Optimal vaccination in a SIRS epidemic model.

Area of Science:

• Epidemiology
• Mathematical Biology
• Public Health

Background:

• SIRS compartmental models are used to study infectious disease dynamics, where immunity is temporary.
• Vaccination campaigns aim to reduce disease burden but incur social and economic costs.
• Optimal control theory provides a framework for resource allocation in public health interventions.

Purpose of the Study:

• To formulate and solve an optimal vaccination problem within a deterministic SIRS model.
• To minimize the social and economic costs associated with infectious diseases.
• To determine long-term vaccination policies for disease control.

Main Methods:

• Development of a deterministic SIRS compartmental model with temporary immunity.
• Application of optimal control theory and Hamilton-Jacobi-Bellman equations.
• Derivation of a non-smooth verification theorem for viscosity solutions.
• Numerical implementation with quadratic cost functions.

Main Results:

• A theoretical framework for optimal vaccination strategies was established.
• Conditions for the well-posedness of the closed-loop system were identified.
• Numerical simulations demonstrated the effectiveness of the optimal policy.
• The optimal policy can achieve zero infections in the long run under certain parameters.

Conclusions:

• Optimal vaccination strategies can effectively control infectious diseases in SIRS models.
• The long-term eradication of infections is feasible with optimal policies when disease parameters are low.
• Mathematical modeling and optimal control are valuable tools for public health intervention planning.