=Paper=
{{Paper
|id=Vol-1638/Paper73
|storemode=property
|title=Study of the dynamical model of HIV
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper73.pdf
|volume=Vol-1638
|authors=Marina A. Lapshova,Elena A. Shchepakina
}}
==Study of the dynamical model of HIV ==
https://ceur-ws.org/Vol-1638/Paper73.pdf
Mathematical Modeling
STUDY OF THE DYNAMICAL MODEL OF HIV
M.A. Lapshova, E.A. Shchepakina
Samara National Research University, Samara, Russia
Abstract. The paper is devoted to the study of the dynamical model of HIV. An
application of the technique of singular perturbation theory allows us to calcu-
late the conditions for the stabilization of the HIV status of the patient depend-
ing on the features that reflect the different nature of the intervention of doctors
in the treatment process.
Keywords: HIV, predator–prey model, stability.
Citation: Lapshova MA, Shchepakina EA. Study of the dynamical model of
HIV. CEUR Workshop Proceedings, 2016; 1638: 600-609. DOI:
10.18287/1613-0073-2016-1638-600-609
Introduction
There are many diseases that cause death nowadays. One of the most widespread
disease is human immunodeficiency virus (HIV). HIV is slowly progressive viral
infection of the immune system that causes weakening of the immune defense against
infections. Acquired immune deficiency syndrome (AIDS) is the last stage of HIV.
Concentration of immune cells so low at that stage, that the body can’t fight the virus.
About 60 million people have been infected HIV and 25 million people have died of
AIDS since the beginning of epidemic. Therefore research of this infection is very
important.
Studying of the mathematical model allows us to determine the dynamics of the virus
and the body's immune response depending on the function that describes medical
intervention.
Model
The following system of ordinary differential equations describes the dynamic of HIV
infection [1]:
x(t ) - xv - dx - f ( y, v) x, (1)
y(t ) xv - ay, (2)
v(t ) ky - uv, (3)
z(t ) v - bz, (4)
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here x(t) is the concentration of healthy CD4+T immune cells, y(t) is the concentra-
tion of infected CD4+T cells, v(t) is the concentration of free virus particles, and z(t)
is the concentration of antigen-specific CTL. Parameter is a rate of generating of
healthy CD4+T cells, d is a rate of dying, free virus particles infect the healthy cell at
a rate , a is a death rate of infected cells, the rates of producing and removing of
free virus particles are described by indexes k and u, respectively. The specific CTL
proliferate at a rate μ and decay at a rate b. All arguments are nonnegative. Function f
describes protection of immune cells against penetration of viral particles and may
depends on concentration of infected CD4+T cells and on concentration of free virus
particles. According to behavior of concavity of this function the dynamics of system
solution are different. The research of principal cases is our goal.
Analysis
The system (1)-(4) is multiscale in time. The equation (3) is a fast stage (cycle corre-
sponds to 12 hours), equation (4) is the fastest stage (cycle corresponds to 3-4 hours).
For this reason we introduce the new small positive parameters 1 and 2 to trans-
form system (1)-(4) into singularly perturbed form:
x(t)= - xv - dx - f(y,v)x, (5)
y (t)= xv - ay, (6)
1 v(t) = k y - uv, (7)
1 2 z(t) = v - bz. (8)
Typically, to investigate a singularly perturbed system a combination of asymptotic
and geometrical techniques of analysis are applied [2, 3]. The essence of this ap-
proach consists in separating out the slow motions of the system under investigation.
Then the order of the differential system decreases, but the reduced system, of a lesser
order, inherits the essential elements of the qualitative behaviour of the original sys-
tem in the corresponding domain and reflect the behaviour of the original models to a
high order of accuracy when the slow integral manifold is attracting. A mathematical
justification of this method can be given by means of the theory of integral manifolds
for singularly perturbed systems [3].
System (5)-(6) is a slow subsystem, and (7)-(8) is a fast subsystem of the full system.
Setting 1 =0 and 2 =0 we obtain the degenerate system:
x(t)= - xv - dx - f(y,v)x, (9)
y (t) = xv - ay, (10)
k y - uv 0, (11)
v - bz 0. (12)
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The equations (11)-(12) describe the so-called slow surface of system (5)-(8) [2, 3].
The slow surface is the zero order approximation (i. e., as 1 0 and 2 0) of
the slow integral manifold. The slow integral manifold is defined as an invariant sur-
face of slow motions (see [3] and references therein).
The Jacobian matrix of (11)-(12)
- u 0
b
has the eigenvalues
1 -u, 2 -b.
Note that 1 and 2 are negative (u>0 and b>0), hence the slow surface is attractive.
For the reason above [4, 5] we can reduce the system with help (9), (10), and instead
of system (5)-(8) we will analyze its projection on the zero order approximation of the
slow integral manifold:
x(t ) - xy - dx - f ( y, v) x, y(t ) xy - ay, (13)
where k / u is the extent of infection.
The following cases are considered in the present paper:
f ( y) y 2 , (14)
cy
f ( y) , (15)
1 y
cy 2
f y . (16)
1 y
It should be noted that we have f(y)=0 if y=0 for all cases (14)-(16).
The equilibrium points of system (13) are determined by the following system:
𝛼 − 𝛽𝑥𝑦 − 𝑑𝑥 − 𝑓(𝑦)𝑥 = 0,
{
𝛽𝑥𝑦 − 𝑎𝑦 = 0.
The equilibrium point / d , 0 corresponds to the case without infected cells. The
Jacobian matrix of (13) at this point is
d
d
.
0
a
d
The characteristic equation
α
(λ + d) (λ − β + a) = 0
d
has the roots
𝛼
𝜆1 = −𝑑, 𝜆2 = 𝛽 − 𝑎.
𝑑
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Note that 1 is always negative. Hence, the equilibrium point / d , 0 is a sink if
a 0, (17)
d
and, in other way, it is a saddle. For example, values of arguments: 0.216 , a 0.8,
0.1, d 0.06 satisfy the inequality (17) with the interior equilibrium at (1.67, 0).
And in this case the concentration of healthy cells increases and stabilizes, see Fig. 1.
Fig. 1. (left) Concentration of healthy cells and (right) the phase portrait for system (13) with
0.216 , a 0.8, 0.1, d 0.06
For the case (14) system (13) takes the form
x t - xy - dx - y 2 x,
(18)
y t xy - ay.
The equilibrium point with coordinates x x* , y y* , where
a * a 4a d 4a a
2 2 2
x* , y ,
2a
corresponds to the case of the infected cells’ presence. Recall that the variables x and
y are positive, then system (18) has an unique positive steady state
a a
, , 4a 2 d 4a a 2 2 .
2a
Moreover, the variable y must be real-valued, thus we get constraint:
4a 2 d 4a a 2 2 0
from which we get ad .
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The characteristic equation at the critical point a / , (a ) / (2a) is
2 2ad 2 0.5a 2 0.5 0. (19)
a
If the discriminant
2
D 4 2ad 2 0.5a 0.5
2
(20)
a
of equation (19) is more than zero then the first root
1 2 2
1 8ad 8 2a 2 2 2
2 a a
is negative. Hence if the second root
1 2 2
2 8ad 8 2a 2 2 2
2 a a
is positive then the critical point a / , (a ) / (2a) is a sink otherwise it is a
saddle. For example, for parameters’ values 3, a 0.3, 0.2, d 0.02 the criti-
cal point is a sink and has the coordinates (0.100, 0.557). Figure 2 shows the solu-
tions’ plots and the phase portrait of system (18) in the neighborhood of the critical
point for this case. From the graphs one can see that the concentrations of healthy (the
solid curve) and infected (the dashed curve) cells increase at first, but then the number
of infected cells continues to grow while the number of healthy cells is reduced. Alt-
hough the status of patient is stabilized but the number of infected cells more than the
number of normal cells. Thus, this case corresponds to the severe disease.
If the discriminant (20) is equal to zero then
1 2 0
2a
and the critical point is a stable degenerate node.
If the discriminant (20) is less than zero, we have
1 2 2
1,2 i 8ad 8 2a 2 2 2 ,
2 a a
and the point a / , (a ) / (2a) is a stable focus. For example, for parameters’
values 3, a 0.8, 0.2, d 0.02 the condition D > 0 is fulfilled and the critical
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point has the coordinates (0.267, 0.226). The graphs of the solutions and the phase
portrait in the neighborhood of the critical point (0.267, 0.226) are presented in Figure
3. Under these conditions, concentration of healthy cells (the solid curve) and infected
cells (the dashed curve) are damped oscillations at the beginning, followed by stabili-
zation. The number of healthy cells exceeds the number of infected cells after the
initial oscillations.
Fig. 2. (left) Concentration of healthy (the solid curve) and infected (the dashed curve) cells
and (right) the phase portrait for system (18) with 3, a 0.3, 0.2, d 0.02
In the case (15) system (13) takes the form
cy
x t - xy - dx - 1 y x,
(21)
y t xy - ay.
Fig. 3. (left) Concentration of healthy (the solid curve) and infected (the dashed curve) cells
and (right) the phase portrait for system (18) with 3, a 0.8, 0.2, d 0.02
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The positive equilibrium state with coordinates
a 1
, ac a ad ,
2 a
4a ad ac a ad ,
2
corresponds to the case of presence of the infected cells. Note that the number of the
cells concentrations must be real under condition
ac a ad 2 4aad .
The characteristic equation at this critical point is
d c c cd c c2 c ad
2
2 2a 2 2a a a 2
ac a c acd ac ac 2 c
0,
2 2 1 2 1 2 1 2 1
2 2 2 2
where
ac a ad
2 1 .
2a
Let us set
d c c cd c c2 c
a1 ,
2 2a 2 2a a a
1 ac a c acd ac ac 2 c
a2 ad .
2 1 1 1 1
2 2 2 2
According to the Routh Hurwitz stability criterion the system is stable at a critical
point if we have
1 a1 0
and
a1 1
2 0.
0 a2
Due to 2 a 2 1 the second inequality implies a 2 0. The graphics of the solu-
tions and the phase portrait in the neighborhood of this critical point are plotted for
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values of the parameters 0.216, a 0.4, 0.2, d 0.02, c 0.1, 2, and
are presented in Figure 4. Note that for these values of the parameters the critical
point has the coordinates (1.852, 0.317) and the stability conditions are fulfilled. The
figure shows that the concentration of healthy cells and infected cells are stabilized at
the same time after several periods of damped oscillations. The number of healthy
cells prevails over the number of infected cells. In this case a patient continues to live
and feels well.
Fig. 4. (left) Concentration of healthy (the solid curve) and infected (the dashed curve) cells
and (right) the phase portrait for system (21) with
0.216, a 0.4, 0.2, d 0.02, c 0.1, 2
In the case (16) system (13) has the form
cy 2
x t - xy - dx - x,
1 y (22)
y t xy - ay.
The positive equilibrium state with coordinates
a a ad
, ,
2ac a
where
4ad ac a a ad 2
corresponds to the case of presence of infected cells. For real values of variable y we
need
a ad 2 4ad ac a .
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The stabilization conditions are found like in the previous case and are
2a c d a c a c a c d ac a c 3ac
2 2 2 2 2 2 2 2 2 2 2 2
a 2 cd 2 2 a 2 cd 2 acd 2 ac 2 2 a 3 2
ad 2 3 2 3 3 ac 2 ac acd ac
2 2 / a c 2 ac a ad 2 2 0
and
2 a 3 4c 2 d d 2 d c 2 2d 5d 2 2
2
a c 5 3 2 3d
2 2 3
2d a 2 4c 2 2 d 3d
d c 2 5d 3d
2ac a ad 2 2 0.
1
Here 2 2 2 2a2c d a 2 4cd d 2 .
For example, for
0.216, a 0.4, 0.2, d 0.02, c 0.1, 2
the critical point has the coordinates (1.852, 0.371) and the plots of the solutions and
the phase portrait in the neighborhood of the critical point are shown at Figure 5. The
critical point is asymptotically stable.
Fig. 5. (left) Concentration of healthy (the solid curve) and infected (the dashed curve) cells
and (right) the phase portrait for system (22) with
0.216, a 0.4, 0.2, d 0.02, c 0.1, 2
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Conclusion
Thus, in the cases considered above the status of a patient infected with the virus may
be stabilized under certain conditions which could be achieved by selection of the
appropriate treatment. It should be noted that the application of the geometric theory
of the singular perturbation made it possible to solve several actual problems concern-
ing multirate processes in biological systems, see, for example, [6-9].
Acknowledgements
This work is supported in part by the Russian Foundation for Basic Research (grant
14-01-97018-p) and the Ministry of Education and Science of the Russian Federation
under the Competitiveness Enhancement Program of Samara University (2013–2020).
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