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Computational methods for Markov series with large state spaces, with application to AIDS modeling

S Yakowitz1

  • 1Systems and Industrial Engineering Department, University of Arizona, Tucson 85721, USA.

Mathematical Biosciences
|May 1, 1995
PubMed
Summary

This study introduces new numerical methods for analyzing large Markov chain models in epidemiology. These techniques offer an alternative to simulation for understanding stochastic epidemics, including complex AIDS models.

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Area of Science:

  • Epidemiology
  • Computational Biology
  • Mathematical Modeling

Background:

  • Epidemiological models predominantly use differential equations or Markov processes.
  • Differential equations often mask variability by tracking only the expected process.
  • Analyzing large Markov chains is computationally challenging, often limiting studies to simulation.

Purpose of the Study:

  • To propose novel numerical techniques for analyzing large Markov chains.
  • To enable the computation of marginal probabilities for Markov chains with millions of states.
  • To provide an alternative to simulation for stochastic epidemic modeling.

Main Methods:

  • Development of numerical techniques for Markov chain state space analysis.
  • Application of these methods to existing AIDS models.

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  • Comparison of numerical techniques with Monte Carlo simulation.

    • Main Results:

    • The proposed numerical methods can handle Markov chains with thousands to millions of states.
    • Demonstrated feasibility on AIDS models previously analyzed only by simulation.
    • Successfully applied to a two-population partition of the San Francisco homosexual epidemic.

    Conclusions:

    • The new numerical techniques offer a viable approach for analyzing complex stochastic epidemics.
    • These methods provide valuable insights into disease dynamics previously masked by simulation.
    • Further refinement of computational methodology is expected to enhance applicability.