Ring vaccination and smallpox control

Abstract

We present a stochastic model for the spread of smallpox after a small number of index cases are introduced into a susceptible population. The model describes a branching process for the spread of the infection and the effects of intervention measures. We discuss scenarios in which ring vaccination of direct contacts of infected persons is sufficient to contain an epidemic. Ring vaccination can be successful if infectious cases are rapidly diagnosed. However, because of the inherent stochastic nature of epidemic outbreaks, both the size and duration of contained outbreaks are highly variable. Intervention requirements depend on the basic reproduction number (R0), for which different estimates exist. When faced with the decision of whether to rely on ring vaccination, the public health community should be aware that an epidemic might take time to subside even for an eventually successful intervention strategy.

Figure 1

A, the transmission probability per contact by day of the infectious period; B, the probability distribution of the number of contacts with susceptible persons per day; C, the probability of remaining undiagnosed but infectious case by day of the infectious period; and D, the mean (solid line) and the 2.5% and 97.5% percentiles (dotted lines) of the number of infected persons for 500 simulation runs for an epidemic without any intervention after the introduction of one index case at the beginning of this incubation period at t=0.

Figure 2

The time course of events in the process of transmission and intervention. The success of intervention is essentially determined by the time between start of the infectious period and diagnosis of the index case, and the time between the start of contact tracing and the vaccination of the contact.

Figure 3

The distribution of A, the total number of infected persons excluding those infected contacts who were vaccinated on time to prevent disease, and B, the time to extinction for 500 simulation runs with the baseline intervention parameter values and a basic reproduction number of 5.23. For a basic reproduction number of 10.46 and an increase of the vaccination coverage in the casual contact ring to 80% in C, the distribution of the total number of infected persons, and in D, the distribution of the time to extinction, is shown for 500 simulation runs.

Figure 4

Results for the sensitivity analyses. The total number of infected persons (excluding successfully vaccinated infected contacts) depends on A, the number of index cases starting the epidemic, and B, the day of the infectious period after which the diagnosis of the first case occurs. The time to extinction is shown for C, different numbers of index cases, and D, the day of the infectious period after which the diagnosis of the first case occurs. The quantiles are taken pointwise for 500 simulation runs.

Figure 5

The effective reproduction number R_υ, that determines the success of intervention is shown as a function of the basic reproduction number R₀ for a vaccination coverage of 50% in the casual contact ring. In A, contacts are not monitored after vaccination; in B, all identified contacts are isolated and cause not further transmission. The different lines in A are for different assumptions about how long it takes to trace and vaccinate those contacts. In B, it does not make a difference whether it takes 1, 2, or 3 days to find the contacts. If R₀ is 5, the intervention will be successful in both cases, if R₀ is 10, 50% coverage is no longer sufficient any longer to curb the epidemic.

Figure 6

Here the critical vaccination coverage in the casual contact ring is shown as a function of the basic reproduction number R₀ for different assumptions about the time it takes to diagnose infectious persons. A, for the baseline assumption, that diagnosis is very quick after the beginning of the infectious period, a low coverage is sufficient if R₀ is 5, but for R₀ around 10 the coverage has to be at least 70% for the intervention to be successful. If the probability of being diagnosed shifts to later days of the infectious period, the situation quickly gets out of control and vaccination can no longer curb the epidemic. In B, the same is shown with the difference that here we assume that vaccinated contacts are successfully monitored such that they can no longer produce any secondary infections, even if their vaccination was too late to prevent them from becoming infectious. In this case a later diagnosis is not that influential, but nevertheless, if R₀ is 10, the vaccination coverage (or the percentage of contacts identified and monitored) must be at least 50% to guarantee success. In C, the critical coverage of the casual contact ring is shown as a function of the average number of contacts per day, again for the situation where vaccination is combined with monitoring of contacts. The average number of contacts was varied by varying the number of daily casual contacts. The effect is the similar to that of varying R₀ through the transmission probability per contact as shown in B.

Figure 7

A, the cumulative number of infected persons (excluding successfully vaccinated infected contacts), and B, the time to extinction are shown for various values of the effective reproduction number _Rυ. The quantiles are taken pointwise for 500 simulation runs.