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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:
60H30Applications of stochastic analysis
92D30Epidemiology
60H35Computational methods for stochastic equations
60H10Stochastic ordinary differential equations
34D05Asymptotic stability of ODE
34F05ODE with randomness
References:
[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) · Zbl 53.0517.01 · doi:10.1098/rspa.1927.0118
[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 · doi:10.1016/S0362-546X(01)00528-4
[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 · doi:10.1016/0895-7177(96)00004-0
[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 · doi:10.1016/j.amc.2007.07.083
[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 · doi:10.1016/j.tpb.2007.12.008
[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 · doi:10.1002/mma.810
[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 · doi:10.1016/j.apm.2007.12.020
[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 · doi:10.1016/j.amc.2007.09.053
[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 · doi:10.1002/bimj.4710210507
[12]Bahar, A.; Mao, X.: Stochastic delay Lotka–Volterra model, J. math. Anal. appl. 292, 364-380 (2004) · Zbl 1043.92034 · doi:10.1016/j.jmaa.2003.12.004
[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 · doi:10.1016/j.jde.2005.06.017
[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 · doi:10.1016/j.jmaa.2009.05.039
[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 · doi:10.1016/j.jmaa.2004.08.027
[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 · doi:10.1016/j.jmaa.2007.08.014
[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 · doi:10.1214/aoap/1015345354
[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 · doi:10.1016/S0022-247X(03)00539-0
[20]Mao, X.; Marion, G.; Renshaw, E.: Environmental noise suppresses explosion in population dynamics, Stochastic process. Appl. 97, 95-110 (2002) · Zbl 1058.60046 · doi:10.1016/S0304-4149(01)00126-0
[21]Mao, X.: Stochastic differential equations and applications, (1997)
[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 · doi:10.1016/S0025-5564(01)00089-X
[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 · doi:10.1016/j.jmaa.2006.01.055
[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 · doi:10.1016/j.jmaa.2007.11.005
[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 · doi:10.1016/j.jmaa.2009.06.050
[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 · doi:10.1016/S0378-4754(97)00106-7
[28]Higham, D. J.: An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM rev. 43, 525-546 (2001) · Zbl 0979.65007 · doi:10.1137/S0036144500378302