Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, Shandong, People's Republic of China.
Department of Mathematics, William & Mary, Williamsburg, VA, 23187-8795, USA.
J Math Biol. 2020 Jun;80(7):2327-2361. doi: 10.1007/s00285-020-01497-8. Epub 2020 May 6.
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).
分析了具有非对称连接矩阵的 SIS 传染病斑块模型的动力学。结果表明，基本再生数 [Formula: see text] 随感染个体的扩散率严格递减。当 [Formula: see text] 时，模型存在唯一的地方病平衡点，并且对小的扩散率刻画了其渐近特征。具体而言，当易感个体的扩散率趋于零时，地方病平衡点收敛于一个限制的无病平衡点，并且在每个低风险斑块上的限制无病平衡点有一个正的易感个体数。此外，提供了一个充分必要条件来刻画在每个高风险斑块上的限制无病平衡点没有正的易感个体数。我们的结果扩展了具有对称连接矩阵的早期结果，为 Allen 等人（SIAM J Appl Math 67(5):1283-1309, 2007）中的一个开放性问题提供了一个肯定的答案。
