# American Institute of Mathematical Sciences

April  2017, 14(2): 559-579. doi: 10.3934/mbe.2017033

## Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model

 1 Department of Mathematics, University of Rochester, Rochester, NY 14627, USA 2 Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164-3113, USA

Received  May 10, 2016 Accepted  September 19, 2016 Published  October 2016

Fund Project: This work was partially supported by a grant from the Simons Foundation (#317047 to Xueying Wang).

We study the global stability issue of the reaction-convection-diffusion cholera epidemic PDE model and show that the basic reproduction number serves as a threshold parameter that predicts whether cholera will persist or become globally extinct. Specifically, when the basic reproduction number is beneath one, we show that the disease-free-equilibrium is globally attractive. On the other hand, when the basic reproduction number exceeds one, if the infectious hosts or the concentration of bacteria in the contaminated water are not initially identically zero, we prove the uniform persistence result and that there exists at least one positive steady state.

Citation: Kazuo Yamazaki, Xueying Wang. Global stability and uniform persistence of the reaction-convection-diffusion cholera epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 559-579. doi: 10.3934/mbe.2017033
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##### References:
Definition of parameters in model (1)
 Parameter Definition $b$ Recruitment rate of susceptible hosts $d$ Natural death rate of human hosts $\gamma$ Recovery rate of infectious hosts $\sigma$ Rate of host immunity loss $\delta$ Natural death rate of bacteria $\xi$ Shedding rate of bacteria by infectious hosts $\beta_{1}$ Direct transmission parameter $\beta_{2}$ Indirect transmission parameter $K$ Half saturation rate of bacteria $U$ Bacterial convection coefficient $K_{B}$ Maximal carrying capacity of bacteria in the environment
 Parameter Definition $b$ Recruitment rate of susceptible hosts $d$ Natural death rate of human hosts $\gamma$ Recovery rate of infectious hosts $\sigma$ Rate of host immunity loss $\delta$ Natural death rate of bacteria $\xi$ Shedding rate of bacteria by infectious hosts $\beta_{1}$ Direct transmission parameter $\beta_{2}$ Indirect transmission parameter $K$ Half saturation rate of bacteria $U$ Bacterial convection coefficient $K_{B}$ Maximal carrying capacity of bacteria in the environment
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