Tenth MSU Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 23 (2016), pp. 197-212.

Qualitative analysis of a mathematical model for malaria transmission and its variation

Zhenbu Zhang, Tor A. Kwembe

Abstract:
In this article we consider a mathematical model of malaria transmission. We investigate both a reduced model which corresponds to the situation when the infected mosquito population equilibrates much faster than the human population and the full model. We prove that when the basic reproduction number is less than one, the disease-free equilibrium is the only equilibrium and it is locally asymptotically stable and if the reproduction number is greater than one, the disease-free equilibrium becomes unstable and an endemic equilibrium emerges and it is asymptotically stable. We also prove that, when the reproduction number is greater than one, there is a minimum wave speed $c^*$ such that a traveling wave solution exists only if the wave speed c satisfies $c\geq c^*$. Finally, we investigate the relationship between spreading speed and diffusion coefficients. Our results show that the movements of mosquito population and human population will speed up the spread of the disease.

Published March 21, 2016.
Math Subject Classifications: 35C07, 35K51, 35K58, 35Q92.
Key Words: Malaria; equilibrium; stability; traveling waves; spreading speed.

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Zhenbu Zhang
Department of Mathematics and Statistical Sciences
Jackson State University
Jackson, MS 39217, USA
email: zhenbu.zhang@jsums.edu
Tor A. Kwembe
Department of Mathematics and Statistical Sciences
Jackson State University
Jackson, MS 39217, USA
email: tor.a.kwembe@jsums.edu

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