Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control

Chenxi Huang; Qianqian Zhang; Sanyi Tang; Chenxi Huang; Qianqian Zhang; Sanyi Tang

doi:10.3934/mbe.2023835

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 10: 18861-18887. doi: 10.3934/mbe.2023835

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Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control

School of Mathematics and Statistics, Shaanxi Normal University, Xi'an 710062, China

Academic Editor: Yang Kuang

Received: 17 July 2023 Revised: 14 September 2023 Accepted: 20 September 2023 Published: 09 October 2023

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.
- SIR model,
- state-dependent feedback control,
- disease-free periodic solution,
- transcritical bifurcation,
- Poincaré map
Citation: Chenxi Huang, Qianqian Zhang, Sanyi Tang. Non-smooth dynamics of a SIR model with nonlinear state-dependent impulsive control[J]. Mathematical Biosciences and Engineering, 2023, 20(10): 18861-18887. doi: 10.3934/mbe.2023835

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Abstract

The classic SIR model is often used to evaluate the effectiveness of controlling infectious diseases. Moreover, when adopting strategies such as isolation and vaccination based on changes in the size of susceptible populations and other states, it is necessary to develop a non-smooth SIR infectious disease model. To do this, we first add a non-linear term to the classical SIR model to describe the impact of limited medical resources or treatment capacity on infectious disease transmission, and then involve the state-dependent impulsive feedback control, which is determined by the convex combinations of the size of the susceptible population and its growth rates, into the model. Further, the analytical methods have been developed to address the existence of non-trivial periodic solutions, the existence and stability of a disease-free periodic solution (DFPS) and its bifurcation. Based on the properties of the established Poincaré map, we conclude that DFPS exists, which is stable under certain conditions. In particular, we show that the non-trivial order-1 periodic solutions may exist and a non-trivial order-$ k $ ($ k\geq 1 $) periodic solution in some special cases may not exist. Moreover, the transcritical bifurcations around the DFPS with respect to the parameters $ p $ and $ AT $ have been investigated by employing the bifurcation theorems of discrete maps.

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