In this paper, we develop a definite tumor-immune model considering Allee effect. The deterministic model is studied qualitatively by mathematical analysis method, including the positivity, boundness, and local stability of the solution. In addition, we explore the effect of random factors on the transition of the tumor-immune system from a stable coexistence equilibrium point to a stable tumor-free equilibrium point. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. And we find that the Allee effect has the greatest impact on the number of cells in the system when the Allee threshold value is within a certain range; the intensity of random factors could affect the likelihood of the system crossing from the coexistence equilibrium to the tumor-free equilibrium.

1. Introduction

With the continuous improvement of living standards, individuals pay more attention to their own health. As one of the most harmful diseases to human health and life, malignant tumor has aroused wide concern in the world. At the same time, it has also become a research hotspot in academia. Thompson [1] discussed strategies to target YAP/TAZ activity in cancer and suggested that YAP and TAZ are important drivers of solid tumor growth, metastasis, and resistance to therapy. Du et al. [2] represented a first-time report of NSCLC-intrinsic PD-1 expression and a potential mechanism by which PD-1 blockade may promote cancer growth. Studies on ascorbic acid, which chemosensitize colorectal cancer cells and synergistically inhibit tumor growth, were presented in [3]. Chaoul et al. [4] found that the lack of MHC class II molecules was beneficial to the infiltration of T-cells into tumors, thus enhancing the control of tumor growth. Villalobos et al. [5] demonstrated the anticancer effect of SOCS1 in prostate cancer and identified its potential inhibitory mechanism. Volz et al. [6] studied a new family of potent TLR9 agonists, EnanDIM, and it can inhibit tumor growth.

Although there are some achievements on finding factors which can effectively inhibit or control tumor growth. What we should not ignore is that the human body itself has a mechanism that can resist tumor cells—the immune system. It can not only protect the body from various internal and external factors against the normal immune system but also defend the body from tumor interference so that the normal operation of the body tissues. This function of the immune system has urged mathematicians and biologists exert to the research on its dynamic properties. Adriana et al. [7] discussed the regulation of angiogenesis by innate immune cells in the tumor microenvironment, specific features, and roles of major players. Key steps for visualizing the complex interaction between malignant tumors and the immune system were proposed and discussed in [8]. Jin et al. [9] summarized the current understanding of autophagy and its regulation in cancer. Medzhitov and Janeway [10] introduced the role of innate immune mechanism in the development of organisms and illustrated that it is the first line of defense of human immunity. The dynamics between tumor and immune cells in microenvironment can be referred to [1114]. Khajanchi [15] described how tumor cells evolve and survive after encounter with the immune system. Terry et al. [16] presented new insights on the role of EMT in tumor-immune escape.

At present, the research on tumor-immune system has obtained above results, and Katrin et al. [17] showed that tumors have the Allee effect. According to the study of literature [18], the Allee effect is a correlation between any component of individual fitness and the number or density of the same individual. Since the Allee effect has been shown to change the optimal control decisions and control costs and estimate the potential risks in ecology, Katrin et al. [17] believed that it was also crucial for tumor growth control. However, so far, it has been neglected in tumor growth and persistence. Therefore, under the excitation of the above research, we attempt to add the Allee effect on the basis of Pillis [19] model, study its dynamic behavior, and explore the influence of Allee effect on tumor cells.

The remainder of the paper is organized as follows. In Section 2, we develop a definite tumor-immune model with the Allee effect and analyze the property of the deterministic model, including the positivity, boundness, and local stability. Section 3 is devoted to discuss the tumor-immune model with the Allee effect, which is disturbed by white noise. We carry out numerical simulations in Section 4. In Section 5, we take a discussion and draw a conclusion.

2. The Deterministic Tumor-Immune Model with Allee Effect

2.1. Establishment of Deterministic Model

In 2003, Pillis and Radunskaya [20, 21] proposed a mathematical model to describe the relationship between the tumor and immune system:

Yet, for the tumor itself, it has its own optimal density in the process of growth and reproduction, whether it is too dense or too thin will have an inhibitory effect on its growth. Based on this, we study the relationship between tumor cells, normal cells, and immune cells under the background of the Allee effect. The mathematical model is as follows:where represents the number of immune cells, is the amount of tumor cells, can be thought of the quantity of normal cells, and represent the resource constant and mortality of effector immune cells, respectively, and is Allee threshold and satisfies the condition of . Since the presence of tumor cells would activate effector immune cells, this term is expressed as by Michaelis–Menten expression. Competition between effector immune cells and tumor cells would lead to the death of two types of cells, which are expressed as and . The maximum growth rate and environmental carrying capacity of normal cells are represented by and , respectively. Similarly, the tumor cells are expressed by and . stand for the cell death rate caused by the competition between normal cells and tumor cells, respectively.

In this study, we assume that the tumor is homogeneous. And according to the physical meaning of each variable, the initial conditions of model (2) are

2.2. Positiveness and Boundedness of Solutions

Theorem 1. For any greater than 0, the solution of system (2) is positive and bounded under the initial conditions (3), when .

Proof. (1)Positiveness: integrating both sides of formula (2), we have the following relations:The initial condition is . When , it can be obtained from the above equation:(2)Boundedness: considering a function when according to equation (2), we can obtain thatThen, the above equation can be written as follows:where and , where is the maximum value obtained by within the range of and is a positive number selected which is greater than the difference between and . We then haveWhen , . This shows that, under initial conditions (3), the solution of system (2) is bounded.

2.3. Classification of Equilibrium Points

According to equation (2), we get the following three kinds of equilibrium points:(1)Tumor-free equilibrium point: .(2)Coexisting equilibrium point: , where satisfies the equationand the coefficients of equation (9) are(3)Abnormal equilibrium point: , where is determined by the equationand the coefficients of above equation are

2.4. Stability of Equilibrium Points

In this section, we analyze the local stability of the equilibrium points. The calculation shows that the number of normal cells at the abnormal equilibrium point is zero, which is not consistent with the practical significance of biology, so we only discuss the local asymptotic stability of tumor-free equilibrium point and coexistence equilibrium point.

The Jacobian matrix of system (2) at the equilibrium point is as follows:

2.4.1. Stability of Tumor-Free Equilibrium Point

The eigenvalues of Jacobian matrix at are

And by calculation, the equilibrium solution is locally asymptotically stable if and only if , while is the saddle point with index two when .

2.4.2. Stability of Coexistence Equilibrium Point

When the equilibrium point is , the characteristic equation of matrix (13) at this point iswhere satisfies equation (9), and

According to Routh–Hurwitz criterion, we know that

The determinant of Hurwitz can be calculated according to the above expressions:

Thus, when are satisfied at the same time, coexistence balance is locally asymptotically stable.

By equation (9), we know that coexistence equilibrium expression is the root of quartic equation, namely, there are at most four coexistence equilibrium points. Considering the meaningful number of equilibrium and its stability is related to parameter selection, and in order to make the results more intuitive, we carry out numerical simulation according to effective range of variables. As a result, the number and stability of the coexistence equilibrium point are shown in Figure 1. In this numerical simulation, the horizontal axis is value from 0 to 2, while the vertical axis is value from −2 to 1.

In order to reveal the role of the Allee effect in the system, we only change the value of . By referring to [20, 21], we make ; then, the time history diagrams of three cells in different are as follows.

Observing Figure 2, we know when , the number of the three types of cells do not change significantly. When belongs to , the cell change trend is the most significant. This indicates that when the Allee threshold of tumor cells is within the range of −1.2 to −0.5, the Allee effect has the greatest influence on the number of cells in the system, and this also proves that the Allee effect has an impact on the tumor-immune system.

3. Stochastic Tumor-Immune Model with Allee Effect

3.1. Establishment of Stochastic Model

Considering that the growth of tumor cells can be affected by various uncertainties, such as vibration and temperature, and these can be simulated with noise, therefore, based on the deterministic model (2), we propose the following stochastic model:

Among them, , , and are three random processes, which represent the number of immune cells, normal cells, and tumor cells, respectively. is the independent Gaussian white noise with zero mean intensity , which acts on the natural growth rate of normal cells and the nature growth rate of tumor , and we assume that both damping and excitation are small. The specific meanings of other parameters are the same as those in model (2).

3.2. The Stochastic Averaging Method

In order to facilitate the application of the stochastic average method, we first simplified model (19), setting

Then, model (19) can be written as follows:

The above equation (21) is converted into the following stochastic differential equation by formula:where and are independent Wiener process and and are the Wong–Zakai amendments. Defining a random process,

Among them, . , , and are positive, . Then, the stochastic differential equation of random process is

Using the random average principle of Stratorovich-Khasminskii, the average stochastic differential equation of is

Whence,and according to the formula,

We have the following expressions:where

Then, with the help of equations (26)–(30), the drift coefficient and diffusion coefficient can be simplified as follows:

3.2.1. Stationary Probability Density

Base on equation (23), we obtain that

Substitute (33) and (34) into (23); then, it can be written as follows:

Due to equations,thus, we obtain that

When the boundary is nonsingular; when it is the second kind of singular boundary. The diffusion index the drift index . Therefore, the nontrivial stationary probability density exists. And its steady-state probability density function satisfies the following FPK equation:


Solving the FPK equation, we then havewhere is a normalization constant, and

Hence, equation (40) can be simplified as follows:

Suppose , the conditional probability density of random process is ; then, the joint probability density of and can be expressed as follows:

As a result, we obtain the stationary probability density of and :

Thus, we get the steady-state probability densities of and , respectively. They are

3.2.2. Mean First-Passage Time

When people are suffering from tumor, we hope that the number of tumor cells go to zero. However, according to the calculation in the above section, we know that the steady-state probability density of tumor cells is present, which indicates that tumor cells will not be extinct under natural state, but the effects of random factors may change this. Therefore, it is practical to study the average first-passage time from the coexistence equilibrium point to the tumor-free equilibrium point, and the Pontryagin equation is satisfied thatwhere

From the above studies, we know that is nonsingular and is natural. According to the constant variation method, can be written as follows:where is the integral constant, and

4. Numerical Simulation

Refer to the experimental data of article [2029] (Table 1) and the analysis of stability of equilibria in the previous section, and we obtain that system (19) of the tumor-free equilibrium is stable, and the corresponding critical state is denoted by . Calculate based on Table 1 and reference to the Figure 1, we know the result of coexisting equilibrium belongs to the red area when the value is −0.8, −0.9, and −1.0. Namely, system (19) has only one stable coexistence equilibrium, and the corresponding critical state is denoted by . As a result, is nonsingular, is natural, and the expression of the first-passage time from to satisfies equation (46). In order to investigate the effect of the Allee threshold and noise intensity on the first-passage time, we numerically simulate the above results and obtain Figures 3 and 4.

As can be seen from Figures 3 and 4, the time of first passage is limited only when the random intensity is within a certain range. This indicates that the existence of random factors does not cause the first-passage to occur, and only when the random intensity is within a certain range will the system cross from the coexistence equilibrium to the tumor-free equilibrium.

5. Conclusions

In this paper, we analyze the property of the deterministic tumor-immune model under the Allee effect, including the positivity and boundness. The results show that, in the condition of , the solution of system (2) is positive and bounded. At the same time, we know that the Allee effect affects the growth and reproduction of cells in the tumor-immune system according to Figure 2. And the influence is most significant when the Allee threshold of tumor cells is in the range of −1.2 to −0.5.

And in the meantime, we are devoted to explore the effect of random factors on the transition of the tumor-immune system from the stable coexistence equilibrium to the stable tumor-free equilibrium. Based on the method of stochastic averaging, we obtain the expressions of the steady-state probability density and the mean first-passage time. Through the numerical simulation about the expression of first-passage time, we know that the existence of random factors will not make the first-passage must happen, in other words, the random strength only within a certain range; the system will pass from the stable coexistence equilibrium point to the tumor-free equilibrium point. Our conclusions provide a possible method to change the number of tumor cells, which is expected to be helpful for the treatment of tumors.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.


This research was funded by the National Natural Science Foundation (11772002), Ningxia Higher Education First-class Discipline Construction Funding Project (NXYLXK2017B09), Major Special Project of North Minzu University (ZDZX201902), Open Project of the Key Laboratory of Intelligent Information and Big Data Processing of NingXia Province (2019KLBD008), “Light of the West” Talent Training Program of Chinese Academy of Sciences (XAB2016AW04), and Graduate Innovation Project of North Minzu University (YCX19130).