A note on global stability for malaria infections  model with latencies

Jinliang Wang; Jingmei Pang; Toshikazu Kuniya

doi:10.3934/mbe.2014.11.995

Article Contents

2014, Volume 11, Issue 4: 995-1001. Doi: 10.3934/mbe.2014.11.995

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A note on global stability for malaria infections model with latencies

1.
School of Mathematical Science, Heilongjiang University, Harbin 150080
2.
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba Meguro-ku, Tokyo 153-8914

Received: May 31, 2013

Revised: September 30, 2013

Published: February 2014

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Abstract

A recent paper [Y. Xiao and X. Zou, On latencies in malaria infections and their impact on the disease dynamics, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.

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Mathematics Subject Classification: Primary: 92D25, 92D30; Secondary: 37G99.

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