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An explicit unconditionally stable scheme: application to diffusive Covid-19 epidemic model.

Yasir Nawaz1, Muhammad Shoaib Arif1, Kamaleldin Abodayeh2

  • 1Department of Mathematics, Air University, PAF Complex E-9, Islamabad, 44000 Pakistan.

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Summary
This summary is machine-generated.

A new numerical method ensures stable and accurate solutions for time-dependent partial differential equations, specifically applied to the COVID-19 epidemic model. This method guarantees positive outcomes for epidemic modeling and demonstrates strong mathematical properties.

Keywords:
Conditionally positivity preservingConvergence conditionsDiffusive COVID-19 modelProposed schemeStability

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Area of Science:

  • Numerical Analysis
  • Mathematical Biology
  • Computational Science

Background:

  • Time-dependent partial differential equations (PDEs) are crucial for modeling complex phenomena.
  • Solving these equations often requires numerical methods that balance accuracy, stability, and computational efficiency.
  • Epidemic modeling, particularly for diseases like COVID-19, relies on accurate mathematical frameworks.

Purpose of the Study:

  • To introduce a novel, unconditionally stable numerical scheme for time-dependent PDEs.
  • To apply this scheme to a COVID-19 epidemic model.
  • To analyze the mathematical properties of the scheme, including stability, consistency, and convergence.

Main Methods:

  • Development of an explicit, unconditionally stable numerical scheme.
  • Application to a mathematical model for COVID-19 transmission.
  • Von Neumann stability analysis for general parabolic equations with source terms.
  • Verification of consistency and discussion of convergence for the epidemic model.

Main Results:

  • The proposed scheme is first-order accurate in time and second-order accurate in space.
  • It ensures a positive solution for the COVID-19 epidemic model.
  • Unconditional stability is proven for parabolic equations with source terms.
  • Consistency and convergence are demonstrated for the specific epidemic model.

Conclusions:

  • The developed numerical scheme offers a robust and reliable tool for solving time-dependent PDEs.
  • Its application to epidemic modeling, like COVID-19, provides accurate and stable predictions.
  • The scheme's proven stability and accuracy make it suitable for a wide range of scientific applications.