Abstract
We answer the stability question of the large scale SIS model describing transmission of highland malaria in Western Kenya in a patchy environment, formulated in [1]. There are two equilibrium states and their stability depends on the basic reproduction number, Ro [2]. If Ro ≤1, the disease-free steady solution is globally asymptotically stable and the disease always dies out. If Ro >1, there exists a unique endemic equilibrium which is globally stable and the disease persists. Application is done on data from Western Kenya. The age structure reduces the level of infection and the populations settle to the equilibrium faster than in the model without age structure.
Highlights
We recall the large scale system developed in [1] reduced into a compact form as ( ) = X diag (1− X ) X + − + X (1) whereX = ( x, y, z), is a vector representing; x is the proportion of infectious children, y is the proportion of infectious adults, and z is the proportion of infectious mosquitoes.How to cite this paper: Wairimu, J., Gauthier, S. and Ogana, W. (2014) Mathematical Analysis of a Large Scale Vector SIS Malaria Model in a Patchy Environment
This implies that J X * is Hurwitz [10] [11]. This completes the proof of the global asymptotic stability of the endemic equilibrium
Highland malaria in Western Kenya remains a source of mortality and morbidity
Summary
We recall the large scale system developed in [1] reduced into a compact form as ( ) = X diag (1− X ) X + − + X (1). The authors used the preceding matrices and the vector X = ( x, y, z) to rewrite Equation (10) in [1] in a compact form as ( ) = X ′ diag (1− X ) X + − + X. This system evolves on the unit cube of 3n. When 0 < 1, the DFE is locally asymptotically stable, and if 0 > 1 the DFE is unstable, see [3] [4]
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