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Global stability of two-group SIR model with random perturbation. (English) Zbl 1184.34064

The authors study a two-group SIR model introduced by H. Guo, M. Y. Li and Z. Shuai [Can. Appl. Math. Q. 14, No. 3, 259–284 (2006; Zbl 1148.34039)] with respect to white noise stochastic perturbations around its positive endemic equilibrium.
They prove a stability result given in Theorem 4.3 which show that the endemic equilibrium is stochastically asymptotically stable in the large. The proof of this theorem is based on a Lyapunov function.
Some suggestive numerical simulations are included.

MSC:

34F05 Ordinary differential equations and systems with randomness
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
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[1] Guo, H. B.; 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
[2] Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics (Part I), Proc. R. Sm. A, 115, 700-721 (1927) · JFM 53.0517.01
[3] Meng, X. Z.; Chen, L. S., The dynamics of a new SIR epidemic model concerning pulse vaccination strategy, Appl. Math. Comput., 197, 528-597 (2008) · Zbl 1131.92056
[4] Zhang, F. P.; Li, Z. Z.; Zhang, F., Global stability of an SIR epidemic model with constant infectious period, Appl. Math. Comput., 199, 285-291 (2008) · Zbl 1136.92336
[5] Hsu, Sze-Bi; Roeger, Lih-Ing W., The final size of a SARS epidemic model without quarantine, J. Math. Anal. Appl., 333, 557-566 (2007) · Zbl 1113.92055
[6] Zhang, Z. H.; Peng, J. G., A SIRS epidemic model with infection-age dependence, J. Math. Anal. Appl., 331, 1396-1414 (2007) · Zbl 1116.35024
[7] Wei, H. M.; Li, X. Z.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, J. Math. Anal. Appl., 342, 895-908 (2008) · Zbl 1146.34059
[8] Roy, M.; Holt, R. D., Effects of predation on host-pathogen dynamics in SIR models, Theor. Popul. Biol., 73, 319-331 (2008) · Zbl 1209.92054
[9] Tchuenche, J. M.; Nwagwo, A.; Levins, R., Global behaviour of an SIR epidemic model with time delay, Math. Methods Appl. Sci., 30, 733-749 (2007) · Zbl 1112.92055
[10] Zhang, T. L.; Teng, Z. D., Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Appl. Math. Model., 33, 1058-1071 (2009) · Zbl 1168.34358
[11] 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
[12] Lajmanovich, A.; York, J. A., A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci., 28, 221-236 (1976) · Zbl 0344.92016
[13] Beretta, E.; Capasso, V., Global stability results for a multigroup SIR epidemic model, (Hallam, T. G.; Gross, L. J.; Levin, S. A., Mathematical Ecology (1986), World Scientific: World Scientific Singapore), 317-342 · Zbl 0684.92015
[14] Hethcote, H. W., An immunization model for a heterogeneous population, Theor. Popul. Biol., 14, 338-349 (1978) · Zbl 0392.92009
[15] Rass, L.; Radcliffe, J., Global asymptotic convergence results for multitype models, Int. J. Appl. Math. Comput. Sci., 10, 63-79 (2000) · Zbl 0978.92026
[16] Koide, C.; Seno, H., Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Math. Comput. Model., 23, 67-91 (1996) · Zbl 0846.92025
[17] Mao, X.; Marion, G.; Renshaw, E., Asymptotic behaviour of the stochastic Lotka-Volterra model, J. Math. Anal. Appl., 287, 141-156 (2003) · Zbl 1048.92027
[18] Mao, X.; Marion, G.; Renshaw, E., Environmental noise suppresses explosion in population dynamics, Stochastic Process. Appl., 97, 95-110 (2002) · Zbl 1058.60046
[19] Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: Horwood Chichester · Zbl 0874.60050
[20] Arnolld, L.; Horsthemke, W.; Stuxki, J. W., The influence of external real and white noise on the Lotka-Volterra model, Biomed. J., 21, 451-471 (1976) · Zbl 0433.92019
[21] Khasminskii, R. Z.; Klerbaner, F. C., Long term behavior of solution of the Lotka-Volterra system under small random perturbations, Ann. Appl. Probab., 11, 952-963 (2001) · Zbl 1061.34513
[22] Bahar, A.; Mao, X., Stochastic delay Lotka-Volterra model, J. Math. Anal. Appl., 292, 364-380 (2004) · Zbl 1043.92034
[23] Jiang, D. Q.; Shi, N. Z., A note on nonautonomous Logistic equation with random perturbation, J. Math. Anal. Appl., 303, 164-172 (2005) · Zbl 1076.34062
[24] Jiang, D. Q.; Shi, N. Z.; Li, X. Y., Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., 303, 164-172 (2008) · Zbl 1076.34062
[25] Gard, T. C., Persistence in stochastic food web models, Bull. Math. Biol., 46, 357-370 (1984) · Zbl 0533.92028
[26] Imhof, L.; Walcher, S., Exclusion and persistence in deterministic and stochastic chemostat models, J. Differential Equations, 217, 26-53 (2005) · Zbl 1089.34041
[27] Dalal, N.; Greenhalgh, D.; Mao, X. R., A stochastic model for internal HIV dynamics, J. Math. Anal. Appl., 341, 1084-1101 (2008) · Zbl 1132.92015
[28] Dalal, N.; Greenhalgh, D.; Mao, X. R., A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325, 36-53 (2007) · Zbl 1101.92037
[29] Tornatore, E.; Buccellato, S. M.; Vetro, P., Stability of a stochastic SIR system, Phys. A, 354, 111-126 (2005)
[30] Carletti, M., On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Math. Biosci., 175, 117-131 (2002) · Zbl 0987.92027
[31] Beretta, E.; Kolmanovskii, V.; Shaikhet, L., Stability of epidemic model with time delays influenced by stochastic perturbations, Math. Comput. Simulation, 45, 269-277 (1998) · Zbl 1017.92504
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