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

Abstract

We present a methodology to connect an ordinary differential 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 differential 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.

The Multi-Compartment SI(RD) Model with Regime Switching: An Application to COVID-19 Pandemic

“…with a multidimensional parameter λ := (ν, η, θ, τ) controlling the main characteristics of the function. Our use of the specific function G in Formula ( 9), in our modelling, stems from the possibility of performing a stability analysis similar to the one presented in [17].…”

“…It is clear that the existence and unicity of a solution for the smooth regime switching model in Section 3 may be dealt either by the usual Cauchy-Lipschitz existence and unicity theorem or by the results of this section, with the necessary adaptations of the definitions of the matrices M(Θ(t), Y(t)) and M ∆ (Θ(t), y 1 , y 2 ). An example of smooth regime switching was studied in [17] with two coupled ODE, having in each equation two evolution regimes in such a way that the transition in time, between these regimes, is achieved by a coupling function similar to the function in Formula (9).…”

“…In [16], the author considers an alternative for the regime switching-of rough type-considered here, but for a SEIR type model in which the coefficients are supposed to take two (or three values) on two (or three) complementary intervals of the time domain. A study of a regime switching models for the evolution of an infected population that influenced this work is [17]. A recent study of a SI(RD) model with regime switching with a logistic function is presented in [18]; the authors use a neural network to detect the regime change points, and a grid search for the estimation of the usual parameters.…”

Open Markov Type Population Models: From Discrete to Continuous Time

“…The continuous time framework has also been addressed, for instance, in [18][19][20], for the case of semi-Markov processes and for nonhomogeneous semi-Markov systems [21]. We may also refer a framework of open Markov chains with finite state space-see in [22] and references therein-that has already seen applications in Actuarial or Financial problems-as, for instance, in [23,24]-but also in population dynamics (see [25]). The weaker formalism open Markov schemes, in discrete time-developed in [26]-allows for influxes of new elements in the population to be given as general time series models.…”

“…Remark 12 (Applying Theorem 6). If we manage to estimate a discrete time Markov chain transition matrix and if we manage to fit some function f -such that lim t→+∞ f (t) = λ ∞ -to the number of new incoming members in the population at a set of non accumulating non-evenly spaced dates (as done with a statistical procedure in [22] or, with a simple fitting in [25]) then, Theorem 6 allows us to get the asymptotic expected number of elements in the transient classes of a sMp having as embedded Markov chain the estimated one.…”