Lu Min, Huang Jicai, Ruan Shigui, Yu Pei
School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, PR China.
Department of Mathematics, University of Miami, Coral Gables, FL 33146, USA.
J Differ Equ. 2019 Jul 15;267(3):1859-1898. doi: 10.1016/j.jde.2019.03.005. Epub 2019 Mar 14.
In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate , 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 for the psychological effect, and two critical values for the infection rate such that: (i) when , or and , the disease will die out for all positive initial populations; (ii) when and , the disease will die out for almost all positive initial populations; (iii) when and , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when and , the disease will persist in the form of multiple positive periodic coexistent oscillations and coexistent steady states for some positive initial populations. Numerical simulations, including the existence of one or two limit cycles and data-fitting of the influenza data in Mainland China, are presented to illustrate the theoretical results.
在本文中，我们研究了一个具有广义非单调饱和发病率的易感 - 感染 - 恢复（SIRS）传染病模型，其中感染函数在一种新的传染病出现时首先增加到最大值，然后由于心理效应而下降，最终由于拥挤效应趋于饱和水平。结果表明，对于不同的参数值，最多存在一个重数为二的弱焦点和一个余维数为二的尖点，并且随着参数的变化，模型经历鞍结分岔、余维数为二的博格达诺夫 - 塔克恩斯分岔、霍普夫分岔和余维数为二的退化霍普夫分岔。结果表明，对于心理效应存在一个临界值，对于感染率存在两个临界值，使得：（i）当 ，或 且 时，对于所有正的初始种群，疾病将灭绝；（ii）当 且 时，对于几乎所有正的初始种群，疾病将灭绝；（iii）当 且 时，对于一些正的初始种群，疾病将以正共存稳态的形式持续存在；（iv）当 且 时，对于一些正的初始种群，疾病将以多个正周期共存振荡和共存稳态的形式持续存在。给出了数值模拟，包括一个或两个极限环的存在以及中国大陆流感数据的数据拟合，以说明理论结果。
