Exact and Approximate Constants of Motion in Stochastic Contact Processes

TL;DR

The paper examines how constants of motion constrain stochastic contact processes, focusing on rumor spreading (Maki–Thompson) and disease spreading (SIR). It constructs exact linear invariants for MT, yielding two independent constants and enabling a full stochastic solution, while linking them to mean-field invariants in the large-$N$ limit. For SIR, only one exact invariant exists, but several approximate conserved quantities arise, including a mean-field-like invariant $K_{ m mf}$, with extensions to networks producing weighted approximations that converge in the appropriate limits. The study further demonstrates nonlinear invariants in modified MT variants and discusses their mean-field correspondence, illustrating how conservation laws can facilitate dimensionality reduction and analytical treatment across stochastic epidemic and information-spreading models.

Abstract

We study a variety of stochastic contact processes -- directly related to models of rumor and disease spreading -- from the viewpoint of their constants of motion, either exact or approximated. Much as in deterministic systems, constants of motion in stochastic dynamics make it possible to reduce the number of relevant variables, confining the set of accessible states, and thus facilitating their analytical treatment. For processes of rumor propagation based on the Maki-Thompson model, we show how to construct exact constants of motion as linear combinations of conserved quantities in each elementary contact event, and how they relate to the constants of motion of the corresponding mean-field equations, which are obtained as the continuous-time, large-size limit of the stochastic process. For SIR epidemic models, both in homogeneous systems and on heterogeneous networks, we find that a similar procedure produces approximate constants of motion, whose average value is preserved along the evolution. We also give examples of exact and approximate constants of motion built as nonlinear combinations of the relevant variables, whose expressions are suggested by their mean-field counterparts.

Figures (2)

• Figure 1: Three realizations of the SIR stochastic process defined by the events of Table \ref{['tab4']}, with $N=100$, $u=0.3$, and initial conditions $S_0=N-1$ and $I_0=1$. In the main panel, curves show the evolution of the number of susceptible and infected agents, $S_n$ and $I_n$, as a function of the evolution step $n$. Each color correspond to a single realization. Note that each realization ends when the number of infected agents reaches zero. Dashed curves represent the solutions of Eqs. (\ref{['meanSI']}) for the mean values $\langle S \rangle_n$ and $\langle I \rangle_n$. The inset shows the evolution of the approximate constant of motion $\widetilde{K}_n$, Eq. (\ref{['Ktilde']}), for the same realizations. The horizontal dashed line is the expected mean value, $\langle \widetilde{K} \rangle_n=S_0+I_0$, and the dashed curves indicate the corresponding standard deviation, $\sigma_n$, given by Eq. (\ref{['sigma']}).
• Figure 2: Three realizations of the SIR stochastic process defined by the events of Table \ref{['tab5']}, with $N=100$, $u=0.2$, and initial conditions $S_0=N-1$ and $I_0=1$. The system evolves on an Erdős-Rényi random network, different for each realization, with an average of $z=10$ neighbors per site. In the main panel, curves show the evolution of the total number of susceptible and infected agents, $S_n$ and $I_n$, as a function of the evolution step $n$. Each color correspond to a single realization. Dashed curves represent the mean values of $S_n$ and $I_n$ averaged over realizations of the evolution and the underlying network. for the same realizations, the inset shows the evolution of the approximate constant of motion $\widetilde{K}_n^{(1)}$, given by Eq. (\ref{['Kg']}) with $\gamma=1$. The horizontal dashed line is the expected mean value, $\langle \widetilde{K}^{(1)} \rangle_n=zN$.