Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix

Shanshan Chen; Junping Shi; Zhisheng Shuai; Yixiang Wu

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arXiv · 1911.02219

Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix

Shanshan Chen·Junping Shi·Zhisheng Shuai·Yixiang Wu

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Abstract

The dynamics of an SIS epidemic patch model with asymmetric connectivity matrix is analyzed. It is shown that the basic reproduction number $R_0$ is strictly decreasing with respect to the dispersal rate of the infected individuals, and the model has a unique endemic equilibrium if $R_0>1$. The asymptotic profiles of the endemic equilibrium for small dispersal rates are characterized. In particular, it is shown that the endemic equilibrium converges to a limiting disease-free equilibrium as the dispersal rate of susceptible individuals tends to zero, and the limiting disease-free equilibrium has a positive number of susceptible individuals on each low-risk patch. Moreover a sufficient and necessary condition is found to guarantee that the limiting disease-free equilibrium has no positive number of susceptible individuals on each high-risk patch. Our results extend earlier results for symmetric connectivity matrix, and we also partially solve an open problem by Allen et al. (SIAM J. Appl. Math., 67: 1283-1309, 2007).

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Shanshan Chen, Junping Shi, Zhisheng Shuai, Yixiang Wu. 2019-11-06. Asymptotic profiles of the steady states for an SIS epidemic patch model with asymmetric connectivity matrix. https://arxiv.org/abs/1911.02219

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