Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model

Weiwei Liu; Jinliang Wang; Yuming Chen

doi:10.3934/dcdsb.2020316

Article Contents

2021, Volume 26, Issue 9: 4867-4885. Doi: 10.3934/dcdsb.2020316

This issue Previous Article On a matrix-valued PDE characterizing a contraction metric for a periodic orbit Next Article A stochastic differential equation SIS epidemic model with regime switching

Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model

1.
School of Mathematical Science, Heilongjiang University, Harbin 150080, China
2.
Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5 Canada

^* Corresponding author: Yuming Chen
^* Corresponding author: Yuming Chen

Received: November 30, 2019

Revised: June 30, 2020

Early access: November 2020

Published: September 2021

This research was partially supported by the Graduate Students Innovation Research Program of Heilongjiang University (No. YJSCX2020-211HLJU) (WL); the National Natural Science Foundation of China (Nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (Nos. LC2018002, LH209A021), Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems (JW); and NSERC of Canada (No. RGPIN-2019-05892) (YC)

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Abstract

Taking account of spatial heterogeneity, latency in infected individuals, and time for shed bacteria to the aquatic environment, we build a delayed nonlocal reaction-diffusion cholera model. A feature of this model is that the incidences are of general nonlinear forms. By using the theories of monotone dynamical systems and uniform persistence, we obtain a threshold dynamics determined by the basic reproduction number $ \mathcal {R}_0 $. Roughly speaking, the cholera will die out if $ \mathcal{R}_0<1 $ while it persists if $ \mathcal{R}_0>1 $. Moreover, we derive the explicit formulae of $ \mathcal{R}_0 $ for two concrete situations.

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Mathematics Subject Classification: Primary: 92D30; Secondary: 35B35, 35B40, 35Q92, 37C75.

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