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<abstract><p>We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.</p></abstract>

<abstract><p>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.</p></abstract>