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Discrete Dynamics in Nature and Society
Volume 2019, Article ID 8959681, 22 pages
https://doi.org/10.1155/2019/8959681
Research Article

Some Formal Results on Positivity, Stability, and Endemic Steady-State Attainability Based on Linear Algebraic Tools for a Class of Epidemic Models with Eventual Incommensurate Delays

1Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, Leioa (Bizkaia), P.O. Box 48940, Spain
2Department of Telecommunications and Systems Engineering, Universitat Autònoma de Barcelona, UAB, 08193-Barcelona, Spain

Correspondence should be addressed to M. De la Sen; sue.uhe@nesaled.leunam

Received 20 December 2018; Revised 20 May 2019; Accepted 16 June 2019; Published 8 July 2019

Academic Editor: Cengiz Çinar

Copyright © 2019 M. De la Sen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A formal description of typical compartmental epidemic models obtained is presented by splitting the state into an infective substate, or infective compartment, and a noninfective substate, or noninfective compartment. A general formal study to obtain the reproduction number and discuss the positivity and stability properties of equilibrium points is proposed and formally discussed. Such a study unifies previous related research and it is based on linear algebraic tools to investigate the positivity and the stability of the linearized dynamics around the disease-free and endemic equilibrium points. To this end, the complete state vector is split into the dynamically coupled infective and noninfective compartments each one containing the corresponding state components. The study is then extended to the case of commensurate internal delays when all the delays are integer multiples of a base delay. Two auxiliary delay-free systems are defined related to the linearization processes around the equilibrium points which correspond to the zero delay, i.e., delay-free, and infinity delay cases. Those auxiliary systems are used to formulate stability and positivity properties independently of the delay sizes. Some examples are discussed to the light of the developed formal study.