Global dynamics of an age-structured malaria model with prevention

Zhong-Kai Guo; Hai-Feng Huo; Hong Xiang; Zhong-Kai Guo; Hai-Feng Huo; Hong Xiang

doi:10.3934/mbe.2019078

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 3: 1625-1653. doi: 10.3934/mbe.2019078

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Global dynamics of an age-structured malaria model with prevention

1.
College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, Gansu, 730050, Peoples Republic of China
2.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, Peoples Republic of China

Received: 18 November 2018 Accepted: 30 January 2019 Published: 26 February 2019

In this paper, we formulate a new age-structured malaria model, which incorporates the age of prevention period of susceptible people, the age of latent period of human and the age of latent period of female Anopheles mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We obtain the basic reproduction number $\mathcal{R}_{0}$ and show that $\mathcal{R}_{0}$ completely determines the global dynamics of the model, that is, if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is globally asymptotically stable, if $\mathcal{R}_{0}>1$, there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief discussion.
- Malaria model,
- prevention age,
- latent age,
- basic reproduction number,
- global stability
Citation: Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Global dynamics of an age-structured malaria model with prevention[J]. Mathematical Biosciences and Engineering, 2019, 16(3): 1625-1653. doi: 10.3934/mbe.2019078

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Abstract

In this paper, we formulate a new age-structured malaria model, which incorporates the age of prevention period of susceptible people, the age of latent period of human and the age of latent period of female Anopheles mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We obtain the basic reproduction number $\mathcal{R}_{0}$ and show that $\mathcal{R}_{0}$ completely determines the global dynamics of the model, that is, if $\mathcal{R}_{0} < 1$, the disease-free equilibrium is globally asymptotically stable, if $\mathcal{R}_{0}>1$, there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs. Finally, we perform some numerical simulations to illustrate our theoretical results and give a brief discussion.

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