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Dynamics of a multigroup SIR epidemic model with stochastic perturbation. (English) Zbl 1244.93154
Summary: We introduce stochasticity into a multigroup Susceptible, Infective, and Recovered (SIR) model. The stochasticity in the model is introduced by parameter perturbation, which is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction number $\cal R_{0}$ is a threshold which completely determines the persistence or extinction of the disease. We carry out a detailed analysis on the asymptotic behavior of the stochastic model, also regarding of the value of $\cal R_{0}$. If $\cal R_{0}\le 1$, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, if $\cal R_{0}>1$, there is a stationary distribution, which means that the disease will prevail.

##### MSC:
 93E03 General theory of stochastic systems 93C73 Perturbations in control systems 92D30 Epidemiology 93A30 Mathematical modelling of systems
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##### References:
 [1] Arnold, L.; Horsthemke, W.; Stucki, J. W.: The influence of external real and white noise on the Lotka--Volterra model, Journal of biomedical 21, 451-471 (1979) · Zbl 0433.92019 · doi:10.1002/bimj.4710210507 [2] Bandyopadhyay, M.; Chattopadhyay, J.: Ratio-dependent predator--prey model: effect of environmental fluctuation and stability, Nonlinearity 18, 913-936 (2005) · Zbl 1078.34035 · doi:10.1088/0951-7715/18/2/022 [3] Beretta, E.; Capasso, V.: Global stability results for a multigroup SIR epidemic model, Mathematical ecology, 317-342 (1986) · Zbl 0684.92015 [4] Berman, A.; Plemmons, R. J.: Nonnegative matrices in the mathematical sciences, (1979) · Zbl 0484.15016 [5] Carletti, M.; Burrage, K.; Burrage, P. M.: Numerical simulation of stochastic ordinary differential equations in biomathematical modelling, Mathematics and computers in simulation 64, 271-277 (2004) · Zbl 1039.65005 · doi:10.1016/j.matcom.2003.09.022 [6] Dalal, N.; Greenhalgh, D.; Mao, X. R.: A stochastic model of AIDS and condom use, Journal of mathematical analysis and applications 325, 36-53 (2007) · Zbl 1101.92037 · doi:10.1016/j.jmaa.2006.01.055 [7] Feng, Z. L.; Huang, W. Z.; Castillo-Chavez, C.: Global behavior of a multi-group SIS epidemic model with age structure, Journal of differential equations 218, 292-324 (2005) · Zbl 1083.35020 · doi:10.1016/j.jde.2004.10.009 [8] Gard, T. C.: Introduction to stochastic differential equations, Introduction to stochastic differential equations 270 (1988) · Zbl 0628.60064 [9] Guo, H. B.; Li, M. Y.; Shuai, Z. S.: Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian applied mathematics quarterly 14, 259-284 (2006) · Zbl 1148.34039 [10] Guo, H. B.; Li, M. Y.; Shuai, Z. S.: A graph-theoretic approach to the method of global Lyapunov functions, Proceedings of the American mathematical society 136, 2793-2802 (2008) · Zbl 1155.34028 · doi:10.1090/S0002-9939-08-09341-6 [11] Hasminskii, R. Z.: Stochastic stability of differential equations, (1980) [12] Huang, W.; Cooke, K. L.; Castillo-Chavez, C.: Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM journal on applied mathematics 52, 835-854 (1992) · Zbl 0769.92023 · doi:10.1137/0152047 [13] 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, Journal of mathematical analysis and applications 359, 482-498 (2009) · Zbl 1190.34064 · doi:10.1016/j.jmaa.2009.05.039 [14] Ji, C. Y.; Jiang, D. Q.; Li, X. Y.: Qualitative analysis of a stochastic ratio-dependent predator--prey system, Journal of computational and applied mathematics 235, 1326-1341 (2011) · Zbl 1229.92076 · doi:10.1016/j.cam.2010.08.021 [15] Koide, C.; Seno, H.: Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease, Mathematical and computer modelling 23, 67-91 (1996) · Zbl 0846.92025 · doi:10.1016/0895-7177(96)00004-0 [16] Li, M. Y.; Shuai, Z. S.: Global-stability problem for coupled systems of differential equations on networks, Journal of differential equations 248, 1-20 (2010) · Zbl 1190.34063 · doi:10.1016/j.jde.2009.09.003 [17] Mao, X. R.: Stochastic differential equations and applications, (1997) · Zbl 0892.60057 [18] Strang, G.: Linear algebra and its applications, (1988) · Zbl 0338.15001 [19] Tornatore, E.; Buccellato, S. M.; Vetro, P.: Stability of a stochastic SIR system, Physica A 354, 111-126 (2005) [20] West, D. B.: Introduction to graph theory, (1996) · Zbl 0845.05001 [21] Zhu, C.; Yin, G.: Asymptotic properties of hybrid diffusion systems, SIAM journal on control and optimization 46, 1155-1179 (2007) · Zbl 1140.93045 · doi:10.1137/060649343