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Disease Extinction Versus Persistence in Discrete-Time Epidemic Models

  • P. van den Driessche
  • Abdul-Aziz Yakubu
Special Issue: Mathematical Epidemiology

Abstract

We focus on discrete-time infectious disease models in populations that are governed by constant, geometric, Beverton–Holt or Ricker demographic equations, and give a method for computing the basic reproduction number, \(\mathcal {R}_{0}\). When \(\mathcal {R}_{0}<1\) and the demographic population dynamics are asymptotically constant or under geometric growth (non-oscillatory), we prove global asymptotic stability of the disease-free equilibrium of the disease models. Under the same demographic assumption, when \(\mathcal {R}_{0}>1\), we prove uniform persistence of the disease. We apply our theoretical results to specific discrete-time epidemic models that are formulated for SEIR infections, cholera in humans and anthrax in animals. Our simulations show that a unique endemic equilibrium of each of the three specific disease models is asymptotically stable whenever \(\mathcal {R}_{0}>1\).

Keywords

Asymptotically constant growth Discrete-time epidemic model Disease extinction or persistence Geometric growth 

Notes

Acknowledgements

We thank the referees for their useful comments and suggestions. This research was partially supported by NSERC, through a Discovery Grant (P.vdD.). A.-A.Y. was partially supported by DHS Center Of Excellence for Command, Control and Interoperability at Rutgers University, NSF Computational Sustainability Grant # CCF-1522054 and NSF Award # DMS-1743144.

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Copyright information

© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of MathematicsHoward UniversityWashingtonUSA

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