Vaccination is one of the most cost-effective and simplest interventions for protecting against infectious-disease epidemics. The COVID-19 (coronavirus disease 2019) outbreak in China demonstrated that physical protection and social isolation are critical to controlling an epidemic in the absence of vaccines or antiviral drugs. Considering isolation and pulse vaccination of the susceptible population subject to a delay in acquiring immunity, we propose a susceptible-infectious-recovered (SIR) model incorporating these features. Using the theory of impulsive differential equations and comparison theorems, we prove that the infection-free periodic solution $(\widetilde{S(t)},0)$ of system $(3.... More >

This paper presents an adaptive block time-stepping algorithm designed for the numerical integration of first-order stiff oscillatory problems, which typically involve multiple time scales and stringent stability conditions. The proposed method integrates a block formulation enabling the simultaneous evaluation of several solution points with an adaptive step size mechanism to effectively regulate local truncation errors. A predictor-corrector strategy is employed, where an explicit predictor generates initial estimates, and an implicit block corrector enhances stability for stiff systems. The resulting nonlinear equations are resolved using Newton's iterative method. The scheme is proven to... More >

Generative artificial intelligence (GenAI) is increasingly embedded in enterprise workflows, yet the models that support these workflows are updated at discrete and sometimes frequent intervals. Each update can improve technical effectiveness while simultaneously invalidating prompts, interfaces, controls, and employee routines. This study develops an impulsive dynamical system to analyze the resulting tension between technical improvement and organizational disruption. The continuous subsystem tracks enterprise process stability, organizational adaptation, and effective model capability between updates. The impulsive subsystem represents the instantaneous effects of a model release on workf... More >

This study develops and analyzes a nonlinear mathematical model to describe the dynamics of marriage, extramarital affairs, marital breakdown, and divorce. The population is divided into five compartments: single $S(t)$, married $M(t)$, individuals in extramarital affairs $A(t)$, broken marriages $B(t)$, and divorced $D(t)$. The model is formulated as a system of nonlinear ordinary differential equations capturing transitions between these states through parameters representing marriage formation, affair initiation, reconciliation, separation, and divorce. Basic qualitative properties, including positivity and boundedness of solutions, are established to ensure the model is well posed. Equil... More >

This paper develops a deliberately expanded analytic program around the three-dimensional incompressible Navier--Stokes equations. The exposition is intentionally heavy in algebra, index notation, multilinear expansions, differentiated hierarchies, and page-filling calculations. The central idea is to organize the continuation problem around a mixed pressure--vorticity--anisotropic functional, while keeping visible every major nonlinear channel: energy, Hessian growth, high-order commutators, vorticity stretching, elliptic pressure reconstruction, dyadic localization, paradifferential splitting, kernel representation, cylindrical-coordinate geometry, and dense symbolic closure systems. The r... More >

Whale optimization algorithm (WOA) is a relatively new population-based metaheuristic optimization method, which has the advantage of fewer control parameters, strong global optimization ability and easy to implement. However, when being used for high-dimensional problems, WOA may be trapped in the local optimum. In this study, we propose an effective whale optimization algorithm called EWOA. Inspired by particle swarm optimization (PSO), a modified position-updated equation by introducing dynamic inertia weight parameter to guide the search of new candidate individuals is presented. In addition, in order to make full use of and balance the exploration and the exploitation of WOA, a nonlinea... More >

This article examines the dynamic behavior of a second-order fuzzy difference equation that models the quantitative changes in a specific biological population: $$ E_{n+1}=\frac{S}{C+E_n+E_{n-1}},\ n\in \mathbb{Z}\ and\ n\ge 0,$$ Here, parameter $S$ represents the carrying capacity of the environment, while $C$ signifies the minimum resources required for population survival. The initial values $E_0$, $E_{-1}$, and parameters $S$ , $C$ are all positive fuzzy numbers. By employing the generalized division (g-division) with respect to fuzzy numbers, we establish the existence, uniqueness, persistence, and boundedness of positive fuzzy solutions to the equation under specified conditions. Furth... More >

A class of multi-age group $SEQ_3IHR$ chickenpox epidemic models with effective vaccination in a different subpopulation size, is considered, in this paper. It is assumed that individual members of the subpopulations who recovered from the disease, may be re-infected by the disease. The models comprise of six infectious classes of subclasses and three distinct infected stages of subclasses. We establish that the global dynamics of the epidemic models are determined entirely by the basic reproduction number. In this paper, we determine that the disease-free and endemic equilibrium states, and carry out a stability analysis of the disease-free as well as the endemic equilibrium points of the m... More >