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Proc Math Phys Eng Sci. 2020 Feb;476(2234):20190673. doi: 10.1098/rspa.2019.0673. Epub 2020 Feb 26.

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

Author information

1
Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88001, USA.
2
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA.
3
Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA.

Abstract

This paper is concerned primarily with constructive mathematical analysis of a general system of nonlinear two-point boundary value problem when an empirically constructed candidate for an approximate solution (quasi-solution) satisfies verifiable conditions. A local analysis in a neighbour- hood of a quasi-solution assures the existence and uniqueness of solutions and, at the same time, provides error bounds for approximate solutions. Applying this method to a cholera epidemic model, we obtain an analytical approximation of the steady-state solution with rigorous error bounds that also displays dependence on a parameter. In connection with this epidemic model, we also analyse the basic reproduction number, an important threshold quantity in the epidemiology context. Through a complex analytic approach, we determine the principal eigenvalue to be real and positive in a range of parameter values.

KEYWORDS:

basic reproduction number; quasi-solution; steady-state solution; two-point boundary value problem

PMID:
32201479
PMCID:
PMC7069490
[Available on 2021-02-01]
DOI:
10.1098/rspa.2019.0673

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