Supplement to: Still protected against smallpox? Estimation of the duration of vaccine-induced immunity against smallpox. Epidemiology
Model description
We assume a population with a vaccination coverage p where vaccination gives complete, but temporary protection. We further assume that there are two subpopulations among the unvaccinated fraction of the total population. Whereas a fraction 1q of the unvaccinated population is randomly mixing with the vaccinated population and experiences the same force infection (FOI)

1
( a ) as the vaccinated fraction, the remaining fraction q lives in separation and experiences a higher FOI

2
( a ) (Fig. S1-1a). The proportion protected by vaccination declines over age at rate

( a ). Vaccinated individuals may experience booster events during their protected stage and acquire life-long immunity at a rate

1
( a ). Fig. S1-1b shows the compartmental diagram of the different states of infection among each subpopulation.
The vaccinated subpopulation splits up into V ( a ) and S v
( a ) (i.e., those who are protected and susceptible due to loss of immunity). Assuming that the immunity loss
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rate

and the FOI

1
are independent of age, the equations for vaccinated individuals can be given by: da v da
 
 
1
 
1 v
( )
(S1)
The equation for S u1
( a ), the susceptible unvaccinated individuals, who experience the same FOI

1
as the vaccinated ones, is given by: dS ( ) u 1
  
S ( )
1 u 1
(S2) da
In the same manner, the equation for S u2
( a ), the susceptible unvaccinated individuals who live in separation, is given by: dS a u 2
( )
  
S ( )
2 u 2
(S3) da whereby

2
>

1
.
Description of parameters
In our sensitivity analysis, we assume that

1
and

2
are independent of age in order to clarify the impact of heterogeneous mixing on the estimation results of key parameters. For unvaccinated individuals, the average ages at infection are given by
A
1
= 1/

1
and A
2
= 1/

2
, respectively. Assuming an average life expectancy L = 70 years, the basic reproduction number is approximated by L / A
1
and L / A
2
, respectively
M1,M2
. We get R
01
= 1.4 and R
02
= 5.6 by choosing

1
= 0.02 /year and

2
=
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0.08 /year, respectively.
The decay rate of immunity

is also assumed to be constant over age. Thus, immunity is lost exponentially and its half-life t m
is given by: t m
 ln(2)
 (S4) where we get t m
= 17.3 years by choosing

= 0.04 /year.
Simulation results
Using a vaccination coverage p = 0.8 and the above given parameter values, we obtain age-specific incidences as shown in Fig. S1-2. Those who were vaccinated show an elevated age at infection compared to those unvaccinated. The pattern of incidence among unvaccinated individuals in the subpopulation with the FOI

2 shows a steeper decline at an earlier age than those who experience the same FOI

1 as vaccinated individuals.
In the following, we investigate how our estimates of the duration of vaccine-induced immunity and of the vaccination coverage are influenced by the fraction q . Fig. S1-3 shows the estimates of t m
and p , resulting from different values of q , based on our assumptions in the main text. The straight lines show the true values used in the simulations. For small q , our estimates appropriately reflect the true
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duration of immunity and vaccination coverage. For larger values of q , we slightly overestimate t m
and p . These differences reflect the impact of the indirect effect (herd immunity) on our estimates
M3
.
References
M1. Dietz K. Transmission and control of arbovirus diseases. In Epidemiology (ed.
Ludwig D, Cooke KL), Philadelphia: Society for Industrial and Applied Mathematics;
1975, pp. 104-121.
M2. Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control .
New York: Oxford University Press; 1991.
M3. Halloran ME, Haber M, Longini IM Jr. Interpretation and estimation of vaccine efficacy under heterogeneity. Am J Epidemiol 1992;136: 328-343.
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Fig. S1-1 a) b)
Fig. S1-1. a) Population with fraction p vaccinated and 1p unvaccinated. The unvaccinated hosts are divided into two different subpopulations: Subpopulation with fraction q and 1q experience different forces of infection. b) Schematic diagram of transmission of smallpox in a hypothetical community as given in (a).
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Fig. S1-2
60
50
40
30
80
70
Vaccinated cases
Unvaccinated cases (total)
Unvaccinated cases living in the vaccinated community
Unvaccinated cases living in separation
20
10
0
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Age [years]
Fig. S1-2. Simulated incidence of vaccinated and unvaccinated cases in a heterogeneously mixing population as described above (where q = 0.3 and the population size N = 10,000).
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Fig. S1-3 a)
25
20
15
10
5
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fraction of those experiencing a higher FOI among unvaccinated ( q ) b)
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0.78
0.77
0.76
0.75
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fraction of those experiencing a higher FOI among unvaccinated ( q )
Fig. S1-3. Estimates of (a) the median duration of vaccine-induced immunity and
(b) the vaccination coverage. Estimates were based on simulations of a heterogeneously mixing population as described above. Horizontal lines show the true values used in the simulations.
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