Journal of Information and Computational Science ISSN ...joics.org/gallery/ics-1391_1.pdf · cancer, brain cancer, lung cancer, and bone cancer. Cancer is caused by chemical substances,

MATHEMATICAL MODELING IN GENE THERAPY USING LOTKA VOLTERRA

MODEL

G.Dhanapratha #1,Dr.K.Selvaraj*2

#1 M.Phil (Scholar) & Mathematics & PRIST University

Vallam,Thanjavur,Tamilnadu,India.

*2 M.Sc.,M.Phil.,B.Ed.,Ph.D.,Assistant Professor & M.Phil(Mathematics) & PRIST University

Vallam,Thanjavur,Tamilnadu,India.

ABSTRACT

This study observed a mathematical model for cancer treatment by using gene therapy.

Started with modelling the gene medical aid for Cancer Treatment then continuing with the

discussion concerning equilibrium purpose, the model was developed by Lotka-Volterra model.

It is very important to investigate the behaviour of equilibrium point such as the stability of that

point in certain conditions. Furthermore, simulation of the model was given by taking some sure

parameter values.

Key words: Cancer, Gene therapy, Lotka Volterra, stability

1. INTRODUCTION

Cancer is a worldwide major cause of death. Cancer arises as a result of the rapid and

uncontrollable growth of abnormal cells in the human body. Cell in the smallest part of the body

and cancer occurs from normal cells. Some types of cancers include breast cancer, cervical

cancer, brain cancer, lung cancer, and bone cancer. Cancer is caused by chemical substances,

alcoholic beverages, excessive solar radiation, and genetic differences [1]. Various methods are

used to cure or inhibit the growth of cancer. The type of treatment, such as surgery, radiation

therapy, chemotherapy, target therapy, immunotherapy, hormonal therapy, angiogenesis

inhibitors, palliative care, and Gene therapy. Each therapy has side effects, eg nausea, vomiting,

pressing the blood production, fatigue, hair lose and mouth sores. Side effects occur, as a results

of therapy medicine not solely kill cancer cells however additionally traditional cells that divide

apace. Such as, cells of skin, gastrointestinal tract, sperm and hair [2]. Therefore, other therapies

developed for the treatment of cancers such as gene therapy.

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Volume 9 Issue 9 - 2019

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A new treatment has to be developed and changed to improve the effectiveness, high

preciseness, additionally assize of survival skills, and improve the standard lifetime of patients.

One of the therapies that are expected to be the long run answer is sequence medical care. This

therapy will specifically against cancer cells, specific area, without distribution normal cells in

the body. Gene therapies performed by replacing or inactivate genes that do not work, adding a

functional gene, or insert a gene into cells to make the normal functioning of cells [3]. In contrast

to the method of immunotherapy treatment that refers to the use of cytokines, namely

interleukin-2, the method of gene therapy, IL-2 eliminating from the immunotherapy model and

replaced

by self-proliferation that is 𝑝1𝐸

𝐸+𝑓.

A modern review summarizes developments in gene treatment and additionally highlights

openings for systems biology and mathematical modeling to synergize action with

experimentalists and clinicians to push cancer analysis forward [4]. A mathematical model is a

description of a system exploitation mathematical ideas and language. The method of developing

a mathematical model is termed as mathematical modeling. Mathematical modeling has been

instrumental within the past fifty years in serving to decipher totally different aspects of

advanced systems in biology. specifically, mathematical modeling has had an impression on our

understanding of cancer biology and treatment (cf. [4, 5] for glorious reviews). As a primary step

to exploring the employment of gene treatment on the tumor-immune interaction throughout

cancer, we tend to apply an easy mathematical model to explore the dynamics of gene therapies,

with the goal of predicting optimum combos of approaches resulting in clearance of a tumour.

we tend to gift the model and its analysis (both analytical and numerical) and supply some

conclusions..

2. Gene therapy:

Gene treatment is associate degree experimental technique that uses genes to treat or stop

sickness. Within the future, this system several permit doctors to treat a disorder by inserting a

gene into a patient’s cells rather than victimisation medication or surgery. Gene remedy is

encouraging treatment option for diversity of diseases (including some styles of cancer,

congenital disease and bound microorganism infections), the method remains unsafe and quiet



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below the study to varieties certain that it'll be harmless and effective. Gene therapy uses vector.

A new gene is injected into a vector, which is used introduced the modified DNA into a human

cell. If the treatment is productive, the new sequence can build a purposeful supermolecule [6].

One will see however this works by staring at the figure one.

Fig 1 Working principle of gene therapy

2.1 The treatment process with gene therapy:

Genes are located on chromosomes inside all of our cells and are made of DNA. Humans

have approximately 35,000 genes. Gene therapy is an experimental treatment currently being

tested in clinical trials that involves introducing additional genetic material (either DNA or

RNA) into cells to fight cancer in a few different ways. There are many sequence treatment

approaches that are being explored. First, scientists are trying to use sequence medical care to

interchange missing or mutated genes with healthy genes (for example, p53, [7]). Second,

scientists are attempting to put genes into tumors that act like suicide bombs once they are turned

on by drugs that are administered to the patient [8]. Similar to the suicide genes, a 3rd approach

is to insert genes that build tumors additional prone to treatments like therapy and radiation

therapy. And finally, gene therapy is being used to improve the immune response to cancers by

enhancing the ability of immune cells, such as T cells, to fight cancer cells [9]. Figure 2

summarizes these different gene therapies.



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Fig. 2. Gene therapy and immunotherapy treatments. As denoted by the numbers in the figure: (1)

Replace missing or mutated genes with healthy genes. (2) Insert genes into tumors that act like suicide

bombs once they are turned on by drugs (3) Genes Insertion that make tumors more vulnerable to

treatments such as radiotherapy and chemotherapy. (4) Augment the reaction to cancers by enhancing the

flexibility of immune cells, like T cells and nerve fibre cells, to fight cancer cells

3. Lotka-volterra model:

The process of gene treatment that has the characteristics of dominant the speed of

cancer cells growth is explicit interest. Seen from the mathematical point of view, gene therapy

for cancer can be modeled mathematically in the form of differential equations system. Based on

the Lotka-volterra model,

Let (𝑡) be the population of predator and (t) is population of its prey. (For example,

one can imagine populations of wolves and rabbits in a forest). Assuming that numbers (𝑡), (𝑡)

are big enough and that the predator and prey populations are homogeneous, one can view them

as continuous functions of time. Let (𝑡)=𝑥(𝑡+ 𝑡)-𝑥(𝑡) and 𝑦(𝑡)=𝑦(𝑡+ 𝑡)-𝑦(𝑡) be small

variations of populations during a certain period of time 𝑡. Taking 𝑡=1 (for example one day)

one can replace (𝑡), (𝑡) by their derivatives, i.e. write �̇�(𝑡),�̇�(𝑡) instead. The Lotka-volterra

equations are given by

{�̇� = 𝒂𝒙 − 𝒃𝒙𝒚 �̇� = −𝒄𝒚 + 𝒅𝒚𝒙

Some positive numbers indicates as 𝑎, 𝑏, 𝑐 and 𝑑.

The linear positive term 𝑎𝑥 in the first equation (prey) corresponds to exponential

growth; the negative predation term, -𝑏𝑥𝑦 describes the rate prey are lost and is proportional to

number of prey and predators in mass-action principle type. In the second equation (predator),

the negative linear term –𝑐𝑦 corresponds to natural death, as prey will not survive without prey,

and +𝑑𝑥𝑦 describes the growth of the predator population proportional to prey and number of

predators. Then simple form of Lotka-volterra (LV) system is remarkable. It allows for

investigation of the quantitative and qualitative behavior for all solutions both analytically and

numerically. First, no chaotic behavior is possible according to Poincare-Bendixon theorem, and,

asymptotically, every non-periodic solution either goes to a fixed point or approaches a limit

cycle. Simple analysis shows that most solution of LV system are periodic, i.e. the population



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numbers 𝑥(𝑡),𝑦(𝑡) are oscillating around a certain equilibrium state 𝑥(𝑡)=𝑥*,𝑦(𝑡)=𝑦*.this stable,

stationary solution is (0,0).

In 1994 Kuznetsov applied Lotka-volterra [10] ideas to cancer modeling, where (𝑡)

represent the effectors immune cells (predator) and (𝑡) the tumor cells (prey). The equations,

which are similar to the LV system, are written as follows:

{�̇� = 𝒔 + 𝒑

𝑬𝑻

𝒈+𝑻− 𝒎𝑬𝑻 − 𝒅𝑬

�̇� = 𝒂𝑻(𝟏 − 𝒃𝑻) − 𝒏𝑬𝑻

Positive parameters are 𝑠, 𝑝, 𝑔, 𝑚, 𝑑, 𝑎, 𝑛 and 𝑏.

Here the exponential growth of T in the second equation, was replaced by more realistic

one in logistic form: 𝑎T(1-𝑏T), where 𝑏−1 is the maximal carrying capacity for tumor cells and 𝑎

is the maximal growth rate. The term –𝑛ET describes the loss of tumor cells due to the presence

of immune cells. In the first equation 𝑠 is normal immune cell growth, which is 𝑛 cells death

with 𝑑 the loss rate; -𝑚ET describes the decay of E cells due to interacting with neoplasm cells

in a very mass-action principle. The term 𝑝𝐸𝑇

𝑔+𝑇 represent Mchaelis-Meten growth of the immune

response in response of tumors.

3.1 Mathematical model for the spread of cancer:

Gene therapy for cancer can be modeled mathematically in the form of differential

equation systems. In 1994 Kuznetsov developed the Lotka-volterra model [10]. Furthermore in

1998 Kirschner and panneta continued to form the KP models established from the Kuznetsov

model by adding cytokines population as physical object communication molecules by the

system. Gene treatment models developed from the 2 previous models created the T-cells induce

each cell to made T-cell receptor (TCR). These cells are transferred back to the cancer patient’s

body and can acknowledge and resist molecules found as neoplasm cells. TCR can activate T-

cells that then attack and kill cancer cells.

This study focused on the growth dynamics of effectors cells and tumor cells because of

the influence of the use of gene therapy. These conditions can be modeled in the form of

mathematical equation. The interaction between effectors cells and tumor cells is described as a



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competition between the two populations; therefore the interaction established mathematical

models based on the system of the predator prey. The plot stars from the formation of the model,

determined the equilibrium point, analyze the local stability and then simulated the model.

Gene therapy model developed in the KP models shown below:

E=𝑐T-𝜇2E+𝑝1𝐸𝐶

𝑔1+𝐶+𝑆1 (1a)

T=𝑐T−𝜇2E+𝑝1𝐸𝐶

𝑔1+𝐶+𝑆1 (1b)

C=𝑝2𝐸𝑇

𝑔3+𝑇 +𝑆2−𝜇3𝐶 (1c)

The notation used is,

𝐸(𝑡) :effectors immune cell; 𝑇(𝑡):cancer cell ; 𝐶(𝑡): effectors molecules ; 𝐶 : antigen

parameters ; 𝜇2 : Pure death parameters ; 𝑆1: Immunotherapy parameters

Table 1. Parameter values for the model (2)



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Equation (1b) shows the supply growth of cancer cells, with 𝑏−1 as a most capacity limit

(maximum carrying capacity). Furthermore, sequence medical aid models obtained by removing

the equation (1c) and replace it with cell proliferation (self proliferation) in equation (1a).

Here are the assumptions needed to get mathematical models:

Tumor cell growth following the logistics growth

Natural death occurred on the effectors cells

The population is not constant

The values of the parameters are largely based on previous research on table. These

parameters are as follow:

The transfer diagram can be seen as follow:

Fig. 1 Gene therapy transfer diagram model

Based on fig.1, the gene therapy model developed in the form of differential equations system

is follows:

𝑑𝐸

𝑑𝑡 = 𝑐T−𝜇2E+𝑝3

𝐸

𝐸+𝑓 +𝑠1

𝑑𝑇

𝑑𝑡 = 𝑟2T (1-𝑏T)-𝑎

𝐸𝑇

𝑇+𝑔2 (2)

With 𝑐, 𝑠1, 𝑟2,𝑎 are constant. Where 𝑠1, 𝑎, 𝑟2, 𝑔2,> and 0<𝜇2, 𝑝3,𝑓, 𝑏, 𝑐 1.

Equilibrium point



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Equation (2) will reach equilibrium point on 𝑑𝐸

𝑑𝑡 =0 and

𝑑𝑇

𝑑𝑡 =0, so the equation can be written as

follow:

𝑐T−𝜇2E+𝑝3𝐸

𝐸+𝑓 +𝑠1=0 (2a)

𝑟2(1-𝑏𝑇)_𝑎𝐸𝑇

𝑇+𝑔2 =0. (2b)

From equation (2b) we obtain:

𝑇1= 0 (3)

𝑇2 = 𝐴1−√𝐵1

2𝑟2𝑏 (4)

𝑇3=𝐴1+√𝐵1

2𝑟2𝑏 (5)

With:

𝐴1= (1−𝑏𝑔2)𝑟2

𝐵1= ((1 − 𝑏𝑔2)𝑟2)2-4(𝑎𝐸−𝑟2𝑔2)𝑏𝑟2.

Furthermore, substitution of each (3), (4), or (5) to (2a) is obtained

𝐸1=𝐴

2−√𝐵2

2𝑟2𝑏 and 𝐸2=

𝐴2+√𝐵2

2𝑟2𝑏

With:

𝐴2=𝑝3+𝑠1−𝜇2𝑓

𝐵2 =(𝑝3+𝑠1−𝜇2𝑓)2 + 4𝜇1𝑠1𝑓.

Thus equilibrium point 𝑃1(𝐸1∗, 𝑇1

∗)𝑎𝑛𝑑 𝑃2(𝐸2∗, 𝑇2

∗) were obtained. It is analog for other following

points:

𝐸3 =𝐴3−√𝐵3

2𝑟2𝑏 and 𝐸4 =

𝐴3+√𝐵3

2𝑟2𝑏

With:

𝐴3= (𝐴1 − √𝐵1)𝑐+ 2𝑟2𝑏(𝑝3 + 𝑠1−𝜇2𝑓

𝐵3= ((𝐴1− √𝐵1) c +2 𝑟2𝑏(𝑝3+𝑠1−𝜇2𝑓))2 + 8(𝜇2𝑟2𝑏) ((𝐴1 − √𝐵1) 𝑐 +2𝑠1𝑟2𝑏)𝑓.

Furthermore, in order to obtain the equilibrium points of

𝑃3(𝐸3∗, 𝑇2

∗)and𝑃4(𝐸4∗, 𝑇2

∗):



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𝐸5=𝐴4−√𝐵4

2𝑟2𝑏 and 𝐸6 =

𝐴4+√𝐵4

2𝑟2𝑏

With:

𝐴4= (𝐴1 + √𝐵1)𝑐+2𝑟2𝑏(𝑝3+𝑠1𝜇2𝑓)

𝐵4=((𝐴1 + √𝐵1)𝑐 + 2𝑟2𝑏(𝑝3+𝑠1𝜇2𝑓))2 +

8(𝜇2𝑟2𝑏) ((𝐴1 + √𝐵1 )𝑐 + 2𝑠1𝑟2𝑏) 𝑓.

Thus the equilibrium points 𝑃(𝐸5∗, 𝑇3

∗) and 𝑃6(𝐸6∗, 𝑇3

∗)were obtained.

𝑃3, 𝑃4,𝑃5and 𝑃6 points know as the equilibrium points of infected cancer. Therefore, those

six equilibrium points, named as:

𝑃1(𝐸1∗, 𝑇1

∗),𝑃2(𝐸2∗, 𝑇1

∗),𝑃3(𝐸3∗, 𝑇2

∗),𝑃4(𝐸4∗, 𝑇2

∗),𝑃5(𝐸5∗, 𝑇3

∗) and 𝑃6(𝐸6∗, 𝑇6

∗)

Analysis of equilibrium point stability

Equation (2) was being linearized to obtain the Jacobian matrix as follows,

𝐽=[−𝜇2 +

𝑝3𝑓

(𝐸+𝑓)2 𝑐

−𝑎𝑇

𝑇+𝑔2(1 − 2𝑏𝑇)𝑟2 −

𝑎𝑔2𝐸

(𝑇+𝑔2)2

] (6)

1) For cancer-free equilibrium point 𝑃1(𝐸1∗, 𝑇1

∗) = 𝑃1(𝐸1∗, 0) , substitute 𝑃1(𝐸1

∗, 0) into

equation (6) to obtain cancer-free Jacobian matrix:

𝐽𝑃1 =[

−𝜇2 +𝑝3𝑓

(𝐸1∗+𝑓)2 𝑐

0 𝑟2 −𝑎𝐸1

∗

𝑔2

] (7)

The characteristic equation of (7) is:

𝜆2-(𝑝3𝑓

𝐸1∗+𝑓)2 − 𝜇2 + 𝑟2 −

𝑎𝐸1∗

𝑔2)λ +

(𝑝3𝑓

(𝐸1∗+𝑓)2 − 𝜇2) (𝑟2 −

𝑎𝐸1∗

𝑔2) =0 (8)

Then, equation (8) can be written as follows:

𝜆2 − 𝑥1𝜆+𝑦1 (9)

With:



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𝑥1 = 𝑝3𝑓

(𝐸1∗+𝑓)2 − 𝜇2 + 𝑟2 −

𝑎𝐸1∗

𝑔2

𝑦1=(𝑝3𝑓

(𝐸1∗+𝑓)2 − 𝜇2) (𝑟2 −

𝑎𝐸1∗

𝑔2)

Eigen values were obtained from equation (8) as 𝑦1is the determination of 𝐽𝑝1 matrix and 𝑥1 is

the trace of the 𝐽𝑝1matrix.

𝜆1,2 = 𝑥1±√𝑥1

2−4𝑦1

2

a) If 𝑦1 > 0 and 𝑥12 − 4𝑦1 ≥ 0 then the equilibrium point 𝑃1(𝐸1,

∗ 0)in the form of nodes, and

if 𝑥1 < 0 then 𝑃1(𝐸1∗, 0) asymptotically stable. If 𝑥1 > 0 then 𝑃1(𝐸1

∗, 0) unstable.

b) If 𝑦1 > 0, 𝑥12 − 4𝑦1 < 0and 𝑥1 ≠ 0 then the equilibrium point𝑃1(𝐸1

∗, 0) in the form of

focus, and if asymptotically stable. If 𝑥1 > 0 then 𝑃1(𝐸1∗, 0) unstable.

2) The analog characteristic equation for cancer-free equilibrium

point𝑃2(𝐸2∗, 𝑇1

∗)=𝑃2(𝐸2∗, 0) is written bellow

𝜆2 − (𝑝3𝑓

(𝐸2∗+𝑓)2 − 𝜇2 + 𝑟2 −

𝑎𝐸2∗

𝑔2) λ +

(𝑝3𝑓

(𝐸2∗+𝑓)2 − 𝜇2) (𝑟2 −

𝑎𝐸2∗

𝑔2) =0 (10)

Equation (10) can be written as follows:

𝜆2 − 𝑥2𝜆+𝑦2 (11)

With:

𝑥2 =𝑝3𝑓

(𝐸2∗+𝑓)2 − 𝜇2 +𝑟2 −

𝑎𝐸2∗

𝑔2

𝑦2 = (𝑝3𝑓

(𝐸2∗+𝑓)2 − 𝜇2) (𝑟2 −

𝑎𝐸2∗

𝑔2)

Eigen values obtained from equation (10) are

𝜆1,2= 𝑥2±√𝑥2

2−4𝑦2

2

𝑦2 Is the determinant of the matrix 𝐽𝑃2 and 𝑥2 is the trace of the matrix 𝐽𝑃2.

a) If 𝑦2 > 0 and 𝑥22 − 4𝑦2 ≥ 0 and then the symmetry point is 𝑃2(𝐸2

∗, 0) in the method of

nodes, and if 𝑥2 < 0 then 𝑃2(𝐸2∗, 0) is asymptotically stable. If 𝑥2 > 0 then 𝑃2(𝐸2

∗, 0)is

unstable.



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b) If 𝑦2 > 0 , 𝑥22 − 4𝑦2 < 0 and 𝑥2 ≠ 0 and then the symmetry point is 𝑃2(𝐸2

∗, 0) in the

formula of focus, and if 𝑥2 < 0 then 𝑃2(𝐸2∗, 0) is asymptotically stable. If 𝑥2 > 0 then 𝑃2(𝐸2

∗, 0)

is unstable.

3) For the infected cancer equilibrium point 𝑃3(𝐸3∗, 𝑇2

∗), substitution of 𝑃3(𝐸3∗, 𝑇2

∗) into (6) can

be used to obtain the characteristic equation as bellow:

𝜆2 (𝑝3𝑓

(𝐸3∗+𝑓)2 −𝜇2 + (1 − 2𝑏𝑇2

∗)𝑟2 −𝑎𝑔2𝐸3

∗

(𝑇2∗+𝑔2)2)λ+ (

𝑝3𝑓

(𝐸3∗+𝑓)2 − 𝜇2) ((1 − 2𝑏𝑇2

∗)𝑟2 −

𝑎𝑔2𝐸3∗

(𝑇2∗+𝑔2)2

) + 𝑐𝑎𝑇2

∗

𝑇2∗+𝑔2

=0 (12)

Then equation (12) can be written as:

𝜆2 − 𝑥3𝜆 + 𝑦3 = 0 (13)

With:

𝑥3=(𝑝3𝑓

(𝐸3∗+𝑓)2 − 𝜇2 + (1 − 2𝑏𝑇2

∗)𝑟2 −𝑎𝑔2𝐸3

∗

(𝑇2∗+𝑔2)2)

𝑦3 = (𝑝3𝑓

(𝐸3∗+𝑓)2 − 𝜇2) ((1 − 2𝑏𝑇2

∗)𝑟2 −𝑎𝑔2𝐸3

∗

(𝑇2∗+𝑔2)2) +

𝑐𝑎𝑇2∗

𝑇2∗+𝑔2

Eigen values obtained from equation (12) is

𝜆1,2 = 𝑥3±√𝑥3

2−4𝑦3

2

𝑦3 is the determinant of the matrix 𝐽𝑃3 and 𝑥3 is the trace of a matrix 𝐽𝑃3

.


∗, 𝑇2∗) in the method of

nodes, and if 𝑥3 < 0 then 𝑃3(𝐸3∗, 𝑇2

∗) is asymptotically stable. If 𝑥3 > 0 then 𝑃3(𝐸3∗, 𝑇2

∗) is

unstable.

b) If 𝑦3 > 0, 𝑥32 − 4𝑦3 < 0 and 𝑥3 ≠ 0 and then the symmetry point is 𝑃3(𝐸3

∗, 𝑇2∗) in the

method of focus, and if 𝑥3 < 0 then 𝑃3(𝐸3∗, 𝑇2

∗) is asymptotically stable. If 𝑥3 > 0 then

𝑃3(𝐸3∗, 𝑇2

∗) is unstable.

Analog for 𝑃4,𝑃5 and 𝑃6 in order to obtain the eigen values of characteristic equation for 𝑃4

is

𝜆1,2=𝑥4±√𝑥4

2−4𝑦4

2


.



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∗, 𝑇2∗) in the method of

nodes, and if 𝑥4 < 0 then 𝑃4(𝐸4∗, 𝑇2

∗) asymptotically stable. If 𝑥4 > 0 then 𝑃4(𝐸4∗, 𝑇2

∗)

unstable.



method of nodes focus, and if 𝑥4 < 0 then 𝑃4(𝐸4∗, 𝑇2

∗) asymptotically stable. If 𝑥4 > 0 then

𝑃4(𝐸4∗, 𝑇2

∗) unstable.

Eigen values of the characteristic equation for the 𝑃5 is

𝜆1,2 = 𝑥5±√𝑥5

2−4𝑦5

2


.



method of nodes, and if 𝑥5 < 0 then 𝑃5(𝐸5∗, 𝑇3


𝑃5(𝐸5∗, 𝑇3

∗) is unstable.



method of nodes focus, and if 𝑥5 < 0 then 𝑃5(𝐸5∗, 𝑇3

∗) is asymptotically stable. If 𝑥5 > 0

then 𝑃5(𝐸5∗, 𝑇3

∗) is unstable.

Eigen values of the characteristic equation for the 𝑃6 is

𝜆1,2=𝑥6±√𝑥6

2−4𝑦6

2

𝑦6 is the determinant of the matrix 𝐽𝑃6 and 𝑥6 is the trace of the matrix 𝐽𝑃6

.


∗, 𝑇3∗) in the method

of nodes, and if 𝑥6 < 0 then 𝑃6(𝐸6∗, 𝑇3


𝑃6(𝐸6∗, 𝑇3

∗) is unstable.


∗, 𝑇3∗) in

the method of nodes focus, and if 𝑥6 < 0 then 𝑃6(𝐸6∗, 𝑇3

∗) is asymptotically stable. If

𝑥6 > 0 then 𝑃6(𝐸6∗, 𝑇3

∗) is unstable.

4. NUMERICAL SIMULATION

Three simulation were being see to see the dynamics of effectors cells and tumor cells. The

initial value used in these simulations are E(0)=T(0)=103. Then table II contained he values of

the parameters for numerical simulations.

Table 2: Parameter values used in numerical simulation [1]



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Simulation

Parameter

𝜇2 𝑝3 𝑓 𝑠1

1 0.03 0.1245 10−3 1

2 0.03 0.1245 10−3 764.5072

3 0.03 0.1245 10−3 100

Simulation

Parameter

𝑐 𝑟2 𝑏 𝑎 𝑔2

1 0.05 0.18 10−9 1 105

2 .3710 0.0023 10−9 38.004 105

3 0.05 0.0523 10−9 2 105

Based on the simulation 1, two equilibrium points 𝑃2(38,0) and 𝑃6(20175,12083) are

eligible in existence. Stability of the simulation 1is 𝑃2 equilibrium point showed unstable result

and has the type of saddle point, while 𝑃6 is asymptotically stable and has a kind of sink

supported the simulation a pair of, there's one purpose of equilibrium 𝑃2(25488,0) is eligible in

presence. Stability condition of the equilibrium purpose 𝑃2 in simulation a pair of is

asymptotically stable and encompasses a reasonably

sink node purpose, means that the effector cell

population can increase whereas the neoplasm

disappear with increasing of 𝑡. cell population can

On the simulation three, there's one purpose of

equilibrium that was eligible living, there is one

point of equilibrium that was eligible in existence 𝑃2(3338,0). Stability conditions of the

equilibrium purpose 𝑃2 in simulation three is asymptotically stable and encompasses a

reasonably skin node purpose, the effector cell population are going to be constant at the

equilibrium purpose, whereas the population of neoplasm cells can disappear with increasing of

𝑡.



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Fig. 2. Numerical solutions of system (Parameters are chosen from the Baseline

column in Table 1) Fig. 2(a), (b), and (c) showed respectively to simulation 1, 2, and 3-point

equilibrium models.

Gene therapy model developed from Lotka-volterra models and Kirschner and panneta

models which the effector molecule is replaced by a self-proliferation of cell on immune effector

cells. Disease-free situations for gene treatment models are cancer-free equilibrium conditions,

T=0. Furthermore the local stability of free cancer equilibrium point and cancer-infected

equilibrium point has to be analyzed. With the parameter price given, the simulation model of

factor medical care is completed.

5. CONCLUSION:

A gene therapy could cure genetic diseases. After obtaining information and views from

different perspectives, I can conclude that gene therapy has the Potential to cure diseases such as

cystic fibrosis and Parkinson's disease; however, it cannot be proven now to be an effective

treatment. Although there is no guarantee that the vector carrying the healthy gene will land in

the place it is intended, future tests may make it a possible guarantee. Of the tests that were done

using gene therapy, the patient was only helped for a short period of time. In order for gene

therapy to become an effective treatment, more tests must be done, and these tests must ensure a

cure that can help for a longer period of time and hopefully, forever.



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As for the other question of whether gene therapy should become a regular procedure in

medical practices regarding genetic diseases, the answer is very controversial. Right now, there

is no chance that this procedure would be used in medical practices as it has lead to ineffective

results. However, if gene therapy does become effective in the future, should we still use it? As I

have mentioned before, a potential risk of this procedure would be finding out the genetic make-

up of an individual. Many feel that this is unethical. The potential is so great that scientists say

there is a possibility of "designer babies", where some feel is unethical and others, a dream come

true. On the other side, the potential of gene therapy can lead to a cure for cystic fibrosis,

Parkinson's disease, and even cancer. If gene therapy proves to be successful, the question left

for the public is if a cure for diseases that have taken many lives is valued more than ethical

beliefs.

BIBLIOGRAPY:

1. Tsygvintsev, S. Marino, and D. E. Kirshner, A Mathematical Model of Gene Therapy for

The Treatment of Cancer, Springer-Verlag, Berlin-Heidelberg- New York, 2013.

2. T. L. Wargasetia, "Gene therapy for cancer," Journal of Public Health, vol. 4, no. 2, 2005

3. N. Bellomo, L. Preziosi, , 2000. Modelling and mathematical problems related to tumor

evolution and its interatoin with the immune system. Math. Comput. Modelling 32,

413452.

4. F.M.Gabhann, Annex BH, A.S. Popel 2010. Gene therapy from the perspective of

systems

biology. Curr Opin Mol Ther. 12(5):570-7.

5. R.A. Gatenby, P.K. Maini 2003. Mathematical oncology: cancer summed up. Nature.

421(6921):321.

6. N. L. P. I. Dharmayanti, "Study of molecular biology: Supressor gene (p53) as gene

target for cancer treatment," Wartazoa, vol. 13, no. 3, 2013.

7. Z. Chang, J. Song, G. Gao and Z. Shen, Adenovirus-mediated p53 gene therapy reverses

resistance of breast cancer cells to adriamycin., Anticancer Drugs. 2011 Jul;22(6):556-62.



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8. Y. Zhao, D.H. Lam, J. Yang, J. Lin, C.K. Tham, W.H. Ng and S. Wang, Targeted suicide

gene therapy for glioma using human embryonic stem cell-derived neural stem cells

genetically modified by baculoviral vectors., Gene Ther. June 2011.

9. H. Rabinowich, M. Banks, Torten E. Reichert, Theodore F. Logan, J. M. Kirkwood and

T. L. Whiteside, Expression and activity of signaling molecules n T lymphocytes

obtained from patients with metastatic melanoma before and after interleukin 2 therapy.,

Clinical Cancer Research, 2 (1996), 12631274.

10. D. Kirschner and J. C. Panetta, "Modeling immunotherapy of the tumor-immune

interaction," Journal of Mathematical Biology, vol. 37, no. 3, pp. 235-252, 1998.



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