Fractional-Order Dynamical Analysis of a New  Malaria–HBV Co-infection Model with Vaccination and Treatment Strategies

Malaria and Hepatitis B virus (HBV) remain major global health challenges, particularly in regions with limited healthcare resources. Their interaction through co-infection further complicates disease management, as it increases both the severity of illness and the difficulty of control. In this study, we proposed a novel fractional-order mathematical modeling approach to examine the impact of treatment strategies and vaccination on the spread of malaria and HBV. Initially, the model was constructed using integer-order nonlinear differential equations to represent scenarios with limited vaccination and treatment coverage. However, given the complex nature of co-infection dynamics, the model was extended by incorporating fractional-order derivatives and power-law expressions. This enhancement allowed for the inclusion of memory-dependent effects, thereby capturing a more comprehensive and realistic representation of disease behavior. The study established rigorous conditions for solution existence and uniqueness within the fractional framework and analyzed the stability of the endemic equilibrium using the Lyapunov function technique. To numerically solve the system, the fractional Adams–Bashforth–Moulton method was employed. Simulations were conducted under various parameter settings and fractional-order values, with visual analyses provided to illustrate the outcomes. The results revealed that strengthening treatment strategies, expanding vaccination coverage, and implementing preventive measures such as mosquito control and timely access to medication significantly reduce both single infections and co-infections. Conversely, increased mosquito contact, low vaccination rates, and ineffective treatments were found to exacerbate disease transmission and severity. Based on these findings, we recommend that policymakers and healthcare planners adopt coordinated approaches that combine vaccination, treatment, and vector control. Such integrated strategies are particularly vital in resource-constrained regions, where optimizing available resources is critical to reducing the overall disease burden and protecting vulnerable populations.