Infectious disease epidemiology and mathematical models in the … · 2014-10-12 · Infectious...

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uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling

Infectious disease epidemiology

and mathematical models

in the context of Public Health

Hans-Peter Duerr

Kurs Infektionsepidemiologie, Studiengang Public Health, Master of Science, Universitätsklinikum Düsseldorf

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Program Slide Lesson 1 - Infectious diseases - how they emerge and disappear

29 Lesson 2 - SARS 2002/2003: why modeling, and what is a mathematical model?

36 Exercise 1 - Design a mathematical model yourself: the bacterial growth curve

42 Exercise 2 - Homemade solution: solve the model numerically (bacterial growth curve with Excel)

45 Lesson 3 - Deterministic models: SIR-model, basic reproduction number R0

55 Exercise 3 - Be professional: simulate an epidemic with professional algorithms

63 Exercise 4 - Harvest: proportion of susceptibles after an epidemic infection

66 Exercise 5 - Think longterm: the influence of time and demography

69 Lesson 4 - Vaccination: final size of an epidemic, critical vaccination coverage

81 Exercise 6 - Predict: how many newborns to vaccinate? (critical vaccination coverage)

93 Lesson 5 - SIR, add-ons: extending SIR to SEIR, SEIRS, etc

99 Exercise 7 - Design: specific models for specific diseases (Extending the SIR to SEIR, SEIRS, etc.)

101 Lesson 6 - Interventions: pandemic influenza preparedness planning using InfluSim

103 Exercise 8 - Be an intervention planner: control an influenza pandemic (using InfluSim)

109 Lesson 7 - Stochastic Models: from theory to reality, the epidemic as a random event

157 Lesson 8 - "Super-"reality: The role of superspreaders

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Literature selection

• Infectious Disease Epidemiology: Theory and Practice. Neil Graham. Jones and Bartlett Publishers, Inc.

• Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (Wiley

Series in Mathematical and Computational Biology), O. Diekmann and J. A. P. Heesterbeek.

• Epidemic Models: Their Structure and Relation to Data (Publications of the Newton Institute). Denis Mollison

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National online ressources

www.promedmail.org www.destatis.de

www.gbe-bund.de

• CDC: Traveler’s Health:

http://www.cdc.gov/travel/

• Emerging Health Threats Forum

(EHTF)

http://www.eht-forum.org/

• GIDEON: Global Infectious

Disease & Epidemiology Network:

http://www.GIDEONonline.com

http://www.promedmail.org/

http://www.promedmail.org/

http://www.destatis.de/

http://www.destatis.de/

http://www.gbe-bund.de/




http://www.cdc.gov/travel/




http://www.gideononline.com/

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WHO ressources

• The Weekly Epidemiological Record

(WER): www.who.int/wer/en/

• WHO Outbreak News:

http://www.who.int/csr/don/en/

• WHO Weekly Epidemiological

Record - WER

http://www.who.int/wer/

• WHO Report on Infectious Diseases

2000: Overcoming Antimicrobial

Resistance

http://www.who.int/infectious-

disease-report/2000/

http://gamapserver.who.int/GlobalAtlas/home.asp

http://www.who.int/csr/outbreaknetwork/en/

http://www.who.int/wer/en/

http://www.who.int/csr/don/en/

http://www.who.int/wer/

http://www.who.int/infectious-disease-report/2000/









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Infectious diseases in ICD-10

Schlüssel Text

A00-B99 Bestimmte infektiöse und parasitäre Krankheiten

C00-D48 Neubildungen

D50-D89 Krankheiten des Blutes und der blutbildenden Organe sowie

bestimmte Störungen mit Beteiligung des Immunsystems

E00-E90 Endokrine, Ernährungs- und Stoffwechselkrankheiten

F00-F99 Psychische und Verhaltensstörungen

G00-G99 Krankheiten des Nervensystems

H00-H59 Krankheiten des Auges und der Augenanhangsgebilde

H60-H95 Krankheiten des Ohres und des Warzenfortsatzes

I00-I99 Krankheiten des Kreislaufsystems

J00-J99 Krankheiten des Atmungssystems

K00-K93 Krankheiten des Verdauungssystems

L00-L99 Krankheiten der Haut und der Unterhaut

M00-M99 Krankheiten des Muskel-Skelett-Systems und Bindegewebes

N00-N99 Krankheiten des Urogenitalsystems

O00-O99 Schwangerschaft, Geburt und Wochenbett

P00-P96 Zustände, die ihren Ursprung in der Perinatalperiode haben

Q00-Q99 Angeborene Fehlbildungen, Deformitäten und

Chromosomenanomalien

R00-R99 Symptome und abnorme klinische und Laborbefunde, die

anderenorts nicht klassifiziert sind

S00-T98 Verletzungen, Vergiftungen und bestimmte andere Folgen

äußerer Ursachen

V01-Y98 Äußere Ursachen von Morbidität und Mortalität

Z00-Z99 Faktoren, die den Gesundheitszustand beeinflussen und zur

Inanspruchnahme des Gesundheitswesens führen

Schlüssel Text

A00-A09 Infektiöse Darmkrankheiten

A15-A19 Tuberkulose

A20-A28 Bestimmte bakterielle Zoonosen

A30-A49 Sonstige bakterielle Krankheiten

A50-A64 Infektionen, die vorwiegend durch

Geschlechtsverkehr übertragen werden

A65-A69 Sonstige Spirochätenkrankheiten

A70-A74 Sonstige Krankheiten durch Chlamydien

A75-A79 Rickettsiosen

A80-A89 Virusinfektionen des Zentralnervensystems

A90-A99 Durch Arthropoden übertragene Viruskrankheiten

und virale hämorrhagische Fieber

B00-B09 Virusinfektionen, die durch Haut- und

Schleimhautläsionen gekennzeichnet sind

B15-B19 Virushepatitis

B20-B24 HIV-Krankheit [Humane Immundefizienz-

Viruskrankheit]

B25-B34 Sonstige Viruskrankheiten

B35-B49 Mykosen

B50-B64 Protozoenkrankheiten

B65-B83 Helminthosen

B85-B89 Pedikulose [Läusebefall], Akarinose [Milbenbefall]

und sonstiger Parasitenbefall der Haut

B90-B94 Folgezustände von infektiösen und parasitären

Krankheiten

B95-B97 Bakterien, Viren und sonstige Infektionserreger als

Ursache von Krankheiten

B99 Sonstige Infektionskrankheiten

ICD-10 Infectious diseases

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Blank page for documentation

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Slides Topic Lesson

Introduction: Infectious diseases - how they emerge and disappear

1

SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)

2

Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case

3

Vaccination: final size of an epidemic, critical vaccination coverage

4

SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5

Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)

6

Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not

7

The role of superspreaders 8

Program

Le

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Infectious diseases:

how they emerge and disappear

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Leishmaniasis

Climate change

Measles

Population growth

SARS

Mobility

Globalisation

Animals

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http://vcolizza.googlepages.com

SARS 2002 / 2003 N

um

be

r in

fecte

d

0

1000

2000

3000

4000

20. March 4. April 19. April 4. May 19. May 3. June 18. June 3. July

China

Hong Kong Taiwan Remaining countries

2003

16:17

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Where did SARS come from?

Literatur: Li W, et al: Bats Are Natural Reservoirs of SARS-Like Coronaviruses. Science 2005.

Hotel Metropole,

Hong Kong

http://www.msnbc.msn.com/id/12371160/

June

2003

March

2003

2005

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Infectious diseases: a matter of opportunities

http://www.msnbc.msn.com/id/12371160/

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Opportunities for emerging infectious diseases

Mobility

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How did we get rid of SARS?

Am

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lock E

Culling of Zibet-cats

Isolation of diseased people, quarantine

Travel restrictions, control

"stay-at-home"-policy,

face masks, etc.

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Diseases come and go

How they come How they go

SARS Bat → Cat → Human Control

AIDS Monkey → Human -

Plague Wild animals → Rodents

→ Fleas → Human

Humans die

(animals stay)

Ebola Bat → Ape, Gorilla → Human Humans die

Smallpox Cattle, Monkey, Cat, Mouse...

→ Human

Eradication by

vaccination

Measles ? Eradication not

yet achieved

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Why are infectious disease there at all?

Measles: Children who have been infected attain life-long immunity.

0 10 20 30 40 50 60 70 Years

susceptible infected immune - - Natural history of disease

another 10 days later Look at this family 10 days later

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Why are infectious disease there?

A kindergarden

10 days later

another 10

days later

some weeks

later

Measles always eliminate themselves

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Computer simulations make us understand

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When did measles attack humans?

Small populations Large populations

…probably in ancient Mesopotamian civilizations 2000–

4000 years BC, when the critical population size for

measeles (about 300 000) has been exceeded. Lit.: Black FL 1966. Measles endemicity in insular populations: critical

community size and its evolutionary implication. J. Theor. Biol. 11, 207–211.

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Causality: Population density

Weltbevölkerung

0

1000

2000

3000

4000

5000

6000

7000

-1000 -500 0 500 1000 1500 2000

Jahr

Mio

. M

en

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Mobility

0

1000

2000

3000

4000

5000

6000

7000

-1000 -500 0 500 1000 1500 2000

Jahr

Mio

. M

enschen

Population growth

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Culex

Phlebotomus

(Sandfly)

Emerging diseases: example Leishmaniasis

Cutaneous Leishmaniasis

Sandflies

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Causality: Climate change D

evia

nce

fro

m a

ver

age

1961

-19

90

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Mobility

Climate change

0

1000

2000

3000

4000

5000

6000

7000

-1000 -500 0 500 1000 1500 2000

Jahr

Mio

. M

enschen

Population growth

Industrialisation

→ CO2 →

Global Warming

Poverty

Mig

ration

Limited

Resources,

Water, Food,

Hygiene

Ecological system

Globalisation

Opportunities

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Slides Topic Lesson


1


2


3


4



6


7


Program

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Am

oy G

ard

en

s B

lock E

Country Cases Deaths Case fatality [%]

China 5327 349 7

Hong Kong 1755 296 17

Taiwan 665 180 27

Kanada 251 41 16

Singapur 238 33 14

Vietnam 63 5 8

USA 33 0 0

Philippinen 14 2 14

Deutschland 9 0 0

8355 906 10.8%

For comparison Malaria: ~300 Mio. Cases / Year,

~1 Mio Deaths / Year (predominantly children)

Example SARS 2002/2003

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Am

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SARS, initial spread

CDC, taken from http://en.wikipedia.org H

ote

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etr

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Spread from Hotel Metropole

Ke

y: B

asic

re

pro

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Problem Global Networks

e.g. between

Chicago and New

York 25.000

Passengers per day

Passengers per day

L. Hufnagel et al. 2004, PNAS 101: 15124-9 Ke

y w

ord

s:

Mo

de

l, N

etw

ork

s

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Prediction

Fig. 2. Global spread of SARS. (A) Geographical

representation of the global spreading of probable

SARS cases on May 30, 2003, as reported by the WHO

and Centers for Disease Control and Prevention. The

first cases of SARS emerged in mid-November 2002 in

Guangdong Province, China (17). The disease was

then carried to Hong Kong on the February 21, 2003,

and began spreading around the world along

international air travel routes, because tourists and the

medical doctors who treated the early cases traveled

internationally. As the disease moved out of southern

China, the first hot zones of SARS were Hong Kong,

Singapore, Hanoi (Vietnam), and Toronto (Canada), but

soon cases in Taiwan, Thailand, the U.S., Europe, and

elsewhere were reported. (B) Geographical

representation of the results of our simulations 90 days

after an initial infection in Hong Kong, The simulation

corresponds to the real SARS infection at the end of

May 2003. Because our simulations cannot describe the

infection in China, where the disease started in

November 2002, we used the WHO data for China.

Ke

y w

ord

: M

od

el p

red

ictio

n

L. Hufnagel et al. 2004, PNAS 101: 15124-9

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What is a mathematical Model?

Model: tBtB 20

Growth from one generation to the next: 12 ii BB

Logari

thm

ic

Lin

ear

Problems of this approach: - only valid for initial growth

- too simple for describing complex processes

Example bacterial groth: bacteria divide 2 times per hour.

Duration between divisions D = 0.5 hours.

Rate of division = 2 per hour D/1

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Exercise 1:

preliminary considerations

E

xerc

ise 1

Draw into each graph the bacterial growth curve you would expect if

• … the before-

mentioned changes

occur simultaneously

• … the generation

time of the bacterium

was not 0.5h but 1h

• … the culture was

started with 10000

bacteria, rather than

1 bacteria

Target: intuitive prediction

of quantitative relations

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Exercise 1: iterative solution

of the model in Excel

Exerc

ise

1 (

File

"0

0_

ba

cte

ria

lGro

wth

.xls

", s

heet "g

en

era

tio

n t

ime")

=A2*tGen 1. Parameter:

"Bakt0"

2. Parameter: "tGen"

3. Parameter "multFaktor"

=Bakt0

=A2*tGen =C2*multFaktor

Complete cells B2 to C32

in file "00_Bakterienwachstum.xls", spreadsheet "generation time"

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Exercise 1

results V

erify

the p

relim

inary

consid

era

tions

of th

e p

revio

us s

lide

Exerc

ise

1 (

File

"0

0_

ba

cte

ria

lGro

wth

.xls

", s

heet "g

en

era

tio

n t

ime")

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Design models with differential equations

tB

dt

tdB~

"The speed by which the

total number of bacteria

B changes over time t…"

"…is proportional

to the individual

rate of division

…"

"…and proportional

to the number of

bacteria reproducing

at time t "

=

teconsttB ~

Total number of bacteria at time t

Example bacterial growth: bacteria divide 2 times per hour.

Duration between divisions D = 0.5 hours.

Rate of division = 2 per hour D/1

Integration Derivative

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Differential equations offer more…

Lin

ear

K

tB1

Previously: the bacterial culture grows indefinitely (unrealistic in a finite world)

Now: the culture cannot exceed a certain capacity K (realistic: test tube)

"The growth rate

approaches zero when

the bacterial culture

approaches the value of

the capacity (B(t)=K)."

teBKB

KBtB

~

00

0

)(

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Summary

• A mathematical Model is just a mathematical way to describe a process.

• Simple processes may be intuitively described "by hand"

• Differential equations are a useful tool to describe complex processes.

• Differential equations allow for describing dynamic processes by means of

interpretable parameters.

Other ways to

model

processes: (Galileos planet model.

In: Universal Dictionary

of Arts, Sciences, and

Literature, Plates, Vol.

IV (London 1820) )

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Exercise 2:


Assume A) a bacterium which, under optimal conditions, reproduces 2 times per

hour (per capita-reproduction rate=2/h) and

B) a volume which can harbour at maximum 1,000,000 bacteria.

• Fill in the numbers

for the upper and

lower bounds of

each axis into the

white boxes

• Draw a qualitative

curve for the per

capita-reproduction

rate and the num-

ber of bacteria over

time, B(t). Where is

the inflection point

of B(t)?

E

xerc

ise 2

Nu

mbe

r of bacte

ria

B(t

)

Per

ca

pita-r

ep

rod

uctio

n r

ate

Aim: intuitive prediction

of a dynamic process

Time

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Exercise 2:

spreadsheet solution

Exerc

ise

2 (

"00_ba

cte

rialG

row

th.x

ls",

sheet "d

iffe

ren

ce

equ

ation

s")

1. Parameter: "deltaT"

2. Parameter: "Bak0"

3. Parameter "lambda"

=lambda

=C2+(A3-A2)*B3

=lambda*C2*(1-C2/kapazitaet)

Complete cells A3 to E1000

in file "00_bacterialGrowth.xls", spreadsheet "difference equations")

4. Parameter "kapazitaet"

=0

=A2+deltaT/60

=Bak0

=Bak0*kapazitaet/

(Bak0+(kapazitaet-Bak0)

*EXP(-lambda*A2))

=B3/C2

Aim: programming an

iterative solution

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Exercise 2: results

Ve

rify

the p

relim

inary

consid

era

tions

of th

e p

revio

us s

lide

Exerc

ise

2 (

"00_ba

cte

rialG

row

th.x

ls",

sheet "d

iffe

ren

ce

equ

ation

s")

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tochastic

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Slides Topic Lesson


1


2


3


4



6


7


Program

Le

sso

n 3

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Polio virus type 1

Extensions of

the SIR-model:

SIRS

SEIR

SEIRS

R S I

Susceptible Immune Infectious Common way of

representing a model:

Compartiments

SIR-Model

& Transititions

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Information needed

• Durations: latent and infectious period

• Rates: contact rate(s)

• Probabilities: P(infection | transmission)

• Demography: birth and death rate,

age structure of the population

• Disease: Proportion of inapparent infections

• ....

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Birth of (susceptible) individuals

dS(t) / dt = m

Dynamic description: Birth

S Proportion susceptibles m per capita birth rate

R

I

S

=1

[ S(t)+I(t)+R(t) ]

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8: S


new Infections

dS(t) / dt = m - bc I(t) S(t)

dI(t) / dt = bc I(t) S(t)

Dynamic description: Infection

S Proportion susceptible

I Proportion infectious

R Proportion immune

c P (infection | contact) m Per capita birth rate

b Contact rate

R

I

S

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Mass action law

The probability of encounterings between

susceptible and infectious individuals

depends on:

• the contact rate b ("Temperature")

• the ratio Susceptible : Infectious

Susceptible Infectious

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Proportion susceptible

P

S*S

2(S*

I)I*I

Sum

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new infections


dI(t) / dt = bc I(t) S(t)

Dynamic description: Infection

R

I

S



R Proportion immune


b Contact rate

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Loss of infectiousity


dI(t) / dt = bc I(t) S(t) - g I(t)

dR(t) / dt = g I(t)

Dynamic description: Loss of infection

R

I

S



R Proportion immune


b Contact rate

g rate of loss of infectiousity

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Mortality

dS(t) / dt = m - bc I(t) S(t) - m S(t)

dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)

dR(t) / dt = g I(t) - m R(t)

Dynamic description: Mortality

R

I

S



R Proportion immune


b Contact rate


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• Initialisation – choose parameter values for b, c, g and m

– choose initial values for S(0), I(0), and R(0)

• First iteration (time = 0) – compute for a short time step D the changes

dS(0)/dt, dI(0)/dt and dR(0)/dt

– extrapolate changes to S(0+D), I(0+D) and R(0+D)

• Following iterations (time = t) 1. compute for a short time step D the changes

dS(t)/dt, dI(t)/dt and dR(t)/dt

2. extrapolate changes to S(t+D), I(t+D) and R(t+D)

t=t+D, goto 1

Numeric solution of the model

A r

ead

y-t

o-u

se

so

ftw

are

of

the S

IR m

ode

l is

ava

ilable

fro

m

ww

w.u

ni-

tue

bin

gen.d

e/m

odelin

g/M

od_P

ub

_S

oft

ware

_S

IR_e

n.h

tml

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7: S

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8: S


Exercise 3:


The SIR model, defined as, … produces qualitatively a graph like



dR(t) / dt = g I(t) - m R(t) Zeit

Zahl der

Infizie

rten

the contact rate between

people increases? (b ↑)

Time

No.

of

infe

cto

us p

eople

Draw your qualitative prediction into the graph on how the course of the

epidemic would change (higher, faster, slower, etc) if

patients recover more

rapidly? (g ↑)

Time

No. of

infe

cto

us p

eople

b and g increase at the

same time

Time

No.

of

infe

cto

us p

eople

Aim: understanding the role

of the parameters in the SIR

Exerc

ise 3

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Exercise 3: solving differen-

tial equations numerically

Exerc

ise 3

(F

ile "10_SIR.txt")

• Aim: quantitative

epidemiology of

infectious diseases –

learning by doing

• Complete file "10_SIR.txt" with the

equations of the SIR-

model (save your work),

and specify the initial

values (INIT S, I, R).

• Copy the text into the

program editor of

Berkeley-Madonna

• Click "Run"

Program here, using

the parameters listed

beyond

{--- Parameters ---}

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Exercise 3: M

ake y

our

slid

ers

in M

enu

Parameters|Define Sliders...

Aim: Performing a

sensitivity analysis

… and verify your preliminary considerations of the previous slide

Exerc

ise 3

(F

ile "10_SIR.txt")

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R0: Basic reproduction number

• Average number of secondary infections

which one infectious individual would cause

in a fully susceptible population

Definition: R0 = b c D

• R0>1: Infection can persist;

an endemic state ist possible

• R0<1: Infection cannot persist; goes extinct

D = 1 / (gm average duration of the infectious period

bc Number of (sufficiently close) contacts per unit of time

Wichtig! Wichtig!

Wichtig!

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R0 for some infectious diseases

Disease

Average

age at

infection

[years]

R0

Critical

vaccination

coverage

pcrit [%]

Measles 5.0 15.6 94

Pertussis 4.5 17.5 94

Mumps 7.0 11.5 91

Rubella 10.2 7.2 86

Polio 10.4 6.1 84

Diphteria 10.4 6.1 84

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bc = 0,5/day, g = 0,1/day, m = 0/day R0 = 5

SIR Model; without births and deaths

Epidemic

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bc = 0,2/Tag, g = 0,1/Tag, m = 0/Tag R0 = 2

SIR Model; without births and deaths

Epidemic

S

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At the end of an Epidemic...

Infectious

... susceptible individuals may remain

Susceptible Resistent

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8: S


Exercise 4: reparameteri-

zation of the SIR model

Exerc

ise 4

(F

ile "11_SIRreparameterized.txt")

• Complete file "11_SIRrepara-meterized.txt" with the parameters at the bottom, and

define how b is derived from R0 and g (save your work as *.txt)

• Copy the text into the program editor of Berkeley-Madonna

• Define sliders as before, in menu Parameters|Define Sliders...

• Click "Run", and evaluate the role of "R0", in particular investigate the the proportion of susceptibles after an epidemic

dependent on R0

Aim: Understanding the

basic reproduction number

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Exercise 4: the proportion of

susceptibles after an epidemic

Exerc

ise 4

(F


Aim: Automation of a

sensitivity analysis

Step 1: check position and upper limits of sliders

(in particular: lifeExpectYears=10000 and

STOPTIME=1000)

Step 2: open menu Parameters|Parameter Plot…

and (try to) tell the software: "please show me on Y the final value of S

for different values of R0 on X"

Step 3: click Run and view results

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0

0.2

0.4

0.6

0.8

1

1 2 3 4 5

Basic reproduction number R0

Pro

port

ion s

usceptib

le

- log (S) = R0 (1 - S)

Proportion S susceptible at the end of the epidemic

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Exercise 5:

endemic infection

Exerc

ise 5

(F


• For R0=15 (Measles-like) simulate an ex-tended period of time by increasing STOP-

TIME from 100 to 5000 days (=13.7 years)

• Simulate a population with a lower life ex-

pectancy (developing countries) by decrea-sing lifeExpectYears from 50 to 30 years.


influence of demography

Parameter Minimum Use Maximum

STOPTIME 0 100 5000

DT 0 0.1 1

DTOUT 0 1 10

iniInfected 0 0.0001 1

lifeExpectYears 0 50 100

durationInfected 0 10 20

R0 0 15 20

• Make sure that slider settings in file "11_SIRreparameterized.mmd"

are as follows

• What is the reason for recurrent epidemics?

• Why reduces the time between epidemics?

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0

0.2

0.4

0.6

0.8

1

0 1000 2000 3000

Time [days]

Pro

po

rtio

ns

suszeptible

infizierte

immune

SIR Modell with demography

bc = 0,5/day, g = 0,1day, m = 0,0005/day R0 = 5

Endemic case

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Summary

• Neglecting births and deaths, – we model an epidemic scenario;

– after the epidemic, a proportion of susceptibles,

which depends on R0, remains

• Considering births and deaths, – we model an endemic scenario

– the model variables (S,I,R, ...) approach the

endemic state (if R0 > 1);

– the equilibirum prevalence depends on R0.

logKurve

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Slides Topic Lesson


1


2


3

Vaccination: complications, contra-intuitive effects, final size of an epidemic, critical vaccination coverage

4



6


7


Program

Le

sso

n 4

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Can vaccinations do harm to the population? (1)

• Typical childhood disease, relatively trivial infection, disease

is often mild or proceeds asymptomatically. Duration of

disease: 1-3 days, fever rarely >38°C. R0: 5-7.

• Epidemiology: worldwide. Humans are the only host, no

animal reservoir. Before vaccination was introduced in 1969,

widespread outbreaks usually occurred every 4-8 years,

mostly affecting children in the 5-9 year old age group.

• Congenital Rubella Syndrome (CRS) if the mother is

infected within the first 20 weeks of pregnancy.

• Vaccination. Two live attenuated virus vaccines can prevent

adult disease. Use in prepubertile females not reliable.

Vaccine is given as part of the MMR vaccine. Schedule: 1st

dose at 12 to 18 months, 2nd dose at 36 months. Pregnant

women are usually tested for immunity to rubella early on.

Women found to be susceptible are not vaccinated until after

the baby is born because the vaccine contains live virus.

• Problem: not eradicated, reintroduction from endemic

countries always possible.

Example & clinical background: Rubelly ("German measles", "Röteln")

© 2

009 N

ucle

us M

edic

al M

edia

, In

c

Countries using Rubella vaccine in National

Immunization Schedule, 2009 (WHO)

Source: NIH

Microcephaly

PDA

Cataracts

Question: Does vaccination always or proportionally decrease CRS?

CRS

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Example Rubella vaccination

Simplifications:

Lit.: K. Dietz & D. Schenzle, 1985: Epidemiologische Auswirkungen von Schutzimpfungen gegen Masern, Mumps und Röteln. In: Schutzimpfungen:

Notwendigkeit, Wirkung/Nebenwirkungen, Impfpolitik. Bericht von der Tagung des Deutschen Grünen Kreuzes, in Verbindung mit der Deutschen

Vereinigung zur Bekämpfung der Viruskrankheiten e.V. Herausgegeben von H. Spiess, Medizinische Verlagsgesellschaft mbH,Marburg/Lahn 1985.

• All mothers give birth to a child at the age of A = 25 years

• Annual Risk of infection with Rubella: R= 0.1 = 10% per year

Incidence of CRS )1()1( pRA

CRS epRI

Vaccination can increase the

risk for higher age groups

0.1 per year ∙ 0.75 ∙ 0.115

= 0.0086 per year

0.1 per year ∙ 0.082

= 0.0082 per year

Incidence of CRS

(ICRS)

e(-25 years∙0.1 per year ∙0.75)

≈ 0.153 (among 75% not vaccinated)

≙0.153 ∙ 0.75 = 0.115 (11.5%) (among all mothers)

e(-25 years∙0.1 per year)

≈ 0.082 (8.2%)

Proportion of

susceptible

mothers at the age

of 25 years

With 25% of newborns

vaccinated (p=0.25)

Without

vaccination

R is reduced

because vacci-

nation has

reduced the

overall rate of

transmission

0

0.5

1

0 10 20 30 40 Age

agee 1.0

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Example Rubella vaccination

Relative Incidence of CRS

)01(

)1(

)01(

)1(

RA

pRA

CRSeR

epRI

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1

Proportion p vaccinated at birth

Rela

tive I

ncid

ence

ICR

S

0

0.02

0.04

0.06

0.08

0.1

0 1 2 3 4 5 6 7 8 9 10

Age [years]

Pro

po

rtio

n a

t risk

Novaccination

25% ofnewborns arevaccinated

Reason for an increased incidence

under (low) vaccination coverage:

Hazard rate is lower

→ hazard increases at higher age.

Oversimplifications so far: R0, Age- and contact-

structure not considered, assumption of constant risk

over time (implies exponential distribtution),

immunization is complete and homgenious, no

immigration, maternal immunity after birth, etc.

Vaccination can increase the

risk for higher age groups

with A =25 years, R =0.1 per year

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Example Rubella vaccination Seperate and

average effects

1) Unvaccinated women

have an increased risk,

(except shortly before

elimination) because the

overall transmission is

reduced and hence, the

average age of infection

will increase.

2) Vaccinated women have

always a low risk,

dependent on vaccine

efficacy

3) The overall effect for the

population is a weighted

average

(Risk∙Proportion)unvaccinated

+ (Risk∙Proportion)vaccinated

Elim

ination

From: Dietz K & Eichner M. The effect of heterogeneous interventions on the spread of infectious diseases. Proceedings 21. Tagung Internationalen Biometrischen Gesellschaft, 2002.

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8: S


Rubella vaccination in Greece: BMJ 1999

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7: S

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8: S


Can vaccinations do harm to the population? (2)

Example & clinical background: Measles

Age-dependent risk factors

after measles infection

Otitis

Pneumonia

Orchitis

Embryopathy




Average age of measles infection

dependent on vaccination coverage

Vaccination reduces the overall risk of infection and hence, increases the

average age of infection. More children may suffer from a "late" infection

if complications after measles infection occur more likely at higher age.

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Example Measles vaccination

Example & clinical background: Measles

Age-dependent risk factors

after measles infection

Otitis

Pneumonia

Orchitis

Embryopathy




Vaccination reduces the overall risk of infection and hence, increases the

average age of infection. More children could suffer from a "late" infection

if complications after measles infection occur more likely at higher age.

Expected changes in the relative

incidence of complications

dependent on vaccination coverage

Orchitis

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Examples Mumps & Rubella vaccination

Mumps Rubella

Congenital

Rubella

Syndrome

(CRS)

Current status of CRS:

Morbidity and Mortality Weekly Report (MMWR)

October 15, 2010 / 59(40);1307-1310. Progress

Toward Control of Rubella and Prevention of

Congenital Rubella Syndrome --- Worldwide, 2009

"...concern that the risk for CRS

might increase if high vacci-

nation coverage could not be

achieved. Low coverage might

result in decreased virus

circulation, which could increase

the average age of rubella

infection for females from child-

hood to the childbearing years."

"...Globally, a total of 165 CRS

cases were reported from 123

member states during 2009,

compared with 157 CRS cases

reported from 75 member states

during 2000" ... "CRS, which

affects an estimated 110,000

infants each year in

developing countries"

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Summary

• Vaccination not necessarily yield benefits

proportional to vaccination coverage. At least

two complications need to be adressed:

• Vaccination decreases the overall virus

circulation and thus can increase the average

age of infection.

• Age-dependently increasing complications of

infection can bring about situations where low

vaccination coverages produce a higher number

of cases (clinical, fatal, etc.).

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Model without vaccination



dR(t) / dt = g I(t) - m R(t)

Dynamic description: SIR without vaccination

R

I

S



R Proportion immune


b Contact rate


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A proportion of newborns will be vaccinated

dS(t) / dt = m ........ - bc I(t) S(t) - m S(t)


dR(t) / dt = ....... + g I(t) - m R(t)

Dynamic description: SIR with vaccination

R

I

S 1-p

m p



R Proportion immune

c P (infection | contact)

p Proportion vaccinated

m Per capita birth rate

b Contact rate


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Exercise 6: Vaccination E

xerc

ise 6

(F



concept of thresholds

• Define slider for "p" in

Menu

Parameters|Define

Sliders... and choose

slider settings as shown in

the screenshot to the right

• Implement parameter "p" for the proportion of vaccinated newborns (see

previous slide) in the equations of file "11_SIRreparameterized.txt"

file and save it as "12_SIRvaccination.txt"

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xerc

ise 6

(F

ile "12_SIRvaccination.txt")



• Technical remark:

1.

2.

For purposes of better inspectation, we change in the

output window axis settings for compartment I to "Auto" –

see below and ask the lecturer.

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xerc

ise 6

(F

ile "12_SIRvaccination.txt")



• For R0=15, increase p up to a value when

there is no epidemic

anymore. This is the

ciritical vaccination

coverage p*. Repeat

the procedure for

R0=10, 5 and 2, and

plot your results in

the graph to the right.

p*

0 5 10 15

0

0.2

0.4

0.6

0.8

1

R0

What is the critical vaccination coverage when the basic repro-duction number tends to values of R0→1

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Endemic equilibrium

no change of model variables in the endemic equilibrium

0 = m 1-p - bc I(t) S(t) - m S(t)

0 = bc I(t) S(t) - g I(t) - m I(t)

0 = m p + g I(t) - m R(t)

R

I

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R Proportion immune




b Contact rate


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Endemic equilibrium

no change of model variables in the endemic equilibrium

0 = m 1-p - bc I S - m S

0 = bc I S - g I - m I

0 = m p + g I - m R

I=...

S=...

R=... R

I

S



R Proportion immune




b Contact rate


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Endemic equilibrium

S = (g + m) / (bc)

I = (1 - 1/R0 - p) m / (g + m)

R = 1 - S - I

R0 = bc / (g+m)

= 1 / R0

Estimate R0 from the

proportion of

susceptibles in the

endemic equilibrium

R

I

S



R Proportion immune




b Contact rate


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Critical vaccination coverage

Parameters of the right hand side are known, except p

I = (1 - 1/R0 - p) m / (g + m)

R

I

S


Basic reproduction number:

R0 = bc / (g+m)

p Proportion vaccinated m Per capita birth rate = death rate


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Parameters of the right hand side are known, except p

0 = (1 - 1/R0 - pcrit) m / (g + m)

pcrit = 1 - 1 / R0

To eliminate a disease, it is not necessary to

vaccinate the whole population

R

I

S

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0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 16 18 20

Basic reproduction number R0

Pro

port

ion v

acc

inate

d

Elimination

Persistence


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Summary (2)

• The proportion of susceptibles in the endemic

equilibrium does not depend on the proportion p of

vaccinated children

• Transmission stops if p pcrit

• The critical vaccination coverage is

• The model can be used for sensitivity analyses into the

effects of different vaccination strategies:

- What is the critical vaccination coverage?

- How does vaccination impact on the prevalence and incidence of the

infection?

- what is the best vaccination strategy (e. g. ring vaccination vs. mass

vaccination)?

pcrit = 1 - 1/R0

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m

g

R0

pcrit

bc

Basic reproduction number 1 / R0 is the endemic prevalence of susceptibles

Per capita birth rate = death rate 1 / m is the life expectancy

Loss-of-infection rate 1 / (gm) is the average duration of the infectious period

Critical vaccination coverage pcrit = 1 - 1 / R0

Effective contact rate bc = R0 (gm)

Estimation of model parameters

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Slides Topic Lesson


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2


3


4



6


7


Program

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Extensions: from SIR to SEIR

SIR

dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)


dR(t) / dt = m p + g I(t) - m R(t)

R

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R Proportion immune




b Contact rate


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Infections with latent stage


Incorporate latent stage


dR(t) / dt = m p + g I(t) - m R(t)

R

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R Proportion immune




b Contact rate


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SEIR


dE(t) / dt = . . . . .


dR(t) / dt = m p + g I(t) - m R(t)

R

I

S

E


E Proportion latent


R Proportion immune




b Contact rate


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SEIR


dE(t) / dt = bc I(t) S(t) - . . . . .

dI(t) / dt = . . . . . - g I(t) - m I(t)

dR(t) / dt = m p + g I(t) - m R(t)

R

I

S

E


E Proportion latent


R Proportion immune




b Contact rate


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SEIR


dE(t) / dt = bc I(t) S(t) - d E(t) - m E(t)

dI(t) / dt = d E(t) - g I(t) - m I(t)

dR(t) / dt = m p + g I(t) - m R(t)

R

I

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E


E Proportion latent


R Proportion immune




b Contact rate


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Exercise 7: SEIR model E

xerc

ise 7

(F

ile "13_SEIRvaccination.txt")

Aim: customize an SIR

model yourself

• Complete file "13_SEIRvacci-nation.txt" with the parameters for the latent stage.

• Copy the text into the program editor of Berkeley-Madonna

• Define sliders as before, in menu Parameters|Define Sliders...

•Run…

Technical skills: What is the value for d that reduces an SEIR model back to an SIR model?

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Slides Topic Lesson


1


2


3


4



6


7


Program

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download: http://www.uni-tuebingen.de/modeling/Mod_Pub_Software_InfluSim_en.html

Lesson 6: InfluSim (Version 1.3)

Parameter panel Output panel

Output

table

Output

graphs

Output for

population with

N=100 000

individuals

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Output tab Output vari-

able (column)

At which stage of

the epidemic?*

R0 =2 R0 1.5

Infection Dead at end* 166 114

Infection Immune at end* 76855 55572

Ressource use Hospital beds at peak* 191 79

Cumulative Work loss [wk] at end* 29059 21647

Costs Total at end* 36130 26708

Hans-Peter Duerr, www.uni-tuebingen.de/modeling

Exercise 8: Influsim E

xerc

ise 8

(F

old

er

"Influsim1.3")

Aim: use a model to evaluate

intervention strategies

• Start Influsim from folder "InfluSim1.3" (screenshot see previous slide).

• Get familiar with using sliders, customizing panel arrangement and switching between tabs

for parameter values and output. First exploration:

Keep the default parameter values (or restart the program to dismiss your changes).

This will produce output based on R0 =2 ("severe" influenza epidemic).

Lookup values in the output table as listed in the table below and transfer the values

predicted by InfluSim into this table under column "R0 =2" (values refer to N=100 000).

Change R0 in parameter tab 'Contagiousness' from the default value of R0 =2 to R0

=1.5 and denote the new output values under column "R0 =1.5" ("moderate" influenza).

* click into the graphics window and/or scroll to the relevant row in the output table.

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Parameter section Slider Value Number

of deaths

Baseline 166

Treatment of severe cases Treatment fraction [%] increase to 100% 117

Treatment of extremely sick cases Treatment fraction [%] increase to 100% 65

Ressources Antiviral availability [%] decrease to 10% 113

Treatment of severe cases Treatment fraction [%] decrease to 0% 93

Treatment of extremely sick cases Begin on day increase to 60 days 161

Treatment of extremely sick cases Begin on day decrease to 30 days 94

Treatment of extremely sick cases Begin on day decrease to 0 days 93

What is your optimal antiviral intervention with 10% antivirals for the population?


xerc

ise 8

(F

old

er

"Influsim1.3")

Aim: explore the effects of

antiviral treatment

• Restart influSim to restore default parameter values (assumes R0 =2).

• Choose parameter tab ''Treatment'

• Change parameter sucessively as listed in the first three columns of the table below and

• denote the predicted number of deaths at the end of the epidemic (output tab 'Infection',

column 'Dead' in output table).

* go to output tab 'Cumulative' and scroll through table column 'Antiviral': this column lists the

number of individuals to whom antivirals have been given. How many?

** why less than before?

*

**

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xerc

ise 8

(F

old

er

"Influsim1.3")

Aim: explore the effects of

social distancing measures

• Restart influSim to restore default parameter values.

• Choose parameter tab ''Social distancing '

• We ignore initially 'Closing of day care centers and schools' and restrict on the two parameter sections

'General reduction of contacts' and

'Cancelling of mass gathering events'

Questions:

• What do you think: A) to which extent can we reduce the general contact rate in the population such

that the country's infrastructure will not be substantially impaired? Denote here: I guess the contact in our population can be reduced by ____________ %. B) guess how many deaths this contact reduction should safe: My guesstimate: this should reduce the number of deaths from 166 per 100 000 to _____________ per 100 000.

Move slider 'Contact reduction by [%]' in section 'General reduction of contacts' to the value you

have assumed under A). Note: For effects to be simulated you must specify how long this

contact reduction will be performed → next point …

Specify with slider 'End on day' in the same section 60 days (produces a scenario with contact

reduction from day 0 to 60 - quite a long time from a perspective of a population): Denote the

result of this simulation: My scenario reduced the number of deaths to _____________ per 100 000.

Move slider 'Contact reduction by [%]' (still in this section 'General reduction of contacts' ) to the unrealistically high value of 50% - why does this increase the number of deaths ??

• Evaluate the same considerations under section 'Cancelling of mass gathering events' - Question:

Which intervention is more efficient: 'General reduction of contacts' or 'Cancelling of mass

gathering events' ?

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Antivirals & social distancing

Sta

y a

t h

om

e

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Exercise 8: Influsim Aim: explore effects of delayed inter-

ventions (here: isolation measures)

Exerc

ise 8

(F

old

er

"Influsim1.3")

• Restart influSim to restore default parameter values (assumes R0 =2).

• Choose parameter tab 'Contagiousness' and specify with slider "Begin on day" a delay of

30 days as baseline delay for isolation measures: "Begin on day" =30 days.

• Vary parameters under 'Partial Isolation': we assume that 50% of Moderately sick cases,

Severe cases (home), or Severe cases (hospital) can be isolated with a delay of 30 days

(Row A1, A2, A3) or with a delay of 60 days (Row B1, B2, B3).

* Columns "Dead" and "Immune" in the ouput table, tab "Infection"

INPUT (tab 'Contagiousness', 'Partial Isolation') OUTPUT Infection tab

Moderately

sick cases

Severe

cases

(home)

Severe

cases

(hospital)

Begin

on day

Number of

deaths*

Number

infected*

No isolation (default) 0 0 0 30 166 76855

A1) Only 'moderate' 50 0 0 30 139 65697

A2) Only 'home' 0 50 0 30 118 58008

A3) Only 'hospital' 0 0 50 30 165 76665

B1) Only 'moderate' + delay 50 0 0 60 165 76278

B2) Only 'home' + delay 0 50 0 60 163 75925

B3) Only 'hospital' + delay 0 0 50 60 166 76837

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Exercise 8: Influsim - Duration of measures E

xerc

ise 8

Don't stop when it seems to be over,

but later.

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Slides Topic Lesson


1


2


3


4



6


7


Program

Le

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Vergleich:

deterministische vs. stochastische Modelle

Deterministische Modelle stochastische Modelle

• werden i.d.R. durch explizite

Formeln (Differenzialgleichungen)

erstellt

• liefern bei gleichen Anfangs-

bedingungen stets identische

Ergebnisse

• ihre Ergebnisse sind meist besser

verallgemeinerbar Zur Planung

von Interventionsmaßnahmen sind

deterministische Modelle oft besser

geeignet

• werden i.d.R. durch (individuen-

basierte) Simulationsprogramme

erstellt

• liefern bei gleichen Anfangs-

bedingungen zufallsbedingt

unterschiedliche Ergebnisse

• ihre Ergebnisse sind meist

realitätsnäher da sie zufällige

Effekte wiedergeben können

(Stochastizität)

• Für die Untersuchung von Effekten

in kleinen Populationen besser

Wichtig! Wichtig!

Wichtig!

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Lesson 7:

• General modeling, SEIR, diseases

• Influenza Vaccination, Antivirals

• SARS Isolation, Contact tracing

Behaviour

• Contact structures Networks

• Conclusions

• ... if time is left Playmulation (Simulator as a playstation)

Interventions in epidemics of influenza-like diseases -

insights from individual-based computer simulations

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The simulator: interface

www.uni-tuebingen.de/modeling

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The simulator: network

www.uni-tuebingen.de/modeling

Contacts of person 1446: Contacts of person 8969:

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Influenza-like SEIR(S)

S susceptible

I infectious

R immune

E latent

S susceptible

• genetic drift in

inluenza

• negligible for short-

term investigations

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Simulation algorithm

Build up network of contacts

Introduce infection

Sample for an infected individual

and each of his/her contacts:

Number of transmitted

infections

Time of infection

Duration

• of latency period • of prodromal phase

• of infectivity D • of immunity

Place events in a priority queue

and execute them consecutively

Scalefree network

8 close contacts

8 remote contacts

10 index cases

R0, close =1

R0, remote=1

Exponential distribution

with mean 1/b = D/R0

Gamma distribution Gamma distribution

Gamma distribution

Binary heap algorithm

R0=2

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Events: dependence & triggering

Demography

(birth of

susceptibles)

Intervention

Contact

structures

Surveillance

Case detection

Contact tracing

Observation

Symptoms Infection process

(Proportions)

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S…

6: In

fluenza

7: S

tochastic

8: S


A 'standard' epidemic

20% have not

been infected

and remain

susceptible

80% have been

infected

and become

immune 10 simulations

latent

infectious

80

29 c

ases

10 index cases initiate at t=0

an epidemic in a fully

susceptible population of

10000 people.

R0 =2.

Latency: 1.9 days.

Infectivity: 4 days.

Lifelong immunity.

SR

eS10

... N index cases

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fluenza

7: S

tochastic

8: S


A 'standard' epidemic Variation: number of index cases

10 simulations




10000 people.

R0 =2.

Latency: 1.9 days.


Lifelong immunity. 80

58 c

ases

Epidemic develops

more rapid

... latency

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EIR

S…

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fluenza

7: S

tochastic

8: S


A 'standard' epidemic Variation: duration of latency period

10 simulations




10000 people.

R0 =2.

Latency: 3.8 days.



61 c

ases

Epidemic delayed

... infectious period

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fluenza

7: S

tochastic

8: S


A 'standard' epidemic Variation: duration of infectious period

10 simulations




10000 people.

R0 =2.

Latency: 1.9 days.



41 c

ases

... but the number

of cases hardly

increases

Epidemic delayed,

appears more severe...

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EIR

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fluenza

7: S

tochastic

8: S


Lesson 7:




Behaviour


• Conclusions




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0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Estimates: Infection

Influenza SARS R0 1.8

close contacts: 1

casual contacts: 0.8

2.6

close contacts: 1


Latency ~2 days (1 day)

Range: 1-3 days

4.5 days (1.5 day)

Range: 3-6 days

Infectivity 4 days (2 days)

Range: 2-6 days

14 days (5 days)

Range: 9-19 days

Immunity forever forever

Symptoms 68% 100%

Case fatality rate 5% 10%

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EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Vaccination

Mass-vaccination Traced vaccination

Ring vaccination

Problems: • Vaccine stock limited,

Production takes time

• Who get's vaccinated ?

• Individual decision

• Which strategy

after stock is used up

• Contact tracing: who was a

contact?

• Where is he/she?

• Individual decision

• Who is a close contact?

• Fraction of contacts traced

Example: • Capacity: how many

persons can be vaccinated

per day?

• Delay between onset of

outbreak and begin of

vaccination

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4: V

accin

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n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Standard epidemics: influenza

General Infection

10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong

... the ultimate vaccination campaign

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EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: Mass-Vaccination (1)

General Infection Vaccination

10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong

Overoptimistic: All people are vaccinated. Vaccine

efficacy: 100%. All vaccinations can be performed

when the first case has been detected.

... Compliance: 50%, vaccine efficacy: 80%

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5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S




10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong

50% of the population are eligible. Vaccine efficacy:

80%. Overoptimistic: All vaccinations can be

performed when the first case has been detected.

... increase compliance, consider limitations

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S




10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong


80%. Vaccination starts without delay (Overoptimistic),

but only 2000 vaccinations per day are feasible.

... vaccination cannot start on day "zero"

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S




10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong


80%. Vaccination starts 2 weeks after the first case

has been detected, 2000 vaccinations per day.

... reduce delays

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IR, R

0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S




10 index cases

R0 = 1.9

Latency: 1.9 days.


Immunity: lifelong


80%. Vaccination starts 1 week after the first case

has been detected, 2000 vaccinations per day.

?

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0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: No intervention vs. Mass vaccination

5331 cases prevented

-4949 vaccinations

382

only 382 cases of

transmission prevented

mass vaccination

no intervention

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EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Conclusions: Vaccination (Influenza)

• Compliance

• Public Health capacities

• Intervention delays

Factors determining the prospects of success of (voluntary) vaccination campaigns:

Given proper compliance and capacities,

reducing intervention delays should be the main goal

For realistic parameter values,

the concomitant profit of vaccination is low

Summary: not very efficient, expensive,

makes voluntary vaccination sense?

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4: V

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EIR

S…

6: In

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7: S

tochastic

8: S



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AR

S

3: S

IR, R

0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Lesson 7:




Behaviour


• Conclusions




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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Treatment / Prophylaxis with antivirals

Prophylaxis Treatment

Questions: • When shall we start taking

prophylaxis and for how long?

• How many people will take

prophylaxis?

• Efficacy: reduction of

susceptibility in non-infected

• How quickly can infection be

diagnosed?

• Efficacy: reduction of

infectiousness in infected

Examples: • Compliance

• Duration

• Delay

• Compliance

• Duration

• Delay

• Shall close contacts take

antivirals prophylactically?

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EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: without Intervention

General Infection Prophylaxis Treatment

10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:

- -

... Population prophylaxis

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n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: Prophylaxis with Antivirals (1)

Useless intervention


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:

- 50% of the population take

antiviral prophylaxis

- (which reduces the

susceptibility to 30%)

- for a period of 2 weeks

-

... prolonge prophylaxis

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tochastic

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"Sometimes useful" intervention


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:






-

epidemics in

4 of 10 simulations

... consider delay

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accin

atio

n

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EIR

S…

6: In

fluenza

7: S

tochastic

8: S



Delays


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:






- but delayed by 10 days

-

epidemics in

all 10 simulations

... Treatment

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: without Intervention


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:

- -

... Treatment

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IR, R

0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: Treatment with Antivirals


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:

- - Cases receive treatment for 7 days

- 1 day after having been detected

- Compliance: 80%

- Efficacy: 80% (reduction infectiousness)

epidemics

in 8 of 10 simulations

... Treatment + Prophylaxis

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4: V

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5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Influenza: Treatment & Prophylaxis


10 index

cases

R0 = 1.9

Latency: 1.9 d

Infectivity: 4 d

Immunity:

- Close contacts take antiviral

prophylaxis (Compliance:

80%), reducing their

susceptibility to 30%


- delayed by 1 day

- Cases receive treatment for 7 days

- 1 day after having been detected

- Compliance: 80%

- Efficacy: 80% (reduction infectiousness)

C

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Conclusions: Antivirals (influenza)

xx

Prophylaxis with of antivirals in the population appears not very efficient

given realistic values for compliance and delays

Major effect: epidemic slows down, but last for much longer

Treating patients is more efficient,

but should be supplemented with

prophylaxis for close contacts (e.g. family)

The delay in prophylaxis for close contacts

is not that critical

A combination

of therapeutic and prophylactic use

seems to be

the most promising approach

Germany: "antivirals for 9% of the population"

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EIR

S…

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tochastic

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AR

S

3: S

IR, R

0

4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Lesson 7:




Behaviour


• Conclusions


Military hospital, flu in Kansas, 1918



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tochastic

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Estimates: Infection

Influenza SARS R0 1.8

close contacts: 1


2.6

close contacts: 1


Latency ~2 days (1 day)

Range: 1-3 days

4.5 days (1.5 day)

Range: 3-6 days

Infectivity 4 days (2 days)

Range: 2-6 days

14 days (5 days)

Range: 9-19 days

Immunity forever forever

Symptoms 68% 100%

Case fatality rate 5% 10%

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Isolation and Contact tracing (SARS)

Problems: • Contact tracing: who is / was a contact?

• What is a close, what is a casual contact?

• Number of traced contact persons per patient

• Limited resources to isolate patients

• Which strategy after resources have been exhausted

• Delay in case detection (Infectivity before showing symptoms?)

Example: • Number of isolation units available

• Detection delay

• Effect of seclusion

• Fraction of asymptomatic (but infectious) infections

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tochastic

8: S


SARS: without intervention

... Isolation

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4: V

accin

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S…

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7: S

tochastic

8: S


SARS: Isolation

only 7 epidemics

in 10 simulations

Isolation: 100 cases (per 10000)

can be isolated at the same time.

Case detection:

• Cases can be detected

within 4 days (2 days).

if occurring,

the size of the epidemic

(number of cases)

is almost the same

... + seclusion


-2951 cases isolated

-224 excess effort

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S…

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fluenza

7: S

tochastic

8: S


SARS: Isolation + Seclusion (1)

if occurring,

the size of the epidemic

(number of cases)

is not much smaller



Case detection:



Seclusion: 100 cases (per 10000) can be

additionally observed at the same time.

only 3 epidemics

in 10 simulations

... 90% symptoms



- 925 cases secluded

3768 cases

of transmission

prevented

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EIR

S…

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fluenza

7: S

tochastic

8: S





Case detection:





Symptoms: 90% of infected subjects

develop detectable symptoms

... improving diagnosis

900 cases prevented


-1538 cases secluded

-4552 excess effort

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accin

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n

5: S

EIR

S…

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fluenza

7: S

tochastic

8: S





Case detection:



• Case detection improves in course of the

epidemic by 5% per day and, finally, cases

will be detected within 2 days (1 day).





... + tracing


- 519 cases isolated

- 140 cases secluded

7526 cases

of transmission

prevented

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fluenza

7: S

tochastic

8: S


SARS: Isolation + Seclusion + Tracing



Case detection:



• Case detection improves in course of the

epidemic by 5% per day and, finally, cases

will be detected within 2 days (1 day).





Contact tracing:

• Contact persons can be traced

within 2.0 days (0.5 day).

• 50% of the contacts of a case can be

traced, at maximum 20 contacts per

case.

• 200 contacts (per 10000) can be

observed at the same time for a period

of 7 days.

C


- 110 cases isolated

- 57 cases secluded

-1052 contacts traced

-1003 contacts observed

6739 cases

of transmission

prevented

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EIR

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fluenza

7: S

tochastic

8: S


Conclusions: Isolation, contact tracing

Isolation of cases is an effective intervention (again: targeted),

but its effectiveness strongly depends on the capacities

of the public health system

Seclusion of cases is an effective complement

As these interventions depend on the detection of cases,

rapid diagnostics becomes more and more important

Improving procedures (e.g. case detection occurs more quickly)

can substantially aid controlling

the spread of infection (again: time)

Contact tracing: very effective ( contact structures...)

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7: S

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8: S


Summary Lesson 7

Successful interventions are immediate & targeted:

• Diagnose quickly

• Act quickly

• target Intervention !

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7: S

tochastic

8: S



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S

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4: V

accin

atio

n

5: S

EIR

S…

6: In

fluenza

7: S

tochastic

8: S


Program

Slides Topic Lesson


1


2


3


4



6


7


Le

sso

n 8

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Networks: Random vs. Scale-free

Barabasi AL, Bonabeau E. Scale-free networks. Sci Am 2003;288(5):60-9

Random network Scale-free network

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The contact network

log(H

äufigkeit)

log(Kontakte)

Scale-free network H

äufig

ke

it

Kontakte

log(H

äufigkeit)

log(Kontakte)

Random network

Häufig

ke

it

Kontakte

log(H

äufigkeit)

log(Kontakte)

Scale-free network

mit Familienstruktur

Häufig

ke

it

Kontakte

"will

mei R

uah"

Fam

ilien

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7: S

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The structure of the contact network & corresponding epidemics

0

100

200

300

400

500

600

0 30 60 90 120 150 180 210 240 270 300

Day

Num

ber

infe

cted

.

IAR: 10%

IAR: 13%

IAR: 21%

1

10

100

1000

10000

1 10 100 1000

Degree

Ab

solu

te f

req

uen

cy

Without

superspreaders

Epidemic: slow

low infection attack

rate (IAR)

0

100

200

300

400

500

600

0 30 60 90 120 150 180 210 240 270 300

Day

Nu

mb

er

infe

cte

d

.

IAR: 26%

IAR: 30%

IAR: 33%

1

10

100

1000

10000

1 10 100 1000

Degree

Ab

solu

te f

req

uen

cy

With

superspreaders

Dmax=479 Dmax=33

Epidemic: fulminant

high infection attack

rate (IAR)

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7: S

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8: S


Attacking a scale-free network

The accidental failure of a

number of nodes in a

random network can

fracture the system into

noncommunicating islands.

Scale-free networks are

robust in the face of such

failures.

...but they are highly

vulnerable to a coordinated

attack against their hubs.

Barabasi AL, Bonabeau E. Scale-free networks. Sci Am 2003;288(5):60-9

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Social networks

Sci Am 2005;292(3):42-49

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7: S

tochastic

8: S


What is a contact?

Kis

sin

g: "h

ello

"

Kis

sin

g: w

ith

in

tention

s

Ha

vin

g s

ex

Ha

ndsh

akin

g

In c

ine

ma

, p

ub

...

Ta

lkin

g to

each

oth

ers

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EIR

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fluenza

7: S

tochastic

8: S


Summary Lesson 7

• The structure of the contact network determines the mode of

transmission of infection – it represents, however, often the most

unknown variable.

• The efficacy of interventions depends on the contact network if

specific groups or individuals are targeted, as is the case for contact

tracing or isolation.

• Stochasticity can change the actual course of an epidemic

decisively.

• Superspreaders can cause profound stochastic fluctuations in the

course of an epidemic, depending on, for instance, when they come

into play or how long they are infectious.

Infectious disease epidemiology and mathematical models in the … · 2014-10-12 · Infectious...

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Transcript of Infectious disease epidemiology and mathematical models in the … · 2014-10-12 · Infectious...