The complexity of cancer has motivated the development of different approaches to understand the dynamics of this large group of diseases. One that may allow us to better comprehend the behavior of cancer cells, in both short and longterm, is mathematical modelling through ordinary differential equations. Several ODE mathematical models concerning tumor evolution and immune response have been formulated through the years, but only a few may exhibit chaotic attractors and oscillations such as stable limit cycles and periodic orbits; these dynamics are not that common among cancer systems. In this paper, we apply the Localization of Compact Invariant Sets (LCIS) method and Lyapunov stability theory to investigate the global dynamics and the main factors involved in tumor growth and immune response for a chaoticcancer system presented by Itik and Banks in 2010. The LCIS method allows us to compute what we define as the
Cancer is a major public health problem; with 8.8 million deaths recorded in 2015, it is considered to be a leading cause of death globally, second only to heart disease, and it should be noticed that several cases are being reported each year. In general, cancer is a complex and large group of diseases with high mortality rates regardless of age, gender, or race. Incidence and mortality statistics from GLOBOCAN and the WHO indicate that breast, colorectal, lung, cervix, and stomach cancer are the most common types among women, while lung, prostate, colorectal, stomach, and liver cancer are the most common in men [
Therefore, the complexity and diversity of cancer demands that specialists of different areas gather together in order to find a feasible way to understand cancer dynamics in the short and longterm. One approach of particular interest is mathematical modelling by ordinary differential equations which may allow us to describe tumor evolution without the need for clinical trials, which can be quite costly [
Mathematical models can help us study the dynamics between cancer tumor growth, the immune system, and different treatments such as chemotherapy or immunotherapy in the short and longterm. Diverse mathematical models have been propounded with this purpose; see, for example, [
Our paper is focused on investigating the global dynamics of a threedimensional chaoticcancer mathematical model formulated by Itik and Banks in [
The remainder of this paper proceeds as follows. In Section
The LCIS method is used to determine a domain on the state space where all compact invariant sets are located. As examples of compact invariant sets, we may recall equilibrium points, periodic orbits, homoclinic and heteroclinic orbits, limit cycles, and chaotic attractors. The relevance of this analysis is due to the fact that it is useful to study the longtime dynamics of the system. Let us consider an autonomous nonlinear system of the form:
Localizing functions are selected by an heuristic process; this means that one may need to analyze several functions in order to find a proper set that will allow fulfilling the General Theorem. If we consider the location of all compact invariant sets inside the domain
This method has already been applied combined with other stability theories to analyze mathematical models that describe cancer tumor growth, the response of the immune system, and the effects of chemotherapy and immunotherapy on cancer evolution. For example, in paper [
Modelling cancer with ordinary differential equations is an approach that allows us to understand the evolution of tumors in diverse scenarios that will be difficult to study in a patient affected by this disease. Hitherto, as we mention in the last section, diverse mathematical models have been propounded with this spirit. In this paper, we analyze the mathematical model presented by Itik and Banks in [
Equation (
Description and values of parameters of the chaoticcancer mathematical model.
Parameter  Description  Value 


Fractional tumor cells killed by healthy cells  1 

Fractional tumor cells killed by effector cells  2.5 

Healthy host cells growth rate  0.6 

Fractional healthy cells killed by tumor cells  1.5 

Maximum effector cells recruitment rate by tumor cells  4.5 

Steepness coefficient of the effector cells recruitment  1 

Fractional effector cells inactivated by tumor cells  0.2 

Death rate of effector cells  0.5 
Further, it should be noticed that dynamics of the nondimensionalized system (
As one can see, all parameters are positive and units of cells were scaled. The maximum carrying capacity for both tumor and healthy cells populations has been normalized to
However, we are interested in one particular dynamic of the system; therefore, we used the values shown in Table
The chaoticcancer mathematical model (
Recently, the global dynamics of model (
The LCIS method is applied in order to define the localizing domain
Dynamics of the chaoticcancer system are directly related to the value of parameter
All simulations illustrated in this section are developed with the following characteristics: the chaotic attractor with
All compact invariant sets of the chaoticcancer system (
It should be noted that the boundaries of the domain
First, let us take a simple linear localizing function that will allow us to compute the upper bound of the tumor population; such function is given by
Now, we compute its Lie derivative as follows:
Solutions to the cancer cells equation (
Further, in order to get the maximum value of the healthy cells population we exploit the next localizing function
Three solutions to the healthy cells equation (
Now, by applying the Iterative Theorem we can establish an upper bound for the effector cells populations by proposing a localizing function with a combination of linear and nonlinear terms as given below
The three solutions of the effector cells population are located inside the bound
The latter three functions allow us to compute all upper bounds. Now, in order to complement our results, we take two nonlinear localizing functions that allow us to improve the localizing domain where all compact invariant sets of the chaoticcancer system (
Another nonlinear localizing function we analyze is given by
Now, in Figure
Phase plane
The localizing domains are defined by the sets
Lyapunov’s direct method can be applied together with the LCIS method to derive sufficient conditions under which the bounded domain
If conditions (
Now, in order to illustrate the previous result, we perform numerical simulations with different initial conditions outside the
The existence of a Bounded Positively Invariant Domain implies that, given any nonnegative initial condition, all trajectories will go inside the domain, as shown with the chaotic attractor, or to its boundary, as one can see with the equilibrium point.
One can see that the
Therefore, to demonstrate Theorem
Biomathematics has been a subject of increasing interest through the last years. Mathematical models can be helpful to comprehend complex biological phenomena such as cancer. These models may provide critical insights concerning the short and longterm evolution of this large group of diseases.
Nonetheless, having a mathematical model that is able to accurately describe tumor growth, the corresponding immune response, the competition between cancer cells with other healthy cells, and even the effects of treatments is only the first step in order to understand cancer evolution on different reallife scenarios. Hence, it is necessary to establish a methodology that allows the scientific community to develop proper treatment schedules that may be able to decrease tumor burden while also diminishing their side effects, thus, improving the overall health and increasing life expectancy of the patient.
Up to date, several approaches have been developed throughout the years to study and control the dynamics of cancer systems described by ordinary differential equations; one may see [
As future work we intend to apply our approach to develop our own models concerning tumor evolution, the immune response, and the application of treatments as control parameters, such as chemotherapy and immunotherapy either alone [
In fact, it was proposed to consider a positive influx of immune cells to further investigate if it is possible to destroy the chaotic regime of the system. The latter has been done in many works, e.g., [
A constant influx of effector cells is able to eliminate the chaotic dynamics on the system (
It is evident that all solutions of the system go to the tumorfree equilibrium point; i.e.,
We study the global dynamics of a chaoticcancer system given by a set of three ordinary differential equations by means of the LCIS method, Lyapunov’s direct method, and LaSalle’s invariance principle. Our main contribution consists in the definition of a Bounded Positively Invariant Domain where all compact invariant sets of the system are located; this
The LCIS method and the Iterative Theorem allow us to determine all upper bounds for the system (
Additionally, by means of Lyapunov stability theory and LaSalle’s invariance principle we were able to derive sufficient conditions for the existence of a global attractor in the nonnegative octant. The latter implies that all trajectories with initial conditions inside the domain
About the biological implications of the results we have determined that global stability can only be achieved if the maximum recruitment rate of effector cells is stronger than its inactivation rate by cancer cells. The latter implies that if the immune system is not able to detect and attack cancer cells, a treatment should be considered to control tumor growth as we briefly elaborated in our discussion section.
Our numerical simulations illustrate in Figure
As future work, we intend to rescale all parameters in order to investigate the time of occurrence of this event (e.g.,
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is fulfilled within the TecNM project number 6575.18P “Ingeniería aplicada mediante el modelizado matemático para comprender la evolución del cáncer, la respuesta inmunológica y el efecto de algunos tratamientos”, and the TecNM project number 6178.17P “Modelos matemáticos para cáncer, VIH y enfisema”.