Endemic Disease Models

  • Fred Brauer
  • Carlos Castillo-Chavez
  • Zhilan Feng
Part of the Texts in Applied Mathematics book series (TAM, volume 69)


In this chapter, we consider models for disease that may be endemic. In the preceding chapter we studied SIS models with and without demographics and SIR models with demographics. In each model, the basic reproduction number \(\mathcal {R}_0\) determined a threshold. If \(\mathcal {R}_0 < 1\) the disease dies out, while if \(\mathcal {R}_0 > 1\) the disease becomes endemic. The analysis in each case involves determination of equilibria and determining the asymptotic stability of each equilibrium by linearization about the equilibrium. In each of the cases studied in the preceding chapter the disease-free equilibrium was asymptotically stable if and only if \(\mathcal {R}_0 < 1\) and if \(\mathcal {R}_0 > 1\) there was a unique endemic equilibrium that was asymptotically stable. In this chapter, we will see that these properties continue to hold for many more general models, but there are situations in which there may be an asymptotically stable endemic equilibrium when \(\mathcal {R}_0 < 1\), and other situations in which there is an endemic equilibrium that is unstable for some values of \(\mathcal {R}_0 > 1\).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Fred Brauer
    • 1
  • Carlos Castillo-Chavez
    • 2
  • Zhilan Feng
    • 3
  1. 1.Department of MathematicsUniversity of British ColumbiaVancouverCanada
  2. 2.Mathematical and Computational Modeling Center (MCMSC), Department of Mathematics and StatisticsArizona State UniversityTempeUSA
  3. 3.Department of MathematicsPurdue UniversityWest LafayetteUSA

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