Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination

Manoj Kumar; Syed Abbas; Abdessamad Tridane; Manoj Kumar; Syed Abbas; Abdessamad Tridane

doi:10.3934/mbe.2023646

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 8: 14438-14463. doi: 10.3934/mbe.2023646

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Research article

Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination

1.
School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, H. P. 175005, India
2.
Department of Mathematical Sciences, United Arab Emirates University, Al Ain 15551, UAE

Received: 19 April 2023 Revised: 04 June 2023 Accepted: 11 June 2023 Published: 30 June 2023

Vaccination programs are crucial for reducing the prevalence of infectious diseases and ultimately eradicating them. A new age-structured SEIRV (S-Susceptible, E-Exposed, I-Infected, R-Recovered, V-Vaccinated) model with imperfect vaccination is proposed. After formulating our model, we show the existence and uniqueness of the solution using semigroup of operators. For stability analysis, we obtain a threshold parameter $ R_0 $. Through rigorous analysis, we show that if $ R_0 < 1 $, then the disease-free equilibrium point is stable. The optimal control strategy is also discussed, with the vaccination rate as the control variable. We derive the optimality conditions, and the form of the optimal control is obtained using the adjoint system and sensitivity equations. We also prove the uniqueness of the optimal controller. To visually illustrate our theoretical results, we also solve the model numerically.
- age-structured model,
- imperfect vaccination,
- optimal control
Citation: Manoj Kumar, Syed Abbas, Abdessamad Tridane. Optimal control and stability analysis of an age-structured SEIRV model with imperfect vaccination[J]. Mathematical Biosciences and Engineering, 2023, 20(8): 14438-14463. doi: 10.3934/mbe.2023646

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Abstract

Vaccination programs are crucial for reducing the prevalence of infectious diseases and ultimately eradicating them. A new age-structured SEIRV (S-Susceptible, E-Exposed, I-Infected, R-Recovered, V-Vaccinated) model with imperfect vaccination is proposed. After formulating our model, we show the existence and uniqueness of the solution using semigroup of operators. For stability analysis, we obtain a threshold parameter $ R_0 $. Through rigorous analysis, we show that if $ R_0 < 1 $, then the disease-free equilibrium point is stable. The optimal control strategy is also discussed, with the vaccination rate as the control variable. We derive the optimality conditions, and the form of the optimal control is obtained using the adjoint system and sensitivity equations. We also prove the uniqueness of the optimal controller. To visually illustrate our theoretical results, we also solve the model numerically.

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