Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Breauer, F.(1987).A class of Volterra integral equations arising in delayed-recruitment population models.Nat.Res.Model. 2,259–278.
Brauer, F.(1990).Models for the spread of universally fatal diseases.J.Math.Biol. 28, 451–462.
Brauer, F.(2003).Disease mortality in epidemic models.Dyn.Contin.Discret.Impulsive Syst. 10,377–387.
Brauer, F.,and Castillo-Chavez, C.(2001).Mathematical Models in Population Biology and Epidemics,Springer-Verlag, Berlin-Heidelberg-New York.
Buenberg, S.,and Cooke, K.L.(1978).Periodic solution of a periodic nonlinear delay differential equation.SIAM J.Appl.Math. 35,704–721.
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.
Cooke, K.L.,and Yorke, J.A.(1973).Some equations modelling growth process and gon-orrhea epidemics.Math.Biosci. 16,75–101.
Hetcote, H.W.(1976).Qualitative analysis for communicable disease models.Math.Bio-sci. 28,335–356.
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.
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.
Hethcote, H.W., Stech, H.W.and van den Driessche, P.(1981a).Nonlinear oscillations in epidemic models.SIAM J.Math.Anal. 40,1–9.
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.
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.
Wilkins, J.E.(1945).The differential-difference equation for the epidemics.B.Math.Bio-phys. 7,149–150.
Wilson, E.B.,and Burke, M.H.(1942).The epidemic curve.Proc.Natl.Acad.Sci.USA 28,361–367.
Wilson, E.B.,and Worcester, J.(1944).A second approximation to Soper 's epidemic curve.Proc.Natl.Acad.Sci.USA 30,37–44.
Wilson, L.O.(1972).An epidemic model involving a threshold.Math.Biosci. 15, 109–121.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brauer, F. The Analysis of Some Characteristic Equations Arising in Population and Epidemic Models. Journal of Dynamics and Differential Equations 16, 441–453 (2004). https://doi.org/10.1007/s10884-004-4287-z
Issue Date:
DOI: https://doi.org/10.1007/s10884-004-4287-z