Transcript Responce to PncPS or PncCRM in children with recurrent

BASICS OF EPIDEMIC
MODELLING
Kari Auranen
Department of Vaccines
National Public Health Institute (KTL), Finland
Division of Biometry, Dpt. of Mathematics and Statistics
University of Helsinki, Finland
Outline
• A simple epidemic model to exemplify
–
–
–
–
–
–
dynamics of transmission
epidemic threshold
herd immunity threshold
basic reproduction number
the effect of vaccination on epidemic cycles
mass action principle
Outline (2)
• The Susceptible - Infected - Removed (SIR) model
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–
–
–
endemic equilibrium
force of infection
estimation of the basic reproduction number R 0
effect of vaccination
• The SIS epidemic model
–
–
R 0 and the choice of the model type
age-specific proportions of susceptibles/infectives
A simple epidemic model (Hamer, 1906)
• Consider an infection that
– involves three states/compartments of infection:
Susceptible
Case
Immune
– proceeds in discrete generations (of infection)
– is transmitted in a homogeneously mixing population
of size N
Dynamics of transmission
• Dependence of generation t+1 on generation t:
C t + 1 = R 0 x C t x St / N
S t+1 = S t - C t+1 + B t
S t = number of susceptibles at time t
C t = number of cases (infectious individuals) at time t
B t = number of new susceptibles (by birth)
Dynamics of transmission
1400
1200
1000
800
600
400
200
0
susceptibles
cases
time period
29
25
21
17
13
9
epidemic
threshold
5
1
numbers of individuals
Dynamics (Ro = 10; N = 10,000; B = 300)
Epidemic threshold : S = N/R
e
0
Epidemic threshold S e
St+1 - S t = - C t+1 + B t
• the number of susceptibles increases when C t+1 < B t
decreases when C t+1 > B t
• the number of susceptibles cycles around
the epidemic threshold S e = N / R 0
• this pattern is sustained as long as transmission is
possible
Epidemic threshold
C t+1 / C t = R 0 x St / N = St / Se
• the number of cases
increases when S > Se
decreases when S < S e
• the number of cases cycles around B t (influx of new
susceptibles)
Herd immunity threshold
• incidence of infection decreases as long as the
proportion of immunes exceeds the herd immunity
threshold
H = 1- S e / N
• a complementary concept to the epidemic threshold
• implies a critical vaccination coverage
Basic reproduction number (R 0 )
• 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 0 = R0 x 1 x N / N = R 0
• herd immunity threshold H = 1 - 1 / R 0
• in the endemic equilibrium: S e = N / R0 , i.e.,
Re0 x Se / N 0= 1
Basic reproduction number (2)
R0 = 3
Basic reproduction number (3)
R0 = 3
endemic equilibrium
R0 x Se / N = 1
Herd immunity threshold and R 0
0,8
0,7
0,6
H = 1-1/R 0
0,5
herd
immunity 0,4
threshold H 0,3
(Assumes homogeneous mixing)
0,2
Ro
5
4
3
2
1
0
0
0,1
Effect of vaccination
Hamer model under vaccination
2000
S
t+1
= S t - C t+1 + B (1- VCxVE)
susc.
1500
cases
1000
epidemic
threshold
500
Vaccine effectiveness (VE)
x
Vaccine coverage (VC) = 80%
time period
36
41
31
21
26
16
11
6
0
1
numbers of individuals
Ro = 10; N = 10,000; B = 300
Epidemic threshold sustained: S = N / R
e
0
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
• in appropriate subpopulations
The SIR epidemic model
• a continuous time model: overlapping generations
• permanent immunity after infection
• the system descibes the flow of individuals between the
epidemiological compartments
• uses a set of differential equations
Susceptiple
Infectious
Removed
The SIR model equations
dS 
dt
 N   I (t) S (t)  S (t)
N
dI 
I (t) S (t)  I (t) I (t)


dt
N
dR 
dt
 I (t)  R(t)
N  S (t)  I (t)  R(t)
 = birth rate
 = rate of clearing infection
 = rate of infectious contacts
by one individual
 = force of infection
Endemic equilibrium (SIR)
1200
susceptibles
1000
800
infectives
600
N = 10,000
 = 300/10000 (per time unit)
 = 10 (per time unit)
epidemic
threshold
400
200
 = 1 (per time unit)
R0     
time
46
7,
00
12
,0
0
19
,0
0
28
,0
0
3.
0
0
0
numbers of individuals
1400
The basic reproduction number (SIR)
• Under the SIR model, Ro given by the ratio of two rates:

R0 =  

= rate of infectious contacts  x
mean duration of infection   )
• R 0 not directly observable
• need to derive relations to observable quantities
Force of infection (SIR)
• the number of infective contacts in the population per
susceptible per time unit:
 (t) = x I(t) / N

• incidence rate of infection:  (t) x S(t)
• endemic force of infection (SIR):  =  x (R 0 - 1)
Estimation of R 0 (SIR)
basic reproduction number
Relation between the average age at infection and R0 (SIR model)
90
80
70

= 1/75 (per year)
 L  1/   75
60
50
40
30
  R0 1)
 /  R 1
R  1  / 
20
10
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15
average age at infection A
R0  1 L / A
A simple alternative formula
• Assume everyone is infected at age A
everyone dies at age L (rectangular age distribution)
Proportion
100 %
Susceptibles
Immunes
Proportion of susceptibles:
Se / N = A / L
A
Age (years)
L
R0 = N / Se = L / A
Estimation of  and Ro from
seroprevalence data
proportion with rubella antibodies
1) Assume equilibrium
2) Parameterise force of infection
100
90
80
70
60
50
40
30
20
10
0
3) Estimate

4) Calculate Ro
observed [8]
model prediction
1
5
10 15 20 25 30
age a (years)
Ex.  constant
Proportion not yet infected:
1 - exp(-  a) ,
estimate  = 0.1 per year gives
reasonable fit to the data
Estimates of R 0
Infection
Location
R0
Measles
Rubella
Poliomyelitis
Hib
England and Wales (1950-68)
England and Wales (1960-70)
USA (1955)
Finland, 70’s and 80’s
16-18*
6-7*
5-6*
1.05
*Anderson and May: Infectious Diseases of Humans, 1991
Indirect effects of vaccination (SIR)
• Vaccinate proportion p of newborns, assume complete
protection against infection
• R vacc = (1-p) x R 0
• If p < H = 1-1/R0 , in the new endemic equilibrium:
S e = N/R 0 ,
 vacc =  (R vacc -1)
» proportion of susceptibles remains untouched
» force of infection decreases
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
Susceptibles
S e / N = (1-p) A’/L
Proportion
1
S e / N = A/ L
=> A’ = A/(1-p)
Immunes
p
A’
L
Age (years)
i.e., increase in the
average age of
infection
Vaccination at age V > 0 (SIR)
• Assume proportion p vaccinated at age V
• Every susceptible infected at age A
• How big should p be to obtain herd immunity threshold H
Proportion
H = 1 - 1/R = 1 - A/L
1
H = p (L-V)/L
=> p = (L-A)/(L-V)
Susceptibles
p
i.e., p bigger than when
vaccination at birth
Immunes
V
A
L
Age (years)
The SIS epidemic model
Susceptible
Immune
• herd immunity threshold : H = 1 - 1/R 0
 R 1)
• endemic force of infection:  
• the proportions of susceptibles and immunes different
from the SIR model
SIS and SIR
R0 and the force of infection
No immunity to infection (SIS)
1,6
Lifelong immunity to infection (SIR)
80
1,4
70
1,2
60
1
50
Ro 0,8
Ro 40
0,6
30
0,4
20
0,2
10
0
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)
birth rate  = 1/75 (per year)
rate of clearing infection  = 2.0 (per year)
0
0
0.2
0.4
0.6
0.8
1.0
force of infection (per year)
birth rate = 1/75(per year)
Extensions of simple models
• 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, DCC’s, schools, etc.
Extensions of simple models (2)
– seasonal patterns in risks of infection
– latency, maternal immunity etc.
– different vaccination strategies
– different models for vaccine effectiveness
• Stochastic models to
– model chance phenomena
– time to eradication
– apply statistical techniques
Example: structured models
Contact structures (WAIFW)
• structure of the Who Acquires Infection From Whom matrix
for varicella , five age groups (0-4, 5-9, 10-14, 15-19, 20-75
years)
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
a
c
d
e
a
b
c
d
e
c
c
c
d
e
d
d
d
d
e
e
e
e
e
e
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 !
References
1
2
3
4
5
6
7
8
Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews, 15, 265-302,1993
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.
Haber M., "Estimation of the direct and indirect effects of vaccination", Statistics in Medicine, 18, 21012109, 1999
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
Levy-Bruhl D., lecture notes in the EPIET course, Helsinki, 1998.
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
Lipsitch M., "Vaccination against colonizing bacteria with multiple serotypes", Population Biology, 94,
6571-6576, 1997
Anderson R.M. and May R.M., ”Infectious Diseases of Humans”; Oxford University Press, 1992.