Fractional-order modeling and analytical study of Hepatitis B with vertical and horizontal transmission dynamics incorporating control strategies via three distinct operators

Hepatitis B remains a major global public health challenge, with both horizontal and vertical (mother-to-child) transmission contributing to its persistence. This study develops a fractional-order Hepatitis B transmission model incorporating vertical transmission using the Caputo, Caputo-Fabrizio, and Atangana–Baleanu fractional derivatives to capture memory effects and the hereditary characteristics of disease progression more effectively than classical integer-order models. The mathematical properties of the model are investigated by establishing the existence and uniqueness of solutions using fixed-point theory, while the disease-free and endemic equilibria are determined and their local stability is analyzed. Numerical simulations are performed to examine the effects of the fractional order, vaccination, treatment, and vertical transmission on disease dynamics. The results show that increasing the fractional order, together with enhanced vaccination coverage and timely treatment, substantially reduces the prevalence of Hepatitis B infection. Furthermore, the inclusion of vertical transmission underscores the importance of maternal screening and neonatal immunization in limiting disease persistence. Overall, the proposed fractional-order model provides a realistic framework for describing Hepatitis B transmission by incorporating memory effects and offers valuable insights for the development of effective vaccination and treatment strategies for disease control.