Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity

Lili Liu; Xi Wang; Yazhi Li; Lili Liu; Xi Wang; Yazhi Li

doi:10.3934/mbe.2023925

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 12: 20914-20938. doi: 10.3934/mbe.2023925

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Research article

Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity

Lili Liu ¹,
Xi Wang ¹,
Yazhi Li ^{2
,
,}

1.
Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Complex Systems Research Center, Shanxi University, Taiyuan 030006, China
2.
School of Mathematics and Statistics, Qiannan Normal University for Nationalities, Duyun 558000, China

Academic Editor: David M. Bortz

Received: 09 August 2023 Revised: 27 October 2023 Accepted: 31 October 2023 Published: 20 November 2023

This paper aims to explore the complex dynamics and impact of vaccinations on controlling epidemic outbreaks. An epidemic transmission model which considers vaccinations and two different infection statuses with different infectivity is developed. In terms of a dynamic analysis, we calculate the basic reproduction number and control reproduction number and discuss the stability of the disease-free equilibrium. Additionally, a numerical simulation is performed to explore the effects of vaccination rate, immune waning rate and vaccine ineffective rate on the epidemic transmission. Finally, a sensitivity analysis revealed three factors that can influence the threshold: transmission rate, vaccination rate, and the hospitalized rate. In terms of optimal control, the following three time-related control variables are introduced to reconstruct the corresponding control problem: reducing social distance, enhancing vaccination rates, and enhancing the hospitalized rates. Moreover, the characteristic expression of optimal control problem. Four different control combinations are designed, and comparative studies on control effectiveness and cost effectiveness are conducted by numerical simulations. The results showed that Strategy C (including all the three controls) is the most effective strategy to reduce the number of symptomatic infections and Strategy A (including reducing social distance and enhancing vaccination rate) is the most cost-effective among the three strategies.
- epidemic model,
- vaccination,
- different infectivity,
- optimal control,
- cost-effectiveness
Citation: Lili Liu, Xi Wang, Yazhi Li. Mathematical analysis and optimal control of an epidemic model with vaccination and different infectivity[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20914-20938. doi: 10.3934/mbe.2023925

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Abstract

This paper aims to explore the complex dynamics and impact of vaccinations on controlling epidemic outbreaks. An epidemic transmission model which considers vaccinations and two different infection statuses with different infectivity is developed. In terms of a dynamic analysis, we calculate the basic reproduction number and control reproduction number and discuss the stability of the disease-free equilibrium. Additionally, a numerical simulation is performed to explore the effects of vaccination rate, immune waning rate and vaccine ineffective rate on the epidemic transmission. Finally, a sensitivity analysis revealed three factors that can influence the threshold: transmission rate, vaccination rate, and the hospitalized rate. In terms of optimal control, the following three time-related control variables are introduced to reconstruct the corresponding control problem: reducing social distance, enhancing vaccination rates, and enhancing the hospitalized rates. Moreover, the characteristic expression of optimal control problem. Four different control combinations are designed, and comparative studies on control effectiveness and cost effectiveness are conducted by numerical simulations. The results showed that Strategy C (including all the three controls) is the most effective strategy to reduce the number of symptomatic infections and Strategy A (including reducing social distance and enhancing vaccination rate) is the most cost-effective among the three strategies.

References

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