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Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

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Abstract

This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number \(\mathfrak {R}_0>1\), there exists a critical wave speed \(c^*>0\), such that for each \(c \ge c^*\) the system admits a nontrivial TWS and for \(c<c^*\) there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behavior of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.

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Acknowledgements

The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions which led to a significant improvement of this work. R Zhang and S Liu were supported by Natural Science Foundation of China (11871179; 11771374), J. Wang was supported by National Natural Science Foundation of China (nos. 12071115, 11871179), Natural Science Foundation of Heilongjiang Province (nos. LC2018002, LH2019A021) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems.

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Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Ran Zhang

  2. School of Mathematical Sciences, Heilongjiang University, Harbin, 150080, China

    Jinliang Wang

  3. School of Mathematical Sciences, Tiangong University, Tianjin, 300387, China

    Shengqiang Liu

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  1. Ran Zhang
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  2. Jinliang Wang
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Correspondence to Shengqiang Liu.

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Communicated by Mary Pugh.

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Zhang, R., Wang, J. & Liu, S. Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model. J Nonlinear Sci 31, 10 (2021). https://doi.org/10.1007/s00332-020-09656-3

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