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Oscillations in a patchy environment disease model.

Fred Brauer1, P van den Driessche, Lin Wang

  • 1Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V8N 3R4.

Mathematical Biosciences
|June 28, 2008
PubMed
Summary
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This study analyzes SIRS models to understand disease dynamics. For a single patch, the disease-free state is stable if the basic reproduction number (R(0)) is less than 1, with complex dynamics emerging when R(0) exceeds 1.

Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Dynamical Systems

Background:

  • SIRS models are crucial for understanding infectious disease dynamics, incorporating temporary immunity.
  • Disease-related deaths and population dynamics (recruitment-death) are key factors in disease spread.
  • Understanding stability and bifurcations in epidemiological models is essential for public health interventions.

Purpose of the Study:

  • To identify the basic reproduction number (R(0)) for a single-patch SIRS model.
  • To analyze the global asymptotic stability of the disease-free equilibrium and local stability of the endemic equilibrium.
  • To investigate Hopf bifurcation and develop a numerical method for delay-dependent coefficients in a two-patch SIRS model with travel.

Main Methods:

  • Mathematical modeling using SIRS (Susceptible-Infected-Recovered-Susceptible) framework.

Related Experiment Videos

  • Analysis of equilibrium stability and Hopf bifurcation theory.
  • Numerical methods for locating bifurcation values in delay-differential equations.

    • Main Results:

    • The disease-free equilibrium is globally asymptotically stable when R(0)<1.
    • For R(0)>1, local stability of the endemic equilibrium and Hopf bifurcation are established.
    • Travel in a two-patch model introduces multiple thresholds, influencing oscillations by reducing, enhancing, or switching them between patches.

    Conclusions:

    • The basic reproduction number (R(0)) is a critical threshold for disease persistence in single-patch SIRS models.
    • Hopf bifurcation analysis reveals complex oscillatory behaviors in endemic states.
    • Inter-patch travel significantly alters disease dynamics, with varied effects on oscillation patterns and stability in multi-patch systems.