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Dynamics and Profiles of a Diffusive Cholera Model with Bacterial Hyperinfectivity and Distinct Dispersal Rates

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Abstract

This paper provides an analysis on global dynamics of a diffusive cholera model. We formulate the model by a reaction–diffusion system with space-dependent parameters and bacterial hyperinfectivity, where susceptible and infectious humans disperse with distinct dispersal rates and the hyper-infectious and lower-infectious vibrios in contaminated water are assumed to be immobile in the domain. We first establish the well-posedness of the model. To cope with the lack of compactness of solution semiflow, we verify the asymptotic smoothness of semiflow implied with \(\kappa \)-contraction condition. The basic reproduction number, \(\mathfrak {R}_0\), is identified as a threshold, predicting whether or not the disease extinction and persistence will occur. \(\mathfrak {R}_0\) is also equivalently characterized by some principle spectral conditions of an associated elliptic eigenvalue problem. The asymptotic profiles of the positive steady state are investigated for the cases when the dispersal rate of the susceptible humans is small and large. Our theoretical results indicate that: (1) cholera epidemics will be extinct through restricting the flow of susceptible humans within some extend; (2) the risk of infection will be underestimated if hyperinfectivity is not considered. In a homogeneous case, we also confirm the global attractivity of a unique positive equilibrium by utilizing the technique of Lyapunov function.

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Acknowledgements

The authors would like to thank the editor and the anonymous reviewer for his/her suggestions that have improved this paper. J. Wang was supported by National Natural Science Foundation of China (Nos. 12071115 and 11871179), Natural Science Foundation of Heilongjiang Province (Nos. LC2018002 and LH2019A021) and Heilongjiang Privincial Key Laboratory of the Theory and Computation of Complex Systems.

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  1. School of Mathematical Science, Heilongjiang University, Harbin, 150080, People’s Republic of China

    Jinliang Wang

  2. Department of Mathematics and Information Science, North Minzu University, Yinchuan, 750021, People’s Republic of China

    Xiaoqing Wu

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  1. Jinliang Wang
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Wang, J., Wu, X. Dynamics and Profiles of a Diffusive Cholera Model with Bacterial Hyperinfectivity and Distinct Dispersal Rates. J Dyn Diff Equat 35, 1205–1241 (2023). https://doi.org/10.1007/s10884-021-09975-3

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