Kinetic models for epidemic dynamics with social heterogeneity

Abstract

We introduce a mathematical description of the impact of the number of daily contacts in the spread of infectious diseases by integrating an epidemiological dynamics with a kinetic modeling of population-based contacts. The kinetic description leads to study the evolution over time of Boltzmann-type equations describing the number densities of social contacts of susceptible, infected and recovered individuals, whose proportions are driven by a classical SIR-type compartmental model in epidemiology. Explicit calculations show that the spread of the disease is closely related to moments of the contact distribution. Furthermore, the kinetic model allows to clarify how a selective control can be assumed to achieve a minimal lockdown strategy by only reducing individuals undergoing a very large number of daily contacts. We conduct numerical simulations which confirm the ability of the model to describe different phenomena characteristic of the rapid spread of an epidemic. Motivated by the COVID-19 pandemic, a last part is dedicated to fit numerical solutions of the proposed model with infection data coming from different European countries.

Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models

“…Extensions of the presented methodology are possible to include disease-related mortality and redistribution operators. Moreover, since a direct comparison of the results of similar compartmental kinetic models—in the case of social aspects related to the transience of the epidemic—outlined a good agreement with the actual data [ 8 , 38 , 39 ], we can assume that the present approach is able to follow the real evolution of the economic parameters of a country over a sufficiently long period of time. Indeed, even though the model introduced here necessarily represents a strong simplification of an extremely complex phenomenon, its qualitative behavior is capable of describing the essential features of the pandemic’s impact on individuals’ wealth.…”

“…Depending on the specific choice of β and using the knowledge on the equilibrium states discussed in Section 3.1 it is possible, through the classical hydrodynamic closure of kinetic theory, to derive epidemic models where the dynamics, instead of being homogeneous as in classical compartmental modeling, is influenced by the heterogeneous wealth status of individuals. We refer to [ 8 , 38 ] for examples in this direction.…”

“…The interplay between epidemic spread and the social economic background is described here as the result of interactions among a large number of individuals, each of which is characterized by the variable , measuring the amount of wealth of a single agent. In this regard, as shown in [ 1 , 8 , 38 , 39 ], the fundamental tools of statistical physics allow the understanding of epidemiological dynamics by linking classical compartmental approaches with a statistical description of economic aspects. Indeed, the multiscale nature of kinetic theory allows for the determination of the macroscopic (or aggregate) and measurable features of disease evolution [ 27 , 40 , 41 ].…”

From agent-based models to the macroscopic description of fake-news spread: the role of competence in data-driven applications

“…The advantage of these methods relies on an arbitrarily accurate description of the steady state distribution of the Fokker-Planck model of interest. Similar approaches have been investigated in a different context also in [12,13].…”

“…Previously, kinetic models that include the role of competence or knowledge had been proposed in [5,29,31]. The behavior of a social system composed by a large number of interacting agents has been studied in the case of opinion formation [3,9,14,15,34] and more recently epidemiological dynamics [1,2,12]. We refer to [28] for an introduction to the subject.…”