1. 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

2. Outline Basic concepts and models dynamics of transmission
herd immunity threshold
basic reproduction number
herd immunity and critical coverage of vaccination
mass action principle
* not a "political" approach, rather a personal view of a data analyst?
* NOT A PRESENTATION OF PRINCIPLES OF MATHEMATICAL MODELLING, RATHER ADD A FEW (PERSONAL) NOTES ON THE THEME
* + STRESS THE IMPORTANCE OF COHERENT, STOCHASTIC MODELLING? (BAYES vs. FREQUENTIST?)

3. 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

4. Basic concepts and models

5. A simple epidemic model (Hamer, 1906)
Consider an infection that
involves three “compartments” of infection:
proceeds in discrete generations (of infection)
is transmitted in a homogeneously mixing (“everyone meets everyone”) population of size N
Susceptible
Case
Immune

6. Dynamics of transmission
Numbers of cases and susceptibles at generation t+1
C = R * C * S / N
S = S - C B
t + 1 t t
t+1 t
t+1 t
S = number of susceptibles at time t (i.e. generation t)
C = number of cases (infectious individuals) at time t
B = number of new susceptibles (by birth) t t t

7. Dynamics of transmission
Epidemic threshold : S = N/R e

8. 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
t+1 t
t+1 t
t+1 t t
t+1 e

9. Epidemic threshold C / C = R x S / N = S / S
the number of cases increases when S > S
decreases when S < S
the number of cases cycles around B (influx of new susceptibles)
t+1 t t t e e e t

10. Herd immunity threshold
incidence of infection decreases as long as the proportion of immunes exceeds the herd immunity threshold
H = 1- S / N
a complementary concept to the epidemic threshold
implies a critical vaccination coverage e

11. 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 = / R
in the endemic equilibrium: S = N / R , i.e., e
R x S / N = 1 e e

12. Basic reproduction number (2)

13. Basic reproduction number (3)
endemic equilibrium
R x S / N = 1 e

14. Herd immunity threshold and RH
= 1-1/R
(Assumes homogeneous mixing)

15. Effect of vaccination Hamer model under vaccination
S = S - C B (1- VCxVE)
Vaccine effectiveness (VE) x
Vaccine coverage (VC) = 80%
t+1 t
t+1
Epidemic threshold sustained: S = N / R e

16. 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)

17. 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

18. The SIR model equations
= birth rate
= rate of clearing infection
= rate of infectious contacts
by one individual
= force of infection

19. Endemic equilibrium (SIR)
= 300/10000 (per time unit)
= 10 (per time unit)
= 1 (per time unit)

20. The basic reproduction number
Under the SIR model, Ro given by the ratio of two rates:
R = = rate of infectious contacts x
mean duration of infection
R not directly observable
need to derive relations to observable quantities

21. 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 - 1)

22. Estimation of R
Relation between the average age at infection and R (SIR model)
= 1/75 (per year)

23. 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:
S / N = A / L
R = N / S = L / A e A L e
Age (years)

24. Estimation of and Ro from seroprevalence data
1) Assume equilibrium
2) Parameterise force of infection
3) Estimate
4) Calculate Ro
Ex constant
Proportion not yet infected:
1 - exp(- a) ,
estimate = 0.1 per year gives
reasonable fit to the data

25. Estimates of R * * *
* PROBLEMS: a) STRUCTURE POPULATIONS?
b) STOCHASTIC MODELS
* MIKÄ ON TARVITTAVA VACCINE COVERAGE (olettaen
homogeneous mixing)? *
Anderson and May: Infectious Diseases of Humans, 1991

26. Critical vaccination coverage to obtain herd immunity
Immunise a proportion p of newborns with a vaccine that offers complete protection against infection
R = (1-p) x R
If the proportion of vaccinated exceeds the herd immunity threshold, i.e., if p > H = 1-1/R ,
infection cannot persist in the population (herd immunity)
vacc

27. Critical vaccination coverage as a function of R0
p = 1 – 1/R0

28. Indirect effects of vaccination
If p < H = 1-1/R , in the new endemic equilibrium:
S = N/R , = (R )
proportion of susceptibles remains untouched
force of infection decreases e
vacc
vacc e

29. 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 / N = (1-p) A’/L
S / N = A/ L
=> A’ = A/(1-p)
i.e., increase in the
average age of
infection
Proportion e 1 e p
Immunes A ’ L
Age (years)

30. 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
H = p (L-V)/L
=> p = (L-A)/(L-V) 1
Susceptibles p
Immunes
i.e., p bigger than when
vaccination at birth V A L
Age (years)

31. Modelling transmission in a
heterogeneously mixing
population

32. 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)

33. 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, 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*I1 + a*I2 + c*I3 + d*I4 + e*I5
I1 = equilibrium number of infectives in age group 0-4, etc.

34. 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

35. POLYMOD contact survey: the mean number of daily contacts
Country
Number of daily contacts
Relative (95% CI) DE
7.95 1 FI
11.06
1.34 ( ) IT
19.77
2.33 ( ) LU
17.46
2.02 ( ) NL
13.85
1.78 ( ) PL
16.31
1.90 (1.79 – 2.01 GB
11.74
1.40 (1.31 – 1.48)

36. POLYMOD contact survey: numbers of daily contacts

37. POLYMOD contact survey: where and for how long

38. 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

39. References
Fine P.E.M, "Herd immunity: History, Theory, Practice", Epidemiologic Reviews, 15, ,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.
3 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