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Algorithmic Approach for a Unique Definition of the Next-Generation Matrix

Author

Listed:
  • Florin Avram

    (Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France
    These authors contributed equally to this work.)

  • Rim Adenane

    (Laboratoire des Equations aux Dérivées Partielles, Algébre et Géométrie Spectrales, Département des Mathématiques, Université Ibn-Tofail, Kenitra 14000, Morocco
    These authors contributed equally to this work.)

  • Lasko Basnarkov

    (Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, 1000 Skopje, North Macedonia
    These authors contributed equally to this work.)

  • Matthew D. Johnston

    (Department of Mathematics, Computer Science Lawrence Technological University, 21000 W 10 Mile Rd., Southfield, MI 48075, USA
    These authors contributed equally to this work.)

Abstract

The basic reproduction number R 0 is a concept which originated in population dynamics, mathematical epidemiology, and ecology and is closely related to the mean number of children in branching processes (reflecting the fact that the phenomena of interest are well approximated via branching processes, at their inception). Despite the very extensive literature around R 0 for deterministic epidemic models, we believe there are still aspects which are not fully understood. Foremost is the fact that R 0 is not a function of the original ODE model, unless we also include in it a certain ( F , V ) gradient decomposition, which is not unique. This is related to the specification of the “infected compartments”, which is also not unique. A second interesting question is whether the extinction probabilities of the natural continuous time Markovian chain approximation of an ODE model around boundary points (disease-free equilibrium and invasion points) are also related to the ( F , V ) gradient decomposition. We offer below several new contributions to the literature: (1) A universal algorithmic definition of a ( F , V ) gradient decomposition (and hence of the resulting R 0 ). (2) A fixed point equation for the extinction probabilities of a stochastic model associated to a deterministic ODE model, which may be expressed in terms of the ( F , V ) decomposition. Last but not least, we offer Mathematica scripts and implement them for a large variety of examples, which illustrate that our recipe offers always reasonable results, but that sometimes other reasonable ( F , V ) decompositions are available as well.

Suggested Citation

  • Florin Avram & Rim Adenane & Lasko Basnarkov & Matthew D. Johnston, 2023. "Algorithmic Approach for a Unique Definition of the Next-Generation Matrix," Mathematics, MDPI, vol. 12(1), pages 1-40, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:27-:d:1305142
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    References listed on IDEAS

    as
    1. Jin, Yu & Wang, Wendi & Xiao, Shiwu, 2007. "An SIRS model with a nonlinear incidence rate," Chaos, Solitons & Fractals, Elsevier, vol. 34(5), pages 1482-1497.
    2. Yang, Hyun Mo & Greenhalgh, David, 2015. "Proof of conjecture in: The basic reproduction number obtained from Jacobian and next generation matrices—A case study of dengue transmission modelling," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 103-107.
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