### Mathematical Biosciences and Engineering

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

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

• 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.

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

### Related Papers:

• 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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