Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate kI(2)S/1+beta I+alpha I-2 in which the infection function first increases to a maximum when a new infectious disease emerges, then decreases due to psychological effect, and eventually tends to a saturation level due to crowding effect. It is shown that there are a weak focus of multiplicity at most two and a cusp of codimension at most two for various parameter values, and the model undergoes saddle-node bifurcation, Bogdanov-Takens bifurcation of codimension two, Hopf bifurcation, and degenerate Hopf bifurcation of codimension two as the parameters vary. It is shown that there exists a critical value alpha = alpha(0) for the psychological effect, and two critical values k = k(0), k(1) (k(0) < k(1)) for the infection rate such that: (i) when alpha > alpha(0), or alpha

在本文中，我们研究了一个具有广义非单调饱和发生率$kI^{2}S/(1 + \beta I + \alpha I^{2})$的易感 - 感染 - 恢复（SIRS）传染病模型，其中当一种新的传染病出现时，感染函数首先增加到最大值，然后由于心理效应而减少，最终由于拥挤效应趋于饱和水平。研究表明，对于各种参数值，存在至多为二阶的弱焦点和至多为二阶的尖点，并且随着参数的变化，该模型经历鞍结分岔、二维的Bogdanov - Takens分岔、Hopf分岔以及二维的退化Hopf分岔。研究表明，对于心理效应存在一个临界值$\alpha = \alpha_{0}$，对于感染率存在两个临界值$k = k_{0}, k_{1}(k_{0} < k_{1})$，使得：（i）当$\alpha > \alpha_{0}$，或者$\alpha$（此处原文似乎不完整）