The dynamics of an SIS epidemic patch model with asymmetric connectivity matrix is analyzed. It is shown that the basic reproduction number [Formula: see text] is strictly decreasing with respect to the dispersal rate of the infected individuals. When [Formula: see text], the model admits a unique endemic equilibrium, and its asymptotic profiles are characterized for small dispersal rates. Specifically, 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. Furthermore, a sufficient and necessary condition is provided to characterize 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, providing a positive answer to an open problem in Allen et al. (SIAM J Appl Math 67(5):1283-1309, 2007).

中文翻译：

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具有不对称连通性矩阵的SIS流行病补丁模型稳态的渐近曲线。

分析了具有不对称连通性矩阵的SIS流行病补丁模型的动力学。结果表明，相对于被感染个体的扩散率，基本繁殖数量[公式：见正文]正在严格降低。当[公式：参见文本]时，该模型承认唯一的地方均衡，并且其渐近曲线的特征在于较小的扩散率。具体而言，当易感个体的扩散率趋于零时，地方病平衡收敛至极限无病平衡，并且在每个低风险斑块上，极限无病平衡具有正数的易感个体。此外，提供了充分必要的条件来表征无疾病的限制平衡，在每个高风险斑块上没有正数的易感个体。我们的结果扩展了对称连接矩阵的早期结果，为Allen等人的开放问题提供了肯定的答案。（SIAM J Appl Math 67（5）：1283-1309，2007）。