Chaos in Tumor Growth Model with Time-Delayed Immune Response

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This Demonstration considers a tumor growth model in which tumor cells play the role of prey, and the immune system is represented by resting and predator cells.


The model consists of two differential equations coupled to a delay differential equation:




with initial history functions . Here , , and represent tumor cells, predator cells, and resting cells, respectively; is time; and is a time delay measured from the time resting cells recognize tumor cells to the time predator cells destroy the prey. The parameters in the model are defined in the reference and are set here to be . When the delay time is small, all three cell populations approach equilibrium values, and the tumor can be considered nonmalignant. For larger delay times, all cell populations coexist in a limit cycle or periodic solution; in this case, the tumor can be termed mildly malignant.

The existence of periodic solutions implies that the tumor levels oscillate around a fixed point even in the absence of treatment; this has been observed clinically and is known as Jeff's phenomenon [2]. With larger , the hunting cells are lethargic in their response and the cell populations grow in an irregular fashion, leading to chaotic attractors; in this case the tumor is said to be malignant, and aggressive treatment is required.


Contributed by: Clay Gruesbeck (July 2013)
Open content licensed under CC BY-NC-SA




[1] M. Saleem and T. Agrawal, "Chaos in a Tumor Growth Model with Delayed Responses of the Immune System," Journal of Applied Mathematics, #891095, 2012 pp. 1–16. doi:10.1155/2012/891095.

[2] R. H. Thomlinson "Measurement and Management of Carcinoma of the Breast," Clinical Radiology, 33(5), 1982 pp. 481–493.

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