epidemiologisten tartuntatautimallien perusteita kari auranen rokoteosasto kansanterveyslaitos...
TRANSCRIPT
Epidemiologisten tartuntatautimallien perusteita
Kari AuranenRokoteosastoKansanterveyslaitos
Matematiikan ja tilastotieteen laitosHelsingin yliopisto
Outline (1)
• A simple epidemic model to exemplify – dynamics of transmission of infectious disease– epidemic threshold– herd immunity threshold
– basic reproduction number R0
– the effect of vaccination on epidemic cycles – mass action principle
Outline (2)
• The Susceptible - Infected - Removed (SIR) model– endemic equilibrium– force of infection– estimation of the basic reproduction number R– effect of vaccination
• The SIS epidemic model– R and the choice of the model type– age-specific proportions of susceptibles/infectives
00
00
A simple epidemic model (Hamer, 1906)
• Consider an infection that– involves three states/compartments of infection:
– proceeds in discrete generations (of infection)– is transmitted in a homogeneously mixing population of
size N
SSusceptibleusceptible CCasease ImmuneImmune
Model equations
• Dependence of generation t+1 on generation t:
C = R x C x S / N
S = S - C + B
t + 1t + 1 00 tt tt
t+1t+1 tt t+1t+1 tt
SS = number of susceptibles at time t = number of susceptibles at time tCC = number of cases (infectious individuals) at time t = number of cases (infectious individuals) at time tBB = number of new susceptibles (by birth) = number of new susceptibles (by birth)
tt
tt
tt
Dynamics of transmission
Dynamics (Ro = 10; N = 10,000; B = 300)
0
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time period
nu
mb
ers
of
ind
ivid
ual
s
susceptibles
cases
epidemicthreshold
Epidemic threshold : S /N = 1/REpidemic threshold : S /N = 1/R00ee
Epidemic threshold S
S - S = - C + B
• the number of susceptibles increases when C < B
decreases when C > B
• the number of susceptibles cycles around the epidemic threshold S = N / R
• this pattern is sustained as long as transmission is possible
ee
t+1t+1 tt t+1t+1 tt
t+1t+1
t+1t+1
tt
tt
ee 00
Epidemic threshold
C / C = R x S / N = S / S
• the number of cases increases when St > S
decreases when St < S
• the number of cases cycles around B (influx of new susceptibles)
t+1t+1 tt 00 tt tt ee
ee
ee
tt
Herd immunity threshold
• incidence of infection decreases as long as the proportion of immunes exceeds the herd immunity threshold
H = 1- Se / N
• a complementary concept to the epidemic threshold
• implies a critical vaccination coverage: what proportion of the population needs to be effectively vaccinated to eliminate infection
ee
Basic reproduction number (R )
• the average number of secondary cases that an infected individual produces in a totally susceptible population during his/her infectious period
• in the Hamer model : R = R x 1 x N / N = R
• herd immunity threshold H = 1 - 1 / R
• in the endemic equilibrium: S = N / R , i.e.,
00
0000
00
00ee
ee 00
00
R x S / N = 1R x S / N = 100 ee
Basic reproduction number (2)
R = 3R = 300
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
CC
Basic reproduction number (3)
R = 3 R = 3 CC
endemic equilibriumendemic equilibrium00
R x S / N = 1R x S / N = 100 ee
CC
CCCC
Herd immunity threshold and R
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
herd immunity
threshold H
0 1 2 3 4 5
Ro
= 1-1/RHH 00
00
(Assumes homogeneous mixing)(Assumes homogeneous mixing)
Effect of vaccination
Ro = 10; N = 10,000; B = 300
0
500
1000
1500
2000
time period
nu
mb
ers
of
ind
ivid
ual
s
susc.
cases
epidemicthreshold
Hamer model under vaccinationHamer model under vaccination
S = S - C + B (1- VCxVE)S = S - C + B (1- VCxVE)
Vaccine efficacy (VE) Vaccine efficacy (VE) xxVaccine coverage (VC) = 80%Vaccine coverage (VC) = 80%
t+1t+1 tt t+1t+1
Epidemic threshold sustained: S = N / R Epidemic threshold sustained: S = N / R ee 00
Mass action principle
• all epidemic/transmission models are variations of the use of the mass action principle which– captures the effect of contacts between individuals– uses the analogy to modelling the rate of chemical reactions – is responsible for indirect effects of vaccination– assumes homogenous mixing
• in the whole population or• in appropriate subpopulations/strata
The SIR epidemic model
• a continuous time model: overlapping generations• permanent immunity after infection• the system desrcibes the flow of individuals between
three epidemiological “compartments” • formally defined through a set of differential equations
SusceptipleSusceptiple RemovedRemovedInfectiousInfectious
The SIR model equations
dtdI )()( tS
NtI )(tI )(tI
dtdS N )()( tS
NtI )(tS
dtdR )(tI )(tR
)()()( tRtItSN
= birth rate= birth rate
= rate of clearing infection= rate of clearing infection
= rate of infectious contacts= rate of infectious contacts
by one individual by one individual
= force of infection= force of infection
λλ(t)(t)
{{
Endemic equilibrium (SIR)
0
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03.
07,
0012
,00
19,0
028
,00 46
time
nu
mb
ers
of
ind
ivid
ual
s
susceptibles
infectives
epidemicthreshold
N = 10,000N = 10,000
= 300/10000 = 300/10000 (per time unit)(per time unit)
= 10 = 10 (per time unit)(per time unit)
= 1 = 1 (per time unit)(per time unit)
0R
The basic reproduction number (SIR)
• Under the SIR model, Ro given by the ratio of two rates:
R = = rate of infectious contacts x
“mean duration” of infection
• R (usually) not directly observable• need to derive relations to observable quantities
00
00
)
Force of infection (SIR)
• the number of infectious contacts in the population per susceptible per time unit:
λ(t) = x I(t) / N
• incidence rate of infection: (t) x S(t)
• endemic force of infection (SIR): = x (R - 1)00
Estimation of R (SIR)
0
10
20
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40
50
60
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80
90
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
average age at infection A
bas
ic r
epro
du
ctio
n n
um
ber
Relation between the average age at infection and R (SIR model)Relation between the average age at infection and R (SIR model)
= 1/75 = 1/75 (per year)(per year)
)10 R
ALR /10
10
/ R /1
0R
00
00
75/1 L
A simple alternative formula
• assume everyone is infected at age A everyone dies at age L (rectangular age distribution)
ImmunesImmunes
AA LL
Age (years)Age (years)
SusceptiblesSusceptibles100 %100 %
Stationary proportion of susceptibles:Stationary proportion of susceptibles: S / N = A / LS / N = A / L
=> R=> R00 = 1/(S = 1/(See/N) = L / A/N) = L / A
ee
ProportionProportion
Estimation of and R0 from seroprevalence data
0
10
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5060
70
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100
1 5 10 15 20 25 30
age a (years)
proportion with rubella antibodies
observed [8]
model prediction
1) Assume equilibrium 1) Assume equilibrium
2) Parameterise force of infection 2) Parameterise force of infection
3) Estimate3) Estimate
4) Calculate R4) Calculate R00
Ex. constant Ex. constant Proportion not yet infected: Proportion not yet infected: 1 - exp(- a) ,1 - exp(- a) , estimate = 0.1 per year gives estimate = 0.1 per year gives reasonable fit to the datareasonable fit to the data
Estimates of R
Infection Location R0
Measles England and Wales (1950-68) 16-18
Rubella England and Wales (1960-70) 6-7
Poliomyelitis USA (1955) 5-6Hib Finland, 70’s and 80’s 1.05
Anderson and May: Infectious Diseases of Humans, 1991*
*
*
**
00
Indirect effects of vaccination (SIR)
• vaccinate proportion p of newborns, assume complete protection against infection
• a new reproduction number R = (1-p) x R0
• if p > H = 1-1/R0 , the infection cannot persist
• if p < H = 1-1/R0 , in the new endemic equilibrium:
S = N/R0 , = (R -1)
» proportion of susceptibles remains untouched(!)» force of infection decreases
vaccvacc
vaccvacc
vaccvacc
Effect of vaccination on average age A’ at infection (SIR)
• life length L; proportion p vaccinated at birth, complete protection • every susceptible infected at age A
SusceptiblesSusceptibles
AA LL
pp
AgeAge
11
S / N = (1-p) A’/L S / N = (1-p) A’/L
S / N = A/ LS / N = A/ L
=> A’ = A/(1-p) => A’ = A/(1-p)
i.e., increase in thei.e., increase in the average age of average age of infectioninfection
ProportionProportion
’’
ee
ee
ImmunesImmunes
with vaccination:with vaccination:
without vaccination:without vaccination:
Vaccination at age V > 0 (SIR)
• assume proportion p vaccinated at age V (instead of at birth)• every susceptible infected at age A• what proportion p should be vaccinated to obtain herd immunity threshold H?
Age Age
ProportionProportion
11
pp
VV LL
H = 1 - 1/RH = 1 - 1/R00 = 1 - A/L = 1 - A/L
proportion immunised proportion immunised by vaccination p (L-V)/L by vaccination p (L-V)/L
=> p = (L-A)/(L-V) => p = (L-A)/(L-V)
i.e., p bigger than when i.e., p bigger than when vaccination at birth vaccination at birth
ImmunesImmunes
SusceptiblesSusceptibles
AA
The SIS epidemic model
• herd immunity threshold still the same: H0 = 1 - 1/R0
• endemic force of infection: • the proportions of susceptibles and immunes different
from the SIR model
SusceptibleSusceptible ImmuneImmune
)10 R
R and the force of infection
0
10
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0 0.2 0.4 0.6 0.8 1.0
force of infection (per year)
Ro
0
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0,8
1
1,2
1,4
1,6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
force of infection (per year)
Ro
birth rate = 1/75 (per year)rate of clearing infection = 2.0 (per year)
birth rate = 1/75(per year)
no immunity to infection (SIS) lifelong immunity to infection (SIR)
00
SIS and SIRSIS and SIR
Extensions of simple models (1)
• so far all models assumed– homogeneous mixing– constant force of infection across age (classes)
• more realistic models incorporate– heterogeneous mixing
• age-dependent contact/transmission rates• social structures: families, day care groups, schools,
neighbourhoods etc.
Extensions of simple models (2)
– seasonal patterns in risks of infection– latency, maternal immunity etc.– different vaccination strategies– different models for the vaccine effect
• stochastic models to– model chance phenomena– time to eradication– apply statistical techniques/inference
Contact structures (WAIFW)
• structure of the Who Acquires Infection From Whom matrix for varicella , five age groups (e.g. 0-4, 5-9, 10-14, 15-19, 20-75 years)
a a c d ea b c d ec c c d ed d d d ee e e e e
table entry = rate of transmission between an infective and a susceptible of respective age groups
e.g., force of infection in age group 0-4: a*I1 + a*I2 + c*I3 + d*I4 + e*I5
I1 = equilibrium number of infectives in age group 0-4, etc.
• WAIFW matrix non-identifiable from age-specific incidence !
Example: structured modelsExample: structured models
References
1 Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews, 15, 265-302,1993
2 Fine P.E.M., "The contribution of modelling to vaccination policy, Vaccination and World Health, Eds. F.T. Cutts and P.G. Smith, Wiley and Sons, 1994.
3 Haber M., "Estimation of the direct and indirect effects of vaccination", Statistics in Medicine, 18, 2101-2109, 1999
4 Halloran M.E., Cochi S., Lieu T.A., Wharton M., Fehrs L., "Theoretical epidemologic and mordibity effects of routine varicella immunization of preschool children in the U.S.", AJE, 140, 81-104, 1994
5 Levy-Bruhl D., lecture notes in the EPIET course, Helsinki, 1998.
6 Nokes D.J., Anderson R.M., "The use of mathematical models in the epidemiological study of infectious diseases and in the desing of mass immunization programmes", Epidemiology and Infection, 101, 1-20, 1988
7 Lipsitch M., "Vaccination against colonizing bacteria with multiple serotypes", Population Biology, 94, 6571-6576, 1997
8 Anderson R.M. and May R.M., ”Infectious Diseases of Humans”; Oxford University Press, 1992.