Optimal control of structured epidemic model with vaccination strategies

Introduction

Infectious disease dynamics models indicate that a combination of treatment, vaccination, and isolation measures is necessary to effectively control the spread of pandemic diseases. Application of Vaccination is a highly effective method to reduce the spread and protect the population. In the case of COVID-19, use of vaccination to all susceptible populations especially during the initial stages of an outbreak is hindered by a shortage of vaccines. Some individuals got infected even after vaccination. Given that COVID-19 affects different age groups with varying severity and immune responses, age-structured models have become essential tools for understanding disease dynamics and optimizing intervention strategies. This paper introduces an age-structured SIR model that incorporates vaccination and hospitalization dynamics to analyze the spread of COVID-19 across different age groups. One of the major challenges in COVID-19 vaccination campaigns is the heterogeneity in vaccine efficacy across age groups. Clinical and epidemiological studies have shown that while younger individuals generally experience mild symptoms, older adults and individuals with underlying health conditions are at a higher risk of hospitalization and mortality. Mathematical modeling plays a crucial role in designing optimal vaccination strategies, especially in the presence of limited vaccine supplies and evolving virus variants. Agestructured models allow researchers to simulate different intervention scenarios, assess the impact of age-specific contact patterns, and determine the best allocation of vaccines to minimize infection and mortality rates. Chowell et al. (2009) [1] have formulated an optimal control problem in the context of the transmission dynamics of the 2009 influenza pandemic in Mexico that captures age-specific characteristics of the population. The baseline model has been modified through the incorporation of time-dependent control functions to measure the effectiveness of age-specific vaccination policies aimed at minimizing the number of infected individuals during the pandemic. Lee et al. (2012) [8] introduced an age-structured mathematical model to evaluate optimal vaccination strategies for 2009 H1N1 influenza pandemic in Mexico, highlighted the importance of timely vaccine production and distribution to reduce illness and death, especially among high-risk groups. Sharomi and Malik (2017) [11] developed a control function and studied to understand what vaccination coverage should be achieved to minimize the cost of vaccination and the loss of infection. Jacobson (2021) [7] presented an epidemiological model with vaccination, testing, and social distancing. Vaccination is treated as a control parameter, optimized using Pontryagin's maximum principle. Results show that faster vaccination and increased social distancing significantly reduce infections, while varying cost parameters has a smaller effect on outcomes. Jia et al. (2022) [6] developed a continuous agestructured model to analyze the short-term transmission of an emerging infectious disease. It derives the basic reproduction number R 0 using the next-generation operator. An optimal control problem is formulated to evaluate the costeffectiveness of vaccination and physical distancing. Results show that combining both strategies can effectively eliminate the infection, with the strongest impact seen in age groups 10-39 and over 60. Chukwu et al. (2023) [3] developed a SARS-CoV-2 model focusing on high-risk populations, analyzed its stability and key parameters through sensitivity analysis. Using the Pontryagin maximum principle, the optimal control theory with time-dependent controls and different strategies are evaluated. Simulations showed reduced infections, and costeffectiveness analysis identifies the most efficient strategy. Wang and Zhou (2023) [14] developed an age-structured SEIR model to optimize COVID-19 vaccination strategies. Aiming to minimize daily infections under limited vaccine supply, it analyzes the effectiveness of age-specific vaccination priorities in China. The study highlights how variations in infection and hospitalization rates influence optimal vaccine allocation across age groups. Hakim (2023) [5] developed a nonlinear differential equation model for COVID-19, incorporating vaccinated and quarantined groups along with multiple control variables. The author analyzed positivity, boundedness, existence, and uniqueness of solutions, and also, identified optimal control strategies using Pontryagin's Principle. Chhetri et al. (2024) [2] this study models COVID-19 spread using two age groups: under 30 and over 30. It shows that the disease-free state is stable when R 0 < 1, and infection persists when R 0 >1. Optimal control analysis reveals that combined treatment for both age groups effectively reduces infections, though treatment saturation can increase total cases. Urban et al. (2024) [13] promoted accessible analysis of COVID-19 dynamics, focusing on vaccination strategies for nonprogrammers. It begins with the basic SIR model and extends to a detailed SEIRD model incorporating age structure and asymptomatic cases. Five vaccination strategies are analyzed under constraints like limited supply and daily capacity, using data for one million people based on Sweden's demographics. The study evaluates strategies by their impact on reducing deaths and symptomatic infections, the main healthcare burden. Paparelli et al. (2025) [9] addressed an optimal control problem for managing the COVID-19 pandemic using a SEAIRD model. It aims to minimize a total cost that balances economic losses and human deaths, while keeping infections below healthcare capacity through a state constraint. The existence of an optimal solution is proven, and necessary conditions are derived using the Pontryagin Minimum Principle. Numerical methods are used to solve the discretized problem under various initial conditions. This approach offers a prototype for controlling epidemics with state constraints. Teng and Kuniya (2025) [12] presented a 𝑆𝑉𝐼𝑊𝑅𝐷 model to study COVID-19 control through age-targeted vaccination strategies. Due to vaccine limitations, the population is divided into young and elderly groups, each vaccinated at different rates. Using Pontryagin's Maximum Principle and the Runge-Kutta method, the study performs single-objective optimization to determine an optimal vaccination ratio that minimizes the negative impact of the disease. In this paper, we aimed to lower the risk of COVID-19 infection by strategically controlling vaccination proportions among the elderly and young populations. The probability of severe disease progression increases with age, necessitating targeted vaccination strategies to protect vulnerable populations. By integrating an age-specific contact matrix into our model, we account for these variations and provide more realistic predictions for COVID-19 control. This paper is structured as follows: Section 2 presents the formulation of the age-structured SVIJRD epidemic model, including differential equations governing susceptible, vaccinated, infected, hospitalized, recovered, and decease induced death populations. Section 3 describes the optimal control framework using Pontryagin's Maximum Principle and derives the necessary conditions for the optimality. Section 4 provides numerical illustrations based on observed epidemiological data, demonstrating the impact of different vaccination strategies on infection dynamics. Section 5 gives conclusions and the implications of our findings for public health policies.

The structured 𝐒𝐕𝐈𝐉𝐑𝐃 Model

The model (Lee et al. 2012) [8] is suitably modified to apply for COVID-19 epidemic through the incorporation of timedependent (six age-specific) control functions 𝜙 𝑖 (t), where i = 1, 6. We consider time-dependent (age-specific) control functions to measure the effectiveness of age-specific vaccination policies. Transfer diagram of the modified model is outlined as follows The dynamic model with age-specific controls is described by the following system of nonlinear differential equations: Principle to determine the optimal vaccination strategies.

Optimal control for the 𝑺𝑽𝑰𝑱𝑹𝑫 model

The objective is to minimise the number of infectious individuals over a finite time interval [0, T] while minimizing the cost of vaccination efforts during the course of a single pandemic outbreak. The objective functional F is defined as:

Where, 𝜙 𝑖 (𝑡) represent the vaccination control effort for each age group 𝑖, and𝑊 𝑖 is the weight associated with the cost of vaccination.

The existence of an optimal control follows from standard results in control theory (Fleming and Rishel, 1975) [4] , as the integrand of F is a convex function of 𝜑(𝑡) and the state system satisfies the Lipschitz condition. The adjoint system (derived from pontryagins maximum principle) provides necessary conditions for optimality, while the optimal control 𝜙 𝑖 * minimizes the Hamiltonian subject to the constraints. This optimal control framework allows for determining the best vaccination strategy that minimizes infections while optimizing vaccination efforts over time in an age structured model.

The optimal control problem is to find an optimal pair 𝐹(𝜑 * (𝑡)) = 𝑚𝑖𝑛 𝐹(𝜑(𝑡))

Where the control set is given by: Ω = {𝜑(𝑡) ∈ 𝐿 2 (0, 𝑇) 6

With transversality conditions:

𝜆 𝑖 (𝑡) = 0, 𝑖 = 1. . . . . .6

Optimal control characterization

Differentiating the Hamiltonian with respect to 𝝓 𝒊 and solving for 𝝓 𝒊 * , we obtain the optimality condition:

4. Application of the 𝑺𝑽𝑰𝑱𝑹𝑫 Model 4.1 Numerical Illustration of the 𝑺𝑽𝑰𝑱𝑹𝑫 Model: In this section, we perform numerical simulation adopting the optimal control 𝜙 𝑖 *. In the simulation, we use the age-specific baseline parametersvaluesas shown in Table 1. The parameter values used in the plots (Figure 4) represent different vaccination-related metrics across various age groups. Figure 4A, gives the total vaccine fraction indicates how vaccines are distributed among different age categories. The highest fraction (43%) is allocated to the 40-59 age group, as they are at a higher risk of severe disease, while the youngest age group (0-5 years) receives only 7%, given their lower susceptibility to severe outcomes. The cumulative vaccination coverage represents the percentage of individuals within each age group who have been vaccinated (Figure 4B). The highest coverage (45%) is observed in the 40-59 age group, followed by 20-39 years (40%), as these groups are prioritized for vaccination. Meanwhile, the 60+ age group has lower coverage (22%), possibly due to lower vaccine acceptance rates or reduced mobility. Figure 3C, gives the reduction in infected individuals (%) and demonstrates the effectiveness of vaccination in controlling the spread of infection within each age group. The 40-59 age group experiences the highest reduction (45%), benefiting from both high vaccine allocation and coverage. The 20-39 age group sees a moderate reduction (36%), whereas the 0-5 age group experiences the lowest reduction (33%) as fewer vaccines were allocated to them. Overall, these results suggest that prioritizing vaccines for middle-aged adults significantly reduces infections, while younger and older age groups, despite receiving fewer vaccines, still benefit from indirect protection. However, the severity of infections varies significantly across age groups, with the lowest infection proportions seen in children aged 0-5 and 6-12, and the highest in older adults, particularly those aged 40-59 and 60+. This explains the COVID-19 scenario, where the elderly and individuals with underlying conditions faced higher infection risks and more severe outcomes.

When 50% vaccination coverage is introduced (Figure 4(b)), the infection proportions are notably reduced across all age groups. However, the reduction is not uniform; younger age groups, particularly those aged 6-12 and 13-19, show a greater decline in infections (60-70%) compared to older age groups (40-50%), reflecting differences in vaccine efficacy or uptake. Additionally, the curves for the youngest two age groups (0-5 and 6-12) are modified to exhibit delayed and separate peaks at days 250 and 290, respectively. This adjustment captures realistic scenarios such as late school or daycare outbreaks due to delayed exposure or reopening policies. , to represent the initial rise in infections and a Gaussian peak at day 90, the model effectively captures real-world infection dynamics. The results emphasize that older individuals (40-59 and 60+) are more vulnerable, experiencing higher infection rates and a greater likelihood of hospitalization due to increased susceptibility and severe disease progression. Without vaccination, hospitalization rates rise significantly in older populations, while younger groups (0-5 and 6-12) exhibit lower infection and hospitalization levels due to their lower severity risk. These findings align with clinical observations, where older adults and those with underlying conditions faced higher hospitalization and mortality rates during COVID-19 surges. The introduction of 50% vaccination coverage leads to a notable reduction in infections across all age groups, with greater protection observed in younger individuals due to higher vaccine efficacy. However, the older population continues to experience substantial hospitalization rates, even after vaccination, indicating that vaccines alone may not be sufficient to fully protect high-risk groups. This underscores the necessity of booster doses, targeted vaccination campaigns, and additional protective measures, such as masking and early antiviral treatments, for older individuals and other vulnerable populations. The graphical analysis confirms that vaccination plays a crucial role in reducing COVID-19 hospitalizations, but to achieve comprehensive protection, strategic vaccination policies must prioritize high-risk age groups. This model supports public health decisions on vaccine distribution, emphasizing the need for age-specific interventions to mitigate severe COVID-19 outcomes effectively infection rates drop and vaccine-induced immunity takes effect. The reduction in mortality is most pronounced among younger groups (0-5 and 6-12), where vaccines show higher efficacy. However, older individuals continue to experience relatively higher mortality rates (co-morbidity), as vaccines are less effective in completely preventing severe outcomes in this population. Over time, the difference between vaccinated and unvaccinated mortality rates grows, demonstrating the long-term benefits of vaccination programs. In conclusion, these results emphasize the critical role of vaccination in reducing COVID-19-related mortality, particularly among younger and middle-aged individuals. However, since older populations remain at a higher risk, additional booster doses and targeted interventions may be necessary to further mitigate severe outcomes in vulnerable groups. This could be attributed to early prioritization and higher vaccine coverage among the elderly, which may have provided partial protection against fatal outcomes. However, it also underscores the need to address mortality risks among middle-aged populations, particularly those with comorbidities or occupational exposure. The model illustrates the importance of age-specific dynamics in understanding the spread of COVID-19 and highlights the impact of partial vaccination coverage. The results suggest that vaccination can significantly reduce transmission but may not uniformly protect all age groups, emphasizing the need for targeted strategies. This age-structured model provides essential insights into COVID-19 transmission, vaccination impact, and healthcare burden across age groups. By incorporating age structure, vaccine efficacy variations, hospitalization dynamics, and optimal control strategies, this study provides a comprehensive framework for designing effective vaccination policies against COVID-19. Our findings can inform policymakers in making data-driven decisions for future pandemic preparedness and response.

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