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Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples

Author

Listed:
  • Florin Avram

    (Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France)

  • Rim Adenane

    (Département des Mathématiques, Université Ibn-Tofail, Kenitra 14000, Morocco)

  • Lasko Basnarkov

    (Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, 1000 Skopje, North Macedonia
    Macedonian Academy of Sciences and Arts, 1000 Skopje, North Macedonia)

Abstract

The fact that the famous basic reproduction number R 0 , i.e., the largest eigenvalue of the next generation matrix F V − 1 , sometimes has a probabilistic interpretation is not as well known as it deserves to be. It is well understood that half of this formula, − V , is a Markovian generating matrix of a continuous-time Markov chain (CTMC) modeling the evolution of one individual on the compartments. It has also been noted that the not well-enough-known rank-one formula for R 0 of Arino et al. (2007) may be interpreted as an expected final reward of a CTMC, whose initial distribution is specified by the rank-one factorization of F . Here, we show that for a large class of ODE epidemic models introduced in Avram et al. (2023), besides the rank-one formula, we may also provide an integral renewal representation of R 0 with respect to explicit “age kernels” a ( t ) , which have a matrix exponential form.This latter formula may be also interpreted as an expected reward of a probabilistic continuous Markov chain (CTMC) model. Besides the rather extensively studied rank one case, we also provide an extension to a case with several susceptible classes.

Suggested Citation

  • Florin Avram & Rim Adenane & Lasko Basnarkov, 2024. "Some Probabilistic Interpretations Related to the Next-Generation Matrix Theory: A Review with Examples," Mathematics, MDPI, vol. 12(15), pages 1-16, August.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:15:p:2425-:d:1449801
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    References listed on IDEAS

    as
    1. Helen J Wearing & Pejman Rohani & Matt J Keeling, 2005. "Appropriate Models for the Management of Infectious Diseases," PLOS Medicine, Public Library of Science, vol. 2(7), pages 1-1, July.
    2. J. A. P. Heesterbeek & K. Dietz, 1996. "The concept of Ro in epidemic theory," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 50(1), pages 89-110, March.
    3. 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.
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