TY - JOUR

T1 - Global Dynamics of a Susceptible-Infectious-Recovered Epidemic Model with a Generalized Nonmonotone Incidence Rate

AU - Lu, Min

AU - Huang, Jicai

AU - Ruan, Shigui

AU - Yu, Pei

N1 - Funding Information:
Research was partially supported by NSFC (Nos. 11871235, 11771168), the Fundamental Research Funds for the Central Universities (CCNU19TS030) and NSERC (No. R2686A02).
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2020

Y1 - 2020

N2 - A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS1+βI+αI2 (β>-2α such that 1 + βI+ αI2> 0 for all I≥ 0 ) is considered in this paper. It is shown that the basic reproduction number R 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< R∗, then the disease-free equilibrium is globally asymptotically stable; (ii) if R= R∗, then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R∗< R< 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≥ 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 α for the psychological effect α, a critical value k for the infection rate k, and two critical values β, β1(β1< β) for β 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.

AB - A susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone incidence rate kIS1+βI+αI2 (β>-2α such that 1 + βI+ αI2> 0 for all I≥ 0 ) is considered in this paper. It is shown that the basic reproduction number R 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< R∗, then the disease-free equilibrium is globally asymptotically stable; (ii) if R= R∗, then there is a unique endemic equilibrium which is a nilpotent cusp of codimension at most three; (iii) if R∗< R< 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≥ 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 α for the psychological effect α, a critical value k for the infection rate k, and two critical values β, β1(β1< β) for β 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.

KW - Backward bifurcation

KW - Bogdanov–Takens bifurcation of codimension three

KW - Degenerate Hopf bifurcation of codimension three

KW - Generalized nonmonotone incidence rate

KW - SIRS epidemic model

KW - Saddle-node bifurcation

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U2 - 10.1007/s10884-020-09862-3

DO - 10.1007/s10884-020-09862-3

M3 - Article

AN - SCOPUS:85087295683

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

SN - 1040-7294

ER -