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Necessary and sufficient condition for extinction and persistence of SIRS system with random perturbation. (English) Zbl 1334.92412
Summary: This paper characterizes the qualitative dynamics of a stochastic SIRS epidemic model. The study shows that the dynamics of the model are determined by a certain threshold quantity \(\mathcal{R}_{\mathcal{S}}\) expressed in terms of the model parameters and the intensity of the noise. If \(\mathcal{R}_{\mathcal{S}} < 1\), the disease will be eliminated from the community; whereas an epidemic occurs if \(\mathcal{R}_{\mathcal{S}} > 1\). Our results recover the known results in the earlier literature as special cases. The presented results are illustrated by numerical simulations.

MSC:
92D30 Epidemiology
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Hethcote, H. W., Qualitative analyses of communicable disease models, Math. Biosci., 28, 335-356, (1976) · Zbl 0326.92017
[2] Tornatore, E.; Buccellato, S. M.; Vetro, P., Stability of a stochastic SIR system, Physica A, 35, 4111-4126, (2005)
[3] Dalal, N.; Greenhalgh, D.; Mao, X., A stochastic model of AIDS and condom use, J. Math. Anal. Appl., 325, 36-53, (2007) · Zbl 1101.92037
[4] Lu, Q., Stability of SIRS system with random perturbations, Physica A, 388, 3677-3686, (2009)
[5] Gray, A.; Greenhalgh, D.; Hu, L.; Mao, X.; Pan, J., A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math., 71, 876-902, (2011) · Zbl 1263.34068
[6] Lahrouz, A.; Omari, L.; Kiouach, D., Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control, 16, 59-76, (2011) · Zbl 1271.93015
[7] Lahrouz, A.; Omari, L., Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Stat. Probab. Lett., 83, 960-968, (2013) · Zbl 1402.92396
[8] Lahrouz, A.; Settati, A., Asymptotic properties of switching diffusion epidemic model with varying population size, Appl. Math. Comput., 219, 11134-11148, (2013) · Zbl 1304.92121
[9] Has’minskii, R. Z., Stochastic stability of differential equations, (1980), Sijthoof & Noordhoof Alphen aan den Rijn, The Netherlands
[10] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46, 1155-1179, (2007) · Zbl 1140.93045
[11] Xia, P.; Zheng, X.; Jiang, D., Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System, Abstract and Applied Analysis, vol. 2013, (2013), Hindawi Publishing Corporation · Zbl 1402.92375
[12] Liu, M.; Wang, K.; Wu, Q., Survival analysis of stochastic competitive models in a polluted environment and stochastic competitive exclusion principle, Bull. Math. Biol., 73, 1969-2012, (2011) · Zbl 1225.92059
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