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This paper aims to develop the mathematical model that explores the immune response to a tumor system as a prey-predator system. A deterministic model defining the dynamics of tumor growth progression and regression has been analyzed. Our analysis indicates the tumor recurring and dormancy on the cellular level in combination with resting and hunting cells. The model considered in the present study is a generalization of El-Gohary (2008) by introducing the Michaelis-Menten function. This function describes the stimulation process of the resting cells by the tumor cells in the presence of tumor specific antigens. Local and global stability analysis have been performed along with the numerical simulation to support our findings.

Cancer is one of the leading causes of death worldwide. Each year, the American Cancer Society projects the number of new cancer cases and death to estimate the contemporary cancer burden. In 2014, there will be an estimated 16 lakh new cancer cases diagnosed and around 5 lakh cancer deaths in the U.S. [

The body is made up of many types of cells. Normally, cells grow and divide to produce new cells in a controlled and orderly manner. Sometimes, however, new cells continue to be produced when they are not needed. As a result, a mass of extra tissue called a tumor may develop. Tumor can be cancerous (malignant) or noncancerous (benign). A benign tumor is a known as tumor cell mass that does not fragment and spread beyond its original area of growth. Generally, benign impact on the body is not harmful and easy to be treated. Benign tumor can be harmful by growing large enough to interfere with normal body functions. Malignant tumors are nonencapsulated growths of tumor cells that are harmful; they have no wall or clear-cut border and may spread or invade other parts of the body normal tissue. In course of time, the cancer cells interfere with the normal functioning of organs via lymph or blood vessels [

Researches are going in two directions to fight with cancer: one is experimental and the other is theoretical, that is, mathematical. Both experimentalists and theoreticians join hands to get rid of deadly disease, that is, cancer [

For understanding the interaction between tumor and immune cells, several researchers used the concept of prey-predator [

Cancer self-remission model using stochastic approach is studied by Sarkar and Banerjee [

In this section, we considered tumor progression and regression as a prey-predator like system. The predator is the immune system which slaughters the tumor cells (prey). In most of the mathematical models of the tumor-immune system, the response of the immune system is considered as a single population of cells, namely, effector cell [

The predator, that is, immune system, is eradicating tumor cells in two stages: one is hunting cells and another is resting cells. Here, we are considering that hunting cells can slaughter tumor cells, but resting cells cannot. The cellular immune response identifies and eliminates the tumor cells from the host because tumor cells produce some antigens on its outer surface. The strength of the immune response depends on the tumor antigenicity [

Considering the above biological mechanism, we have produced a mathematical model of tumor development in immune response. The model involves certain assumptions as follows:

logistic growth function is assumed for the growth of tumor cells in the absence of hunting CTL cells;

the tumor cells and hunting cells are being eradicated at a rate proportional to the densities of tumor cells and hunting predator cells according to the law of mass action;

the resting predator cells are converted to the hunting cells, either by direct contact with them or by contact with a fast diffusing substance (cytokines) produced by the hunting cells;

resting cells also follow logistic growth in absence of tumor cells;

once a hunting T cell has been converted, it will never return to the resting stage;

resting cells also were stimulated due to the presence of tumor cells, and this is considered by the Michaelis-Menten function.

Let

Define the following dimensionless variables to reduce the number of the system parameters:

To find the equilibrium point, we put the time rate of change as zero. Therefore, the system of (

The solution of (

The equilibrium point lies on the boundary on the positive octant; that is,

This will always exist.

The equilibrium point

where

The equilibrium point

Coexisting equilibrium point is

Furthermore, eliminating

where

The local asymptotic stability of each nonnegative equilibrium point has been studied by computing the variational matrix and finding the eigenvalues. For the stability of equilibrium points, the real parts of eigenvalues of variational matrix must be negative.

The variational matrix due to linearization of the system equations (

The local dynamical behavior of equilibrium points are investigated and obtained results by computing the variational matrices corresponding to each equilibrium point. The local asymptotic stability for each equilibrium point has been analyzed as follows

The variational matrix at

The eigenvalues of the

(ii) The variational matrix at

The eigenvalues of this matrix are

If

(iii) The variational matrix at

As

Which is only possible when

As

If

The system considered in this paper is determined by three nonnegative equilibrium points

If the equilibrium point

Define Dulac function

Now,

If the equilibrium point

Consider the following Lyapunov function around

This section is devoted to the study of a mathematical model presented by the system of (

Tumor cells, hunting cells, and resting cells with time for

Phase portraits corresponding to the system (

From a numerical simulation, we discovered that by introducing the simulating function in resting cell population due to the presence of tumor; while increase the growth rate of resting cells can control the progression of tumor. In Figure

An interaction between tumor cells as prey and immune system which consists of resting and hunting cells as predator has been studied. We analyze the model with regard to local and global stability of equilibrium points. Global stability of steady state

The authors have no conflict of interests related to the conduct and reporting of this research. They have no financial relationships with any organization.