Systems Biology (Methods in Molecular Biology)

dx

dt

¼xðÞ 2 Nx Hxz,

dy

dt

¼yðÞþ 4  0 : 14 y 0 : 5 x^2 Iy 0 : 5 Hyzþ 0 : 001 z^2 ,

dz

dt

¼Izþ 0 : 07 y^2 þ 0 : 5 Hyz 0 : 002 z^2 ,

ð 24 Þ

The numerical values of the constants were taken from the

model of chemical network studied in a previous work

[83]. Obtaining fixed points, stability analysis and bifurcation was

performed using the standard procedure [24, 51, 52] using as a

control parameter the population of immune cells

I(T lymphocytes, CTL and natural killer, NK, cells [39]).

Lyapunov exponents were calculated using the classical Wolf

algorithm, FORTRAN language [53]. In relation to the calculation

of the Lyapunov exponents, note that:

1. It is important to choose an appropriate algorithm, usually the
   Wolf algorithm [53] that yields good results.


2. The length of the time series used must contain at least 2000
   values to achieve good results [14].


3. In the calculation it must be verified that there is no appreciable
   difference between the sum of the Lyapunov exponents and
   the divergence of the flow.

The LZ complexity [54, 55] was calculated using the proposed

algorithm by Lempel and Ziv, Lyapunov dimensionDL, also known

as dimension Kaplan-York [56], it was evaluated across the spec-

trum of Lyapunov exponentsλjas

DL¼jþ

Pj

i¼ 1 λi

λjþ 1

, ð^25 Þ

wherejis the largest integer number for whichλ 1 +λ 2 þþ

λj0. The results are summarized in Table1.

Fig. 3Chemical network for the model for cancer growth

138 Sheyla Montero et al.