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November  2004, 4(4): 1173-1202. doi: 10.3934/dcdsb.2004.4.1173

Modelling the dynamics of endemic malaria in growing populations

G.A. Ngwa 1,

1. 

The Abdus Salam International Centre for Theoretical Physics, Trieste 34100, Italy

Received  January 2003 Revised  March 2004 Published  August 2004

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: disease related deaths, Threshold parameter, diffusion approximation..
Mathematics Subject Classification: 92D30, 60J2.
Citation: G.A. Ngwa. Modelling the dynamics of endemic malaria in growing populations. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1173-1202. doi: 10.3934/dcdsb.2004.4.1173
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