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Global stability of multi-group epidemic models with distributed delays. (English) Zbl 1175.92046
Summary: We investigate a class of multi-group epidemic models with distributed delays. We establish that the global dynamics are completely determined by the basic reproduction number $\cal R_0$. More specifically, we prove that, if $\cal R_0\leqslant 1$, then the disease-free equilibrium is globally asymptotically stable; if $\cal R_0>1$, then there exists a unique endemic equilibrium and it is globally asymptotically stable. Our proof of global stability of the endemic equilibrium utilizes a graph-theoretical approach to the method of Lyapunov functionals.

MSC:
92D30Epidemiology
34K20Stability theory of functional-differential equations
34D23Global stability of ODE
05C90Applications of graph theory
34K60Qualitative investigation and simulation of models
34D05Asymptotic stability of ODE
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References:
[1] Atkinson, F. V.; Haddock, J. R.: On determining phase spaces for functional differential equations, Funkcial. ekvac. 31, 331-347 (1988) · Zbl 0665.45004
[2] Beretta, E.; Capasso, V.: Global stability results for a multigroup SIR epidemic model, Mathematical ecology, 317-342 (1988) · Zbl 0684.92015
[3] Beretta, E.; Capasso, V.; Rinaldi, F.: Global stability results for a generalized Lotka -- Volterra system with distributed delays: applications to predator-prey and to epidemic systems, J. math. Biol. 26, 661-688 (1988) · Zbl 0716.92020 · doi:10.1007/BF00277730
[4] Beretta, E.; Hara, T.; Ma, W.; Takeuchi, Y.: Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear anal. 47, 4107-4115 (2001) · Zbl 1042.34585 · doi:10.1016/S0362-546X(01)00528-4
[5] Beretta, E.; Takeuchi, Y.: Global stability of an SIR epidemic model with time delays, J. math. Biol. 33, 250-260 (1995) · Zbl 0811.92019 · doi:10.1007/BF00169563
[6] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences, (1979) · Zbl 0484.15016
[7] Diekmann, O.; Heesterbeek, J. A. P.; Metz, J. A. J.: On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. math. Biol. 28, 365-382 (1990) · Zbl 0726.92018 · doi:10.1007/BF00178324
[8] 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
[9] 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 · doi:10.1090/S0002-9939-08-09341-6
[10] Hale, J. K.; Lunel, S. M. V.: Introduction to functional differential equations, Appl. math. Sci. 99 (1993) · Zbl 0787.34002
[11] Hethcote, H. W.: An immunization model for a heterogeneous population, Theor. popul. Biol. 14, 338-349 (1978) · Zbl 0392.92009
[12] 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 · doi:10.1016/0025-5564(85)90038-0
[13] Hino, Y.; Murakami, S.; Naito, T.: Functional differential equations with infinite delay, Lecture notes in math. 1473 (1991) · Zbl 0732.34051
[14] 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 · doi:10.1137/0152047
[15] Lajmanovich, A.; Yorke, J. A.: A deterministic model for gonorrhea in a nonhomogeneous population, Math. biosci. 28, 221-236 (1976) · Zbl 0344.92016 · doi:10.1016/0025-5564(76)90125-5
[16] Lasalle, J. P.: The stability of dynamical systems, Reg. conf. Ser. appl. Math. (1976) · Zbl 0364.93002
[17] 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 · doi:10.1017/S0334270000008900
[18] Ma, W.; Takeuchi, Y.; Hara, T.; Beretta, E.: Permanence of an SIR epidemic model with distributed time delays, Tohoku math. J. 54, 581-591 (2002) · Zbl 1014.92033 · doi:10.2748/tmj/1113247650
[19] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay -- distributed or discrete, Nonlinear Anal. Real World Appl., in press · Zbl 1185.37209 · doi:10.1016/j.nonrwa.2008.10.014
[20] Mccluskey, C. C.: Global stability for an SEIR epidemiological model with varying infectivity and infinity delay, Math. biosci. Eng. 6, 603-610 (2009) · Zbl 1190.34108 · doi:10.3934/mbe.2009.6.603
[21] J.W. Moon, Counting Labelled Trees, Canadian Mathematical Congress, Montreal, 1970 · Zbl 0214.23204
[22] Röst, G.; Wu, J.: SEIR epidemiological model with varying infectivity and finite delay, Math. biosci. Eng. 5, 389-402 (2008) · Zbl 1165.34421 · doi:10.3934/mbe.2008.5.389 · http://aimsciences.org/journals/redirecting.jsp?paperID=3256
[23] Thieme, H. R.: Local stability in epidemic models for heterogeneous populations, Lecture notes biomath. 57, 185-189 (1985) · Zbl 0584.92020
[24] Thieme, H. R.: Mathematics in population biology, (2003) · Zbl 1054.92042
[25] Den Driessche, P. Van; Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci. 180, 29-48 (2002) · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6
[26] Wang, W.; Zhao, X. -Q.: An epidemic model with population dispersal and infection period, SIAM J. Appl. math. 66, 1454-1472 (2006) · Zbl 1094.92055 · doi:10.1137/050622948