一个具有暂时免疫和非线性接触率的SIS流行病模型的分析

摘要

本文研究一个具有时滞,一般接触率,常数出生和疾病引起死亡的流行病模型.假设时滞表示暂时免疫期,即恢复者再次变成易感者所需要的时间,同时在模型中考虑了对易感者和恢复者的接种.本文得到了基本再生数R0.分析了模型的无病平衡点和地方病平衡点的存在性.通过Hurwitz准则,研究了无病平衡点和地方病平衡点的局部渐近稳定性.通过Liapunov泛函和Lasalle不变原理,证明了无病平衡点的全局渐近稳定性及在双线性接触率的情况下地方病平衡点的全局渐近稳定性.研究结果表明:R0与对易感者的有效接种率P有关,并且通过增加接种率P可以根除疾病.最后给出了数值模拟.%Studied in this paper is an SIS epidemic model with a time delay, a general contact rate, constant recruitment and disease-caused death. It is assumed that the time delay represents the temporary immunity period, i.e., the time from recovered to becoming the susceptible again, which incorporates the infected recovery and the recovery of the susceptible due to vaccinating. The basic reproduction number is found. The existence of disease-free and endemic equilibria is analyzed. By the Hurwitz criterion, the local asymptotic stability of disease-free and endemic equilibria is investigated. By means of the Liapunov functional and Lasalle's invariant principle, we prove the global stability of the disease-free equilibrium and the endemic equilibrium in a special case that the incidence rate is bilinear. It is shown that the reproduction number is related with the rate of efficient vaccination for the susceptible, and the disease may be eradicated by increasing the efficient vaccination rate.Numerical simulations support our analytical conclusions.