Vaccination Control in a Stochastic SVIR Epidemic Model

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Analysis

Background:

• Stochastic differential equations (SDEs) are crucial for modeling infectious disease dynamics, incorporating random fluctuations.
• The SIR (Susceptible-Infectious-Recovered) and its variants like SVIR (Susceptible-Vaccinated-Infectious-Recovered) are fundamental epidemiological models.
• Understanding disease-free equilibrium stability is key to assessing disease eradication potential.

Purpose of the Study:

• To analyze the stability of the disease-free equilibrium in a stochastic SVIR epidemic model with vaccination.
• To investigate optimal control strategies for managing epidemics in both stochastic and deterministic SVIR models.
• To develop a numerical approximation for solving the stochastic optimal control problem.

Main Methods:

• Utilizing stochastic differential equations to model the SVIR epidemic.
• Applying stability theory for stochastic systems to prove almost sure exponential stability.
• Formulating and analyzing optimal control problems for both deterministic and stochastic models.
• Employing numerical approximation techniques based on deterministic model solutions.

Main Results:

• Demonstrated almost sure exponential stability of the disease-free equilibrium for the stochastic SVIR model when the basic reproduction number (ℛ(0)) is less than 1.
• Developed an optimal control framework applicable to both the deterministic and stochastic SVIR models.
• Established a numerical method for approximating solutions to the stochastic optimal control problem.

Conclusions:

• The vaccination strategy in the stochastic SVIR model can effectively lead to disease eradication under specific conditions (ℛ(0) < 1).
• The study provides a robust mathematical framework for analyzing and controlling stochastic epidemic models.
• Numerical approximations based on deterministic models offer a viable approach for practical implementation.