Modeling of Within-Host Immune Response to Mycobacterium tuberculosis Infection Dynamics Using Fractional-Order Differential Equations

In this study, a fractional-order differential equation model was proposed as a tool to study the within-host immune response to Mycobacterium tuberculosis (Mtb) 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). Different analytical methods were employed for fractional-order model. The positivity of future solution of the model, the invariant region, infection-free equilibrium (IFE) points, and endemic equilibrium (EE) points were studied. The basic reproduction number was obtained using the next generation matrix to study the stability of the equilibria. The global stability conditions of both infection-free and EE points were demonstrated by constructing suitable Lyapunov functions. A 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. Numerical simulations indicated that the infection progresses as the average number of Mb released and the infection rate increase while other parameters remain constant. The Mi and Mb were rapidly reduced at low fractional orders as a result of an increased rate of T activation. Moreover, it was observed that as the activation rate of B rises, the total Mb rapidly decreases at low fractional orders. We also showed that the presence of memory explained by the fractional-order model activates adaptive immune responses and reduces the concentration of Mb in the host compared with classical models without memory. However, the immune system alone cannot eliminate the infection unless drug therapy is used to effectively manage the infection.