The size of a major epidemic of a vector-borne disease

Abstract

Based on a simple model due to Dietz, it is shown that the size of a major epidemic of a vector-borne disease with basic reproduction ratio R 0 > 1 is dominated by the size of a standard SIR (susceptible-infected-removed) epidemic with direct host-to-host transmission of disease and the same R 0 . Further bounds and numerical illustrations are provided, broadly spanning situations where the size of the epidemic is short of infecting almost all those susceptible. The total size is moderately sensitive to changes in the population parameters that contribute to R 0 , so that the fluctuating behaviour in 'annual' epidemics is not surprising.Keywords: Epidemic; dengue; basic reproduction ratio; epidemic total size; branching process; deterministic epidemic model; seasonal infection 2010 Mathematics Subject Classification: Primary 92D30 Secondary 62P10 PreambleMy (DJD's) first contact with Søren Asmussen was indirect, through a note that I had written years earlier establishing the criticality condition for a simple two-sex branching process modelled in 1967 much as I heard it from Geoff Watterson while visiting Melbourne University. Søren, then a student in Göteborg, discussed the criticality condition for related models by a much simpler martingale argument, and some years later he came on an extended visit to Canberra. Now an epidemic model can be construed as a branching process that takes place on a pre-existing population, as distinct from a biological population that may grow. Models for epidemic processes are arguably better formulated within a population setting, where at the micro-level we describe what actions and interactions may occur as affecting individual members of the population. Analysis of the model usually describes aggregate behaviour over many individuals, and the stochastic element is largely confined to the model construction stage. The ensuing discussion is no exception.