patient acquires MRSA on their hands or gloves during the contact. An accu-
rate estimate of the transmission rate of MRSA is essential for development
of realistic mathematical models of transmission which can be used to pre-
dict outcomes of interventions and their cost-effectiveness (see, for example
Cooper et al. (1999) and Austin et al. (1999) and references therein).
To date, as far as the authors are aware, there have been few attempts to
measure contact transmission rates for resistant bacteria. Most studies
on bacterial transmission have relied on self-inoculation (Foster, 1960)
4.2 Methods 107
artificial contamination (Rotter, 2001) and random sampling of staff on
wards (Casewell and Phillips, 1977) to determine transmission of organisms
to healthcare workers’ hands. One investigation examined nurses’ hands
after a clinical care episode (Trick et al., 2003) although this study did not
look specifically at MRSA-positive patients.
Other important epidemiological determinants of MRSA transmission are
the effects of different infection control interventions on the contact trans-
mission rate. Again, few studies have addressed this specifically. While it
has been shown that the use of a combination of infection control measures
can reduce the incidence of MRSA (Srinivasan et al., 2002) there have been
no reports to date that have demonstrated the efficacy of glove use alone in
reducing MRSA transmission.
4.2 Methods
4.2.1 Hand sampling
All healthcare workers on duty at Princess Alexandra Hospital (Brisbane, Aus-
tralia) between July 2003 and December 2003 entering the room of a patient
with MRSA were eligible for the study. Healthcare workers were intercepted
following a patient care episode and asked to participate in the study. Once
consent was obtained, a modified glove juice hand culture was performed on
the healthcare worker in accordance with published methodologies (Larson
et al., 2002). The hands of any healthcare worker were sampled no more
than twice throughout the study. A pre-wash sample was obtained from
one hand and a post-wash sample from the other hand. The choice of hand
was determined by a coin toss. Participants inserted one hand into a sterile
polyethylene bag with 50ml of sampling solution, containing 0.3% lecithin,
0.1% polysorbate 80 and 0.1% sodium thiosulphate. A timed one minute
hand massage was performed through the wall of the bag. The HCW was
then asked to remove both gloves and wash and dry hands as normal. A
second identically-conducted one minute hand massage was performed
on the other hand. The liquid from the hand massages was transferred to
separate sterile jars before transport to the microbiology laboratory.
Data were also collected regarding glove use, wearing of rings, type of pa-
tient contact (direct contact with patient skin, body fluids, patient transfer,
intravenous line care, tracheostomy care, wound care) and role of healthcare
worker. Hand washing agent was recorded (Microshield TM and Microshield
108 Chapter 4. Contact transmission of MRSA
2TM were available) and hand wash was timed. Information was recorded on
the patient, including most recent MRSA positive culture, presence of infec-
tion or colonisation and site of colonisation or infection.
Additionally, 100 gloves taken from cubicles around patients with MRSA were
tested for the presence of MRSA, using the same glove juice formula as was
used for the post-contact samples.
4.2.2 Laboratory technique
Dilutions : 500µL of the original sample was used to prepare three 10-fold di-
lutions, in glove juice sampling medium, described elsewhere.10 A sample of
100µL from each dilution (including the original sample) was plated to man-
nitol salt agar with oxacillin and incubated in air at 350C for 48hrs. Colony
counts were performed, selecting plates with 10-100 colonies (except undi-
luted sample where plates with 1-100 colonies were counted). Colony counts
were expressed in colony forming units per mL of glove juice.
MRSA Identification : the specimens were plated onto oxacillin-impregnated
agar at four different dilutions. Colony counts were performed for all growth
and a coagulase test was used to establish the presence of S. aureus. Sus-
pected S. aureus colonies were sub-cultured onto colistin-naladixic acid agar
and incubated in air at 350C for 24hr. Catalase and slide coagulase positive
colonies were tested using the Bio Merieux VitekTM gps-431 gram-positive
susceptibility card, to confirm methicillin resistance. An antibiogram was
performed on each of the MRSA isolates. Non-multiresistant MRSA was de-
fined as a S. aureus isolate which was resistant to penicillin, methicillin, (and
therefore cephalothin) and was resistant to no more than two of the remain-
ing antibiotics tested, namely to gentamicin, tetracycline, erythromycin,
ciprofloxacin, fusidic acid, rifampicin, and clindamycin.
4.2.3 Data analysis
The Pearson correlation coefficient was used to compare hand washing time
with post- and pre-wash counts. A Mann-Whitney test was used to compare
discrete variables such as glove use, presence of rings and type of hand wash-
ing liquid, against continuous data such as colony counts. For all analyses of
discrete variables and discrete outcomes, such as compliance with glove use
versus MRSA acquisition, χ2 tests were used. Analysis of hand washing time
4.3 Results 109
versus type of healthcare worker colony counts utilised analysis of variance
(ANOVA) techniques.
4.3 Results
4.3.1 Detection of MRSA
Samples were taken from 129 healthcare workers. Of these 36 (28%) were not
wearing gloves. Of the 93 staff members who put gloves on prior to the con-
tact, 12 (13%) tested positive for MRSA in the post-contact pre-wash sample.
When healthcare workers who had no contact with the patient, bed or pa-
tient or bed clothes were excluded from the analysis 12 out of 70 tested posi-
tive leading to an estimate of a transmission rate of 17% (CI 9-25%). Of those
who had positive pre-wash samples, 11 of 12 were negative for MRSA in the
post-wash sample. One staff member who initially tested negative for MRSA
in the pre-wash sample, tested positive following removal of gloves and hand
washing. Of the 36 HCW who did not wear gloves, 5 (14%) tested positive in
the pre-wash for MRSA and two of these also had positive post-wash results.
The relative risk of MRSA following hand washing if gloves were not worn
compared with those who wore gloves in this study was 5.2, however this was
not statistically significant (p=0.3).
Sampling Episode
(n=129)
No gloves worn
(n=36)
Gloves worn
(n=93)
MRSA+
Pre-wash
(n=5)
MRSA-
Pre-wash
(n=31)
MRSA-
Pre-wash
(n=81)
MRSA+
Pre-wash
(n=12)
MRSA+
Post-wash
(n=2)
MRSA-
Post-wash
(n=3)
MRSA+
Pre-wash
(n=0)
MRSA-
Pre-wash
(n=31)
MRSA+
Pre-wash
(n=1)
MRSA-
Pre-wash
(n=11)
MRSA+
Pre-wash
(n=1)
MRSA-
Pre-wash
(n=80)
Figure 4.1: Flow diagram of study participants and results of MRSA testing.
110 Chapter 4. Contact transmission of MRSA
4.3.2 Pre-handwash sample: oxacillin-resistant colony
counts
A number of factors were investigated, looking for an association with
pre-wash colony count. The only factor that had a significant association
with colony count was glove use. The group that used gloves (N=93) had a
median colony count of 30 colonies/ml whereas those without gloves (N=36)
had a median colony count of 930 colonies/ml. The difference between the
two groups was highly significant (p=0.0005). All other factors investigated
including; healthcare worker, presence of rings on the pre-wash hand and
type of contact had no statistically significant association with prewash
colony count.
4.3.3 Post-handwash sample: oxacillin-resistant colony
counts
There was a trend towards lower post wash colony counts when Microshield
2TM was used compared with Microshield TM with mean post wash counts
of 5030 colonies/mL and 12010 colonies/mL respectively (p=0.063). Hand
washing time, glove use, type of healthcare worker and presence of rings
on the post-wash hand did not have significant effects on post-wash colony
counts.
4.3.4 Compliance with infection control procedures
The median hand washing time was 26s, but there was considerable variation
as shown in Figure 4.2. Nurses and ward assistants had the highest median
hand washing time, with food assistants and cleaners having the lowest.
Table 4.1 shows the compliance rate for glove use in each of the healthcare
worker groups. Compliance with glove use was 75% or more in all healthcare
worker groups except doctors in whom it was 27% (p=0.003).
The difference in hand washing time amongst different healthcare groups is
statistically significant (p=0.019 using one way ANOVA).
4.4 Discussion 111
Figure 4.2: Boxplot of time taken to wash hands, based on type of healthcareworker.
Type of Healthcare Worker Compliancewith glove use(%)
Nurse 76Doctor 27Physiotherapist 83Ward Assistant 91Food Services 75Cleaner 75
Table 4.1: Compliance with glove use amongst different healthcare workergroups.
4.4 Discussion
This is the first study, as far as the authors are aware, that has attempted to
measure the rate of transmission of MRSA from patient to healthcare worker
during a single routine contact between a healthcare worker and an MRSA
positive patient prior to infection control procedures.
Trick et al. (2003) performed cultures of samples of nurses hands after a
patient contact, aiming to determine the prevalence of contamination of
healthcare workers’ hands with a variety of organisms. The investigation did
not aim to determine the specific transmission probability following a single
contact as gloves were removed prior to pre-wash testing. In that study,
gloves were only worn in 55 of the 282 health workers sampled, making it
difficult to attribute the presence of organisms to the particular contact.
The current study attempts to measure the transmission of MRSA during a
112 Chapter 4. Contact transmission of MRSA
single contact. To achieve this, healthcare workers were intercepted after
a patient contact and cultures were taken from their gloved hands, before
hand washing had occurred. A potential source of contamination in this
study were the gloves worn by healthcare workers. To assess the likelihood
of glove contamination, 100 gloves were taken from the cubicles of MRSA
colonised patients and cultured for MRSA. None of gloves tested positive.
The rate of contamination of healthcare workers gloved hands with MRSA
following a contact with patient, patient’s clothes or patient’s bed was esti-
mated to be 17%. Interestingly, the type of healthcare worker, type of contact
and use of gloves did not have an impact on transmission rate. There was
a trend towards persistence of MRSA after hand washing if gloves were not
worn, but this did not reach statistical significance. A future study with larger
numbers is needed to confirm the association.
The only factor that significantly correlated with number of oxacillin
colonies on pre-wash specimens was glove use. Those wearing gloves had
lower colony counts than those not wearing gloves. Following glove removal
and hand washing, there was no difference between the two groups.
There was a trend toward fewer colonies in the post-wash sample when
Microshield 2TM was used rather than Microshield ∗TM . All other factors
tested were not associated with differences in post-wash colony counts.
Trick et al. (2003) found an association between ring-wearing and presence
of organisms on nurses hands for all organisms tested except methicillin-
resistant coagulase-negative staphylococci. The results of the current study
were consistent with these findings by Trick et al. (2003) as in the current
study only oxacillin resistant flora were measured.
In summary, the results of this study show that about 17%(9%-25%) of
contacts between a healthcare worker and a MRSA-colonised patient leads
to transmission of MRSA from a patient the gloves of a healthcare worker.
In addition, different healthcare workers exhibited different behaviour
with adherence to infection control measures. Non-contact staff (cleaners,
food-services) had the shortest hand washing times. In this study, glove use
compliance rates were above 75% in all healthcare worker groups except
doctors whose compliance was only 27 %.
4.5 Acknowledgement 113
4.5 Acknowledgement
This work was partially supported by a grant under the Australian Research
Council Linkage Scheme (LP347112).
114 Chapter 4. Contact transmission of MRSA
Bibliography
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Casewell, M. W., Phillips, I., 1977. Hands as a route of transmission for klebsiellaspecies. Brit Med J 2, 1315–1317.
Cooper, B. S., Medley, G. F., Scott, G. M., 1999. Preliminary analysis of the transmis-sion dynamics of nosocomial infections: stochastic and management effects. JHosp Infect 43 (2), 131–47.
Ericksen, K., Erichsen, I., 1963. Clinical occurrence of methicillin-resistant strains ofStaphylococcus aureus. Ugeskr Laeger 125, 1234–40.
Foster, W., 1960. Experimental staphylococcal infections in man. Lancet ii, 1373–1376.
Larson, E., Gomez-Duarte, C., Lee, L., Della-Latta, P., Kain, D. J., Keswick, B. H., 2002.Microbial flora on the hands of homemakers. Am J Infect Control 31, 72–79.
Pittet, D., Hugonnet, S., Harbarth, S., Mourouga, P., Sauvan, V., Touveneau, S., Per-neger, T. V., 2000. Effectiveness of a hospital-wide programme to improve compli-ance with hand hygiene. Infection Control Programme. Lancet 356 (9238), 1307–12.
Rotter, M., 2001. Arguments for alcoholic hand disinfection. J Hosp Infect 48 (supplA), s4–8.
Srinivasan, A., Song, X., Ross, T., Merz, W., Brower, R., Perl, T. M., 2002. A prospectivestudy to determine whether cover gowns in addition to gloves decrease nosoco-mial transmission of vancomycin-resistant enterococci in an intensive care unit.Infect Control Hosp Epidemiol 23 (8), 424–8.
Trick, W., Vernon, M., Hayes, R., Nathan, C., Rice, T., Peterson, B. J., Segreti, J., Welbel,S., Solomon, S., Weinstein, R. A., 2003. Impact of ring wearing on hand contami-nation and comparison of hygiene agents in a hospital. Clin Infect Dis 36 (1 June),1383–1390.
CHAPTER 5
Characterising outbreaks of
vancomycin-resistant enterococci using
statistical methods
Statement of joint authorshipEmma McBryde wrote the manuscript, developed the mathematical model,
designed the analysis and acted as corresponding author.
Tony Pettitt assisted with the analysis of data, proof read and critically re-
viewed the manuscript.
Sean McElwain assisted with the development of the mathematical model
and proof read and critically reviewed the manuscript.
Ben Cooper proof read and critically reviewed the manuscript. Assisted with
design of the analysis.
118 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
AbstractBackground Antibiotic-resistant nosocomial pathogens, such as van-
comycin resistant enterococci (VRE), can arise in epidemic clusters or
sporadically. Genotyping, to determine the number of colonising strains,
is commonly used to distinguish epidemic from sporadic VRE. We aim to
develop a statistical method to determine the transmission characteristics of
VRE.
Methods and Findings A structured continuous-time hidden Markov model
(HMM) was developed. The hidden states were the number of VRE-colonised
patients (both detected and undetected) at a series of time points. The in-
put for this study was weekly prevalence data; 157 weeks of VRE prevalence
observations from an Australian teaching hospital. We estimated 2 parame-
ters; one to quantify the cross-transmission of VRE (epidemic component)
and one to quantify the level of VRE colonisation from sources other than
cross-transmission (sporadic component). We compared the results to those
obtained by concomitant genotyping and phenotyping.
We estimated that 89% of transmissions were due to ward cross-transmission
while 11% were sporadic. This concordes with the findings that 90% were
identical with respect to glycopeptide resistance genotype and 84% were
identical or nearly identical on Pulsed-Field Gel Electrophoresis (PFGE). We
were also able to estimate the underlying colonisation prevalence (including
those not detected). There was some evidence, based on model selection
criteria, that the cross-transmission parameter changed throughout the
study period. The model that allowed for a change in transmission just
prior to the outbreak and again at the peak of the outbreak was superior
to other models. This model estimated that cross-transmission increased
at week 120 and declined after week 135, coinciding with environmental
decontamination.
Significance We found that HMMs can be applied to serial prevalence data
to estimate the characteristics of acquisition of nosocomial pathogens and
distinguish between epidemic and sporadic acquisition. Our methodology
required only serial prevalence and length of hospital stay data. This model
was able to estimate transmission parameters despite imperfect detection of
the organism. The results of this model were validated against PFGE and gly-
copeptide resistance genotype data and produced very similar results. Ad-
ditionally, HMMs can provide information about unobserved events such as
undetected colonisation.
5.1 Introduction 119
5.1 Introduction
There has been an alarming world-wide increase in the rate of infection
from vancomycin-resistant enterococci (VRE) in the last 15 years (Murray,
2006). Enterococci are part of the normal gastrointestinal flora and VRE
colonisation often is asymptomatic and undetected. However, in patients
with compromised immune systems and breached integument, enterococci
can become pathogenic, causing, for example, urinary tract infection, bac-
teraemia, and endocarditis. Large teaching hospitals and intensive care units
have the highest rate of infection with VRE (Weinstein, 2005). Infection with
enterococci harbouring a vancomycin resistance gene is associated with
higher mortality (Lodise et al., 2002) and many strains of VRE are resistant to
all known antibiotics.
Acquisitions of VRE colonisation can be broadly grouped into those that
come from cross-transmission within the ward, which we call transmitted,
and VRE that comes from other sources, which we call sporadic. Ward
transmission of multi-resistant organisms (MROs) is believed to be pre-
dominantly from patient to patient via the transiently contaminated hands
of health care workers (Boyce, 2001). The sources of sporadic VRE include
patients gastrointestinal tract, prior colonisation with VRE and transmission
from outside the ward. The presence of VRE on admission is often initially
not detected owing to infrequent swabbing, poor sensitivity of swabs or un-
detectable quantities of organism. VRE may exist in sub-detectable numbers
in human gut so that exposure of patients to antibiotics which facilitate VRE
growth (Donskey et al., 2002) may lead to an apparently new case of VRE.
VRE is also known to spread from other hospital wards via patient and staff
movements (Trick et al., 1999).
To select the most appropriate infection control interventions, one needs
to be able to estimate how much of the new acquisition is transmitted and
how much is sporadic. Restricting antibiotic exposure is thought to control
sporadic VRE, by reducing selection pressure in patients endogenous flora,
while hand hygiene, cohorting, patient isolation and limiting admission of
colonised patients are thought to impact on transmitted VRE.
Outbreak investigation often involves time intensive methods to characterise
the mode of VRE acquisition. Genotyping techniques such as pulsed-field gel
electrophoresis (PFGE), distinguish clonal outbreaks, which are presumed to
be due to transmitted VRE, from multiple new strain introductions, which
120 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
are presumed to be due to sporadic VRE. There are occasions when this tech-
nique breaks down, when horizontal transfer of the resistance gene, vanA or
vanB, can lead to several different genotypes being detected when in fact a
single transposon is being transmitted (Suppola et al., 1999; Weinstein, 2005;
Bradley, 2002).
Attempts have been made to distinguish between the two processes of
colonisation based on statistical analysis of surveillance data. Pelupessy
et al. (2002) used a Markov model, without hidden states, to estimate
transmission parameters, finding estimates were similar to those using
full event data and genotyping (PFGE). Cooper and Lipsitch (2004) used
structured and unstructured hidden Markov models (HMMs) to describe
infection incidence time series data, and to estimate transmission para-
meters. Collinearity between parameter estimates, failure of convergence
and computational difficulties were identified as potential problems using
HMMs for sparse data such as is typically found in time series infection
control data. Forrester and Pettitt (2005) compared background rates to
cross-transmission rates of methicillin-resistance Staphylococcus aureus,
finding background rates were larger than cross-transmission rates. Esti-
mating transmission coefficients using hospital infection control data has
a number of challenges. There are unobserved processes occurring; the
time of new acquisition of colonisation is not observed. Additionally, when
relying on routine swabs to determine the number of colonised patients, the
sensitivity of swabs is less than 100%.
This study uses an epidemic model structure to characterise transmission
of vancomycin-resistant enterococci during an outbreak at an 800 bed Aus-
tralian teaching hospital. The current paper extends the work by Pelupessy
et al. (2002) by estimating epidemiological parameters in the presence of
suboptimal swab sensitivities. It also allows delays in detection of VRE. We
use a hidden Markov model structure to estimate transmission in the face of
incomplete datasets and unobserved events. This framework distinguishes
between rates of transmitted and sporadic VRE acquisitions. This study also
considers that the transmission rates may change over time. Section 5.2.1
describes the data used to estimate VRE epidemic determinants. Section
5.2.4 describes the model of VRE transmission, while Section 5.2.5 describes
the HMM and the methodology behind it. Section 5.3 gives the results of the
parameter estimates, comparison of model estimates and genotyping data
and model selection.
5.2 Methods 121
5.2 Methods
5.2.1 Description of outbreak and infection control inter-
ventions
VRE was first isolated at the Princess Alexandra Hospital in October 1996 and
a VRE screening programme commenced in January 1997, the beginning
of the data collection period for this study. Data used in this study are VRE
colonisation data from the Intensive Care Unit (ICU), Renal and Infectious
Diseases Units. VRE colonised patients were identified by clinical isolates,
weekly routine screening and contact tracing swabs. Infection control
interventions introduced from the start of the study period were restriction
of vancomycin and third-generation cephalosporin use and isolation of
colonised patients. From week 125 of this study, infection control teams
were aware of an increased prevalence of VRE and further measures were
taken. Dedicated equipment was used in patient rooms and patients were
cohorted. VRE patients requiring haemodialysis used a dialysis facility
within the infection control unit. Medical and nursing staff wore disposable
aprons and latex gloves for patient contacts. An environmental audit was
performed in August 1999, approximately week 135 of the study period, and
an aggressive cleaning programme was instituted (Bartley et al., 2001).
5.2.2 Serial surveillance data used for statistical analysis
Input data for the statistical model in this study were
• weekly prevalence data for VRE colonisation
• mean length of stay of colonised patients; 15 days.
• the total number of beds in the wards; N = 68.
The data were collected from 1st January, 1997 until 31st December, 1999. The
weekly prevalence data are shown in Figure 5.1.
5.2.3 Data used for cluster analysis
Microbiological and clinical data were collected, including admission dates
and discharge dates of VRE colonised patients, as well as date of first positive
isolate. Additionally, we had information on the colonisation status on ad-
mission of three of the patients transferred from other hospitals. Genotype
122 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
0 50 100 1500
2
4
6
8
10
12
14
16
18
Week of Study
Week 120 Week 135
Figure 5.1: Prevalence data for VRE over 157 weeks. Arrows show times inwhich changes in transmission rates may have taken place.
data, both Pulsed Field Gel Electrophoresis (PFGE) and glycopeptide resis-
tance genotyping, were compared with the results of the statistical analysis
as part of the study validation. Presumptive VRE colonies were identified us-
ing standard techniques. Speciation (distinguishing E.fecium and E.fecalis)
was initially achieved by carbohydrate fermentation reactions of arabinose,
mannose and raffinose then confirmed by a multiplex PCR assay based on
specific detection of genes encoding D-alanine: D-alanine ligases (Bartley
et al., 2001). VRE phenotype was identified based on vancomycin and teicho-
planin MICs (mean inhibitory concentrations) using the E-test method. This
presumptively distinguishes vanA VRE, resistant to both vancomycin and te-
ichoplanin, from vanB VRE, resistant to vancomycin but sensitive to teicho-
planin. This presumptive phenotype result was confirmed by glycopeptide
resistance genotyping, achieved through a modified multiplex PCR assay, de-
scribed in detail in Bartley et al. (2001).
In the study on this outbreak by Bartley et al. (2001), isolates were also charac-
terised using PFGE. Electrophoretic band patterns were analysed according
to the criteria established by Tenover et al. (1995). Computer comparison us-
ing Gel Compar version 4.1 (Applied Maths Kortrijk, Belgium) was based on
the algorithm of the unweighted pair group method for arithmetic averages
and using the Dice coefficient with 1.5% band tolerance (Bartley et al., 2001).
This information was used to estimate the proportion of isolates that were
from the same strain.
5.2.4 Model of transmission
We base our ward transmission model on the Susceptible-Infected (SI) model
with migration, described by Bailey (1975). Modified versions of this model
5.2 Methods 123
have been used previously to analyse nosocomial transmission data (Pelu-
pessy et al., 2002; Cooper and Lipsitch, 2004; Forrester and Pettitt, 2005).
A schematic of the model is shown in Figure 5.2. The rate of cross-
transmission of VRE colonisation (per colonised per susceptible patient per
day) is denoted by β. It is assumed that the ward is of fixed size, N , hence the
number of uncolonised patients is N −C. Colonised patients are assumed to
remain colonised for their entire hospital stay, therefore transition from the
colonised to uncolonised compartments occurs via discharge of a colonised
patient and replacement with an uncolonised patient, which occurs at a rate
µC. Duration of stay of colonised patients was available from the dataset.
Acquisition of VRE that is transmitted is described by the mass-action
term, βC(N − C). VRE acquisition that is sporadic can arise through ward
admission of a colonised patient or any other process that is not related to
the number of colonised patients, and occurs at a rate, ν(N −C). Each of the
processes that lead to sporadic acquisition (for example prior colonisation
or colonisation from out-of-ward sources, endogenous gastrointestinal
colonisation) can reasonably be assumed to be independent of the number
of colonised patients in the ward.
CN-C
( ) ( )C N C N Cβ υ− + −
Cµ
Figure 5.2: The transmission of bacterial pathogens in the hospital ward.
The probability of a change in the number of colonised patients, C, in a short
time period, h, is given by
Pr[C(t + h) = i + 1|C(t) = i] = βi(N − i)h + ν (N − i)h + o(h),
P r[C(t + h) = i− 1|C(t) = i] = µih + o(h),
P r[C(t + h) = i|C(t) = i] = 1− βi(N − i)h− ν (N − i)h−µih + o(h),
P r[C(t + h) = j (j 6= i− 1, i, i + 1)|C(t) = i] = o(h). (5.1)
124 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
The number of colonised patients in the ward at time t, C(t), forms a Markov
process on state space 0, ..., N , where N is the number of patients on the
ward. Reflecting boundaries occur at states i = 0 and i = N , provided ν > 0,
otherwise 0 is an absorbing state, and provided µ > 0, otherwise N is an
absorbing state.
5.2.5 Hidden Markov model
We aim to estimate parameters associated with sporadic colonisation, ν, and
the colonisation caused by ward transmission, β, using the structured HMM
illustrated in Figure 5.3.
C1
Y1
C2
Y2
C4
Y4
C3
Y3
Figure 5.3: Hidden Markov model. Here C represents the number ofcolonised patients in the ward (detected or undetected), Y represents thenumber of patients detected at each time point. The horizontal arrows rep-resent the transition from one state to the next, and the vertical arrows rep-resent the relationship between the hidden state and the corresponding ob-servation.
Our hidden Markov model (HMM) consists of: observations, Y , the num-
ber of patients detected at each time point; underlying hidden states, C, the
number of colonised patients in the ward; a transition model linking each
hidden state with its adjacent states, represented by horizontal lines in Fig-
ure 5.3; an observation model linking the data with the hidden state, repre-
sented by the vertical lines in Figure 5.3. There is one hidden state for each
observation, denoted C1, C2, ..., Cn.
The full conditional probability of any node depends only on neighbouring
nodes to which it is connected directly. The observation component of the
HMM, denoted by Y , consists of 157 data inputs of weekly VRE prevalence
taken over 3 years and the vector of time points, t = t1, ..., tn, corresponding
to each observation time. The vector C consists of the n = 157 hidden states.
The transition probability matrix, giving the relationship between the hid-
den states, is described in Section 5.2.6. The observation model, giving the
relationship between the observed and hidden states, is described in Section
5.2.7.
5.2 Methods 125
The parameters used in the model are given in Table 5.1.
Parameter Symbol value sourceNumber of patients N 68 directly from data setRemoval rate of colonised pt µ 1/15 day−1 directly from data setTransmission rate β 1.0× 10−3 fitted using HMMSporadic acquisition rate ν 2.0× 10−4 fitted using HMMDetection probability d 0.58-0.97 literature review
Table 5.1: Parameters used in the model. Fitted values are discussed in sec-tion 5.3.
Model assumptions
The model makes the following assumptions
1. The ward is of fixed size, N .
2. The model parameters are time invariant (this assumption is relaxed
later in the study).
3. Each observation is conditionally independent given the corresponding
hidden state.
4. The hidden states follow a first order time homogenous Markov
process, that is Pr(C(tk)|C(t1), ..., C(tk−1)) = Pr(C(tk)|C(tk−1)) =
Pr(C(tk − tk−1)|C(0)).
5. Homogenous mixing of patients takes place.
These assumptions are discussed in Section 5.4.
5.2.6 Constructing a transition probability matrix
Following the theory of Cox and Miller (1965), we developed a transition
probability matrix, Γ(tk−tk−1). The ijth element of Γ(tk−tk−1) gives the proba-
bility of having j colonised patients on the ward at time tk, given that there
were i colonised patients on the ward at time tk−1.
To construct the transition probability matrix for an arbitrary time interval,
firstly we developed a discrete time transition probability matrix, A, for a
small time interval, h. Let A be the matrix in which the ijth element is given
by Pr(C(t + h) = j|C(t) = i). A is given using the system of equations 5.1.
Here, i and j are the number of patients colonised in the ward and can take
on values 0, ..., N .
126 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
Let p(t) be the (N +1) vector of probabilities of the number colonised at time
t. The generator matrix, G is a square, (N + 1)× (N + 1), matrix that has the
property that
dp(t)
dt= Gp(t). (5.2)
The ijth element of the generator matrix, G, is the instantaneous rate of
change of probability of being in state j, given a beginning in state i. Then G
is given by
G = limh→0
1
h(A− I). (5.3)
Following from Expression 5.2, we have
p(tk+1) = p(tk)e(tk+1−tk)G, (5.4)
in general. Specifically, after a time interval tk+−tk, the probability of being in
state j having begun in state i is the ijth element of the transition probability
matrix, given by
Γ(tk+1−tk)ij= Pr(Ck+1 = j|Ck = i) = (e(tk+1−tk)G) ij. (5.5)
Cox and Miller (1965, Chap 4.5) and MacDonald and Zucchini (1997) give an
expanded explanation. The matrix exponential e(tk−tk−1)G was calculated us-
ing the MatlabTM “expm” function.
5.2.7 Observation Model
The probability, d, of being known to be colonised (and therefore being in-
cluded in the prevalence data) given that a patient is colonised was unknown.
Literature sources regarding the sensitivity of rectal swabs in detecting VRE
were used to develop an expression for the uncertainty in this parameter. Es-
timates of the sensitivity range from 0.58 (D’Agata et al., 2002) to 0.97 (Reisner
et al., 2000) with values in between (Lemmen et al., 2001; Trick et al., 2004).
We allowed for the uncertainty regarding the probability of detection by as-
signing a uniform[0.58, 0.97] prior distribution to d.
The probability relationship between the states and the data is described
by the binomial distribution Yk ∼ Bin(Ck, d), where Yk is the kth observed
5.2 Methods 127
colonisation prevalence and Ck is the actual number of colonised patients,
the hidden state. This assumes that the probability, d, remains constant over
the study period (for each iteration) and the probability of detection of each
colonised patient is independent of the number of other colonised patients.
Alternative observation models with greater dispersion could have been
used. For example, the Poisson or negative binomial distribution could have
been chosen, had we been dealing with incidence rather than prevalence
data. We chose the Binomial distribution because it has a sound probabilis-
tic basis (assuming fixed detection) and, unlike the Poisson, ensures that
the hidden state (number colonised) is always larger than the observation
(number detected), a necessary result when using prevalence data.
5.2.8 Bayesian framework
The parameters for transmitted VRE, β, and sporadic VRE, ν, were estimated
using a Bayesian framework. Let θp = β, ν, d be the vector of model
parameters. Baum et al.’s recursion formula, summarised in Appendix 5.A,
was used to determine the likelihood of the data, L(Y |θp). Uniform U[0,
0.1] prior probability distributions were assigned to β and ν, because little
was known about these parameters other than that negative values or values
higher than 0.1 were completely implausible. The posterior probability
distribution is given by
p(θp|Y ) ∝ π(θp)L(Y |θp), (5.6)
where π(θp) is the prior probability distribution of θp. This was estimated us-
ing a Monte-Carlo Markov chain algorithm, described in Appendix 5.B.
The Bayesian framework can provide estimates (and full posterior probabil-
ity density) of any function of model parameters including functions which
depend upon knowledge of hidden states. Let θh be the vector of n inferred
hidden states C1, ..., Cn and let θ = θp, θh. The proportion of VRE acquisi-
tions due to ward transmission, f(θ), is given by:
f(θ) =
∑nk=1 βCk(N − Ck)∑n
k=1 βCk(N − Ck) + ν(N − Ck). (5.7)
We evaluate the expectation, E[f(θ)|Y ], by drawing samples θk, k = 1, ..., m
128 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
from p(θ|Y ) and using the approximation of Gilks et al. (1996, Chapt1)
E[f(θ)|Y ] ≈ 1
m
m∑
k=1
f(θk). (5.8)
The algorithm for this Monte Carlo integration is given in Appendix 5.B.
5.2.9 Comparison of cluster analysis results using genotyp-
ing with statistical analysis
A genotyping study was performed on the VRE isolates by Bartley et al. (2001).
Of the 49 isolates available for analysis, 44 were found to be E.fecium vanA
using glycopeptide resistance genotyping. The estimated number of isolates
having identical or closely related patterns on PFGE using the criteria of Ten-
over et al. (1995) was 41 of 49.
Cluster analysis based on genotypic relatedness
We compared the proportion of “identical isolates” (presumed to be part of
a cluster) with the estimated proportion of transmitted VRE derived from the
HMM and prevalence data. The posterior probability distribution of the pro-
portion of VRE cases that are identical can readily be derived using a Bayesian
framework and conjugate prior distribution (see Gelman et al., 2004). De-
note the parameter of interest, the proportion of VRE acquisitions that are
identical, by p. Assume the form Beta(1, 1) for the prior distribution for the
proportion; this is the same as the uniform[0,1] prior. The probability of the
data is given by given by the Binomial Bin(a; (a + b), p), where a is the num-
ber of identical isolates and b is the number of non-identical isolates, as de-
tected by the laboratory methods. The posterior probability density of p is
Beta(1 + a, 1 + b).
5.3 Results
5.3.1 Parameter estimation
The estimated value for the transmission coefficient, β was 10×10−4 (CI957.9×10−4, 13× 10−4) and the sporadic acquisition rate ν was 2.0× 10−4 (CI950.85×
5.3 Results 129
10−4, 3.8×10−4 ). The coefficient of correlation between β and ν was estimated
to be -0.24.
The basic reproduction ratio, R0, is “the average number of persons directly
infected by an infectious case during its entire infectious period, after enter-
ing a totally susceptible population” (Giesecke, 1994). In this model it can be
shown to be R0 = βNµ
. The basic reproduction ratio is estimated to be 1.07
(CI95 0.78-1.34).
The mean value for the estimated detection rate was 0.75 with a 95% credible
interval of 0.59 to 0.93.
5.3.2 Comparison of statistical model and genotyping data
The proportion of VRE acquisitions due to transmission, was estimated to be
89% (CI95=78-95%), using Bayesian inference applied to the hidden Markov
model structure. This compares with 84% (41/49) of isolates observed to be
identical or nearly identical using PFGE genotyping and 90% (44/49) using
glycopeptide resistance genotyping. The posterior distribution of the esti-
mated proportion of colonisations due to ward transmission compared with
those found to be identical by glycopeptide resistance genotype and PFGE
methods are displayed in Figure 5.4.
5.3.3 Model selection and validation
The values of the Deviance information criterion (DIC) were used to assess
the optimum model to fit the data (Gelman et al., 2004). Results are given in
Table 5.2.
Several models were explored. Setting either β or ν to zero led to much higher
values for the DIC, giving substantial statistical support to a mixed model, in
which VRE colonisation arose both from cross transmission in the ward and
sporadically. The model in which β changed after week 120 was a superior fit
to the model with time-invariant parameters. Allowing for a further change
in β after week 135 provided the best fit of those models investigated. The
effective number of parameters in a latent variable model depends on the
collinearity of the parameters and the influence of the latent variables.
Internal validation of the model was achieved using a parametric bootstrap
analysis. Data were simulated using the time-invariant model (one β one
130 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
Proportion of colonisation due to cross−transmission
Pos
terio
r pr
obab
ility
den
sity
statistical resultPFGE dataglycopeptideresistance data
Figure 5.4: Posterior distribution of proportion of VRE acquisitions that aredue to ward transmission. The histogram gives the posterior distributionfrom the Bayesian analysis of the hidden Markov model, the solid curvegives the posterior distribution based on the observed proportion of identicalstrains using PFGE genotype data and the broken line gives the posterior dis-tribution based on observed proportion of identical strains using glycopep-tide resistance phenotype and genotype data (Bartley et al., 2001).
ν). The posterior distribution ν and β were estimated from simulated data,
demonstrating that the method achieves an unbiased estimate of the trans-
mission parameters.
5.4 Discussion
The aim of this study was to characterise transmission of VRE using sta-
tistical methods and simple serial surveillance data. We included a term
for sporadic colonisation because of evidence that new acquisitions of
VRE can occur through means other than within-ward patient to patient
cross-transmission. Sources of sporadic colonisation have been labelled in
the past as endogenous, spontaneous (Pelupessy et al., 2002) or background
(Forrester and Pettitt, 2005). Our statistical methods were designed to
distinguish between these two sources. Previous attempts have encountered
difficulties especially with identifiability of variables (Cooper and Lipsitch,
2004).
Full patient histories, PFGE and glycopeptide resistance genotype data were
used for validation but were not included in the statistical analysis in this
5.4 Discussion 131
Model Estimate ofβ (95%CI)× 10−4
Estimate ofν(95%CI)×10−4
DIC Pd
One value for ν and threevalues for β with changepoints at the end of week120 and 135
β13.4(0.28− 8.8)β215.3(13.5− 17.1)β310.9(7.1− 13.0)
2.2(0.96-4.0) 251 4.0
One value for ν and twovalues for β with changepoint at the end of week120
β13.4(0.28− 8.7)β211.9(10.2− 13.5)
2.2 (0.96-4.0) 253 2.3
One value for ν and onevalue for β
10(7.9-13) 2.0(0.85-3.8) 261 2.6
One value for ν and twovalues for β with changepoint at the end of week135
β111(7.6− 14.6)β29.6(7.9− 11.4)
2.0(0.88-3.7) 261 2.6
β = 0 and one value for ν 0 9.7(7.7-11.7) 393 1.2ν = 0 and one value for β 8.7(6.9-10.1) 0 531 1.5
Table 5.2: Comparison of different models using the Deviance InformationCriterion. Key Pd: effective number of parameters.
study. Estimates of the proportion of VRE resulting from cross-transmission
based on statistical methods (hidden Markov models) in this study were very
similar to those based on vancomycin resistance genotype data.
The proportion of identical isolates based on PFGE analysis was lower
than both the vancomycin resistance genotype data and the statistical
cluster analysis. This could be due to horizontal transfer of resistance
gene to new strains of enterococci, which has been reported previously
(Suppola et al., 1999; Weinstein, 2005). If horizontal transfer of resistance
genes occurs during an outbreak, cross-transmitted strains have identical
glycopeptide-resistance genotypes but different PFGE patterns, hence PFGE
under-estimates clustering.
Using a structured hidden Markov model, one can estimate the hidden
states behind the data, the number of patients colonised on the ward (both
detected and undetected). We used this to estimate the ward reproduction
ratio, which was 1.07. This value is just above the threshold value of unity,
which portends endemic VRE. We were able to make estimates of trans-
mission in the face of imperfect datasets in which transmission times and
patient histories were unknown and swab sensitivity was considerably less
than 100%.
132 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
For simplicity, this study assumed homogenous mixing of staff and patients.
Future studies could extend this model to include ward coupling, however
dividing the data to incorporate ward structure would lead to reduced preci-
sion in parameter estimates and increased model complexity.
The model presented in this study postulated that VRE acquisition arose
from both cross-transmission and sporadic sources. Model comparison
techniques found this model to be a far superior fit to the data compared
to models which relied on either cross-transmission or sporadic sources of
VRE acquisition alone, strongly supporting both modes of acquisition were
taking place.
We investigated changes in transmission over time using a structured epi-
demic model. Model comparison showed that there was evidence support-
ing the conclusion that there was an increase in cross-transmission just prior
to the outbreak. There was also evidence that the cross-transmission rate re-
duced after the epidemic peak at week 135, coinciding with the environmen-
tal cleaning intervention. Future studies using larger surveillance datasets
could extend the methodology presented to consider more models in which
parameters are time dependent. One approach to this would be to use the re-
versible jump Monte Carlo Markov chain method (Green, 1995) or the birth
death Markov process model (Stephens, 2000).
Inaccuracies in PFGE cluster analysis can arise from the horizontal transfer of
resistance genes. Glycopeptide resistance genotype analyses are not subject
to inaccuracies due to gene transfer but cannot distinguish different strains
that might all be of the same resistance genotype. Statistical methods are not
subject to these problems and have the additional advantage that they are
not resource intensive. They also have the potential to be used in real time,
within a control-chart outbreak alert system.
The model presented here can be applied to the surveillance of other
bacterial pathogens in small scale settings of healthcare institutions, such
as methicillin-resistant Staphylococcus aureus (MRSA), extended spectrum
beta-lactamase (ESBL) producing and other multi-resistant Gram-negative
pathogens.
Acknowledgements
This work was partially supported by a grant under the Australian Research
Council Linkage Scheme (LP0347112) and NHMRC scholarship number
5.A Likelihood computation 133
290541. The authors would like to thank Dr Mike Whitby for providing data
and Dr Paul Bartley for helpful comments.
Appendix
5.A Likelihood computation
The probability of the full dataset and a particular sequence of hidden states,
C1, C2, ..., Cn is given by
Pr(Y1, ..., Yn, C1, ..., Cn| β, ν) = Pr(C1)Pr(Y1|C1)n∏
k=2
ΓCk−1 CkPr(Yk|Ck), (5.9)
with ΓCk−1 Ckas defined in Section 5.2.6.
The likelihood calculation of this single permutation of hidden states
requires 2n computations even after the matrix exponential has been
evaluated. The full likelihood of the data over all the states is
Pr(Y1, ..., Yn| β, ν) =N+1∑C1=1
, ...,
N+1∑Cn=1
Pr(Y1, ..., Yn, C1, ..., Cn| β, ν) (5.10)
which requires 2n(N + 1)n computations for one likelihood evaluation
(Le Strat and Carrat, 1999). This intractable calculation (with n = 157 and
N = 68) can be simplified using Baum’s recursion technique (Baum et al.,
1970), as shown below.
The forward recursion involves simplifying the likelihood computations by
considering a partial observation sequence and a single state sequence. Let
φk(i) be the probability of the partial observation sequence (Y1, Y2, ..., Yk) pro-
duced by all possible state sequences that end in state i. The probability is
given by
φk(i) = L(Y1, ..., Yk, Ck = i| ν, β), k ≤ n. (5.11)
Let δ be the (size N + 1) vector of probabilities of the first state, (δi = Pr(C1 =
i)). In the forward recursion method of likelihood computation, the value of
δ needs to be determined in the absence of data. The stationary distribution
134 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
of the transition matrix can be used for this (MacDonald and Zucchini, 1997).
The probability of the first state and first observation, Y1, is given by
φ1(i) = δi Pr(Y1|C1 = i). (5.12)
The forward recursion formula is then applied. We multiply every state prob-
ability, φk−1(i), by the transition probability Γij and by the probability of the
kth data point given the hidden state j. This results in a vector of probabilities
which is then summed to determine φk(j). Thus the probability of subse-
quent states is given by
φk(j) =
[N∑
i=0
φk−1(i)Γij
]Pr(Yk|Ck = j). (5.13)
At each step in the forward recursion, the procedure can be terminated and
the probability of the partial observation sequence determined by
Pr(Y1, ..., Yk| ν, β) = ΣNi=0φk(i). (5.14)
The likelihood of the data can then be determined by
Pr(Y1, ..., Yn|β, ν) = ΣNi=0φn(i). (5.15)
See Petrushin (2000) for a detailed discussion of the forward and backward
recursion formulae.
5.B Monte Carlo Markov chain algorithm
The algorithm for this Monte-Carlo integration used to estimate the propor-
tion of VRE acquisitions due to ward cross-transmission, f(θ), is given below.
The MCMC algorithm has the following steps:
1. Assume the prior probability for β and ν, to be (U[0, .1]). These priors
were used as little prior information was known except that negative
values and values greater than 0.1 are completely implausible.
2. Initialise β and ν and d′.
5.B Monte Carlo Markov chain algorithm 135
3. Assign the prior probability of the hidden states. A discrete uniform dis-
tribution on (0, ..., N) was used.
4. Initialise each hidden state using its corresponding observation and the
(binomial) observation model Yk ∼ Bin(Ck, d).
5. Determine the probability of the data and sequence of hidden states
using Equation 5.9.
6. Propose a new β′ using a simple random walk, the step size ∼N(0, 10−4).
7. Accept β′ using a Metropolis-Hastings step with the acceptance proba-
bility
a = min1, π(β′)Pr(Y , C|β′)q(β′ → β)
π(β)Pr(Y , C|β)q(β → β′), (5.16)
where q(β → β′) is the proposal probability for β′ from β which is the
normal density for β′ with mean β and variance 10−4.
8. Repeat for ν ′ and d′.
9. Update each hidden state using a Gibbs update, drawing from the dis-
tributions given by the conditional probability of the states, determined
by neighbouring states and observations, as described below.
10. Determine f(θ) for the particular sequence of hidden states and para-
meters β and ν using Expression (5.7).
11. Iterate by returning to step 4.
12. Burn in using 50 000 iterations. Use the following 50 000 updates to es-
timate the posterior probability distribution (using the ergodic average)
of the hidden states (C1, ..., Cn) and f(θ).
13. Repeat steps 2-12 to construct 10 such Markov chains each with differ-
ent initial values. Convergence tests showed that 50 000 updates were
sufficient to get precise estimates of the parameters (R = 1.02 for esti-
mates of logit(proportion)) (Gelman et al., 2004, Chapter 11.6).
14. Use 10× 50000 updates to determine the posterior probability densities
of the model parameters.
The Gibbs update involves determining the full conditional probability of the
hidden states (given everything else). The assumption that the hidden states
136 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
are a first order Markov process means that the conditional probability of the
hidden states is based only on neighbouring states and the corresponding da-
tum. The full conditional probability of the hidden state, Ck (k = 2, ..., n− 1),
is given by
Pr(Ck = i|C\k,y) ∝ Pr(Ck+1 = j|Ck = i)Pr(Ck = i|Ck−1 = h)Pr(Yk|Ck = i)
(5.17)
where C\k is the set of all states other than Ck and i is the proposed value of
the kth hidden state and h and j are the current values of the hidden states
k − 1 and k + 1, respectively.
The first and last state depend only on a single neighbour and the data asso-
ciated with that state. That is
Pr(C1 = i|C\1,Y ) ∝ Pr(C2 = j|C1 = i)Pr(Y1|C1 = i), (5.18)
and
Pr(Cn = i|C\n,Y ) ∝ Pr(Cn = i|Cn−1 = h)Pr(Yn|Cn = i). (5.19)
The conditional probability of the states can be determined and this becomes
the sampling distribution for the hidden state. Each of the n states can be up-
dated in a forward, backward or random manner. To estimate values of ν and
β, we do not need to infer hidden states. The simplified MCMC algorithm has
the following steps:
1. Assign the prior probability for β and ν using (U[0, .1]).
2. Initialise β and ν and d′.
3. Determine the likelihood of the data using Baum’s recursion formula.
4. Propose a new β′ using a simple random walk, the step size ∼N(0, .0001).
5.B Monte Carlo Markov chain algorithm 137
5. Accept β′ using a Metropolis-Hastings step with the acceptance proba-
bility
a = min1, π(β′)l(Y |β′)q(β′ → β)
π(β)l(Y |β)q(β → β′). (5.20)
6. Repeat for ν.
7. Iterate as above.
138 Chapter 5. Characterising outbreaks of VRE using statisticalmethods
Bibliography
Aslanidou, H., Dey, D., Sinha, D., 1998. Bayesian analysis of multivariate survival datausing Monte Carlo methods. The Canadian Journal of Statistics 26 (1), 33–48.
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Bailey, N., 1975. The Biomathematics of Malaria. Charles Griffin, London.
Bartley, P. B., Schooneveldt, J. M., Looke, D. F., Morton, A., Johnson, D. W., Nimmo,G. R., 2001. The relationship of a clonal outbreak of enterococcus faecium VanA tomethicillin-resistant Staphylococcus aureus incidence in an Australian hospital. JHosp Infect 48 (1), 43–54.
Baum, L., Petrie, T., Soules, G., Weiss, N., 1970. A maximisation technique occurringin the statistical analysis of probabilistic functions of Markov chains. Annals ofMathematical Statistics 41, 164–171.
Becker, N., 1989. Analysis of Infectious Diseases Data. Chapman and Hall/CRC.
Boyce, J. M., 2001. MRSA patients: proven methods to treat colonization and infec-tion. J Hosp Infect 48 Suppl A, S9–14.
Bradley, S.J., Kaufmann, M.E., Happy, C., Ghori, S., Wilson, A.L., Scott, G.M., 2001.The epidemiology of glycopeptide-resistant enterococci on a haematology unit:analysis by pulsed-field gel electrophoresis. Epidemiology and Infection 129 (1),57-64.
CDC Guidelines, 1995. Recommendations for preventing the spread of vancomycinresistance. Hospital Infection Control Practices Advisory Committee (HICPAC).Infect Control Hosp Epidemiol 16 (2), 105–13.
Cooper, B., Lipsitch, M., 2004. The analysis of hospital infection data using HiddenMarkov Models. Biostatistics 5 (2), 223–37.
Cooper, B. S., Medley, G. F., Scott, G. M., 1999. Preliminary analysis of the transmis-sion dynamics of nosocomial infections: stochastic and management effects. JHosp Infect 43 (2), 131–47.
Cox, D. R., Miller, H. D., 1965. The theory of stochastic processes. Methuen, London.
140 BIBLIOGRAPHY
D’Agata, E. M., Gautam, S., Green, W. K., Tang, Y. W., 2002. High rate of false-negativeresults of the rectal swab culture method in detection of gastrointestinal coloniza-tion with vancomycin-resistant enterococci. Clin Infect Dis 34 (2), 167–72.
Donskey, C. J., Hoyen, C. K., Das, S. M., Helfand, M. S., Hecker, M. T., 2002. Recur-rence of vancomycin-resistant enterococcus stool colonization during antibiotictherapy. Infect Control Hosp Epidemiol 23 (8), 436–40.
Forrester, M., Pettitt, A. N., 2005. Use of stochastic epidemic modeling to quantifytransmission rates of colonization with methicillin-resistant Staphylococcus au-reus in an intensive care unit. Infect Control Hosp Epidemiol 26 (7), 598–606.
Gelman, A., Carlin, J., Stern, H., Rubin, D. B., 2004. Bayesian data analysis, 2nd Edi-tion. Texts in statistical science. Chapman & Hall/CRC, Boca Raton, Fla.
Giesecke, J., 1994. Modern Infectious Disease Epidemiology. Edward Arnold, Lon-don.
Gilks, W., Richardson, S., Spiegelhalter, D., 1996. Markov Chain Monte Carlo in Prac-tice. Chapman and Hall.
Green, P., 1995. Reversible Jump Markov Chain Monte Carlo Computation andBayesian Model Determination. Biometrika 82 (4), 711–732.
Grundmann, H., Hori, S., Winter, B., Tami, A., Austin, D. J., 2002. Risk factors for thetransmission of methicillin-resistant Staphylococcus aureus in an adult intensivecare unit: fitting a model to the data. J Infect Dis 185 (4), 481–8.
Le Strat, Y., Carrat, F., 1999. Monitoring epidemiologic surveillance data using hiddenMarkov models. Statistics in Medicine 18, 3463–3478.
Lemmen, S. W., Hafner, H., Zolldann, D., Amedick, G., Lutticken, R., 2001. Com-parison of two sampling methods for the detection of Gram-positive and Gram-negative bacteria in the environment: moistened swabs versus rodac plates. Int JHyg Environ Health 203 (3), 245–8.
Lodise, T. P., McKinnon, P. S., Tam, V. H., Rybak, M. J., 2002. Clinical outcomes forpatients with bacteremia caused by vancomycin-resistant enterococcus in a level1 trauma center. Clin Infect Dis 34 (7), 922–9.
MacDonald, I., Zucchini, W., 1997. Hidden Markov Models for Discrete Valued TimeSeries. Chapman and Hall, London.
Murray, B., 2005. Overview of enterococci. In: UpToDate, Rose, BD(Ed), UpToDate,MA, 2006.
Pearman, J., May 2006. 2004 lowbury lecture: the Western Australian experience withvancomycin-resistant enterococci - from disaster to ongoing control. J Hosp Infect63 (1), 14–26.
Pelupessy, I., Bonten, M. J., Diekmann, O., 2002. How to assess the relative impor-tance of different colonization routes of pathogens within hospital settings. ProcNatl Acad Sci U S A 99 (8), 5601–5.
BIBLIOGRAPHY 141
Petrushin, V., 2000. Hidden Markov Models: Fundamentals and applications. Part2Discrete and continuous hidden Markov models. Online Symposium for Electron-ics Engineers. http://www.techonline.com/osee/.
Reisner, B. S., Shaw, S., Huber, M. E., Woodmansee, C. E., Costa, S., Falk, P. S., May-hall, C. G., 2000. Comparison of three methods to recover vancomycin-resistantenterococci (VRE) from perianal and environmental samples collected during ahospital outbreak of VRE. Infect Control Hosp Epidemiol 21 (12), 775–9.
Scott, S., 2002. Bayesian methods for Hidden Markov Models: Recursive computingin the 21st century. J. Amer. Statist Assoc. 97, 337–351.
Stephens, M., 2000. Bayesian analysis of mixture models with an unknown num-ber of components-an alternative to reversible jump methods. Annals of Statistics28 (1), 40–74.
Suppola, J. P., Kolho, E., Salmenlinna, S., Tarkka, E., Vuopio-Varkila, J., Vaara,M., 1999. vanA and vanB incorporate into an endemic ampicillin-resistantvancomycin-sensitive Enterococcus faecium strain: effect on interpretation ofclonality. J Clin Microbiol 37 (12), 3934–9.
Tenover, F., Arbeit, R., Goering, R., , Mickelsen, P., Murray, B., Persing, D., Swami-nathan, B., 1995. Interpreting chromosomal DNA restriction patterns producedby pulsed field gel electrophoresis: criteria for bacterial strain typing. Journal ofClinical Microbiology 33, 22332239.
Tenover, F. C., Weigel, L. M., Appelbaum, P. C., McDougal, L. K., Chaitram, J., McAl-lister, S., Clark, N., Killgore, G., O’Hara, C. M., Jevitt, L., Patel, J. B., Bozdogan, B.,2004. Vancomycin-resistant Staphylococcus aureus isolate from a patient in Penn-sylvania. Antimicrob Agents Chemother 48 (1), 275–80.
Trick, W. E., Kuehnert, M. J., Quirk, S. B., Arduino, M. J., Aguero, S. M., Carson,L. A., Hill, B. C., Banerjee, S. N., Jarvis, W. R., 1999. Regional dissemination ofvancomycin-resistant enterococci resulting from interfacility transfer of colonizedpatients. J Infect Dis 180 (2), 391–6.
Trick, W. E., Paule, S. M., Cunningham, S., Cordell, R. L., Lankford, M., Stosor, V.,Solomon, S. L., Peterson, L. R., 2004. Detection of vancomycin-resistant entero-cocci before and after antimicrobial therapy: use of conventional culture and poly-merase chain reaction. Clin Infect Dis 38 (6), 780–6.
Weinstein, J., 2005. Hospital-acquired (nosocomial) infections with vancomycin-resistant enterococci In: UpToDate, Rose, BD(Ed), UpToDate, MA, 2006.
CHAPTER 6
A Mathematical Model Investigating the
Impact of an Environmental Reservoir on
Prevalence and Control of
Vancomycin-Resistant Enterococci
This chapter consists of a publication in the form of a correspondence article which
was necessarily brief, followed by an elaboration of the model described in the pub-
lication.
Statement of joint authorshipEmma McBryde wrote the manuscript, developed the mathematical model,
wrote code for model extensions including interventions and acted as corre-
sponding author.
Sean McElwain assisted with the development of mathematical model and
proof read and critically reviewed the manuscript.
6.A Publication
To the Editor-
In an article recently published in Journal of Infectious Diseases, D’Agata
et al. (2005), present a mathematical model of transmission of vancomycin-
resistant enterococci (VRE). We developed an extension of that model that
incorporates an environmental reservoir for VRE. While our model (which
we call the environment model) supports many of the findings of D’Agata et
al., we predict different outcomes for some infection control interventions.
VRE is known to contaminate environmental surfaces (Bonten et al., 1996;
Smith et al., 1998) and case control studies suggest that this can contribute to
144 Chapter 6. Environmental reservoir model for VRE
VRE acquisition (Martinez et al., 2003). This has led many to speculate that
environment plays an important role in patient acquisition of VRE.
Mathematical models can give insight into the likely consequences of infec-
tion control practices such as hand hygiene, patient cohorting (Austin et al.,
1999a; Cooper et al., 1999), staff patient ratios (Austin et al., 1998) and an-
tibiotic restriction (D’Agata et al., 2005). However, mathematical models will
only deliver results based on the assumptions underlying them. All mod-
els published to date on nosocomial transmission of VRE have assumed that
there is no transmission due to environmental contamination. It is important
to estimate how environmental contamination could influence the outcomes
of infection control interventions.
The environment model uses the structure and assumptions of the model
presented by D’Agata et al. adding a new (environment) compartment. It
is assumed that the environment is saturable and that colonised patients
and healthcare workers contribute to environmental contamination. In
turn, the contaminated environment can cause contamination of healthcare
workers, indirectly leading to patient colonisation. The new model requires
the addition of three transmission parameters; βeh(0.15), the transmission
from healthcare workers to the environment, βe0(0.4) the transmission
from colonised patients not exposed to antibiotics to the environment and
βe1(4), the transmission from colonised, antibiotic exposed patients to the
environment. Parameters were chosen so that the rate of environmental
contamination was 25% that of patient or healthcare worker contamination.
Following the findings of Wendt et al. (1998) and Noskin et al. (2000), that
VRE persists in the environment for at least one week, we assumed that
VRE persists an average of 10 days in the environment (decontamination
parameter, κ=0.1). In order to make the equilibrium colonisation prevalence
the same as D’Agata et al., the “fitted” parameter, βp1, was 0.0074 in this
model. All other parameters followed D’Agata et al..
The system of ordinary differential equations describing the environment
model is:
6.A Publication 145
dPu0
dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0 (6.1)
dPu1
dt= Λu1 − (σu1 + γu1)Pu1 + τu0Pu0 − αp1βp1(1− η)Pu1Hc/Nh
dPc0
dt= Λc0 + σc1Pc1 − (τc0 + γc0)Pc0
dPc1
dt= Λc1 − (σc1 + γc1)Pc1 + τc0Pc0 + αp1βp1(1− η)Pu1Hc/Nh
dHc
dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np + αp1ρβh1E)(Nh −Hc)− µHc
dE
dt= (βe0
Pc0
Np
+ βe1Pc1
Np
+ βehHc
Nh
)(1− E)− κE
Please see D’Agata et al. for an explanation of parameters not given in the
text.
Our model predicted that, in the presence of an environmental reservoir, the
direction of the impact of infection control interventions is the same as the
predictions in the model by D’Agata et al. (which we call the original model)
but the magnitude is altered. The environment model predicts that improv-
ing hand hygiene compliance from 40% to 60% leads to a reduction in coloni-
sation prevalence by 17% for the environment model compared with 23% for
the original model. Increasing staff-patient ratios from 1:4 to 1:2 leads to a
reduction in colonisation prevalence by 24% in the environment model com-
pared with 32%, as predicted by the original model. Reducing the length of
stay of colonised patients from 28 days to 14 days led to a reduction in coloni-
sation prevalence of 51% in the environment model and 64% in the original
model.
Selective isolation of colonised patients (presuming 80% efficacy) led to
a predicted 44% reduction in colonisation prevalence in the environment
model compared with 42% reduction in the original model.
A significant prediction of the environment model is that even if colonised
patients are prevented from entering the ward, VRE remains endemic at 5.3%,
as illustrated in Figure 6.1. This differs from the conclusion by D’Agata et al..
The model predicts that the presence of an environmental reservoir reduces
the predicted efficacy of some interventions (compliance with hand hygiene,
increased staffing levels, reduced length of stay) yet marginally increases the
predicted efficacy of others (selective isolation). The results of the model with
regard to different effects due to interventions are not intuitively obvious. We
146 Chapter 6. Environmental reservoir model for VRE
show that under certain circumstances, an environmental reservoir for VRE
could lead to endemic VRE transmission even if admission of VRE colonised
patients ceases.
These results suggests that, in the presence of an environmental reservoir,
VRE may be harder to eradicate, and infection control interventions less ef-
fective, with the exception of patient isolation which remains effective as pre-
dicted by this model.
0 100 200 300 400 500 600 7000
0.02
0.04
0.06
0.08
0.1
0.12
Time (days)
Pro
po
rtio
n c
olo
niz
ed
Figure 6.1: Model predictions of the prevalence of VRE over time. Both theenvironment and the original model begin with VRE at an endemic steady-state level of 12%. On day 200 further colonized patients are prevented fromentering the ward. In the environment model, a new equilibrium is estab-lished at 5.3%.
6.B Elaboration of Environment Model
6.B.1 Models
Standard Model
The model uses a four compartment model published previously (Austin and
Anderson, 1999; Austin et al., 1999b; Cooper et al., 1999). The compartments
in that model are; uncolonised patients, colonised patients, uncontaminated
healthcare workers and contaminated healthcare workers. The patient com-
partments are extended so that patients can be either antibiotic-exposed, or
not. This 6 compartment model is referred to as the “standard model” in this
6.B Elaboration of Environment Model 147
article and follows the model published by D’Agata et al., and adopts its as-
sumptions.
Environment Model
In the environment model, there is an additional compartment for the envi-
ronment. It is assumed that the environment can become contaminated by
both healthcare workers and patients and but that only healthcare workers
can acquire VRE from the environment. The interaction of these compart-
ments is shown in Figure 6.2.
Uncolonised patients
not receiving
antibiotics
Pu0
0uτ
1 1(1 ) /
p p c hH Nα β η ρ−
1cσ
Uncolonised patients
receiving antibiotics
Pu1
Colonised patients not
receiving antibiotics
Pc0
Colonised patients
receiving antibiotics
Pc1
0cτ
1uσ
1u∆0u∆
0c∆1c∆
0uγ
0cγ
1cγ
1uγ
0 0 0
1 1 1
/
( /
p h c p
p h c p
P N
P N
α β ρ
α β ρ
+
Uncontaminated HCW
Hu
Contaminated HCW
Hc
µ
)E+
1 1/e c pP Nβ
0 0/e c pP Nβ
/eh c h
H Nβ
κ
Environmental
contamination
E
Figure 6.2: Environmental model of VRE transmission in the hospital setting.The impact of the environment on colonisation of patients and contamina-tion of HCWs is indicated by E. The contamination of the environment arisesfrom colonised patients and contaminated HCWs.
Model assumptions
A summary of model assumptions and justification are given in Table 6.1. An-
tibiotic exposure in the colonised patient group is assumed to confer greater
148 Chapter 6. Environmental reservoir model for VRE
likelihood of transmission of VRE to a healthcare worker per contact. This as-
sumption is based on the reduced VRE density in stool found following cessa-
tion of antibiotics in a mouse model (Donskey et al., 1999) and human cases
(Donskey et al., 2000). Additionally, only those not exposed to antibiotics are
able to revert to the uncolonised state. In the uncolonised group, it is as-
sumed that only those exposed to antibiotics are able to acquire colonisation.
Isolation can be applied to colonised patients, or to all patients. The effect of
each of these isolation strategies was examined separately.
6.B.2 Methods
Parameters
The parameters used in the standard model are those used by D’Agata
et al. Four additional model parameters were required to model environ-
mental contamination. These include transmission rate from patients and
healthcare workers to the environment, and transmission rate from the
environment to the hands of healthcare workers. At default values, environ-
mental contamination contributed to 25% of new VRE acquisitions, similar
to the proportion of VRE acquisitions in which environmental contamina-
tion was potentially implicated in the study by Bonten et al. (1996). In order
to begin with a prevalence level equal to that of the standard model (12%)
the transmission parameter, βp1, was refitted in the environmental model,
with the value 0.0074.
Equations governing models
The standard mathematical model taken from D’Agata et al. (2005) can be de-
scribed by a system of 5 ordinary differential equations. The compartments
include 4 patient compartments who can be either colonised or uncolonised,
(Pc, Pu) and exposed to antibiotics or not (Pc1, Pc0, Pu1, Pu0). The healthcare
workers also can be either colonised or uncolonised (Hc, Hu).
The nonlinear terms represent interactions between contaminated HCW and
uncolonised patient, αp1βp1(1− η)ρpu1Hc/Nh or colonised patient and uncon-
taminated HCW, (αp0βh0ρPc0/Np + αp1βh1ρPc1/Np)(Nh −Hc).
The environmental model incorporates a seventh compartment, E. The
6.B Elaboration of Environment Model 149
environment is modelled as a saturable compartment taking values be-
tween zero and one. Additional terms in the environment model repre-
sent colonised patients’ contribution to environmental contamination,
(βe0Pc0
Np+ βe1
Pc1
Np)(1−E), HCW contribution to environmental contamination,
βehHc
Nh(1 − E) and the environmental contribution to HCW contamination,
αp1ρβh1E.
The system of ordinary differential equations that govern the model by
D’Agata et al. (2005) is given by
dPu0
dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0
dPu1
dt= Λu1 − (σu1 + γu1)Pu1 + τu0Pu0 − αp1βp1(1− η)Pu1Hc/Nh
dPc0
dt= Λc0 + σc1Pc1 − (τc0 + γc0)Pc0
dPc1
dt= Λc1 − (σc1 + γc1)Pc1 + τc0Pc0 + αp1βp1(1− η)Pu1Hc/Nh
dHc
dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np)(Nh −Hc)− µHc. (6.2)
The system of ordinary differential equations that govern the extended model
is given by
dPu0
dt= Λu0 + σu1Pu1 − (τu0 + γu0)Pu0
dPu1
dt= Λu1 − (σu1 + γu1)Pu1 + τu0Pu0 − αp1βp1(1− η)Pu1Hc/Nh
dPc0
dt= Λc0 + σc1Pc1 − (τc0 + γc0)Pc0
dPc1
dt= Λc1 − (σc1 + γc1)Pc1 + τc0Pc0 + αp1βp1(1− η)Pu1Hc/Nh
dHc
dt= (αp0ρβh0Pc0/Np + αp1ρβh1Pc1/Np + αp1ρβh1E)(Nh −Hc)− µHc
dE
dt= (βe0
Pc0
Np
+ βe1Pc1
Np
+ βehHc
Nh
)(1− E)− κE. (6.3)
Simulations
Deterministic simulations were run using the MatlabT M ODE45 function.
Initial conditions that all patients were uncolonised and not exposed to an-
tibiotics (PU0 = Np). Interventions were predicted by incrementally changing
150 Chapter 6. Environmental reservoir model for VRE
the value of one parameter while keeping each of the other parameters
constant at their default values.
The impact of restricting VRE admission was simulated using default para-
meters and initial conditions PU0 = Np for both models. In both models the
proportion of colonised patients reached equilibrium values of 12%. On day
200, we changed the conditions such that no colonised patients were admit-
ted to the ward (ΛC1 = 0, ΛC0 = 0).
6.B.3 Further discussion
This model is the first of its type, addressing the issue of environmental reser-
voirs for nosocomial pathogens. It was developed in response to the repeated
observation of antibiotic-resistant pathogens being retrieved from the envi-
ronment and anecdotal reports of failure to eliminate some pathogens fol-
lowing ward closure. Since the publication of the article (McBryde and McEl-
wain, 2006), an interrupted time series study has been published (Hayden
et al., 2006). In that study, environmental decontamination was associated
with a decline in VRE acquisition, lending further support for the role of envi-
ronment in transmission of VRE. Data from the study by Hayden et al. (2006)
could be incorporated into an environmental model to quantify further the
role of the environment in transmission.
The authors chose to allow 25% of transmission to be due to the environ-
ment at default values in this model to take a conservative approach. The
level could in fact be much higher. Environmental contamination was
assumed only to influence healthcare worker contamination not patient
colonisation directly. Such a model is appropriate for a setting where pa-
tients are relatively immobile and unlikely to make frequent contacts with
the environment, such as in an Intensive Care Unit. Additionally, many of
the objects on which pathogens have been found are equipment used by
healthcare workers, for example ward telephones or computer keyboards.
Future studies could incorporate direct environment to patient colonisation.
6.B Elaboration of Environment Model 151
Assumption Support from literature Reference
Antibiotic exposureof colonised patientsincreases patient toHCW transmission
Increased stool density of VREin animal model followingantibiotic exposure Increasedstool density of VRE in humancases.
Donskey et al.(1999)Donskey et al.(2000)
Antibiotic exposurenecessary foruncolonisedpatients to acquireVRE.
Increased risk in case controlstudies for acquisition whenantibiotic exposed.
Carmeli et al.(2002)
This study incorporated thisassumption into the model toremain consistent withprevious modelling studies, forthe sake of comparison.
D’Agata et al.(2005)
Colonised patientscontribute toenvironmentalcontamination.
Rooms of colonised patientsmore likely to haveenvironmental contamination.
Trick et al.(2002)
Environmentalcontaminationcontributes tocolonisation.
23% of patients with roomspositive for VRE subsequentlyacquired colonisation.
Bonten et al.(1996)
Recurrence of VRE afterrecrudescence blamed onenvironmental point source.
Falk et al.(2000)
Patients colonised with VREmore likely to have beenexposed to contaminated roomin case-control study.
Martinez et al.(2003)
Patientdecolonisation doesnot occur inhospital.
Duration of colonisation ismuch longer (months) thanmean length of stay (weeks).
Lai et al. (1997);Byers et al.(2002)
Table 6.1: Model assumptions and their justifications.
152 Chapter 6. Environmental reservoir model for VRE
Parameter Symbol Default ValuePatient→ environment transmission rate (per day)*
Antibiotic exposed βe1 4Unexposed βe0 0.4
Probability of patient colonisation per HCW contact βp1 0.0074HCW→ environment transmission rate* βeh 0.3Environment decontamination rate* κ 0.1Number of patients Np 400Ratio of patients/HCW ρ 4Uncolonised patients admitted (per day)
Antibiotic exposed Λu1 60Unexposed Λu0 3
Colonised patients admitted per dayAntibiotic exposed Λc1 0.4Unexposed Λc0 0.6
Length of hospital stayUncolonised patients
Unexposed to antibiotics 1γu0
14Antibiotic exposed 1
γu15
Colonized patientsUnexposed to antibiotics 1
γc028
Antibiotic exposed 1γc1
28HCW contact rate
Unexposed colonised patients αp0 8Antibiotic exposed colonised patients αp1 10
Probability of HCW contamination per contactUnexposed colonised patients βh0 0.05Antibiotic exposed colonised patients βh1 0.4
Antibiotics started per dayUncolonised patients τu0 0.15Colonised patients τc0 0.16
Antibiotics stopped per dayUncolonised patients σu1 0.15Colonised patients σc1 0.04
Hand hygiene compliance η 0.4Healthcare worker decontamination rate µ 93
Table 6.2: Table of parameters, their symbols and default values. The addi-tional parameters introduced for the environment model are indicated withan asterisk.
Bibliography
Austin, D. J., Anderson, R. M., 1999. Studies of antibiotic resistance within the pa-tient, hospitals and the community using simple mathematical models. PhilosTrans R Soc Lond B Biol Sci 354 (1384), 721–38.
Austin, D. J., Bonten, M. J., Weinstein, R. A., Slaughter, S., Anderson, R. M., 1999a.Vancomycin-resistant enterococci in intensive-care hospital settings: transmis-sion dynamics, persistence, and the impact of infection control programs. ProcNatl Acad Sci U S A 96 (12), 6908–13.
Austin, D. J., Kristinsson, K. G., Anderson, R. M., 1999b. The relationship between thevolume of antimicrobial consumption in human communities and the frequencyof resistance. Proc Natl Acad Sci U S A 96 (3), 1152–6.
Austin, D. J., White, N. J., Anderson, R. M., 1998. The dynamics of drug action on thewithin-host population growth of infectious agents: melding pharmacokineticswith pathogen population dynamics. J Theor Biol 194 (3), 313–39.
Bonten, M. J., Hayden, M. K., Nathan, C., van Voorhis, J., Matushek, M., Slaughter,S., Rice, T., Weinstein, R. A., 1996. Epidemiology of colonisation of patients andenvironment with vancomycin-resistant enterococci. Lancet 348 (9042), 1615–9.
Byers, K. E., Anglim, A. M., Anneski, C. J., Farr, B. M., 2002. Duration of colonizationwith vancomycin-resistant enterococcus. Infect Control Hosp Epidemiol 23 (4),207–11.
Carmeli, Y., Eliopoulos, G. M., Samore, M. H., 2002. Antecedent treatment with dif-ferent antibiotic agents as a risk factor for vancomycin-resistant enterococcus.Emerg Infect Dis 8 (8), 802–7.
Cooper, B. S., Medley, G. F., Scott, G. M., 1999. Preliminary analysis of the transmis-sion dynamics of nosocomial infections: stochastic and management effects. JHosp Infect 43 (2), 131–47.
D’Agata, E. M., Webb, G., Horn, M., 2005. A mathematical model quantifying the im-pact of antibiotic exposure and other interventions on the endemic prevalence ofvancomycin-resistant enterococci. J Infect Dis 192 (11), 2004–11.
Donskey, C., Chowdhry, T., Hecker, M., Hoyen, C., Hanrahan, J., Hujer, A., Hutton-Thomas, R., Whalen, C., Bonomo, R., Rice, L., 2000. Effect of antibiotic therapy onthe density of vancomycin-resistant enterococci in the stool of colonized patients.N Engl J Med 343 (I), 1925–1932.
154 BIBLIOGRAPHY
Donskey, C. J., Hanrahan, J. A., Hutton, R. A., Rice, L. B., 1999. Effect of parenteral an-tibiotic administration on persistence of vancomycin-resistant enterococcus fae-cium in the mouse gastrointestinal tract. J Infect Dis 180 (2), 384–390.
Falk, P. S., Winnike, J., Woodmansee, C., Desai, M., Mayhall, C. G., 2000. Outbreak ofvancomycin-resistant enterococci in a burn unit. Infect Control Hosp Epidemiol21 (9), 575–82.
Hayden, M., Bonten, M., Blom, D., Lyle, E., van de Vijver, D., Weinstein, R., Jun 2006.Reduction in acquisition of vancomycin-resistant enterococcus after enforcementof routine environmental cleaning measures. Clinical Infectious Diseases 42 (11),1552–1560.
Lai, K. K., Fontecchio, S. A., Kelley, A. L., Melvin, Z. S., Baker, S., 1997. The epidemi-ology of fecal carriage of vancomycin-resistant enterococci. Infect Control HospEpidemiol 18 (11), 762–5.
Martinez, J. A., Ruthazer, R., Hansjosten, K., Barefoot, L., Snydman, D. R., 2003. Roleof environmental contamination as a risk factor for acquisition of vancomycin-resistant enterococci in patients treated in a medical intensive care unit. Arch In-tern Med 163 (16), 1905–12.
McBryde, E. S., McElwain, D. L., 2006. A mathematical model investigating the im-pact of an environmental reservoir on the prevalence and control of vancomycin-resistant enterococci. J Infect Dis 193 (10), 1473–4.
Noskin, G. A., Bednarz, P., Suriano, T., Reiner, S., Peterson, L. R., 2000. Persistentcontamination of fabric-covered furniture by vancomycin-resistant enterococci:implications for upholstery selection in hospitals. Am J Infect Control 28 (4), 311–3.
Smith, N. P., Nelson, M. R., Azadian, B., Gazzard, B. G., 1998. An outbreak ofmethicillin-resistant Staphylococcus aureus (MRSA) infection in HIV-seropositivepersons. Int J STD AIDS 9 (12), 726–30.
Trick, W. E., Temple, R. S., Chen, D., Wright, M. O., Solomon, S. L., Peterson, L. R.,2002. Patient colonization and environmental contamination by vancomycin-resistant enterococci in a rehabilitation facility. Arch Phys Med Rehabil 83 (7), 899–902.
Wendt, C., Wiesenthal, B., Dietz, E., Ruden, H., 1998. Survival of vancomycin-resistant and vancomycin-susceptible enterococci on dry surfaces. J Clin Micro-biol 36 (12), 3734–6.
CHAPTER 7
Bayesian Modelling of an epidemic of
Severe Acute Respiratory Syndrome
Statement of joint authorshipEmma McBryde wrote the manuscript, constructed the dataset, developed
the mathematical model, developed the code for the data analysis and acted
as corresponding author.
Gavin Gibson assisted with the analysis of data, Bayesian inference and
piecewise hazard model and proof read and critically reviewed the manu-
script.
Tony Pettitt assisted with the analysis of data, Bayesian inference and piece-
wise hazard model and proof read and critically reviewed the manuscript.
Y.Zhang and B.Zhao initiated the concept for the manuscript, constructed
the dataset and reviewed the manuscript.
Sean McElwain assisted with the development of mathematical model and
analysis and proof read and critically reviewed the manuscript.
156 Chapter 7. Bayesian modelling of an epidemic of SARS
Abstract
This paper analyses data arising from a SARS epidemic in Shanxi Province of
China involving a total of 354 people infected with SARS-CoV between late
February and late May, 2003. Using Bayesian inference, we have estimated
critical epidemiological determinants. The estimated mean incubation pe-
riod was 5.3 days (95%CI 4.2-6.8 days), mean time to hospitalisation was 3.5
days (95%CI 2.8-3.6 days), mean time from symptom onset to recovery was
26 days (95%CI 25-27 days) and mean time from symptom onset to death was
21 days (95%CI 16-26 days). The reproduction ratio was estimated to be 4.8
(95%CI 2.2-8.8) in the early part of the epidemic (February and March, 2003)
reducing to 0.75 (95%CI 0.65-0.85) in the later part of the epidemic (April and
May, 2003). The infectivity of symptomatic SARS cases in hospital and in the
community was estimated. Community SARS cases caused transmission to
others at an estimated rate of 0.4 per infective per day during the early part
of the epidemic, reducing to 0.2 in the later part of the epidemic. For hos-
pitalised patients, the daily infectivity was approximately 0.15 early in the
epidemic, but this fell to 0.0006 in the later part of the epidemic. Despite
the lower daily infectivity level for hospitalised patients, the long duration of
the hospitalisation led to a greater number of transmissions within hospitals
compared with the community in the early part of the epidemic, as estimated
by this study. This study investigated the individual infectivity profile dur-
ing the symptomatic period, with an estimated peak infectivity on the ninth
symptomatic day.
7.1 Introduction
Severe acute respiratory syndrome, (SARS), caused a perplexing epidemic
with propensity for hospital transmission, rapid worldwide spread and
markedly different epidemic curves in different countries (Wallinga and
Teunis, 2004). Beginning in November 2002 in the Guangdong province of
China, the SARS epidemic spread to Hong Kong, Viet Nam and Singapore by
March, 2003 and eventually to 29 countries around the world (Poon et al.,
2004). The World Health Organisation (WHO) issued a global alert on March
12, 2003 regarding a cluster of cases of severe atypical pneumonia and 3
days later gave a case definition and name to the condition (WHO, 2003c). A
novel coronavirus, named SARS-CoV, was identified as the infectious agent
responsible for SARS in April 2003 (Peiris et al., 2003b; Drosten et al., 2003;
7.1 Introduction 157
Ksiazek et al., 2003). In total, 8098 SARS infections and 774 deaths were
reported in the 2002/2003 epidemic of SARS (Gumel et al., 2004). The largest
outbreaks occurred in mainland China, in which there were 5327 infections
and 349 deaths reported (WHO, 2003a). Despite the initial worldwide spread
and early predictions of high case numbers, the 2003 SARS epidemic was
contained relatively rapidly with no further spread reported after July, 2003
(Donnelly et al., 2004).
SARS-CoV is likely to have an animal reservoir, possibly the palm civet cat,
Paguma lavata, (Guan et al., 2003; Webster, 2004), and further epidemics
are anticipated. Laboratory associated infections in Singapore (Lim et al.,
2004), Taiwan (Orellana, 2004) and China (WHO, 2004), the latter involving
onward transmission (Normille, 2004), remind us that further outbreaks of
SARS could occur. To help contain future epidemics of SARS, it is essential to
have an understanding of the infectivity, incubation period and likely course
of the illness.
Nosocomial transmission was a prominent feature of SARS epidemiology.
Early in the SARS pandemic, a majority of cases arose from hospital trans-
mission in many places, including Toronto (Booth et al., 2003), Hong Kong
(Wong et al., 2004; Riley et al., 2003) and Singapore (Gopalakrishna et al.,
2004). Later in the course of the epidemic, hospitals were effective sites
of containment of SARS (Gopalakrishna et al., 2004). Factors believed to
be important in reducing nosocomial transmission of SARS include hand
washing and wearing of masks, while contact with respiratory secretions is
highly correlated with SARS transmission (Teleman et al., 2004). Thorough
contact tracing and quarantine of exposed cases led to reduced transmission
in Singapore (Gopalakrishna et al., 2004). In this study, we compare the
estimated infectivity of SARS cases in the community and in hospitals. We
also examine how this changes over time.
Mathematical models of the SARS epidemic have the potential to give
insights into the disease process, to estimate critical epidemiological deter-
minants and ultimately to predict outcomes of public health interventions.
Models of SARS transmission published to date have already been useful
tools for designing control strategies; estimating the incubation period
(Donnelly et al., 2003), the infectivity (Lipsitch et al., 2003; Riley et al., 2003;
Wallinga and Teunis, 2004) and the potential impact of interventions (Riley
et al., 2003). Models have been used to predict the effect of public health
measures on the SARS epidemic in many countries including Canada (Choi
and Pak, 2003; Chowell et al., 2003), Hong Kong (Chowell et al., 2003; Lee
158 Chapter 7. Bayesian modelling of an epidemic of SARS
et al., 2003; Riley et al., 2003), Singapore (Chowell et al., 2003; Lipsitch et al.,
2003), Taiwan (Hsieh et al., 2004), and mainland China (Wang and Ruan,
2004).
For transmission models to be realistic and predictive, accurate measures of
the various transition times are required, including the incubation period,
and the time from symptom onset to removal (isolation, recovery or death).
Estimates of infectivity, particularly those based on the early behaviour of an
epidemic, are sensitive to the estimate of the incubation period. Models are
also sensitive to the full distribution of the transition periods (Lloyd, 2001),
such that summary measures (mean, median) alone are often inadequate in
modelling the behaviour of the epidemic.
In order to design effective and safe interventions, public health practitioners
also need an accurate estimate of the incubation period. Decisions regard-
ing quarantine time require estimates of the mean incubation period and the
probability of outliers. Therefore, the full probability distributions of the in-
cubation and symptomatic periods are required.
The general aims of this study are to estimate accurately the full distribution
of the transition times; the incubation period, time from symptom onset to
hospitalisation, recovery and death, to determine the infectivity of SARS in-
cluding the relative infectivity of symptomatic SARS cases in and out of hos-
pital and early and late in the epidemic, and to estimate the individual infec-
tivity profile over the course of SARS infection.
This study makes some unique contributions to the study of SARS transmis-
sion. Firstly, it uses a Bayesian framework to infer transmission times and
calculate the incubation period. In doing so, it investigates three different
models of viral transmission. Secondly, it compares the infectiousness of
SARS cases in the community and in hospital and during different times
of the epidemic. Thirdly, this study considers three different models for
individual infectivity profiles over time, using model selection criteria to
determine the optimal model. The current study investigates a database
from mainland China which has not been published previously.
7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model 159
S E RI
Figure 7.1: The schematic of the SEIR model.
7.2 Susceptible-Exposed-Infectious-Removed
(SEIR) model
The model used in this study is an extension of the stochastic version of the
compartmental Susceptible-Exposed-Infectious-Removed (SEIR) model (see
Figure 7.1) used extensively in infectious disease modelling literature (see for
example, Kermack and McKendrick (1927)). In the SEIR model individuals in
a population begin as susceptible (S) and move to the exposed (E) state fol-
lowing transmission of a contagion. This occurs at a rate that is proportional
to the number of infectious (I) and the proportion of susceptible people in
the community, SN
, (the mass action effect) so that in a small time interval, dt,
the probability of a transmission occurring is given by
Pr(S(t + dt) = i− 1, E(t + dt) = j + 1 | S(t) = i, E(t) = j) =β S(t) I(t) dt
N(t), (7.1)
where β is a constant. In the simplest version of the SEIR model, transition
between subsequent model compartments occurs at a constant rate, becom-
ing infectious as they move into the I compartment and being neither infec-
tious nor susceptible after being Removed (see Figure 7.1). This leads to
Pr(E(t + dt) = j− 1, I(t + dt) = k + 1 |E(t) = j, I(t) = k) = δ E(t) dt (7.2)
Pr(I(t + dt) = k− 1 | I(t) = k) = γ I(t) dt, (7.3)
where δ and γ are constants.
The assumption of a constant transition rate in the basic SEIR model,
adopted for ease of calculation, leads to an exponential distribution of the
probability density function for the time to transition. Other distributions,
parametric or non-parametric, could be used to model sojourn times (Diek-
mann and Heesterbeek, 2000). In the case of SARS, the incubation period,
time to hospitalisation and time from hospital admission to discharge have
been shown not to be exponentially distributed (Donnelly et al., 2003).
160 Chapter 7. Bayesian modelling of an epidemic of SARS
Assuming an exponentially distributed incubation period, with a mode of
zero, (when in fact the mode of the incubation period is considerably greater
than zero) leads to under-estimation of infectivity inferred from the early
epidemic growth curve.
In the current study, we implemented an alternative parameterisation of the
transition times. Following Donnelly et al. (2003), the Gamma distribution
was used. Other distributions could also be utilised to approximate the incu-
bation period, such as the Weibull distribution, used by Lipsitch et al. (2003).
In this study we use Gamma(α, β) notation, where α is the shape parameter
and β is the reciprocal of the scale parameter, such that
p(x) =βαxα−1e−βx
Γ(α)(x > 0, α > 0, β > 0)
and
Γ(α) =
∫ ∞
0
tα−1e−t dt.
S E H
R
D
I
Figure 7.2: The schematic of the extended SEIHRD model used in this study.The heavy arrows represent the transitions that were observed or inferred inthe current study. The thin arrows represent events that probably occur, butwith a low frequency relative to other transitions and therefore are not con-sidered in the current study.
The current study extends the SEIR model by considering two infectious
groups and two removed groups. As shown in Figure 7.2, in this model the
patients can either be infectious and in the community, I, or infectious and
hospitalised H. Removal can represent either recovery, R, or death, D. This
model, similar to that used by Riley et al. (2003) and Lipsitch et al. (2003), will
be referred to as the SEIHRD model.
In addition to dividing the infectious compartments into 2 groups, commu-
nity and hospitalised, the study also examines infectivity early and late in the
7.2 Susceptible-Exposed-Infectious-Removed (SEIR) model 161
epidemic. Hence there are 4 infectious groups to consider (a) early commu-
nity (b) early hospitalised (c) late community (d) late hospitalised.
In this study, three different models of individual infectivity profiles are con-
sidered:
Uniform transmission model: Constant infectivity within each of the 4
groups of patients (a)-(d), but different between groups.
Model with transmission proportional to viral load: Infectivity is modelled
as a triangular distribution, with zero infectivity on day 0 and day 20
and a peak at day 10, following the viral load as described by Peiris et al.
(2003a). This is also influenced by the group (a)-(d) into which the pa-
tient falls.
Model with transmission given by a Gamma distribution: Infectivity takes
on values given by the Gamma distribution, the shape and scale para-
meters of which are inferred. Again this is modified by the co-efficient
of infectivity based on the group into which the patient falls (a)-(d).
This study assumes that the proportion of the population who are suscepti-
ble, S/N , remains at unity throughout the epidemic. The authors justify this
by the large number of people in the region investigated in this study, with
the largest city in Shanxi Province having a population of around 3 million
people, compared with the small number (354) of SARS cases observed in the
epidemic in the region. The full description of the database used in this paper
is given in the next section.
Other assumptions implicit in the current model are that there is homoge-
neous mixing of the population and that SARS cases were only infectious dur-
ing the symptomatic period. Early contact tracing studies suggest that infec-
tivity is indeed low during the incubation period (Poutanen et al., 2003). The
current study also assumes that sub-clinical SARS cases (not recorded in the
database) did not contribute significantly to the epidemic. This assumption
is supported by the finding of a very low proportion of asymptomatic con-
tacts who were SARS antibody positive (0.2%)(Leung et al., 2004).
162 Chapter 7. Bayesian modelling of an epidemic of SARS
7.3 Severe Acute Respiratory Syndrome Data from
Shanxi Province
The data used in this study come from Shanxi province in China. On 23rd
April, the WHO travel warning to China was extended to include Beijing
and Shanxi province (WHO, 2003d). The travel warning was removed on
13th June, 2003, after it was concluded no further chains of transmission
were occurring (WHO, 2003b). The Shanxi province epidemic began when
a person returned to the province while incubating SARS after visiting
Beijing in February, 2003. There were 354 reported cases of SARS during the
epidemic which began in late February 2003 and ended late-May 2003.
Appendix 7.A gives the full Gantt chart of the epidemic in Shanxi province.
Figure 7.3 shows the daily number of hospital admissions of SARS cases in
the Shanxi province. It can be seen that the peak incidence of SARS cases
admitted to hospital in Shanxi province was in mid to late April, 2003.
0
5
10
15
20
25
2-M
ar
9-M
ar
16-M
ar
23-M
ar
30-M
ar
6-A
pr
13-A
pr
20-A
pr
27-A
pr
4-M
ay
11-M
ay
Figure 7.3: Histogram of daily admissions to hospital.
Data recording the duration of exposure to another person with SARS were
available in 85 cases. Exposure-time, recorded by calendar day, ranged from
zero to a maximum of 26 days as shown in Figure 7.4. The mean time from
the day of first known exposure to the day of symptom onset (inclusive) was
7.3 SARS Data from Shanxi Province 163
8.5 days using the discrete dataset. The time from the end of exposure to the
symptomatic period had a mean of 2.9 days. This places an upper and lower
limit on estimates of the mean incubation period.
0 5 10 15 20 25 300
2
4
6
8
10
12
14
Time from first exposure to symptom onset (days)
Nu
mb
er o
f ca
ses
Figure 7.4: Histogram of time from first exposure to another SARS case tosymptom onset.
The time from symptom onset to hospitalisation was recorded in 351 of the
354 cases. In two cases, the recorded hospital admission day preceded the
recorded time of symptom onset. This was due to quarantining of exposed
individuals during the incubation period. These patients were excluded from
the analysis of time from symptom onset to hospitalisation, leaving 349 avail-
able patient records. Figure 7.5 shows a histogram of the time from symptom
onset to hospitalisation. It is an approximately exponential distribution and
the majority of SARS cases reached hospital within 4 days. It is widely dis-
persed, however, with some people taking more than 10 days to reach hospi-
tal. There is a clear outlier among these data with one SARS case reporting 44
days of symptoms prior to hospitalisation. This is also evident on the Gantt
chart, shown in Figure 7.16. It seems most likely that the date of onset of
symptoms is erroneous and this case has been excluded from the remainder
of the analysis.
Of the 354 cases in the epidemic, 344 had a recorded outcome (recovery or
death), of whom 20 died and the remainder were discharged from hospital
following recovery. The time from symptom onset to recovery was available
in all 324 cases and the time from symptom onset to death was available in 18
of the 20 cases. The distributions of symptom onset to recovery and symptom
164 Chapter 7. Bayesian modelling of an epidemic of SARS
0 5 10 15 20 25 30 35 40 450
10
20
30
40
50
60
70
80
90
100
Time from symptom onset to hospital admission (days)
Nu
mb
er o
f ca
ses
Figure 7.5: Recorded time interval from symptom onset to hospitalisation.
onset to death are shown in Figures 7.6 and 7.7, respectively.
0 10 20 30 40 50 60 700
5
10
15
20
25
30
35
Time from symptom onset to recovery (days)
Nu
mb
er o
f ca
ses
Figure 7.6: Recorded time interval from symptom onset to recovery.
7.3 SARS Data from Shanxi Province 165
0 20 40 60 800
1
2
3
4
Time from symptom onset to death (days)
Nu
mb
er o
f ca
ses
Figure 7.7: Recorded time interval from symptom onset to death.
166 Chapter 7. Bayesian modelling of an epidemic of SARS
7.4 Challenges and specific aims of the study
A major challenge of the study was to estimate the distribution of the incuba-
tion period of SARS. The time of transmission of SARS is unobservable, such
that estimates of the incubation period are necessarily based on inference.
A Bayesian inference framework was used in this study as described in Sec-
tion 7.5. Only a limited number of cases have recorded known symptomatic
SARS contacts and these are used to infer transmission times and thereby es-
timate the incubation period. The cases with the shortest contact periods are
most informative. Section 7.6 describes the methodology used to parame-
terise the distributions of time to hospitalisation, recovery and death. This is
more straightforward as the times are observed and recorded.
In Section 7.7 we estimate the infectivity of the two compartments assumed
to be infectious, the symptomatic patients in the community and in hospi-
tal. This requires inference regarding missing data and transmission times.
Extending the SEIR model to include two infectious compartments, allows
us to estimate the relative impact of hospitalised and community SARS cases
on the epidemiology. Additionally, we can compare how infectivity changed
over time in each group, reflecting the effects of interventions. In this sec-
tion we also estimate the changepoint; the date that marked the transition
from high to relatively low infectivity. Finally, this study explores individual
infectivity profiles over the course of SARS illness, in Section 7.8.
7.5 Estimation of time to transmission and incu-
bation period
The incubation period was estimated only from those cases who had known
contact with another SARS case, and when there was a single contact of
known duration. In the Shanxi database this included 85 cases. It was
assumed that transmission occurred from the known contact during the
contact period and that the rate of transmission, given the contact was
independent of the state of the epidemic. The required times of exposure for
transmission to occur for the 85 cases under consideration was assumed to
be a set of independent random variables. Incubation periods of the SARS
cases are also assumed to be independent.
The model assumes that during periods of exposure to symptomatic SARS
cases, susceptible individuals acquire the disease at a fixed daily hazard rate,
7.5 Estimation of time to transmission and incubation period 167
λ. This constant hazard model is compared with two other models, a model
assuming immediate transmission and a model in which the probability
of transmission is uniform across the contact period. Following transmis-
sion, there is an incubation period that occurs before patients become
symptomatic. This period is assumed to be drawn from a Gamma(αL, βL)
distribution.
7.5.1 Bayesian approach to estimating incubation period
A Bayesian approach was used to estimate the incubation period:
π(λ, αL, βL|data) ∝ π(λ, αL, βL)L(data|λ, αL, βL), (7.4)
where π(λ, αL, βL) is the prior probability of the parameters, L(data|λ, αL, βL)
is the likelihood of the data given the parameters and π(λ, αL, βL|data) is
the posterior probability distribution of the parameters. Explanation of
the choice of prior probability distributions for the parameters, use of
augmented data and determination of likelihood of the data are given in this
section. Details of computations are given in Appendix 7.B.
Choice of prior probability distributions
Gamma priors were chosen for the three parameters. Vague prior distrib-
utions, Gamma(0.001, 0.001), were chosen for λ, αL, and βL because little is
known about the transmission rate.
Likelihood of the data given the parameters
The data used for estimation of the incubation period are the durations of ex-
posure to another SARS case, denoted vi for each individual, i, and the time
from first exposure to onset of symptoms, denoted si for each individual i. If
N is the total number of cases, the vector of the N exposure times is denoted
by v and the vector of N times to symptom onset is denoted by s.
The time that each individual in the dataset acquired SARS-CoV is not known.
It is assumed to be during the period of exposure to another symptomatic
SARS case. The time to transmission, denoted by ui, was estimated and in-
cluded in the model as an auxiliary variable. The remaining time to onset of
symptoms (si − ui) is the incubation period.
168 Chapter 7. Bayesian modelling of an epidemic of SARS
In the dataset available, all patients developed SARS, so we are considering
the probability density of ui conditional on transmission having occurred
(therefore ui < vi). Assuming a constant hazard of transmission throughout
the contact period, the conditional probability density of ui is a truncated
exponential distribution given by
fi(ui, λ) =λe−λui
1− e−λvi(0 < ui < vi ). (7.5)
The likelihood of u is also dependent on the probability density of the incu-
bation period, si − ui. The distribution, g, of the incubation period, given ui,
is determined by the Gamma(αL, βL) distribution, so that
g((si − ui)|αL, βL) ∼ Gamma(αL, βL). (7.6)
Assuming the observations are independent, the likelihood of the augmented
data (observations plus auxiliary variables, u) is given by
L(u, s|λ, αL, βL) =N∏
i=1
fi(ui|λ) g((si − ui)|αL, βL). (7.7)
The likelihood of the full set of N observations is given by
L(s |λ, αL, βL) =N∏
i=1
∫ vi
0
fi(ui|λ)g((si − ui)|αL, βL) dui.
(7.8)
Because the integral (7.8) is not straight forward to compute, a Markov chain
Monte Carlo (MCMC) algorithm, given in Appendix 7.B, was used to deter-
mine the posterior probability distributions of the parameters.
7.5.2 Results: Time to transmission and incubation period
The posterior distribution of the hazard of transmission, λ, had a maxi-
mum density close to zero and a mean of 0.18 per day, see Figure 7.8. The
inferred mean time from exposure to transmission was 2.5 days (95%CI
0.19-4.4). The estimated incubation period is shown in Figure 7.9. It follows
a Gamma(1.4, 0.26) distribution. The standard deviation for the incubation
period was 4.5 days (95%CI 3.4-5.9 days) and mean was 5.3 days (95% CI
7.5 Estimation of time to transmission and incubation period 169
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
λ
Po
ster
ior
pro
bab
ility
den
sity
Figure 7.8: Posterior distribution for the hazard of transmission, λ.
4.2-6.8 days). The median is 4.2 days, shorter than that reported by Lee et al.
(2003), 6 days, but similar to that reported by Donnelly et al. (2003), 3.8 and
Meltzer (2004), 4 days.
Appendix 7.C.1 compares the sensitivity of the results for the incubation pe-
riod to the value of λ and to model choice, showing that the conclusions re-
garding the incubation period are robust to these.
0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
Incubation period (days)
Pro
abili
ty d
ensi
ty f
un
ctio
n
Figure 7.9: Estimated distribution of the incubation period based on max-imum posterior probability density estimates for the shape and scale para-meters of the Gamma distribution.
170 Chapter 7. Bayesian modelling of an epidemic of SARS
7.5.3 Discussion: Time to transmission and incubation pe-
riod
Estimation of the incubation period for SARS-CoV has proven to be a consid-
erable challenge. Numerous studies have attempted to make estimates (see
Donnelly et al. (2004) for a review). In papers in which interval censoring
methodology is outlined, a common strategy to deal with censored data is
to assume a uniform probability of transmission across the exposure period,
(see, for example Donnelly et al. (2003) and Meltzer (2004)). An alternative is
to assume immediate transmission upon exposure to a known symptomatic
SARS case, (see, for example Lee et al. (2003)).
The methodology used to estimate the incubation period in the current study
was to assume a constant hazard of transmission within the contact period.
The estimated incubation period, for a given dataset, using this model would
be expected to be longer than the estimations using the uniform probability
model, but shorter than the estimates based on the assumption of immediate
transmission.
The constant hazard model has the advantage that it has a biologically plau-
sible basis. However, because the estimated value of the hazard of transmis-
sion, λ, had a large probability mass near zero in this study, it would be rea-
sonable to use a uniform probability density function for time to transmis-
sion as an approximation. Figure 7.18 illustrates the estimated incubation
period based on the two different models. There is little difference between
the result of the incubation period assuming a constant hazard and assuming
uniform probability of infection during the exposure period, and the subse-
quent conclusions of the model are robust to the estimates of λ. Figure 7.18
also gives the expected value of the incubation period assuming instanta-
neous transmission at the time of contact, which is considerably longer than
the estimated incubation period in the constant hazard or uniform transmis-
sion models.
Determining the incubation period following point exposure avoids the as-
sumptions required to infer transmission times. Olsen et al. (2003) investi-
gated cases following a 3 hour in-flight exposure to a symptomatic SARS case
and found an incubation period of 4 (2-8) days. The numbers in that study
were small (22 cases), and the rapid transmission may reflect a large inocu-
lum which could impact on incubation period. Studies using larger datasets
of fully observed exposure times would be useful.
7.6 Estimation of other transition periods 171
A deficiency in the current study is that there is only weak information on
hazard of transmission, λ, since only those known to be infected with SARS-
CoV are included in the dataset. This leads to the posterior probability den-
sity for λ taking on values similar to the prior probability. In future studies,
more informative estimates of λ could be obtained by incorporating knowl-
edge about those who had exposure to a SARS case but did not become in-
fected. Alternatively, the number of contacts per infectious patient per day
could be incorporated into the model. This would provide a direct relation-
ship between the daily hazard of transmission for a single contact and the
infectivity per patient per day, which is estimated from the large scale behav-
iour of the epidemic (see Section 7.7).
7.6 Estimation of other transition periods
A Bayesian framework was also used to estimate the other transition periods
in the SEIHRD model: time from symptom onset to hospitalisation, time
from hospital admission to recovery and time from hospital admission to
death. The transition periods were assumed to be drawn from Gamma(α, β)
distributions. The parameters of the Gamma distributions were given vague
prior probability densities (π(α, β) ∼ Gamma(0.001, 0.001)). All observations
for transition periods were assumed to be independent. The posterior
probability densities of the Gamma distribution parameters (α, β) were
determined for each of the transition periods using
π(α, β|z) ∝ π(α, β)L(z|α, β), (7.9)
where z is the vector of observations for each of the transition period and
L(z|α, β) is the likelihood given by
L(z|α, β) =N∏
i=1
g(zi|α, β), (7.10)
where N is the number of observations. The calculations were performed
using Metropolis-Hastings steps in a manner similar to that described in Ap-
pendix 7.B.
172 Chapter 7. Bayesian modelling of an epidemic of SARS
7.6.1 Results: Estimation of other transition periods
Figure 7.10 gives the parameterised posterior probability distribution of the
time interval from symptom onset to hospitalisation, with the recorded dis-
crete data in the background. The distribution is approximately exponential,
with a mean of 3.5 days and a median of 2.9 days.
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
Pro
bab
ility
den
sity
fu
nct
ion
Time from symptom onset to hospital admission (days)
Figure 7.10: Estimated best fit Gamma distribution for time from symptomonset to hospitalisation, based on maximum posterior probability density es-timates for the shape and scale parameters. A histogram of recorded discretetimes is also shown.
Figure 7.11 shows the parameterised distribution of the time from symptom
onset to recovery. The mean time from symptom onset to death was 26 days
with a standard deviation of 11 days. Figure 7.12 shows the parameterised
distribution of the time from symptom onset to death. The distribution is
widely dispersed, with a mean of 21 days and standard deviation of 9.4 days.
Table 7.1 gives the means and standard deviation for the duration of each of
the stages of infection. Appendix 7.D gives the estimated values of the shape
and scale parameters of the inferred Gamma distributions.
7.7 Model for estimating coefficients of infectivity
The extended SEIHRD model was used to estimate the infectivity of SARS
cases. Coefficients of infectivity were defined in this study as the expected
number of new transmissions per infectious case per day. The infectious
group was divided into community, I, and hospitalised, H symptomatic
7.7 Model for estimating coefficients of infectivity 173
0 10 20 30 40 50 60 700
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
Time from symptom onset to recovery (days)
Pro
bab
ility
den
sity
fu
nct
ion
Figure 7.11: Estimated best fit Gamma distribution for time from symptomonset to recovery, based on maximum posterior probability density estimatesfor the shape and scale parameters. A histogram of recorded discrete times isalso shown.
0 10 20 30 40 50 60 70 800
0.01
0.02
0.03
0.04
0.05
0.06
Time from symptom onset to death (days)
Pro
bab
ility
den
sity
fu
nct
ion
Figure 7.12: Estimated best fit Gamma distribution for time from symptomonset to death, based on maximum posterior probability density estimatesfor the shape and scale parameters. A histogram of recorded discrete times isalso shown.
SARS cases. The epidemic was assumed to begin on February 28, 2003 when
the first introduced SARS case became symptomatic. Following the SEIHRD
model outlined in Section 7.2, the rate of new transmissions was assumed to
be proportional to the number of infectious patients at that time and their
infectivity.
174 Chapter 7. Bayesian modelling of an epidemic of SARS
Mean 95% CI Standard Deviation 95% CISymptoms to hospitalisation 3.5 3.2-3.9 3.2 2.8-3.6Symptoms to recovery 26 25-27 11 10-12Symptoms to death 21 15-29 15 9.7-23Hospitalisation to recovery 23 21-24 11 10-12Hospitalisation to death 17 11-28 16 6.9-31
Table 7.1: The posterior mean and standard deviation (in days) of the timesto hospitalisation, hospital discharge and death.
Two different states, community and hospitalised, and two different time
periods, early and late in the epidemic, were investigated. The time of
change from high to low infectivity was also estimated. The change-point
was considered an additional parameter, and its posterior probability was
investigated.
7.7.1 Bayesian approach to estimation of the transmission
coefficients
The parameters of interest in this part of the model are the coefficients of
infectivity of symptomatic community SARS cases (prior to hospitalisation)
early and late in the epidemic, denoted by x1, and x2, respectively, and the co-
efficients of infectivity of hospitalised patients early and late in the epidemic,
denoted by y1, and y2, respectively. Also of interest is the change-point, de-
noted by C.
We did not assume a fixed change-point because there were a number
of stages of intervention in the Shanxi epidemic. Firstly, the global alert
occurred on March 12, 2003. This was followed by a concerted public
health campaign in early April. It was not until April 23, 2003 that WHO
included Shanxi province on its travel warning. We therefore included the
change-point as an unknown parameter, requiring estimation.
7.7.2 Prior specification
Gamma(0.001, 0.001), were used for the four coefficients of infectivity. A dis-
crete, uniform U [1, n] distribution was used as the prior for the change-point,
where n is the number of days of the epidemic.
7.7 Model for estimating coefficients of infectivity 175
7.7.3 Likelihood estimation
Following the SEIHRD model and assuming constant infectivity within each
of the 4 groups of symptomatic SARS cases, the transmission pressure, ρj , on
day, j, is given by
ρj = xiI(j) + yiH(j), (7.11)
where i = 1 (j < C) and i = 2 (j ≥ C). I(j) is the number of symptomatic
community patients and H(j) is the number of symptomatic hospitalised pa-
tients.
The likelihood of Tj transmissions occurring on day j is assumed to be drawn
from the Poisson distribution:
k(Tj, Hj, Ij|x, y) ∼ Poisson(ρj). (7.12)
In a small scale epidemic, if the number of susceptibles were known, the
Binomial probability distribution could be used. In this epidemic in which
there are approximately 3 million susceptibles, the Poisson approximation
is reasonable, although it may underestimate the dispersion of the offspring
distribution (as would the binomial distribution), particularly if there is
marked heterogeneity of spreading (for example super-spreaders). An
alternative parameterisation with higher dispersion would be the Negative
Binomial distribution.
With all data included it is straightforward to find the full likelihood of the
data given the parameters:
L(T,H, I|x, y, C) =C∏
j=1
k(Tj, Hj, Ij|x1, y1)n∏
j=C+1
k(Tj, Hj, Ij|x2, y2), (7.13)
where n is the number of days of the epidemic and T ,H and I, represent
the vectors of n values of daily transmissions, community case numbers and
hospitalised case numbers, respectively.
Because the times of transmission are unknown, and there are some missing
values in the hospitalisation and recovery and death times, missing data and
unobserved data need to be inferred. The simulated data are drawn from
the distributions of the incubation period, time to hospitalisation and time
to recovery and discharge estimated in the first part of the study. The likeli-
hood estimation of the day of transmission for each individual was based on
the parameterised incubation period and for cases with known contacts the
176 Chapter 7. Bayesian modelling of an epidemic of SARS
truncated exponential distribution given in Expression (7.5). The state of the
epidemic was not considered in the likelihood computation. The techniques
used for data augmentation and computation are given in Appendix 7.E.
7.7.4 Results: Change point Estimation
The epidemic was measured from the day of symptom onset of patient 1,
which was February 28, 2003. Figure 7.13 shows the posterior distribution
for the estimated time of the change in infectivity (change-point). The maxi-
mum density is taken to be the end of day 29 of the epidemic, corresponding
to the beginning of March 29, 2003. Following this, the estimates of the co-
efficients of infectivity were performed assuming a change point at midnight
March 28/29.
20 40 60 80 100 1200
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
Days from start of epidemic
Pos
terio
r pr
obab
ility
den
sity
Figure 7.13: Posterior distribution for change-point.
Figure 7.13 demonstrates that there is considerable uncertainty with this
estimate, with the posterior probability also giving some support to an
earlier change point time. The posterior weight rapidly declines for times
after March 29th (day 30 from the start of the epidemic), suggesting later
times are unlikely.
7.7.5 Results: Coefficients of Infectivity
The estimated means of the four coefficients of SARS-CoV transmission,
representing the mean number of new infections per infectious case per
7.7 Model for estimating coefficients of infectivity 177
day, are given in Table 7.2. The relative infectivity of community compared
to hospitalised SARS cases increases markedly after the change point, with
x1/y1 = 5.1 (95% CI 0.8-17), and x2/y2 = 350 (95% CI 95-1400), where xi
refers to symptomatic community SARS cases (prior to hospitalisation) and
yi refers to hospitalised patients.
Parameter Mean 95% CredibleInterval
x1 0.41 0.24-0.59y1 0.15 0.023-0.34x2 0.21 0.18-0.24y2 0.0006 0.000018-
0.0022Ra 4.8 2.2-8.8Rb 0.75 0.65-0.85
Table 7.2: Table of transmission coefficients (mean number of transmissions perinfective per day) x1: symptomatic community cases before March 29, x2: symp-tomatic community cases after March 29 y1: symptomatic hospitalised cases beforeMarch 29, y2: symptomatic hospitalised cases after March 29. March 29 was thechange-point with the highest posterior probability density.
178 Chapter 7. Bayesian modelling of an epidemic of SARS
7.7.6 Results: Reproduction ratio
The basic reproduction ratio, R0, is defined as the expected number of
secondary cases per primary case in a fully susceptible population (Ander-
son and May, 1991; Diekmann and Heesterbeek, 2000). As the epidemic
progresses, the reproduction ratio could be modified both by a decrease in
the number of susceptible cases or a change in infectivity (for example, due
to infection control interventions). In this study, we estimated the effective
reproduction ratio before and after the change point.
The effective reproduction ratio can be deduced from the inferred coeffi-
cients and the known data. The mean time from symptom onset to hospital
admission is 3.5 days and the mean time from hospital admission to either
recovery or death is 22.2 days. The posterior probability distribution of the
effective reproduction ratio can be calculated using
Ra = x1X + y1Y , (7.14)
where Ra is the reproduction ratio prior to the change point, X is the mean
duration of symptoms prior to hospitalisation and Y is the mean duration of
symptoms in hospital. Similarly Rb, the reproduction ratio after the change
point can be calculated using
Rb = x2X + y2Y . (7.15)
Ra is estimated to be 4.8 (95%CI 2.2-8.8 )and Rb is estimated to be 0.75 (95%CI
0.65-0.85). The distributions for Ra and Rb are displayed in Figure 7.14. The
greatest impact on the reproduction ratio was the change in infectivity of the
hospitalised group.
During the first part of the epidemic prior to March 29, the expected number
of transmissions resulting from each symptomatic SARS case is 1.4 during
the community period, and 3.4 during the hospitalised period. For the SARS
cases from March 29 onwards, the expected number of transmissions result-
ing from each symptomatic SARS case is 0.73 during the community period,
and 0.013 during the hospitalised symptomatic period. The ratio of infec-
tivity in the community to infectivity following hospitalisation is 5.1, simi-
lar to Riley et al. (2003)’s estimate of 5. After March 29 however, this figure
was much higher, owing to very much reduced estimated infectivity in hos-
pitalised patients.
7.8 Individual Infectivity profiles 179
1 2 3 4 5 6 7 8 9 100
1
2
3
4
5
6
7
Reproduction ratio
Po
ster
ior
pro
bab
ility
den
sity
Figure 7.14: Posterior distribution for the reproduction ratios prior to (white)and after (black) March 29, 2003.
7.8 Individual Infectivity profiles
A preliminary analysis compares three models of individual infectivity over
the course of SARS-CoV infection. The first is the uniform transmission model
in which infectiousness within the 4 groups (out of hospital early, in hospi-
tal early, out of hospital late, in hospital late) is uniform over the course of
illness. In the second model is the model with transmission proportional to
viral load. In this model infectivity takes on a triangular distribution peaking
on day 10, following the results for viral load described by Peiris et al. (2003a).
In the third model, the model with transmission given by a Gamma distribu-
tion; shape and scale parameters were inferred.
Using the Akaike information criterion (AIC) (Akaike, 1974), the model with
transmission given by a Gamma distribution is superior (AIC=320) to the uni-
form transmission model (AIC=328). The model model with transmission pro-
portional to viral load performs the worst (AIC=356). With reference to the
model with transmission given by a Gamma distribution, the inferred Gamma
distribution of the infectivity profile is shown in Figure 7.15. The peak infec-
tivity is estimated to be on the 9th day following symptom onset in the Gamma
model. The infectivity follows the Gamma(3.9, 0.36) distribution.
This finding is based on an initial exploration of the dataset and the analysis
can be extended. In particular, the infectivity profiles could inform the trans-
mission times. In this study, as a simplification, the unobserved transmission
180 Chapter 7. Bayesian modelling of an epidemic of SARS
0 10 20 30 400
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Day since symptom onset
Est
imat
ed in
fect
ivit
y
Figure 7.15: Infectivity profile versus time since symptom onset.
times were inferred using the uniform transmission model only.
7.9 Discussion and Conclusions
Important conclusions regarding the infectivity of SARS-CoV can be drawn
from this analysis. The estimated daily infectivity of the hospitalised patients
was lower than for community patients. Despite this, it was estimated that
early in the epidemic, a larger number of secondary cases resulted from hos-
pitalised patients because people remained in this stage for a longer time (an
average of 22.2 days symptomatic in hospital compared with 3.5 days prior to
hospitalisation). Later in the epidemic, the transmission rate of symptomatic
community SARS cases decreased to around 50% of previous levels, whereas
the decline in transmission rate for those SARS patients in hospital reduced
more dramatically to around 0.4% of previous levels.
These results support the conclusion that interventions were effective at con-
trolling the SARS epidemic in Shanxi province, particularly those interven-
tions directed at hospital isolation. However, other possible causes for these
results need to be considered. The relatively high infectivity of the commu-
nity SARS cases could be due to their earlier stage in the course of SARS-CoV
infection. Much of the time spent in hospital is associated with the conva-
lescent stages of the illness and it could be argued that SARS patients would
be less infectious during this period. On the other hand, Peiris et al. (2003a)
showed that viral shedding peaks around day 10, suggesting that for many
people the most infectious stage of the illness occurs following hospitalisa-
tion.
7.9 Discussion and Conclusions 181
The reduction of infectivity over time could be partly explained by the pro-
portion of contacts who are susceptible decreasing as an epidemic proceeds.
While depletion of susceptibles undoubtedly occurs in widespread viral epi-
demics, the authors believe that the transmission of SARS-CoV to 354 people
in a population of over 3 million would not account for a significant drop in
the proportion of contacts who are susceptible, assuming homogenous pop-
ulation mixing. In the hospital setting and in families, in which contacts tend
to cluster, depletion of susceptibles may account for some of the change in
the reproduction ratio. This could be further explored using a network or
household model.
Another reason for the difference in infectivity before and after the March 29
could be seasonal. It is possible that SARS-CoV , like many other respiratory
viruses, is transmitted more efficiently in winter. However, this would result
in a general decline in infectivity, which does not explain the much greater re-
duction in infectivity of hospitalised SARS cases compared with community
SARS cases, observed in this study.
The estimated date on which the infectivity of SARS declined (the change
point) predated the peak incidence of admission of SARS cases to hospital.
Both the incubation period and the delay between symptom onset and hos-
pitalisation contributed to this lag. It is a lesson for future epidemics, that
even after appropriate interventions are successful in reducing transmission,
we can expect a further increase in infection notifications.
The reproduction ratio late in the Shanxi epidemic is very similar to those
estimated by Wallinga and Teunis (2004) in Singapore and Hong Kong, both
estimated to be 0.7. Wallinga and Teunis (2004) studied 4 countries (Singa-
pore, Viet Nam, Hong Kong and Canada) and found that although the epi-
demic curves initially were markedly different, following interventions, the
estimated reproduction ratio was very similar in 3 of the 4 countries exam-
ined in that study. Although it is reassuring that in most cases (all except
Canada), a reproduction ratio of less than one was achieved, it was only fol-
lowing the implementation of stringent control measures. It could be pre-
dicted that if complacency occurs in future epidemics, it may be difficult to
achieve a reproduction ratio of less than one for SARS.
Three different models of infectivity profiles over the course of SARS-CoV in-
fection were considered in this study. The model considering a Gamma shape
for infectivity appeared statistically slightly superior to the model assuming
uniform infectivity. Of interest is that the estimated peak infectivity occurs
182 Chapter 7. Bayesian modelling of an epidemic of SARS
on the ninth day following symptom onset. This is consistent with specimen
positivity in the lower and upper respiratory tract and gut reported by Cheng
et al. (2004). Additionally, (Peiris et al., 2003a) measured nasopharyngeal as-
pirate viral loads of 14 SARS cases on day 5, 10 and 15 following symptom
onset and found that day 10 was consistently the highest of these measure-
ments. The concordance between viral load data and infectivity inferred in
this study warrants further investigation. A larger dataset in which contact
times are fully observed would be useful in elucidating infectivity profile.
There are several ways in which the current model can be extended. This
study assumed Gamma distributions for transition times. Other distributions
could be considered including the Weibull and non-parametric approaches.
A mixture model may be particularly useful for estimating susceptibility,
infectiousness and duration of infectivity. The possibility of more than one
change point or a gradual transition could also be explored. Reversible
jump MCMC would be a useful tool in determining this (Green, 1995). SARS
models to date including the current study have assumed zero infectivity
during the incubation period. Infectivity of SARS cases during the incubation
period could be estimated by extending the Bayesian inference model. While
there were no clearly identified super-spreaders in the Shanxi epidemic,
heterogeneity of infectivity was a major feature of the epidemiology of
SARS in Singapore and Hong Kong (Li et al., 2004). This could be further
investigated using the current dataset, however a dataset containing detailed
information on transmission trees would be more informative.
Appendix
7.A Gantt chart of Shanxi epidemic
Figure 7.16 displays a visual depiction of the epidemic. Each individual is
represented as a horizontal line, with the colour code indicating the stage of
SARS-CoV infection for that individual. It can be seen from Figure 7.16 that
in mid to late April, the daily number of new cases began to decline.
7.B Computations for time to transmission and incubation period183
Figure 7.16: Gantt chart of epidemic. The time of exposure to another SARScase is mid-blue, the time that a patient is asymptomatic following exposureis light blue, the time of symptoms prior to hospitalisation is yellow, the timeof hospitalisation is orange and the time of discharge or death, maroon. Pa-tients are ordered according to hospital admission date.
7.B Computations for time to transmission and
incubation period
Computations were performed using a Markov chain Monte Carlo (MCMC)
algorithm.
1. Initialise the parameters λ, αL and βL.
2. For each patient i, propose a new ui by drawing u′i randomly from the
distribution described in Expression (7.5)
3. Accept u′i using the acceptance probability,
Pacc = min
1,
g((si − u′i)|αL, βL)
g((si − ui)|αL, βL)
(7.16)
4. Propose λ′ using a simple random walk step such that λ′ = λ + ε, where
184 Chapter 7. Bayesian modelling of an epidemic of SARS
ε is drawn from the N(0, 100) distribution. In this paper, we follow the
Bayesian notation where N(0, 100) is used for a normal distribution with
a mean of zero and a precision of 100 and hence a variance of 0.01. The
precision of the proposal distribution was chosen as a balance of the need
to have rapid mixing and the desire to improve acceptance probability
5. Accept λ′ with a probability Pacc given by
Pacc = min
1,
∏Ni=1 f(ui|λ′)π(λ′)∏Ni=1 f(ui|λ)π(λ)
, (7.17)
6. Update αL, proposing a new value α′L using a simple random walk, each
step is drawn from a random normal distribution N(0, 100). Accept α′Lwith probability Pacc given by
Pacc = min
1,
∏Ni=1 g((si − ui), α
′L, βL)π(α′L)∏N
i=1 g((si − ui), αL, βL)π(αL)
(7.18)
7. Update βL using a Gibbs step. A conjugate prior, π(βL) ∼ Gamma(l,m),
is assigned to βL, making the full conditional posterior for βL
βL|(s− u), αL ∼ Gamma(l + αLN,m +N∑
i=1
(si − ui)), (7.19)
which enables a Gibbs update of βL by drawing a value randomly from
this distribution.
Steps 2 to 7 constitute a single iteration of the algorithm.
The “burn-in” period was 10 000 iterations. The posterior probability distri-
butions of u, λ, αL, βL were determined by taking the next 90 000 iterations.
7.C Diagnostics: Convergence and Sensitivity
analysis
Visual inspection of the trace plots showed that the chains for all parameters
appeared to converge within 1000 iterations. A number of different initial
values were considered for the parameters and the results were essentially
unchanged. Figure 7.17, for example, shows values of αL plotted against iter-
ation number for 6 different initial values. The plots show that the estimates
7.C Diagnostics: Convergence and Sensitivity analysis 185
of αL settle down well before the end of the 10 000 iteration burn-in.
500 1000 1500 2000 2500 3000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Iterations
α
Figure 7.17: Output from one series of Markov chains. Six different initial val-ues of the shape parameter of the incubation period, αL all settle down to thesame distribution after a few hundred iterations.
7.C.1 Sensitivity of estimate of incubation period to model
choice and hazard of transmission parameter
Sensitivity analysis was performed on the choice of model used to estimate
the incubation period. The current study assumed that during a contact the
hazard of transmission remained constant, leading to an exponential proba-
bility density function for time to transmission. Two alternative approaches
would be
1. to assume that the probability of transmission was constant throughout
the contact period, a uniform probability density for time to transmis-
sion, effectively putting λ = 0
2. to assume transmission coincides with onset of infection challenge, ef-
fectively putting λ = ∞.
The posterior probability density of the incubation period was estimated us-
ing these models and compared with the estimation in the current study as
summarised in Table 7.3 and illustrated in Figure 7.18.
In the model used in the current study (the assumption of constant hazard),
the maximum posterior density for the daily hazard of transmission, λ was
186 Chapter 7. Bayesian modelling of an epidemic of SARS
Incubation period Mean 95% CI Standard Deviation 95 % CIConstant hazard 5.3 4.2-6.8 4.5 3.4-5.9Uniform 5.1 4.1-6.3 4.4 3.4-5.6Immediate Transmission 7.9 6.9-9.0 4.9 4.1-5.9
Table 7.3: The estimated mean and standard deviation (in days) of the in-cubation period comparing the estimates using the assumption of constanthazard, used by the current study, and the assumptions of uniform probabil-ity and immediate transmission.
0 5 10 15 20 25 300
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Incubation period (days)
Pro
bab
ility
den
sity
fu
nct
ion
constant hazarduniform probabilityimmediate transmission
Figure 7.18: Comparison of the incubation period as estimated in the currentstudy using the constant hazard model, the uniform probability model andthe immediate transmission model. The constant hazard model used in thisstudy leads to a similar result to the uniform probability model.
close to zero. Little information was available in the data-set regarding λ,
therefore λ took on a distribution similar to its prior probability, with a large
probability mass near zero and a long tail. This effectively makes the model
in which a constant hazard is assumed equivalent to the model of uniform
probability, the model that suggests infection is equally likely at any stage
during the exposure period. Even at the extreme values of λ, the effects of the
estimate of λ on incubation period shown in Figure 7.18 are relatively small.
Therefore the conclusions of the subsequent components of the model are
robust to the choice of model for transmission and the value of λ.
7.D Estimated values of shape and scale parameters for the Gammadistributions 187
7.D Estimated values of shape and scale parame-
ters for the Gamma distributions
Table 7.4 gives the estimated values of the parameters of the Gamma distrib-
utions applied in the model. These values can be used along with the coef-
ficients of infectivity to reconstruct the epidemic and explore the large scale
effect of interventions, including reduced time to isolation, quarantine, and
more effective isolation.
shape parameter scale−1 parameterIncubation period 1.4 0.26Symptom onset to hospitalisation 1.3 0.37Symptom onset to recovery 5.6 0.22Symptom onset to death 2.1 0.11Hospital admission to recovery 4.1 0.18Hospital admission to death 1.2 0.068Individual infectivity 3.9 0.36
Table 7.4: Estimated values (based on maximum posterior density) for theshape and scale−1 parameters of the Gamma distributions fitted to the data.
7.E Techniques used for data augmentation and
computation to determine coefficients of in-
fectivity and change point
7.E.1 Augmented data
There were missing values for the time of symptom onset hospitalisation
times and time to recovery (1, 2 and 10 missing values respectively out of
the 354 SARS cases in the database). Missing data were simulated for each
iteration of the Markov chain using the inferred distributions of transition
times. The likelihood of the data given the parameters is given by
L(d|θ) =
∫L(d, s|θ) ds, (7.20)
where d is the known data and s is the simulated data.
Because the integral above is not straightforward, L(d|θ) was inferred by
drawing s using the known times and the parameterised distributions,
estimated in Section 7.6. For example, where recovery times were missing,
188 Chapter 7. Bayesian modelling of an epidemic of SARS
these were inferred from the hospitalisation date and the parameterised time
to recovery distribution.
The date of each individual’s transmission of SARS-CoV also became an aux-
iliary variable in the model. The times were inferred from
1. the known date of onset of symptoms (taken directly from the database)
2. the parameterised incubation period,
so that transmission date = date of symptom onset - incubation period, where
the incubation period was drawn randomly from the Gamma(αL, βL) distrib-
ution. If the time of exposure to another SARS case was known, the proposed
transmission time (ti) was drawn from a distribution based on the joint prob-
ability of (a) time to transmission, calculated using Expression 7.5 and (b) the
incubation period, with Gamma(αL, βL) distribution.
7.E.2 Computations to determine posterior distributions of
the coefficients of infectivity and the change point
For each iteration of the model, the auxiliary variables were firstly de-
termined using Gibbs sampling of the parameterised distributions. The
likelihood of the augmented data was calculated using Expression (7.13).
Coefficients of infectivity were proposed and accepted according to:
Pacc = min
1,
C∏i=1
k(Tj, Hj, Ij, x′1, y1)p(x′1)prop(x′1 → x1)
k(Tj, Hj, Ij, x1, y1)p(x1)prop(x1 → x′1)
, (7.21)
where C is the date of the change point and prop(x′1 → x1) is the proposal
probability of x1 from x′1. Similarly, x′2 is updated by:
Pacc = min
1,
n∏i=C+1
k(Tj, Hj, Ij, x′2, y2)p(x′2)prop(x′2 → x2)
k(Tj, Hj, Ij, x2, y2)p(x2)prop(x2 → x′2)
, (7.22)
where n is the number of days of the epidemic. Acceptance equations were
similarly constructed for y1 and y2.
The change-point day was updated as follows:
1. For each iteration a new change-point day was proposed drawn as an
integer from the U [1, n] distribution, where the epidemic begins on day
1 and ends on day n
7.E Statistical inference used to estimate infectivity and change points189
2. The change-point day was updated using a Metropolis step based on
the full likelihood given by:
Pacc = min
1,
p(C ′)∏C′
j=1 k(Tj, x1, y1)∏n
j=C′+1 k(Tj, x2, y2)
p(C)∏C
j=1 k(Tj, x1, y1)∏n
j=C+1 k(Tj, x2, y2)
. (7.23)
The process was iterated 100 000 times and the first 10 000 iterations were
used as a burn-in period. The following 90 000 updates of x1, x2, y1, y2 and C
were used to determine the posterior distribution.
190 Chapter 7. Bayesian modelling of an epidemic of SARS
Bibliography
Akaike, H., 1974. A new look at the statistical model identification. IEEE Transactionson Automatic Control 19 (6), 716–723.
Anderson, R. M., May, R. M., 1991. Infectious diseases of humans : dynamics andcontrol. Oxford University Press, Oxford ; New York.
Booth, C. M., Matukas, L. M., Tomlinson, G. A., Rachlis, A. R., Rose, D. B., Dwosh,H. A., Walmsley, S. L., Mazzulli, T., Avendano, M., Derkach, P., Ephtimios, I. E.,Kitai, I., Mederski, B. D., Shadowitz, S. B., Gold, W. L., Hawryluck, L. A., Rea, E.,Chenkin, J. S., Cescon, D. W., Poutanen, S. M., Detsky, A. S., 2003. Clinical featuresand short-term outcomes of 144 patients with SARS in the greater Toronto area.Jama 289 (21), 2801–9.
Cheng, P. K., Wong, D. A., Tong, L. K., Ip, S. M., Lo, A. C., Lau, C. S., Yeung, E. Y.,Lim, W. W., 2004. Viral shedding patterns of coronavirus in patients with probablesevere acute respiratory syndrome. Lancet 363 (9422), 1699–700.
Choi, B., Pak, A., 2003. A simple approximate mathematical model to predict thenumber of severe acute respiratory syndrome cases and deaths. J Epi CommHealth 57, 831–835.
Chowell, G., Fenimore, P. W., Castillo-Garsow, M. A., Castillo-Chavez, C., 2003. SARSoutbreaks in Ontario, Hong Kong and Singapore: the role of diagnosis and isola-tion as a control mechanism. J Theor Biol 224 (1), 1–8.
Diekmann, O., Heesterbeek, J., 2000. Mathematical Epidemiology of Infectious Dis-eases: Model Building, Analysis and Interpretation. John Wiley and Son, LTD.
Donnelly, C., Fisher, M., Fraser, C., Ghani, A., Riley, S., Ferguson, N., Anderson, R.,2004. Epidemiological and genetic analysis of severe acute respiratory syndrome.Lancet Infectious Diseases 4 (11), 672–83.
Donnelly, C. A., Ghani, A. C., Leung, G. M., Hedley, A. J., Fraser, C., Riley, S., Abu-Raddad, L. J., Ho, L. M., Thach, T. Q., Chau, P., Chan, K. P., Lam, T. H., Tse, L. Y.,Tsang, T., Liu, S. H., Kong, J. H., Lau, E. M., Ferguson, N. M., Anderson, R. M., 2003.Epidemiological determinants of spread of causal agent of severe acute respira-tory syndrome in Hong Kong. Lancet 361 (9371), 1761–6.
Drosten, C., Gunther, S., Preiser, W., van der Werf, S., Brodt, H. R., Becker, S.,Rabenau, H., Panning, M., Kolesnikova, L., Fouchier, R. A., Berger, A., Burguiere,A. M., Cinatl, J., Eickmann, M., Escriou, N., Grywna, K., Kramme, S., Manuguerra,J. C., Muller, S., Rickerts, V., Sturmer, M., Vieth, S., Klenk, H. D., Osterhaus, A. D.,Schmitz, H., Doerr, H. W., 2003. Identification of a novel coronavirus in patientswith severe acute respiratory syndrome. N Engl J Med 348 (20), 1967–76.
192 BIBLIOGRAPHY
Gopalakrishna, G., Choo, P., Leo, Y. S., Tay, B. K., Lim, Y. T., Khan, A. S., Tan, C. C.,2004. SARS transmission and hospital containment. Emerg Infect Dis 10 (3), 395–400.
Green, P., 1995. Reversible Jump Markov chain Monte Carlo computation andBayesian model determination. Biometrika 82 (4), 711–732.
Guan, Y., Zheng, B. J., He, Y. Q., Liu, X. L., Zhuang, Z. X., Cheung, C. L., Luo, S. W.,Li, P. H., Zhang, L. J., Guan, Y. J., Butt, K. M., Wong, K. L., Chan, K. W., Lim, W.,Shortridge, K. F., Yuen, K. Y., Peiris, J. S., Poon, L. L., 2003. Isolation and characteri-zation of viruses related to the SARS coronavirus from animals in southern China.Science 302 (5643), 276–8.
Gumel, A. B., Ruan, S., Day, T., Watmough, J., Brauer, F., van den Driessche, P.,Gabrielson, D., Bowman, C., Alexander, M. E., Ardal, S., Wu, J., Sahai, B. M., 2004.Modelling strategies for controlling SARS outbreaks. Proc R Soc Lond B Biol Sci271 (1554), 2223–32.
Hsieh, Y. H., Chen, C. W., Hsu, S. B., 2004. SARS outbreak, Taiwan, 2003. Emerg InfectDis 10 (2), 201–6.
Kermack, W., McKendrick, A., 1927. Contributions to the mathematical theory of epi-demics:part 1. Proceedings of the Royal Society of London A 115, 700–721.
Ksiazek, T. G., Erdman, D., Goldsmith, C. S., Zaki, S. R., Peret, T., Emery, S., Tong, S.,Urbani, C., Comer, J. A., Lim, W., Rollin, P. E., Dowell, S. F., Ling, A. E., Humphrey,C. D., Shieh, W. J., Guarner, J., Paddock, C. D., Rota, P., Fields, B., DeRisi, J., Yang,J. Y., Cox, N., Hughes, J. M., LeDuc, J. W., Bellini, W. J., Anderson, L. J., 2003. Anovel coronavirus associated with severe acute respiratory syndrome. N Engl JMed 348 (20), 1953–66.
Lee, N., Hui, D., Wu, A., Chan, P., Cameron, P., Joynt, G. M., Ahuja, A., Yung, M. Y.,Leung, C. B., To, K. F., Lui, S. F., Szeto, C. C., Chung, S., Sung, J. J., 2003. A ma-jor outbreak of severe acute respiratory syndrome in Hong Kong. N Engl J Med348 (20), 1986–94.
Leung, G. M., Chung, P. H., Tsang, T., Lim, W., Chan, S. K., Chau, P., Donnelly, C. A.,Ghani, A. C., Fraser, C., Riley, S., Ferguson, N. M., Anderson, R. M., Law, Y. L., Mok,T., Ng, T., Fu, A., Leung, P. Y., Peiris, J. S., Lam, T. H., Hedley, A. J., 2004. SARS-coVantibody prevalence in all Hong Kong patient contacts. Emerg Infect Dis 10 (9),1653–6.
Li, Y., Yu, I., Xu, P., Lee, J., Wong, T., Ooi, P., Sleigh, A., 2004. Predicting super spread-ing events during the 2003 severe acute respiratory syndrome epidemics in HongKong and Singapore. Am J Epi 160 (8), 719–728.
Lim, P. L., Kurup, A., Gopalakrishna, G., Chan, K. P., Wong, C. W., Ng, L. C., Se-Thoe,S. Y., Oon, L., Bai, X., Stanton, L. W., Ruan, Y., Miller, L. D., Vega, V. B., James, L.,Ooi, P. L., Kai, C. S., Olsen, S. J., Ang, B., Leo, Y. S., 2004. Laboratory-acquired se-vere acute respiratory syndrome. N Engl J Med 350 (17), 1740–5.
Lipsitch, M., Cohen, T., Cooper, B., Robins, J. M., Ma, S., James, L., Gopalakrishna, G.,Chew, S. K., Tan, C. C., Samore, M. H., Fisman, D., Murray, M., 2003. Transmissiondynamics and control of severe acute respiratory syndrome. Science 300 (5627),1966–70.
BIBLIOGRAPHY 193
Lloyd, A. L., 2001. Destabilization of epidemic models with the inclusion of realisticdistributions of infectious periods. Proc R Soc Lond B 268, 985–993.
Meltzer, M., 2004. Multiple contact dates and SARS incubation periods. EmergingInfectious Diseases 10 (2), 207–209.
Normille, D., April 2004. Mounting lab accidents raise SARS fears. Science 304, 659–661.
Olsen, S. J., Chang, H. L., Cheung, T. Y., Tang, A. F., Fisk, T. L., Ooi, S. P., Kuo, H. W.,Jiang, D. D., Chen, K. T., Lando, J., Hsu, K. H., Chen, T. J., Dowell, S. F., 2003.Transmission of the severe acute respiratory syndrome on aircraft. N Engl J Med349 (25), 2416–22.
Orellana, C., 2004. Laboratory-acquired SARS raises worries on biosafety. Lancet In-fect Dis 4 (2), 64.
Peiris, J. S., Chu, C. M., Cheng, V. C., Chan, K. S., Hung, I. F., Poon, L. L., Law, K. I.,Tang, B. S., Hon, T. Y., Chan, C. S., Chan, K. H., Ng, J. S., Zheng, B. J., Ng, W. L., Lai,R. W., Guan, Y., Yuen, K. Y., 2003a. Clinical progression and viral load in a commu-nity outbreak of coronavirus-associated SARS pneumonia: a prospective study.Lancet 361 (9371), 1767–72.
Peiris, J. S., Lai, S. T., Poon, L. L., Guan, Y., Yam, L. Y., Lim, W., Nicholls, J., Yee, W. K.,Yan, W. W., Cheung, M. T., Cheng, V. C., Chan, K. H., Tsang, D. N., Yung, R. W., Ng,T. K., Yuen, K. Y., 2003b. Coronavirus as a possible cause of severe acute respiratorysyndrome. Lancet 361 (9366), 1319–25.
Poon, L. L., Guan, Y., Nicholls, J. M., Yuen, K. Y., Peiris, J. S., 2004. The aetiology, ori-gins, and diagnosis of severe acute respiratory syndrome. Lancet Infect Dis 4 (11),663–71.
Poutanen, S. M., Low, D. E., Henry, B., Finkelstein, S., Rose, D., Green, K., Tellier, R.,Draker, R., Adachi, D., Ayers, M., Chan, A. K., Skowronski, D. M., Salit, I., Simor,A. E., Slutsky, A. S., Doyle, P. W., Krajden, M., Petric, M., Brunham, R. C., McGeer,A. J., 2003. Identification of severe acute respiratory syndrome in Canada. N Engl JMed 348 (20), 1995–2005.
Riley, S., Fraser, C., Donnelly, C. A., Ghani, A. C., Abu-Raddad, L. J., Hedley, A. J., Le-ung, G. M., Ho, L. M., Lam, T. H., Thach, T. Q., Chau, P., Chan, K. P., Lo, S. V., Leung,P. Y., Tsang, T., Ho, W., Lee, K. H., Lau, E. M., Ferguson, N. M., Anderson, R. M.,2003. Transmission dynamics of the etiological agent of SARS in Hong Kong: im-pact of public health interventions. Science 300 (5627), 1961–6.
Teleman, M. D., Boudville, I. C., Heng, B. H., Zhu, D., Leo, Y. S., 2004. Factors asso-ciated with transmission of severe acute respiratory syndrome among health-careworkers in Singapore. Epidemiol Infect 132 (5), 797–803.
Wallinga, J., Teunis, P., 2004. Different epidemic curves for severe acute respiratorysyndrome reveal similar impacts of control measures. Am J Epidemiol 160 (6), 509–16.
Wang, W., Ruan, S., 2004. Simulating the SARS outbreak in Beijing with limited data.J Theor Biol 227 (3), 369–79.
194 BIBLIOGRAPHY
Webster, R., 2004. Wet markets: a continuous source of severe acute respiratory syn-drome and influenza? Lancet 363, 234–236.
WHO, 2003a. Summary of probable SARS cases with onset of illness from 1 Novem-ber 2002 to 31 July 2003. Technical Report
WHO, 2003b. Update 80 - change in travel recommendations forparts of China, situation in Toronto. (World Wide Web URL:http://www.who.int/csr/don/2003 06 13/en/).
WHO, 2003c. Update 95-SARS: chronology of a serial killer. Geneva, Switzerland.(World Wide Web URL: http://www.who.int/csr/don/2003 07 04/en).
WHO, 2003d. WHO extends its SARS-related travel advice to Beijing andShanxi province in China and to Toronto, Canada. (World Wide Web URL:http://www.who.int/mediacentre/news/notes/2003/np7/en/).
WHO, 2004. Investigation into China’s recent SARS outbreak yields im-portant lessons for global public health. (World Wide Web URL:http://www.wpro.who.int/sars/docs/update/update 07022004.asp).
Wong, T. W., Lee, C. K., Tam, W., Lau, J. T., Yu, T. S., Lui, S. F., Chan, P. K., Li, Y., Bre-see, J. S., Sung, J. J., Parashar, U. D., 2004. Cluster of SARS among medical studentsexposed to single patient, Hong Kong. Emerg Infect Dis 10 (2), 269–76.
CHAPTER 8
Conclusions and suggestions for future
work
Mathematical models add value to the study of infectious diseases. Models are
required both to form a basis of valid statistical inference and, in the absence of ade-
quate epidemiological data, produce some evidence for the efficacy of interventions
through simulation. Knowledge of the transmission characteristics of contagions
can be incorporated into statistical models to improve statistical inference. Such
structured models have an advantage over standard statistical techniques in that
they give a meaningful interpretation to the value of the estimated parameters. Epi-
demic models, founded on biologically plausible assumptions, can be an extremely
useful predictive tool. One can explore some “what if?” scenarios, most importantly
the predicted effect of infection control interventions. Such models are crucial in the
study of infectious diseases epidemics because the standard frequentist statistical
analysis involving repeated trials is infeasible in outbreak settings and randomised
controlled trials of infection control interventions for bacterial pathogens are
logistically challenging. Additionally cost-effectiveness studies can be teamed with
such models to ensure optimal resource utilisation.
The Bayesian approach was adopted in this thesis because the main questions posed
by the studies were How does the information, provided in this single dataset, mod-
ify my belief regarding the transmission of the organism? Such a question does not
have meaning in a frequentist context. In the case of a single pandemic, such as
SARS, there is no opportunity for repeated measurements. Additionally, the Markov
chain Monte-Carlo algorithm is a very convenient tool for numerical integration of
the complex expressions derived from incorporation of latent variables into trans-
mission models. Bayesian inference also allows a researcher to incorporate prior
information into models. In models developed in this thesis, priors probabilities
were vague. Mostly, this was because little was known about the model parameters.
In the cases where a small amount of independent data were available, these were
used to independently validate model conclusions (in Chapters 5 and 7) rather than
incorporated as priors. The results of the studies could be used to develop prior
196 Chapter 8. Conclusions and suggestions for future work
probabilities for subsequent studies.
8.1 What has been achieved?
This thesis used previously unpublished datasets to develop pathogen-specific mod-
els of infectious disease transmission. Using these models, the studies in this thesis
have quantified the cross-transmission rate of three pathogens and explored poten-
tial alternative sources of pathogen acquisition. By allowing the transmission para-
meters to be time-dependent, the studies in Chapters 5 and 7 assessed the impact of
infection control interventions that took place during the data collection period. For
the other studies, the potential outcome of infection control measures were mod-
elled and predictions were made regarding their effect on transmission.
8.1.1 Estimation of basic reproduction ratio and cross-
transmission rates
In Chapters 3, 5 and 7, statistical inference was used to estimate infectivity and the
basic reproduction ratio. Time dependence in the parameters for infectivity was in-
corporated into the study in Chapters 5 and 7, to assess the impact of interventions
that occurred during the data collection period.
The study in Chapter 3 estimated the reproduction ratio to be below unity, and finds
that methicillin-resistant Staphylococcus aureus (MRSA) was endemic to the ward
due to continued importation of new cases through admissions of patients already
colonised. The study in Chapter 5 found that the majority of VRE acquisition in
the institute occurred through cross-transmission on the ward. The study also
concluded that there is some evidence that the infectivity changed just prior to the
hospital outbreak and following infection control interventions.
The study in Chapter 7 estimated the reproduction ratio for SARS-CoV before and
after infection control interventions. It concluded that people were less infectious in
hospital than in the community in Shanxi province, and that the difference increased
after infection control interventions were put in place.
8.1.2 Development of new models
Chapter 6 developed and explored a model that has not been described previously.
The study examined the hypothesis that an environmental reservoir would affect
the impact of infection control interventions and the endemic level of colonisation.
Chapter 7 considered models both for the incubation period of SARS and infectivity
profile over time that gave insight into the transmission dynamic of SARS.
8.1 What has been achieved? 197
8.1.3 Using of models to inform health policy
Chapter 3 examined the predicted impact of a number of infection control interven-
tions. Hand hygiene was predicted to be the most effective intervention. Increasing
staff/patient ratio was predicted to increase MRSA transmission if no cohorting took
place, a finding that contradicts a number of other studies. The findings suggest that
caution should be taken when a ward implements a policy to increase staffing lev-
els. Increasing the number of staff in the setting of cohorting is predicted to reduce
transmission. This study also demonstrates that stochastic model predictions are of-
ten different from deterministic model predictions on the scale of the hospital ward
setting. The changes in ward size leading to differences in transmission cannot be
predicted using deterministic models. Some of the model predictions are not intu-
itive, for example, that very small increments in hand hygiene lead to large reduc-
tions in transmission, and that increasing staff could lead to increased transmission.
Chapter 6 suggests that an environmental reservoir should be considered for
pathogens known to survive in the environment as such a reservoir is predicted to
reduce the efficacy of many infection control interventions.
8.1.4 Methodological framework for future studies
Methods that allowed for serial dependence in infection control data are used
throughout this thesis, namely structured models that allow for changes in coloni-
sation or infection pressure. Censored transmission data are accounted for by
inferring transmission times using latent variables in a Bayesian framework. Like-
lihood estimates are based on piecewise constant hazard formulae. Monte-Carlo
Markov chain integration is used to simplify the intractable integrals that result from
the latent variable models.
The study in Chapter 5 employed a method that could be applied to simple serial
surveillance data with no information on event histories. A hidden Markov model
is used in which the transition component is based on a structured Susceptible-
Infectious (SI) model. By following this methodology, we are able to estimate the
transmission characteristics of VRE without assuming full or immediate detection
of transmission events. This model could be applied to a number of datasets in the
future. Serial surveillance is a common way of measuring the status of hospitals
with regard to nosocomial pathogen containment. These datasets are, therefore,
available for analysis in many hospitals.
198 Chapter 8. Conclusions and suggestions for future work
Models developed in this thesis can form the foundation of future statistical mod-
els of nosocomial transmission. Models of pathogens need to consider both cross-
transmission and independent sources of colonisation. Such a model could be in-
corporated into conventional statistical approaches using a Cox proportional haz-
ards model with colonisation pressure included in the model as a time dependent
covariate, for example. The possibility of an environmental reservoir could be in-
corporated into future analyses of transmission of agents such as Acinetobacter spp.,
other Gram negative bacteria and norovirus.
Chapter 7 uses a Bayesian framework to estimate incubation period and infectious-
ness of SARS in mainland China. There are many emerging threats including the
H5N1 strain of influenza to which similar methodology could be applied.
8.1.5 Model comparison
Several different models are compared using the Deviance Information Criterion
(DIC) in Chapter 5. The model that suggested VRE colonisation arose both from
cross-transmission and sporadically was superior to the models that included only
one of these. Additionally, comparison of models with a time dependent cross-
transmission parameter suggested there is some evidence that the transmission
changed at the time of the interventions. Chapter 7 compares different individual
infectivity profile models for SARS using the Akaike Information Criterion (AIC).
8.1.6 Model diagnostics
The study described in Chapters 3 and 5 used parametric bootstrap technique to test
the model. The data were simulated using estimated model parameters and the pre-
cision with which the model was able to estimate parameters was measured. Chap-
ter 5 also used genotyping data as an external comparison with model results. The
study described in Chapter 7 compared the individual infectivity profile estimated by
the study with virological data, finding a close relationship with this external source
of information.
8.2 Limitations of the approach adopted in this
thesis and opportunities for extensions
Chapter 3
Chapter 3 used a four compartment model to quantify the transmission of MRSA in
the hospital intensive care unit. The model can readily be adapted to incorporate
8.2 Limitations and opportunities for extensions 199
covariates in future studies. Time-dependence could be incorporated into parame-
ters; for example, in the context of a planned interrupted time series studies with the
aim of investigating the impact of infection control interventions.
A major draw-back of the model described in Chapter 3 is that it assumed only one
mode of transmission of contagion, namely, via the hands of healthcare workers.
There was no exploration of alternative models. Additionally, the inclusion of four
compartments led to a number of parameters, to which the model outcomes are
often highly sensitive, as explored in Chapter 3. A more parsimonious model con-
sisting of only two compartments, was developed and presented in Chapter 5. The
study in Chapter 3 also assumes perfect swab sensitivity. This is addressed in Chap-
ter 5.
Chapter 5
The model in Chapter 5 investigated two possible sources of vancomycin-resistant
enterococci (VRE); acquisition arising from cross-transmission in the hospital en-
vironment and acquisition which is sporadic. Other possibilities could be consid-
ered. For example, one could model a hyper-endemic period that might arise from a
point-source outbreak, by including a parameter and a time-dependent indicator.
The model described in Chapter 5 used serial prevalence data. Often, however, lon-
gitudinal datasets involve incidence data. An obvious extension of this model is to
use incidence data in a HMM framework. One could use an approximate relation-
ship between underlying hidden state (prevalence of colonised patients) and obser-
vations (incidence of detection/infection), as was done in the study by Cooper and
Lipsitch (2004). An alternative method would be to derive a direct relationship be-
tween the incidence of transmission and the colonisation pressure over a time in-
terval. Colonisation pressure over time could be estimated by integrating across the
hidden states.
Chapter 5 assumes that there was homogenous mixing of patients and staff and that
the rate of cross-transmission was density dependent (Reed-Frost assumption). Al-
ternative models, based on the Greenwood assumption (Becker, 1989), for example,
could be explored. Here, the maximum risk of cross-transmission is achieved by a
single colonised patient (a saturation effect), with no further risk as more colonised
patients are in the ward. Larger databases could give information on networks or
mixing across wards and relax the assumption of homogenous mixing.
A weak prior probability distribution was used for the parameter values estimated in
Chapter 5. We could have incorporated the data from the genotype study as a more
informative prior probability distribution. Instead these data were used to validate
the model.
200 Chapter 8. Conclusions and suggestions for future work
There was some evidence of a change in transmissibility in this study. The cause of
this change is unclear, but given the mounting evidence for environmental contam-
ination and its contribution to transmission, a model incorporating an environmen-
tal reservoir needs to be considered. This model was developed in Chapter 6.
Chapter 6
The model presented in Chapter 6 is not based on a data series and has not been
validated against other methods. It is therefore a theoretical model with the aim of
generating hypotheses. Future studies may compare models with an environmental
compartment with those without and use model comparison techniques to estab-
lish the evidence for this reservoir compartment. Alternatively, intervention studies
that quantify environmental contamination, compare environmental strains with
human pathogens and aim to decontaminate the environment could be used to
validate the model.
Chapter 7
This study relaxed the assumption of homogeneity of transmission by sub-dividing
the infectious compartment (early and late, hospitalised and community). However,
within each compartment homogeneity was assumed. While there were no clearly
identified super-spreaders in the Shanxi epidemic, heterogeneity of infectivity was
a major feature of the epidemiology of SARS in Singapore and Hong Kong (Li et al.,
2004). This was not explored in the study as detailed chains of transmission were not
available. Additionally it was assumed that all people were equally susceptible.
This study assumed Gamma distributions for sojourn times. Other distributions
could be considered including the Weibull, lognormal and non-parametric ap-
proaches. The study investigates the time of change from a rising epidemic to a
declining epidemic. The possibility of more than one change point or a gradual
transition could also be explored. Reversible jump Markov chain Monte Carlo would
be a useful tool in determining this. SARS models to date, including the current
study, have assumed zero infectivity during the incubation period. The possibility
of infectivity of SARS cases during the incubation period could be investigated by
extending the Bayesian inference model.
8.3 Future work
The motivation for this work is the high mortality, morbidity and cost of healthcare
associated infection. The reduction of healthcare associated infections is a complex
8.3 Future work 201
process, dependent on human perceptions, motivations and ultimately behaviour.
For example, while hand hygiene is widely acknowledged to be a critical factor in
transmission of diseases in hospitals, hand hygiene compliance continues to be poor
(Whitby et al., 2006). In addition to knowledge; motivation, intervention, measure-
ment and feedback are required to change behaviour.
This thesis has focussed principally on valid measurement of the rate of healthcare
associated transmission of infectious agents, and the manner in which these organ-
isms are acquired and estimating the effect of interventions through measurement
and simulation. Methods described in this thesis could be incorporated into con-
trol charts to deliver real-time response to infection control interventions, providing
greater incentive for behaviour change.
The models developed in Chapters 3, 5 and 7 have application to other emerging
threats such as antibiotic resistant Gram-negative bacteria, some of which are found
in the environment, for example carbapenem resistant Acinetobacter baumanii. The
models in Chapter 7 have application to emerging community infectious diseases
such as H5N1 influenza.
Further adaptation and utilisation of models can be made as technology and knowl-
edge improves. The model in Chapter 3 could be readily adapted to investigate
the effect of patient isolation and early patient detection when technology for
more rapid and sensitive detection of colonised patients is achieved and when the
protective effect of isolation is known. As well as predicting the impact of emerg-
ing infection control initiatives, the models help structure study design. Models
developed in this thesis could be components of economic models for cost-utility
analysis.
This thesis confined itself to available hospital datasets and as such did not consider
issues of control of hospital pathogens at a regional level. Chapter 3 showed that
MRSA persisted in the Intensive Care Ward despite a reproduction ratio within the
ward of well below unity. The cause for this was introduction of MRSA via patients
colonised on admission. Cooper et al. (2004) showed that readmission of patients
from the community with MRSA may lead to an effective reproduction ratio greater
than unity despite a ward reproduction ratio less than unity.
For pathogens that are carried long term and have reached significant levels in the
community, infection control must be conducted by a higher stratum than the hos-
pital ward and models comparing regional control strategies are essential. Network
models and structured community models could be used to predict large scale epi-
demic behaviour at this level.
Multi-resistant bacteria, previously the domain of hospitals, especially intensive care
units, are now seen increasingly in the community. Two examples are the emergence
202 Chapter 8. Conclusions and suggestions for future work
of community acquired MRSA and the spread of VRE in the community in Europe,
linked to the use of avoparcin prior to its removal from the market in 1997 (Ridwan
et al., 2002). Future models of multi-resistant pathogens will need to address the
complex interaction between community, long term healthcare facilities and hospi-
tals.
Bibliography
Becker, N., 1989. Analysis of Infectious Diseases Data. Chapman and Hall/CRC.
Cooper, B., Lipsitch, M., 2004. The analysis of hospital infection data using hiddenMarkov models. Biostatistics 5 (2), 223–37.
Cooper, B. S., Medley, G. F., Stone, S. P., Kibbler, C. C., Cookson, B. D., Roberts, J. A.,Duckworth, G., Lai, R., Ebrahim, S., 2004. Methicillin-resistant Staphylococcus au-reus in hospitals and the community: stealth dynamics and control catastrophes.Proc Natl Acad Sci U S A 101 (27), 10223–8.
Li, Y., Yu, I., Xu, P., Lee, J., Wong, T., Ooi, P., Sleigh, A., 2004. Predicting super spread-ing events during the 2003 severe acute respiratory syndrome epidemics in HongKong and Singapore. Am J Epi 160 (8), 719–728.
Ridwan, B., Mascini, E., van der Reijden, N., Verhoef, J., Bonten, M., 16 March 2002.What action should be taken to prevent spread of vancomycin resistant entero-cocci in European hospitals? Brit Med J 324, 666–668.
Whitby, M., McLaws, M., Ross, M., 2006. Why healthcare workers don’t wash theirhands: a behavioral explanation. Infect Control Hosp Epidemiol. 27 (5), 484–492.