Ergodic stationary distribution and practical application of a hybrid stochastic cholera transmission model with waning vaccine-induced immunity under nonlinear regime switching

Considering the effect of stochasticity including white noise and colored noise, this paper aims to study a hybrid stochastic cholera epidemic model with waning vaccine-induced immunity and nonlinear telegraph perturbations. First, we derive a critical value ${?}_0^C$ related to the basic reproduction number ${?}_0$ of the deterministic model. The key aim of this paper is to generalize the θ-stochastic criterion method proposed by the recent work (Han et al. in Chaos Solit Fract 140:110238, 2020) to eliminate nonlinear telegraph perturbations. Next, via constructing several θ-stochastic Lyapunov functions and using the generalized method, we further prove that the stochastic model have a unique ergodic stationary distribution under ${?}_0^C>1$. Results show that the prevention and control of cholera epidemic depend on low transmission rate and small telegraph perturbations. Finally, the corresponding numerical simulations are performed to illustrate our analytical results and a practical application on the Somalia cholera outbreak is shown at the end of this paper.     

Vanadium(V) Oxo and Imido Calix[8]arene Complexes: Synthesis,Structural Studies,and Ethylene Homo/Copolymerisation Capability

Interaction of p‐tert‐butylcalix[8]areneH₈ (L⁸H₈) with [NaVO(OtBu)₄] (formed in situ from VOCl₃) afforded the complex [Na(NCMe)₅][(VO)₂L⁸H]?4 MeCN ( 1 ?4 MeCN). Increasing [NaVO(OtBu)₄] to 4 equiv led to [Na(NCMe)₆]₂[(Na(VO)₄L⁸)(Na(NCMe))₃]₂?10 MeCN ( 2 ?10 MeCN). With adventitious oxygen, reaction of 4 equiv of [VO(OtBu)₃] with L⁸H₈ afforded the alkali‐metal‐free complex [(VO)₄L⁸(μ³‐O)₂] ( 3 ); solvates 3 ?3 MeCN and 3 ?3 CH₂Cl₂ were isolated. For the lithium analogue, the order of addition had to be reversed such that lithium tert‐butoxide was added to L⁸H₈ and then treated with 2 equiv of VOCl₃; crystallisation afforded [(VO₂)₂Li₆[L⁸](thf)₂(OtBu)₂(Et₂O)₂]?Et₂O ( 4 ?Et₂O). Upon extraction into acetonitrile, [Li(NCMe)₄][(VO)₂L⁸H]?8 MeCN ( 5 ?8 MeCN) was formed. Use of the imido precursors [V(NtBu)(OtBu)₃] and [V(Np‐tolyl)(OtBu)₃] and L⁸H₈, afforded [tBuNH₃][{V(p‐tolylN)}₂L⁸H]?3 1/2 MeCN ( 6 ?3 1/2 MeCN). The molecular structures of 1 to 6 are reported. Complexes 1 , 3 , and 4 were screened as precatalysts for the polymerisation of ethylene in the presence of cocatalysts at various temperatures and for the copolymerisation of ethylene with propylene. Activities as high as 136 000 g (mmol(V) h)^?1 were sometimes achieved; higher molecular weight polymers could be obtained versus the benchmark [VO(OEt)Cl₂]. For copolymerisation, incorporation of propylene was 7.1–10.9 mol % (compare 10 mol % for [VO(OEt)Cl₂]), although catalytic activities were lower than [VO(OEt)Cl₂].     

TBAI‐Catalyzed/Water‐Assisted Double C−S Bond Formations: An Efficient Approach to Sulfides through Metal‐Free Three‐Component Reactions

An aqueous catalytic method for double C?S bond formations that involves alcohol derivatives, organic halides, and sodium thiosulfate has been developed. A diverse range of functionalized sulfides, including pharmaceutical and biological derivatives, can be obtained in an efficient and eco‐friendly manner under air. The mechanistic studies revealed that this tetrabutylammonium‐iodide‐catalyzed/water‐assisted reaction generated a mercaptan species as the key intermediate.     

Cubic C8: An Observable Allotrope of Carbon?

Ab initio and DFT calculations are used to investigate the structure, electronic properties, spectra and reactivity of cubic C₈, which is predicted to be aromatic according to Hirsch′s rule. Although highly strained and with a small amount of diradical character, the carbon cube represents a surprisingly deep minimum and should therefore be observable as an isolated molecule. It is, however, predicted to be very reactive, both with itself and triplet oxygen. Calculated IR, Raman, and UV/Vis spectra are provided to aid identification of cubic C₈ should it be synthesized.     

CH Oxidation of Ingenanes Enables Potent and Selective Protein Kinase C Isoform Activation

Ingenol derivatives with varying degrees of oxidation were prepared by two‐phase terpene synthesis. This strategy has allowed access to analogues that cannot be prepared by semisynthesis from natural ingenol. Complex ingenanes resulting from divergent C? H oxidation of a common intermediate were found to interact with protein kinase C in a manner that correlates well with the oxidation state of the ingenane core. Even though previous work on ingenanes has suggested a strong correlation between potential to activate PKCδ and induction of neutrophil oxidative burst, the current study shows that the potential to activate PKCβII is of key importance while interaction with PKCδ is dispensable. Thus, key modifications of the ingenane core allowed PKC isoform selectivity wherein PKCδ‐driven activation of keratinocytes is strongly reduced or even absent while PKCβII‐driven activation of neutrophils is retained.