In this work, we investigate the existence and uniqueness of the solutions for a class of nonlinear fractional differential equations with nonlocal integral boundary conditions. Our analysis relies on the fixed point index theory and a $u_{0}$ -positive operator. Examples are discussed for the illustration of the main work.

In this paper, we formulate and analyze a class of discrete state-structured epidemic models that spread through both horizontal and vertical transmissions on networks, where infected individuals can move from one infected state to any other state so that our models include all possible state-transfers (disease deterioration and amelioration) among different states. Many epidemic transmissions with or without vertical transmission in nature can be analyzed by referring to our models, such as HIV-1, viral hepatitis, and Covid-19. We derive the basic reproduction number $ \mathcal{R}_0 $ = $ \mathcal{R}_h $+ $ \mathcal{R}_v $, and prove that the global dynamics are completely determined by the basic reproduction number: if $ \mathcal{R}_0\leq 1 $, the disease-free equilibrium is globally asymptotically stable and the disease always dies out; if $ \mathcal{R}_0>1 $, the disease-free equilibrium is unstable, and there exists a unique endemic equilibrium that is globally asymptotically stable, and the disease persists at a positive level in the population. It also implies that vertical transmission has an impact on maintaining infectious diseases when horizontal transmission cannot sustain the disease on its own. The proof of global stability is based on the graph-theoretic approach and answer the open problem left in [1]. Finally, numerical simulations are performed to illustrate the theoretical results.

With the ongoing evolution of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and its increasing adaptation to humans, several variants of concern (VOCs) and variants of interest (VOIs) have been identified since late 2020. These include Alpha, Beta, Gamma, Delta, Omicron parent lineage, and other variants. These variants may show distinct levels of virulence, antigenicity, and infectivity, which require specific defense and control measures. In this study, we propose an SI1…InR infectious disease model to simulate the spread of SARS-CoV-2 variants among the human population. We combine the proposed epidemic model and reported infected data of variants with physical information neural networks (PINNs) to develop a novel mechanism called VOCs-informed neural network (VOCs-INN). In our experiments, we found that this algorithm can accurately fit the reported data of the British Columbia (BC) province and its five internal health agencies in Canada. Furthermore, it can simulate observed or unobserved dynamics, infer time-dependent parameters, and enable short-term predictions. The experimental results also reveal variations in the intensity of control strategies implemented across these regions. VOCs-INN performs well in fitting and forecasting when analyzing long-term or multi-wave data.

This study identifies a behavior-driven instability mechanism for spatial pattern formation in ecology, demonstrating that repulsive prey-taxis, a phenomenon where predators avoid well-defended prey, can operate as a primary driver of spatiotemporal complexity through mechanisms distinct from classical diffusion-driven Turing instability. We develop a diffusive predator–prey model incorporating prey-taxis and a Crowley–Martin functional response to capture the interplay between directed movement and predator interference. Our analysis establishes the global existence, uniqueness, and boundedness of classical solutions, ensuring the model’s biological well-posedness. We prove that the unique positive equilibrium is globally asymptotically stable under weak attractive prey-taxis, while deriving explicit, ecologically interpretable thresholds for instability. A key finding is that sufficiently strong repulsive prey-taxis (χ0) exerts a consistent stabilizing effect. Extensive numerical simulations confirm these predictions and unveil a rich spectrum of patterns, ranging from stationary spots, stripes, and labyrinths to dynamic spiral waves and chaos, all of which align with observable ecological phenomena. Our results fundamentally expand the theory of biological pattern formation by establishing prey-taxis, particularly in its repulsive form, as a versatile and potent mechanism for spatial self-organization beyond the effects of pure diffusion.