Positivity-Preserving Numerical Method and Relaxed Control for Stochastic Susceptible-Infected-Vaccinated Epidemic...

Area of Science:

• Epidemiology
• Computational Mathematics
• Stochastic Processes

Background:

• Stochastic Susceptible-Infected-Vaccinated (SIV) models are crucial for understanding infectious disease dynamics.
• Analytical solutions for SIV models are often intractable due to nonlinear terms.
• Existing numerical methods like Euler-Maruyama (EM) have limited convergence rates (1/2 order).

Purpose of the Study:

• To develop a novel numerical scheme for the stochastic SIV model with improved convergence.
• To ensure positive and stable numerical solutions for epidemic modeling.
• To investigate optimal control strategies for infectious diseases using advanced approximation methods.

Main Methods:

• Construction of a logarithmic truncated Euler-Maruyama (EM) scheme.
• Proof of the existence of an invariant measure for the stochastic SIV model with Markov switching.
• Application of the Markov chain approximation method for relaxed controls.

Main Results:

• The proposed logarithmic truncated EM scheme achieves order-1 convergence.
• The scheme guarantees positive numerical solutions for the stochastic SIV model.
• The Markov chain approximation method converges to optimal control strategies as mesh size decreases.

Conclusions:

• The developed logarithmic truncated EM scheme offers a significant improvement over existing methods for SIV modeling.
• The study provides a robust framework for predicting disease dynamics and optimizing intervention strategies.
• Numerical examples validate the theoretical findings and the efficacy of the new scheme.