Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect

Xiaofen Lin; Hua Liu; Xiaotao Han; Yumei Wei; Xiaofen Lin; Hua Liu; Xiaotao Han; Yumei Wei

doi:10.3934/mbe.2023129

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 2750-2775. doi: 10.3934/mbe.2023129

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Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect

1.
School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
2.
Experimental Teaching Deparment, Northwest Minzu University, Lanzhou 730030, China

Academic Editor: Sanling Yuan

Received: 09 August 2022 Revised: 29 October 2022 Accepted: 13 November 2022 Published: 28 November 2022

In this paper, an SIR model with a strong Allee effect and density-dependent transmission is proposed, and its characteristic dynamics are investigated. The elementary mathematical characteristic of the model is studied, including positivity, boundedness and the existence of equilibrium. The local asymptotic stability of the equilibrium points is analyzed using linear stability analysis. Our results indicate that the asymptotic dynamics of the model are not only determined using the basic reproduction number ${R_0}$. If ${R_0} < 1$, there are three disease-free equilibrium points, and a disease-free equilibrium is always stable. At the same time, the conditions for other disease-free equilibrium points to be bistable were determined. If ${R_0} > 1$ and in certain conditions, either an endemic equilibrium emerges and is locally asymptotically stable, or the endemic equilibrium becomes unstable. What must be emphasized is that there is a locally asymptotically stable limit cycle when the latter happens. The Hopf bifurcation of the model is also discussed using topological normal forms. The stable limit cycle can be interpreted in a biological significance as a recurrence of the disease. Numerical simulations are used to verify the theoretical analysis. Taking into account both density-dependent transmission of infectious diseases and the Allee effect, the dynamic behavior becomes more interesting than when considering only one of them in the model. The Allee effect makes the SIR epidemic model bistable, which also makes the disappearance of diseases possible, since the disease-free equilibrium in the model is locally asymptotically stable. At the same time, persistent oscillations due to the synergistic effect of density-dependent transmission and the Allee effect may explain the recurrence and disappearance of disease.
- strong Allee effect,
- density-dependent transmission,
- SIR epidemic model,
- bistability,
- Hopf bifurcation
Citation: Xiaofen Lin, Hua Liu, Xiaotao Han, Yumei Wei. Stability and Hopf bifurcation of an SIR epidemic model with density-dependent transmission and Allee effect[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2750-2775. doi: 10.3934/mbe.2023129

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Abstract

In this paper, an SIR model with a strong Allee effect and density-dependent transmission is proposed, and its characteristic dynamics are investigated. The elementary mathematical characteristic of the model is studied, including positivity, boundedness and the existence of equilibrium. The local asymptotic stability of the equilibrium points is analyzed using linear stability analysis. Our results indicate that the asymptotic dynamics of the model are not only determined using the basic reproduction number ${R_0}$. If ${R_0} < 1$, there are three disease-free equilibrium points, and a disease-free equilibrium is always stable. At the same time, the conditions for other disease-free equilibrium points to be bistable were determined. If ${R_0} > 1$ and in certain conditions, either an endemic equilibrium emerges and is locally asymptotically stable, or the endemic equilibrium becomes unstable. What must be emphasized is that there is a locally asymptotically stable limit cycle when the latter happens. The Hopf bifurcation of the model is also discussed using topological normal forms. The stable limit cycle can be interpreted in a biological significance as a recurrence of the disease. Numerical simulations are used to verify the theoretical analysis. Taking into account both density-dependent transmission of infectious diseases and the Allee effect, the dynamic behavior becomes more interesting than when considering only one of them in the model. The Allee effect makes the SIR epidemic model bistable, which also makes the disappearance of diseases possible, since the disease-free equilibrium in the model is locally asymptotically stable. At the same time, persistent oscillations due to the synergistic effect of density-dependent transmission and the Allee effect may explain the recurrence and disappearance of disease.

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