×

zbMATH — the first resource for mathematics

Asymptotic properties of switching diffusion epidemic model with varying population size. (English) Zbl 1304.92121
Summary: The purpose of this work is to investigate the asymptotic properties of a stochastic version of the classical SIS epidemic model with standard incidence and varying population size. The stochastic model studied here includes white vector noise and telegraph noise modeled by Markovian switching. We established conditions for extinction both in probability one and in \(p\)th moment. We also established the persistence of disease under different conditions on the intensities of noises and the parameters of the model. Furthermore, we showed the existence of a stationary distribution and derive expressions for its mean and variance. The presented results are demonstrated by numerical simulations.

MSC:
92D30 Epidemiology
34F05 Ordinary differential equations and systems with randomness
PDF BibTeX Cite
Full Text: DOI
References:
[1] World Health Organization, The global burden of disease: 2004 update, 2008. <www.who.inthealthinfoglobal_burden_diseaseGBD_report_2004update_full.pdf>.
[2] Capasso, V., Mathematical Structures of Epidemic Systems, (2008), Springer Heidelberg, Corrected 2nd printing · Zbl 1141.92035
[3] Zhou, J.; Hethcote, H. W., Population size dependent incidence in models for diseases without immunity, J. Math. Biol., 32, 809-834, (1994) · Zbl 0823.92027
[4] Beretta, E.; Takeuchi, Y., Global stability of a SIR epidemic model with time delay, J. Math. Biol., 33, 250-260, (1995) · Zbl 0811.92019
[5] Gray, A.; Greenhalgh, D.; Mao, X.; Pan, J., The SIS epidemic model with Markovian switching, J. Math. Anal. Appl., 394, 496-516, (2012) · Zbl 1271.92030
[6] 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
[7] Lahrouz, A.; Omari, L.; Kiouach, D.; Belmaati, A., Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Appl. Math. Comput., 218, 6519-6525, (2012) · Zbl 1237.92054
[8] Kermack, W. O.; McKendrick, A. G., Contribution to mathematical theory of epidemics, P. Roy. Soc. Lond. A Math., 115, 700-721, (1927) · JFM 53.0517.01
[9] Anderson, R. M.; May, R. M., Population biology of infectious diseases I, Nature, 180, 361-367, (1979)
[10] Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160, 191-213, (1999) · Zbl 0974.92029
[11] Herbert, H. W.; Yorke, J. A., Gonorrhea transmission dynamics and control, Lecture Notes in Biomathematics, 56, (1984), Springer-Verlag Berlin · Zbl 0542.92026
[12] Anderson, R. M.; May, R. M., The population dynamics of microparasites and their invertebrate hosts, Trans. R. Philos. Soc. B, 291, 451-524, (1981)
[13] May, R. M., Stability and Complexity in Model Ecosystems, (1973), Princeton University
[14] 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
[15] Liu, M.; Wang, K., Stationary distribution, ergodicity and extinction of a stochastic generalized logistic system app, Math. Lett., 25, 1980-1985, (2012) · Zbl 1261.60057
[16] 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
[17] Slatkin, M., The dynamics of a population in a Markovian environment, Ecology, 59, 249-256, (1978)
[18] Takeuchi, Y.; Du, N. H.; Hieu, N. T.; Sato, K., Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment, J. Math. Anal. Appl., 323, 938-957, (2006) · Zbl 1113.34042
[19] Du, N. H.; Kon, R.; Sato, K.; Takeuchi, Y., Dynamical behaviour of Lotka-Volterra competition systems: non autonomous bistable case and the effect of telegraph noise, J. Comput. Appl. Math., 170, 399-422, (2004) · Zbl 1089.34047
[20] Luo, Q.; Mao, X., Stochastic population dynamics under regime switching, J. Math. Anal. Appl., 334, 69-84, (2007) · Zbl 1113.92052
[21] Zhu, C.; Yin, G., On competitive Lotka-Volterra model in random environments, J. Math. Anal. Appl., 357, 154-170, (2009) · Zbl 1182.34078
[22] Li, X.; Jiang, D.; Mao, X., Population dynamical behavior of Lotka-Volterra system under regime switching, J. Comput. Appl. Math., 232, 427-448, (2009) · Zbl 1173.60020
[23] Liu, M.; Li, W.; Wang, K., Persistence and extinction of a stochastic delay logistic equation under regime switching, App. Math. Lett., 26, 140-144, (2013) · Zbl 1270.34188
[24] Han, Z.; Zhao, J., Stochastic SIRS model under regime switching, NONRWA, 14, 352-364, (2013) · Zbl 1267.34079
[25] Zhu, C.; Yin, G., Asymptotic properties of hybrid diffusion systems, SIAM J. Control Optim., 46, 1155-1179, (2007) · Zbl 1140.93045
[26] Yang, Q.; Jiang, D.; Shi, N.; Ji, C., The ergodicity and extinction of stochastically perturbed SIR and SEIR epidemic models with saturated incidence, J. Math. Anal. Appl., 388, 248-271, (2012) · Zbl 1231.92058
[27] Lahrouz, A.; Omari, L., Extinction and stationary distribution of a stochastic SIRS epidemic model with non-linear incidence, Statist. Probab. Lett., 83, 960-968, (2013) · Zbl 1402.92396
[28] Rudnicki, R.; Pichór, K., Influence of stochastic perturbation on prey-predator systems, Math. Biosci., 206, 108-119, (2007) · Zbl 1124.92055
[29] Mao, X., Stationary distribution of stochastic population systems, Syst. Control Lett., 60, 398-405, (2011) · Zbl 1387.60107
[30] Hu, G.; Liu, M.; Wang, K., The asymptotic behaviours of an epidemic model with two correlated stochastic perturbations, Appl. Math. Comput., 218, 10520-10532, (2012) · Zbl 1250.92038
[31] Mao, X., Stochastic differential equations and applications, (1997), Horwood Publishing Limited Chichester · Zbl 0884.60052
[32] Khasminskii, R. Z.; Zhu, C.; Yin, G., Stability of regime-switching diffusions, Stochastic Process Appl., 117, 1037-1051, (2007) · Zbl 1119.60065
[33] Anderson, W. J., Continuous-time Markov chains, (1991), Springer Berlin
[34] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations, (1992), Springer · Zbl 0925.65261
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.