×

Asymptotic behaviors of stochastic epidemic models with jump-diffusion. (English) Zbl 1481.92166

Summary: In this paper, we classify the asymptotic behavior for a class of stochastic SIR epidemic models represented by stochastic differential systems where the Brownian motions and Lévy jumps perturb to the linear terms of each equation. We construct a threshold value between permanence and extinction and develop the ergodicity of the underlying system. It is shown that the transition probabilities converge in total variation norm to the invariant measure. Our results can be considered as a significant contribution in studying the long term behavior of stochastic differential models because there are no restrictions imposed to the parameters of models. Techniques used in proving results of this paper are new and suitable to deal with other stochastic models in biology where the noises may perturb to nonlinear terms of equations or with delay equations.

MSC:

92D30 Epidemiology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
60J76 Jump processes on general state spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brauer, F.; Chavez, C. C., Mathematical Models in Population Biology and Epidemiology (2012), Springer-Verlag: Springer-Verlag New York · Zbl 1302.92001
[2] Capasso, V., Mathematical Structures of Epidemic Systems (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0798.92024
[3] Keeling, M. J.; Rohani, P., Modeling Infectious Diseases in Humans and Animals (2008), Princeton University Press: Princeton University Press Princeton and Oxford · Zbl 1279.92038
[4] Anderson, R. M.; May, R. M., Population biology of infectious diseases, part I, Nature, 280, 361-367 (1979)
[5] Dieu, N. T.; Nguyen, D. H.; Du, N. H.; Yin, G., Classification of asymptotic behavior in a stochastic SIR model, SIAM J. Appl. Dyn. Syst., 15, 2, 1062-1084 (2016) · Zbl 1343.34109
[6] Ji, C. Y.; Jiang, D. Q.; Shi, N. Z., The behavior of an SIR epidemic model with stochastic perturbation, Stoch. Anal. Appl., 30, 755-773 (2012) · Zbl 1272.60035
[7] Liu, Q.; Jiang, D.; Shi, N.; Hayat, T., Dynamics of a stochastic delayed SIR epidemic model with vaccination and double diseases driven by Lévy jumps, Phys. A, 492, 2010-2018 (2018) · Zbl 1514.92148
[8] Zhang, X.; Wang, K., Stochastic SIR model with jumps, Appl. Math. Lett., 26, 8, 867-874 (2013) · Zbl 1308.92107
[9] Zhou, Y.; Zhang, W., Threshold of a stochastic SIR epidemic model with Lévy jumps, Physica A, 446, 204-216 (2016) · Zbl 1400.92566
[10] Liu, M.; Wang, K., Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps, Nonlinear Anal., 85, 204-213 (2013) · Zbl 1285.34047
[11] Bao, J.; Yuan, C., Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116, 2, 119-132 (2011) · Zbl 1230.34068
[12] Xi, F., Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Process. Appl., 119, 2198-2221 (2009) · Zbl 1191.60091
[13] Stettner, L., On the Existence and Uniqueness of Invariant Measure for Continuous Time Markov Processes, Lcds report no. 86-16 (1986), Brown University, Providence
[14] Meyn, S. P.; Tweedie, R. L., Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Adv. Appl. Prob., 25, 518-548 (1993) · Zbl 0781.60053
[15] Khas’minskii, R. A., Ergodic properties of recurrent diffusion processes and stabilization of the cauchy problem for parabolic equations, Theory Probab. Appl., 5, 179-196 (1960) · Zbl 0093.14902
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.