This paper analyzes a quasilinear elliptic system modeling a general Gause-type cross-diffusion prey–predator model in a spatially heterogeneous environment. Treating [Formula: see text] as a bifurcation parameter, we establish the global bifurcation structure of positive solutions emanating from the semi-trivial solution branch, utilizing bifurcation theory coupled with the method of upper and lower solutions. We prove that when the prey birth rate [Formula: see text] is relatively high (i.e. [Formula: see text]) and [Formula: see text] lies within a moderate interval (i.e. between [Formula: see text] and [Formula: see text]), coexistence of prey and predator occurs, that is, the system admits positive solutions. Furthermore, we derive explicit conditions determining the direction of this bifurcation. Crucially, we demonstrate that the spatial distribution of prey resources, [Formula: see text], significantly influences the existence of positive solutions for the system. Our results obtained in this paper can be applied to prey–predator models with Holling-type II functional response and non-monotonic functional response.

This paper develops a dynamic model of two-stage recovery on scale-free networks to analyze the infection pathways of Human Papillomavirus (HPV) leading to cervical cancer, incorporating heterogeneous contact patterns. The framework stratifies the demographic cohort across four distinct compartments: susceptible [Formula: see text], HPV-infected [Formula: see text], precancerous lesions [Formula: see text] and cervical cancer [Formula: see text], with degree-based differential equations governing transmission dynamics. Theoretical analysis demonstrates that the basic reproduction number [Formula: see text] serves as a critical threshold: when [Formula: see text], the disease-free equilibrium is globally asymptotically stable, ensuring disease eradication; when [Formula: see text], the dynamical system asymptotically approaches a globally stable endemic equilibrium, with proven permanence. Numerical simulations also validate the theoretical results. Research indicates that under scale-free network structures, priority should be given to implement targeted screening high-contact groups, leveraging their central role in transmission dynamics to block HPV diffusion. The results of this study emphasize the critical role of early screening, targeted interventions for high-contact groups, and optimized treatment protocols to mitigate HPV-related cervical cancer.

Based on the effectiveness of vaccine inoculation and hospital treatment in controlling infectious diseases, and accounting for the complexity of epidemic transmission mechanisms and environmental uncertainties in population dynamics, in this article, we investigate the dynamics of a stochastic SAIR model with vaccination and saturated treatment, where the transmission rate is governed by the logarithmic Ornstein-Uhlenbeck (O-U) process. First, we demonstrate that for any given initial value, there exists a global positive solution. Next, the stationary distribution of the system is obtained through developing complex Lyapunov functions and applying the ergodicity theory of the O-U process. Then, to uncover the statistical properties, the precise expression of the probability density function close to the quasi-equilibrium is given by applying the Routh-Hurwitz criterion and solving the four-dimensional Fokker-Planck equation. We also define sufficient criteria for disease extinction at the local level. Lastly, computer simulations verify that the O-U process has a substantial effect on the dynamics of the model. Notably, our results demonstrate that higher mean-reversion rates coupled with lower volatility intensities substantially mitigate infection outbreaks, potentially informing future therapeutic strategies.

This paper investigates the emergence of time-periodic solutions and spatio-temporal patterns in a reaction–diffusion–ODE system, inspired by biological models such as FitzHugh–Nagumo for neuronal activity. We focus on a variant where the activator lacks diffusion, a feature known to support discontinuous patterns. By employing multiple time scale analysis, we rigorously establish the existence of relaxation oscillations, which are fundamental to rhythmic phenomena in biology, such as neuronal spiking. For distinct parameter regimes, we also derive regular periodic solutions via Hopf bifurcation theory. By extending the analysis to the spatial domain, we characterize the Hopf bifurcation that gives rise to both spatially homogeneous and inhomogeneous periodic solutions, elucidating the mechanism behind spatio-temporal pattern formation. Our theoretical findings are corroborated by numerical simulations, which reveal a rich variety of dynamics, including discontinuous stationary states and quasi-periodic patterns. This work provides a mathematical framework for understanding how coupling between non-diffusive and diffusive components can generate complex rhythms and patterns relevant to electrophysiology and ecology.

Multiple Myeloma (MM) is a hematologic malignancy characterized by abnormal plasma cell proliferation and bone destruction. Increasing evidence underscores the immune system’s critical role in MM initiation, therapy and recurrence. Mathematical modeling offers a quantitative framework for analyzing such complex interactions. Existing MM models have provided insights into tumor dynamics but largely overlooked immune regulation. To address this gap, we develop a novel model explicitly incorporating Natural Killer (NK) cells, Cytotoxic T Lymphocytes (CTLs), Regulatory T cells (Tregs) and Cytokines Interleukin-2 (IL-2) and Interleukin-10 (IL-10). The model first integrates two immune evasion mechanisms, antigen escape and an immunosuppressive microenvironment, alongside competition among MM subtypes for limited resources. It also simulates Chimeric Antigen Receptor-T cell (CAR-T) therapy to examine how immunosuppression influences outcomes. Simulations reproduce immune–tumor interactions in the bone microenvironment, reveal that antigen loss exerts a stronger effect on immune escape than immunosuppression alone, and show that increased immune pressure accelerates evolution toward resistant subtypes. Enhancing IL-2 and IL-10 critically modulates CAR-T cell abundance and efficacy. Model predictions agree with experimental data, improving invasion kinetics accuracy over models without immune components. This framework offers a robust basis for studying MM-immune dynamics and advancing immunotherapy modeling.

This paper investigates the synchronization and control of memristive neural networks (MNNs) with time-varying and distributed delays. Firstly, the existence of a unique equilibrium point for MNNs is proven using the homeomorphism theory. By employing inequality technique, a new lemma is proposed which is used to prove MNNs are exponentially stable. Then, the error of the drive-response system is studied through Lyapunov method. By utilizing feedback controllers and adaptive controllers, we have derived the conditions necessary for achieving synchronization in the drive-response system of Memristive Neural Networks (MNNs). These conditions are presented in the form of Linear Matrix Inequalities (LMIs). Additionally, through comparative experiments, the feedback controller designed in this paper enables faster convergence of MNNs, and the adaptive controller designed is applicable to a larger range of delays. Finally, three numerical examples are provided to demonstrate the effectiveness of the theoretical results and their applicability to secure communication.

This study investigates an age-structured SI epidemic model with a general infection function [Formula: see text]. We establish well-posedness of the model, showing that the solutions generate a unique semiflow that admits a global compact attractor. A sharp threshold dynamic governed by the basic reproduction number [Formula: see text] is established: when [Formula: see text], the disease-free equilibrium is globally asymptotically stable, ensuring disease elimination; when [Formula: see text], the system exhibits uniform persistence and the unique endemic equilibrium is globally asymptotically stable. Numerical simulations confirm the theoretical findings, demonstrating the robustness and relevance of the model.

Brucellosis, a persistent zoonotic disease, presents significant challenges in understanding its transmission dynamics and in designing effective control strategies due to the complex interaction of biological and environmental factors. In this study, we develop a delayed differential equation model incorporating discrete time delays to represent both disease latency and the delayed effects of interventions, as well as environmental transmission mechanisms. These elements allow for a more accurate representation of the epidemiological processes that sustain brucellosis in both livestock and human populations. We derive threshold conditions for controlling disease prevalence and verify the existence of equilibrium points, analyzing their global asymptotic stability using Lyapunov functions. A key contribution of this work is the formulation and analysis of a mixed state-control delay optimal control problem, which explicitly considers the impact of delays on both state variables and control measures. This framework provides new insights into the timing of interventions and the optimal allocation of resources. Optimal strategies, including vaccination, culling, environmental disinfection and public awareness campaigns, are derived using the delay Pontryagin maximum principle. The study balances the direct costs of interventions with the indirect economic and societal impacts of the disease, offering actionable, cost-effective recommendations. Sensitivity analysis identifies key parameters influencing transmission and control effectiveness. The model is validated with epidemiological data from Inner Mongolia, demonstrating its applicability to endemic settings. Our findings highlight the critical role of mixed state-control delays in shaping optimal control strategies, showing that neglecting these factors can lead to ineffective or suboptimal results. This research advances both the theoretical and practical understanding of mixed delay optimal control problems, offering a valuable framework for managing brucellosis and other delay-sensitive infectious diseases.

We introduce and analyze an infectious disease model that combines the Crowley–Martin incidence rate with the Holling-type II treatment rate. This model aims to capture the transmission dynamics of infectious diseases and takes into account the potential for reinfections. We thoroughly examine the behavior and characteristics of this model to gain insights into the spread and control of infectious diseases. We obtained the case of transcritical and backward bifurcations conditionally. The global stability of a unique endemic steady state is obtained under appropriate parametric conditions, as demonstrated using a geometric approach. When the basic reproduction number exceeds one, the system exhibits the coexistence of numerous endemic steady states. These steady states give birth to complex and sophisticated dynamics because they display a variety of bifurcations, oscillations (via Hopf bifurcation), bistability, and hysteresis. Through the use of sensitivity analysis, the parameters that are most sensitive to the basic reproduction number are identified. The theoretical results are accompanied by their corresponding numerical simulations. This work highlights the importance of ensuring proper access to effective treatment and preventing reinfection to achieve the ultimate goal of disease eradication.

A promising avenue for the control of mosquito-borne diseases such as dengue fever, malaria and Zika is to sterilize female mosquitoes by releasing the males carrying the bacterium Wolbachia through a mechanism cytoplasmic incompatibility (CI). In this work, we develop a model of delay differential equations, incorporating larval and adult stages, to assess the delicate impact of the incomplete CI, weak mating competitiveness of infected males and immigration of wild mosquitoes from surrounding uncontrolled areas on suppression efficiency. Our analyses identify the thresholds of CI intensity and immigration number to ensure that the model displays globally asymptotically stable dynamics. We quantify the maximum suppression efficiency by the maximum relative reduction of wild adult mosquitoes, which increases almost linearly in the CI intensity and decreases almost linearly in the immigration number when it is no more than [Formula: see text] of the carrying capacity. To reduce [Formula: see text] of wild adult mosquitoes in the peak season within 7 weeks, we find that the least release number would be reduced effectively by increasing the CI intensity or decreasing the immigration number.