一类具有非线性感染率的SIRS传染病模型的定性分析
Qualitative analysis of a class of SIRS infectious disease models with nonlinear infection rate
AI总结：
该研究针对总人口恒定下的一类带非线性感染率与治疗率的SIRS传染病模型，通过微分方程定性理论分析非负平衡点稳定性，用分岔理论证明自然恢复增长率为鞍结分岔参数并给出存在条件，完善了该类模型的定性理论结果。
摘要：
本文在总人口数恒定的条件下，研究了一类具有非线性感染率与治疗率的SIRS传染病模型若干非负平衡点的存在性与局部稳定性。研究采用微分方程定性理论，证明该系统的地方病平衡点在不同情形下可为稳定平衡点、不稳定平衡点或退化平衡点，随后分析了系统非负平衡点的局部稳定性。最后，利用分岔理论证明该系统以自然恢复增长率为鞍结分岔参数，并给出了模型鞍结分岔的存在条件。
英文摘要：
The existence and local stability of some non-negative equilibrium points of a class of SIRS infectious disease models with non-linear infection and treatment rates are investigated under the condition that the total population is a constant. The qualitative theory of differential equations was used to demonstrate that the endemic equilibrium point of the system is either a stable equilibrium, an unstable equilibrium or a degenerate equilibrium under different circumstances. Subsequently, the local stability of the non-negative equilibrium point of the system is analyzed. Finally, the bifurcation theory is used to prove that the system takes the natural recovery growth rate as the parameter of the saddle-node branching, and the conditions for the existence of the model saddle-node branching are given.
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