带有治疗函数及免疫损失率的SIRS流行病模型的动力学分析

Analysis of SIRS Epidemic Models with Treatment Function and Immunity Loss Rate

本文研究一类带有非线性发病率、新生儿垂直传播、疫苗接种及治疗能力的SIRS传染病模型的动力学行为,该模型充分考虑了医疗资源的局限性,可用医疗资源的供应效率,当感染的数量低于容量时,治疗率与感染的数量成正比,而当感染数量达到容量时,治疗率为常数.在一定条件下,证明了该模型存在后向分支,这意味着边界平衡点与一个正平衡点共存.在这种情况下,控制基本再生数R0小于1不足以控制和根除这种疾病,需要采取额外的措施来确保其解趋近于边界平衡点.当基本再生数R0大于1时,由于治疗、疫苗接种、免疫损失和其他参数的影响,该模型可能存在多个正平衡点.本文分析了该模型平衡点的存在性和稳定性,得到了Hopf分支以及BT分支的存在性,进而发现不稳定极限环及同宿轨道的存在性,并且通过数值模拟来验证所得结果.

A susceptible-infectious-recovered-susceptible(SIRS)epidemic model with a nonlinear incidence rate,newborn vertical transmission,vaccination,treatment ability and immunity loss rate are analyzed in this paper.The model fully considers the limitation of medical resources and the supply efficiency of health available resources and assumes that when the number of infections below capacity,treatment rate proportional to the number of infection,when infection number reaches capacity,treatment rate is constant.Under certain conditions,we prove that the model has backward bifurcation,which means that the disease-free equilibrium and the endemic equilibrium coexist.In this case,the basic reproduction number less than unity is not enough to control and eradicate the disease,some additional measures need to be taken to ensure that the solutions approach the disease-free equilibrium.As a result of treatment,vaccination,immune loss and the influence of other parameters,the model may have multiple endemic equilibria when the basic reproduction number is greater than unity.The existence and stability of equilibria are analyzed and obtained the existence of Hopf bifurcation and BT bifurcation.And the found the existence of the unstable limit cycle and the homoclinic orbit.Finally,some numerical simulations are presented to illustrate the theoretical results.