Considering the time delay originating from a certain incubation period or asymptomatic state, we propose a delayed epidemic system within the noisy environment of the hepatitis B virus to analyze the mechanism of disease transmission and elucidate how to control it by applying the strategy of vaccinating and treatment. Applying stochastic Lyapunov functional theory, we first construct an integral Lyapunov function coupling the time delay and stochastic fluctuation to investigate whether there exists a unique global solution to the model. Next, we yield the threshold condition for controlling disease extinction, and persistence, as well as its stationary distribution. Governed by these sufficient conditions, we study the existence of optimal control solutions in deterministic and stochastic scenarios to uncover how to accelerate disease extinction through vaccination and treatment. The results indicate that the time delay will prolong the duration of the disease for the original system but suppress the peak value of HBV in the controlled system. Finally, we verify the versatility of theoretical results by numerical simulations. These results will effectively decipher the importance of the time delay in the control of hepatitis B.

We describe finite soluble nonnilpotent groups in which every minimal nonnilpotent subgroup is abnormal. We also show that if G is a nonsoluble finite group in which every minimal nonnilpotent subgroup is abnormal, then G is quasisimple and is cyclic of order .

In this paper, G is a finite group and σ a partition of the set of all primes ℙ, that is, σ = {σi | i ∈ I}, where ℙ = ∪i∈I σi and σi ∩ σj = Ø for all i ≠ j; σ(G) = {σi | σi ∩ π(G) ≠ Ø}.We say that ...