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Modelling the dynamics of endemic malaria in growing populations. (English) Zbl 1052.92048

Summary: 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 exists and is globally stable when \(R_0\) is below 1.
A peturbation 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.

MSC:

92D30 Epidemiology
34D10 Perturbations of ordinary differential equations
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
34D23 Global stability of solutions to ordinary differential equations
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
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