Modelling the dynamics of endemic malaria in growing populations

A mathematical model for endemic malaria involving variable human and mosquito populations is analysed. A threshold parameter $R_0$ exists and the disease can persist if and only if $R_0$ exceeds $1$. $R_0$ is seen to be a generalisation of the basic reproduction ratio associated with the Ross-Macdonald model for malaria transmission. The disease free equilibrium always exist and is globally stable when $R_0$ is below $1$. A perturbation analysis is used to approximate the endemic equilibrium in the important case where the disease related death rate is nonzero, small but significant. A diffusion approximation is used to approximate the quasi-stationary distribution of the associated stochastic model. Numerical simulations show that when $R_0$ is distinctly greater than $1$, the endemic deterministic equilibrium is globally stable. Furthermore, in quasi-stationarity, the stochastic process undergoes oscillations about a mean population whose size can be approximated by the stable endemic deterministic equilibrium.