具有时滞和治疗的丙肝模型分析

Analysis of Hepatitis C Virus model with time delay and treatment

为了进一步研究丙型肝炎病毒的传播机理及其治疗的有效方法,针对丙型肝炎病毒潜伏期比较长,且在整个患病期内均具有传染性的特性,在已有的模型基础上引入时滞来反映丙肝潜伏期的存在。为研究时滞和药物治疗对丙肝传染病的影响,建立了具有常数输入和时滞的丙肝传染病模型,利用常微分方程定性理论知识分析其对应平衡点的存在,构造Liapunov函数讨论无病平衡点的全局稳定性,并且分析了时滞对系统的影响。通过分析可知,当时滞大于零且基本再生数小于1时,系统在无病平衡点处是全局渐近稳定的。给出了正平衡点稳定、不稳定与系统产生Hopf分支分别对应的前提条件,通过对阈值的分析给出了控制丙肝传播的措施建议。数值模拟验证了结果的正确性,其结果对具有时滞和治疗的丙肝模型的研究是有意义的,可为减少丙肝的流行提供新的思路。

In order to further study the mechanism of Hepatitis C Virus(HCV)transmission and effective methods for its treatment,aiming at the characteristics that the latent period of HCV is quite long and HCV is an infectious virus throughout the whole disease period,based on the existing model,the time delay is used to reflect the existence of the latent period of HCV.In order to study the effect of the time delay and the medical treatment on HCV,a model with the constant input and the time delay is established.The existence of equilibria is evaluated using the qualitative theory of ordinary differential equation,the global stability of the disease-free equilibrium is got by constructing the suitable Liapunov function,and the influence of the time delay on the system is also discussed.The analysis results show that if the time delay is greater than zero and the reproduction number is less than 1,the system is globally asymptotically stable at the disease-free equilibrium;the stable and unstable conditions of the endemic equilibrium are given,respectively,and the system can generate a Hopf bifurcation under the condition.Finally,the measures to control HCV are put forward by analyzing the threshold.The numerical simulation verifies the corresponding results.Therefore,it is significant to study the HCV model with the time delay and the medical treatment,and it can provide new ideas for reducing the prevalence of Hepatitis C.