Mathematical Biosciences and Engineering (MBE)

Global dynamics of a vector-host epidemic model with age of infection

Pages: 1159 - 1186, Volume 14, Issue 5/6, October/December 2017      doi:10.3934/mbe.2017060

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Yan-Xia Dang - Department of Public Education, Zhumadian Vocational and Technical College, Zhumadian 463000, China (email)
Zhi-Peng Qiu - School of Science, Nanjing University of Science and Technology, Nanjing 210094, China (email)
Xue-Zhi Li - Department of Mathematics and Physics, Anyang Institute of Technology, Anyang 455000, China (email)
Maia Martcheva - Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL 32611-8105, United States (email)

Abstract: In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The reproductive number $\mathcal R_0$ is derived. By using the method of Lyapunov function, the global dynamics of the PDE model is further established, and the results show that the basic reproduction number $\mathcal R_0$ determines the transmission dynamics of vector-borne diseases: the disease-free equilibrium is globally asymptotically stable if $\mathcal R_0\leq 1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The results suggest that an effective strategy to contain vector-borne diseases is decreasing the basic reproduction number $\mathcal{R}_0$ below one.

Keywords:  Age structure, reproduction number, global stability, vector-borne disease, Lyapunov function.
Mathematics Subject Classification:  Primary: 92D30.

Received: July 2016;      Accepted: December 2016;      Available Online: May 2017.