具有一般传染率的 SIRS 年龄结构模型的分支研究

考虑到年龄在一些传染病流行过程中的重要影响，建立了一个具有一般传染率的 SIRS 年龄结构仓室模型。通过将模型改写为抽象柯西问题并利用 Hille-Yosida 算子相关定理，分析了模型的动力学性态，讨论了平衡点的稳定性以及平衡点失稳时产生 Hopf 分支的条件。结果表明，当基本再生数小于 1 时，免疫年龄不影响无病平衡点的全局稳定性；当基本再生数大于 1 时，免疫年龄扰动导致地方病平衡点的稳定性改变，从而产生 Hopf 分支。同时，数值模拟验证了理论结果并显示了免疫年龄对模型动力学性态的影响。

Bifurcation Analysis of a SIRS Age-structured Model with General Incidence Function

Considering the impact of age in the transmission of infectious diseases, we present an age-structured model with SIRS type and general incidence function. By reformulating the model as an abstract Cauchy problem and applying theorems related with the Hille-Yosida operator, we investigated the dynamic properties, including stability of equilibria and the condition for Hopf bifurcation due to the destabilization of endemic equilibrium. The results reveal that, if the basic reproductive number is less than 1, the infection-free equilibrium is globally stable, without being influenced by the immune age. Conversely, if the basic reproductive number is larger than 1, the endemic equilibrium may be destabilized by the perturbation of the immune age and a Hopf bifurcation can occur. Meanwhile, numerical simulations are conducted to ill-ustrate the theoretical results and to show the influence of the immune age on the dynamical behaviors of the model.