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The generalized linear chain trick (GLCT) enables the creation of ODE models from stochastic assumptions, extending the linear chain trick (LCT) to phase-type distributions. This method simplifies complex model building and speeds up computations.

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Coxian distributionErlang distributionLinear chain trickgamma chain trickphase-type distribution

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Area of Science:

  • Mathematical modeling
  • Computational biology
  • Systems ecology

Background:

  • The linear chain trick (LCT) is a method for deriving ordinary differential equation (ODE) models with gamma-distributed passage times from stochastic processes.
  • Gamma distributions are sums of exponentially distributed random variables, representing sequential transitions through sub-states.
  • Phase-type distributions offer a broader family of distributions, including exponential, Erlang, hypoexponential, and Coxian, for modeling absorption times in continuous-time Markov chains (CTMCs).

Purpose of the Study:

  • To review continuous-time Markov chains (CTMCs) and phase-type distributions.
  • To demonstrate the application of the generalized linear chain trick (GLCT) for constructing ODE models from stochastic assumptions.
  • To introduce novel generalized predator-prey and SEIR models using the GLCT.

Main Methods:

  • Review of CTMCs and phase-type distributions.
  • Application of the generalized linear chain trick (GLCT) to derive ODE models.
  • Generalization of the Rosenzweig-MacArthur predator-prey model and the SEIR model using GLCT.
  • Illustrative examples showcasing model complexity and computational speed-up.

Main Results:

  • The GLCT extends the LCT to the comprehensive family of phase-type distributions.
  • Novel ODE models were developed by generalizing established ecological and epidemiological models (Rosenzweig-MacArthur, SEIR) using the GLCT.
  • GLCT-based model formulations significantly accelerate the computation of numerical solutions compared to traditional methods.

Conclusions:

  • The GLCT provides an intuitive and powerful framework for deriving ODE models from first principles and stochastic models.
  • This approach facilitates the incorporation of complex waiting time distributions into ODE models.
  • The GLCT enhances computational efficiency for numerical simulations of complex systems.