The risk index for an SIR epidemic model and spatial spreading of the infectious disease

Min Zhu; Xiaofei Guo; Zhigui Lin; Min Zhu; Xiaofei Guo; Zhigui Lin

doi:10.3934/mbe.2017081

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1565-1583. doi: 10.3934/mbe.2017081

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The risk index for an SIR epidemic model and spatial spreading of the infectious disease

1.
School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
2.
Department of Mathematics, Anhui Normal University, Wuhu 241000, China

Received: 05 May 2016 Accepted: 19 September 2016 Published: 01 October 2017
MSC : Primary: 35K51, 35R35; Secondary: 35B40, 92D25

In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number $R_0^{DA}$ for an associated model with Dirichlet boundary condition, we introduce the risk index $R^F_0(t)$ for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if $R^F_0(t_0)≥q 1$ for some $t_0$ and the disease is vanishing if $R^F_0(∞) \lt 1$, while if $R^F_0(0) \lt 1$, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.
- SIR model,
- risk index,
- free boundary,
- advection
Citation: Min Zhu, Xiaofei Guo, Zhigui Lin. The risk index for an SIR epidemic model and spatial spreading of the infectious disease[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1565-1583. doi: 10.3934/mbe.2017081

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Abstract

In this paper, a reaction-diffusion-advection SIR model for the transmission of the infectious disease is proposed and analyzed. The free boundaries are introduced to describe the spreading fronts of the disease. By exhibiting the basic reproduction number $R_0^{DA}$ for an associated model with Dirichlet boundary condition, we introduce the risk index $R^F_0(t)$ for the free boundary problem, which depends on the advection coefficient and time. Sufficient conditions for the disease to prevail or not are obtained. Our results suggest that the disease must spread if $R^F_0(t_0)≥q 1$ for some $t_0$ and the disease is vanishing if $R^F_0(∞) \lt 1$, while if $R^F_0(0) \lt 1$, the spreading or vanishing of the disease depends on the initial state of infected individuals as well as the expanding capability of the free boundary. We also illustrate the impacts of the expanding capability on the spreading fronts via the numerical simulations.

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