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Asymptotic profiles of a diffusive SIRS epidemic model with standard incidence mechanism and a logistic source

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Abstract

The asymptotic profiles of the endemic equilibrium play an important role in determining whether or not eliminating the infectious disease when considering the dispersal rates of populations approach to zero or infinity. This paper aims to provide a qualitative analysis of a reaction–diffusion Susceptible–Infective–Recovered–Susceptible epidemic model with standard incidence mechanism and a logistic source in a spatially heterogeneous environment. For this purpose, we first estimate the uniform bounds of the solution of the model. By the theory of uniform persistence, we explore the threshold-type result of the model in terms of the basic reproduction number \(\Re _0\). Compared to the results in Han et al. (Z Angew Math Phys 71:190, 2020) where the growth of susceptible population is adopt by linear source, our theoretical results reveal that controlling the dispersal rates of population cannot eradicate the disease, and the infection component of the steady state solution approaches to nonzero level when the dispersal rates of populations approach to zero or infinity. We also obtain that varying total population enhances persistence of infectious disease, the results are consistent with Li et al. (Z Angew Math Phys 68:96, 2017).

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Authors and Affiliations

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

    Yifei Pan & Jinliang Wang

  2. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Siyao Zhu

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  1. Yifei Pan
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  2. Siyao Zhu
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  3. Jinliang Wang
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Correspondence to Jinliang Wang.

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J. Wang was supported by National Natural Science Foundation of China (Nos. 12071115, 11871179) and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems. S. Zhu was supported by the National Natural Science Foundation of China (Nos. 12171234,11631006, 11790272), the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and the Fundamental Research Funds for the Central Universities.

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Pan, Y., Zhu, S. & Wang, J. Asymptotic profiles of a diffusive SIRS epidemic model with standard incidence mechanism and a logistic source. Z. Angew. Math. Phys. 73, 36 (2022). https://doi.org/10.1007/s00033-021-01667-8

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  • DOI: https://doi.org/10.1007/s00033-021-01667-8

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