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Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
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The link model is a fundamental pharmacokinetic-pharmacodynamic (PK–PD) approach to account for delayed drug responses when the observed effect does not immediately correlate with the drug's plasma concentration peak. This delay is mathematically addressed by introducing an effect compartment concentration, Ce, which is kinetically linked to the plasma concentration, Cp, via a first-order rate constant, ke0. The linkage allows for a more accurate prediction of drug effects over time. A...
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Quantitative Analysis of Random Migration of Cells Using Time-lapse Video Microscopy
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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
PubMed
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.

Keywords:
Asymptotic stabilityDistributed delayOrnstein-Uhlenbeck processStationary distributionStochastic migrationTwo-patch epidemic model

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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.