Modelling activated macrophages and T-cells in a within-host tuberculosis model via numerical simulation

This study extends a classical within-host tuberculosis model by incorporating activated macrophages and T cells, thereby capturing key immune-regulatory processes during Mycobacterium tuberculosis infection. The resulting integer-order system is analysed for positivity, boundedness, disease-free equilibrium, local stability, and the basic reproduction number. The model is then reformulated as a Caputo fractional-order system and solved semi-analytically with the Laplace-Adomian Decomposition Method (LADM). Numerical simulations show that lower fractional orders strengthen memory effects, increase uninfected and activated macrophage levels, accelerate the decline of infected macrophages, and suppress bacterial growth. The simulations further show that stronger T-cell activity and higher macrophage activation improve bacterial control. These results indicate that fractional-order modelling provides a biologically richer system for studying immune memory, latency, and persistence in within-host tuberculosis dynamics.