Exploring syphilis transmission dynamics with congenital infection and disability compartments

This study proposes a novel sex-structured syphilis transmission model that incorporates key biological features such as congenital infection and disability due to long-term complications. While previous works have explored syphilis dynamics using fractional calculus, we extend this framework by introducing two additional compartments: congenitally infected newborns ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>C</mml:mi></mml:math> ) and individuals with syphilis-induced disabilities ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi></mml:math> ), allowing for a more realistic representation of vertical transmission and chronic disease outcomes. The model is formulated using the Atangana-Baleanu-Caputo (ABC) fractional derivative to capture memory effects and non-local dynamics often observed in infectious disease spread. We rigorously analyze the positivity and boundedness of solutions, and establish the existence and uniqueness using Lipschitz continuity and Banach space theory. The basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> is derived to characterize disease dynamics, and both local and global stability of the disease-free equilibrium are proven. A detailed sensitivity analysis identifies the most influential parameters affecting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi> <mml:mn>0</mml:mn></mml:msub> </mml:math> , such as transmission probabilities, treatment rates, and the vertical transmission factor. Numerical simulations based on the Adams-Bashforth and Newton polynomial schemes validate the theoretical findings and provide insights into the potential effectiveness of various intervention strategies. Overall, this work contributes an enhanced modeling framework that integrates mathematical rigor with public health relevance, offering valuable guidance for targeted strategies to control syphilis and reduce its congenital and long-term burden.