BCG and IL − 2 model for bladder cancer treatment with fast and slow dynamics based on SPVF method—stability analysis

Abstract

In this study, we apply the method of singularly perturbed vector field (SPVF) and its application to the problem of bladder cancer treatment that takes into account the combination of Bacillus Calmette–Guérin vaccine (BCG) and interleukin (IL)-2 immunotherapy (IL−2). The model is presented with a hidden hierarchy of time scale of the dynamical variables of the system. By applying the SPVF, we transform the model to SPS (Singular Perturbed System) form with explicit hierarchy, i.e., slow and fast sub-systems. The decomposition of the model to fast and slow subsystems, first of all, reduces significantly the time computer calculations as well as the long and complex algebraic expressions when investigating the full model. In addition, this decomposition allows us to explore only the fast subsystem without losing important biological/ mathematical information of the original system.The main results of the paper were that we obtained explicit expressions of the equilibrium points of the model and investigated the stability of these points.

Keywords

• mathematical modeling
• therapy schedule
• impulse differential equations
• dirac deltafunction
• gamma distribution function
• bcg and il-2 combined therapy