Kinetic SIR equations and particle limits

Propagation of chaos for a stochastic particle system modelling epidemics

“…In this paper we show that, in the limit N → ∞, the N -particle model introduced in Ciallella et al (2021b) reduces to kinetic equations ((1.9) below) for the oneparticle marginals of the probability measure describing the statistical behaviour of the system. In particular as a crucial step, we prove propagation of chaos, namely the asymptotic statistical independence of the agents.…”

“…However, at variance with the SIR model, we intend to allow spatially inhomogeneous distributions of the populations. A simple proposal for such a model has been given in Ciallella et al (2021b), which we reconsider in the present paper. Our purpose is not to provide realistic modelling of spatial patterns (even though this can be important in applications), but rather follow a natural approach, inspired by the kinetic theory of rarefied gases; see for instance Bellomo et al (2020); Albi et al (2021) and references therein.…”

Kinetic modelling of epidemic dynamics: social contacts, control with uncertain data, and multiscale spatial dynamics

“…Thus, formally as 𝑡 → ∞ in the Fokker-Planck system (33) we get that the stationary states 𝑓 ∞ 𝑆 (𝑤) and 𝑓 ∞ 𝑅 (𝑤) are given by two inverse Gamma densities…”

“…Next, we compare the evolution of the wealth distribution of the system under more realistic hypotheses about the dependence of the risk coefficient 𝜎 on the epidemic spread. We consider the kinetic model (33) in the case of the following two infectious-dependent market risk coefficients…”

“…See for instance the recent monographs and collections [91,95]. Similarly, the use of kinetic theory has proven to be very useful in designing feedback controlled models in various fields of social sciences [3,4,48] and in modeling the movement dynamics of individuals at different scales [2,11,14,33,107].…”