# zbMATH — the first resource for mathematics

Global stability of a stage-structured epidemic model with a nonlinear incidence. (English) Zbl 1172.92027
Summary: A stage-structured epidemic model with a nonlinear incidence with a factor $$S^p$$ is investigated. By using limit theory of differential equations and a theorem of S. Busenberg and P. van den Driessche [J. Math. Biol. 28, No. 3, 257–270 (1990; Zbl 0725.92021)] the global dynamics of the model is rigorously established. We prove that if the basic reproduction number $$\operatorname{Re} _0$$ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease dies out; if $$\operatorname{Re} _0$$ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations support our analytical results and illustrate the effect of $$p$$ on the dynamic behavior of the model.

##### MSC:
 92D30 Epidemiology 34D05 Asymptotic properties of solutions to ordinary differential equations 34D23 Global stability of solutions to ordinary differential equations 37N25 Dynamical systems in biology
##### Keywords:
epidemic model; stage structure; global stability
Full Text:
##### References:
 [1] Anderson, R.M.; May, R.M., Infectious diseases of humans, (1991), Oxford University Press London [2] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033 [3] Levin, S.A.; Hallam, T.G.; Gross, L.J., Applied mathematical ecology, (1989), Springer New York [4] Velasco-Hernandez, J.X., An epidemiological model for the dynamics of chagas’ disease, Biosystems, 26, 127-134, (1991) [5] Hyman, J.M.; Li, J.; Stanley, E.A., The differential infectivity and staged progression models for the transmission of HIV, Math. biosci., 155, 77-109, (1999) · Zbl 0942.92030 [6] Guo, H.; Li, M.Y., Global dynamics of a staged progression model for infectious diseases, Math. biosci. eng., 3, 513-525, (2006) · Zbl 1092.92040 [7] Li, J.; Zhou, Y.; Ma, Z.; Hyman, J.M., Epidemiological models for mutating pathogens, SIAM J. appl. math., 65, 1-23, (2004) · Zbl 1088.34045 [8] Martcheva, M.; Castillo-Chavez, C., Disease with chronic stage in a population with varying size, Math. biosci., 182, 1-25, (2003) · Zbl 1012.92024 [9] Moghadas, S.M.; Gumel, A.B., Global stability of a two-stage epidemic model with generalized non-linear incidence, Math. comput. simulat., 60, 107-118, (2002) · Zbl 1005.92031 [10] Martcheva, M.; Castillo-Chavez, C., Diseases with chronic stage in a population with varying size, Math. biosci., 182, 1-25, (2003) · Zbl 1012.92024 [11] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical models and dynamics of infectious diseases, (2004), China Sciences Press Beijing [12] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014 [13] Liu, W.M.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023 [14] Wilson, E.B.; Worcester, J., The law of mass action in epidemiology, Proc. natl. acad. sci. USA, 31, 1, 24-34, (1945) [15] Wilson, E.B.; Worcester, J., The law of mass action in epidemiology II, Proc. natl. acad. sci. USA, 31, 4, 109-116, (1945) [16] Yorke, J.A.; London, W.P., Recurrent outbreaks of measles, chickenpox and mumps II, Am. J. epidemiol., 98, 469-482, (1973) [17] d’Onofrio, Alberto, Biomathematical analysis and extension of the new class of epidemic models proposed by Satsuma et al. (2004), Appl. math. comput., 170, 125-134, (2005) · Zbl 1075.92047 [18] Hethcote, H.W.; van den Driessche, P., Some epidemiological models with nonlinear incidence, J. math. biol., 29, 271-287, (1991) · Zbl 0722.92015 [19] Ruan, S.; Wang, W., Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. differ. equat., 188, 135-163, (2003) · Zbl 1028.34046 [20] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J. math. biol., 40, 525-540, (2000) · Zbl 0961.92029 [21] Alexander, M.E.; Moghadas, S.M., Periodicity in an epidemic model with a generalized nonlinear incidence, Math. biosci., 189, 75-96, (2004) · Zbl 1073.92040 [22] Li, M.Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math. biosci., 125, 155-164, (1995) · Zbl 0821.92022 [23] Dulac, H., Recherche des cycles limites, C.R. math. acad. sci. Paris, 204, 1703-1706, (1973) · Zbl 0016.40003 [24] Li, M.Y.; Muldowney, J.S., A geometric approach to global-stability problems, SIAM J. math. anal., 27, 1070-1083, (1996) · Zbl 0873.34041 [25] Li, M.Y.; Muldowney, J.S., On bendixsons criterion, J. differ. equat., 106, 27-39, (1993) [26] Li, M.Y.; Muldowney, J.S.; van den Driessche, P., Global stability of SEIRS models in epidemiology, Can. appl. math. quart., 7, 4, 409-425, (1999) · Zbl 0976.92020 [27] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. appl. math., 62, 1, 58-69, (2001) · Zbl 0991.92029 [28] Busenberg, S.; van den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. math. biol., 28, 257-270, (1990) · Zbl 0725.92021 [29] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29-48, (2002) · Zbl 1015.92036 [30] McCluskey, C.C.; van den Driessche, P., Global analysis of two tuberculosis models, J. dynam. differ. equat., 16, 139-166, (2004) · Zbl 1056.92052 [31] Feng, Z.; Iannelli, M.; Milner, F., A two-strain tuberculosis model with age of infection, SIAM J. appl. math., 62, 1634-1656, (2002) · Zbl 1017.35066 [32] Thieme, H.R., Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. math. anal., 24, 2, 407-435, (1993) · Zbl 0774.34030 [33] Li, J.; Ma, Z., Qualitative analyses of SIS epidemic model with vaccination varying total population, Math. comput. model., 35, 1235-1243, (2002) · Zbl 1045.92039 [34] Perko, L., Differential equations and dynamical systems, (1998), Springer-Verlag New York
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.