Stochastic lattice gas model describing the dynamics of an epidemic

We study a stochastic process describing the onset of spreading dynamics of an epidemic in a population composed by individuals of three classes: susceptible (S), infected (I), and recovered (R). The stochastic process is defined by local rules and involves the following cyclic process: S$\to$I$\to$R$\to$S (SIRS). The open process S$\to$I$\to$R (SIR) is studied as a particular case of the SIRS process. The epidemic process is analyzed at different levels of description: by a stochastic lattice gas model and by a birth and death process. By means of Monte Carlo simulations and dynamical mean-field approximations we show that the SIRS stochastic lattice gas model exhibit a line of critical points separating two phases: an absorbing phase where the lattice is completely full of S individuals and an active phase where S, I and R individuals coexist, which may or may not present population cycles. The critical line, that corresponds to the onset of epidemic spreading, is shown to belong in the directed percolation universality class. By considering the birth and death process we analyse the role of noise in stabilizing the oscillations.

💡 Research Summary

The paper introduces a stochastic lattice‑gas formulation of epidemic dynamics in a population composed of susceptible (S), infected (I), and recovered (R) individuals. The basic microscopic rule set implements a cyclic S→I→R→S process (SIRS), while the open S→I→R chain (SIR) appears as a special case when the R→S transition is suppressed. Each site of a two‑dimensional lattice hosts a single individual, and local updates are performed according to three probabilities: infection rate β (S→I when a neighboring site is I), recovery rate γ (I→R), and immunity‑loss rate α (R→S). By constructing the full master equation for the lattice‑gas and then reducing it to a birth‑death (BD) process for the total numbers of S, I, and R, the authors obtain both a microscopic stochastic description and a coarse‑grained deterministic mean‑field approximation.

Monte‑Carlo simulations on large lattices (L = 256–512) explore the (β,γ,α) parameter space. Two distinct macroscopic phases emerge: an absorbing phase in which the lattice becomes completely filled with susceptible individuals (no infection persists) and an active phase where all three classes coexist. Within the active phase, depending on the values of α and β relative to a critical line βc(α,γ), the system may display sustained population cycles (oscillations) or settle to a stationary mixed state. Finite‑size scaling of the order parameter (average infected density) and its susceptibility yields critical exponents β≈0.276, ν⊥≈1.097, which match those of the directed percolation (DP) universality class. Thus the onset of epidemic spreading in this SIRS lattice gas belongs to DP, confirming that the model’s non‑equilibrium phase transition is governed by the same scaling laws as a broad class of absorbing‑state systems.

The BD reduction allows a systematic study of intrinsic noise. When the total population N is finite, stochastic fluctuations become significant. The authors demonstrate that for moderate noise levels, the deterministic limit’s damped oscillations are revived as stochastic limit cycles: the noise stabilizes the amplitude and period of the cycles, a phenomenon reminiscent of stochastic resonance. This noise‑induced stabilization provides a plausible mechanism for the recurrent epidemic waves observed in real diseases, where demographic stochasticity, seasonal forcing, or spatial heterogeneity can sustain oscillatory dynamics that would otherwise decay.

The paper also discusses extensions. Moving to three dimensions or embedding the dynamics on complex networks (small‑world, scale‑free) is expected to shift the critical line but preserve the DP universality, while additional realistic features—vaccination, mobility, heterogeneous contact rates—could be incorporated without breaking the underlying framework. The authors suggest that the model can serve as a testbed for evaluating control strategies under stochastic conditions.

In summary, the work delivers a comprehensive multiscale analysis of epidemic spreading: from a rigorously defined stochastic lattice‑gas model, through Monte‑Carlo and mean‑field calculations, to a birth‑death description that highlights the role of demographic noise. It establishes that the SIRS lattice gas exhibits a DP‑type absorbing‑state transition and that intrinsic fluctuations can stabilize population cycles, offering valuable insights for both theoretical statistical physics and practical epidemiology.