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\title{Analysis of global dynamics based on localization of compact
invariant sets of \ some cancer tumor growth models }
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\author{ \twlsfb Konstantin E. Starkov
\affiliation{
Departamento de Control\\
CITEDI-IPN\\
Tijuana, B.C\\
konst@citedi.mx, konstarkov@hotmail.com
}
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\begin{document}
\maketitle
\begin{abstract}
In this work we describe recent results of dynamical analysis of two cancer growth models with help of the localization method of compact invariant sets. These models are the Owen-Sherratt system and the Kirschner-Panetta model.
\end{abstract}
\begin{keywords}
Polynomial system, cancer tumor growth model, localization, compact
invariant set, global stability
\end{keywords}
\section{ Introduction}
Nowadays it is well-recognized that the cancer is one of the most dangerous killers of the humankind. Many dynamic models are elaborated in order to study the progress of the cancer tumor and comprehensively apply efficient treatment strategies.
Dynamical analysis of models describing interactions between the cancer
tumor and the immune system of a patient is one of topics in mathematical
biology which attracts the attention of many researchers, see the review
paper \cite{Eftime} and \cite{dOnofrio}. For example, such mathematical models as the Kirschner-Panetta model, \cite{Kirsh}; Kuznetsov et. al. model \cite{kuz} and the dePillis- Radunskaya
models \cite{Pillis,Pilliss} are examined in many publications which can be easily seen from
web pages in Scholar Google. \ The goal of this work is to review main
results on dynamical analysis of two cancer tumor models which were very
recently published\ by the author and his colleagues, \cite{starpo,starco}.
\section{ The 1st model: the Owen- Sherratt system}
In 1997 Owen and Sherratt in \cite{owen} obtained the spatially independent model
describing the macrophage- tumor interactions:
\begin{equation}
\begin{array}{l}
\dot{x}_{1}=\frac{a_{1}x_{1}x_{4}}{a_{2}+x_{1}+x_{2}+x_{3}}%
+a_{3}+a_{4}x_{4}-a_{5}x_{1}x_{2}x_{4} \\
+a_{6}x_{5}-a_{7}x_{1}, \\
\dot{x}_{2}=\frac{a_{8}x_{2}}{a_{2}+x_{1}+x_{2}+x_{3}}%
-a_{9}x_{2}-a_{5}x_{1}x_{2}x_{4}, \\
\dot{x}_{3}=\frac{a_{10}x_{3}}{a_{2}+x_{1}+x_{2}+x_{3}}-a_{9}x_{3}, \\
\dot{x}_{4}=a_{11}x_{2}-a_{12}x_{4}, \\
\dot{x}_{5}=a_{5}x_{1}x_{2}x_{4}-a_{13}x_{5}.%
\end{array}
\label{CancerModel}
\end{equation}
In these equations by $x_{1}(t),...,x_{5}(t)$ we denote vectors of densities
of the macrophages, the mutant cells, the normal-tissue cells, the chemical
concentration and the macrophage- mutant complexes.
In (\ref{CancerModel}) we exploit parameters $a_{j},j=1,...,13,$ which
depend on biological parameters given in \cite{owen} by the following formulae%
\begin{eqnarray}
a_{1} &=&\alpha (N+N_{e});a_{2}=N;a_{3}=I;a_{4}=I\sigma; \nonumber \\
a_{5} &=& k_{1};a_{6}=k_{2};a_{7}=\delta _{l}; \label{parameters} \\
a_{8} &=& \xi \delta (N+N_{e});a_{9}=\delta ;a_{10}=\delta(N+N_{e}); \nonumber \\
a_{11} &=& \beta ;a_{12}=\delta _{f};a_{13}=k_{2}+\delta _{c}, \nonumber
\end{eqnarray}
Here: 1) $k_{1},k_{2}$ are two positive parameters characterizing the lysis
process in the two upper equations of the system (\ref{CancerModel}); 2) $N$
is a measure of the crowding response; 3) $N_{e}$ is the total equilibrium
density of all cell types in the normal tissue; 4) $\xi $ is the scaling
parameter of the mutant cell growth rate, $\xi >1$; 5) $\delta $ is the
growth rate in the normal tissue; 6) $\beta $ is the constant secretion rate
per unit of mutant-cell density; 7) $\delta _{f}$ is the rate of a linear
natural decay of the mutant density; 8) $\delta _{c}$ is the rate of a
linear natural decay of the complex density; 9) $\delta _{l}$ is the rate of
a linear natural decay of the macrophage density; 10) $\alpha $ is parameter
characterizing the proliferation term in the first equation in (\ref%
{CancerModel}); 11) $I$ and $\sigma $ are two positive parameters
characterizing the constant influx of tissue macrophages.
Despite of the existence of more than one hundred papers with references on
these three articles \ up to now dynamic properties of the system (\ref%
{CancerModel}) are not well investigated.
The proof of the existence of a bounded positively invariant domain (BPID)\
inside the positive orthant of biologically feasible states for cancer
growth models is a typical problem which is investigated in a number of
publications; some of them will be mentioned below. \ This is due to the
fact that ultimate upper and lower bounds for trajectories have a biological
sense characterizing the "ultimate" health condition of a patient and
efficiency of applied drug treatment and immunotherapy. Therefore finding of
bounds for a BPID is of essential interest as well. These bounds are
expressed as functions of the model parameters. If some of these parameters
can be manipulated by biologists the latter have chance to regulate global
dynamics via the process of a variation of parameters admissible for
manipulations. In this case this type of model parameters may be considered
as control parameters. In this relation one may mention the rate of tumor
cell lysis and the density of engineered macrophages in the blood vessel.
Further, if we may decrease sufficiently the upper bound for the density of
tumor cells by manipulating of some control parameters the health condition
of a patient may be characterized as a cancer remission. Not only upper
bounds but some lower bounds for the ultimate dynamics may be essential for
a characterization of health conditions because they provide conservative
estimates of densities of cells populations. For example, it concerns the
density of the macrophages population because macrophages are important for
general homeostasis, see comment e.g. in \cite{owen}.
However, up to the author knowledge, computing explicit upper and lower
bounds for the BPID is a problem which is not well-studied for the most of
cancer tumor growth models. For example, we mention that
1) for the de Pillis- Radunskaya model, see \cite{Pilliss}, the upper bound respecting
the ultimate value for effector immune cells has not \ been derived in \cite{Itik};
no lower bounds have not been found there as well;
2) for the Kirschner- Panetta model only \ the upper bound respecting the
ultimate value for the density of tumor cells and the lower bound respecting
the ultimate value for the concentration of effector molecules were given in
\cite{Kirshh};
3) Nani and Freedman established in \cite{nani} the existence of a bounded
positively invariant domain is established for one four-dimensional cancer
growth model; however, values for its bounds were not computed explicitly.
In some publications concerning cancer growth models, see e.g. \cite{Pilliss,Kirshh,lej,nani} one can meet results related to the study of $\omega -$ limit sets of
trajectories in the BPID. If the structure of these $\omega -$ limit sets is
not complex as it occurs in low- dimensional models, for example, if they
contain only equilibrium points then the ultimate health condition of a
patient may be characterized by the complete cancer clearance equilibrium
point, or the "small" tumor size equilibrium point, or the "death"
equilibrium point.
Below by the positive orthant $\mathbf{R}_{+}^{n}=\{x=(x_{1},...,x_{n})\in
\mathbf{R}^{n},x_{i}>0,i=1,...,n\}.$ Let $\mathbf{R}_{+,0}^{n}$ be the
closure of $\mathbf{R}_{+}^{n}.$
\section{Some preliminaries and notations}
The principal idea of the localization method of compact invariant sets is
to study extrema of some differentiable functions specially constructed and
to use the fact that they are bounded on trajectories taken from compact
invariant sets. This idea is realized with help of exploiting the first
order extremum conditions and in some cases of the high order extremum
conditions, see e.g. in \cite{Krishko,starkov}.
Let us introduce some objects and recall useful assertions, see in \cite{Krishko}. We
consider a $C^{\infty }-$ differentiable system
\begin{equation}
\dot{x}=F(x), \label{main-system-pol}
\end{equation}%
with $x\in \mathbf{R}^{n}$, $F(x)=(F_{1}(x),\dots ,F_{n}(x))^{T}$ and $%
F_{i}(x)\in C^{\infty }(\mathbf{R}^{n})$, $i=1,\dots ,n.$
Let $h(x)\in C^{\infty }(\mathbf{R}^{n})$ be a function such that $h$ is not
the first integral of the system (\ref{main-system-pol}). The function $h$
is used in the solution of 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 some set $N\subset
\mathbf{R}^{n}$ where $N$ \ is an invariant set for the system (\ref%
{main-system-pol}) or a domain. By $S(h)$ we denote the set $\{x\in \mathbf{R%
}^{n}:L_{F}h(x)=0\},$ where $L_{F}h(x)$ is a Lie derivative with respect to $%
F$. Further, we define
\begin{eqnarray*}
h_{\inf }(U) &:&=\inf \{h(x)\mid x\in U\cap S(h)\}; \\
h_{\sup }(U) &:&=\sup \{h(x)\mid x\in U\cap S(h)\}.
\end{eqnarray*}%
\textbf{Assertion 1. }\textit{1. For any }$h(x)\in C^{\infty }(R^{n})$%
\textit{\ all compact invariant sets of the system (\ref{main-system-pol})
located in }$N$\textit{\ are contained in the set defined by the formula }$%
K(N)=\{x\in N:h_{\inf }(N)\leq h(x)\leq h_{\sup }(N)\}$\textit{\ as well. 2.
If }$N\cap S(h)=\emptyset $\textit{\ then the system (\ref{main-system-pol})
has no compact invariant sets located in }$N$\textit{.}
Any of sets $K(N)$\ is called a localization set. Now we remind another
result called the iteration theorem:
\textbf{Assertion 2.}\textit{\ Let }$h_{m}(x),m=1,2,\dots $\textit{\ be a
sequence of functions from }$C^{\infty }(R^{n})$\textit{. Sets }%
\[
K_{1}=K_{h_{1}},\quad K_{m}=K_{m-1}\cap K_{m-1,m},\quad m>1,
\]%
\textit{with}
\[
\begin{array}{l}
K_{m-1,m}=\{x:h_{m,\inf }\leq h_{m}(x)\leq h_{m,\sup }\}, \\
h_{m,.\sup }=\sup\limits_{S_{h_{m}}\cap K_{m-1}}h_{m}(x), \\
h_{m,\inf }=\inf\limits_{S_{h_{m}}\cap K_{m-1}}h_{m}(x),%
\end{array}%
\]%
\textit{contain all compact invariant sets of of the system (\ref%
{main-system-pol}) and }$K_{1}\supseteq K_{2}\supseteq \dots \supseteq
K_{m}\supseteq \dots \;.$
Below by $f$ we denote the vector field which corresponds the system under
investigation.
\section{Compact localization set for the system (\protect\ref{CancerModel})}
Below in this section we derive the compact localization set containing all
compact invariant sets in $\mathbf{R}_{+,0}^{5}$ of the system (\ref%
{CancerModel}). It is easy to see that the domain $\mathbf{R}_{+,0}^{5}$ is
positively invariant which means that each trajectory in $\mathbf{R}%
_{+,0}^{5}$ cannot escape from $\mathbf{R}_{+,0}^{5}.$
Firstly, we present
\textbf{Theorem 1.}\textit{\ All compact invariant sets of the system (\ref%
{CancerModel}) are placed in the set }$\cap _{i=1}^{4}K(h_{i}),$\textit{\
with}
\begin{eqnarray*}
K(h_{1}) &=&\{0\leq x_{3}\leq x_{3\max }:=\frac{a_{10}-a_{2}a_{9}}{a_{9}}\};
\\
K(h_{2}) &=&\{0\leq x_{2}\leq x_{2\max }:=\frac{a_{8}-a_{2}a_{9}}{a_{9}}\};
\\
K(h_{3}) &=&\{0\leq x_{4}\leq x_{4\max }:=\frac{(a_{8}-a_{2}a_{9})a_{11}}{%
a_{9}a_{12}}\}; \\
K(h_{4}) &=&\{0\leq x_{1}+x_{5}\leq h_{4\max }: \\
&=&(a_{3}+
\\
&&\frac{(a_{1}+a_{4})(a_{8}-a_{2}a_{9})a_{11}}{a_{9}a_{12}})(\frac{1%
}{a_{7}}+\frac{1}{a_{13}-a_{6}})\}.
\end{eqnarray*}
Examining formulae for $L_{f}h_{i},i=1,...,4,$ one can see that all
trajectories in $\mathbf{R}_{+}^{5}$ enter into some compact domain and
remain there. More precisely, we establish
\textbf{Theorem 2. }\textit{The domain \ }$\cap _{i=1}^{4}K(h_{i})$\textit{\
is a positively invariant polytope for the system (\ref{CancerModel}).}
\section{Lower bounds}
Here we prove
\textbf{Proposition 1.}\textit{\ \ All compact invariant sets of (\ref%
{CancerModel}) are located in the set}%
\[
\{x_{1}+x_{5}>\frac{a_{3}}{M_{1}}\},
\]
\textit{with }$M_{1}:=\max \{a_{7};a_{13}-a_{6}\}.$
\textbf{Proposition 2.} \textit{All compact invariant sets of (\ref%
{CancerModel}) are located in the set}%
\[
\{x_{1}+x_{2}+2x_{5}>\frac{a_{3}}{M_{2}}\},
\]
\textit{with }$M_{2}:=\max \{a_{7};a_{9};a_{13}-\frac{1}{2}a_{6}\}.$
\textbf{Proposition 3.} \textit{All compact invariant sets of (\ref%
{CancerModel}) are located in the set}
\[
K(h_{6})=\{x_{1}\geq x_{1\min }:=\frac{a_{3}}{a_{5}x_{2\max }x_{4\max }+a_{7}%
}\}.
\]
\textbf{Proposition 4 }\textit{All compact invariant sets of (\ref%
{CancerModel}) are located in the set}%
\[
K(h_{7})=\{x_{1}-x_{2}\geq (a_{9}-a_{7})a_{9}^{-1}x_{1\min
}+(a_{3}-a_{8})a_{9}^{-1}\}
\]
\textit{provided }$a_{9}\geq a_{7}$\textit{\ \ or in }%
\[
K(h_{7})=\{x_{1}-x_{2}\geq (a_{9}-a_{7})a_{9}^{-1}x_{1\max
}+(a_{3}-a_{8})a_{9}^{-1}\}
\]
\textit{provided }$a_{9}\eta
_{1}>a_{10} $\textit{\ holds. Then all compact invariant sets located in }$%
\mathbf{R}_{+,0}^{5} $\textit{\ are contained in the set }%
\[
\{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cup
\{x_{3}=0;x_{1},x_{2},x_{4},x_{5}\geq 0\}.
\]%
\textit{2. If }$\eta _{1}>a_{8}$\textit{\ holds then all compact invariant
sets located in }$\mathbf{R}_{+,0}^{5}$\textit{\ are contained in the set}%
\[
\{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}.
\]
\textbf{Corollary 1}\textit{. In the first case of Theorem 3 the }$\omega -$%
\textit{\ limit set }$\omega (x)$\textit{\ of any trajectory }$\varphi (x,t)$%
\textit{\ in }$\mathbf{R}_{+,0}^{5}$\textit{\ is contained in the set}
\begin{eqnarray*}
\{\{x_{2} &=&x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cup
\{x_{3}=0;x_{1},x_{2},x_{4},x_{5}\geq 0\}\} \\
&&\cap K(h_{2})\cap K(h_{3})\cap K(h_{6})
\end{eqnarray*}%
\textit{while in the second case of Theorem 3 the }$\omega -$\textit{\ limit
set }$\omega (x)$\textit{\ of any trajectory }$\varphi (x,t)$\textit{\ in} $%
\mathbf{R}_{+,0}^{5}$ \textit{is located in the set}
\begin{equation}
\{x_{2}=x_{5}=0;x_{1},x_{3},x_{4}\geq 0\}\cap K(h_{2})\cap K(h_{3})\cap
K(h_{6}) \label{40}
\end{equation}
\section{ The Kirschner- Panetta model}
In what follows, we describe recent results concerning dynamical analysis of
the Kirschner- Panetta model, see [6]. The paper [6] has more than 180
citations and is of essential interest for biologists and, in particular,
for specialists in mathematical biology. Though dynamics \ of the Kirschner-
Panetta model is rich and may include e.g. periodic orbits; homoclinic
orbits, see [6], its rigorous analysis is far from being completed, [7].
This model is written in the form
\begin{eqnarray}
\dot{x} &=&cy-\mu _{2}x+\frac{p_{1}xz}{g_{1}+z}+s_{1}; \nonumber \\
\dot{y} &=&r_{2}y\left( 1-by\right) -\frac{axy}{g_{2}+y}; \label{Cancer Eq}
\\
\dot{z} &=&\frac{p_{2}xy}{g_{3}+y}+s_{2}-\mu _{3}z \nonumber
\end{eqnarray}
where by $x(t);y(t);z(t)$ we denote vectors of densities of immune
(effector) cells; tumor cells; the concentration of effector molecules
correspondingly at the moment $t$ in the single tumor-site compartment.
Dynamics of (\ref{Cancer Eq}) is biologically feasible on the positive
orthant $\mathbf{R}_{+}^{3}.$ All parameters are supposed to be positive
excepting $s_{1}$ and $s_{2}.$ In more details, the parameter $c$ is known
as the \textquotedblright antigenicity\textquotedblright\ which
characterizes the strength of the tumor to generate effector immune cells; $%
\mu _{2}$ is the death rate of immune cells; $p_{1}$ is the proliferation
rate of immune cells; $g_{1}$ is the half saturation for proliferation term;
$r_{2}$ is the cancer growth rate; $b$ is the logistic growth of cancer
capacity; $a$ is the cancer clearance term; $g_{2}$ is the half saturation
for cancer clearance; $p_{2}$ is the production rate of immune molecule; \ $%
g_{3}$ is the half saturation of production; $\mu _{3}$ is the half-life of
effector molecule. Parameters $s_{1}$ and $s_{2}$ are considered as
%TCIMACRO{\U{b4}}%
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controls%
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. They correspond to applying two kinds of medical treatment and it is
supposed that they may be zeroes but not simultaneously. More precisely, $%
s_{1}$ represent a treatment term where by a physician administers effector
cells that have been taken from a patient, stimulated to a large degree, and
then subsequently infused back into the patient. The term $s_{2\text{ }}$ is
a treatment term that represents administration of the cytokine
interleukin-2 (manufactured) by a physician to a patient, to stimulate again
\ effector cell growth and proliferation, see [6,7].
\section{Upper and lower bounds for the localization polytope}
In this section we give results on computations of upper and lower bounds
for a polytope containing all compact invariant sets of the system (\ref%
{Cancer Eq}) by using the localization method of compact invariant sets.
\textbf{Lemma 1}. 1. \textit{All compact invariant sets in }$\mathbf{R}%
_{+}^{3}$\textit{\ are located in the set }$K\left( h_{1}\right) =\left\{
0\leq y\leq b^{-1}\right\} \cap \mathbf{R}_{+}^{3}.$ \textit{2. If }$s_{1}>0$
\textit{then} \textit{all compact invariant sets in }$\mathbf{R}_{+}^{3}$%
\textit{\ are located in the set }$K_{1}\left( h_{2}\right) =\left\{ x\geq
x_{\min }:=s_{1}\mu _{2}^{-1}\right\} \cap \mathbf{R}_{+}^{3}.$ 3. \textit{%
If }$s_{2}>0$\textit{\ then all compact invariant sets in }$\mathbf{R}%
_{+}^{3}$\textit{\ are located in }$K\left( h_{3}\right) =\left\{ z\geq
\frac{s_{2}}{\mu _{3}}\right\} \cap \mathbf{R}_{+}^{3}.$
Further, we have
\textbf{Proposition 6. }\textit{All compact invariant sets are located in
the domain}%
\[
K_{\ast }\left( h_{2}\right) =\{x\geq x_{\min \ast }:=\frac{%
s_{1}(s_{2}+g_{1}\mu _{3})}{\mu _{2}g_{1}\mu _{3}+s_{2}(\mu _{2}-p_{1})}\}
\]%
\textit{where there are no additional conditions on }$s_{1};s_{2}$\textit{\
if }$p_{1}\leq \mu _{2},$\textit{\ and satisfy}%
\[
s_{2}<\frac{g_{1}\mu _{2}\mu _{3}}{p_{1}-\mu _{2}},\text{ if }p_{1}>\mu _{2}.
\]
Next, we refine the localization set $K\left( h_{1}\right) $ as follows:
\textbf{Proposition 7. }\textit{\ Let }$bg_{2}>1.$ \textit{If }%
\begin{equation}
00;s_{2}\geq 0.$
\textbf{Theorem 4.} \ 1.\textit{\ Let\ }$g_{3}\geq g_{2}.$\textit{\ Then we
get the localization set defined by}
\begin{equation}
K(h_{5})=\{z\leq z_{\max }:=\frac{s_{2}}{\mu _{3}}+\frac{(\mu
_{3}+r_{2})^{2}p_{2}}{4abr_{2}\mu _{3}}\}. \label{zmax}
\end{equation}
2.\textit{\ Let \ }$g_{3}\geq g_{2};$\textit{\ }$\mu _{2}>p_{1}$\textit{\
and (\ref{30}) holds. Then we get the localization set defined by}
\[
K_{up}(h_{2})=\{x\leq x_{\max }:=\frac{1}{\mu _{2}-p_{1}}(cy_{\max }+s_{1}-%
\frac{p_{1}g_{1}x_{\min }}{g_{1}+z_{\max }})\}.
\]
\textbf{Corollary 2. }\textit{Suppose that }$g_{3}\geq g_{2};bg_{2}\geq 1$%
\textit{\ and 1) }$p_{1}=\mu _{2}$\textit{\ or 2) }$p_{1}>\mu _{2}$\textit{\
and }%
\[
z_{\max }<\frac{\mu _{2}g_{1}}{p_{1}-\mu _{2}}.
\]
\textit{In addition, we suppose that parameters }$s_{1}>0$ and $s_{2}$%
\textit{\ satisfy conditions of Proposition 6. Then all compact invariant
sets are located in the domain}
\begin{eqnarray*}
K_{\ast up}(h_{2}) &=& \{x\leq x_{\max \ast }\}, \\
x_{\max \ast }:&=& \frac{(g_{1}+z_{\max})(cy_{\max }+s_{1})}{p_{1}g_{1}},\\
\text{ \textit{in case 1);}} \\
K_{\ast \ast up}(h_{2}) &=& \{x\leq x_{\max \ast \ast }\} , \\
x_{\max \ast \ast } :&=& \frac{(g_{1}+z_{\max})(cy_{\max }+s_{1})}{\mu _{2}(g_{1}+z_{\max })-p_{1}z_{\max }}\},\\
\text{\textit{in case 2).}}
\end{eqnarray*}
\section{The existence and bounds for the BPID}
In this section we describe sufficient conditions under which the system has
a positively invariant polytope which is contained in $\mathbf{R}_{+,0}^{3}$
:
\textbf{Theorem 5. }\textit{\ Suppose that parameters of (\ref{Cancer Eq})
satisfy }%
\begin{equation}
r_{2}g_{2}>a. \label{23}
\end{equation}%
\textit{and }$s_{i}\geq 0,i=1,2$\textit{.} \textit{Then the polytope }%
\[
K_{BPID}:=K_{\ast }(h_{1})\cap K_{1}(h_{2})\cap K_{0,up}(h_{2})\cap
K(h_{3})\cap K(h_{5}),
\]%
\textit{\ with }$K_{0,up}(h_{2}):=K_{up}(h_{2})$\textit{\ or }$K_{\ast
up}(h_{2})$\textit{\ or }$K_{\ast \ast up}(h_{2})$\textit{\ in corresponding
cases, is a BPID. \ }
\section{Conditions of the tumor free limit behavior}
In this section we describe one assertion in which sufficient conditions of
the tumor clearance limit dynamics are given. Its statement is expressed in
terms of inequalities imposed on parameters of the system and on treatment
terms under which $y(t)\longrightarrow 0,$ with $t\longrightarrow \infty .$
Let $s_{1};s_{2}>0.$ The tumor-free equilibrium point containing in $\mathbf{%
R}_{+,0}^{3}\cap \{y=0\}$ is given by\textit{\ }%
\[
O_{1}=(O_{11},O_{12},O_{13})=(\frac{s_{1}(g_{1}\mu _{3}+s_{2})}{\mu
_{2}(g_{1}\mu _{3}+s_{2})-p_{1}s_{2}},0,\frac{s_{2}}{\mu _{3}}).
\]
Here%
\begin{eqnarray}
0 &<&s_{2}<\frac{\mu _{2}\mu _{3}g_{1}}{p_{1}-\mu _{2}},\text{ with }%
p_{1}>\mu _{2}, \label{38} \\
\text{or }s_{2} &>&0,\text{ with }p_{1}\leq \mu _{2}. \nonumber
\end{eqnarray}
The point $O_{1}$ taken from $\mathbf{R}_{+,0}^{3}\cap \{y=0\}$ is locally
asymptotically stable if, in addition, we claim that%
\begin{equation}
s_{1}>\frac{r_{2}g_{2}[\mu _{2}(g_{1}\mu _{3}+s_{2})-p_{1}s_{2}]}{a(\mu
_{3}g_{1}+s_{2})}. \label{39}
\end{equation}
Now we state
\textbf{Theorem 6.} \textit{Suppose that }
\textit{1) conditions (\ref{38})-(\ref{39}) are fulfilled and}
\textit{2) inequalities } \textit{\ }%
\begin{equation}
\frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{ab}\geq s_{1}>\max \{\frac{\mu
_{2}g_{2}^{2}r_{2}b}{a};\frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{4ab}\},
\label{49b}
\end{equation}
\textit{hold, or instead of (\ref{49b}),}
\textit{3) inequality }%
\begin{equation}
\frac{\mu _{2}(bg_{2}+1)^{2}r_{2}}{ab}0;z>0\}$\textit{\ tend to the tumor-free
equilibrium point }$O_{1}.$
\textbf{Corollary 3.} \textit{Assume that conditions of Theorems 5 and 6
hold. Then all trajectories in }$\mathbf{R}_{+,0}^{3}\cap \{x>0;z>0\}$%
\textit{\ tend to }$O_{1}.$
\section{Concluding remarks}
In this work we demonstrate for that the localization method of compact
invariant sets [9] is helpful in studies of global aspects of population
dynamics, especially for finding upper and lower bounds of ultimate
densities of specific types of interacting populations.
The biological significance of compact invariant sets is connected with the
fact that they carry information about the long-time behavior of our model
in future. With help of finding bounds for compact invariant sets we compute
bounds for the bounded positively invariant domain. The goal of this
computation is two-fold. From the first side, the bounded positively
invariant domain consists of states whose evolution in future always
satisfies bounds for the bounded positively invariant domain; this makes
tumor growth dynamics predictable in the sense that the density of mutant
cells has a more accurate and predictable estimate of changes in a tumor
size.
From another side, these bounds are used in statements and proofs of
assertions relating to a location of $\omega $- limit sets and conditions of
the attractivity to the tumor-free equilibrium point.
Main results concerning mutant-cells-free limit behavior are contained in
Theorem 3 and Corollary 1 for the Owen-Sherratt system and concerning
tumor-free limit behavior are contained in Theorem 6 and Corollary 3 for
the Kirchner-Panetta system.
Finally, we indicate that mathematical models describing the cancer tumor
growth contain a lot of parameters which cannot be measured with an
acceptable accuracy. It takes place because these models are usually derived
in the situation of the deficiency of clinical results/ experimental
observations. Therefore our results on global dynamics of the cancer tumor
growth models expressed in terms of simple algebraic conditions may be
helpful for biological applications. In particular, these conditions give
thresholds of the uncertainty of model parameters under which the given type
of global dynamics preserves.
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\bibitem[Krishchenko and Starkov, 2006] {Krishko} A.P. Krishchenko and K.E. Starkov (2006). Localization of compact invariant sets of the Lorenz system. {\em Phys Lett A},vol.353, pp.~383--388.
\bibitem[Kirschner and Panetta, 1998] {Kirsh} D. Kirschner and J. Panetta (1998). Modelling immunotherapy of the tumor- immune interaction,. {\em J Math Biol},vol.37, pp.~235--252.
\bibitem[Kirschner and Tsygvintsev, 2009] {Kirshh} D. Kirschner and A.V. Tsygvintsev (2009). On the global dynamics of a model for tumor immunotherapy. {\em Math Biosci Eng},vol.6, pp.~579--583.
\bibitem[Nani and Freedman, 2000] {nani} F. Nani and H.I. Freedman (2000). A mathematical model of cancer treatment by immunotherapy. {\em Math Biosci},vol.163, pp.~159--199.
\bibitem[Starkov, 2011] {starkov} K.E. Starkov (2011). Compact invariant sets of the Bianchi VIII and Bianchi IX Hamiltonian systems. {\em Phys Lett A},vol.375, pp.~3184--3187.
\bibitem[Starkov and Pogromsky, 2013] {starpo} K.E. Starkov and A. Pogromsky (2013). On the global dynamics of the Owen-Sherratt model describing the tumor- macrophage interactions. {\em Int J Bifurcat Chaos},vol.23, pp.~9.
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\bibitem[dePillis and Radunskaya, 2001] {Pillis} L. dePillis and A. Radunskaya (2001). A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. {\em Comput Math Methods Med 3}, pp.~78--100.
\bibitem[dePillis and Radunskaya, 2003] {Pilliss} L. dePillis and A. Radunskaya (2003). The dynamics of the optimally controlled tumor model: a case study. {\em Math Comput Model},vol.37, pp.~78--100.
\bibitem[Itik and Banks, 2010] {Itik} M. Itik and S. Banks (2010). Chaos in a three-dimensional cancer model. {\em Int J Bifurcat Chaos},vol.20, pp.~71--79.
\bibitem[Owen and Sherratt, 1998] {owen} M.R. Owen and J.A. Sherratt (1998). Modelling the macrophage invasion of tumors: Effects on growth and composition. {\em IMA J Math Appl Med Biol},vol.15, pp.~165--185.
\bibitem[Lejeune,Chaplain and Akili, 2008] {lej} O. Lejeune, M.A.J. Chaplain and I. El. Akili (2008). Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumors. {\em Math Comput Model},vol.47, pp.~649--662.
\bibitem[Eftimie,Bramson and Earn, 2011] {Eftime} R. Eftimie, J.L. Bramson and D.J.D Earn (2011). Interactions between the immune system and cancer: a brief review of non-spatial mathematical models. {\em Bull Math Biol},vol.73, pp.~2--32.
\bibitem[Kuznetsov,Makalkin,Taylor and Perelson, 1994] {kuz} V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor and A.S. Perelson (1994). Nonlinear dynamics of immunogenic tumors: Parameters estimation and global bifurcation analysis. {\em Bull Math Biol},vol.56, pp.~295--321.
\end{thebibliography}
\end{document}