Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion <em>SVIR</em> epidemic model with relaps

Cheng-Cheng Zhu; Jiang Zhu; Xiao-Lan Liu; Cheng-Cheng Zhu; Jiang Zhu; Xiao-Lan Liu

doi:10.3934/mbe.2019295

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 5897-5922. doi: 10.3934/mbe.2019295

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Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relaps

1.
School of Science, Jiangnan University, Wuxi, Jiangsu 214122, P.R. China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P.R. China
3.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, P.R. China
4.
School of Arts and Science, Suqian College, Suqian, Jiangsu 223800, P.R. China

Received: 27 February 2019 Accepted: 27 May 2019 Published: 24 June 2019

In this paper by adding the factors of disease relapse and vaccination in the space hetero-geneous environment, we establish and discuss a class of reaction-diffusion SVIR model with relapse and a varying external source in spatial heterogeneous environment. By applying a different method than the Lyapunov function, we study the long-term dynamic behavior of this model by means of global exponential attractor theory and gradient flow method. The global asymptotic stability and the persistence of epidemic are proved. To test the validity of our theoretical results, we choose some specific epidemic disease with some more practical and more definitive official data to simulate the global stability and exponential attraction of the model. The simulation results showed that the factors of disease relapse, vaccination and spatial heterogeneity had a great influence on the persists uniformly of the disease.
- reaction-diffusion; global exponential attract set; gradient flow; spatial heterogeneous environment; relapse
Citation: Cheng-Cheng Zhu, Jiang Zhu, Xiao-Lan Liu. Influence of spatial heterogeneous environment on long-term dynamics of a reaction-diffusion SVIR epidemic model with relaps[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5897-5922. doi: 10.3934/mbe.2019295

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Abstract

In this paper by adding the factors of disease relapse and vaccination in the space hetero-geneous environment, we establish and discuss a class of reaction-diffusion SVIR model with relapse and a varying external source in spatial heterogeneous environment. By applying a different method than the Lyapunov function, we study the long-term dynamic behavior of this model by means of global exponential attractor theory and gradient flow method. The global asymptotic stability and the persistence of epidemic are proved. To test the validity of our theoretical results, we choose some specific epidemic disease with some more practical and more definitive official data to simulate the global stability and exponential attraction of the model. The simulation results showed that the factors of disease relapse, vaccination and spatial heterogeneity had a great influence on the persists uniformly of the disease.

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