The dynamic behavior of alien mussels interacting with algae after arriving in a new environment has long been a focus of invasion ecology research. This paper extends and analyzes a classical mussel-algae model by incorporating a time delay in mussel filter feeding and accounting for environmental variability. We theoretically study the stochastic dynamics, including the global existence and uniqueness of the positive solution, the existence of a unique stationary distribution, and mussel extinction, using tools from stochastic analysis. Furthermore, we derive an explicit expression for the probability density function around the quasi-stable equilibrium by solving the corresponding Fokker-Planck equation. Our theoretical and numerical results indicate that: (a) larger environmental disturbances or artificial removal can effectively prevent the survival of alien mussels in novel habitats, (b) a decreased filter feeding rate leads to an accelerated extinction rate of mussels, and (c) an increased consumption constant c decelerates the transition rate of mussels from the initial state to the extinction state, as analyzed through the mean first passage time of mussels. These findings highlight the complex interaction between intrinsic and extrinsic factors in influencing the invasion dynamics of alien mussels.

Multi-scale models improve the prediction of key epidemic factors and dis- ease prevalence by capturing the interaction between individual immune responses and the spread of diseases among the population, helping to implement targeted con- trol strategies for disease management. In this paper, we develop a novel size-structu- red influenza model based on the nested rules, aiming to explore how the replication of influenza virus affects its transmission at a population level. We calculate the basic reproduction numbers separately at the individual and population levels and rigor- ously prove the conditions under which the feasible equilibrium exists and is stable. Then, by evaluating the effectiveness of four measures consisting of individual antivi- ral treatment and population vaccination, we can determine an optimal treatment to minimize both the influenza cases and the total expenditure on influenza prevention. Numerical results reveal the complex interactions between the two interventions and the progression of the epidemic.

We develop a stochastic human immunodeficiency virus type 1 (HIV-1) in- fection model to analyze combination antiretroviral therapy (cART) dynamics in the brain microenvironment, explicitly accounting for two infected cell states: (1) produc- tively infected and (2) latently infected populations. The model introduces two key epidemiological thresholds $-\overline{\mathcal{R}}_{c 1}$ (productive infection) and $-\overline{\mathcal{R}}_{c 2}$ (latent infection) – and defines the stochastic control reproduction number as $\overline{\mathcal{R}}_{c}=\max \left\{\overline{\mathcal{R}}_{c 1}, \overline{\mathcal{R}}_{c 2}\right\}$. Our analysis reveals three distinct dynamical regimes: (1) viral extinction $\left(\overline{\mathcal{R}}_{c}<1\right)$: the in- fection clears exponentially with probability one; (2) latent reservoir dominance $\left(\overline{\mathcal{R}}_{c}=\overline{\mathcal{R}}_{c 2}>1\right)$: the system almost surely converges to a purely latent state, characterizing stable viral reservoir formation; (3) persistent productive infection $\left(\overline{\mathcal{R}}_{c}=\overline{\mathcal{R}}_{c 1}>1\right)$: the infection persists indefinitely with a unique stationary distribution, for which we de- rive the exact probability density function. And numerical simulations validate these theoretical predictions, demonstrating how environmental noise critically modulates HIV-1 dynamics in neural reservoirs. Our results quantify the stochastic balance be- tween productive infection, latency establishment, and cART efficacy, offering mecha- nistic insights into viral persistence in the brain.

Coral reefs have evolved over hundreds of millions of years and are now considered one of the most critical yet vulnerable ecosystem on earth. Environmental changes, including variations in light availability, nutrient levels, and fishing pres- sure, can significantly impact the health of coral reefs, potentially leading to ecosys- tem degradation. A coral-macroalgae-herbivorous fish model is developed based on ecological stoichiometry to investigate the dynamics of coral-algae phase shifts. The positivity, invariance, and dissipativity of the model are carefully established, the exis- tence and stability of equilibria are rigorously demonstrated, and rich dynamics such as bistability and various types of periodic oscillations are numerically explored. Fur- thermore, the effects of environmental factors (e.g., light intensity, nutrient levels, and fishing pressure) on the system’s dynamics are investigated. The main findings high- light that environmental variations are key drivers of ecological phase transitions in coral reef ecosystems, providing more insights into the mechanisms underlying coral- algae dynamics and offering implications for the sustainable management of coral reefs.

Changes in personal protective behaviors driven by media influence con- tribute to epidemic prevention and control, whereas limited medical resources con- strain the effectiveness of such interventions. In this study, we propose a novel in- fluenza transmission model with the media impact and limited medical resources. Theoretical and numerical analyses reveal some complex nonlinear dynamics, includ- ing saddle node bifurcations, forward and backward bifurcations, and both subcritical and supercritical Hopf bifurcations. Besides, two types of bistable scenarios are iden- tified: bistability of a disease-free equilibrium and an endemic equilibrium, and bista- bility of two different endemic equilibria. In addition, we fit the model to the monthly new influenza reported case data from August 2023 to October 2024 in Jiangsu, China, where the fitting results successfully capture the observed epidemic trends. The es- timated basic reproduction number R₀ = 1.2183 > 1 implies sustained transmission. Finally, sensitivity analysis suggests that effectively controlling influenza transmission can be achieved by decreasing the population input rate, transmission rate, aware- ness loss rate, and medical saturation constant, as well as increasing the awareness transmission rate, media response intensity, and maximum recovery rate. These find- ings highlight the critical role of combining public awareness initiatives with enhanced medical resource allocation to strengthen influenza prevention and control efforts.

We consider a two-patch Susceptible-Infected-Recovered epidemic model that incorporates awareness-driven behavioural changes in both contact and mobility patterns. Individuals modify their behaviour in response to a perceived risk of infection, which is modelled through two awareness variables that depend either on the current or past disease prevalence in each patch. We qualitatively analyse the model through stability and bifurcation theory and derive threshold conditions that determine the existence and stability of the biologically relevant equilibria. We find that awareness-induced behavioural changes in contact and mobility can destabilise mixed equilibria - where the disease persists in one patch only - and contribute to the emergence of stable co-endemic states. When awareness depends on past epidemic values, the stability analysis shows that mixed equilibria may lose stability via Hopf bifurcations, depending on the sign of some awareness-related parameters. Finally, the impact of the behaviour-related parameters on the epidemic dynamics is investigated through numerical simulations.

Brucellosis is a significant zoonotic disease that has a high incidence rate in sheep, particularly in the Inner Mongolia region of China. To better investigate its transmission dynamics, this paper proposes a sheep brucellosis model incorporating acute and chronic infections, a saturated incidence rate describing environmental transmission and a time delay representing the incubation period. By constructing Lyapunov functionals and using LaSalle’s invariance principle, it is shown that the global dynamics of the disease is completely determined by the basic reproduction number: If R₀<1, the brucellosis always dies out; if R₀>1, a unique endemic equilibrium exists and is globally asymptotically stable. Using data of sheep brucellosis cases in Inner Mongolia from 2016 to 2020, unknown parameters and the basic reproduction number (R₀=1.544) are estimated via the Markov chain Monte Carlo method. Numerical simulations show that reducing the transmission rate of acutely infected sheep while increasing the culling rate of symptomatic infected sheep is the most effective strategy to control the spread of brucellosis in Inner Mongolia. Notably, decreasing the proportion of acutely infected sheep or increasing acute-to-chronic conversion rate may lead to a significant short-term increase in chronically infected sheep.