No abstract

In the mathematical epidemiology community, there is an increasing interest in shaping the complex interplay between human behaviour and disease spreading. We give a contribution in this direction by illustrating a method to derive behavioural change epidemic models from a stochastic particle description by the means of kinetic equations. We consider an Susceptible–Infected–Removed–like model where contact rates depend on the behavioural patterns adopted across the population. The selection of the social behaviour happens during the interactions between individuals adopting alternative strategies and it is driven by an imitation game dynamics. Agents have a double microscopic state: a discrete label, that denotes the epidemiological compartment to which they belong, and the degree of flexibility of opinion, that is a measure of the personal attitude to change opinion and, hence, to switch between the alternative social contact patterns. We derive kinetic evolution equations for the distribution functions of the degree of flexibility of opinion of the individuals for each compartment, whence we obtain macroscopic equations for the densities and average flexibilities of opinion. After providing the basic properties of the macroscopic model, we numerically investigate it by focusing on the impact of the flexibility of opinion on the epidemic course and on the consequent behavioural responses.

This paper is devoted to the derivation and mathematical analysis of new thermostatted kinetic theory frameworks for the modeling of nonequilibrium complex systems composed by particles whose microscopic state includes a vectorial state variable. The mathematical analysis refers to the global existence and uniqueness of the solution of the related Cauchy problem. Specifically, the paper is divided in two parts. In the first part the thermostatted framework with a continuous vectorial variable is proposed and analyzed. The framework consists of a system of partial integro-differential equations with quadratic type nonlinearities. In the second part the thermostatted framework with a discrete vectorial variable is investigated. Real world applications, such as social systems and crowd dynamics, and future research directions are outlined in the paper.

A differential equation-based framework is suitable for the modeling of nonequilibrium complex systems if its solution is able to reach, as time goes to infinity, the related nonequilibrium steady states. The thermostatted kinetic theory framework has been recently proposed for the modeling of complex systems subjected to an external force field. The present paper is devoted to the mathematical proof of the convergence of the solutions of the thermostatted kinetic framework towards the related nonequilibrium stationary states. The proof of the main result is gained by employing the Fourier transform and distribution theory arguments.