# Exact and Approximate Constants of Motion in Stochastic Contact Processes

## TL;DR

This paper explores constants of motion in stochastic processes related to rumor and disease spreading, showing how they simplify analysis and predict system behavior.

## Contribution

The paper introduces exact and approximate constants of motion for stochastic contact processes, linking them to mean-field models.

## Key findings

- Exact constants of motion can be constructed for Maki–Thompson rumor propagation models.
- Approximate constants of motion are derived for SIR epidemic models on homogeneous and heterogeneous networks.
- Nonlinear combinations of variables yield new constants of motion inspired by mean-field equations.

## 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. As in deterministic systems, constants of motion in stochastic dynamics make it possible to reduce the number of relevant variables, thereby confining the set of accessible states and facilitating 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.