Modeling and analysis of the fractional-order epidemic model to investigate mutual influence in HIV/HCV co-infection

Based on the Caputo fractional-order derivative, this work investigates the dynamics of a newly developed co-infection model of Human immunodeficiency virus (HIV) and Hepatitis C virus (HCV). Due to their ability to take into account memory history and heritability, Caputo fractional-order derivatives are a natural candidate to study the HIV/HCV co-infection where these two properties are critical to study how infections spread. Furthermore, applying the Caputo fractional-derivative to the co-infection model helps forecast disease progression and offers optimal treatment strategies for understanding complex HIV/HCV interactions and co-evolutionary dynamics. Mathematical analysis of the co-infection model reveals two equilibria, one without sickness and the other with sickness. The next-generation matrix approach is employed to calculate the basic reproduction number for the cases of HIV and HCV only respectively, and the co-infection model of HIV and HCV, jointly that demonstrates the mutual influence of the two diseases. Using the reproduction numbers, the Lyapunov functional method, and the Routh-Hurwitz criterion, we establish the global dynamics of the model. To validate theoretical predictions, the fractional Adams Method (FAM), a popular numerical technique with a predictor-corrector structure, is utilized to compute the model’s numerical solutions. Finally, numerical simulations confirm the theoretical findings, elucidating the high degree of agreement between the theoretical analysis and the numerical results. Different from the existing literature using the L1 scheme, we incorporated a memory trace (MT) procedure in our paper that captures and amalgamates the historical dynamics of the system to evoke the memory effect in detail. One of the novel results obtained from this study is that the memory trace starts to come into existence once fractional power \(\zeta \) $\zeta$ starts to increase from 0 to 1 and completely disappears when \(\zeta \) $\zeta$ becomes 1. Upon increasing the fractional-order \(\zeta \) $\zeta$ from 0, the memory effect exploits a nonlinear proliferation starting from zero. This observed memory effect emphasizes the difference between the integer and non-integer order derivatives and thus claims the existence of memory effects of fractional-order derivatives. The findings of the paper will contribute to a better understanding of the disease outbreak, as well as aid in the development of future predictions and control strategies.