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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.

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Hopf bifurcation in delay differential equations has been a central topic in the study of complex dynamical behaviors in biological and ecological systems. In this review, we revisit Hopf bifurcation phenomena in single-species models that incorporate time delays, emphasizing recent progress in both ordinary and partial differential equation frameworks. We present a comprehensive overview of classic and contemporary models, such as Wright’s equation, Nicholson’s blowflies equation, and diffusive logistic models, highlighting criteria for local and global bifurcations, the geometric and analytical methods used to determine critical values, and the stability of emerging periodic solutions. The review also covers structured models with age, stage, advection, and spatial effects, as well as equations with multiple delays. Through this survey, we aim to consolidate theoretical insights and provide a unified understanding of delay-induced oscillations in population models, laying the groundwork for future developments in delay-driven dynamics.

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A mathematical model is proposed that describes the adaptive spatial movement of prey towards higher population density to reduce predation risk. The model admits the increased nonlinearity and the global existence of solutions of the system is established in Sobolev space through analytical estimates. The conditions for the Turing instability from a coexistence steady state are obtained, and sharp conditions for the asymptotical stability of the positive equilibrium in a large region are established with the help of a Lyapunov function. Numerical simulations are presented to support the theoretical results and demonstrate the versatility of spatial models.

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We develop a stochastic human immunodeficiency virus type 1 (HIV-1) infection model to analyze combination antiretroviral therapy (cART) dynamics in the brain microenvironment, explicitly accounting for two infected cell states: (1) productively infected and (2) latently infected populations. The model introduces two key epidemiological thresholds $–\overline{\mathcal{R}}_{c1}$ (productive infection) and $\overline{\mathcal{R}}_{c2}$ (latent infection) – and defines the stochastic control reproduction number as $\overline{\mathcal{R}}_c={\rm max} \{\overline{\mathcal{R}}_{c1},\overline{\mathcal{R}}_{c2}\}.$ Our analysis reveals three distinct dynamical regimes: (1) viral extinction $(\overline{\mathcal{R}}_c <1):$ the infection clears exponentially with probability one; (2) latent reservoir dominance $(\overline{\mathcal{R}}_c=\overline{\mathcal{R}}_{c2}>1):$ the system almost surely converges to a purely latent state, characterizing stable viral reservoir formation; (3) persistent productive infection $(\overline{\mathcal{R}}_c =\overline{\mathcal{R}}_{c1} >1):$ the infection persists indefinitely with a unique stationary distribution, for which we derive 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 between productive infection, latency establishment, and cART efficacy, offering mechanistic insights into viral persistence in the brain.

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Personal diverse interests and many group interactions naturally create partially overlapping groups within a population. Those who belong to multiple groups play a crucial role in spreading of infectious diseases across the whole population. We develop an algorithm to decompose the microscopic overlap structure of groups with representing a population of partially overlapping groups as a hypergraph of partially overlapping hyperedges, and characterize it using a newly defined overlap matrix. We formulate a specific multi-group SIR epidemic model, and address a one-time preventive vaccine allocation problem aimed at effectively reducing the basic reproduction number. By leveraging perturbation theory, we derive a principled ranking index to measure the vaccination priority of different groups, and establish a ranking vaccination strategy, which usually outperforms random vaccination strategies as verified by a series of numerical examples. These results offer a theoretical foundation for public health decision-making to develop effective vaccination allocation plans.

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Multi-scale models improve the prediction of key epidemic factors and disease prevalence by capturing the interaction between individual immune responses and the spread of diseases among the population, helping to implement targeted control strategies for disease management. In this paper, we develop a novel size-structured 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 rigorously prove the conditions under which the feasible equilibrium exists and is stable. Then, by evaluating the effectiveness of four measures consisting of individual antiviral 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.

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During chronic viral infection, sustained antigen stimulation leads to exhaustion of virus-specific CD$8^+$ T cells, characterized by elevated expression of inhibitory receptors and progressive functional impairment, including loss of cytokine production, reduced cytotoxicity, and diminished proliferative capacity. In this paper, to investigate how T cell exhaustion influences viral persistence, we developed a within-host mathematical model integrating viral infection dynamics with adaptive immune responses. The model demonstrates three non-trivial equilibria: infection-free equilibrium $(S_1),$ uncontrolled-infection state $(S_2),$ and immune-controlled equilibrium $(S_3).$ Through dynamical systems analysis, we established the local stability of all states $(S_1-S_3)$ and prove global stability for both $S_1$ (complete viral clearance) and $S_2$ (chronic infection). Notably, the system exhibits Hopf bifurcations at $S_2$ and $S_3,$ with distinct critical thresholds governing oscillatory dynamics. Numerical simulations reveal that successful immune-mediated control of viral load and infected cell levels requires maintenance of low CD$8^+$ T cell exhaustion rates.

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Changes in personal protective behaviors driven by media influence contribute to epidemic prevention and control, whereas limited medical resources constrain the effectiveness of such interventions. In this study, we propose a novel influenza transmission model with the media impact and limited medical resources. Theoretical and numerical analyses reveal some complex nonlinear dynamics, including saddle node bifurcations, forward and backward bifurcations, and both subcritical and supercritical Hopf bifurcations. Besides, two types of bistable scenarios are identified: bistability of a disease-free equilibrium and an endemic equilibrium, and bistability 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 estimated basic reproduction number $\mathcal{R}_0 = 1.2183 > 1$ implies sustained transmission. Finally, sensitivity analysis suggests that effectively controlling influenza transmission can be achieved by decreasing the population input rate, transmission rate, awareness loss rate, and medical saturation constant, as well as increasing the awareness transmission rate, media response intensity, and maximum recovery rate. These findings highlight the critical role of combining public awareness initiatives with enhanced medical resource allocation to strengthen influenza prevention and control efforts.

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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 pressure, can significantly impact the health of coral reefs, potentially leading to ecosystem 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 existence and stability of equilibria are rigorously demonstrated, and rich dynamics such as bistability and various types of periodic oscillations are numerically explored. Furthermore, the effects of environmental factors (e.g., light intensity, nutrient levels, and fishing pressure) on the system’s dynamics are investigated. The main findings highlight 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.

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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.

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Under the assumption of the autopilot model, after antigen stimulation exceeds a threshold, the proliferation and effector function of CD4+ T cells are self-sustained and do not need further antigen stimulation. However, CD4+ T cell proliferation is driven by their production of IL-2, which then binds to cells and triggers proliferation. Without regulation, this autocrine process forms a positive feedback loop that causes uncontrolled proliferation. This study mathematically modeled the regulatory mechanisms of the CD4+ T cell response after infection, focusing on the role of IL-2 self-regulation and Treg in this mechanism. We performed a phase-space analysis to study the long-term behavior of the proliferation process. Our results show that IL-2 self-regulation alone is not sufficient to fully inhibit CD4+ T cell response, and that the involvement of Treg cells is essential to regulate the immune response effectively. In particular, when the rate of CD4+ T cell proliferation is controlled by the rate of IL-2-mediated CD4+ T cell removal, Treg cells control CD4+ T cell proliferation by releasing immunosuppressive cytokines such as IL-10 and TGF-$β,$ thus inhibiting the unregulated immune response.

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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 $\mathcal{R}_0<1,$ the brucellosis always dies out; if $\mathcal{R}_0>1,$ 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 $(\mathcal{R}_0 = 1.544)$ 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.

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Dispersal strategies that lead to the ideal free distribution (IFD) were shown to be evolutionarily stable in various ecological models. In this paper, we investigate this phenomenon in time-periodic environments where $N$ species – identical except for dispersal strategies – compete. We extend the notions of IFD and joint IFD, previously established in spatially continuous models, to time-periodic and spatially discrete models and derive sufficient and necessary conditions for IFD to be feasible. Under these conditions, we demonstrate two competitive advantages of ideal free dispersal: if there exists a subset of species that can achieve a joint IFD, then the persisting collection of species must converge to a joint IFD for large time; if a unique subcollection of species achieves a joint IFD, then that group will dominate and competitively exclude all the other species. Furthermore, we show that ideal free dispersal strategies are the only evolutionarily stable strategies. Our results generalize previous work by construction of Lyapunov functions in multi-species, time-periodic setting.

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In this paper, we propose an age-structured epidemic model with strain mutation and age-based vaccination. We define the reproduction numbers of both the original and mutant strains ($R^1_0$ and $R^2_0).$ If the reproduction number $R_0 < 1,$ the disease-free steady state is locally asymptotically stable. If the reproduction number $R^2_0 >1,$ there exists a dominant steady state of the mutant strain. Conditions for local stability of this dominant steady state are also obtained. If both reproduction numbers $R^1_0$ and $R^2_0$ are greater than 1, a coexistence steady state may occur. Finally, the uniform persistence of the disease described by our age structured model is strictly proved when the reproduction number $R^1_0 > 1.$ By using the data of the COVID-19 epidemic in Wuhan and the theoretical results obtained in this paper, some numerical calculations are carried out to prove the effect of the age-based vaccination strategy.

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