Liu Luju, Cai Weiyun, Wu Yusen
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471003 P.R. China.
Adv Differ Equ. 2012;2012(1):131. doi: 10.1186/1687-1847-2012-131. Epub 2012 Aug 1.
An epidemiological model with suscptibles dispersal between two patches is addressed and discussed. The basic reproduction numbers and are defined as the threshold parameters. It shows that if both and are below unity, the disease-free equilibrium is shown to be globally asymptotically stable by using the comparison principle of the cooperative systems. If is above unity and is below unity, the disease persists in the first patch provided . If is above unity, is below unity, and , the disease persists in the second patch. And if and are above unity, and further and are satisfied, the unique endemic equilibrium is globally asymptotically stable by constructing the Lyapunov function. Furthermore, it follows that the susceptibles dispersal in the population does not alter the qualitative behavior of the epidemiological model.
研究并讨论了一个易感者在两个斑块间扩散的流行病学模型。基本再生数(R_0^1)和(R_0^2)被定义为阈值参数。结果表明，如果(R_0^1)和(R_0^2)都小于1，通过合作系统的比较原理可知无病平衡点是全局渐近稳定的。如果(R_0^1)大于1且(R_0^2)小于1，当(R_0^1R_0^2 > 1)时疾病在第一个斑块中持续存在。如果(R_0^1)大于1，(R_0^2)小于1，且(R_0^1R_0^2 < 1)，疾病在第二个斑块中持续存在。并且如果(R_0^1)和(R_0^2)都大于1，且进一步满足(R_0^1R_0^2 < 1)，通过构造李雅普诺夫函数可知唯一的地方病平衡点是全局渐近稳定的。此外，由此可知人群中易感者的扩散不会改变流行病学模型的定性行为。
