This paper presents an analysis and numerical simulation of financial crime population dynamics using fractional order calculus and Newton's polynomial. The dynamics of financial crimes are modeled as a fractional-order system, which is then solved using numerical methods based on Newton's polynomial. The results of the simulation provide insights into the behavior of financial crime populations over time, including the stability and convergence of the systems. The study provides a new approach to understanding financial crime populations and has potential applications in developing effective strategies for combating financial crimes. Fractional derivatives are commonly applied in many interdisciplinary fields of science because of its effectiveness in understanding and analyzing complicated phenomena. In this work, a mathematical model for the population dynamics of financial crime with fractional derivatives is reformulated and analyzed. A fractional-order financial crime model using the new Atangana-Baleanu-Caputo (ABC) derivative is introduced. The reproduction number for financial crime is calculated. In addition, the relative significance of model parameters is also determined by sensitivity analysis. The existence and uniqueness of the solution in consideration of the ABC derivative are discussed. A number of conditions are established for the existence and Ulam-Hyers stability of financial crime equilibria. A numerical scheme is presented for the proposed model, starting with the Caputo-Fabrizio fractional derivative, followed by the Caputo and Atangana-Baleanu fractional derivatives. Finally, we solve the models with fractal-fractional derivatives.

In this study, we investigate two mathematical models, formulated using delay differential equations, to capture HIV-1 transmission dynamics. Both models incorporate CD4 ${}^{+}$T cells and macrophages as target cells and saturated non-linear infection transmission terms. The first model introduces a discrete time delay, while the second employs a distributed delay to reflect more realistic biological insights. Our theoretical analysis explores key properties such as the basic reproduction number, non-negativity, boundedness, and the existence of equilibria. The global stability analysis of the disease-free and endemic equilibrium are analysed using the LaSalle invariance principle by proposing Lyapunov functions. We conducted sensitivity analysis to find the significant parameters related to the infection dynamics. Numerical simulations are performed to validate our theoretical results and visualize the behavior of the system under different parametric conditions. This study shows the critical role of time delays in shaping HIV-1 infection dynamics. Incorporation of such delays into mathematical models are essential for accurately capturing the progression of the disease and proposing effective intervention strategies for HIV-1.