A comprehensive numerical investigation of memory effects and treatment impact in HIV-HPV co-infection dynamics

A surge in virus-associated malignancies, especially in areas where human immunodeficiency virus is prevalent, is underscoring the need for an understanding of the complex interplay between HIV and human papillomavirus co-infections. Both infections have been studied individually in great detail, but few studies have included both memory-dependent transmission dynamics and dual treatment effects in a single mathematical model that is calibrated using real epidemiology data. In this work, we construct a novel HIV-HPV co-infection model with fractional-order derivatives which incorporates treatment classes for both HIV and HPV and incorporates the memory effects inherent in chronic infections. Analyses are performed to assess the qualitative behavior of the model. The co-infection model reproduction number is calculated via the next-generation method. Furthermore, the model's disease-free and endemic points are also calculated to examine the complex dynamics and determine whether the disease persists in the population or not. The novel feature of this paper is the incorporation of the treatment classes for both diseases. To make the discussion more comprehensive, the real-world data of Zimbabwe (1987-2003) is used for the parameter estimation to evaluate the parameters and for the simulations. Local stability of the disease-free equilibrium is determined using the Routh-Hurwitz criterion, while global stability is proven by constructing a Lyapunov function and applying LaSalle's invariance principle. The results highlight the impact of transmission rate on co-infection dynamics. Meanwhile, the numerical simulations are performed in MATLAB using the Adams-Bashforth predictor-corrector method. The results show that all state variables converge to their equilibrium points for different fractional-order values, confirming the robustness of the model. Fractional-order dynamics suggest lower growth rates of epidemics than classical integer-order dynamics, which capture biological memory effects that are realistic. The results highlight the importance of long-term treatment coverage and the need for multi-intervention approaches in high burden countries to control HIV-HPV co-infection. The results also indicate that improving the quality of treatment could curb the co-infection. Latin hypercube sampling and partial rank correlation were both used to assess the sensitivity of the basic reproduction numbers. Coefficient was ran in more than 1500 runs. The results shows that the transmission rate [Formula: see text] and the progression rates [Formula: see text] are the most influencing parameters spreading the disease. Furthermore, the study highlights the factors that contribute to the ongoing presence of co-infection and enhances our insight into its behavior. It also stresses the necessity of strategic interventions and effective resource management to mitigate the impact of potential future outbreaks.