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Dynamics of a two-patch epidemic model with deterministic/stochastic migration and distributed delays
Ting Kang1,2, Boqiang Cao1,2, Zhenfeng Shi3
1School of Mathematics and Statistics, Ningxia University, Yinchuan, 750021, China.
Infectious Disease Modelling
|April 9, 2026
Summary
This study models epidemic spread across two patches with migration, delays, and environmental noise. Findings show migration randomness and delays significantly alter infection persistence and distribution.
Area of Science:
- Epidemiology
- Mathematical Biology
- Dynamical Systems
Background:
- Understanding epidemic dynamics in spatially structured populations is crucial.
- Incorporating migration, distributed delays, and environmental stochasticity provides a more realistic model.
Purpose of the Study:
- To analyze the global dynamics of a two-patch epidemic model with Ornstein-Uhlenbeck modulated migration and Erlang-distributed delays.
- To establish conditions for disease extinction and persistence in both deterministic and stochastic frameworks.
- To investigate the impact of migration randomness and delays on long-term infection burden and prevalence.
Main Methods:
- Analysis of a deterministic two-patch epidemic model using basic reproduction number thresholds.
- Development of a stochastic model incorporating environmental stochasticity and distributed delays.
- Construction of Lyapunov functions and exploitation of Metzler structures for stochastic analysis.
- Numerical simulations to validate theoretical results and explore parameter effects.
Main Results:
- Deterministic model shows disease extinction for R0 < 1 and persistence for R0 > 1.
- Stochastic model provides conditions for almost sure exponential extinction.
- A stationary distribution exists for the stochastic model when the stochastic threshold R0s > 1, indicating persistent random fluctuations.
- Migration noise and mean-reversion rates can redistribute infection burden between patches.
Conclusions:
- The study provides a comprehensive analysis of a complex spatio-temporal epidemic model.
- Migration-driven randomness fundamentally reshapes spatial epidemic patterns, influencing infection burden and prevalence.
- The findings highlight the importance of considering migration dynamics and environmental stochasticity in epidemic control strategies.
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