A kinetic framework under the action of an external force field: Analysis and application in epidemiology

Markov Chains and Kinetic Theory: A Possible Application to Socio-Economic Problems

“…The above reconstruction of the equations of Kinetic Theory in terms of a random process {(X t , χ t )} t∈[0,T) coupling a stationary Markov Chain and a Bernoullian process is very useful to extend our model for many-particle systems to cover the case of systems interacting with the external world. In fact, as we stressed in Sections 4 and 5, Equation (19) holds for isolated many-particle systems. These equations are based on the assumption that the instantaneous variation in the distribution of particles over the state space is uniquely due to mutual interactions between particles, and that these interactions always modify the states of the involved particles in the same way, independently of the time at which they occur.…”

“…1. As the above deduction procedure shows, system (19) is actually originated by coupling a stationary vector Markov Chain with an additional vector random process; 2. This latter process modifies the final form of the transition matrix, but preserves its stationarity (that is, its independence on time); 3.…”

“…As already pointed out, the literature about the Kinetic Theory for Active Particles and other kinetic models offers a wide number of papers treating particular examples and interesting applications of system (19). In particular, in the socio-economic framework, the reader can find interesting applications in [21][22][23][24].…”

“…This has led to flourishing research devoted to treating socio-economic problems in the framework of the Kinetic Theory (see, e.g., [20][21][22][23][24]). However, as far as we are aware, with the only exception of [18,19], where socio-economic problems are proposed as possible interpretations of a more general scheme, the influence of environmental conditions has not been considered, and also in the above quoted papers such influence is described by introducing an "external force" acting directly on the distribution of states: till now, no studies have been conducted about external influences only indirectly modifying the distribution of the states, by producing direct variations only in transition probabilities and in interaction rates (see the references above and Section 7). The aim of the present paper is just to fill this gap, by showing how the evolution of a many-particle system can always be described as a vector time-continuous random process {(X t , χ t , E t )} t∈[0,T) , where {X t } is a vector Markov Chain, {χ t } is a suitably defined vector Bernoulli process, and {E t } is a vector (or scalar) random process expressing the variation in time of the environment.…”

“…Nevertheless, in the literature about the kinetic-theoretic models Markov Chains are never explicitly mentioned (at least to our knowledge). So, the question spontaneously arises whether exploiting their role could improve our understanding of the terms of the equations and suggest new ways to achieve a better and more effective description of the behavior of systems interacting with the external world (in this connection, see [16,17], where any implicit use of the Principle of Inertia is renounced, and [18,19], where suitable «forcing» terms are introduced in the equations). For this connection, we want to show that the interpretation of the evolution of a many-particle system as an n-tuple of joined stochastic processes, one of which is a vector Markov Chain [15], allows us to describe interactions with the external world by assuming that the interaction rates and the transition matrices undergo stochastic variations at each step [15].…”