Individual-Based Foundation of SIR-Type Epidemic Models: mean-field limit and large time behaviour

TL;DR

The paper develops a kinetic, mean-field framework for SIR-type epidemic dynamics with reinfection by modeling distributions $f_J(x,t)$ over activity $x$ and deriving macroscopic equations for $m_J(t)$. It connects microscopic binary interactions to a macro-level SIR system and, in the grazing limit, produces a coupled system of Fokker-Planck equations whose large-time behavior is analyzed via energy-distance convergence in Sobolev spaces. A key finding is the dichotomy between reinfection and no-reinfection: with reinfection the system loses memory of initial distributions and converges to stationary local equilibria, while with no reinfection the initial state exerts a lasting influence on the asymptotic profiles. Numerically, structure-preserving schemes corroborate the theoretical predictions, illustrating the link between kinetic descriptions and classical epidemic models and providing a framework for multi-scale analysis and control strategies.

Abstract

We introduce a kinetic framework for modeling the time evolution of the statistical distributions of the population densities in the three compartments of susceptible, infectious, and recovered individuals, under epidemic spreading driven by susceptible-infectious interactions. The model is based on a system of Boltzmann-type equations describing binary interactions between susceptible and infectious individuals, supplemented with linear redistribution operators that account for recovery and reinfection dynamics. The mean values of the kinetic system recover a SIR-type model with reinfection, where the macroscopic parameters are explicitly derived from the underlying microscopic interaction rules. In the grazing collision regime, the Boltzmann system can be approximated by a system of coupled Fokker-Planck equations. This limit allows for a more tractable analysis of the dynamics, including the large-time behavior of the population densities. In this context, we rigorously prove the convergence to equilibrium of the resulting mean-field system in a suitable Sobolev space by means of the so-called energy distance. The analysis reveals the dissipative structure of the dynamics and the role of the interaction terms in driving the system toward a stable equilibrium configuration. These results provide a multi-scale perspective connecting kinetic theory with classical epidemic models.

Figures (4)

• Figure 1: Evolution of $m_J(t)$ and dynamics towards equilibrium (top row). Evolution of $V_J(t)$ and dynamics towards equilibrium point (bottom row). Parameters: $\beta = 2/10$, $\gamma = 1/21$, $\theta = 2$, $\sigma = 1/10$, $m_S(0) = 0.9$, $m_I(0) = m_R(0) = 0.05$, $v_J(0) = 0.01$, $J \in\mathcal{C}$.
• Figure 2: Evolution of the kinetic distributions $f_S(x,t)$ (left), $f_I(x,t)$ (center) and $f_R(x,t)$ (right), over the time horizon $[0,100]$.
• Figure 3: Evolution of $m_J(t)$ and $V_J(t)$ from the solution of the system of Fokker-Planck equations and from the closed system for means \ref{['eq:mSIR']} and variances \ref{['eq:variance_SIR_PDE']}. We report also the evolution of the mean and variance of the quasi-equilibrium densities $f_J^q(x,t)$, $J \in \mathcal{C}$ to highlight the consistency of the large time behaviour.
• Figure 4: Evolution of the energy distance $\mathcal{E}^p(f_J,f_J^q)$ for several weights $p = 5/8,3/4,7/8$.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Remark 3.1
• Remark 3.2
• Remark 4.1
• Remark 4.2
• Remark 4.3
• Remark 4.4
• Remark 4.5
• Remark 5.1