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Analysis of a Reaction–Diffusion Cholera Model with Distinct Dispersal Rates in the Human Population

  • Jinliang WangEmail author
  • Jing Wang
Article
  • 22 Downloads

Abstract

In this paper, we investigate the long term behaviors of a reaction-diffusion cholera model in bounded spatial domain with zero-flux boundary condition. The parameters in the model involving space are typical space-dependent due to spatial heterogeneity. We consider the case that the dispersal rates of the susceptible and infected hosts are different, no diffusion term in the cholera equation and bilinear incidence infection mechanism. The existence of global solution, uniform boundedness of solution, asymptotic smoothness of semiflows and existence of global attractor are also addressed. We define the basic reproduction number \(\mathfrak {R}_0\) for the model for the disease transmission in spatially homogeneous environment and establish a threshold type result for the disease eradication or uniform persistence. Considering the cases that either the dispersal rate of the susceptible individuals or the dispersal rate of the infected individuals approaches zero, we investigate the asymptotical profiles of the endemic steady state. Our results suggest that: cholera can be eliminated by limiting the movement of the susceptible individuals, while limiting the mobility of the infected hosts, the infected individuals concentrate on certain points in some circumstances.

Keywords

Spatial heterogeneity Asymptotic profile Basic reproduction number Reaction–diffusion cholera model 

Notes

Acknowledgements

The authors would like to thank the editor and the anonymous reviewer for his/her suggestions that have improved this paper. J.L. Wang was supported by National Natural Science Foundation of China (No. 11871179), Natural Science Foundation of Heilongjiang Province (No. LC2018002) and Heilongjiang Privincial Key Laboratory of the Theory and Computation of Complex Systems.

Compliance with Ethical Standards

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

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© Springer Science+Business Media, LLC, part of Springer Nature 2020

Authors and Affiliations

  1. 1.School of Mathematical ScienceHeilongjiang UniversityHarbinPeople’s Republic of China

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