Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates. (English) Zbl 1223.92057

Summary: We introduce a basic reproduction number for a multigroup SEIR model with nonlinear incidence of infection and nonlinear removal functions between compartments. Then, we establish that the global dynamics are completely determined by the basic reproduction number \(R_{0}\). This shows that the basic reproduction number \(R_{0}\) is a global threshold parameter in the sense that if it is less than or equal to one, the disease free equilibrium is globally stable and the disease dies out; whereas if it is larger than one, there is a unique endemic equilibrium which is globally stable and thus the disease persists in the population. Finally, two numerical examples are also included to illustrate the effectiveness of the proposed result.


92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
65C20 Probabilistic models, generic numerical methods in probability and statistics
Full Text: DOI


[1] Lajmanovich, A.; York, J.A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci., 28, 221-236, (1976) · Zbl 0344.92016
[2] Anderson, R.M.; May, R.M., Population biology of infectious diseases I, Nature, 280, 361-367, (1979)
[3] Huang, W.; Cooke, K.L.; Castillo-Chavez, C., Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. appl. math., 52, 835-854, (1992) · Zbl 0769.92023
[4] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. biosci., 160, 191-213, (1999) · Zbl 0974.92029
[5] Thieme, H.R., Mathematics in population biology, (2003), Princeton University Press Princeton · Zbl 1054.92042
[6] Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Math. biosci., 208, 419-429, (2007) · Zbl 1119.92042
[7] Guo, H.; Li, M.Y.; Shuai, Z., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. appl. math. Q, 14, 259-284, (2006) · Zbl 1148.34039
[8] Guo, H.; Li, M.Y.; Shuai, Z., A graphtheoretic approach to the method of global Lyapunov functions, Proc. am. math. soc., 136, 2793-2802, (2008) · Zbl 1155.34028
[9] Li, M.Y.; Shuai, Z.S.; Wang, C.C., Global stability of multi-group epidemic models with distributed delays, J. math. anal. appl., 361, 38-47, (2010) · Zbl 1175.92046
[10] Li, M.Y.; Shuai, Z.S., Global-stability problem for coupled systems of differential equations on networks, J. differ. eqn., 248, 1-20, (2010) · Zbl 1190.34063
[11] McCluskey, C.C., Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear anal. real world appl., 11, 55-59, (2010) · Zbl 1185.37209
[12] Yuan, Z.H.; Wang, L., Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear anal. real world appl., 11, 995-1004, (2010) · Zbl 1254.34075
[13] Yuan, Z.H.; Zou, X.F., Global threshold property in an epidemic model for disease with latency spreading in a heterogeneous host population, Nonlinear anal. real world appl., 11, 3479-3490, (2010) · Zbl 1208.34134
[14] Korobeinikov, A., Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. math. biol., 71, 75-83, (2009) · Zbl 1169.92041
[15] Georgescu, P.; Hsieh, Y.H.; Zhang, H., A Lyapunov functional for a stage structured predator – prey model with nonlinear predation rate, Nonlinear anal. real world appl., 11, 3653-3665, (2010) · Zbl 1206.34065
[16] Gao, S.J.; Teng, Z.D.; Xie, D.H., The effects of pulse vaccination on SEIR model with two time delays, Appl. math. comput., 201, 282-292, (2008) · Zbl 1143.92024
[17] Kyrychko, Y.N.; Blyuss, K.B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal. real world appl., 6, 495-507, (2005) · Zbl 1144.34374
[18] Wang, X.; Tao, Y.D.; Song, X.Y., Pulse vaccination on SEIR epidemic model with nonlinear incidence rate, Appl. math. comput., 210, 398-404, (2009) · Zbl 1162.92323
[19] Cai, L.M.; Lin, X.Z., Analysis of a SEIV epidemic model with a nonlinear incidence rate, Appl. math. model., 33, 2919-2926, (2009) · Zbl 1205.34049
[20] Freedman, H.I.; Tang, M.X.; Ruan, S.G., Uniform persistence and flows near a closed positively invariant set, J. dyn. differ. eqn., 6, 583-600, (1994) · Zbl 0811.34033
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.