### Mathematical Biosciences and Engineering

2014, Issue 4: 995-1001. doi: 10.3934/mbe.2014.11.995

# A note on global stability for malaria infections model with latencies

• Received: 01 June 2013 Accepted: 29 June 2018 Published: 01 March 2014
• MSC : Primary: 92D25, 92D30; Secondary: 37G99.

• 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.

Citation: Jinliang Wang, Jingmei Pang, Toshikazu Kuniya. A note on global stability for malaria infections model with latencies[J]. Mathematical Biosciences and Engineering, 2014, 11(4): 995-1001. doi: 10.3934/mbe.2014.11.995

### Related Papers:

• 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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沈阳化工大学材料科学与工程学院 沈阳 110142

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