Deterministic and Fractional-Order Modeling of Influenza Transmission with Control Law

Abstract

Standard integer-order models struggle to capture the complex dynamics of influenza spread. These steps are affected by memories and treatments. The purpose of this study is to develop and evaluate deterministic and fractional-order model of influenza spread to improve control assessment and epidemiological accuracy. The main goal is to examine how memory-based dynamics affect disease progression, stability traits, and the effectiveness of control methods. A detailed study is conducted on a deterministic compartmental influenza model to determine whether it is positive, limited, in equilibrium, and to determine the basic reproduction number. The model is then expanded into a framework that employs a difference operator similar to Caputo's and allows for fractional-order discrete-time to account for memory effects. We do local and global stability assessments, examine the sensitivity of the reproduction number, and employ forward bifurcation analysis and Lyapunov exponents to investigate nonlinear dynamics. A feedback treatment control approach was created to help decrease the effects of epidemics. Using real influenza data to test numerical models, we find that fractional-order dynamics provide a clearer picture of how the virus spreads, revealing more complex patterns such as chaotic oscillations and delayed responses. One approach to show this is to give a more in-depth look at how transmission patterns work. The results indicate that fractional-order modeling provides superior forecasts and supports more effective influenza management strategies than traditional deterministic approaches.

J. Bangladesh Acad. Sci. 50(3); 447–478: September 2026