### Mathematical Biosciences and Engineering

2019, Issue 5: 5346-5379. doi: 10.3934/mbe.2019267
Research article

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

• Received: 21 November 2018 Accepted: 21 February 2019 Published: 10 June 2019
• 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.

Citation: OPhir Nave, Shlomo Hareli, Miriam Elbaz, Itzhak Hayim Iluz, Svetlana Bunimovich-Mendrazitsky. BCG and IL − 2 model for bladder cancer treatment with fast and slow dynamics based on SPVF method—stability analysis[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5346-5379. doi: 10.3934/mbe.2019267

### Related Papers:

• 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.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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