Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission
Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many...
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AIMS Press
2009-12-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2010.7.51 |
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| author | Wenzhang Huang Maoan Han Kaiyu Liu |
| author_facet | Wenzhang Huang Maoan Han Kaiyu Liu |
| author_sort | Wenzhang Huang |
| collection | DOAJ |
| description | Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained. |
| format | Article |
| id | doaj-art-59907d1611c24211bc200fba005d4bcb |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2009-12-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-59907d1611c24211bc200fba005d4bcb2025-01-24T02:00:16ZengAIMS PressMathematical Biosciences and Engineering1551-00182009-12-0171516610.3934/mbe.2010.7.51Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmissionWenzhang Huang0Maoan Han1Kaiyu Liu2Department of Mathematics, Shanghai Normal University, Shanghai, 200234Department of Mathematics, Shanghai Normal University, Shanghai, 200234Department of Mathematics, Shanghai Normal University, Shanghai, 200234Recently an SIS epidemic reaction-diffusion model with Neumann (or no-flux) boundary condition has been proposed and studied by several authors to understand the dynamics of disease transmission in a spatially heterogeneous environment in which the individuals are subject to a random movement. Many important and interesting properties have been obtained: such as the role of diffusion coefficients in defining the reproductive number; the global stability of disease-free equilibrium; the existence and uniqueness of a positive endemic steady; global stability of endemic steady for some particular cases; and the asymptotical profiles of the endemic steady states as the diffusion coefficient for susceptible individuals is sufficiently small. In this research we will study two modified SIS diffusion models with the Dirichlet boundary condition that reflects a hostile environment in the boundary. The reproductive number is defined which plays an essential role in determining whether the disease will extinct or persist. We have showed that the disease will die out when the reproductive number is less than one and that the endemic equilibrium occurs when the reproductive number is exceeds one. Partial result on the global stability of the endemic equilibrium is also obtained.https://www.aimspress.com/article/doi/10.3934/mbe.2010.7.51stability.epidemic modelreaction-diffusion equations |
| spellingShingle | Wenzhang Huang Maoan Han Kaiyu Liu Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission Mathematical Biosciences and Engineering stability. epidemic model reaction-diffusion equations |
| title | Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission |
| title_full | Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission |
| title_fullStr | Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission |
| title_full_unstemmed | Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission |
| title_short | Dynamics of an SIS reaction-diffusion epidemic model for diseasetransmission |
| title_sort | dynamics of an sis reaction diffusion epidemic model for diseasetransmission |
| topic | stability. epidemic model reaction-diffusion equations |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2010.7.51 |
| work_keys_str_mv | AT wenzhanghuang dynamicsofansisreactiondiffusionepidemicmodelfordiseasetransmission AT maoanhan dynamicsofansisreactiondiffusionepidemicmodelfordiseasetransmission AT kaiyuliu dynamicsofansisreactiondiffusionepidemicmodelfordiseasetransmission |