From ODE to Open Markov Chains, via SDE: an application to models for infections in individuals and populations

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Abstract

We present a methodology to connect an ordinary dierential equation (ODE) model of interacting entities at the individual level, to an open Markov chain (OMC) model of a population of such individuals, via a stochastic diferential equation (SDE) intermediate model. The ODE model here presented is formulated
as a dynamic change between two regimes; one regime is of mean reverting type and the other is of inverse logistic type. For the general purpose of defining an OMC model for a population of individuals, we associate an Ito processes, in the form of SDE to ODE system of equations, by means of the addition of Gaussian noise terms which may be thought to model non essential characteristics of the phenomena with small and undifferentiated influences. The next step consists on discretizing the SDE and using the discretized trajectories computed by simulation to define transitions of a finite valued Markov chain; for that, the state space of the Ito processes is partitioned according to some rule. For the example proposed for illustration, the state space of the ODE system referred – corresponding to a model of a viral infection – is partitioned into six infection classes determined by some of the critical points of the ODE system; we detail the evolution of some infected population in these infection classes.
Original languageEnglish
Pages (from-to)180-197
Number of pages18
JournalComputacional and Mathematical Biophysics
Volume8
Issue number1
DOIs
Publication statusPublished - 17 Dec 2020

Keywords

  • Infection modeling
  • population dynamics
  • Ordinary differentila equations
  • Stochastic differential equations
  • Markov chains

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