This paper studied the group consensus problem in heterogeneous multi-agent systems (HMASs) subject to input delays, denial of service (DoS) attacks, and external disturbances. The agents interact within a cooperative-competitive network, where first- and second-order dynamics coexist. To address the challenges introduced by disruptions in communication and system heterogeneity, an intermittent control protocol was designed. This protocol operates based on a time-varying binary signal that reflects the availability of communication. Control actions are suspended during DoS intervals and resume when communication is restored. The proposed method incorporates virtual velocity estimation to handle mixed-order agents and employs frequency-domain analysis, specifically the Nyquist stability criterion, to derive algebraic conditions that ensure consensus. These conditions relate the maximum allowable delay to system topology, attack patterns, and disturbance levels. Numerical simulations demonstrate that consensus can be achieved under both directed and undirected network structures, even in the presence of constrained DoS disruptions and bounded disturbances.

This article addresses the hostile-based bipartite containment control of nonlinear fractional multi-agent systems (FMASs) with input delays. Several fundamental algebraic criteria have been offered by the use of signed graph theory. To make the controller design more realistic, we assumed that the controller was under some disturbance. For the analysis of bipartite containment control, we used a fixed and switching signed network. The commonly used Lyapunov function approach and the Razumikhin technique were used. The use of these techniques can conquer the challenge brought on by switching, temporal delays, and fractional mathematics. To better elucidate the theoretical results, two examples are provided.

In this paper, we deal with a Lévy noise-driven epidemic model reflecting the dynamics of measles infection subject to the effect of vaccination. After model formulation, the feasibility of the system was studied by using the underlying existence and uniqueness theory. Moreover, we discussed the behavior of solution around the infection-free and disease-present steady states. To check the persistence and extinction of the infection, we calculated the threshold parameter ${\mathbf{R}}_{\mathbf{s}}$ and it was determined that the disease vanishes whenever ${\mathbf{R}}_{\mathbf{s}}<1$. From January to October 2019, the reported measles cases in Pakistan wear used and the model was fitted against this data by using the well-known fitting techniques. The values of the parameter were estimated and future behavior of the infection was predicted by simulating the model. The model was further simulated and theoretical findings of the study were validated.

This research provides an improved theoretical framework of the Kermack-McKendrick system. By considering the general interference function and the polynomial perturbation, we give the sharp threshold between two situations: the disappearance of the illness and the ergodicity of the higher-order perturbed system. Obviously, the ergodic characteristic indicates the continuation of the infection in the population over time. Our study upgrades and enhances the work of Zhou et al. (2021) and suggests a new path of research that will serve as a basis for future investigations. As an illustrative application, we discuss some special cases of the polynomial perturbation to examine the precision of our outcomes. We deduce that higher order fluctuations positively affect the illness extinction time and lead to its rapid disappearance.