The influence of time delay in a chaotic cancer model

Abstract

The tumor-immune interactive dynamics is an evergreen subject that continues to draw attention from applied mathematicians and oncologists, especially so due to the unpredictable growth of tumor cells. In this respect, mathematical modeling promises insights that might help us to better understand this harmful aspect of our biology. With this goal, we here present and study a mathematical model that describes how tumor cells evolve and survive the brief encounter with the immune system, mediated by effector cells and host cells. We focus on the distribution of eigenvalues of the resulting ordinary differential equations, the local stability of the biologically feasible singular points, and the existence of Hopf bifurcations, whereby the time lag is used as the bifurcation parameter. We estimate analytically the length of the time delay to preserve the stability of the period-1 limit cycle, which arises at the Hopf bifurcation point. We also perform numerical simulations, which reveal the rich dynamics of the studied system. We show that the delayed model exhibits periodic oscillations as well as chaotic behavior, which are often indicators of long-term tumor relapse.

Cancer: A Turbulence Problem

“…The results were recently reconfirmed for a cancer model with time-delay as the bifurcation parameter. It exhibited periodic oscillations as well as chaotic behavior (strange attractors), which are indicators of long-term tumor relapse (72). These findings are some of the many emerging data suggesting cancer cells are aberrant, unsteady attractors in the network state-space governed by nonlinear dynamics (73).…”

Chaos in a nonautonomous eco-epidemiological model with delay

“…It is well documented that many biological and/or environmental processes, such as Allee effect [56] , omnivory [50] , latency in biological processes [47,57] , large turn over of resource [58] , coupling of incommensurate oscillations [59] , seasonal forcing [19,20] , noise [60,61] , etc., may produce chaotic dynamics in simple predator-prey systems; out of which the presence of time delay and seasonal forcing can be observed in many biological processes. Recently, Khajanchi et al [62] showed that the incorporation of realistic time delays in cancer models exhibits deterministic chaos, and significantly increase the dynamical complexity of the tumor-immune competitive system. In eco-epidemiology, presence of delay and seasonal forcing is very common.…”

Critical behavior of spatial networks as a model of paracrine signaling in tumorigenesis

“…An excellent review of these approaches can be found for example in the work of Perc and Szolnoki (2010). In addition, applications of chaos theory (Khajanchi et al 2018), seek to model the fine balance between the immune system and cancer evolution by analyzing instabilities in the systems of coupled differential equations that model this fine balance. As such these techniques directly introduce the concept of an unstable equilibrium that can tip between a stable benign tumor and an unstably growing malignant one.…”