Modelling trypanosomiasis prevalence and periodic epidemics and epizootics

Existing mathematical models of trypanosomiasis epidemiology and epizootiology are extended by including some relevant biology of the disease vector, the tsetse fly. Rickettsia-like organisms, or RLO, are a vertically transmitted symbiont of tsetse, which confer an increased susceptibility to trypanosomiasis infection. Tsetse populations are also limited by density-dependent starvation. Modelling leads to the prediction of a stable dimorphism with a fraction of tsetse possessing RLO. The equilibrium prevalence of trypanosomiasis in the vertebrate hosts is no longer in RLO models determined simply by such traditional parameters as vectorial capacity. Only the RLO-positive tsetse carry infection, and their number is itself regulated by trypanosomiasis prevalence. The result of a naive model is that controlling tsetse numbers does not decrease prevalence until all tsetse are RLO-positive. However, under the density-dependent starvation model derived in this paper, the relative mortality of RLO-positive flies is greater at lower tsetse numbers. This tips the balance towards lower equilibrium prevalence of trypanosomiasis as tsetse numbers are decreased. The presence of RLO also gives rise to long-term oscillations in trypanosomiasis prevalence in humans and animals. However, when another mechanism that can also cause periodic epizootics (of shorter periodicity) is included, namely host immunity, the two epizootic processes combine to produce periodic epizootics (and therefore epidemics) at a single frequency. There are two decaying modes, one in which the tsetse population size quickly reaches equilibrium in a few weeks, and a second very slowly decaying mode in which host immunity and RLO effects interact. The equilibrium reached is shown to be asymptotically stable. In view of the seeming importance of RLO in trypanosomiasis epidemiology, it is important that field biologists enable RLO models to be validated by measuring the proportion of tsetse with RLO, in conjunction with vector density and trypanosomiasis prevalence and incidence in tsetse and vertebrate hosts.