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Modelling as a tool for planning
vaccination policies
Kari Auranen
Department of Vaccines
National Public Health Institute, Finland
Department of Mathematics and Statistics
University of Helsinki, Finland
Outline
• Basic concepts and models
–
–
–
–
–
dynamics of transmission
herd immunity threshold
basic reproduction number
herd immunity and critical coverage of vaccination
mass action principle
Outline continues
• Heterogeneously mixing populations
– more complex models and new survey data to learn about routes of
transmission
• Use of models in decision making
– example: varicella vaccinations in Finland
• Ude of models in planning contaiment strategies
– example: a simulation tool for pandemic influenza
Basic concepts and models
A simple epidemic model (Hamer, 1906)
• Consider an infection that
– involves three “compartments” of infection:
Susceptible
Case
Immune
– proceeds in discrete generations (of infection)
– is transmitted in a homogeneously mixing (“everyone meets
everyone”) population of size N
Dynamics of transmission
• Numbers of cases and susceptibles at generation t+1
Ct + 1 = R 0 * C t * S t / N
S t+1 = S t - C t+1 + B t
S t = number of susceptibles at time t (i.e. generation 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
R0 = 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 an 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 (defined by usually by age
categories)
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
epidemic
threshold
400
200
N = 10,000
 = 300/10000 (per time unit)
 = 10 (per time unit)
 = 1 (per time unit)
R     
0
time
46
7,
00
12
,0
0
19
,0
0
28
,0
0
0
3.
0
0
numbers of individuals
1400
The basic reproduction number
• 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
• 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
 =  x (R 0 - 1)
Estimation of R 0
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
Critical vaccination coverage to obtain
herd immunity
• Immunise a proportion p of newborns with a vaccine
that offers complete protection against infection
• R vacc = (1-p) x R 0
• If the proportion of vaccinated exceeds the herd
immunity threshold, i.e., if p > H = 1-1/R 0 ,
infection cannot persist in the population (herd
immunity)
Critical vaccination coverage as a
function of R0
1
Critical vaccination coverage
0.9
0.8
0.7
p = 1 – 1/R0
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
4
5
6
7
8
9
10
11
Basic reproduction number
12
13
14
15
Indirect effects of vaccination
• If p < H = 1-1/R0 , in the new endemic equilibrium:
S e = N/R 0 ,
 vacc=  (Rvacc -1)
e
» 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)
Modelling transmission in a
heterogeneously mixing
population
More complex mixing patterns
• So far we have assumes (so called) homogeneous mixing
– “everyone meets everyone”
• More realistic models incorporate some form of heterogeneity
in mixing (“who meets whom”)
– e.g. individuals of the same age meet more often each other
than individual from other age classes
(assortative mixing)
Example: WAIFW matrix
• 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
a*I1 + a*I2 + c*I3 + d*I4 + e*I5
I1 = equilibrium number of infectives in age group 0-4, etc.
c
c
c
d
e
d
d
d
d
e
e
e
e
e
e
POLYMOD contact survey
• Records the number of daily conversations in study
participants in 7 European countries
• Use the number of contacts between individuals from
different age categories as a proxy for chances of
transmission
• Is currently being used to aid in modelling the impact of
varicella vaccination in Finland
POLYMOD contact survey:
the mean number of daily contacts
Country
DE
FI
IT
LU
NL
PL
GB
Number of daily
contacts
7.95
11.06
19.77
17.46
13.85
16.31
11.74
Relative (95% CI)
1
1.34 (1.26-1.42)
2.33 (2.19-2.48)
2.02 (1.90-2.14)
1.78 (1.63 -1.95)
1.90 (1.79 – 2.01
1.40 (1.31 – 1.48)
POLYMOD contact survey:
numbers of daily contacts
Finland
70+
45-49
40-44
35-39
30-34
25-29
Age of contact
65-69
60-64
55-59
50-54
00-04
50-54
55-59
60-64
65-69
70+
45-49
25-29
30-34
35-39
40-44
15-19
20-24
00-04
05-09
10-14
20-24
15-19
10-14
05-09
Age of participant
0.00-0.31
0.31-0.63
0.63-0.94
1.56-1.88
1.88-2.19
2.19-2.50
0.94-1.25
1.25-1.56
POLYMOD contact survey:
where and for how long
Duration
100%
100%
90%
90%
80%
80%
70%
70%
60%
60%
50%
non-physical
physical
40%
30%
non-physical
physical
50%
40%
30%
20%
20%
10%
10%
0%
0%
<5 min
5-15 min
15-60 min
1-4 h
4+ h
100%
home
school
leisure
work
transport
other
multiple
100%
90%
90%
non-physical
physical
80%
70%
80%
60%
60%
50%
50%
40%
4+ h
70%
1-4 h
15-60 min
5-15 min
40%
30%
<5 min
30%
20%
20%
10%
10%
0%
daily
weekly
monthly
less often
first time
0%
daily
weekly
monthly
less often
first time
Use of models in policy making
• Large-scale vaccinations usually bring along indirect effects
– the mean age at disease increases
– population immunity changes
• Population-level experiments are impossible
Need for mathematicl modelling
– to predict indirect effects of vaccination
– to summarise the epidemiology of the infection
– to identify missing data or knowledge about the natural
history of the infection
References
1 Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews,
2
3
4
5
6
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.
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
Anderson R.M. and May R.M., ”Infectious Diseases of Humans”; Oxford
University Press, 1992.
Mossong J et al, Social contacts and mixing patterns relevant to the spread of
infectious diseases: a multi-country population-based survey, Plos Medicine, in
press
Duerr et al, Influenza pandemic intervention planning using InfluSim, BMC Infect
Dis, 2007