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Analytical methods for predicting the behaviour of population models with general spatial interactions.

J A N Filipe1, M M Maule

  • 1Biomathematics and Statistics Scotland, The King's Buildings (JCMB), EH9 3JZ, Edinburgh, UK. jf263@cam.ac.uk

Mathematical Biosciences
|February 27, 2003
PubMed
Summary

Analytical methods offer an alternative to computer simulations for spatial, stochastic, individual-based population models. A modified Kirkwood approximation provides stable and accurate solutions, outperforming moment closure for epidemic models.

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Area of Science:

  • Ecology
  • Mathematical Biology
  • Epidemiology

Background:

  • Biologists increasingly use spatial, stochastic, individual-based population models.
  • Analytical methods offer an alternative to computationally expensive simulations for model analysis.
  • Existing mean-field methods are being extended to incorporate spatial correlations.

Purpose of the Study:

  • To investigate an analytical approach for spatial and stochastic models with distance-dependent interactions.
  • To test the accuracy of different closure approximations, including moment closure (MC) and Kirkwood approximation.
  • To evaluate these methods in the context of a susceptible-infected-susceptible epidemic model.

Main Methods:

  • Developed an analytical approach for discrete spatial models with generic interaction functions.

Related Experiment Videos

  • Compared the performance of moment closure (MC) and Kirkwood approximations.
  • Applied methods to a susceptible-infected-susceptible epidemic model with power-law interactions.

    • Main Results:

    • Moment closure (MC) showed numerical instability and inaccuracy in certain parameter regions.
    • A modified Kirkwood approximation yielded stable and accurate transient and long-term solutions.
    • Derived a generalized basic reproduction ratio (R(0)) for spatial models.

    Conclusions:

    • The modified Kirkwood approximation is a viable and accurate substitute for MC in spatial models.
    • This analytical approach can be applied to both lattice and continuous-space models.
    • The study provides a generalized R(0) for spatial epidemic models.