Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity

Xiaoyan Zhang; Yuxiang Zhang

doi:10.3934/dcdsb.2018124

Article Contents

2018, Volume 23, Issue 6: 2625-2640. Doi: 10.3934/dcdsb.2018124

This issue Previous Article Fink type conjecture on affine-periodic solutions and Levinson's conjecture to Newtonian systems Next Article

Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity

Xiaoyan Zhang^1, and
Yuxiang Zhang^2, ,

1.
School of Mathematics, Shandong University, Jinan, Shandong 250100, China
2.
School of Mathematics, Tianjin University, Tianjin 300350, China

^* Corresponding author: Yuxiang Zhang
^* Corresponding author: Yuxiang Zhang

Received: May 31, 2017

Revised: September 30, 2017

Early access: April 2018

Published: June 2018

X. Zhang is partially supported by the NSF of China (No. 11571200,11425105), and Y. Zhang is supported in part by the NSF of China (No. 11701415,11601386).

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Abstract

This work is devoted to study the spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. In the case of the spatial domain is bounded and heterogeneous, we assume some key parameters in the model explicitly depend on spatial location. We first define the basic reproduction number $\mathcal{R}_0$ for the disease transmission, which generalizes the existing definition of $\mathcal{R}_0$ for the system in spatially homogeneous environment. Then we establish a threshold type result for the disease eradication ($\mathcal{R}_0 <1$) or uniform persistence ($\mathcal{R}_0>1)$. In the case of the domain is linear, unbounded, and spatially homogenerous, we further establish the existence of traveling wave solutions and the minimum wave speed $c^*$ for the disease transmission. At the end of this work, we characteristic the minimum wave speed $c^*$ and provide a method for the calculation of $c^*$.

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Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 37N25, 92D25.

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Table 1. Biological interpretations for parameters in model (2)

Symbols	Interpretations
$N_0$	Total population size at time $t=0$
$d_i$	Diffusion coefficients for $i=1, 2, 3, 4$
$\mu$	Birth/death rate
$\sigma$	Recovery rate
$\mu_B$	Loss rate of bacteria
$\pi_B$	Growth rate of bacteria
$\beta(x)$	Contact rate with contaminated water at location $x$
$e(x)$	Contribution of each infected person to the population of V. cholerae

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