This article gives bounds on a substantial number of BID (bond incident degree) indices for connected graphs in terms of their order, size, and maximum degree. The considered BID indices include, among others, the Sombor index (together with its reduced version), atom-bond sum-connectivity index, symmetric division deg index, sum-connectivity index, harmonic index, and Randi´c index. All the graphs that attain the obtained bounds are also characterized. All the established bounds are valid also for molecular graphs. A graph of order n and size m is called an (n,m)-graph. The obtained bounds provide a partial solution to the problem of finding graphs with extremum (considered) BID indices over the class of all connected (n,m)-graphs with a fixed maximum degree under certain constraints.

This research work employs a powerful analytical method known as the Riccati Modified Extended Simple Equation Method (RMESEM) to investigate and analyse chaotic soliton solutions of the (1 + 1)-dimensional Complex Quintic Swift–Hohenberg Equation (CQSHE). This model serves to describe complex dissipative systems that produce patterns. We have found that there exist numerous chaotic soliton solutions with periodic and axial perturbations to the intended CQSHE, provided that the coefficients are constrained by certain conditions. Furthermore, by applying a sophisticated transformation, the provided transformative approach RMESEM transforms CQSHE into a set of Nonlinear Ordinary Differential Equations (NODEs). The resulting set of NODEs is then transformed into an algebraic system of equations by incorporating the extended Riccati NODE to assume a series form solution. The soliton solutions to this system of equations can be found as periodic, hyperbolic, exponential, rational-hyperbolic, and rational families of functions. A variety of 3D and contour visuals are also provided to graphically illustrate the axially and periodically perturbed dynamics of these chaotic soliton solutions and the formation of fractals. Our findings are noteworthy because they shed light on the chaotic nature of the framework we are examining, enabling us to better understand the dynamics that underlie it.

In this work, we investigate the use of two new methods, residual power series natural transform method (RPSNTM) and new iteration natural transform method (NINTM), to tackle the fractional-order Korteweg-de Vries (KdV) equation. Many wave effects in fluid dynamics, plasma physics and traffic flow are described with the help of the fractional-order KdV equation. Tradi-tional techniques used to solve equations generally are not adapted to deal with the complications caused by fractional derivatives. The RPSNTM is presented as a valuable method to approximate how the solution will change with time by converting the problem into a step-by-step series. The NINTM is also used to make the solution more effective and reliable gradually. Applying both approaches offers a strong way to find approximate analytical answers to the KdV fractional-orderequation. Numerical data is used to reveal that the presented methods work better and faster than existing methods in terms of precision and speed of convergence. These results create new opportunities to apply fractional calculus in analyzing nonlinear waves.preparation.

A macroscopic model for nonhomogeneous traffic is introduced that incorporates relaxation time and lateral time headway, and accounts for driver reaction time. Driver reaction is based on models of real-world nonhomogeneous traffic flow in Pakistan and Iran. A lateral time headway model is obtained using lane change data to characterize traffic during lane changes. The performance of the proposed model is compared with the well-known Payne–Whitham (PW) model on a 3000 m circular road using the FORCE numerical scheme. The results show that the initial multi-cluster density distribution evolves more realistically and accurately with the proposed model. Thus, it can be used to aid in traffic congestion mitigation for on-ramps and off-ramps.

Human Immunodeficiency Virus/Acquired Immune Deficiency Syndrome (HIV/AIDS) remains a global health crisis characterized by significant inherent uncertainties in transmission and intervention effectiveness, which are poorly captured by deterministic models. This study aims to develop a stochastic Susceptible-Exposed-Infected-Treated-AIDS (SEITA) model to quantify the impact of environmental and demographic noise on HIV/AIDS dynamics. This paper presents a stochastic extension of the classical SEITA model for HIV/AIDS transmission, incorporating treatment and vertical transmission pathways. Environmental and demographic variability are modeled through Ito-type Stochastic Differential Equations (SDEs). A novel stochastic reproduction number, , is derived, generalizing the deterministic threshold . We rigorously establish that the infection dies out almost surely when < 1 (that is, ≈ 0.87 in a representative scenario), and persists with positive probability when > 1 (that, ≈ 2.08). Lyapunov-based stability analysis is conducted without relying on the deterministic endemic equilibrium. Numerical simulations, implemented via the Milstein method, demonstrate how stochastic fluctuations significantly affect long-term behavior; for instance, high noise intensities (σ = 0.2) can cause infected population fluctuations exceeding deterministic levels. These findings highlight the critical importance of incorporating stochasticity into epidemic modeling to improve prediction accuracy and inform robust public health strategies, particularly in resource-constrained settings where noise is inherent.