Stability and Hopf bifurcation of a tumor–immune system interaction model with an immune checkpoint inhibitor

Shujing Shi, Jicai Huang, Yang Kuang, Shigui Ruan

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


In this paper we study a three-dimensional tumor–immune system interaction model consisted of tumor cells, activated T cells, and immune checkpoint inhibitor anti-PD-1. Based on the uncontrollable character of tumor cells in the absence of immune response and treatment, the growth of tumor cells is assumed to be exponential. We discuss the distribution of equilibria qualitatively and study the stability of all possible equilibria with and without anti-PD-1 drug. When no drug is applied, the model has a tumor-free equilibrium and at most one tumorous equilibrium. Biologically, there exists a threshold dT1 for the death rate dT of T cells: when dT≥dT1 tumor cells will keep growing; when dT<dT1 tumor cells may be eradicated for some positive initial values and keep growing for some other positive initial values. For the case with anti-PD-1 treatment, the model has at most five tumor-free equilibria and two interior equilibria. Our analysis indicates that there exists a threshold γA1 for the intravenous continuous injection γA: when γA≤γA1 the fate of tumor cells is the same as the case with no drug applied; when γAA1 the model may exhibit bistable phenomena and periodic orbits. Furthermore, we establish the existence of local Hopf bifurcation around the interior equilibrium and determine the stability of the bifurcating periodic orbits. Our simulations show that the model exhibits a stable periodic orbit which implies the long term coexistence and balance of the tumor and immune system.

Original languageEnglish (US)
Article number106996
JournalCommunications in Nonlinear Science and Numerical Simulation
StatePublished - Apr 2023


  • Anti-PD-1
  • Hopf bifurcation
  • Immunotherapy
  • Periodic orbit
  • Stability
  • Tumor–immune system interaction model

ASJC Scopus subject areas

  • Numerical Analysis
  • Modeling and Simulation
  • Applied Mathematics


Dive into the research topics of 'Stability and Hopf bifurcation of a tumor–immune system interaction model with an immune checkpoint inhibitor'. Together they form a unique fingerprint.

Cite this