Numerical Analysis of BIMs for Stochastic SIR and SIS Models with Variable Contact Diffusion Rates

Area of Science:

• Mathematical epidemiology
• Computational mathematics
• Stochastic modeling

Background:

• Stochastic differential equations (SDEs) are crucial for modeling population dynamics with inherent randomness.
• Existing numerical methods may not adequately capture the qualitative properties of SDEs in epidemiological contexts.
• Variable contact rates and non-constant population sizes introduce complexities in SIR and SIS models.

Purpose of the Study:

• To investigate the qualitative properties of numerical methods for SDEs in mathematical epidemiology.
• To assess the suitability of Balanced Implicit Methods (BIMs) for simulating SIS and SIR models with variable parameters.
• To demonstrate the convergence and reliability of BIMs for these complex models.

Main Methods:

• Analysis of qualitative properties including positivity, invariance, and stability.
• Proof of mean and mean square consistency.
• Demonstration of local uniform boundedness and mean square contractivity.
• Application of Balanced Implicit Methods (BIMs) to SDEs.

Main Results:

• Convergence of numerical approximations using BIMs is proven.
• Key properties like positivity, invariance, stability, and boundedness are established.
• Mean and mean square consistency are demonstrated, ensuring simulation accuracy.
• Mean square Hölder continuity and contractivity confirm method robustness.

Conclusions:

• Balanced Implicit Methods (BIMs) provide adequate and reliable numerical simulations for epidemiological models.
• The proven properties ensure the fidelity of SDE simulations for SIS and SIR models.
• The findings support the use of BIMs for analyzing population dynamics with variable contact rates and sizes.