### Abstract

In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate [Formula presented], 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 α=α
_{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 α>α
_{0}
, or α≤α
_{0}
and k≤k
_{0}
, the disease will die out for all positive initial populations; (ii) when α=α
_{0}
and k
_{0}
<k≤k
_{1}
, the disease will die out for almost all positive initial populations; (iii) when α=α
_{0}
and k>k
_{1}
, the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α<α
_{0}
and k>k
_{0}
, 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.

Original language | English (US) |
---|---|

Journal | Journal of Differential Equations |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Bogdanov-Takens bifurcation
- Degenerate Hopf bifurcation
- Hopf bifurcation
- Saddle-node bifurcation
- SIRS epidemic model

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate.** / Lu, Min; Huang, Jicai; Ruan, Shigui; Yu, Pei.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

AU - Lu, Min

AU - Huang, Jicai

AU - Ruan, Shigui

AU - Yu, Pei

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate [Formula presented], 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 α=α 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 α>α 0 , or α≤α 0 and k≤k 0 , the disease will die out for all positive initial populations; (ii) when α=α 0 and k 0 <k≤k 1 , the disease will die out for almost all positive initial populations; (iii) when α=α 0 and k>k 1 , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α<α 0 and k>k 0 , 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.

AB - In this paper, we study a susceptible-infectious-recovered (SIRS) epidemic model with a generalized nonmonotone and saturated incidence rate [Formula presented], 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 α=α 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 α>α 0 , or α≤α 0 and k≤k 0 , the disease will die out for all positive initial populations; (ii) when α=α 0 and k 0 <k≤k 1 , the disease will die out for almost all positive initial populations; (iii) when α=α 0 and k>k 1 , the disease will persist in the form of a positive coexistent steady state for some positive initial populations; and (iv) when α<α 0 and k>k 0 , 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.

KW - Bogdanov-Takens bifurcation

KW - Degenerate Hopf bifurcation

KW - Hopf bifurcation

KW - Saddle-node bifurcation

KW - SIRS epidemic model

UR - http://www.scopus.com/inward/record.url?scp=85062706020&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062706020&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2019.03.005

DO - 10.1016/j.jde.2019.03.005

M3 - Article

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -