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Asymptotic behavior of global positive solution to a stochastic SIR model. (English) Zbl 1225.60114

Summary: We explore a stochastic SIR model and show that this model has a unique global positive solution. Furthermore, we investigate the asymptotic behavior of this solution. Finally, numerical simulations are presented to illustrate our mathematical findings.

MSC:

60H30 Applications of stochastic analysis (to PDEs, etc.)
92D30 Epidemiology
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34D05 Asymptotic properties of solutions to ordinary differential equations
34F05 Ordinary differential equations and systems with randomness
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[1] Kermack, W.O.; McKendrick, A.G., Contributions to the mathematical theory of epidemics (part I), Proc. R. soc. lond. ser. A, 115, 700-721, (1927) · JFM 53.0517.01
[2] Anderson, R.M.; May, R.M., Population biology of infectious diseases, part I, Nature, 280, 361-367, (1979)
[3] 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
[4] Guo, H.B.; Li, M.Y.; Shuai, Z.S., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can. appl. math. Q., 14, 259-284, (2006) · Zbl 1148.34039
[5] Koide, C.; Seno, H., Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Math. comput. modelling, 23, 67-91, (1996) · Zbl 0846.92025
[6] 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
[7] Roy, M.; Holt, R.D., Effects of predation on host-cpathogen dynamics in SIR models, Theor. popul. biol., 73, 319-331, (2008) · Zbl 1209.92054
[8] 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
[9] 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
[10] 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
[11] 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
[12] Bahar, A.; Mao, X., Stochastic delay lotka – volterra model, J. math. anal. appl., 292, 364-380, (2004) · Zbl 1043.92034
[13] Gard, T.C., Persistence in stochastic food web models, Bull. math. biol., 46, 357-370, (1984) · Zbl 0533.92028
[14] Imhof, L.; Walcher, S., Exclusion and persistence in deterministic and stochastic chemostat models, J. differential equations, 217, 26-53, (2005) · Zbl 1089.34041
[15] Ji, C.Y.; Jiang, D.Q.; Shi, N.Z., Analysis of a predator – prey model with modified leslie – gower and Holling-type II schemes with stochastic perturbation, J. math. anal. appl., 359, 482-498, (2009) · Zbl 1190.34064
[16] 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
[17] 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., 340, 588-597, (2008) · Zbl 1140.60032
[18] 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
[19] 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
[20] Mao, X.; Marion, G.; Renshaw, E., Environmental noise suppresses explosion in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046
[21] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050
[22] 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
[23] 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
[24] 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
[25] Tornatore, E.; Buccellato, S.M.; Vetro, P., Stability of a stochastic SIR system, Physica A, 354, 111-126, (2005)
[26] Yu, J.J.; Jiang, D.Q.; Shi, N.Z., Global stability of two-group SIR model with random perturbation, J. math. anal. appl., 360, 235-244, (2009) · Zbl 1184.34064
[27] 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
[28] Higham, D.J., An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev., 43, 525-546, (2001) · Zbl 0979.65007
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