A robust study of the transmission dynamics of syphilis infection through non-integer derivative

One of the most harmful and widespread sexually transmitted diseases is syphilis. This infection is caused by the Treponema Palladum bacterium that spreads through sexual intercourse and is projected to affect $ 12 $ million people annually worldwide. In order to thoroughly examine the complex and all-encompassing dynamics of syphilis infection. In this article, we constructed the dynamics of syphilis using the fractional derivative of the Atangana-Baleanu for more accurate outcomes. The basic theory of non-integer derivative is illustrated for the examination of the recommended model. We determined the steady-states of the system and calculated the $ \mathcal{R}_{0} $ for the intended fractional model with the help of the next-generation method. The infection-free steady-state of the system is locally stable if $ \mathcal{R}_{0} < 1 $ through jacobian matrix method. The existence and uniqueness of the fractional order system are investigate by applying the fixed-point theory. The iterative solution of our model with fractional order was then carried out by utilising a newly generated numerical approach. Finally, numerical results are computed for various values of the factor $ \Phi $ and other parameters of the system. The solution pathways and chaotic phenomena of the system are highlighted. Our findings show that fractional order derivatives provide more precise and realistic information regarding the dynamics of syphilis infection.