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

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  • Atiqur Chowdhury
    Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88001, USA
  • Saleh Tanveer
    Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
  • Xueying Wang
    Department of Mathematics and Statistics, Washington State University, Pullman, WA 99164, USA

Abstract

<jats:p> 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 ( <jats:italic>quasi-solution</jats:italic> ) satisfies verifiable conditions. A local analysis in a neighbour- hood of a <jats:italic>quasi-solution</jats:italic> 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. </jats:p>

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