Extinction times and phase transitions for spatially structured closed epidemics

Area of Science:

• Epidemiology
• Mathematical Biology
• Stochastic Processes

Background:

• Stochastic epidemic models are crucial for understanding disease dynamics.
• Previous research established rigorous results for extinction times.
• Extending these models to structured populations is essential for realistic scenarios.

Purpose of the Study:

• To analyze the time to extinction in a stochastic SEIR epidemic model.
• To investigate the impact of population structure (patches) and mixing on extinction time.
• To provide a theoretical framework for understanding epidemic spread and fade-out in metapopulations.

Main Methods:

• Utilized a stochastic SEIR (Susceptible-Exposed-Infectious-Recovered) model without replacement.
• Employed an approximating deterministic system for heuristic explanations and analysis.
• Extended the model to a metapopulation structure with nearest-neighbor mixing.
• Derived analytical expressions for expected extinction time based on patch size and connectivity.

Main Results:

• The expected time to extinction follows a logarithmic relationship with patch size (N) for N > Nc (critical patch size).
• Extinction time can be decomposed into time within a patch plus time for inter-patch transmission.
• Derived expressions for critical patch size and extinction time coefficients.
• Numerical results for phocine distemper virus in seals show good agreement with the model for large and small N.
• Observed transitional behavior and non-monotonicity in extinction time near the critical patch size.

Conclusions:

• The study provides a theoretical understanding of epidemic extinction in structured populations.
• The findings offer insights into disease persistence and fade-out dynamics in metapopulations.
• The model's predictions align well with empirical data, validating its utility for real-world applications like wildlife disease management.