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A non-standard finite-difference-method for a non-autonomous epidemiological model: analysis, parameter

Benjamin Wacker1,2, Jan Christian Schlüter2,3

  • 1Department of Engineering and Natural Sciences, University of Applied Sciences Merseburg, Eberhard-Leibnitz-Str. 2, D-06217 Merseburg, Germany.

Mathematical Biosciences and Engineering : MBE
|July 28, 2023
PubMed
Summary

We developed a new numerical method for the susceptible-infected-recovered (SIR) model, proving its non-negativity and linear convergence. A parameter identification algorithm was also introduced for enhanced SIR model analysis.

Keywords:
COVID-19SIR modelconvergenceepidemiologynon-negativitynon-standard finite-difference-method

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Area of Science:

  • Mathematical modeling
  • Computational epidemiology
  • Numerical analysis

Background:

  • The susceptible-infected-recovered (SIR) model is fundamental in epidemiology.
  • Accurate numerical solutions are crucial for understanding disease dynamics.
  • Existing methods may face challenges with non-autonomous and time-continuous models.

Purpose of the Study:

  • To introduce a novel non-standard finite-difference method for the time-continuous non-autonomous SIR model.
  • To establish theoretical guarantees for the numerical solution's properties.
  • To develop and validate a parameter identification algorithm for the SIR model.

Main Methods:

  • Development of a new non-standard finite-difference scheme.
  • Mathematical proofs for non-negativity preservation of the numerical solution.
  • Convergence analysis to demonstrate linear convergence to the exact solution.
  • Introduction of a parameter identification algorithm.

Main Results:

  • The proposed finite-difference method ensures non-negativity of the numerical solution.
  • Linear convergence of the time-discrete solution to the time-continuous solution is proven.
  • A functional parameter identification algorithm for the SIR model is presented.

Conclusions:

  • The novel numerical method provides a reliable and accurate approach for solving non-autonomous SIR models.
  • The proven convergence and non-negativity properties enhance the method's applicability.
  • The parameter identification algorithm aids in fitting the SIR model to real-world data.