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Long-time behavior of a regime-switching susceptible-infective epidemic model with degenerate diffusion. (English) Zbl 1444.37075

Summary: In this paper, we consider a stochastic SI epidemic model with regime switching. The Markov semigroup theory is employed to obtain the existence of a unique stable stationary distribution. We prove that if \(\mathcal{R}^{s}<0\), then the disease becomes extinct exponentially; whereas if \(\mathcal{R}^{s}>0\) and \(\beta(i)>\alpha(i)\), \(i\in\mathbb{S}\), then the densities of the distributions of the solution can converge in \(L^{1}\) to an invariant density.

MSC:

37N25 Dynamical systems in biology
92D30 Epidemiology
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[1] Mao, X: Stochastic Differential Equations and Applications. Horwood, Chichester (1997) · Zbl 0892.60057
[2] Zhang, X, Jiang, D, Alsaedib, A, et al.: Stationary distribution of stochastic SIS epidemic model with vaccination under regime switching. Appl. Math. Lett. 59, 87-93 (2016) · Zbl 1343.60095 · doi:10.1016/j.aml.2016.03.010
[3] Liu, Q: The threshold of a stochastic Susceptible-Infective epidemic model under regime switching. Nonlinear Anal. Hybrid Syst. 21, 49-58 (2016) · Zbl 1358.92092 · doi:10.1016/j.nahs.2016.01.002
[4] Settati, A, Lahrouz, A: Asymptotic properties of switching diffusion epidemic model with varying population size. Appl. Math. Comput. 219, 11134-11148 (2013) · Zbl 1304.92121
[5] Han, Z, Zhao, J: Stochastic SIRS model under regime switching. Nonlinear Anal., Real World Appl. 14, 352-364 (2013) · Zbl 1267.34079 · doi:10.1016/j.nonrwa.2012.06.008
[6] Bell, DR: The Malliavin Calculus. Dover, New York (2006) · Zbl 1099.60041
[7] Aida, S.; Kusuoka, S.; Strook, D.; Elworthy, KD (ed.); Ikeda, N. (ed.), On the support of Wiener functionals, No. 284, 3-34 (1993), Harlow
[8] Arous, GB, Léandre, R: Décroissance exponentielle du noyau de la chaleur sur la diagonale (II). Probab. Theory Relat. Fields 90, 377-402 (1991) · Zbl 0734.60027 · doi:10.1007/BF01193751
[9] Stroock, DW; Varadhan, SRS, On the support of diffusion processes with applications to the strong maximum principle, 333-360 (1972), Berkeley
[10] Xi, F: On the stability of jump-diffusions with Markovian switching. J. Math. Anal. Appl. 341, 588-600 (2008) · Zbl 1138.60044 · doi:10.1016/j.jmaa.2007.10.018
[11] Lasota, A, Mackey, MC: In: Chaos, Fractals and Noise. Stochastic Aspects of Dynamics. Springer Applied Mathematical Sciences, vol. 97 (1994) New York · Zbl 0784.58005 · doi:10.1007/978-1-4612-4286-4
[12] Rudnicki, R: Long-time behaviour of a stochastic prey-predator model. Stoch. Process. Appl. 108, 93-107 (2003) · Zbl 1075.60539 · doi:10.1016/S0304-4149(03)00090-5
[13] Khasminskii, RZ, Zhu, C, Yin, G: Stability of regime-switching diffusions. Stoch. Process. Appl. 117, 1037-1051 (2007) · Zbl 1119.60065 · doi:10.1016/j.spa.2006.12.001
[14] Pichór, K, Rudnicki, R: Stability of Markov semigroups and applications to parabolic systems. J. Math. Anal. Appl. 215, 56-74 (1997) · Zbl 0892.35072 · doi:10.1006/jmaa.1997.5609
[15] Rudnicki, R: On asymptotic stability and sweeping for Markov operators. Bull. Pol. Acad. Sci., Math. 43, 245-262 (1995) · Zbl 0838.47040
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