一类具有线性传染力的SIRS传染病动力系统的分析与控制

Analysis and control of an SIRS epidemic system with linear infection rate

面对传染病对人类身体健康的严重威胁,医学工作者以及数学工作者在使用数学模型研究传染病的传染规律、趋势预测以及预防控制等方面做了大量的研究工作。主要对一类具有线性传染力的SIRS传染病模型进行了平衡点的稳定性分析,并将控制理论应用于传染病的控制,即对染病者施加控制以使疾病最终消除,对于具有线性传染力的SIRS传染病的控制有一定的实际应用意义。首先,根据疾病消除平衡点和地方病平衡点存在的条件定义阈值;然后,用Lyapunov方法和LaSalle不变原理证明了不同阈值下疾病消除平衡点和地方病平衡点的全局渐近稳定性,根据系统对染病者施加有效的控制以使疾病消除平衡点全局渐近稳定,并对控制器的设计加以解释;最后,以数字例子进行系统仿真以说明控制的有效性。

Facing the threaten of infectious disease to the health of man, doctors and mathematicians do a lot research with mathematic models in this area, such as the rule of epidemics, the forecast of epidemic trends and the prevention and control of epidemics. An SIRS epidemic model with linear infection is discussed in this paper. We analyzed the stability of equilibriums and with the control theory controlled the system to eliminate the disease. It is significant to apply the results obtained in this paper to the epidemic represented by SIRS models with linear infection rate. We first defined threshold according to the existing conditions for disease-free equilibriums and endemic equilibriums. The globally asymptotical stabilities of the equilibriums under different thresholds were then proved by Lyapunov method and LaSalle Theory and the controllers based on the realities were given to make disease-free equilibriums globally asymptotically stable. A numerical example was also given to illustrate the effectiveness of the control at last.