Hyers Ulam stability and bifurcation control of leptospirosis disease dynamics and preventations: Modeling with singular and non-singular kernels.

Due to its various uses, the dynamical system is a significant research area in the field of mathematical biology. The model is first developed by applying the usual derivative with combined recovery measures of humans as well as animals for leptospirosis transmission and then converted into a generalized form of the fractal fractional sense with power law kernel, exponential law kernel, and Mittag-Leffler kernel. We verify all the fundamental characteristics of the newly developed model for the validation analysis of the system such as equilibrium points, local stability, positivity of solutions, reproductive number, and existence of a unique solution. Also, bifurcation analysis has been used for newly developed systems to observe the impact of each sub-compartment with the effect of different parameters. The results on Hyers Ulam stability are established by utilizing different kernels to observe its stable state. We used a numerical scheme based on the Lagrange polynomials for all three cases of fractal fractional derivatives having different kernels. The efficiency of the fractional operators with comparative analysis of different kernels is shown in simulation form to verify the validity and real behavior of leptospirosis transmission for humans as well as animals. he graphical explanation of our model's solution depicts the effectiveness of our techniques applied and this study helps for future predictions and developing better control strategies.