Exact and Approximate Constants of Motion in Stochastic Contact Processes

TL;DR

This paper investigates exact and approximate constants of motion in stochastic contact processes, including rumor spreading and epidemic models, to simplify analysis and understand their dynamics.

Contribution

It introduces methods to construct exact and approximate constants of motion in stochastic contact processes, linking them to mean-field models and nonlinear combinations.

Findings

Exact constants of motion for rumor models derived from contact events.

Approximate constants of motion in SIR models on networks.

Mean-field analogs help identify conserved quantities.

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.