A vertically transmitted epidemic model with two state-dependent pulse controls

Xunyang Wang; Canyun Huang; Yuanjie Liu; Xunyang Wang; Canyun Huang; Yuanjie Liu

doi:10.3934/mbe.2022651

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 12: 13967-13987. doi: 10.3934/mbe.2022651

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Research article

A vertically transmitted epidemic model with two state-dependent pulse controls

1.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
2.
State Grid Gansu Electric Power Research Institute, Lanzhou 730070, China

Academic Editor: Yang Kuang

Received: 01 August 2022 Revised: 09 September 2022 Accepted: 15 September 2022 Published: 22 September 2022

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the $ Poincar\acute{e}\; map $, an analogue of $ Poincar\acute{e} $ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.
- SIR epidemic model,
- vertical transmission,
- state-dependent pulses,
- orbital stability,
- periodic solution
Citation: Xunyang Wang, Canyun Huang, Yuanjie Liu. A vertically transmitted epidemic model with two state-dependent pulse controls[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13967-13987. doi: 10.3934/mbe.2022651

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Abstract

Vertical transmission refers to the process in which a mother transmits bacteria or viruses to her offspring through childbirth, and this phenomenon takes place commonly in nature. This paper formulates an SIR epidemic model where the impact of vertical transmission and two state-dependent pulse controls are both taken into consideration. Using the $ Poincar\acute{e}\; map $, an analogue of $ Poincar\acute{e} $ criterion and the method of related qualitative analysis, the existence and the stability of a positive order-1 or order-2 periodic solution for the epidemic model are proved. Furthermore, phase diagrams are demonstrated by means of numerical simulations, illustrating the feasibility and correctness of our main results. It can be further implied that the epidemic can be controlled to a certain extent, with vertical transmission reduced and timely state-dependent pulse controls carried out.

References

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