Epidemics in heterogeneous populations: aspects of optimal vaccination policies

Three models for epidemics among m groups with applications to human disease, especially socially and geographically heterogeneous populations, are considered. It is shown that the initial growth of each is an increasing function of the Frobenius root of R0, the matrix of reproductive ratios. A new way of looking at optimal vaccination is presented by linking policies to the growth rate of a new epidemic. Aspects of how to minimize the initial growth rate are analysed. In particular, we see that, when R0 has positive eigenvalues, we can find an explicit solution for the final optimal policy, and that there exists an extension of this policy which always gives the least possible growth rate at all stages of vaccination.