含有疫苗接种项离散传染病模型SIVS的稳定性和分岔分析  

Stability and Bifurcation Analysis of DiscreteSIVS Epidemic Model with Vaccination Items

作　　者：王轲 雷策宇 韩晓玲 WANG Ke;LEI Ceyu;HAN Xiaoling(College of Mathematics and Statistics,Northwest Normal University,Lanzhou 730070,China)

机构地区：[1]西北师范大学数学与统计学院,兰州730070

出　　处：《吉林大学学报(理学版)》2025年第2期331-339,共9页Journal of Jilin University:Science Edition

基　　金：国家自然科学基金(批准号:12161079);西北师范大学研究生科研资助项目(批准号:2023KYZZ-S116)。

摘　　要：首先,用下代矩阵的方法求解所建立的传染病模型SIVS(Susceptible-Infectious-Immune-Susceptible)的基本再生数R_(0),通过该阈值得到无病平衡点一直存在,地方病平衡点仅当R_(0)>1时才存在,进而确定疾病消亡和持久的条件.其次,借助Jacobi矩阵的性质、Jury判据以及Lyapunov函数的构造等方法证明该模型在平衡点处的稳定性以及产生分岔的情形,结果表明:当R_(0)≤1时,无病平衡点全局渐近稳定,且在R_(0)=1时产生翻转分岔;当R_0>1时,地方病平衡点局部渐近稳定,如果忽略现实中对接触率β的限制,模型在地方病平衡点处会产生倍周期分岔甚至混沌现象.最后,结合数值模拟的图像和敏感性指数法验证理论分析结果,并得出提高疫苗接种率和恢复率可有效降低疾病发病率的结论.Firstly,the basic reproduction number R_(0) of the SIVS(Susceptible-Infectious-Immune-Susceptible)epidemic model is solved by the method of next generation matrix,through the threshold,the disease-free equilibrium point always exists,and the endemic equilibrium point only exists when R_(0)>1,furthermore,the conditions of extinction and persistence of the disease are determined.Secondly,the stability and the bifurcation situations of the model at the equilibrium point are proved by the properties of Jacobian matrix,Jury criterions and the construction of Lyapunov function.The results show that when R_(0)<1,the disease-free equilibrium point is globally asymptotically stable,and the transcritical bifurcation occurs when R 0=1.When R_(0)>1,the endemic equilibrium point is locally asymptotically stable,if the limitation on contact rateβin reality is ignored,the model will produce the period-doubling bifurcation and even chaotic phenomena at the endemic equilibrium point.Finally,numerical simulation and sensitivity index method are used to verify the theoretical analysis results,it is concluded that improving the vaccination rate and recovery rate can effectively reduce the incidence of the disease.

关 键 词：离散模型 模型SIVS 疫苗接种 稳定性 分岔 

分 类 号：O175.1[理学—数学]