Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission. (English) Zbl 1254.92085

Summary: The dynamics of multi-group SEIR epidemic models with distributed and infinite delay and nonlinear transmission are investigated. We derive the basic reproduction number \(R_{0}\) and establish that the global dynamics are completely determined by the values of \(R_{0}\): if \(R_{0}\leq 1\), then the disease-free equilibrium is globally asymptotically stable; if \(R_{0}>1\), then there exists a unique endemic equilibrium which is globally asymptotically stable. Our results contain those for single-group SEIR models with distributed and infinite delays. In the proof of global stability of the endemic equilibrium, we exploit a graph-theoretical approach to the method of Lyapunov functionals. The biological significance of the results is also discussed.


92D30 Epidemiology
37N25 Dynamical systems in biology
Full Text: DOI


[1] Lloyd, A.L.; Jansen, V.A.A., Spatiotemporal dynamics of epidemics: synchrony in metapopulation models, Math. biosci., 188, 1-16, (2004) · Zbl 1036.92029
[2] Lloyd, A.L.; May, R.M., Spatial heterogeneity in epidemic models, J. theoret. biol., 179, 1-11, (1996)
[3] Thieme, H.R., Mathematics in population biology, (2003), Princeton University Press Princeton · Zbl 1054.92042
[4] Lajmanovich, A.; York, J.A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci., 28, 221-236, (1976) · Zbl 0344.92016
[5] Beretta, E.; Capasso, V., Global stability results for a multigroup SIR epidemic model, (), 317-342 · Zbl 0684.92015
[6] Hethcote, H.W., An immunization model for a heterogeneous population, Theor. popul. biol., 14, 338-349, (1978) · Zbl 0392.92009
[7] Hethcote, H.W.; Thieme, H.R., Stability of the endemic equilibrium in epidemic models with subpopulations, Math. biosci., 75, 205-227, (1985) · Zbl 0582.92024
[8] Lin, X.; So, J.W.-H., Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations, J. aust. math. soc. ser. B, 34, 282-295, (1993) · Zbl 0778.92020
[9] 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
[10] 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
[11] Guo, H.; Li, M.Y.; Shuai, Z., A graph-theoretic approach to the method of global Lyapunov functions, Proc. amer. math. soc., 136, 2793-2802, (2008) · Zbl 1155.34028
[12] Li, M.Y.; Shuai, Z.; Wang, C., Global stability of multi-group epidemic models with distributed delays, J. math. anal. appl., 361, 38-47, (2010) · Zbl 1175.92046
[13] Liu, S.; Wang, S.; Wang, L., Global dynamics of delay epidemic models with nonlinear incidence rate and relapse, Nonlinear anal. RWA, 12, 119-127, (2011) · Zbl 1208.34125
[14] Ding, D.; Ding, X., Global stability of multi-group vaccination epidemic models with delays, Nonlinear anal. RWA, 12, 1991-1997, (2011) · Zbl 1215.92038
[15] Kuniya, T., Global stability analysis with a discretization approach for an age-structured multigroup SIR epidemic model, Nonlinear anal. RWA, 12, 2640-2655, (2011) · Zbl 1219.35330
[16] Zhou, L.; Fan, M., Dynamics of an SIR epidemic model with limited medical resources revisited, Nonlinear anal. RWA, 13, 312-324, (2012) · Zbl 1238.37041
[17] Wang, L.; Chen, L.; Nieto, J.J., The dynamics of an epidemic model for pest control with impulsive effect, Nonlinear anal. RWA, 11, 1374-1386, (2010) · Zbl 1188.93038
[18] Gao, S.; Liu, Y.; Nieto, J.J.; Andrade, H., Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission, Math. comput. simul., 81, 1855-1868, (2011) · Zbl 1217.92066
[19] Röst, G.; Wu, J., SEIR epidemiological with varying infectivity and infinite delay, Math. biosci. eng., 5, 389-402, (2008) · Zbl 1165.34421
[20] Atkinson, F.V.; Haddock, J.R., On determining phase spaces for functional differential equations, Funkcial. ekvac., 31, 331-347, (1988) · Zbl 0665.45004
[21] Hino, Y.; Murakami, S.; Naito, T., ()
[22] Hale, J.; Verduyn Lunel, S., ()
[23] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio \(R_0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[24] Haddock, J.R.; Terjéki, J., Liapunov – razumikhin functions and an invariance principle for functional-differential equations, J. differential equations, 48, 95-122, (1983) · Zbl 0531.34058
[25] Haddock, J.R.; Krisztin, T.; Terjéki, J., Invariance principles for autonomous functional-differential equations, J. integral equations, 10, 123-136, (1985) · Zbl 0592.34047
[26] Ruess, W.M.; Summers, W.H., Linearized stability for abstract differential equations with delay, J. math. anal. appl., 198, 310-336, (1996) · Zbl 0861.34046
[27] Li, M.Y.; Graef, J.R.; Wang, L.C.; Karsai, J., Global dynamics of a SEIR model with a varying total population size, Math. biosci., 160, 191-213, (1999) · Zbl 0974.92029
[28] Bhatia, N.P.; Szegö, G.P., ()
[29] Smith, H.L.; Waltman, P., The theory of the chemostat: dynamics of microbial competition, (1995), Cambridge University Press Cambridge · Zbl 0860.92031
[30] Li, M.Y.; Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, J. differential equations, 248, 1-20, (2010) · Zbl 1190.34063
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