×

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 \(\mathcal R_{0}\) 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 \(\mathcal R_{0}\). If \(\mathcal R_{0}\leq 1\), the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, if \(\mathcal R_{0}>1\), there is a stationary distribution, which means that the disease will prevail.

MSC:

93E03 Stochastic systems in control theory (general)
93C73 Perturbations in control/observation systems
92D30 Epidemiology
93A30 Mathematical modelling of systems (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

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
[2] Bandyopadhyay, M.; Chattopadhyay, J., Ratio-dependent predator-prey model: effect of environmental fluctuation and stability, Nonlinearity, 18, 913-936 (2005) · Zbl 1078.34035
[3] Beretta, E.; Capasso, V., Global stability results for a multigroup SIR epidemic model, Mathematical Ecology, 317-342 (1986)
[4] Berman, A.; Plemmons, R. J., Nonnegative matrices in the mathematical sciences (1979), Academic Press: Academic Press New York · 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
[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
[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
[8] Gard, T. C., Introduction to stochastic differential equations, vol. 270 (1988), Madison Avenue: Madison Avenue New York · Zbl 0682.92018
[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
[11] Hasminskii, R. Z., Stochastic stability of differential equations (1980), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn, The Netherlands · Zbl 0419.62037
[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
[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
[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
[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
[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
[17] Mao, X. R., Stochastic differential equations and applications (1997), Horwood: Horwood Chichester
[18] Strang, G., Linear algebra and its applications (1988), Thomson Learning: Inc
[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), Prentice Hall: Prentice Hall Upper Saddle River · 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.