Analysis of a diffusion epidemic SIR model with saturated treatment in a heterogeneous environment

Zhengran Zhan; Chunxiao Zhang; Zhengran Zhan; Chunxiao Zhang

doi:10.3934/era.2025277

Electronic Research Archive

2025, Volume 33, Issue 10: 6267-6297. doi: 10.3934/era.2025277

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Research article

Analysis of a diffusion epidemic SIR model with saturated treatment in a heterogeneous environment

Zhengran Zhan ¹,
Chunxiao Zhang ^{2
,
,}

1.
Department of Basic Course Teaching, Baoding University of Technology, Baoding 071002, China
2.
China Center for Human Capital and Labor Market Research, Central University of Finance and Economics, Beijing 100081, China

Received: 19 July 2025 Revised: 14 September 2025 Accepted: 24 September 2025 Published: 24 October 2025

The spread of infectious diseases is profoundly influenced by spatial heterogeneity and the availability of medical resources. While reaction-diffusion epidemic models have been extensively studied, there has been little research on SIR models that incorporate a spatially heterogeneous standard incidence rate and a saturated treatment function, which is crucial for modeling the limited capacity of health care systems. To address this, we proposed a novel diffusive SIR epidemic model in a heterogeneous environment. Methodologically, we defined the basic reproduction number $ R_0 $ and employed Lyapunov functions, the theory of monotone dynamical systems, and asymptotic analysis to investigate the dynamics. Our key results showed that if $ R_0 < 1 $, the disease-free equilibrium is globally asymptotically stable, implying epidemic extinction. If $ R_0 > 1 $, the disease becomes uniformly persistent, and an endemic equilibrium exists. Furthermore, we derived the asymptotic profiles of the endemic equilibrium as the diffusion rates of susceptible or infected populations approached zero. Numerical simulations not only validated our theoretical findings but also demonstrated that increasing medical resources or reducing spatial heterogeneity can effectively lower the infection peak and help control the disease. These results provide theoretical guidance for designing effective public health policies.
- diffusive epidemic system,
- basic reproduction number,
- spatial heterogeneity,
- the saturated treatment function
Citation: Zhengran Zhan, Chunxiao Zhang. Analysis of a diffusion epidemic SIR model with saturated treatment in a heterogeneous environment[J]. Electronic Research Archive, 2025, 33(10): 6267-6297. doi: 10.3934/era.2025277

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Abstract

The spread of infectious diseases is profoundly influenced by spatial heterogeneity and the availability of medical resources. While reaction-diffusion epidemic models have been extensively studied, there has been little research on SIR models that incorporate a spatially heterogeneous standard incidence rate and a saturated treatment function, which is crucial for modeling the limited capacity of health care systems. To address this, we proposed a novel diffusive SIR epidemic model in a heterogeneous environment. Methodologically, we defined the basic reproduction number $ R_0 $ and employed Lyapunov functions, the theory of monotone dynamical systems, and asymptotic analysis to investigate the dynamics. Our key results showed that if $ R_0 < 1 $, the disease-free equilibrium is globally asymptotically stable, implying epidemic extinction. If $ R_0 > 1 $, the disease becomes uniformly persistent, and an endemic equilibrium exists. Furthermore, we derived the asymptotic profiles of the endemic equilibrium as the diffusion rates of susceptible or infected populations approached zero. Numerical simulations not only validated our theoretical findings but also demonstrated that increasing medical resources or reducing spatial heterogeneity can effectively lower the infection peak and help control the disease. These results provide theoretical guidance for designing effective public health policies.

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