Disease Extinction Versus Persistence in Discrete-Time Epidemic Models

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


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 \(\mathcal {R}_{0}>1\).


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



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.


  1. Allen L (1994) Some discrete-time SI, SIR and SIS epidemic models. Math Biosci 124:83–105CrossRefzbMATHGoogle Scholar
  2. Allen L, van den Driessche P (2008) The basic reproduction number in some discrete-time epidemic models. J Differ Equ Appl 14(10–11):1127–1147MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bacaer N, Ait Dads EH (2012) On the biological interpretation of a definition for the parameter \(\cal{R}_{0}\) in periodic models. J Math Biol 65:601–621MathSciNetCrossRefzbMATHGoogle Scholar
  4. Barton JT (2016) An introduction to discrete mathematical modeling with Microsoft Office Excel. Wiley, HobokenzbMATHGoogle Scholar
  5. Best J, Castillo-Chavez C, Yakubu A-A (2003) Hierarchical competition in discrete time models with dispersal. Fields Inst Commun 36:59–86MathSciNetzbMATHGoogle Scholar
  6. Brauer F, Feng Z, Castillo-Chavez C (2010) Discrete epidemic models. Math Biosci Eng 7(1):1–15MathSciNetCrossRefzbMATHGoogle Scholar
  7. Castillo-Chavez C, Yakubu A-A (2001) Dispersal, disease and life-history evolution. Math Biosci 173:35–53MathSciNetCrossRefzbMATHGoogle Scholar
  8. Codeço CT (2001) Endemic and epidemic dynamics of cholera: the role of aquatic reservoir. BMC Infect Dis 1:1CrossRefGoogle Scholar
  9. Cushing JM, Diekmann O (2016) The many guises of \(\cal{R}_{0}\) (a diadactic note). J Theor Biol 404:295–302CrossRefzbMATHGoogle Scholar
  10. Cushing JM, Yicang Z (1994) The net reproductive value and stability in matrix population models. Nat Resour Model 8:297–333CrossRefGoogle Scholar
  11. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and computation of the basic reproduction ratio \(\cal{R}_{0}\) in models for infectious diseases in heterogeneous populations. J Math Biol 28:365–382MathSciNetCrossRefzbMATHGoogle Scholar
  12. Eisenberg MC, Robertson SL, Tien JH (2013) Identifiability and estimation of multiple transmission pathways in cholera and waterborne disease. J Theor Biol 324:84–102MathSciNetCrossRefzbMATHGoogle Scholar
  13. Elaydi N (2000) Discrete chaos. Chapman & Hall/CRC, Boca RatonzbMATHGoogle Scholar
  14. Franke J, Yakubu A-A (1996) Extinction and persistence of species in discrete competitive systems with a safe refuge. J Math Anal Appl 203:746–761MathSciNetCrossRefzbMATHGoogle Scholar
  15. Friedman A, Yakubu A-A (2013) Anthrax epizootic and migration: persistence or extinction. Math Biosci 241:137–144MathSciNetCrossRefzbMATHGoogle Scholar
  16. Furniss PR, Hahn BD (1981) A mathematical model of an anthrax epizootic in the Kruger National Park. Appl Math Model 5:130–136CrossRefGoogle Scholar
  17. Hahn BD, Furniss PR (1983) A mathematical model of anthrax epizootic: threshold results. Ecol Model 20:233–241CrossRefGoogle Scholar
  18. Hofbauer J, So JW-H (1987) Uniform persistence and repellors for maps. Proc Am Math Soc 107:1137–1142MathSciNetCrossRefzbMATHGoogle Scholar
  19. La Salle JP (1976) The stability of dynamical systems. SIAM, PhiladelphiaCrossRefGoogle Scholar
  20. Lewis MA, Rencławowicz J, van den Driesschen P, Wonham M (2006) A comparison of continuous and discrete-time West Nile virus models. Bull Math Biol 68:491–509MathSciNetCrossRefzbMATHGoogle Scholar
  21. Li C-K, Schneider H (2002) Applications of Perron–Frobenius theory to population dynamics. J Math Biol 44:450–462MathSciNetCrossRefzbMATHGoogle Scholar
  22. Mace KE, Arguin PM (2017) Malaria surveillance—United States, 2014. Surveill Summ 66(12):1–24.
  23. Martcheva M (2015) An introduction to mathematical epidemiology, vol 61. Springer texts in applied mathematics. Springer, New YorkzbMATHGoogle Scholar
  24. Saad-Roy CM, van den Driessche P, Yakubu A-A (2017) A mathematical model of anthrax transmission in animal populations. Bull Math Biol 79(2):303–324MathSciNetCrossRefzbMATHGoogle Scholar
  25. Shuai Z, van den Driessche P (2013) Global stability of infectious disease models using Lyapunov function. SIAM J Appl Math 70(1):1513–1532MathSciNetCrossRefzbMATHGoogle Scholar
  26. Tien JH, Earn DJD (2010) Multiple transmission pathways and disease dynamics in a water borne pathogen model. Bull Math Biol 72(6):1506–1533MathSciNetCrossRefzbMATHGoogle Scholar
  27. van den Driessche P (2017) Reproduction numbers of infectious disease models. Infect Dis Model 2(3):288–303Google Scholar
  28. Yakubu A-A (2010) Introduction to discrete-time epidemic models. DIMACS Ser Discrete Math Theor Comput Sci 75:83–109MathSciNetCrossRefzbMATHGoogle Scholar
  29. Zhao X-Q (2003) Dynamical systems in population biology. CMS books in mathematics. Springer, BerlinCrossRefGoogle Scholar

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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