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Updated: Oct 29, 2025

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A unifying nonlinear probabilistic epidemic model in space and time.

Roberto Beneduci1,2, Eleonora Bilotta3, Pietro Pantano3

  • 1Department of Physics, University of Calabria, Rende, CS, 87036, Italy. roberto.beneduci@unical.it.

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|July 6, 2021
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Summary

This study introduces a novel non-linear probabilistic model for epidemic forecasting. This flexible mathematical framework unifies various deterministic and stochastic models, aiding disease control strategies.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Statistical Mechanics

Background:

  • The COVID-19 pandemic highlighted the critical role of mathematical modeling in public health decision-making.
  • Existing epidemic models, both deterministic and stochastic, have limitations in capturing complex disease dynamics.
  • Integrating diverse modeling approaches is crucial for robust epidemic forecasting and control.

Purpose of the Study:

  • To propose a unified continuous space-time non-linear probabilistic model for epidemic dynamics.
  • To demonstrate the derivation of existing epidemic models (SI, SIR, Fisher-Kolmogorov) from this unified framework.
  • To explore the connection between non-linear probabilistic models and non-linear deterministic models, including anomalous diffusion.

Main Methods:

  • Development of a continuous space-time non-linear probabilistic model.
  • Derivation of various deterministic and stochastic epidemic models (e.g., SI, SIR, Fisher-Kolmogorov) as special cases.
  • Analysis of the relationship between probabilistic and deterministic models, including generalized Fisher-Kolmogorov equations.

Main Results:

  • The proposed model successfully unifies a wide range of existing epidemic models.
  • A generalized Fisher-Kolmogorov equation with a time-dependent diffusion coefficient (anomalous diffusion) is derived.
  • The model offers a framework for epidemic forecasting, identifying regions of increasing infection probability.

Conclusions:

  • The unified non-linear probabilistic model provides a powerful and flexible tool for understanding and predicting epidemic spread.
  • The derived anomalous diffusion term captures essential non-linear probabilistic information.
  • This approach enhances the capabilities of mathematical modeling in supporting governmental decisions for disease control.