Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Modelling the dynamics of endemic malaria in growing populations

Pages: 1173 - 1202, Volume 4, Issue 4, November 2004      doi:10.3934/dcdsb.2004.4.1173

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G.A. Ngwa - The Abdus Salam International Centre for Theoretical Physics, Trieste 34100, Italy (email)

Abstract: 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.

Keywords:  Threshold parameter, disease related deaths, diffusion approximation.
Mathematics Subject Classification:  92D30, 60J27.

Received: January 2003;      Revised: March 2004;      Published: August 2004.