# Stability Results for a Class of Differential Equation and Application in Medicine

Authors: Qingyi Zhan; Xiangdong Xie; Zhifang Zhang
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

AChemostat system incorporating hepatocellular carcinomas
is discussed. The model generalizes the classical Chemostat model, and it assumes that the Chemostat is an increasing function of the concentration. The
asymptotic behavior of solutions is determined. Sufficient conditions for the
local and global asymptotic stability of equilibrium and numerical simulation
are obtained, which is used to select the disease control tactics.


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## Body

## 1. Introduction

As we know,   the stability of ecological systems and the persistence of species within them are fundamental concerns in ecology. Mathematical models of ecological systems, reflecting these concerns, have been used to investigate the stability of a variety of systems. For example, see [1–7]. The dynamic relationship between predator and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [4], and many good achievements have been reached [1–5, 8–10]. But few works about hepatocellular carcinomas cell Chemostat model have been done.According to some medical knowledge, under certain condition, the growth rate of tumor is in proportion to the volume on the timet. Let V(t) denote this volume, such differential equations take the form
(1.1)dV(t)dt=(K(t)-Q(t))V(t),
where K(t) and Q(t) denote the cell growth and death rate, respectively.Some research on tumor cell shows that the growth rate is not invariable, but in reciprocal proportion to the factort. Therefore, it becomes
(1.2)K'(t)=-L(t)K(t).So the growth of the tumor disciplinarian can be denoted as follows, see [2],
(1.3)V'(t)=(K(t)-Q(t))V(t),K'(t)=-L(t)K(t).Following the accumulation of experimental data, it became evident that the system (1.3) requires modification. Another research shows that the growth speed of the tumor cell is continuously decreased, which can be denoted by cube root function.In fact, different kinds of tumor cell have different growth mode which behave exponential function, cube root function, or linear equation. It has been well established in experimental literature, so it becomes(1.4)V'(t)=(K(t)-Q(t))Vα(t).The data presented by clinical experience shows that the liver tumor tissue is similar to entity ball. Because the volume of the liver on the timet may be regarded as invariable, it is treated as Chemostat model.In this paper, we modify the modeling approach developed in some paper [2–7]. It extends the above model by assuming that Q(t) is constant.The symbol in paper [7] is used as follows: x(t) and y(t) denote tumor and normal cell consistency, respectively; Q(t) and P(t) denote the death and growth rate of tumor cell, and a typical choice for Q(t) and P(t) are P(t)=μx/(ky+x)δ, Q(t)=Q respectively; x0 is the growth speed of normal cell, 1/δ is the consume rate of normal cell. The system (1.3) becomes, see [6, 7]
(1.5)x'(t)=Q(x0-x)-μx(ky+x)δy2/3,y'(t)=y2/3[μxky+x-Q],x(0)≥0,y(0)>0,
where realistic meaning is the same as the paper in [7, 11].The study of the system (1.5) shows that it can be global and local asymptotic stability. Specifically, we demonstrate that stability change occurs only when the coefficient varies thus correcting the previously published results.This paper has the following structure. The basic properties of its solutions and equilibrium are given in Section2.  The stability of the system (1.5) is discussed in Section 3. Applications of the theorem, numerical simulation, and control strategy are presented in Section 4, two examples of oscillatory coexistence are also presented.


## 2. Equilibrium

In this section, we will study all possible equilibriums of the system (1.5).For simplicity, the system (1.5) is rescaled with substitutions(2.1)x=x0x¯,y=x0δy¯,t=1Qt¯,x¯, y¯, t¯are still replaced by x, y, t, respectively, then the new system is written as
(2.2)x'(t)=1-x-bmx(ay+x)δy2/3,y'(t)=by2/3[mxay+x-1],x(0)≥0,y(0)>0,
where a=kδ,b=(x0δ)-1/3,m=μ/Q.Substitutingz=1-x-y,λ=a/(m-1) into the system (2.2), it takes the form
(2.3)z'(t)=-z-y-by2/3,y'(t)=b(m-1)y2/31-(1+λ)y-z1+(a-1)y-z,z(0)≥0,y(0)≥0.By straightforward computing:(2.4)-z-y-by2/3=0,b(m-1)y2/31-(1+λ)y-z1+(a-1)y-z=0,two equilibriums of the system (2.3) are obtained, E1(0,0) and E2(z*,y*), where
(2.5)z*=(1+λ)y*-1,1-λy*-by*2/3=0.Obviously, according to the first equation of the system (2.3), if y=0, then z(t)=z0e-t. Consequently, if t→0, then z(t)→0. That is, the system (2.3) W limit set of all positive solutions are in the point E1(0,0).We assume that if the liver volume does not obviously change, we call it health equilibrium, otherwise we named it disease equilibrium. Therefore,E1 is the former and E2 is the latter.A necessary condition for existence of positive equilibrium in the system (2.3) is that
(2.6)(H1):R0=m+a-1m-1>0.Lemma 2.1.
The system (2.3) has positive equilibrium if and only if (H1) is hold.


## 3. Stability

People always expect that any disease can be cured no matter what stage it is, that is to say that this differential equation is asymptotically stable.Following Section2, we apply Lyapunov's stability theorem to analyse the two equilibriums of the system (1.5).Theorem 3.1.
IfR0≤0, then the system (2.3) has only one equilibrium, and it is locally asymptotically stable.Proof.
IfR0≤0, then E2(z*,y*) does not exist, the system has only E1. Therefore, we obtain the  Jacobic matrix
(3.1)JE1=J(0,0)=(-1,-10,0).
BecauseT=-1,D=0, E1 is a stable crunode, and it is locally asymptotically stable. This completes the proof.Theorem 3.2.
IfR0>0 and m<1 hold, the positive equilibrium E2 of the system (2.3) is globally asymptotically stable.Proof.
Substitutingξ=y-y* into the system (2.3), we get
(3.2)dξdt=b(ξ+y*)2/3(m-1)1-(1+λ)(ξ+y*)-z1+(a-1)(ξ+y*)-z.
ConstructLyapunov  function:
(3.3)V(ξ)=ξ-y*lnξ+y*y*,
therefore, the derived function of V(ξ) about the system (2.3) take the form
(3.4)dVdt=ξξ+y*dξdt=b(m-1)ξ(ξ+y*)-1/31-(1+λ)(ξ+y*)-z1+(a-1)(ξ+y*)-z.
According to the formal conclusion:m<1, b>0, ξ(ξ+y*)-1/3>0, we obtain
(3.5)dVdt<0.
Therefore, ifR0>0 and m<1 hold, E2 is globally asymptotically stable. This completes the proof.Theorem 3.3.
IfR0>0 and m<1 hold, the positive equilibrium E0 of the system (2.2) is globally asymptotically stable.


## 4. Numerical Simulation and Control Policy

In this section,  we will select proper parameter and present numerical computer simulation to obtain control policy.


### 4.1. WhenR0≤0

Then(m+a-1)(m-1)<0. To illustrate the result of this subsection numerically, we fix the initialization points (0.03,0.1), (0.1,0.15) and (0.01,0.2), (0.5,1), respectively and present computer simulations of the system (1.5), which based on the experiment and clinic date. The critical parameters δ,k,μ,Q,x0 are as follows: δ=10,k=0.03,μ=0.08,Q=0.1,x0=0.2, where R0=-0.5; δ=10,k=-0.03,μ=0.12,Q=0.1,x0=0.2, where R0=-0.5, respectively. Figures 1 and 2 show that it is asymptotically stable.Figure 1
Solution curves forR0=-0.5 with initial value (0.03,0.1), (0.1,0.15).Figure 2
Solution curves forR0=-0.5 with initial value (0.01,0.2), (0.5,1).Therefore, the system has only one equilibrium in which the volume of liver does not obviously change. This probably is in the delitescence or no disease in which the liver has changed at functionality at most, but not organic. If proper measure is taken which will control the growth of the tumor, the development of tumor will be stagnancy.We suppose that if the volume of liver does not obviously change, disease will not exacerbate, not to mention proper measure is adopted; therefore, hepatocellular carcinomas early diagnosis is very important which can control the development of the disease. Therefore we advocate early detection, early diagnosis, and early treatment.


### 4.2. WhenR0>0

The system has one equilibrium on which the volume of liver has obviously changed, that is to say it is in serious condition.Therefore,m+a-1>0,m<1. Based on the experiment and clinic date, we fix the initialization points of the system (1.5): (0.03,0.1), (0.1,0.15), the critical parameters are as follows: δ=10,k=0.03,μ=0.06,Q=0.1,x0=0.2, where R0=0.25. It is shown in Figures 3 and 4.Figure 3
Solution curves forR0=0.25 with initial value (0.03,0.1), (0.1,0.15).Figure 4
Solution curves forR0=0.25 with initial value (0.03,0.1), (0.1,0.15).Otherwise, we fix the initialization points of the system (2.3): (1,2), the critical parameters are as follows: m=1.2,a=-0.3,λ=-1.5,b=2-1/3, where R0=0.25. Figure 5 shows that such system may exhibit a stable periodic solution.Figure 5
Solution curves forR0=0.25 with initial value (1,2).We suppose that when the volume of liver has obviously changed, if proper measure such as chemotherapy or radiology is adopted, disease will not exacerbate at least, then the development of the tumor is in the phase of stagnation; therefore, hepatocellular carcinomas diagnosis is very important as it can control the development of disease. So we prefer medication treatment.


## 4.1. WhenR0≤0

Then(m+a-1)(m-1)<0. To illustrate the result of this subsection numerically, we fix the initialization points (0.03,0.1), (0.1,0.15) and (0.01,0.2), (0.5,1), respectively and present computer simulations of the system (1.5), which based on the experiment and clinic date. The critical parameters δ,k,μ,Q,x0 are as follows: δ=10,k=0.03,μ=0.08,Q=0.1,x0=0.2, where R0=-0.5; δ=10,k=-0.03,μ=0.12,Q=0.1,x0=0.2, where R0=-0.5, respectively. Figures 1 and 2 show that it is asymptotically stable.Figure 1
Solution curves forR0=-0.5 with initial value (0.03,0.1), (0.1,0.15).Figure 2
Solution curves forR0=-0.5 with initial value (0.01,0.2), (0.5,1).Therefore, the system has only one equilibrium in which the volume of liver does not obviously change. This probably is in the delitescence or no disease in which the liver has changed at functionality at most, but not organic. If proper measure is taken which will control the growth of the tumor, the development of tumor will be stagnancy.We suppose that if the volume of liver does not obviously change, disease will not exacerbate, not to mention proper measure is adopted; therefore, hepatocellular carcinomas early diagnosis is very important which can control the development of the disease. Therefore we advocate early detection, early diagnosis, and early treatment.


## 4.2. WhenR0>0

The system has one equilibrium on which the volume of liver has obviously changed, that is to say it is in serious condition.Therefore,m+a-1>0,m<1. Based on the experiment and clinic date, we fix the initialization points of the system (1.5): (0.03,0.1), (0.1,0.15), the critical parameters are as follows: δ=10,k=0.03,μ=0.06,Q=0.1,x0=0.2, where R0=0.25. It is shown in Figures 3 and 4.Figure 3
Solution curves forR0=0.25 with initial value (0.03,0.1), (0.1,0.15).Figure 4
Solution curves forR0=0.25 with initial value (0.03,0.1), (0.1,0.15).Otherwise, we fix the initialization points of the system (2.3): (1,2), the critical parameters are as follows: m=1.2,a=-0.3,λ=-1.5,b=2-1/3, where R0=0.25. Figure 5 shows that such system may exhibit a stable periodic solution.Figure 5
Solution curves forR0=0.25 with initial value (1,2).We suppose that when the volume of liver has obviously changed, if proper measure such as chemotherapy or radiology is adopted, disease will not exacerbate at least, then the development of the tumor is in the phase of stagnation; therefore, hepatocellular carcinomas diagnosis is very important as it can control the development of disease. So we prefer medication treatment.


## 5. Concluding Remarks

In this paper we have considered aChemostat system incorporating hepatocellular carcinomas. We obtained a more realistic model by incorporating hepatocellular carcinomas in system (1.5). We have proved that local asymptotic stability of the positive equilibrium implies that it is global asymptotic stability. All the results indicated that medication treatment had a stabilizing effect on hepatocellular carcinomas development. We have given a numerical simulation to verify some of the key results we have obtained.



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*Source: 187021-2009-06-14.xml*