Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation

Zimeng Lv; Jiahong Zeng; Yuting Ding; Xinyu Liu; Zimeng Lv; Jiahong Zeng; Yuting Ding; Xinyu Liu

doi:10.3934/era.2023050

Electronic Research Archive

2023, Volume 31, Issue 2: 1004-1030. doi: 10.3934/era.2023050

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Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation

College of Science, Northeast Forestry University, Harbin 150040, China

Received: 30 September 2022 Revised: 04 December 2022 Accepted: 06 December 2022 Published: 09 December 2022

As the COVID-19 continues threatening public health worldwide, when to vaccinate the booster shots becomes the hot topic. In this paper, based on the characteristics of COVID-19 and its vaccine, an $ SAIR $ model associated with temporary immunity is proposed to study the effect on epidemic situation. Second, we theoretically analyze the existence and stability of equilibrium and the system undergoes Hopf bifurcation when delay passes through some critical values. Third, we study the dynamic properties of Hopf bifurcation and derive the normal form of Hopf bifurcation to determine the stability and direction of bifurcating periodic solutions. After that, numerical simulations are carried out to demonstrate the application of the theoretical results. Particularly, in order to ensure the validity, statistical analysis of data is conducted to determine the values for model parameters. Next, we study the impact of the infection rates on booster vaccination time to simulate the mutants, and the results are consistent with the facts. Finally, we predict the mean time of completing a round of vaccination worldwide with the help fitting and put forward some suggestions by comparing with the critical time of booster vaccination.
- SARS-CoV-2019,
- booster vaccination,
- stability,
- time delay,
- Hopf bifurcation
Citation: Zimeng Lv, Jiahong Zeng, Yuting Ding, Xinyu Liu. Stability analysis of time-delayed SAIR model for duration of vaccine in the context of temporary immunity for COVID-19 situation[J]. Electronic Research Archive, 2023, 31(2): 1004-1030. doi: 10.3934/era.2023050

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Abstract

As the COVID-19 continues threatening public health worldwide, when to vaccinate the booster shots becomes the hot topic. In this paper, based on the characteristics of COVID-19 and its vaccine, an $ SAIR $ model associated with temporary immunity is proposed to study the effect on epidemic situation. Second, we theoretically analyze the existence and stability of equilibrium and the system undergoes Hopf bifurcation when delay passes through some critical values. Third, we study the dynamic properties of Hopf bifurcation and derive the normal form of Hopf bifurcation to determine the stability and direction of bifurcating periodic solutions. After that, numerical simulations are carried out to demonstrate the application of the theoretical results. Particularly, in order to ensure the validity, statistical analysis of data is conducted to determine the values for model parameters. Next, we study the impact of the infection rates on booster vaccination time to simulate the mutants, and the results are consistent with the facts. Finally, we predict the mean time of completing a round of vaccination worldwide with the help fitting and put forward some suggestions by comparing with the critical time of booster vaccination.

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