A model for the spatial spread of an epidemic

Area of Science:

• Epidemiology
• Mathematical Biology
• Dynamical Systems

Background:

• Understanding the spatial and temporal dynamics of epidemic spread is crucial for public health interventions.
• Deterministic models are valuable tools for analyzing disease transmission patterns.

Purpose of the Study:

• To develop and analyze a deterministic model for the spatial spread of epidemics.
• To determine the long-term behavior and final state of an epidemic using mathematical modeling.
• To generalize existing pandemic threshold theorems.

Main Methods:

• Formulation of a deterministic epidemic spread model based on a nonlinear integral equation.
• Analysis of the unique solution to the integral equation.
• Investigation of the temporally asymptotic limit of the solution.
• Characterization of the minimal solution to a related nonlinear integral equation.

Main Results:

• The model's unique solution converges to a temporally asymptotic limit representing the epidemic's final state.
• This final state is identified as the minimal solution to a secondary nonlinear integral equation.
• The asymptotic behavior of the minimal solution at distances from the origin was described.
• Generalization of D. G. Kendall's pandemic threshold theorem (1957) was achieved.

Conclusions:

• The developed nonlinear integral equation model accurately predicts the final state of spatial epidemic spread.
• The study provides a mathematical framework for understanding epidemic dynamics and long-term spatial distribution.
• The generalization of Kendall's theorem offers new insights into epidemic control and prediction.