We use the epidemic threshold parameter, \({{\mathcal {R}}}_{0}\), and invariant rectangles to investigate the global asymptotic behavior of solutions of the density-dependent discrete-time SI epidemic model where the variables \(S_{n}\) and \(I_{n}\) represent the populations of susceptibles and infectives at time \(n = 0,1,\ldots \), respectively. The model features constant survival “probabilities” of susceptible and infective individuals and the constant recruitment per the unit time interval \([n, n+1]\) into the susceptible class. We compute the basic reproductive number, \({{\mathcal {R}}}_{0}\), and use it to prove that independent of positive initial population sizes, \({{\mathcal {R}}}_{0}<1\) implies the unique disease-free equilibrium is globally stable and the infective population goes extinct. However, the unique endemic equilibrium is globally stable and the infective population persists whenever \({{\mathcal {R}}}_{0}>1\) and the constant survival probability of susceptible is either less than or equal than 1/3 or the constant recruitment is large enough.

我们使用流行病阈值参数\({{\mathcal {R}}}_{0}\)和不变矩形来研究密度相关离散时间 SI 流行病模型解的全局渐近行为，其中变量\(S_{n}\)和\(I_{n}\)分别代表时间\(n = 0,1,\ldots \)的易感人群和感染人群。该模型具有易感和感染个体的恒定生存“概率”以及每单位时间间隔\([n, n+1]\)不断招募到易感类。我们计算基本再生数\({{\mathcal {R}}}_{0}\)，并用它来证明独立于正的初始种群大小，\({{\mathcal {R}}}_{0}<1\)意味着独特的无病平衡是全局稳定的，感染性种群灭绝。然而，独特的地方病平衡是全局稳定的，只要\({{\mathcal {R}}}_{0}>1\)且易感者的恒定生存概率小于或等于 1/ ，感染群体就会持续存在。 3 还是恒招聘够大。