Global properties of infectious disease models with nonlinear incidence. (English) Zbl 1298.92101

Korobeinikov, Andrei

Bull. Math. Biol. 69, No. 6, 1871-1886 (2007).

Summary: 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.

MSC:

92D30	Epidemiology
34D20	Stability of solutions to ordinary differential equations

Keywords:

direct Lyapunov method; Lyapunov function; endemic equilibrium state; global stability; nonlinear incidence

References:

[1]	Anderson, R.M., May, R.M., 1991. Infectious Diseases in Humans: Dynamics and Control. Oxford University Press, Oxford.
[2]	Barbashin, E.A., 1970. Introduction to the Theory of Stability. Wolters–Noordhoff, Groningen. · Zbl 0198.19703
[3]	Briggs, C.J., Godfray, H.C.J., 1995. The dynamics of insect-pathogen interactions in stage-structured populations. Am. Nat. 145(6), 855–887.
[4]	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.
[5]	Busenberg, S., Cooke, K., 1993. Vertically Transmitted Diseases: Models and Dynamics. Springer, Berlin. · Zbl 0837.92021
[6]	Capasso, V., Serio, G., 1978. A generalisation of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42, 43–61. · Zbl 0398.92026
[7]	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. · Zbl 1126.34337
[8]	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. · Zbl 0991.92028
[9]	Hethcote, H.W., 2000. The mathematics of infectious diseases. SIAM Rev. 42(4), 599–653. · Zbl 0993.92033
[10]	Hethcote, H.W., van den Driessche, P., 1991. Some epidemiological models with nonlinear incidence. J. Math. Biol. 29, 271–287. · Zbl 0722.92015
[11]	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. · Zbl 0714.92021
[12]	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. · Zbl 1334.92410
[13]	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. · Zbl 1062.92061
[14]	Korobeinikov, A., Maini, P.K., 2005. Nonlinear incidence and stability of infectious disease models. Math. Med. Biol. A J. IMA 22, 113–128. · Zbl 1076.92048
[15]	La Salle, J., Lefschetz, S., 1961. Stability by Liapunov’s Direct Method. Academic, New York. · Zbl 0098.06102
[16]	Li, M.Y., Muldowney, J.S., van den Driessche, P., 1999. Global stability of SEIRS models in epidemiology. Can. Appl. Math. Quort., 7. · Zbl 0976.92020
[17]	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. · Zbl 0621.92014
[18]	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. · Zbl 0582.92023
[19]	Lyapunov, A.M., 1992. The General Problem of the Stability of Motion. Taylor & Francis, London. · Zbl 0786.70001
[20]	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. · Zbl 1015.92036