一个具有饱和增殖率的抗体免疫HIV模型的动力学与最优控制  

Dynamics and Optimal Control of an Antibody Immune HIV Model with a Saturated Proliferation Rate

作　　者：陈冲 周英告[1] 周凯 Chen Chong;Zhou Yinggao;Zhou Kai(School of Mathematics and Statistics,Central South University,Changsha 410083,China)

机构地区：[1]中南大学数学与统计学院,长沙410083

出　　处：《数学理论与应用》2021年第2期1-18,共18页Mathematical Theory and Applications

基　　金：湖南省自然科学基金项目(2019JJ40354);湖南省研究生学位与教育改革项目(2020JGYB031)。

摘　　要：本文建立一个具有饱和增殖率的抗体免疫的HIV模型,并通过线性化方法和构造Lyapunov函数的方法分别获得无病平衡点、无免疫地方病平衡点和免疫地方病平衡点的局部稳定和全局稳定的条件.此外,受最新抗AIDS疫苗在动物试验中取得成功的激励,论文将抗体免疫与药物治疗的双重作用引入上述动力学模型,构成使感染细胞浓度和病毒浓度、以及控制成本最低的最优控制问题.运用Pontryagin极大值原理,获得该最优控制问题的最优性条件.最后,在获取模型参数的情况下,分别对双控制问题与单控制问题进行数值模拟试验.试验结果表明,有效的疫苗可以迅速降低感染细胞和病毒的浓度,在跟双控制效果进行比较之后还发现免疫控制的效果几乎与双控制的效果一样,这说明疫苗在控制AIDS的有效性方面成效显著,可以预期在疫苗正式投入临床实践后,将会极大地改变AIDS蔓延的现状,甚至可期最终消灭AIDS.In this paper,an antibody immune HIV model with a saturated proliferation rate is established,and some conditions of local stability and global stability for the disease-free equilibrium,the no-immune endemic equilibrium and the immune endemic equilibrium are obtained by the linearization method and the Lyapunov function method,respectively.Moreover,motivated by the success of the latest anti-AIDS vaccine in some animal experiments,the dual action of antibody immunity and drug therapy is builded into the above kinetic model,which forms a optimal control problem to minimize the concentration of infected cells and virus and the cost of control.Using the Pontryagin maximum principle,the optimality conditions for the optimal control problem are gotten.After obtaining the parameters of the model,numerical simulation tests are carried out for the double control problem and the single control problem respectively.Experimental results show that the concentration of the infected cells and viruses can quickly reduce under the condition of the effective vaccine,and the immune control effect is almost as well as the effect of the double control by comparing with double control effect,which illustrates the vaccine is fairly effective in controlling AIDS.It can be expected that vaccine will greatly change the current situation of AIDS spread and may even eliminate AIDS eventually,after the vaccine is put into clinical practice.

关 键 词：HIV模型 LYAPUNOV函数 稳定性 接种 最优控制 

分 类 号：R512.91[医药卫生—内科学] O232[医药卫生—临床医学]