Stochastic multitype epidemics in a community of households: estimation of threshold parameter ... and secure vaccination coverage

This paper is concerned with estimation of the threshold parameter $R_{\ast}$ for a stochastic model for the spread of a susceptible . infective . removed epidemic among a closed, finite population that contains several types of individual and is partitioned into households. It turns out that $R_{\ast}$ cannot be estimated consistently from final outcome data, so a Perron-Frobenius argument is used to obtain sharp lower and upper bounds for $R_{\ast}$, which can be estimated consistently. Determining the allocation of vaccines that reduces the upper bound for $R_{\ast}$ to its threshold value of one, thus preventing the occurrence of a major outbreak, with minimum vaccine coverage is shown to be a linear programming problem. The estimates of $R_{\ast}$, before and after vaccination, and of the secure vaccination coverage, i.e. the proportion of individuals that have to be vaccinated to reduce the upper bound for $R_{\ast}$ to 1 assuming an optimal vaccination scheme, are equipped with standard errors, thus yielding conservative confidence bounds for these key epidemiological parameters. The methodology is illustrated by application to data on influenza outbreaks in Tecumseh, Michigan.