MODELING OF WITHIN-HOST IMMUNE RESPONSE TO MYCOBACTERIUM TUBERCULOSIS INFECTION DYNAMICS USING FRACTIONAL-ORDER DIFFERENTIAL EQUATIONS

Abstract

In this study, fractional-order differential equations model was proposed as a tool to study the within-host immune response to mycobacterium tuberculosis infection dynamics. The formulated model describes the interaction of uninfected macrophages (𝑀𝑢), infected macrophages (𝑀𝑖), extracellular mycobacterium tuberculosis (𝑀𝑏), T-lymphocytes (𝑇), and B-lymphocytes(𝐵). Firstly, we develop the integer order model of within-host immune response to the mycobacterium tuberculosis infection dynamics and then we extend it into fractional order due to the reason that fractional order possesses memory in modeling real-life phenomena. Different analytical methods were employed to analyses both integer-order and fractional-order model. The positivity of future solution of the model, the invariant region, and infection-free equilibrium point, and endemic equilibrium points were studied. A basic reproduction number was obtained using the next generation matrix to study the stability of the equilibria. Global stability conditions of both infection-free and endemic equilibrium points were demonstrated by constructing suitable Lyapunov functions. Sensitivity analysis was performed using a forward-normalized sensitivity index approach. Sensitivity analyses revealed that infection rate, average number of bacteria released, and adaptive immune cell activation were the most influential factors during infection. As shown by numerical simulations, tuberculosis infection progresses as the average number of released bacteria and the infection rate increase while other parameters remain constant. Additionally, it was observed that as the activation rate of T and B lymphocytes rises, the total infections classes and extracellular bacteria rapidly decreases at low fractional orders respectively. We also showed that the presence of activates adaptive immune responses with memory explained by the fractional-order model reduces the concentration of tuberculosis infection in the host compared to classical models without memory. Therefore, by focusing on the most sensitive parameters 𝛽,𝑟,𝑝, and 𝑞 medical and experimental research may be conducted to promote successful therapies for TB patients.