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

Florin Avram (), Rim Adenane, Lasko Basnarkov and Matthew D. Johnston
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Florin Avram: Laboratoire de Mathématiques Appliquées, Université de Pau, 64000 Pau, France
Rim Adenane: Laboratoire des Equations aux Dérivées Partielles, Algébre et Géométrie Spectrales, Département des Mathématiques, Université Ibn-Tofail, Kenitra 14000, Morocco
Lasko Basnarkov: Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje, 1000 Skopje, North Macedonia
Matthew D. Johnston: Department of Mathematics, Computer Science Lawrence Technological University, 21000 W 10 Mile Rd., Southfield, MI 48075, USA

Mathematics, 2023, vol. 12, issue 1, 1-40

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.

Keywords: deterministic epidemic model; disease-free equilibrium; stability threshold; basic reproduction number; ( F , V ) gradient decomposition; next-generation matrix; Jacobian approach; CTMC stochastic model associated to a deterministic epidemic model; probability of extinction; rational univariate representation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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