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.

In this work, we examine a kinetic framework for modeling the time evolution of size distribution densities of two populations governed by predator–prey interactions. The model builds upon the classical Boltzmann-type equations, where the dynamics arise from elementary binary interactions between the populations. The model uniquely incorporates a linear redistribution operator to quantify the birth rates in both populations, inspired by wealth redistribution operators. We prove that, under a suitable scaling regime, the Boltzmann formulation transitions to a system of coupled Fokker–Planck-type equations. These equations describe the evolution of the distribution densities and link the macroscopic dynamics of their mean values to a Lotka–Volterra system of ordinary differential equations, with parameters explicitly derived from the microscopic interaction rules. We then determine the local equilibrium of the Fokker–Planck system, which are Gamma-type densities, and investigate the problem of relaxation of its solutions toward these kinetic equilibrium, in terms of their moments’ dynamics. The results establish a bridge between kinetic modeling and classical population dynamics, offering a multiscale perspective on predator–prey systems.

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.

This paper is devoted to a mathematical proof of the continuous dependence on the initial data for the discrete thermostatted kinetic framework, for all T > 0. This is a versatile model for describing the time-evolution of a biological complex system which is composed by a large number of interacting entities, called active particles, and is subjected to an external force field due to the environment. A thermostat term is introduced in order to keep the 2nd-order moment of the system, corresponding to the physical global activation energy, constant in time. This model is expressed by a system of nonlinear ordinary differential equations with quadratic nonlinearity.