Mathematical analysis of tumor-free equilibrium in BCG treatment with effective IL-2 infusion for bladder cancer model

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We present a theoretical study of bladder cancer treatment with Bacillus Calmette-Guerin (BCG) and interleukin 2 (IL-2) using a system biology approach to translate the treatment process into a mathematical model. We investigated the influence of IL-2 on effector cell proliferation, presented as a distributed feedback control in integral form. The variables in the system of Ordinary Differential Equations (ODE) are the main participants in the immune response after BCG instillations: BCG, immune cells, tumor cells infected with BCG, and non-infected with BCG. IL-2 was involved in the tumor-immune response without adding a new equation. We use the idea of reducing the system of integro-differential equations (IDE) to a system of ODE and examine the local stability analysis of the tumor-free equilibrium state of the model. A significant result of the model analysis is the requirements for the IL-2 dose and duration, depending on the treatment regimen and tumor growth. We proved that the BCG+IL-2 treatment protocol is more effective in this model, using the spectral radius method. Moreover, we introduced a parameter for individual control of IL-2 in each injection using the Cauchy matrix for the IDE system, and we obtained conditions under which this system would be exponentially stable in a tumor-free equilibrium.

Original languageEnglish
Pages (from-to)16388-16406
Number of pages19
JournalAIMS Mathematics
Issue number9
StatePublished - 2022


  • combined BCG + IL-2 treatment
  • exponential stability
  • feedback control, Cauchy matrix
  • system of integro-differential equations

All Science Journal Classification (ASJC) codes

  • General Mathematics


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