The risk index for an SIR epidemic model and spatial spreading of the infectious disease
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 Dir...
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AIMS Press
2017-09-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2017081 |
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| author | Min Zhu Xiaofei Guo Zhigui Lin |
| author_facet | Min Zhu Xiaofei Guo Zhigui Lin |
| author_sort | Min Zhu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-aed5f748518a46f686a069eb43daae36 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2017-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-aed5f748518a46f686a069eb43daae362025-01-24T02:40:31ZengAIMS PressMathematical Biosciences and Engineering1551-00182017-09-01145&61565158310.3934/mbe.2017081The risk index for an SIR epidemic model and spatial spreading of the infectious diseaseMin Zhu0Xiaofei Guo1Zhigui Lin2School of Mathematical Science, Yangzhou University, Yangzhou 225002, ChinaDepartment of Mathematics, Anhui Normal University, Wuhu 241000, ChinaSchool of Mathematical Science, Yangzhou University, Yangzhou 225002, ChinaIn 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.https://www.aimspress.com/article/doi/10.3934/mbe.2017081sir modelrisk indexfree boundaryadvection |
| spellingShingle | Min Zhu Xiaofei Guo Zhigui Lin The risk index for an SIR epidemic model and spatial spreading of the infectious disease Mathematical Biosciences and Engineering sir model risk index free boundary advection |
| title | The risk index for an SIR epidemic model and spatial spreading of the infectious disease |
| title_full | The risk index for an SIR epidemic model and spatial spreading of the infectious disease |
| title_fullStr | The risk index for an SIR epidemic model and spatial spreading of the infectious disease |
| title_full_unstemmed | The risk index for an SIR epidemic model and spatial spreading of the infectious disease |
| title_short | The risk index for an SIR epidemic model and spatial spreading of the infectious disease |
| title_sort | risk index for an sir epidemic model and spatial spreading of the infectious disease |
| topic | sir model risk index free boundary advection |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2017081 |
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