In this paper, a stochastic model (with random noise transmission) is designed. The model possesses substantial potential to describe the dynamical behavior of the Hepatitis B (HBV) virus and it’s control by applying the strategy of vaccinating an offspring. The number of basic reproductive is calculated and proved that the system holds some sharp threshold properties. It is investigated that the model has a bounded, unique and positive solution subject to initial positive data. Furthermore, the stability of the investigated system has been presented by using stochastic Lyapunov functional theory. Stationary distribution and extinction of the infection are examined by providing sufficient conditions. To control the spread of the disease through some external measures, we used optimal control theory and analyzed stochastic as well as deterministic control problems. For further verification of the obtained analytical results, additional graphical solutions have been presented for the ease of understanding. This study may provide a strong theoretical basis for understanding worldwide chronic infectious diseases.

By introducing the function 1/(min{x, y}), we establish several new inequalities similar to Hilbert's type inequality. Moreover, some further unification of Hardy-Hilbert's and Hardy-Hilbert's type integral inequality and its equivalent form with the best constant factor are proved, which contain the classic Hilbert's inequality as special case.

In this paper, we study the existence and uniqueness of solution (EUS) as well as Hyers-Ulam stability for a coupled system of FDEs in Caputo’s sense with nonlinear p-Laplacian operator. For this purpose, the suggested coupled system is transferred to an integral system with the help of four Green functions $\mathcal{G}^{\alpha_{1}}(t,s)$ , $\mathcal{G}^{\beta_{1}}(t,s)$ , $\mathcal{G}^{\alpha_{2}}(t,s)$ , $\mathcal{G}^{\beta_{2}}(t,s)$ . Then using topological degree theory and Leray-Schauder’s-type fixed point theorem, existence and uniqueness results are proved. An illustrative and expressive example is given as an application of the results.