A dynamically consistent computational method to solve numerically a mathematical model of polio propagation with spatial diffusion.
Basic reproductive number
Diffusive mathematical model
Global stability analysis
Neumann stability analysis
Positivity-preserving scheme
Journal
Computer methods and programs in biomedicine
ISSN: 1872-7565
Titre abrégé: Comput Methods Programs Biomed
Pays: Ireland
ID NLM: 8506513
Informations de publication
Date de publication:
May 2022
May 2022
Historique:
received:
24
11
2021
revised:
16
02
2022
accepted:
21
02
2022
pubmed:
3
3
2022
medline:
13
4
2022
entrez:
2
3
2022
Statut:
ppublish
Résumé
In this work, a mathematical model based on differential equations is proposed to describe the propagation of polio in a human population. The motivating system is a compartmental nonlinear model which is based on the use of ordinary differential equations and four compartments, namely, susceptible, exposed, infected and vaccinated individuals. In this manuscript, the mathematical model is extended in order to account for spatial diffusion in one dimension. Nonnegative initial conditions are used, and we impose homogeneous Neumann conditions at the boundary. We determine analytically the disease-free and the endemic equilibria of the system along with the basic reproductive number. We establish thoroughly the nonnegativity and the boundedness of the solutions of this problem, and the stability analysis of the equilibrium solutions is carried out rigorously. In order to confirm the validity of these results, we propose an implicit and linear finite-difference method to approximate the solutions of the continuous model. The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and that it is capable of preserving unconditionally the positivity of the approximations. For illustration purposes, we provide some computer simulations that confirm some theoretical results derived in the present manuscript.
Sections du résumé
BACKGROUND AND OBJECTIVE
OBJECTIVE
In this work, a mathematical model based on differential equations is proposed to describe the propagation of polio in a human population. The motivating system is a compartmental nonlinear model which is based on the use of ordinary differential equations and four compartments, namely, susceptible, exposed, infected and vaccinated individuals.
METHODS
METHODS
In this manuscript, the mathematical model is extended in order to account for spatial diffusion in one dimension. Nonnegative initial conditions are used, and we impose homogeneous Neumann conditions at the boundary. We determine analytically the disease-free and the endemic equilibria of the system along with the basic reproductive number.
RESULTS
RESULTS
We establish thoroughly the nonnegativity and the boundedness of the solutions of this problem, and the stability analysis of the equilibrium solutions is carried out rigorously. In order to confirm the validity of these results, we propose an implicit and linear finite-difference method to approximate the solutions of the continuous model.
CONCLUSIONS
CONCLUSIONS
The numerical model is stable in the sense of von Neumann, it yields consistent approximations to the exact solutions of the differential problem, and that it is capable of preserving unconditionally the positivity of the approximations. For illustration purposes, we provide some computer simulations that confirm some theoretical results derived in the present manuscript.
Identifiants
pubmed: 35235894
pii: S0169-2607(22)00094-3
doi: 10.1016/j.cmpb.2022.106709
pii:
doi:
Types de publication
Journal Article
Langues
eng
Sous-ensembles de citation
IM
Pagination
106709Informations de copyright
Copyright © 2022 Elsevier B.V. All rights reserved.
Déclaration de conflit d'intérêts
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.