Nonlinear dynamics in tumor-immune system interaction models with delays

Research output: Contribution to journalReview articlepeer-review

6 Scopus citations


In this paper, we review some recent results on the nonlinear dynamics of delayed differential equation models describing the interaction between tumor cells and effector cells of the immune system, in which the delays represent times necessary for molecule production, proliferation, differentiation of cells, transport, etc. First we consider a tumor-immune system interaction model with a single delay and present results on the existence and local stability of equilibria as well as the existence of Hopf bifurcation in the model when the delay varies. Second we investigate a tumor-immune system interaction model with two delays and show that the model undergoes various possible bifurcations including Hopf, Bautin, Fold-Hopf (zero-Hopf), and Hopf-Hopf bifurcations. Finally we discuss a tumor-immune system interaction model with three delays and demonstrate that the model exhibits more complex behaviors including chaos. Numerical simulations are provided to illustrate the nonlinear dynamics of the delayed tumor-immune system interaction models. More interesting issues and questions on modeling and analyzing tumor-immune dynamics are given in the discussion section.

Original languageEnglish (US)
Pages (from-to)541-602
Number of pages62
JournalDiscrete and Continuous Dynamical Systems - Series B
Issue number1
StatePublished - Jan 2021


  • Bautin bifurcation
  • Chaos
  • Delay differential equations
  • Fold-Hopf (zero-Hopf) bifurcation
  • Hopf bifurcation
  • Hopf-Hopf bifurcation
  • Tumor-immune system interaction

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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