Stochastic diffusion using mean-field limits to approximate master equations

Area of Science:

• Mathematical Biology
• Computational Ecology
• Epidemiology

Background:

• Stochastic diffusion models the spread of epidemics and species ranges, crucial for pandemic preparedness and climate change adaptation.
• Current modeling relies on computationally expensive simulations or inaccurate deterministic tools that neglect randomness.
• Accurate modeling of spatial dynamics is vital for understanding ecological and epidemiological processes.

Purpose of the Study:

• Introduce 'mean-FLAME' models for tracking stochastic dispersion using approximate master equations.
• Provide a locally exact modeling approach that bridges the gap between detailed stochastic simulations and simplified deterministic models.
• Improve the accuracy of modeling extinction events, epidemic heterogeneity, and spatial diffusion speeds.

Main Methods:

• Developed 'mean-FLAME' models based on approximate master equations to track probability distributions over system states.
• Applied models to predator-prey systems, disease dynamics, and nonlinear dispersal scenarios.
• Investigated the accuracy of tracking key states near absorbing states for extinction modeling.

Main Results:

• Mean-FLAME models accurately predict extinction in predator-prey systems by tracking essential states.
• Classic mean-field models underestimate epidemic heterogeneity, highlighting the importance of stochasticity.
• Deterministic tools fail to capture the true speed of spatial diffusion in nonlinear models.

Conclusions:

• Mean-FLAME models offer a significant improvement over traditional methods for modeling stochastic diffusion.
• Accurate modeling of stochasticity is critical for understanding processes in marginal areas, such as species range edges and small-population epidemics.
• This approach enhances preparedness for pandemics and adaptation to climate-driven range shifts.