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Global asymptotic stability of endemic equilibria for a diffusive SIR epidemic model with saturated incidence and logistic growth

The fourth author is supported by JSPS KAKENHI Grant Number JP21K03278

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  • This paper deals with the diffusive epidemic model with saturated incidence and logistic growth,

    $ \begin{align*} \begin{cases} \dfrac{\partial S}{\partial t} = d_S \Delta S - \dfrac{\beta S I}{1+\alpha I} + rS\left(1- \dfrac{S}{K} \right), &x \in \Omega, \ t>0, \\ \dfrac{\partial I}{\partial t} = d_I \Delta I + \dfrac{\beta S I}{1+\alpha I} - \gamma I, &x \in \Omega, \ t>0, \end{cases} \end{align*} $

    where $ \Omega \subset \mathbb{R}^N $ $ (N \in \mathbb{N}) $ is a bounded domain with smooth boundary and $ d_S, d_I, K, r, \alpha, \beta, \gamma >0 $ are constants. Setting $ \mathcal{R}_0: = \frac{K \beta}{\gamma} $, Avila-Vales et al. [1] succeeded in showing that if $ \mathcal{R}_0\leq1 $, then the disease-free equilibrium $ (K, 0) $ of the model with saturated treatment is globally asymptotically stable, whereas in the case $ \mathcal{R}_0>1 $ the model admits a constant endemic equilibrium $ (S^*, I^*) $ ($ S^*, I^*>0 $), and it is unknown whether $ (S^*, I^*) $ is globally asymptotically stable or not. The purpose of this paper is to establish that the constant endemic equilibrium of the above model is globally asymptotically stable by constructing a strict Lyapunov functional. The construction is carried out by optimizing a function of two real variables through straightforward calculations, division into some cases and arrangement of several conditions. Moreover, to show that the functional is strict, some auxiliary function is introduced.

    Mathematics Subject Classification: Primary: 35K40; Secondary: 35B40, 92D30.

    Citation:

    \begin{equation} \\ \end{equation}
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  • Figure 4.1.  The solution of (1.1) with the parameters given in (4.1). The distributions of $ S, I $ become milder with time. Also, the asymptotic distributions of $ S, I $ are stable when $ t \to \infty $

    Figure 4.2.  The solution of (1.1) with the parameters given in (4.2). Even though the parameters do not satisfy the condition (1.3), the asymptotic distributions of $ S, I $ converge to constant values

    Table 1.  Known results

    Local asymptotic stability Global asymptotic stability
    The DFE
    ($ \mathcal{R}_0 \le 1 $)
    Pérez et al. [14],
    Avila-Vales et al. [1]
    Avila-Vales et al. [1]
    The EE
    ($ \mathcal{R}_0> 1 $)
    Pérez et al. [14] Unknown
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  • [1] E. Avila-Vales, G. E. García-Almeida and Á. G. C. Pérez, Qualitative analysis of a diffusive SIR epidemic model with saturated incidence rate in a heterogeneous environment, J. Math. Anal. Appl., 503 (2021), Paper No. 125295, 35 pp. doi: 10.1016/j.jmaa.2021.125295.
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    [4] R. Cui, Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 2997-3022.  doi: 10.3934/dcdsb.2020217.
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    [7] G. Guan and Z. Guo, Bifurcation and stability of a delayed SIS epidemic model with saturated incidence and treatment rates in heterogeneous networks, Appl. Math. Model., 101 (2022), 55-75.  doi: 10.1016/j.apm.2021.08.024.
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