# American Institute of Mathematical Sciences

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October  2018, 15(5): 1099-1116. doi: 10.3934/mbe.2018049

## An age-structured vector-borne disease model with horizontal transmission in the host

 1 College of Mathematics and Information Science, Xinyang Normal University, Xinyang 464000, Henan, China 2 Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario, N2L 3C5, Canada

1Correspondence email: xywangxia@163.com(X. Wang)

Received  April 04, 2017 Accepted  March 22, 2018 Published  May 2018

We concern with a vector-borne disease model with horizontal transmission and infection age in the host population. With the approach of Lyapunov functionals, we establish a threshold dynamics, which is completely determined by the basic reproduction number. Roughly speaking, if the basic reproduction number is less than one then the infection-free equilibrium is globally asymptotically stable while if the basic reproduction number is larger than one then the infected equilibrium attracts all solutions with initial infection. These theoretical results are illustrated with numerical simulations.

Citation: Xia Wang, Yuming Chen. An age-structured vector-borne disease model with horizontal transmission in the host. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1099-1116. doi: 10.3934/mbe.2018049
##### References:

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##### References:
When $R_0<1$, the infection-free equilibrium $E^0$ of (2) is globally asymptotically stable. Here since $E_h(t)$ converges to $0$ very fast, we use the time interval $[0, 100]$ different from the interval $[0, 1000]$ for other components
When $R_0>1$, the infected equilibrium $E^{\ast}$ of (2) is globally asymptotically stable
Biological meanings of parameters in (1)
 Parameter Meaning $\lambda_h$ Per capita host birth rate $\mu_h$ Host death rate $\beta_1$ Rate of horizontal transmission of the disease $\beta_2$ Rate of a pathogen carrying mosquito biting susceptible host $\alpha_h$ Inverse of host latent period $\delta_h$ Disease related death rate of host $\gamma_h$ Recovery rate of host $\lambda_v$ Per capita vector birth rate $k$ Biting rate of per susceptible vector per host per unit time $\mu_v$ Vector death rate $\alpha_v$ Inverse of vector latent period $\delta_v$ Disease related death rate of vectors
 Parameter Meaning $\lambda_h$ Per capita host birth rate $\mu_h$ Host death rate $\beta_1$ Rate of horizontal transmission of the disease $\beta_2$ Rate of a pathogen carrying mosquito biting susceptible host $\alpha_h$ Inverse of host latent period $\delta_h$ Disease related death rate of host $\gamma_h$ Recovery rate of host $\lambda_v$ Per capita vector birth rate $k$ Biting rate of per susceptible vector per host per unit time $\mu_v$ Vector death rate $\alpha_v$ Inverse of vector latent period $\delta_v$ Disease related death rate of vectors
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