This paper investigates a stochastic SIRS epidemic model with an incidence rate that is sufficiently general and that covers many incidence rate models considered to date in the literature. We classify the extinction and permanence by introducing $\lambda$, a real-valued threshold. We show that if $\lambda<0$, then the disease will eventually disappear (i.e., the disease-free state is globally asymptotically stable); if the threshold value $\lambda>0$, the epidemic becomes strongly stochastically permanent. This result substantially generalizes and improves the related results in the literature. Moreover, the mathematical development in this paper is interesting in its own right. The essential difficulties lie in that the dynamics of the susceptible class depend explicitly on the removed class resulting in a three-dimensional system rather than a two-dimensional system. Consequently, the methodologies developed in the literature are not applicable here. One of the main ingredients in the analyses is this: Though it is not possible to compare solutions in the interior and on the boundary for all $t\in[0,\infty)$, approximation in a long but finite interval $[0,T]$ can be carried out. Then, using the ergodicity of the solution on the boundary and exploiting the mutual interplay between the distance of solutions in the interior and solutions on the boundary and the exponential decay or growth (depending on the sign of the Lyapunov exponent), one can classify the behavior of the system. The convergence to the invariant measure is established under the total variation norm together with the corresponding rate of convergence. To demonstrate, some numerical examples are provided to illustrate our results.
This work takes up the challenges of utility maximization problem when the market is indivisible and the transaction costs are included. First there is a so-called solvency region given by the minimum margin requirement in the problem formulation. Then the associated utility maximization is formulated as an optimal switching problem. The diffusion turns out to be degenerate and the boundary of domain is an unbounded set. One no longer has the continuity of the value function without posing further conditions due to the degeneracy and the dependence of the random terminal time on the initial data. This paper provides sufficient conditions under which the continuity of the value function is obtained. The essence of our approach is to find a sequence of continuous functions locally uniformly converging to the desired value function. Thanks to continuity, the value function can be characterized by using the notion of viscosity solution of certain quasi-variational inequality.
This paper considers multidimensional jump type stochastic differential equations with super linear growth and non-Lipschitz coefficients. After establishing a sufficient condition for nonexplosion, this paper presents sufficient non-Lipschitz conditions for pathwise uniqueness. The non confluence property for solutions is investigated. Feller and strong Feller properties under non-Lipschitz conditions are investigated via the coupling method. Sufficient conditions for irreducibility and exponential ergodicity are derived. As applications, this paper also studies multidimensional stochastic differential equations driven by Levy processes and presents a Feynman-Kac formula for Levy type operators.
As one of the most innovative cement-based materials, ultra-high performance concrete (UHPC), with excellent durability and mechanical properties, has been widely used in strengthening existing bridges. In this study, in-situ four-point bending tests were carried out to investigate the flexural behavior of precast reinforced concrete (RC) hollow slab beams in service for 15 years strengthened with UHPC. Among them, three hollow slab beams were strengthened with UHPC, and the interface treatment was chiseling, planting rebars, and a combination of chiseling and planting rebars, respectively. The remaining one without any strengthening treatment was used as the control specimen. To evaluate the enhancement effect of different interface treatments on UHPC-strengthened beams, the cracking load, ultimate load, crack development and failure modes of UHPC-strengthened beams were analyzed. Results indicated that the stiffness, deflection capacity and flexural capacity of UHPC-strengthened beams was significantly improved. Meanwhile, the stiffness of UHPC-strengthened beams in the pre-damage stage was increased by 49%–94%, when compared with the unstrengthened beam. Correspondingly, the ultimate flexural capacity was increased by 29%–38%. The interface chiseling treatment was more favorable to enhance the deformation capacity of UHPC-strengthened beams. The interface planting rebar treatment was more favorable to enhancing the ductility of UHPC-strengthened beams. The crack development was effectively suppressed by the interface chiseling and planting rebars together. This contributes to a higher load capacity reserve for UHPC-strengthened beams. The bearing capacity under serviceability limit state of the UHPC-strengthened beams was increased by 1.25, 2, and 2.5 times through the interface treatments of chiseling, planting rebars, and a combination of both, respectively.
'/%3e%3cuse xlink:href='%23a' transform='rotate(23.036 .093 25.536)'/%3e%3cuse xlink:href='%23a' transform='rotate(45.87 1.273 16.18)'/%3e%3cuse xlink:href='%23a' transform='rotate(69.945 .996 12.078)'/%3e%3cuse xlink:href='%23a' transform='rotate(20.66 -19.689 31.932)'/%3e%3c/svg%3e)