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Nonlinear two-point boundary value problems: applications to a cholera epidemic model.

Atiqur Chowdhury1, Saleh Tanveer2, Xueying Wang3

  • 1Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88001, USA.

Proceedings. Mathematical, Physical, and Engineering Sciences
|March 24, 2020
PubMed
Summary

This study introduces a novel mathematical method for analyzing nonlinear boundary value problems using quasi-solutions. The approach guarantees solution existence and uniqueness, providing error bounds for approximate solutions, particularly for epidemic models.

Keywords:
basic reproduction numberquasi-solutionsteady-state solutiontwo-point boundary value problem

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Area of Science:

  • Mathematical Analysis
  • Epidemiology
  • Applied Mathematics

Background:

  • Nonlinear two-point boundary value problems are common in applied sciences.
  • Approximate solutions often lack rigorous error bounds.
  • Mathematical modeling of epidemic dynamics requires robust analytical tools.

Purpose of the Study:

  • To develop a constructive mathematical analysis for nonlinear boundary value problems.
  • To establish verifiable conditions for approximate solutions (quasi-solutions).
  • To apply this method to a cholera epidemic model for analytical approximation and error bound determination.

Main Methods:

  • Local analysis in the neighborhood of a quasi-solution.
  • Verifiable conditions for existence and uniqueness of solutions.
  • Complex analytic approach to determine eigenvalues.

Main Results:

  • The quasi-solution method assures existence, uniqueness, and provides error bounds for approximate solutions.
  • An analytical approximation of the steady-state solution for a cholera model was obtained with rigorous error bounds.
  • The basic reproduction number and principal eigenvalue were analyzed, showing real and positive values within specific parameter ranges.

Conclusions:

  • The developed constructive analysis provides a rigorous framework for nonlinear boundary value problems.
  • The method is effective for modeling epidemic dynamics, offering reliable approximations and error bounds.
  • The analysis of the basic reproduction number offers insights into disease transmission parameters.