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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.