离散的SI和SIS传染病模型的研究

Study of Some Discrete SI and SIS Epidemic Models

为了描述个体的死亡、染病者的恢复以及疾病的传染,引入了相应的概率.基于总种群中个体数量为常数的假设,根据染病者能否恢复分别建立了具有生命动力学的离散SI和SIS传染病模型.所得到的结果显示:它们具有与相应连续模型相同的动力学性态,并确定了各自的阈值.在它们的阈值之下,传染病最终将灭绝;在它们的阈值之上,传染病将会发展成为地方病,染病者的数量将趋向于一确定的正常数.

The probability is introduced to formulate the death of individuals, the recovery of the infected individuals and incidence of epidemic disease. Based on the assumption that the number of individuals in population is a constant, discrete-time SI and SIS epidemic models with vital dynamics are established respectively corresponding to the case that the infectives can recover from the disease or not. For these two models, the results obtained show that there is the same dynamical behavior as their corresponding continuous ones. And the threshold determining its dynamical behavior is found. Below the threshold the epidemic disease dies out eventually. Above the threshold the epidemic disease becomes an endemic eventually. The number of the infectives approaches a positive constant.