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Disease Extinction Versus Persistence in Discrete-Time Epidemic Models.

P van den Driessche1, Abdul-Aziz Yakubu2

  • 1Department of Mathematics and Statistics, University of Victoria, Victoria, BC, V8W 2Y2, Canada.

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Summary

This study introduces a method for calculating the basic reproduction number in discrete-time disease models. It proves disease-free equilibrium stability and disease persistence based on demographic factors and the reproduction number.

Keywords:
Asymptotically constant growthDiscrete-time epidemic modelDisease extinction or persistenceGeometric growth

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Population Dynamics

Background:

  • Discrete-time infectious disease models are crucial for understanding disease spread.
  • Demographic factors significantly influence disease dynamics.
  • The basic reproduction number is a key metric for disease control.

Purpose of the Study:

  • To develop a method for computing the basic reproduction number in discrete-time population models.
  • To analyze the stability of disease-free equilibria and the persistence of diseases under various demographic conditions.
  • To apply these theoretical findings to specific infectious disease models.

Main Methods:

  • Utilized discrete-time population dynamics equations (constant, geometric, Beverton-Holt, Ricker).
  • Developed a method for calculating the basic reproduction number ([Formula: see text]).
  • Applied stability analysis and persistence theory to disease models.

Main Results:

  • Proved global asymptotic stability of the disease-free equilibrium when [Formula: see text] and demographics are non-oscillatory.
  • Proved uniform persistence of disease when [Formula: see text] under similar demographic assumptions.
  • Simulations confirmed the asymptotic stability of unique endemic equilibria for SEIR, cholera, and anthrax models when [Formula: see text].

Conclusions:

  • The basic reproduction number is a critical determinant of disease dynamics in discrete-time models.
  • Demographic stability influences the long-term behavior of infectious diseases.
  • The findings provide valuable insights for managing and controlling specific infectious diseases like cholera and anthrax.