Ningxia University

We study the global dynamics of a two-patch epidemic model that integrates spatial migration, Erlang-distributed delays, and environmental stochasticity. The two patches are coupled by migration whose intensities are modulated by an Ornstein-Uhlenbeck (OU) process, capturing stochasticity but mean-reverting fluctuations in population movement, while the infection progression is described by a distributed delay structure represented through a linear-chain formulation. For the deterministic system, we establish threshold-type results that characterize disease extinction and persistence: the disease-free equilibrium is globally asymptotically stable when the basic reproduction number __-mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"-__ R 0 < 1 , whereas infection persists when __-mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"-__ R 0 > 1 . For the corresponding stochastic model, we derive explicit sufficient conditions for almost sure exponential extinction by constructing Lyapunov functions and exploiting the Metzler structure of the infected subsystem. Moreover, we prove that the stochastic system admits a stationary distribution when stochastic threshold __-mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"-__ R 0 s > 1 , which serves as a stochastic analogue of the endemic equilibrium and describes persistent random fluctuations of infection levels. Numerical simulations are provided to validate the theoretical findings and to quantify how distributed delays and OU-driven migration randomness shape long-term infection burden and stationary prevalence across patches. Notably, adjusting the noise intensity and the mean-reversion rate can redistribute the long-term infection burden between patches, which highlights how migration-driven randomness fundamentally reshapes spatial epidemic patterns.

The significantly shortened shelf life of reduced-salt frankfurters necessitates the development of effective, non-chemical preservatives. This study aimed to optimise the ratio of nisin, ε-polylysine (ε-PL), and chitosan (CTS) and elucidate its effects on the bacterial community, physicochemical properties, and shelf life of reduced-salt frankfurters during refrigeration. The results indicated that the optimal composition of the ternary biological preservative (TBP) was 0.16, 0.18, and 0.47 g/kg of nisin, ε-PL, and CTS, respectively. The TBP significantly reduced bacterial diversity and inhibited spoilage bacteria such as Brochothrix thermosphacta, decreasing its relative abundance from 40.34% in the control group to 23.02% by day 70, while regulating the overall bacterial community composition. Moreover, the TBP-enhanced reduced-salt frankfurters maintained superior moisture and colour characteristics during storage at 4 °C, minimised pH fluctuations, decelerated fat oxidation and protein degradation, and ultimately delayed texture loss and flavour deterioration. A validated predictive model confirmed a 102-day shelf life for the TBP-enhanced reduced-salt frankfurters, surpassing that of preservative-free samples by 54.55% and that of samples with commercially available preservatives (CAP) by 4.08%. Therefore, our findings demonstrate that the TBP represents an efficient biological preservative strategy, offering a robust solution for developing non-chemical, high-quality, shelf-stable reduced-salt emulsified meat products.

Application of combined indicators in quantitative analysis is relatively uncommon. This study introduced the entropy weighting method (EWM) combined with fluorescence hyperspectral imaging (FHSI) to develop a polycyclic aromatic hydrocarbon (PAHs) composite index (PCI). Using generalized and heterogeneous two-dimensional correlation spectroscopy, characteristic fluorescence peaks of trace components were resolved and the sequence of spectral changes was tracked. With PCI as the external perturbation condition, using PCI as the outer winding condition, the fluorescence intensity of PAHs varied sequentially from 465 nm to 442 nm, 433 nm, and finally 420 nm, suggesting that the energy level transition of pyrene occurs prior to that of fluoranthene during the roasting process. The CARS-PLSR model achieved an R²_C of 0.9472 and R²_P of 0.9350, with RMSEC and RMSEP values of 1.4998 μg/kg and 2.0710 μg/kg, respectively. Model performance was satisfactory, and other modeling approaches also yielded R² values above 0.7. Furthermore, spatial visualization of PCI in roasted lamb was achieved, confirming the accuracy and reliability of the proposed method for PAHs content prediction.