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A dynamical study on stochastic reaction diffusion epidemic model with nonlinear incidence rate.

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Summary

This study introduces two numerical schemes for stochastic epidemic models, finding the implicit finite difference scheme accurately predicts disease dynamics and preserves positivity. The implicit scheme offers a more robust solution for stochastic SIR models.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Computational Science

Background:

  • Traditional epidemic models struggle to capture the non-deterministic behavior observed in real-world disease spread.
  • Stochastic epidemic models are crucial for accurately predicting infectious disease dynamics in wildlife and human populations.
  • Incorporating stochastic processes presents a significant challenge in epidemiological modeling.

Purpose of the Study:

  • To numerically solve a stochastic reaction-diffusion SIR epidemic model using two novel schemes.
  • To analyze the influence of stochastic processes on disease dynamics.
  • To evaluate the convergence, stability, and positivity-preserving properties of the proposed numerical methods.

Main Methods:

  • The study considers a stochastic SIR (Susceptible-Infected-Recovered) epidemic model with a time noise term as the stochastic source.
  • Two numerical schemes are proposed: a stochastic backward Euler scheme and a stochastic implicit finite difference scheme (IFDS).
  • Stability is proven using Von-Neumann criteria, demonstrating unconditional stability for both schemes.

Main Results:

  • The stochastic backward Euler scheme converges to a disease-free equilibrium but exhibits negative behavior and fails to converge to an endemic equilibrium.
  • The stochastic IFDS converges to both disease-free and endemic equilibria, crucially preserving the positivity of solutions.
  • Graphical analysis shows the stochastic SIR model behaves similarly to the classical model when noise intensity approaches zero.

Conclusions:

  • The proposed stochastic IFDS is a reliable and effective numerical method for solving stochastic epidemic models, outperforming the stochastic backward Euler scheme.
  • The IFDS accurately captures disease dynamics, converges to biologically relevant equilibria, and maintains solution positivity.
  • This work provides a robust computational tool for understanding and predicting infectious disease spread in complex, real-world scenarios.