Stability analysis and backward bifurcation on an SEIQR epidemic model with nonlinear innate immunity

Xueyong Zhou; Xiangyun Shi; Xueyong Zhou; Xiangyun Shi

doi:10.3934/era.2022178

Electronic Research Archive

2022, Volume 30, Issue 9: 3481-3508. doi: 10.3934/era.2022178

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Research article

Stability analysis and backward bifurcation on an SEIQR epidemic model with nonlinear innate immunity

Xueyong Zhou ^,,
Xiangyun Shi

School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, China

Received: 02 June 2022 Revised: 18 July 2022 Accepted: 20 July 2022 Published: 27 July 2022

Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.
- innate immunity,
- SEIQR epidemic model,
- backward bifurcation,
- stability
Citation: Xueyong Zhou, Xiangyun Shi. Stability analysis and backward bifurcation on an SEIQR epidemic model with nonlinear innate immunity[J]. Electronic Research Archive, 2022, 30(9): 3481-3508. doi: 10.3934/era.2022178

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Abstract

Infectious diseases have a great impact on the economy and society. Dynamic models of infectious diseases are an effective tool for revealing the laws of disease transmission. Quarantine and nonlinear innate immunity are the crucial factors in the control of infectious diseases. Currently, there no mathematical models that comprehensively study the effect of both innate immunity and quarantine. In this paper, we propose and analyze an SEIQR epidemic model with nonlinear innate immunity. The boundedness and positivity of the solutions are discussed. Employing the next-generation matrix, we compute the expression of the basic reproduction number. Under certain conditions, the phenomenon of backward bifurcation may occur. That is to say, the stable disease-free equilibrium point and the stable endemic equilibrium point coexist when the basic reproduction ratio is less than one. And the basic reproduction number is no longer the threshold value to determine whether the disease breaks out. We investigate the globally asymptotical stability of the disease-free equilibrium point for the system by constructing Lyapunov function. Also, we research the global stability of the endemic equilibrium by using geometric approach. Numerical simulations are carried out to reveal the theoretical results and find some complex dynamics (for example, the existence of Hopf bifurcation) of the system. Both theoretical and numerical results indicate that the nonlinear innate immunity may cause backward bifurcation and Hopf bifurcation, which makes more difficult to eliminate the disease.

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