On the Dynamics of Immune-Tumor Conjugates in a Four-Dimensional Tumor Model

Abstract

We examine the ultimate dynamics of the four-dimensional model describing interactions between host cells, immune cells, tumor cells, and immune-tumor conjugate cells proposed by Abernethy and Gooding in 2018. In our paper, the ultimate upper bounds for all variables of this model are obtained. Formulas for positively invariant sets are deduced. Using these results, we establish conditions for the existence of the global attractor, derive formulas for its location, and present conditions under which immune and immune-tumor conjugate cells asymptotically die out. Next, we study equilibrium points, including the stability property for most of the equilibrium points. We discuss the existence of very low cancer-burden equilibrium points. Next, parametric conditions are derived under which the derivative of the density of the immune-tumor conjugate cell population eventually tends to zero; this mathematically rigorously confirms the correctness of the application of model reduction for this model in studies of its ultimate dynamics. In the final section, we summarize the results of this work and outline how to continue this study.

1. Introduction

Cancer is a group of diseases that is characterized by the uncontrollable proliferation and spread of tumor cells that elicit a host immune response. This reaction has a very complex form and has been investigated in mathematical oncology for 30 years. In 1994, Kuznetsov et al. [1] published an innovative work devoted to the study of interactions of cytotoxic T-lymphocytes with immunogenic tumor cells. The two-dimensional model described there is based on the classical “prey–predator” scheme taken from the theory of population dynamics, where populations of tumor cells/immune cells are prey/predators correspondingly. In 1998, Kirschner and Panetta [2], employing results of Kuznetsov et al. [1], proposed the three-dimensional model in which they also included the dynamics of cytokines apart from tumor and immune cells and termed modeling the application of immunotherapy. The important outcome of this work is related to studies of cases in which tumor cell eradication is possible, at least at the local level. Another extension of the Kuznetsov et al. [1] model was described by dePillis and Radunskaya [3], where dynamics of the host, tumor, and immune cell populations under the application of chemotherapy were examined.

Various studies relating to the existence of chaotic dynamics of the dePillis–Radunskaya model [3] have been performed by Itik and Banks [4], Duarte et al. [5], Letellier et al. [6], Galindo et al. [7], Abernethy and Gooding [8], Das et al. [9]. Results of studies of a location of chaotic attractor and other compact invariant sets have been given by authors of this work in [10]. The four-dimensional extension of the model [3] by including the endothelial population was proposed in [11], with the goal of studying angiogenic switch. The ultimate dynamics of the latter model have been investigated by one of the authors in [12], where various global cancer-eradication conditions and endothelial cell-eradication conditions (the case of localized cancer) have been derived.

The four-dimensional extension of the model [3] by including the endothelial population has been proposed in [11] with the goal of studying angiogenic switch. The ultimate dynamics of the latter model were investigated by one of the authors in [12], where various global cancer-eradication conditions and endothelial cell-eradication conditions (the case of localized cancer) were derived. It is worth mentioning that, currently, papers have been published in which the dynamics of cancer models with time delay have been explored; see, e.g., [13,14].

The problem of the existence and computing-invariant manifolds of nonlinear systems described by ordinary differential equations is closely connected with model-reduction methods aimed at simplifying the analysis of complex/multidimensional dynamics. In the life sciences, the quasi-steady-state approximation (QSSA) method, see, e.g., the book of Hoppensteadt [15], and also, [16], is widely used. For example, on this subject, see articles [8,17,18,19,20] in the area of mathematical oncology, paper [21] in the area of mathematics of infectious diseases, etc. Some other model-reduction methods have been presented or discussed in [22,23,24,25,26,27].

The problem of searching/computing a surface containing all $\omega$-limit sets arises naturally in problems of studying the ultimate dynamics of a nonlinear system. Such a surface may or may not be invariant and appears in research in the context of applications of the LaSalle theorem. In models of mathematical oncology, dynamics under which all $\omega$-limit sets are contained in the tumor cell-free plane corresponds to the asymptotic cancer-eradication process.

In this work, we will consider the cancer model introduced earlier in the paper [8] (see system (6a)–(6d) there),

$$\begin{array}{ccc}\hfill \frac{dH}{dt}& =& {\tilde{\rho }}_{1}H(1-\frac{H}{b_1})-{\tilde{\alpha }}_{13}HT,\hfill \\ \hfill \frac{dI}{dt}& =& \frac{{\tilde{\rho }}_{2}IT}{\tilde{g}+T}-k_{1}IT+(k_{-1}+k_2)C-\tilde{\delta }I,\hfill \\ \hfill \frac{dT}{dt}& =& {\tilde{\rho }}_{3}T(1-\frac{T}{b_2})-k_{1}IT+(k_{-1}+k_3)C-{\tilde{\alpha }}_{31}HT,\hfill \\ \hfill \frac{dC}{dt}& =& k_{1}IT+(k_{-1}+k_2+k_3)C.\hfill \end{array}$$

(1)

Here by H/I/T/C we denote the density of host/immune/tumour/immune-tumour conjugate cell populations respectively. This model is derived from the model of Kuznetsov et al. [1] and takes into account not only immune and tumour cells, but also immune-tumour conjugate cell population, which is the result of the interaction of immune and tumour cells with each other, and, in addition, interactions of tumour cells with host cells. Here pairs of rates ${\tilde{\rho }}_1;b_1$ and ${\tilde{\rho }}_3;b_2$ characterize the logistic growth law for host cells and tumour cells respectively; the production of immune cells is taken in the Michaelis–Menten form and is characterized by ${\tilde{\rho }}_2;g$; further, ${\tilde{\alpha }}_{13};{\tilde{\alpha }}_{31}$ are rates of interactions between host-tumour cells; immune-tumour cells respectively arising owing to competition for resources and space; $\tilde{\delta }$ is the death rate of immune cells. Parameters $k_{-1};k_1;k_2;k_3$ are rates of biochemical reactions:

$$\begin{array}{ccc}\hfill I+T& \rightleftarrows & { }_{k_{-1}}^{k_1}C,\hfill \\ \hfill C& \to & { }^{k_2}I+T^{*},\hfill \\ \hfill C& \to & { }^{k_3}{I}^{*}+T.\hfill \end{array}$$

Here, $I^{*}$ and $T^{*}$ represent the density of inactivated immune cells and lethally hit tumor cells correspondingly; $k_1$ is the rate of forming conjugates from immune and tumor cells; $k_{-1}$ is the rate of destruction of immune-tumor conjugates; $k_2$ is the rate of tumor cell death; $k_3$ is the rate of immune cell inactivation. All parameters of the system (1) are positive.

The system (1) in scaled variables

$$x_1=\frac{H}{b_1},x_2=\frac{I}{b_2},x_3=\frac{T}{b_2},x_4=\frac{C}{b_2},t^{\prime }=t{\tilde{\rho }}_3$$

has the following form:

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& a_1{x}_1(1-x_1)-a_2{x}_1{x}_3,\hfill \\ \hfill {\dot{x}}_2& =& \frac{a_3{x}_2{x}_3}{g+x_3}-a_4{x}_2{x}_3+a_5{x}_4-a_6{x}_2,\hfill \\ \hfill {\dot{x}}_3& =& x_3(1-x_3)-a_4{x}_2{x}_3+a_7{x}_4-a_8{x}_1{x}_3,\hfill \\ \hfill {\dot{x}}_4& =& a_4{x}_2{x}_3-a_9{x}_4,\hfill \end{array}$$

(2)

where $(x_1;x_2;x_3;x_4)=x\in {\mathbb{R}}_{+,0}^4=\{x\ge 0\}$, $\dot{\{.\}}=d\{.\}/d{t}^{\prime },$

$$a_1=\frac{{\tilde{\rho }}_1}{{\tilde{\rho }}_3},a_2=\frac{{\tilde{\alpha }}_{13}{b}_2}{{\tilde{\rho }}_3},a_3=\frac{{\tilde{\rho }}_2}{{\tilde{\rho }}_3},g=\frac{\tilde{g}}{b_2},a_4=\frac{k_1{b}_2}{{\tilde{\rho }}_3},a_5=\frac{k_{-1}+k_2}{{\tilde{\rho }}_3},$$

$$a_6=\frac{\tilde{\delta }}{{\tilde{\rho }}_3},a_7=\frac{k_{-1}+k_3}{{\tilde{\rho }}_3},a_8=\frac{{\tilde{\alpha }}_{31}{b}_1}{{\tilde{\rho }}_3},a_9=\frac{k_{-1}+k_2+k_3}{{\tilde{\rho }}_3}.$$

Obviously,

$$a_5<a_9,a_7<a_9.$$

(3)

In papers [1,8], the dynamics of the system (2) were explored using the following equation:

$$a_4{x}_2{x}_3-a_9{x}_4=0$$

(4)

with the goal of model reduction by the QSSA. As a result, dynamic analysis in [8] has been carried out for 3D system corresponding to ${\dot{x}}_i$-equations, $i=1,2,3$. This was explained by the fact that the influx of immune cells occurs very slowly. However, there was no strict mathematical justification for such a model reduction. Moreover, it is known that the use of the QSSA can lead to the loss of many properties of the original system, as shown in [28].

Our approach is based on using the localization method of compact invariant sets (LMCIS) [29] and the LaSalle theorem.

The purpose of this work is as follows:

(1)

We study the ultimate dynamics of four-dimensional Equation (2). In particular,

(a)

we calculate the parameters of the polytopes containing all compact invariant sets;

(b)

these polytopes contain an attractor of (2);

(c)

we find equilibrium points;

(d)

we obtain conditions for the location of $\omega$–limit sets in the invariant plane denoted by ${\mathrm{\Pi }}_{24}:x_2=0$; $x_4=0$;

(e)

we study the asymptotic stability of equilibrium points belonging to the set ${\mathrm{\Pi }}_{24}$.

(2)

Next, it is demonstrated in the example of the system (2) how the reduction of the model can be justified. For this, we describe conditions under which all solutions are attracted to the surface (4) at an exponential rate. This justifies the model-reduction applied in papers [1,8] for exploring (2) in the case indicated in Theorem 5. As a result, assertions contained there have been rigorously validated. To say it in another way, we find conditions under which the derivative of the density of immune-tumor conjugates tends to zero, i.e., the surface (4) characterizes ultimate dynamics.

The structure of this paper is as follows. In Section 2, helpful statements used in this work are included. Furthermore, in Section 3, the ultimate upper bounds are obtained for all state variables; it is shown that all trajectories are defined for all $t>0$; we give conditions on parameters under which we describe the bounded positively invariant localization domain containing the attractor of the system. We find conditions under which the system (2) has only convergence dynamics. It is worth pointing out that the restriction of (2) on the invariant plane ${\mathrm{\Pi }}_{24}$ is a classical prey–predator model, for which we will describe its dynamics for the sake of completeness. We describe four cases of convergent dynamics when inner trajectories tend to equilibrium points located in ${\mathrm{\Pi }}_{24}$. In Section 4, using purely analytical considerations, we find conditions under which the $\omega$-limit set of each trajectory is contained in the surface (4). In Section 5, we provide formulas and conditions of existence for all equilibrium points in ${\mathbb{R}}_{+,0}^4$. Next, we explore stability/instability for equilibrium points lying in ${\mathrm{\Pi }}_{24}$. Furthermore, in Section 5, we discuss the existence problem of equilibrium points located outside ${\mathrm{\Pi }}_{24}$, including very low cancer-burden equilibrium point. Section 6 contains some concluding remarks.

2. Some Preliminaries

We consider a nonlinear system

$$\dot{x}=F\left(x\right),$$

(5)

where $x\in {\mathbb{R}}^n$, $F\left(x\right)$ is a differentiable vector field. Below, we use the following frequently encountered definition of an attractor of a system (5). Specifically, it is an invariant compact attracting set.

Let $h\in C^1\left({\mathbb{R}}^n\right)$ be a function such that h is not the first integral of the system (5). The function h is used to solve the localization problem of compact invariant sets and is called a localizing function. Suppose that we are interested in the localization of all compact invariant sets located in the set $U\subset {\mathbb{R}}^n$. By $S(h,U)$ ($S\left(h\right)$), we denote the set $\{\dot{h}\left(x\right)=0\}\cap U$ ($\{x\in {\mathbb{R}}^n|\dot{h}\left(x\right)=0\}$), where $\dot{h}\left(x\right)$ is a derivative of h with respect to the system (5). Furthermore, we define

$$h_{inf}\left(U\right):=inf\{h\left(x\right)\mid x\in S(h,U)\};h_{sup}\left(U\right):=sup\{h\left(x\right)\mid x\in S(h,U)\}.$$

Assertion 1 

([29]). For any $h\left(x\right)\in C^1\left({\mathbb{R}}^n\right)$, all compact invariant sets of the system (5) located in U are contained in the sets $\{h\left(x\right)\le h_{sup}\left(U\right)\}\cap U$, $\{h_{inf}\left(U\right)\le h\left(x\right)\}\cap U$ and $K(h,U)=\{h_{inf}\left(U\right)\le h\left(x\right)\le h_{sup}\left(U\right)\}\cap U$ as well.

By ${\psi |}_U$, we denote the restriction of a function $\psi \in C^1\left({\mathbb{R}}^n\right)$ to the set U. In what follows, all objects will be considered only within ${\mathbb{R}}_{+,0}^4.$

3. Localizations of Compact Invariant Sets for the System (2)

3.1. Ultimate Upper Bounds for System Variables and Formulas for Positively Invariant Domains

At the beginning, we note that ${\mathbb{R}}_{+,0}^4$ is a positively invariant set of the system (2) since all components of the vector field take nonnegative values on the boundary of ${\mathbb{R}}_{+,0}^4$.

To derive ultimate upper bounds, we begin from the variable $x_1.$

Lemma 1. 

For any $\tau \ge 0$, all compact invariant sets are contained in the positively invariant set

$$K_1\left(\tau \right)=\{x_1\le 1+\tau \}\cap {\mathbb{R}}_{+,0}^4.$$

Proof. 

Applying $h_1=x_1$, we obtain the set

$$S(h_1,{\mathbb{R}}_{+,0}^4)=\{x_1(a_1(1-x_1)-a_2{x}_3)=0\}\cap {\mathbb{R}}_{+,0}^4;$$

the extreme values

$$h_{1inf}\left({\mathbb{R}}_{+,0}^4\right)=0,h_{1sup}\left({\mathbb{R}}_{+,0}^4\right)=1$$

and find the localization set

$$K(h_1,{\mathbb{R}}_{+,0}^4)=\{0\le x_1\le 1\}\cap {\mathbb{R}}_{+,0}^4=K_1.$$

The derivative ${\dot{h}}_1$ is negative in the set ${\mathbb{R}}_{+,0}^4\setminus K_1=\{x:x_1>1\}\cap {\mathbb{R}}_{+,0}^4$. Therefore, the set $K_1\left(\tau \right)$ is positively invariant for any $\tau \ge 0$. □

Lemma 2. 

For any $\nu \ge 0$, all compact invariant sets are contained in the positively invariant set

$$K_2\left(\nu \right)=\{x_3+x_4\le a+\nu \}\cap {\mathbb{R}}_{+,0}^4,a=\frac{{(1+a_9-a_7)}^2}{4(a_9-a_7)}.$$

Proof. 

Let us use the function $h_2=x_3+x_4$ and we obtain that

$${\dot{h}}_2=x_3(1-x_3)+a_7{x}_4-a_8{x}_1{x}_3-a_9{x}_4.$$

Therefore, in the set $S(h_2,{\mathbb{R}}_{+,0}^4)$, the inequality

$$(a_9-a_7)h_2\le -x_3^2+(1+a_9-a_7)x_3$$

is fulfilled. Under the condition of (3), we obtain inequality

$$h_2{|}_{S(h_2,{\mathbb{R}}_{+,0}^4)}\le \frac{-x_3^2+(1+a_9-a_7)x_3}{a_9-a_7}\le \underset{x_3\ge 0}{max}\frac{-x_3^2+(1+a_9-a_7)x_3}{a_9-a_7}=a$$

and find that $h_{2sup}\left({\mathbb{R}}_{+,0}^4\right)\le a$, $h_{2inf}\left({\mathbb{R}}_{+,0}^4\right)=0$. Now, we come to the localization set

$$K(h_2,{\mathbb{R}}_{+,0}^4)\subset \{x_3+x_4\le a\}\cap {\mathbb{R}}_{+,0}^4=K_2.$$

The derivative ${\dot{h}}_2$ is negative in the set ${\mathbb{R}}_{+,0}^4\setminus K_2=\{x:h_2>a\ge h_{2sup}\left({\mathbb{R}}_{+,0}^4\right)\}\cap {\mathbb{R}}_{+,0}^4$ because if $h_2=x_3+x_4=a+d$, $d>0$, i.e., $x_4=a-x_3+d$, then

$${\dot{h}}_2\le x_3(1-x_3)+(a_7-a_9)(a-x_3+d)=-x_3^2+(1-a_7+a_9)x_3+(a_7-a_9)(a+d)\le$$

$$\le \frac{{(1-a_7+a_9)}^2}{4}-\frac{{(1+a_9-a_7)}^2}{4}+(a_7-a_9)d=(a_7-a_9)d<0.$$

Therefore, the set $K_2\left(\nu \right)$ is positively invariant. □

Let

$$\xi \left(x_3\right):=a_6+a_4{x}_3-\frac{a_3{x}_3}{g+x_3},{\xi }_1(x_3,\nu )=\frac{a_5(a+\nu -x_3)}{\xi \left(x_3\right)}.$$

It is easy to see that the following statement holds:

Proposition 1. 

If the inequality

$$\sqrt{a_3}<\sqrt{a_{4}g}+\sqrt{a_6}$$

(6)

is fulfilled, then $\xi \left(x_3\right)>0$ for $x_3\ge 0$ and $b\left(\nu \right)={max}_{x_3\ge 0}{\xi }_1(x_3,\nu )<+\infty$, $m>0$, where

$$m=\underset{x_3\ge 0}{min}\xi \left(x_3\right)=\left\{\begin{array}{cc}{a}_6,\hfill & a_3<a_{4}g,\hfill \\ a_6-{(\sqrt{a_3}-\sqrt{a_{4}g})}^2,\hfill & a_3\ge a_{4}g.\hfill \end{array}\right.$$

Proof. 

We have that $\xi \left(0\right)=a_6>0$, ${lim}_{x_3\to +\infty }\xi \left(x_3\right)=+\infty$ and

$${\xi }^{\prime }\left(x_3\right)=a_4-\frac{a_{3}g}{{(g+x_3)}^2}=0$$

for

$$x_3=x_{31}=\sqrt{\frac{a_{3}g}{a_4}}-g;x_3=x_{32}=-\sqrt{\frac{a_{3}g}{a_4}}-g.$$

If $\sqrt{\frac{a_{3}g}{a_4}}-g<0$, i.e., $\sqrt{a_3}<\sqrt{a_{4}g}$, then ${\xi }^{\prime }\left(x_3\right)>0$ for $x_3>0$ and ${min}_{x_3\ge 0}\xi \left(x_3\right)=\xi \left(0\right)=a_6>0$.

If $\sqrt{\frac{a_{3}g}{a_4}}-g\ge 0$, i.e., $\sqrt{a_3}\ge \sqrt{a_{4}g}$, then ${min}_{x_3\ge 0}\xi \left(x_3\right)=\xi \left(x_{31}\right)$ and $\xi \left(x_{31}\right)>0$ if

$$\begin{array}{ccc}\hfill (a_6-a_{4}g-a_3)\sqrt{\frac{a_{3}g}{a_4}}+2a_{3}g& =& \sqrt{\frac{a_{3}g}{a_4}}(a_6-a_{4}g-a_3+2\sqrt{a_3{a}_{4}g})\hfill \\ & =& \sqrt{\frac{a_{3}g}{a_4}}(a_6-{(\sqrt{a_3}-\sqrt{a_{4}g})}^2)>0,\hfill \end{array}$$

i.e., $\sqrt{a_3}<\sqrt{a_{4}g}+\sqrt{a_6}$. □

Lemma 3. 

If Inequality (6) is performed, then, for any $\nu \ge 0$, $\eta \ge 0$, all compact invariant sets are contained in the positively invariant set

$$K_3(\nu ,\eta )=\{x:x_2\le b\left(\nu \right)+\eta \}\cap K_2\left(\nu \right).$$

Proof. 

We utilize the function $h_3=x_2$ and find that

$$S\left(h_3\right)=\{a_5{x}_4-x_2\xi \left(x_3\right)=0\},$$

where $\xi \left(x_3\right)>0$ for $x_3\ge 0$. In the set $S(h_3,K_2\left(\nu \right))$ we obtain that

$$h_3=x_2=\frac{a_5{x}_4}{\xi \left(x_3\right)}\le \frac{a_5(a+\nu -x_3)}{\xi \left(x_3\right)}={\xi }_1(x_3,\nu )\le b\left(\nu \right).$$

Therefore, we obtain the extreme values

$$h_{3inf}\left(K_2\left(\nu \right)\right)=0,h_{3sup}\left(K_2\left(\nu \right)\right)\le b\left(\nu \right).$$

Therefore, we come to the localization set

$$K(h_3,K_2\left(\nu \right))=\{x_2\le h_{3sup}\left(K_2\left(\nu \right)\right)\}\cap K_2\left(\nu \right)\subset \{x_2\le b\left(\nu \right)\}\cap K_2\left(\nu \right)=K_3\left(\nu \right).$$

The derivative ${\dot{h}}_3$ is negative in the set

$$K_2\left(\nu \right)\setminus K_3\left(\nu \right)=\{x:x_2>b\left(\nu \right)\ge h_{3sup}\left(K_2\left(\nu \right)\right)\}\cap K_2\left(\nu \right).$$

Therefore, for any $\nu \ge 0$, $\eta \ge 0$ the set $K_3(\nu ,\eta )$ is positively invariant. □

As a result, we come to

Theorem 1. 

If Inequality (6) is performed, then, for any $\tau \ge 0$, $\nu \ge 0$, $\eta \ge 0$ all compact invariant sets of the system (2) are contained in the positively invariant compact set

$$K(\tau ,\nu ,\eta )=K_1\left(\tau \right)\cap K_3(\nu ,\eta )=\{x_1\le 1+\tau ;x_2\le b\left(\nu \right)+\eta ;x_3+x_4\le a+\nu \}\cap {\mathbb{R}}_{+,0}^4.$$

Corollary 1. 

If Inequality (6) is fulfilled, then, for any point $x_{(}0)=(x_{10};x_{20};x_{30};x_{40})\in {\mathbb{R}}_{+,0}^4$ exists a positively invariant compact set containing it.

Proof. 

Consider the positively invariant compact set $K(\tau ,\nu ,\eta )$ with $\tau ={\tau }_0=max\{0;x_{10}-1\}$, $\nu ={\nu }_0=max\{0;x_{30}+x_{40}-a\}$, $\eta ={\eta }_0=max\{0;x_{20}-b\left({\nu }_0\right)\}$. Obviously, $x\left(0\right)\in K({\tau }_0,{\nu }_0,{\eta }_0)$. □

Corollary 2. 

If Inequality (6) is fulfilled, then, any solution $\gamma (t,x_0)$, $\gamma (0,x_0)=x_0$ of the system (2) can be extended to the interval $[0,+\infty )$.

Proof. 

It is known [30] that the solution of the autonomous $C^1$- system with an initial value from a compact set of the phase space continues forward either infinitely or up to the boundary of this compact set.

We apply this fact in the case of a positively invariant compact set that contains the initial state $x_0$. Let $x_0\in K({\tau }_0,{\nu }_0,{\eta }_0)$ then the solution $\gamma (t,x_0)$ does not reach the boundary of the positively invariant compact set $K({\tau }_0,{\nu }_0,{\eta }_0)$ in finite time, and therefore, it continues forward infinitely. □

Corollary 3. 

If Inequality (6) is fulfilled, then any solution $\gamma (t,x_0)$, $\gamma (0,x_0)=x_0$ of the system (2) is bounded.

Proof. 

Let $x_0\in K({\tau }_0,{\nu }_0,{\eta }_0)$, then the trajectory $\gamma (t,x_0)$ cannot leave the positively invariant compact set $K({\tau }_0,{\nu }_0,{\eta }_0)$ and, therefore, it is bounded. □

Corollary 4. 

If Inequality (6) is fulfilled, then the positively invariant compact set $K(0,0,0)$ contains the attractor of the system (2).

Proof. 

All trajectories of the system are bounded, and therefore, they tend to their $\omega$-limit sets, which, as invariant compact sets, are contained in the compact set $K(0,0,0)$. The closure of the union of all invariant compact sets of the system is the attractor of the system in $K(0,0,0)$. □

3.2. Additional Localizations and Extinction Conditions of $x_2$;$x_4$-Cell Populations

The positively invariant sets $K_1\left(\tau \right)$ and $K_2\left(\nu \right)$ are found for all parameter values, while the set $K_3(\nu ,\eta )$ is found when executing Inequality (6). As a result, this inequality falls into the conditions of Theorem 1 and its corollaries 1–4.

Here, we construct a positively invariant set while fulfilling other constraints on parameter values, which complement the statements of corollaries 1–4 and are used to obtain extinction conditions of $x_2$;$x_4$-cell populations.

First, we establish the following general property characterizing interactions between immune cells and immune-tumor conjugates.

Lemma 4. 

Suppose that

$$a_6>a_9.$$

(7)

Then, for any $\omega \ge 0$ all compact invariant sets are in the positively invariant set

$$K_4\left(\omega \right)=\{x_2-{\xi }_{*}{x}_4\le \omega \}\cap {\mathbb{R}}_{+,0}^4$$

with

$${\xi }_{*}>max\{\frac{a_3-a_{4}g}{a_{4}g};\frac{a_5}{a_6-a_9}\}.$$

(8)

Proof. 

We take $h_4=x_2-\xi x_4$ and find that

$$S\left(h_4\right)=\{{\dot{h}}_4=x_2{x}_3(-a_4+\frac{a_3}{g+x_3}-\xi a_4)+x_4(\xi a_9+a_5)-a_6{x}_2=0\}.$$

Then

$$x_2-\frac{\xi a_9+a_5}{a_6}{x}_4\le \frac{x_2{x}_3}{a_6}(-a_4+\frac{a_3}{g}-\xi a_4)$$

on the set $S(h_4,{\mathbb{R}}_{+,0}^4)$. Let us consider the conditions

$$\frac{a_3}{g}-a_4\le \xi a_4;\xi \ge \frac{\xi a_9+a_5}{a_6}.$$

They are satisfied with the choice of $\xi ={\xi }_{*}$ in (8). Let us take the function $h_{4*}=x_2-{\xi }_{*}{x}_4$. Now, we find that

$$h_{4*}=x_2-{\xi }_{*}{x}_4\le x_2-\frac{{\xi }_{*}{a}_9+a_5}{a_6}{x}_4\le \frac{x_2{x}_3}{a_6}(-a_4+\frac{a_3}{g}-{\xi }_{*}{a}_4)\le 0$$

in ${\mathbb{R}}_{+,0}^4$; $h_{4*sup}\left({\mathbb{R}}_{+,0}^4\right)=0$ on the set $S(h_{4*},{\mathbb{R}}_{+,0}^4)$ and the set $\{x_2-{\xi }_{*}{x}_4\le 0\}\cap {\mathbb{R}}_{+,0}^4$ contains all compact invariant sets.

The set $K_4\left(\omega \right)$ is positively invariant because the derivative ${\dot{h}}_{4*}$ is negative in the set $\{x_2-{\xi }_{*}{x}_4>\omega \}\cap {\mathbb{R}}_{+,0}^4$. Indeed, for $h_{4*}=x_2-{\xi }_{*}{x}_4=d$, $d>0$ we get that

$${\dot{h}}_{4*}\le x_2{x}_3(-a_4+\frac{a_3}{g+x_3}-{\xi }_{*}{a}_4)+x_4{a}_6{\xi }_{*}-a_6(d+{\xi }_{*}{x}_4)\le -a_{6}d<0$$

because $a_5<(a_6-a_9){\xi }_{*}$. □

As in the case of Theorem 1 and its corollaries, we arrive at the following result.

Theorem 2. 

If Inequality (7) is performed, then (1) for any $\tau \ge 0$, $\nu \ge 0$, $\omega \ge 0$ all compact invariant sets of the system (2) are contained in the positively invariant compact set

$$L(\tau ,\nu ,\omega )=K_1\left(\tau \right)\cap K_2\left(\nu \right)\cap K_4\left(\omega \right)=\{x_1\le 1+\tau ;x_2-{\xi }_{*}{x}_4\le \omega ;x_3+x_4\le a+\nu \}\cap {\mathbb{R}}_{+,0}^4;$$

(2) any solution of the system (2) can be extended to the interval $[0,+\infty )$ and is bounded; and (3) the positively invariant compact set

$$L(0,0,0)=\{x_1\le 1;x_2-{\xi }_{*}{x}_4\le 0;x_3+x_4\le a\}\cap {\mathbb{R}}_{+,0}^4$$

contains the attractor of the system (2).

In the following result, we describe one specific type of ultimate behavior of (2) under which both populations of immune cells and immune-tumor conjugates die out.

Theorem 3. 

Assume that

$$a_6>a_3.$$

(9)

Then, (1) for any $\tau \ge 0$, $\nu \ge 0$, $\rho \ge 0$ all compact invariant sets of the system (2) are contained in the positively invariant compact set

$$M(\tau ,\nu ,\rho )=\{x_1\le 1+\tau ;x_2+x_4\le \rho ;x_3+x_4\le a+\nu \}\cap {\mathbb{R}}_{+,0}^4;$$

(2) any solution of the system (2) can be extended over the entire interval $[0,+\infty )$ and is bounded; and (3) the positively invariant compact set

$$M(0,0,0)=\{x_1\le 1;x_2=0;x_3\le a;x_4=0\}\cap {\mathbb{R}}_{+,0}^4$$

contains the attractor of the system (2).

Proof. 

We take the function $h_5=x_2+x_4.$ Then we have within the positively invariant set $K_2\left(\nu \right)$

$${\dot{h}}_5=\frac{a_3{x}_2{x}_3}{g+x_3}+a_5{x}_4-a_6{x}_2-a_9{x}_4\le (a_3-a_6)x_2+(a_5-a_9)x_4\le 0;$$

${\dot{h}}_5=0$ iff $x_2=0$, $x_4=0$ and $h_{5sup}\left(K_2\left(\nu \right)\right)=0$. Therefore, the localization set $\{x_2+x_4\le 0\}\cap K_2\left(\nu \right)$ contain all invariant compact sets.

In the set $\{x_2+x_4>0\}\cap K_2\left(\nu \right)$ the derivative ${\dot{h}}_5$ is negative because for any $x_2\ge 0$, $x_4\ge 0$, $x_2+x_4=d$, $d>0$ we arrive to two cases $x_2\ge d/2$, or $x_4\ge d/2$. In the first case

$${\dot{h}}_5=\frac{a_3{x}_2{x}_3}{g+x_3}+a_5{x}_4-a_6{x}_2-a_9{x}_4\le (a_3-a_6)\frac{d}{2}<0$$

and in the second case

$${\dot{h}}_5=\frac{a_3{x}_2{x}_3}{g+x_3}+a_5{x}_4-a_6{x}_2-a_9{x}_4\le (a_5-a_9)\frac{d}{2}<0.$$

Therefore, the set $\{h_5=x_2+x_4\le \rho \}\cap K_2\left(\nu \right)$ and the compact set

$$\{h_5=x_2+x_4\le \rho \}\cap K_2\left(\nu \right)\cap K_1\left(\tau \right)=M(\tau ,\nu ,\rho )$$

are positively invariant. □

Theorem 4. 

Assume that

$$a_6>a_9,a_4{\xi }_{*}a<a_9.$$

(10)

Then, the positively invariant set $M(0,0,0)$ contains the attractor of the system (2).

Proof. 

The positively invariant bounded set $L(0,0,0)$ contains the attractor. For the function $h_6=x_4$, we have

$${\dot{h}}_6{|}_{L(0,0,0)}\le x_4(a_4{\xi }_{*}{x}_3-a_9){{|}_{L(0,0,0)}\le x_4(a_4{\xi }_{*}a-a_9)|}_{L(0,0,0)}\le 0;$$

${\dot{h}}_6{|}_{L(0,0,0)}=0$ iff $h_6=0$. Therefore, the attractor of the system (2) lies in the localization set

$$\{x_4\le 0\}\cap L(0,0,0)=\{x_4=0\}\cap L(0,0,0)=M(0,0,0).$$

□

3.3. Attraction to Equilibrium Points in the Invariant Plane

The system (2) has the invariant plane $x_2=0$; $x_4=0$ denoted by ${\mathrm{\Pi }}_{24}$. If one of the conditions (9) or (10) is fulfilled, then the system (2) has an attractor lying in the set $M(0;0;0)\subset {\mathrm{\Pi }}_{24}$. Consequently, the ultimate dynamics of the system (2) in this case is determined by the dynamics of the system (2) restricted on the invariant plane ${\mathrm{\Pi }}_{24}$

$$\begin{array}{ccc}\hfill {\dot{x}}_1& =& x_1(a_1(1-x_1)-a_2{x}_3)\hfill \\ \hfill {\dot{x}}_3& =& x_3(1-x_3-a_8{x}_1).\hfill \end{array}$$

(11)

This system is a classical predator–prey model, and its dynamics are well known [31]; see also [32]. Please note that if $\widehat{E}(x_{1*};x_{3*})$ is an equilibrium point of system (11), then it corresponds to the equilibrium point $E(x_{1*};0;x_{3*};0)$ in the invariant plane of the system (2) and vice versa.

The system (11) has three boundary equilibrium points: ${\widehat{E}}_0(0;0)$ is an unstable node; ${\widehat{E}}_1(1;0)$ is a stable node for $a_8>1$ and a saddle for $a_8<1$; ${\widehat{E}}_2(0;1)$ is a stable node for $a_1<a_2$ and a saddle for $a_1>a_2$.

Conditions of the feasibility of the inner equilibrium point ${\widehat{E}}_3(e_{31};e_{33})$, $e_{31}>0$; $e_{33}>0$, where

$$e_{31}=\frac{a_1-a_2}{a_1-a_2{a}_8};e_{33}=\frac{a_1(1-a_8)}{a_1-a_2{a}_8}$$

are

$$a_1>a_2;a_8<1,$$

(12)

when ${\widehat{E}}_3$ is a stable node, and

$$a_1<a_2;a_8>1,$$

(13)

when ${\widehat{E}}_3$ is a saddle.

Using the Dulac function $\zeta =x_1^{-1}{x}_3^{-1}$, we see that there are no closed invariant orbits in the virtue of the Dulac theorem. Therefore, each trajectory of (11) tends to one of the equilibrium points given above. More precisely, we have four cases of qualitatively different dynamics of the system (11), i.e., four cases of different ultimate dynamics of the system (2) in ${\mathbb{R}}_{+}^4$.

• In case (12), the stable node ${\widehat{E}}_3$ attracts all inner trajectories in ${\mathbb{R}}_{+}^2=\{x_1;x_3>0\}$. The rest of the equilibrium points are unstable. The equilibrium point ${\widehat{E}}_0(0;0)$ is an unstable node; both points ${\widehat{E}}_2(0;1)$ and ${\widehat{E}}_1(1;0)$ are saddles.
  Therefore, in case (12), the equilibrium point $E_3(e_{31};0;e_{33};0)$ attracts all trajectories of the system (2) in the set ${\mathbb{R}}_{+}^4$.
• In case (13), the point ${\widehat{E}}_3$ is a saddle; ${\widehat{E}}_0(0;0)$ is an unstable node; points ${\widehat{E}}_2(0;1)$ and ${\widehat{E}}_1(1;0)$ are stable nodes. The stable manifold $W^{+}$ of ${\widehat{E}}_3$ splits ${\mathbb{R}}_{+}^2$ in two regions. In one of them, trajectories go to ${\widehat{E}}_1(1;0)$, and, in the other, they tend to ${\widehat{E}}_2(0;1)$.
  In this case, the equilibrium point $E_3(e_{31};0;e_{33};0)$ of the system (2) is a saddle. Its stable manifold splits ${\mathbb{R}}_{+}^4$ in two domains. In one of them, trajectories of the system (2) go to $E_1(1;0;0;0)$, and, in the other, they tend to $E_2(0;0;1;0)$.
• If $a_1>a_2$; $a_8>1$, then there is no equilibrium point of the system (11) in ${\mathbb{R}}_{+}^2$. The points ${\widehat{E}}_0(0;0)$ and ${\widehat{E}}_2(0;1)$ are unstable, but ${\widehat{E}}_1(1,0)$ is stable and inner trajectories of the system (11) tend to ${\widehat{E}}_1(1;0).$
  In this case, the equilibrium point $E_1(1;0;0;0)$ attracts all inner trajectories of the system (2).
• If $a_1<a_2$; $a_8<1$, then there is no inner equilibrium point in ${\mathbb{R}}_{+}^2$. The points ${\widehat{E}}_0(0;0)$ and ${\widehat{E}}_1(1;0)$ are unstable, but ${\widehat{E}}_2(0;1)$ is stable and inner trajectories of the system (11) tend to ${\widehat{E}}_2(0;1).$
  In this case, the equilibrium point $E_2(0;0;1;0)$ attracts all inner trajectories of the system (2).

4. Justification of the Model Reduction to the 3D System

Here, we find conditions under which all $\omega$-limit sets are in the surface $a_4{x}_2{x}_3-a_9{x}_4=0$. This means that for sufficiently large T, the function ${\dot{x}}_4\left(t\right)\approx 0$$,t>T,$ and this gives algebraic parametric conditions that analytically substantiate the correctness of using the model reduction in [8] to analyze ultimate dynamics.

Lemma 5. 

Suppose that the inequalities

$$a_6\ge 1,g(a_4+1)-a_3>\frac{g{a}_4{a}_5}{a_9}$$

(14)

are fulfilled. Then, all compact invariant sets are contained in the positively invariant localization set

$$K_5=\{a_4{x}_2{x}_3-a_9{x}_4\le 0\}\cap {\mathbb{R}}_{+,0}^4.$$

Proof. 

We apply the function $h_7=a_4{x}_2{x}_3-a_9{x}_4$ and compute that the set $S\left(h_7\right)$ is defined by the equality

$$\begin{array}{ccc}\hfill {\dot{h}}_7& =& a_4{x}_3[\frac{a_3{x}_2{x}_3}{g+x_3}-a_4{x}_2{x}_3+a_5{x}_4-a_6{x}_2]\hfill \\ & +& a_4{x}_2[x_3-x_3^2-a_4{x}_2{x}_3+a_4{x}_4-a_8{x}_1{x}_3]-a_9{h}_7\hfill \\ & =& (h_4+a_9{x}_4)[\frac{a_3{x}_3}{g+x_3}-a_4{x}_3-a_6+1-x_3-a_4{x}_2-a_8{x}_1]+a_4{x}_4(a_5{x}_3+a_4{x}_2)-a_9{h}_7\hfill \\ & =& h_7\alpha \left(x\right)+\beta \left(x\right)=0,\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill \alpha \left(x\right)& =& -a_9-a_6+1-x_3(\frac{-a_3}{g+x_3}+a_4+1)-a_4{x}_2-a_8{x}_1,\hfill \\ \hfill \beta \left(x\right)& =& x_4[(-a_6+1)a_9+x_3(\frac{a_3{a}_9}{g+x_3}+(a_5-a_9)a_4-a_9)-a_8{a}_9{x}_1+(a_7-a_9)a_4{x}_2],\hfill \end{array}$$

and we utilize $a_4{x}_2{x}_3=h_7+a_9{x}_4$. Now, it follows from conditions of this theorem that

$$\begin{array}{ccc}\hfill -a_6+1& \le & 0;\hfill \\ \hfill (a_4+1)g-a_3& >& 0;\hfill \\ \hfill a_3{a}_9+g(-a_4{a}_9+a_4{a}_5-a_9)& <& 0.\hfill \end{array}$$

Therefore, $\alpha \left(x\right)<0$, $\beta \left(x\right)\le 0$ in ${\mathbb{R}}_{+,0}^4$. Thus, we get that $h_7\le 0$ on the set $S(h_7,{\mathbb{R}}_{+,0}^4)$ and ${sup}_{S(h_7,{\mathbb{R}}_{+,0}^4)}{h}_7=0$. Hence, all compact invariant sets are in the localization set $K_5$. This set is positively invariant because ${\dot{h}}_7\left(x\right)<0$ in the set ${\mathbb{R}}_{+,0}^4\setminus \{a_4{x}_2{x}_3-a_9{x}_4\le 0\}=\{h_7>0\}\cap {\mathbb{R}}_{+,0}^4$. □

Theorem 5. 

Suppose that conditions (6) and (14) are fulfilled. Then, all ω-limit sets are in the set

$$K_6=\{a_4{x}_2{x}_3-a_9{x}_4=0\}\cap {\mathbb{R}}_{+,0}^4.$$

Proof. 

All solutions are bounded (Corollary 3). Using the function $h_6=x_4$ with respect to the set $K_5$. We conclude that in virtue of the LaSalle theorem, all $\omega$-limit sets are in the set $\{{\dot{h}}_6=0\}\cap K_5=K_6$. □

Remark 1. 

It follows from the formula for $\alpha \left(x\right)$ that the function $h_7$ is decreasing on trajectories at the exponential rate $-a_9-a_6+1<0$.

5. Equilibrium Points of the System (2)

5.1. Existence Conditions

Here, we find equilibrium points of the system (2) by solving the system of equations

$$\begin{array}{ccc}\hfill & & a_1{x}_1(1-x_1)-a_2{x}_1{x}_3=0,\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill & & {\displaystyle \frac{a_3{x}_2{x}_3}{g+x_3}-a_4{x}_2{x}_3+a_5{x}_4-a_6{x}_2=0,}\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill & & x_3(1-x_3)-a_4{x}_2{x}_3+a_7{x}_4-a_8{x}_1{x}_3=0,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill & & a_4{x}_2{x}_3-a_9{x}_4=0,\hfill \end{array}$$

(18)

in the nonnegative orthant.

In Section 3.3, we have indicated the four equilibrium points $E_0(0;0;0;0)$; $E_1(1;0;0;0)$; $E_2(0;0;1;0)$ and $E_3(e_{31};0;e_{33};0)$ in the invariant plane ${\mathrm{\Pi }}_{24}$. Conditions of feasibility of $E_3$ are (12) and (13).

Let us find the equilibrium points outside ${\mathrm{\Pi }}_{24}$.

Consider the case $x_2>0$. From Equation (18), we find

$$x_4=\frac{a_4{x}_2{x}_3}{a_9}$$

and substitute it into Equation (16). As a result, we come to the quadratic equation

$$y\left(x_3\right)=a_4(a_5-a_9)x_3^2+d_1{x}_3-a_6{a}_{9}g=0,d_1=a_9(a_3-a_6)+a_{4}g(a_5-a_9).$$

(19)

If $D_1=d_1^2+4a_4(a_5-a_9)a_6{a}_{9}g$, then Equation (19) has:

(1)

two positive roots for $d_1>0$, $D_1>0$;

(2)

one positive root for $d_1>0$, $D_1=0$;

(3)

no nonnegative roots in other cases.

For $x_3>0$, Equations (15) and (17) form the system

$$\begin{array}{ccc}\hfill & \hfill & x_1(a_1(1-x_1)-a_2{x}_3)=0,\hfill \\ \hfill & \hfill & 1-x_3-a_4{x}_2+\frac{a_7{a}_4}{a_9}{x}_2-a_8{x}_1=0\hfill \end{array}$$

(20)

for each positive root of Equation (19).

If $x_1=0$ in the system (20) then

$$a_4(\frac{a_7}{a_9}-1)x_2=x_3-1$$

and we obtain

Proposition 2. 

If $0<e_{43}<1$, where $e_{43}$ is the positive root of Equation (19), then the system (2) has the equilibrium point $E_4(0;e_{42};e_{43};e_{44})$; here,

$$e_{42}=\frac{a_9(e_{43}-1)}{a_4(a_7-a_9)},e_{44}=\frac{a_4{e}_{42}{e}_{43}}{a_9}.$$

If $x_1>0$ in the system (20), then

$${\displaystyle x_1=1-\frac{a_2}{a_1}{x}_3,a_4(\frac{a_7}{a_9}-1)x_2=x_3-1+a_8{x}_1}$$

and we obtain

Proposition 3. 

If inequalities

$$0<e_{53}<\frac{a_1}{a_2};(a_1-a_2{a}_8)e_{53}<a_1(1-a_8)$$

(21)

are fulfilled where $e_{53}$ is the positive root of Equation (19), then the system (2) has the equilibrium point $E_5(e_{51};e_{52};e_{53};e_{54})$, where

$$e_{51}=1-\frac{a_2}{a_1}{e}_{53},e_{52}=\frac{a_9(1-e_{53}-a_8{e}_{51})}{a_4(a_9-a_7)}=\frac{a_9(1-a_8-(1-\frac{a_2}{a_1}{a}_8)e_{53})}{a_4(a_9-a_7)},e_{54}=\frac{a_4{e}_{52}{e}_{53}}{a_9}.$$

Let us find the conditions when the system (2) may or may not have the inner equilibrium point $E_5$ with a very low cancer burden.

The roots of Equation (19) depend on the of parameters $A_1=\{a_3,a_4,a_5,a_6,a_6,a_9,g\}$, but the inequalities (21) depend on the values of other parameters $A_2=\{a_1,a_2,a_8\}$. Consider the inequalities (21). If

$$a_8>1;a_8>\frac{a_1}{a_2},$$

then we have

$$0<\frac{a_1(1-a_8)}{a_1-a_2{a}_8}<e_{53}<\frac{a_1}{a_2}$$

(22)

and if

$$\frac{a_1}{a_2}<a_8<1,$$

(23)

then, we obtain that

$$e_{53}<\frac{a_1}{a_2}.$$

Inequality (22) means that system (2) for any values of parameters $A_1$ does not have the point $E_5$ with the value of $e_{53}$ less than $a_1(1-a_8)/(a_1-a_2{a}_8)$. In the case of (23), the coordinate $e_{53}$ of the equilibrium point $E_5$ will be less than $a_1/a_2$ including the possible case when

$$e_{53}<<\frac{a_1}{a_2}.$$

5.2. Stability Conditions for Equilibrium Points Lying in ${\mathrm{\Pi }}_{24}$

First, we calculate the Jacobian matrix at any point:

$$\left(\begin{array}{cccc}{a}_1-2a_1{x}_1-a_2{x}_3& 0& -a_2{x}_1& 0\\ 0& \frac{a_3{x}_3}{g+x_3}-a_6-a_4{x}_3& \frac{g{a}_3{x}_2}{{(g+x_3)}^2}-a_4{x}_2& a_5\\ -a_8{x}_3& -a_4{x}_3& 1-a_8{x}_1-a_4{x}_2-2x_3& a_7\\ 0& a_4{x}_3& a_4{x}_2& -a_9\end{array}\right).$$

Next, we calculate the Jacobian matrix at $E_0(0;0;0;0):$

$$J\left(E_0\right)=\left(\begin{array}{cccc}{a}_1& 0& 0& 0\\ 0& -a_6& 0& a_5\\ 0& 0& 1& a_7\\ 0& 0& 0& -a_9\end{array}\right)$$

and discover $E_0$ is always a saddle point.

Furthermore, we establish that the Jacobian matrix at $E_1(1;0;0;0)$ is

$$J\left(E_1\right)=\left(\begin{array}{cccc}-a_1& 0& -a_2& 0\\ 0& -a_6& 0& a_5\\ 0& 0& 1-a_8& a_7\\ 0& 0& 0& -a_9\end{array}\right)$$

and discover that $E_1$ is asymptotically stable if $1-a_8<0.$

Furthermore, we establish that the Jacobian matrix at $E_2(0;0;1;0)$ is

$$J\left(E_2\right)=\left(\begin{array}{cccc}{a}_1-a_2& 0& 0& 0\\ 0& \frac{a_3}{g+1}-a_6-a_4& 0& a_5\\ -a_8& -a_4& -1& a_7\\ 0& a_4& 0& -a_9\end{array}\right)$$

and discover that $E_2$ is asymptotically stable if $a_1-a_2<0$, $(\frac{a_3}{g+1}-a_6-a_4)a_9+a_5{a}_4<0$.

At last, we calculate the Jacobian matrix at $E_3(e_{31};0;e_{33};0):$

$$J\left(E_3\right)=\left(\begin{array}{cccc}-a_1{e}_{31}& 0& -a_2{e}_{31}& 0\\ 0& \alpha & 0& a_5\\ -a_8{e}_{33}& -a_4{e}_{33}& -e_{33}& a_7\\ 0& a_4{e}_{33}& 0& -a_9\end{array}\right),$$

where

$$\alpha =\frac{a_3{e}_{33}}{g+e_{33}}-a_6-a_4{e}_{33}.$$

The characteristic polynomial of the matrix $J\left(E_3\right)$ is

$$\chi \left(\lambda \right)={\chi }_1\left(\lambda \right){\chi }_2\left(\lambda \right),$$

where

$$\begin{array}{ccc}\hfill {\chi }_1\left(\lambda \right)& =& {\lambda }^2+(a_1{e}_{31}+e_{33})\lambda +e_{31}{e}_{33}(a_1-a_2{a}_8),\hfill \\ \hfill {\chi }_2\left(\lambda \right)& =& {\lambda }^2+(a_9-\alpha )\lambda -(a_9\alpha +a_4{a}_5{e}_{33}).\hfill \end{array}$$

Theorem 6. 

The equilibrium point $E_3$ is asymptotically stable in the case of (12) if

$$a_9\alpha +a_4{a}_5{e}_{33}<0.$$

The equilibrium point $E_3$ is unstable in the case (13) and in the case (12) if

$$a_9\alpha +a_4{a}_5{e}_{33}>0\mathrm{or}{a}_9<\alpha .$$

The stability conditions for the equilibrium points $E_{4,5}$ look cumbersome and are not given here in general form. We note that for the stability of the equilibrium point $E_4$, it is necessary to fulfill the inequality $e_{43}>a_1/a_2$.

6. Concluding Remarks

In this article, we studied the ultimate dynamics of the four-dimensional cancer-growth model that was taken from [8].

Our main results are as follows:

• We find in Theorems 1 and 2 parameters of the positively invariant polytope containing the attractor of the system (2). These parameters depend on the ultimate upper bounds of host cells, tumor cells, immune cells, and immune-tumor conjugates.
• We have found the condition

  $$\sqrt{a_6}>min\{\sqrt{a_3}-\sqrt{a_{4}g};\sqrt{a_9};\sqrt{a_3}\},$$

  (24)

  under which the system (2) has an attractor. Inequality (24) follows from Corollary 4 and Theorems 2 and 3, in the proof of which the LMCIS was used to construct families of localization sets. Each of these families consists of positively invariant compact sets and fills the orthant $R_{+,0}^4$. The condition (9) is weaker than (6), but under condition (9), the attractor is contained in the invariant plane ${\mathrm{\Pi }}_{24}$. The same follows from Theorem 4 but under different conditions (10).
  In Theorem 4, we find two conditions under which all $\omega$-limit sets lay in the plane ${\mathrm{\Pi }}_{24}$. Curiously, they depend on only some subsets of parameters. In particular, the extinction of immune and immune-tumor conjugate cell populations occurs for any value of the parameter $a_8$ characterizing the competition between host cells and tumor cells in the equation for cancer dynamics provided (9), or the condition (10) holds.
• Since there are no healthy equilibrium points for this cancer model, it is important to consider various types of equilibrium points. We note that attracting very low cancer-burden equilibrium points may exist, but finding corresponding local attraction conditions is a challenging task that is not addressed in this work.
• We find algebraic conditions in Theorem 5, under which the surface (4) used by Abernethy and Gooding for model reduction of the system (2) is attractive. This provides the purely analytical validation of the correctness of using model reduction for (2).
  In the proof of Theorem 5, we employ the LMCIS and LaSalle theorem and the property of boundedness of all trajectories in ${\mathbb{R}}_{+,0}^4$. As a result, we can choose the positively invariant set $K_5$ corresponding to the function $h_7$ in which $h_7$ is nonpositive. In this case, our choice of the function $h_7$ is suggested by Equation (4), which defines the QSSA. The conditions of the LaSalle theorem for the function $h_6$ are satisfied. Hence, it is established that all trajectories in the nonnegative orthant tend to their $\omega$-limit sets located in the QSSA surface (4). All conditions under which these localization sets are constructed and the LaSalle theorem is applied to substantiate the use of model reduction to analyze the ultimate dynamics of the system (2). In our opinion, the use of LaSalle and LMCIS-type arguments in practical applications of model reduction is of significant interest.
• We believe that the inclusion of a drift term in the differential equation for immune cells, which corresponds to the application of a constant influx of immune cells can lead to the extinction dynamics of tumor and immune-tumor conjugate cells if we impose an appropriate restriction on the rate of influx. This could be a topic for future research on this issue.