Likelihood-based estimation of dynamic transmissionmodel parameters for seasonal influenza by fitting to

age and season specific ILI data

Michael Waithaka

September 18, 2014

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Outline

1 Introduction

2 Study objective

3 Data

4 Methodology

5 Results

6 Conclusions and Recommendations

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Introduction Mathematical models

Introduction

Mathematical models

! A mathematical model is a description of a system using mathematicalconcepts and language.

! Such models are said to be dynamic if they account for time-dependentchanges in the state of the system.

! They are widely used in the design of infection control programmes.

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Introduction Seasonal influenza

Introduction cont’d

Seasonal influenza

! Seasonal influenza is a contagious respiratory illness transmitted mainlythrough social interactions and strikes every year.

! In Europe, influenza epidemics usually occur between week 40 of thecurrent year and week 20 of the following year.

! Vaccination is the most common and most effective public healthresponse.

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Study objective

Study Objective

! The project aimed at directly estimating the parameters of a dynamictransmission model using likelihood-based estimation methods.

→ This was achieved by fitting the model to age-specificinfluenza-like-illness (ILI) incidence over multiple influenza seasons.

Why is parameter estimation important?

1 There exists considerable uncertainty over the most appropriatevalues for parameters of such models.

2 Projections based on the mathematical models heavily rely on theassumed input parameter values.

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Data

Data

! ILI incidence data from week 40 of year 2003 to week 35 of year 2009.

→ The data had been collected from a network of general practitionersin Belgium.

! Belgian demographic data for year 2009 obtained from Eurostat.

! Daily rates of close contacts >15 minutes estimated by Goeyvaerts et al.(2010) were also used.

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Methodology The dynamic model structure

The dynamic model structure

Figure 1: Age-stratified SEIRS model with vaccination

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Methodology Model parameters

The model parameters

→ Seasonal force of infection considered given by:

λa(t) = Z(t)∑

a′βa,a′Ia′(t),

where

Z(t) = 1 + δ ∗ sin(

2π(t− t0)

365

).

Z(t) reflects the relative change of the basic reproduction number attime t from the average basic reproduction number measured atreference time t0.

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Methodology Estimation of the model parameters

Estimation of the model parameters

Weighted Least Squares Approach

X The epidemiological system was assumed to be described by thedynamic model.

X The observed data were assumed to arise from some deviation ofthe output of this system by observational errors.

4∑j=1

∑i

vaj (wi)(Yaj (wi)− αZaj (wi)

)2

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Methodology Estimation of the model parameters

Estimation of the model parameters

Maximum Likelihood approach

X ML estimation seek the values of the parameters that maximizethe likelihood function.

X Maximizing the likelihood function determines the parametersthat are most likely to produce the observed data.

X The observations were assumed to be negative binomialdistributed.

To account for overdispersion in the data.

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Results Exploratory results

Exploratory results

(a) Daily close contact rates (b) Observed ILI incidence rates for thetotal population

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Results Exploratory results

Exploratory results cont’d

(a) 0− 4 years (b) 5− 14 years

(c) 15− 64 years (d) ≥65 years

Figure 2: Observed ILI incidence rates stratified by age groupsMichael Waithaka September 18, 2014 12 / 19

Results Parameters estimation results

Weighted least squares estimation

SeasonParameters

δ wv = wi α t0 tseed R0

2003 - 2004 0.201 0.440 0.212 Oct 05 Sept 21 5.0022004 - 2005 ′′ ′′ ′′ Sept 17 Sept 14 3.5302005 - 2006 ′′ ′′ ′′ Sept 01 Oct 11 2.9422006 - 2007 ′′ ′′ ′′ Sept 30 Sept 02 4.1192007 - 2008 ′′ ′′ ′′ Sept 07 Nov 15 3.2362008 - 2009 ′′ ′′ ′′ Oct 26 Sept 04 4.707

Table 1: Weighted least squares estimates for the dynamic transmission modelparameters

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Results Parameters estimation results

Weighted least squares estimation

(a) 0− 4 years (b) 5− 14 years

(c) 15− 64 years (d) ≥65 years

(e) Total population

Figure 3: Observed incidence rates & corresponding model-based estimates

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Results Parameters estimation results

Maximum likelihood estimation

SeasonParameters

δ wv = wi α t0 tseed R0

2003 - 2004 0.210 0.439 0.230 Oct 05 Sept 24 4.9432004 - 2005 ′′ ′′ ′′ Sept 13 Sept 27 3.4132005 - 2006 ′′ ′′ ′′ Sept 02 Oct 29 2.9422006 - 2007 ′′ ′′ ′′ Oct 05 Sept 03 4.1192007 - 2008 ′′ ′′ ′′ Sept 11 Dec 05 3.2362008 - 2009 ′′ ′′ ′′ Oct 26 Sept 05 4.737

Table 2: Maximum likelihood estimates for the dynamic transmission modelparameters

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Results Parameters estimation results

Maximum likelihood estimation

(a) 0− 4 years (b) 5− 14 years

(c) 15− 64 years (d) ≥65 years

(e) Total population

Figure 4: Observed incidence rates & corresponding model-based estimates

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Conclusions and Recommendations

Conclusions and Recommendations

ConclusionsSince the parameter estimates obtained using the two approaches donot differ much, the choice between the two estimation methods havetrivial consequences.

Future studyUse of Bayesian approaches such as Markov Chain Monte Carlotechniques.

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Thank you for your attention!

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References

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Accounting for methodological, structural, and parameter uncertainty in decision-analytic models: apractical guide.Medical Decision Making, 31(4):675–692, 2011.

Eurostat.

Population table for Belgium, 2009.Eurostat: Luxembourg, 2010.

N. Goeyvaerts, N. Hens, B. Ogunjimi, M. Aerts, Z. Shkedy, P. van Damme, and P. Beutels.

Estimating infectious disease parameters from data on social contacts and serological status.Journal of the Royal Statistical Society: series C: applied statistics / Royal Statistical Society [London]- ISSN 0035-9254, 59(2):255–277, 2010.

N. Goeyvaerts, L. Willem, K. V. Kerckhove, Y. Vandendijck, G. Hanquet, P. Beutels, and N. Hens.

Estimating dynamic transmission model parameters for seasonal influenza by fitting to age andseason specific Influenza-Like-Illness.Unpublished, 2014.

E. Vynnycky, R. Pitman, R. Siddiqui, N. Gay, and W. J. Edmunds.

Estimating the impact of childhood influenza vaccination programmes in England and Wales.Vaccine, 26(41):5321–5330, September 2008.

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