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A New Stochastic Split-Step θ-Nonstandard Finite Difference Method for the Developed SVIR Epidemic Model with

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Summary

This study analyzes an SVIR epidemic model with temporary immunity, proving disease extinction and persistence conditions. Novel numerical methods, SSTM and SSSNSFD, validate theoretical results and explore model dynamics under various parameters.

Keywords:
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Area of Science:

  • Mathematical Biology
  • Epidemiology
  • Numerical Analysis

Background:

  • Analysis of epidemic models is crucial for understanding disease dynamics.
  • Temporary immunity and general incidence rates present complex modeling challenges.
  • Existing analytical solutions are often difficult to obtain for stochastic epidemic models.

Purpose of the Study:

  • To construct and analyze a stochastic SVIR (Susceptible-Infectious-Vaccinated-Recovered) epidemic model with temporary immunity.
  • To establish conditions for disease extinction and persistence using Lyapunov functions.
  • To develop and validate novel numerical methods for approximating the model's solution.

Main Methods:

  • Theoretical analysis using Lyapunov functions to prove existence, uniqueness, and stability of solutions.
  • Development of two numerical schemes: split-step θ-Milstein (SSTM) and stochastic split-step θ-nonstandard finite difference (SSSNSFD).
  • Simulation and graphical comparison of numerical methods to validate theoretical findings and analyze parameter effects.

Main Results:

  • Sufficient conditions for disease extinction and persistence were mathematically derived.
  • The SSSNSFD method was proven to preserve positivity, boundedness, and stability.
  • Numerical simulations supported theoretical results and illustrated the impact of immunity duration, incidence rates, and noise on epidemic spread.

Conclusions:

  • The developed SVIR model provides a robust framework for studying epidemic dynamics with temporary immunity.
  • The SSSNSFD method offers a reliable approach for simulating such complex stochastic models.
  • Parameter variations, including noise and immunity periods, significantly influence disease persistence and spread.