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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

Author

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  • M. De la Sen
  • R. Nistal
  • S. Alonso-Quesada
  • A. Ibeas

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.

Suggested Citation

  • M. De la Sen & R. Nistal & S. Alonso-Quesada & A. Ibeas, 2019. "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," Discrete Dynamics in Nature and Society, Hindawi, vol. 2019, pages 1-22, July.
  • Handle: RePEc:hin:jnddns:8959681
    DOI: 10.1155/2019/8959681
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    Cited by:

    1. Shanshan Chen & Yijun Ran & Hebo Huang & Zhenzhen Wang & Ke-ke Shang, 2022. "Epidemic Dynamics of Two-Pathogen Spreading for Pairwise Models," Mathematics, MDPI, vol. 10(11), pages 1-18, June.
    2. Florin Avram & Rim Adenane & Lasko Basnarkov & Gianluca Bianchin & Dan Goreac & Andrei Halanay, 2023. "An Age of Infection Kernel, an R Formula, and Further Results for Arino–Brauer A , B Matrix Epidemic Models with Varying Populations, Waning Immunity, and Disease and Vaccination Fatalities," Mathematics, MDPI, vol. 11(6), pages 1-21, March.
    3. Carmen Legarreta & Manuel De la Sen & Santiago Alonso-Quesada, 2024. "On the Properties of a Newly Susceptible, Non-Seriously Infected, Hospitalized, and Recovered Subpopulation Epidemic Model," Mathematics, MDPI, vol. 12(2), pages 1-34, January.

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