Abstract
This paper proposes a new nonlinear stochastic SIVS epidemic model with double epidemic hypothesis and Lévy jumps. The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model. By using the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Finally, this paper describes the results of numerical simulations investigating the dynamical effects of stochastic disturbance. Our results significantly improve and generalize the corresponding results in recent literatures. The developed theoretical methods and stochastic inequalities technique can be used to investigate the high-dimensional nonlinear stochastic differential systems.
Highlights
Mathematical inequalities are widely used in many fields of mathematical analysis, especially differential systems [ – ]
Motivated by the above works, in this paper, we propose a stochastic SIVS model with double epidemic diseases and Lévy jumps under vaccination as follows:
The main purpose of this paper is to investigate the threshold dynamics of the stochastic SIVS epidemic model
Summary
Mathematical inequalities are widely used in many fields of mathematical analysis, especially differential systems [ – ]. In model ( ), the authors discussed in detail the conditions for persistence in mean and extinction of each epidemic disease. Motivated by the above works, in this paper, we propose a stochastic SIVS model with double epidemic diseases and Lévy jumps under vaccination as follows:. By using the Lyapunov method and the technique of a series of stochastic inequalities, we obtain sufficient conditions for the persistence in mean and extinction of the stochastic system and the threshold which governs the extinction and the spread of the epidemic diseases. Applying the Burkholder-Davis-Gundy inequality, integrating equation ( ) from to t, and for an arbitrarily small positive constant δ, one has. Applying the Borel-Cantelli lemma [ ], for almost all ω ∈ , one has sup X (t) ≤ (kδ) +κX kδ≤t≤(k+ )δ holds for all but finitely many k. For any initial value (S( ), I ( ), I (t), V ( )) ∈ R +, model ( ) has a unique positive solution (S(t), I (t), I (t), V (t)) ∈ R + on t ≥ with probability
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