Exact and Approximate Constants of Motion in Stochastic Contact Processes

Area of Science:

• Mathematical modeling
• Statistical physics
• Epidemiology

Background:

• Stochastic contact processes model real-world phenomena like disease and rumor spread.
• Constants of motion simplify complex dynamical systems by reducing variables.
• Understanding these processes is crucial for public health and information dissemination.

Purpose of the Study:

• To investigate exact and approximate constants of motion in stochastic contact processes.
• To explore the relationship between constants of motion in stochastic models and their mean-field counterparts.
• To demonstrate how constants of motion can facilitate the analytical treatment of rumor and epidemic models.

Main Methods:

• Analysis of stochastic contact processes, including Maki-Thompson rumor models and SIR epidemic models.
• Construction of exact constants of motion as linear combinations of conserved quantities.
• Derivation of approximate constants of motion for epidemic models on homogeneous and heterogeneous systems.
• Comparison with constants of motion from corresponding mean-field equations.

Main Results:

• Exact constants of motion were constructed for rumor propagation models.
• Approximate constants of motion were identified for SIR epidemic models, with preserved average values.
• The relationship between stochastic and mean-field constants of motion was established.
• Nonlinear combinations of variables were explored as potential constants of motion.

Conclusions:

• Constants of motion, both exact and approximate, are valuable tools for analyzing stochastic contact processes.
• The methodology used for rumor models can be extended to epidemic models, albeit with approximations.
• This work provides a framework for simplifying complex models of spreading phenomena.