In this paper we analyze the stochastic version of a minimalistic multi-strain model, which captures essential differences between primary and secondary infections in dengue fever epidemiology, and investigate the interplay between stochasticity, seasonality and import. The introduction of stochasticity is needed to explain the fluctuations observed in some of the available data sets, revealing a scenario where noise and complex deterministic skeleton...

We consider the problem of the existence of solutions of the random set-valued equation: (I) $D_H{X}_t=F(t,X_t)P.1$, t ∈ [0,T] -a.e.; X₀ = U p.1 where F and U are given random set-valued mappings with values in the space $K_c\left(E\right)$, of all nonempty, compact and convex subsets of the separable Banach space E. Under certain restrictions on F we obtain existence of solutions of the problem (I). The connections between solutions of (I) and solutions of random differential inclusions are investigated.

We present the concepts of set-valued stochastic integrals in a plane and prove the existence of a solution to stochastic integral inclusions of the form $z_{s,t}\in {\phi }_{s,t}+{\int }_0^s{\int }_0^t{F}_{u,v}\left(z_{u,v}\right)dudv+{\int }_0^s{\int }_0^t{G}_{u,v}\left(z_{u,v}\right)d{w}_{u,v}$

A generalization of the Poisson driven stochastic differential equation is considered. A sufficient condition for asymptotic stability of a discrete time-nonhomogeneous Markov process is proved.

We consider non-degenerate SDEs with a β-Hölder continuous and bounded drift term and driven by a Lévy noise L which is of α-stable type. If β > 1 - α/2 and α ∈ [1,2), we show pathwise uniqueness and existence of a stochastic flow. We follow the approach of [Priola, Osaka J. Math. 2012] improving the assumptions on the noise L. In our previous paper L was assumed to be non-degenerate, α-stable and symmetric. Here we can also recover relativistic and truncated stable processes and some classes...

By using successive approximation, we prove existence and uniqueness result for a class of neutral functional stochastic differential equations in Hilbert spaces with non-Lipschitzian coefficients

This paper proves a version for stochastic differential equations of the Lie–Scheffers theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie–Scheffers result. We show that the stochastic analog...

This paper concerns the synchronization of time-delayed systems with periodic on-off coupling. Based on the stability theory and the comparison theorem of time-delayed differential equations, sufficient conditions for complete synchronization of systems with constant delay and time-varying delay are established. Compared with the results based on the Krasovskii-Lyapunov method, the sufficient conditions established in this paper are less restrictive. The theoretical results show that two time-delayed...

The work deals with non-Markov processes and the construction of systems of differential equations with delay that describe the probability vectors of such processes. The generating stochastic operator and properties of stochastic operators are used to construct systems that define non-Markov processes.