Dear Siavash !

I work in another field of research, but also am involved in modeling nonequilibrium processes. I am not an expert in biophysics, so I can be mistaken in my assumptions. But, perhaps, my assumptions will be useful to you

Your model is represented by a system of nonlinear equations. Thanks to its nonlinearity, a variety of phase curves is provided, the trajectories of which change with the slightest change in external parameters. In such systems, even asymptotic stability cannot be considered absolute.

Unfortunately, in your question you did not indicate the meaning of the parameters T(t) and E(t) of your system. Therefore, I cannot identify the obtained singular points with the specific states of the system. However, I can make such assumptions:

1) The stable node to which the epsilon 3 curve arrives probably characterizes the critical tumor size at which death occurs. The phase trajectory of this curve determines the duration of the evolution of the system, and, accordingly, the growth of this tumor to a critical size. To know this duration, you can construct evolutionary curves in dimensional form.

2) Curve epsilon 2 falls into an unstable node. Although I think that this point has a saddle character. This condition shows that the tumor is not stable, and can both grow, leading to the phase trajectory getting to a stable node, and decrease, depending on external influences. The system cannot stay at this state for a long time, because the internal energy of the system must get its minimum. Remember that a minimal change in external conditions significantly affects the evolution of the system. Therefore, curve 2 must continue its evolution to tha stable state.

3) The epsilon 3 curve shows that with such parameters, the tumor will not develop and will remain in some germinal state. You indicate this state as an unstable node. But judging by your graphs, it looks more like a special point such as a steady focus with a limit cycle inside.

I apologize if I did not understand your question exactly.

With best regards,

Dr. Olga Mazur.

Dear O. Yu. Mazur

Thank you for your answer regarding my question, so I can understand the details of the figure better.

In this equation, E(t) is the effector cells, T(t) is tumour cells , I(t) is Immune cells, by the way, if you want to know the detail of the work, I can send you the article or I can explain more detail in order to understand my question better.

Dear Siavash Mohammadi!

Your work is very interesting, and I will be glad to study your articles.

After your answer, I realized that your equations describe the process of ordering of two types of cells (immune and tumor). Effector cells act as activators for immune cells. Therefore, phase portraits of E–T represent the evolution of the development and growth of tumor and immune cells. In this case, purely for analytical reasons, we can assume that the phase portrait should have 3 singular points:

1) a stable node characterizing the development of a tumor to a critical size – death. You have this point (epsilon curve 1). It is seen that tumor cells significantly exceed the volume of immune cells. Indeed, if a person dies of a tumor, his body will still have a small local volume of immune cells. Therefore, in this stable node, the number of immune cells you have is slightly more than zero.

2) steady focus – remission. This point characterizes the balance at which immune cells defeat tumor cells. This condition can conditionally be called recovery, since your curve (epsilon 3) has reached the limit cycle. This condition cannot be characterized by a stable node, since a sick person always has a chance of a relapse of the disease. This is evidenced by the nonzero value of T cells in this region. If a full recovery were supposed, then on your graph on the abscissa axis a stable node with coordinates (T = 0, E> 0) would appear.

3) a saddle point characterizing the stage of cell ordering, at which the spatial volumes of all types of cells are in a state of metastability. This is observed on the epsilon 2 curve. In this state, the probability of developing a tumor is 50%, as well as the probability of its collapse. Near this point, the system stays for a while. Metastable cell balance will not last long, and as a result, one type of cell will begin to prevail over another. After this, the system will continue to evolve to one of the stable states - a stable node (death) or a stable focus (remission). What condition the system will come to after kinetic inhibition near the saddle point depends on the initial conditions (general human health, genetic predisposition to cancer, age, medical history, etc.).

A numerical solution is, in fact, a game with parameters. It allows you to make assumptions under many different conditions. But for a better understanding of the laws of evolution from these parameters, I would advise you to look for the analytical roots of the system, even in the asymptotic approximation. So you can determine not only the coordinates and types of singular points, but also see the area of attraction of the characteristic states of the system. If you are interested, do not hesitate to contact.

With best regards,

Dr. Olga Mazur.

Dear Dr. Olga Mazur !

Thanks for your suggestions regarding my work.

The following file which I sent to you, is the article that I am working in my PhD program.

Kind regards,

SIAVASH MOHAMMADI.

Dear Siavash Mohammadi!

Thank you for the article. I am very interested to read it.

With best regatds,

Dr. Olga Mazur.