This study investigates the dynamics and control of an infectious disease using an age-structured Susceptible-Vaccinated-Infected (SVI) model, incorporating imperfect vaccination and therapeutic treatment. We identify a backward bifurcation phenomenon, driven by treatment rates, which allows disease persistence even when the basic reproduction number R0 < 1, complicating eradication e orts. To address this, we formulate an optimal control problem with time-dependent vaccination and treatment rates to minimize infected individuals and control costs. Using bifurcation theory and integrated semigroup methods, we establish the model's well-posedness, equilibria stability, and bistability conditions. First-order optimality conditions are derived to characterize optimal controls. Numerical simulations,solved via the Forward-Backward Sweep method, demonstrate that combined vaccination and treatment signi cantly reduces disease prevalence, with early intervention being critical. These ndings underscore the importance of strategic resourceallocation in managing complex epidemic dynamics.

The problem of monitoring an electric power system by placing Phasor Measurement Units in the system as possible is closely related to the well-known power domination problem in graphs. In this paper, we compute, 2-power domination number for fully connected cubic networks and also developed k-power domination algorithm for specific networks and evaluates its performance across these networks. Furthermore, the algorithm’s outputs are used for various other applications such as fault detection and localization, state identification, accuracy improvement, and cyber security enhancement, thereby making it a valuable tool for utilities and grid operators.

This paper analytically investigates unsteady, laminar, magnetohydrodynamic (MHD) mixed convective oscillatory flow of two immiscible viscous fluids through a horizontal two-phase channel. The channel consists of a porous upper region (Region~I, $0 \le y \le h$) and a non-porous lower region (Region~II, $-h \le y \le 0$), separated by a permeable interface, with a uniform transverse magnetic field $B_0$ applied normal to the flow. The governing equations incorporate thermal buoyancy ($Gr$), concentration buoyancy ($Gc$), Darcy porous resistance, thermal radiation (Rosseland approximation), volumetric heat generation/absorption, first-order destructive chemical reaction, and the Soret effect in both regions. An oscillatory suction/injection velocity is assumed at the walls, and a perturbation expansion in the small parameter $\varepsilon \ll 1$ decouples the system into zeroth-order (steady) and first-order (oscillatory) ODEs, solved analytically. Six interface conditions enforce continuity of velocity, shear stress, temperature, heat flux, concentration, and mass flux. Closed-form expressions are derived for velocity, temperature, and concentration profiles, as well as skin friction $\tau$, Nusselt number $Nu$, and Sherwood number $Sh$. Parametric studies show that increasing the Soret number $Sr$ enhances velocity by 15-20% in the porous region, while increasing the Hartmann number $M$ retards flow by up to 40% via the Lorentz force. Thermal radiation augments both $Nu$ and the thermal field, and destructive chemical reaction increases $Sh$ by 25-30%. Results are validated against limiting cases including hydrodynamic flow ($M=0$), no Soret effect ($Sr=0$), and the single-phase limit.

Recent studies highlight how stochastic e ects, such as environmental noise and resource constraints, alter epidemic dynamics, diverging from deterministic predictions. Unlike classical models with bifurcations, stochastic models capture more realistic outbreak scenarios. Research suggests that limited medical resources, coupled with stochastic uctuations, signi cantly in uence disease spread and control strategies. Numerical simulations support these ndings, emphasizing the need for randomness in epidemic modeling to improve predictive accuracy and public health interventions.Understanding these stochastic in uences can aid in developing more adaptive policies for epidemic mitigation. Future research aims to re ne these models by incorporating spatial heterogeneity, population diversity, and pathogen evolution for better epidemic control.