The Analysis of Some Characteristic Equations Arising in Population and Epidemic Models

  • Fred Brauer


Epidemic models with a general infective period distribution are formulated as functional differential equations, as are population models with a general life span distribution. The analysis of the local stability properties of equilibria of such models leads to a characteristic equation involving the Laplace transform of the infective period (or life span) distribution. We obtain conditions under which all roots of the characteristic equation are in the left half plane, implying asymptotic stability of equilibrium, for every infective period distribution. We also consider the converse problem of describing when instability can occur for specific infective period distributions.

Characteristic equations epidemic models infective period distributions 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Breauer, F.(1987).A class of Volterra integral equations arising in delayed-recruitment population models.Nat.Res.Model. 2,259–278.Google Scholar
  2. 2.
    Brauer, F.(1990).Models for the spread of universally fatal diseases.J.Math.Biol. 28, 451–462.Google Scholar
  3. 3.
    Brauer, F.(2003).Disease mortality in epidemic models.Dyn.Contin.Discret.Impulsive Syst. 10,377–387.Google Scholar
  4. 4.
    Brauer, F.,and Castillo-Chavez, C.(2001).Mathematical Models in Population Biology and Epidemics,Springer-Verlag, Berlin-Heidelberg-New York.Google Scholar
  5. 5.
    Buenberg, S.,and Cooke, K.L.(1978).Periodic solution of a periodic nonlinear delay differential equation.SIAM J.Appl.Math. 35,704–721.Google Scholar
  6. 6.
    Busenberg, S.,and Cooke, K.L.(1980).The effect of integral conditions in certain equa-tions modeling epidemics and population growth.J.Math.Biol.13–22.Google Scholar
  7. 7.
    Cooke, K.L.,and Yorke, J.A.(1973).Some equations modelling growth process and gon-orrhea epidemics.Math.Biosci. 16,75–101.Google Scholar
  8. 8.
    Hetcote, H.W.(1976).Qualitative analysis for communicable disease models.Math.Bio-sci. 28,335–356.Google Scholar
  9. 9.
    Hethcote, H.W.(1989).Three basic epidemiological models.In Levin, S.A., Hallam, T.G., and Gross, L.J.(eds.),Applied Mathematical Ecology,Biomathematics Vol.18,Springer-Verlag, Berlin-Heidelberg-New York,pp.119–144.Google Scholar
  10. 10.
    Hethcote, H.W.,and Levin, S.A.(1989).Periodicity in epidemic models.In Levin S. A., Hallam,T. G.,and Gross, L.J.(eds.),Applied Mathematical Ecology,Biomathematics Vol.18,Springer-Verlag, Berlin-Heidelberg-New York,pp.193–211.Google Scholar
  11. 11.
    Hethcote, H.W., Stech, H.W.and van den Driessche, P.(1981a).Nonlinear oscillations in epidemic models.SIAM J.Math.Anal. 40,1–9.Google Scholar
  12. 12.
    Hethcote, H.W., Stech, H.W.and van den Driessche, P.(1981b).Stability analysis for models of diseases without immunity.J.Math.Biol. 13,185–198.Google Scholar
  13. 13.
    Hethcote, H.W., Stech,H.W.,and van den Driessche, P.(1981c).Periodicity and stabil-ity in epidemic models:A survey.In Busenberg, S.N.,and Cooke, K.L.(eds.),Differen-tial Equations and Applications in Ecology,Epidemics and Population Problems,pp.65–82.Google Scholar
  14. 14.
    Wilkins, J.E.(1945).The differential-difference equation for the epidemics.B.Math.Bio-phys. 7,149–150.Google Scholar
  15. 15.
    Wilson, E.B.,and Burke, M.H.(1942).The epidemic curve.Proc.Natl.Acad.Sci.USA 28,361–367.Google Scholar
  16. 16.
    Wilson, E.B.,and Worcester, J.(1944).A second approximation to Soper 's epidemic curve.Proc.Natl.Acad.Sci.USA 30,37–44.Google Scholar
  17. 17.
    Wilson, L.O.(1972).An epidemic model involving a threshold.Math.Biosci. 15, 109–121.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2004

Authors and Affiliations

  • Fred Brauer
    • 1
  1. 1.Department of MathematicsUniversity of Brtish ColumbiaVancouverCanada

Personalised recommendations