Mathematical modeling and stability analysis of the time-delayed $ SAIM $ model for COVID-19 vaccination and media coverage

Xinyu Liu; Zimeng Lv; Yuting Ding; Xinyu Liu; Zimeng Lv; Yuting Ding

doi:10.3934/mbe.2022294

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 6296-6316. doi: 10.3934/mbe.2022294

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Research article

Mathematical modeling and stability analysis of the time-delayed $ SAIM $ model for COVID-19 vaccination and media coverage

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received: 29 January 2022 Revised: 31 March 2022 Accepted: 31 March 2022 Published: 19 April 2022

Since the COVID-19 outbreak began in early 2020, it has spread rapidly and threatened public health worldwide. Vaccination is an effective way to control the epidemic. In this paper, we model a $ SAIM $ equation. Our model involves vaccination and the time delay for people to change their willingness to be vaccinated, which is influenced by media coverage. Second, we theoretically analyze the existence and stability of the equilibria of our model. Then, we study the existence of Hopf bifurcation related to the two equilibria and obtain the normal form near the Hopf bifurcating critical point. Third, numerical simulations based two groups of values for model parameters are carried out to verify our theoretical analysis and assess features such as stable equilibria and periodic solutions. To ensure the appropriateness of model parameters, we conduct a mathematical analysis of official data. Next, we study the effect of the media influence rate and attenuation rate of media coverage on vaccination and epidemic control. The analysis results are consistent with real-world conditions. Finally, we present conclusions and suggestions related to the impact of media coverage on vaccination and epidemic control.
- COVID-19 epidemic,
- media coverage,
- vaccination,
- time-delay,
- Hopf bifurcation,
- normal form,
- multiple time scales method
Citation: Xinyu Liu, Zimeng Lv, Yuting Ding. Mathematical modeling and stability analysis of the time-delayed $ SAIM $ model for COVID-19 vaccination and media coverage[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 6296-6316. doi: 10.3934/mbe.2022294

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Abstract

Since the COVID-19 outbreak began in early 2020, it has spread rapidly and threatened public health worldwide. Vaccination is an effective way to control the epidemic. In this paper, we model a $ SAIM $ equation. Our model involves vaccination and the time delay for people to change their willingness to be vaccinated, which is influenced by media coverage. Second, we theoretically analyze the existence and stability of the equilibria of our model. Then, we study the existence of Hopf bifurcation related to the two equilibria and obtain the normal form near the Hopf bifurcating critical point. Third, numerical simulations based two groups of values for model parameters are carried out to verify our theoretical analysis and assess features such as stable equilibria and periodic solutions. To ensure the appropriateness of model parameters, we conduct a mathematical analysis of official data. Next, we study the effect of the media influence rate and attenuation rate of media coverage on vaccination and epidemic control. The analysis results are consistent with real-world conditions. Finally, we present conclusions and suggestions related to the impact of media coverage on vaccination and epidemic control.

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