Kinetic formulation of compartmental epidemic models

TL;DR

The paper introduces a kinetic, multi-class epidemic model that couples spatial movement with disease dynamics, enabling transmission when susceptible and infectious individuals share both position and velocity and thus remain in contact. Through adimensionalization and a formal $\varepsilon$-expansion, it derives a mesoscopic drift-diffusion-advection PDE for class densities $\rho_S,\rho_I,\rho_R$, which reduces to the classical SIRS system in spatially homogeneous settings, expressed as $\begin{cases} S' = \bar{\alpha}R - \bar{\beta}\dfrac{SI}{S+I+R},\\ I' = \bar{\beta}\dfrac{SI}{S+I+R} - \bar{\gamma}I,\\ R' = \bar{\gamma}I - \bar{\alpha}R. \end{cases}$ The authors prove global existence and uniqueness for the kinetic model in a simplified regime without direction-variance coupling, connect the mesoscopic limit to replicator dynamics from evolutionary game theory, and illustrate the framework with numerical simulations and applications such as regional risk variation and infectious-confinement scenarios. This kinetic view provides a principled way to incorporate mobility and local interactions into epidemic modeling, potentially informing spatially targeted interventions and vaccination strategies. The work also outlines future directions, including generalised interaction kernels, convergence to the mesoscopic limit, and the integration of behaviour-driven dynamics.

Abstract

We introduce a kinetic model that couples the movement of a population of individuals with the dynamics of a pathogen in the same population. We consider that transmission occurs when a susceptible and an infectious individual are sufficiently close for a sufficiently long time. We show that the model is formally compatible with the well-known SIRS model in mathematical epidemiology. Namely, after identifying an appropriate dimensionless variable and considering the limit when that variable is small, we introduce a partial differential equation model of advection-drift-diffusion type (mesoscopic model), which for spatially homogeneous solutions reduces to the SIRS model. We prove the existence and uniqueness of solutions in appropriate spaces for particular instances of the model. We finish with some examples and discuss possible applications and generalisation of this modelling approach, linking kinetic models, evolutionary game theory, and mathematical epidemiology.

Figures (3)

• Figure 1: Illustration of the kinetic models for three classes of particles moving in a two-dimensional space. Each circle represents an individual, and the arrow indicates his or her velocity vector. Movements are composed of regression toward the mean velocity perturbed by a density-dependent drift. Blue individuals are of S type, while red and green individuals are of I and R types, respectively. In the highlight, we see an encounter in phase space (i.e., same position in space and same velocity) of an S and an I individual, possibly resulting in two I individuals, according to the prescribed transition rate. Note that when two individuals have the same position and velocity, there is more opportunity (i.e., time) for the transmission to happen than if only the same position were required.
• Figure 2: Example of a movement of a particle from point A (red) to point B (blue). The particle travels at instantaneous velocity $v$ with trajectory given by the grey line, with typical modulus $v_0$. The direction of movement changes periodically, according to the given kernels. After an observation time $t_0$, the typical displacement (black arrow) is given by $x$, with modulus $x_0$, resulting in a average (macroscopic) velocity of $\frac{x_0}{t_0}\ll v_0$. Typical densities are not represented in the figure, which focus on the movement of a single particle.
• Figure 3: Representation of the mesoscopic density $\rho_S$ (blue, left), $\rho_I$ (red, middle), and $\rho_R$ (right, green). More intense colors indicate higher concentrations. We consider the initial condition given by S and I individuals in the center of a square lattice of size $50\times 50$, with random initial velocities. Initial conditions are far from the boundary which are not reached for times considered. Initially, there are 10000 S individuals and 50 I individuals. We plot the configuration after 3, 6, 9, 12, and 15 steps, each step including the update of all particles. Class-transition parameters were given by $\alpha=0$ (i.e., immunity is permanent), $\beta=0.75$ (each encounter between S and I individuals has a probability of $3/4$ to generate a new I individual), and $\gamma=0.5$ (i.e., in each time step, each I individual has a probability $1/2$ to became R. The regression toward the mean parameter is $\lambda=0.5$, and we did not consider preferred directions, i.e., $\mathcal{Q}^{(P)}_j\equiv0$.

Theorems & Definitions (10)

• Remark 1
• Theorem 1: Lax-Milgram Theorem Ramaswamy_1980Showalter_1997Bisi_etal
• Corollary 1
• proof
• Remark 2
• Theorem 2
• proof
• Theorem 3
• proof
• Remark 3