Mathematical Modeling of the Tumor–Immune System with Time Delay and Diffusion

Abstract

This paper proposes a partial differential equation model based on the model introduced by V. A. Kuznetsov and M. A. Taylor, which explains the dynamics of a tumor–immune interaction system, where the immune reactions are described by a Michaelis–Menten function. In this work, time delay and diffusion process are considered in order to make the studied model closer to reality. Firstly, we analyze the local stability of equilibria and the existence of Hopf bifurcation by using the delay as a bifurcation parameter. Secondly, we use the normal form theory and the center manifold reduction to determine the normal form of Hopf bifurcation for the studied model. Finally, some numerical simulations are provided to illustrate the analytic results. We show how diffusion has a significant effect on the dynamics of the delayed interaction tumor–immune system.

Optimal control of combination immunotherapy for a virtual murine cohort in a glioblastoma-immune dynamics model

“…Many mathematical models of tumor-immune dynamics have been developed (Eladdadi et al (2014); Sardar et al (2024); Cherraf et al (2023); Song et al (2021); Liu et al (2021); Yin et al (2019); Nikolopoulou et al (2018); Lai and Friedman (2017); de Pillis et al (2005); De Pillis and Radunskaya (2003)), including a few that focus specifically on GBM (Anderson et al (2023); Santurio and Barros (2022); Khajanchi (2021); Storey et al (2020)). Here, we extend the ODE model of Anderson et al (2023), which incorporates tumor cells, T cells, and MDSCs, to include treatment with anti-PD-1 and a CCR2 antagonist and subsequently apply optimal control theory.…”