Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate
Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate
复制标题
具有广义非单调发病率的易感感染-恢复流行病模型的全局动态
DOI：
10.1007/s10884-020-09862-3
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发表时间：
2020-06-29
影响因子：
1.3
通讯作者：
Yu, Pei
中科院分区：
文献类型：
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作者：
Lu, Min;Huang, Jicai;Yu, Pei
研究了一类具有广义非单调传染率kIS/1+beta I+alpha I-2（beta > -2 root alpha，使得1 + beta I + alpha I-2 > 0，对所有I >= 0）的传染病模型.结果表明，基本再生数R-0不再作为疾病传播的阈值，存在一个次阈值R-*（< 1），使得：（i）若R-0 < R-*，则无病平衡点全局渐近稳定;（ii）若R-0 = R-*，则存在唯一的地方病平衡点，它是余维至多为3的幂零尖点;（iii）若R-* < R-0 < 1，则存在两个地方病平衡点，一个是重数至少为3的弱焦点，另一个是鞍点;（iv）若R-0 >= 1，则又存在唯一的地方病平衡点，它是重数至少为3的弱焦点。随着参数的变化，模型经历了鞍结分支、后向分支、余维3的Bogdanov-Takens分支、Hopf分支和余维3的退化Hopf分支。此外，还发现存在一个临界值α（0），一个临界值k（0），一个临界值k（0），以及两个临界值β（0），β（1）（β（1）<β（0）），它们决定了疾病在不同初始种群下是以正周期共存振荡或共存稳态的形式消亡还是持续存在。数值模拟证明了存在一个，两个或三个极限环。
A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS/1+beta I+alpha I-2 (beta > -2 root alpha such that 1 + beta I + alpha I-2 > 0 for all I >= 0) is considered in this paper. It is shown that the basic reproduction number R-0 does not act as a threshold value for the disease spread anymore, and there exists a sub-threshold value R-*(< 1) such that: (i) if R-0 < R-*, then the disease-free equilibrium is globally asymptotically stable; (ii) if R-0 = R-*, then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R-* < R-0 < 1, then there are two endemic equilibria, one is a weak focus of multiplicity at least three, the other is a saddle; (iv) if R-0 >= 1, then there is again a unique endemic equilibrium which is a weak focus of multiplicity at least three. As parameters vary, the model undergoes saddle-node bifurcation, backward bifurcation, Bogdanov-Takens bifurcation of codimension three, Hopf bifurcation, and degenerate Hopf bifurcation of codimension three. Moreover, it is shown that there exists a critical value alpha(0) for the psychological effect alpha, a critical value k(0) for the infection rate k, and two critical values beta(0), beta(1)(beta(1) < beta(0)) for beta that will determine whether the disease dies out or persists in the form of positive periodic coexistent oscillations or coexistent steady states under different initial populations. Numerical simulations are given to demonstrate the existence of one, two or three limit cycles.