Abstract
We consider global properties for the classical SIR, SIRS and SEIR models of infectious diseases, including the models with the vertical transmission, assuming that the horizontal transmission is governed by an unspecified function f(S,I). We construct Lyapunov functions which enable us to find biologically realistic conditions sufficient to ensure existence and uniqueness of a globally asymptotically stable equilibrium state. This state can be either endemic, or infection-free, depending on the value of the basic reproduction number.
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References
Anderson, R.M., May, R.M., 1991. Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford.
Barbashin, E.A., 1970. Introduction to the Theory of Stability. Wolters–Noordhoff, Groningen.
Briggs, C.J., Godfray, H.C.J., 1995. The dynamics of insect-pathogen interactions in stage-structured populations. Am. Nat. 145(6), 855–887.
Brown, G.C., Hasibuan, R., 1995. Conidial discharge and transmission efficiency of Neozygites floridana, an Entomopathogenic fungus infecting two-spotted spider mites under laboratory conditions. J. Invertebr. Pathol. 65, 10–16.
Busenberg, S., Cooke, K., 1993. Vertically Transmitted Diseases: Models and Dynamics. Springer, Berlin.
Capasso, V., Serio, G., 1978. A generalisation of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 43–61.
Derrick, W.R., van den Driessche, P., 2003. Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population. Discret. Contin. Dyn. Syst. Ser. B 3, 299–309.
Feng, Z., Thieme, H.R., 2000. Endemic models with arbitrarily distributed periods of infection I: fundamental properties of the model. SIAM J. Appl. Math. 61(3), 803–833.
Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653.
Hethcote, H.W., van den Driessche, P., 1991. Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287.
Hethcote, H.W., Lewis, M.A., van den Driessche, P., 1989. An epidemiological model with delay and a nonlinear incidence rate. J. Math. Biol. 27, 49–64.
Korobeinikov, A., 2006. Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol. 68(3), 615–626.
Korobeinikov, A., Maini, P.K., 2004. A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math. Biosci. Eng. 1(1), 57–60.
Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. A J. IMA 22, 113–128.
La Salle, J., Lefschetz, S., 1961. Stability by Liapunov’s Direct Method. Academic, New York.
Li, M.Y., Muldowney, J.S., van den Driessche, P., 1999. Global stability of SEIRS models in epidemiology. Can. Appl. Math. Quort., 7.
Liu, W.M., Hethcote, H.W., Levin, S.A., 1987. Dynamical behaviour of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359–380.
Liu, W.M., Levin, S.A., Iwasa, Y., 1986. Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models. J. Math. Biol. 23, 187–204.
Lyapunov, A.M., 1992. The General Problem of the Stability of Motion. Taylor & Francis, London.
van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48.
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Korobeinikov, A. Global Properties of Infectious Disease Models with Nonlinear Incidence. Bull. Math. Biol. 69, 1871–1886 (2007). https://doi.org/10.1007/s11538-007-9196-y
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DOI: https://doi.org/10.1007/s11538-007-9196-y