Journal of Applied Mathematics & Bioinformatics, vol.3, no.4, 2013, 269-284
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2013
Numerical computation and series solution for
mathematical model of HIV/AIDS
Deborah O. Odulaja1, L.M. Erinle-Ibrahim1 and Adedapo C. Loyinmi 2
Abstract
In this paper, a mathematical model of HIV/AIDS model was examined and of
particular interest is the stability of equilibrium solutions. The characteristic
equation which gives the Eigen values was examined. By series solution method
the behaviour of the viruses and CD4+Tcells was looked into. It was shown that if
the recovery rate is high enough, the healthy CD4+Tcell may never die out
completely. Hence the patient that test HIV positive, may never develop into
full-blown AIDS.
Keyword: CD4+Tcell
1
2
Department of Mathematics, Tai Solarin University of Education, Ogun State. Nigeria.
Department of Mathematical Sciences, University of Liverpool, England.
E-mail: a.c.loyinmi@liv.ac.uk
Article Info: Received : August 26, 2013. Revised : October 26, 2013.
Published online : December 10, 2013.
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Computation and series solution for mathematical model of HIV/AIDS
1 Introduction
AIDS is caused by Human Immunodeficiency Virus (HIV) infection and is
characterized by a severe reduction in CD4+ Tcells, which means an infected
person develops a very weak immune system and becomes vulnerable to
contracting life-threatening infection (such as pneumocysticcarinii pneumonia).
AIDS (Acquired immunodeficiency syndrome) occurs late in HIV disease. The
first cares of AIDS were reported in the United States in the spring of 1981. By
1983 HIV had been isolated. Several mathematicians have proposed models to
describe the dynamics of the HIV/AIDS infection of CD4+Tcells. In particular
Ayeni etal [1,2] proposed the following model:
dX
= −(d1 + k1V* )X + µY − k1T* Z
dt
(1.1)
dY
= k1V* X − (d 2 + µ )Y + k1T* Z
dt
(1.2)
dZ
= k 2Y − CZ
dt
(1.3)
where:
T = Population of CD4+T cells; Ti =Population of infected CD4+ T cells;
V = Virus; π = Production rate of CD4+T cells; d1 = National death rate of healthy
CD4+T cells; d 2 = Death rate of infected CD4+T cells; k1 = Viral infection rate of
CD4+T cells; k2 = Viral production rate for CD4+T cells; C = Viral clearance rate.
Clearance
Td1 → Death of normal CD4+T cells; Ti d 2 → Death of infected CD4+T cells
VC → Viral clearance rate.
Ayeni (2010) replaced equation (1.1) by
k TV
dT
= π − d1T − 1
+ µTi
1 + αV
dt
and (1.2) by
(1.4)
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
dTi
k TV
− d 2Ti − µTi
= 1
dt 1 + αV
271
(1.5)
where α = Disease related to death rate .
The mathematical model of (1.1) – (1.3) and (1.4) and (1.5) have not been
fully established in literature. So the research goes on and on this basis we propose
the following model:
2 Mathematical Formulation
A model of HIV infection similar to (1.1) and (1.2) but using
k1TV
for infection CD4+T cells is proposed .
1 + αV
Thus the model is
k TV
dT
=
+ µTi ,
π − d1T − 1
dt
1 + αV
dTi
k1TV
=
− d 2Ti − µTi ,
dt 1 + αV
dV
=
k2Ti − CV ,
dt
T ( 0) =
T0
Ti ( 0=
) Ti(0)
V ( 0) =
V0
3 Models
3.1 Method of Solution
To obtain the critical point, we set in infected free equilibrium then,
dT dTi dV
= = = 0.
dt
dt
dt
Equation (2.1) becomes
(2.1)
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Computation and series solution for mathematical model of HIV/AIDS
π − d1T −
k1TV
+ µTi =
0
1 + αV
k1TV
− d 2Ti − µTi =
0
1 + αV
k2Ti − CV =
0
(3.1)
when there is no CD4+T cells infection then V= T=
0.
i
And equation (3.1) becomes π − d1T = 0 , with T =
So the un- infected equilibrium is (T , 0, 0) = (
π
d1
π
d1
.
, 0, 0) .
The infected equilibrium when there is CD4+Tcells infections is V ≠0 ,TI≠ 0.
k TV
dT
= π − d1T − 1
+ µTi
1 + αV
dt
T (0 ) = T0
(3.1a)
dTi
k TV
= 1
− d 2Ti − µTi
dt 1 + αV
Ti (0 ) = Ti (0 )
(3.1b)
dV
= k 2Ti − CV
dt
V (0 ) = V0
(3.1c)
Then equation (3.1c) becomes
CV = k2Ti
(3.2)
k2Ti
C
V=
Substituting (3.2) in (3.1b)
k T 
k1T  2 i 
 C  − d T − µT =
0.
2 i
i
 k2Ti 
1+ α 

 C 
Then
T=
(d 2 + µ )(C + αk 2Ti )
k 2 k1
Substituting (3.2) and (3.3) in equation (3.1a)
(3.3)
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
273
 ( d + µ )( C + α k2Ti )   k2Ti
k1  2

k2 k1
 ( d + µ )( C + α k2Ti ) 

 C
π − d1  2
−

k2 k1
k T 


1+ α  2 i 
 C 



0.
+ µTi =
Then the equation becomes
Ti =
π k2 k1 − Cd1 ( d 2 + µ )
.
α d1k2 ( d 2 + µ ) + k2 k1d 2
Then
V=
π k2 k1 − Cd1 ( d 2 + µ )
.
α Cd1 ( d 2 + µ ) + Ck1d 2
Now T becomes
T
=
( d 2 + µ ) C + α k2 ( d 2 + µ )  π k2 k1 − Cd1 ( d 2 + µ )  .


k2 k1
k2 k1
 α d1k2 ( d 2 + µ ) + k2 k1d 2 
Then the infected equilibrium is
 ( d 2 + µ ) C α k2 ( d 2 + µ )  π k2 k1 − Cd1 ( d 2 + µ )  π k2 k1 − Cd1 ( d 2 + µ )
,
+


 ,
 k2 k1
k2 k1
 α d1k2 ( d 2 + µ ) + k2 k1d 2  α d1k2 ( d 2 + µ ) + k2 k1d 2

π k2 k1 − Cd1 ( d 2 + µ ) 

α Cd1 ( d 2 + µ ) + Ck1d 2 
3.2 Reduction to origin
Let
x = T − T*
T = x + T*
y = Ti − Ti*
Ti* = y + Ti*
z = V − V*
V = z + V*
where (T* , Ti* , V* ) is the infected equilibrium such that
274
Computation and series solution for mathematical model of HIV/AIDS
π − d1T* −
k1T*V*
0
+ µTi* =
1 + αV*
k1T*V*
− d 2Ti* − µTi* =
0
1 + αV*
(3.2.1)
0
k2Ti* − CV* =
Then
dx dT dy dTi dz dV
=
=
, =
,
dt
dt dt
dt dt
dt
(3.2.2)
Substituting (3.2. 1) and (3.2.2) in (3.1)
k ( x + T* )( z + V* )
dx
=π − d1 ( x + T* ) − 1
+ µ ( y + Ti* )
dt
1 + α ( z + V* )
dy k ( x + T* )( z + V* )
= 1
− d 2 ( y + Ti* ) − µ ( y + Ti* )
dt
1 + α ( z + V* )
dV
= k2 ( y + Ti* ) − C ( z + V* )
dt
Then equation (3.2.3) becomes,
kV 
kTz
dx 
=  − d1 + 1 *  x + µy − 1 * + nonlinearterms
dt 
1 + αV* 
1 + αV*
kV
kTz
dy
= 1 * x − (d 2 + µ ) y + 1 * + nonlinearterms
1 + αV*
dt 1 + αV*
dz
= k 2 y − Cz + nonlinearterms
dt
So
 dx 
 dt 
 
 dy 
=
 dt 
 
 dz 
 dt 
Then

k1V* 
k1T* 
µ
  −d1 +


1 + αV* 
1 + αV* 


k1V*
k1T* 
− ( d2 + µ )

 + ( nonlinearterms )
1 + αV*
1 + αV* 


0
k2
−C 






(3.2.3)
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
 dx 
 dt 
 x
 
dy 
 
=
A  y  + ( nonlinearterms )
 dt 
z
 
 
dz
 
 dt 
A − λI = 0
where

k1V* 
k1T* 
µ
  −d1 +


1 + αV* 
1 + αV* 


k1V*
k1T* 
=
− ( d2 + µ )
A 

1 + αV*
1 + αV* 


−C 
0
k2







k1V* 
 −d1 +
−λ
1 + αV* 

k1V*
=
A − λI
1 + αV*
µ
k1T*
1 + αV*
− ( d2 + µ ) − λ
k1T*
1 + αV*
k2
−C − λ
0
Let π= 50, d=
d=
0.01, k=
k=
0.03, C= 0.01, µ= 0.01 and α = 2 .
1
2
1
2
Substituting the parameters in (3.4).
Then
(T* , Ti* , V* ) = (2857.24, 2142.76, 6428.28).
So
−0.0249 − λ
=
A − λI
0.0149
0
0.01
−0.02 − λ
0.03
−0.0067
=
0.0067
0
−0.01 − λ
𝜆3 + 0.0549 𝜆2 + 0.000597𝜆 + 0.000001255 =0
The eigenvalues of the system are check by MATLAB function and it gives
275
276
Computation and series solution for mathematical model of HIV/AIDS
𝜆1 = -0.002774, 𝜆2 = -0.041125, 𝜆3 = - 0.0110004.
Therefore the Eigenvalues of this system are 𝜆 = - 0.0028, - 0.0411 and – 0.011,
hence this system is asymptotically stable.
The conclusion of this system is similar to
Theorem (Derrick and Grossman 1976)
Let V ( x1 , x2 , x3 ) be a lyapunov function the system
x11 =
f1 ( x1 , x2 , x3 )
x12 =
f 2 ( x1 , x2 , x3 )
x31 = f 3 ( x1 , x2 , x3 )
Then,
V1 = (x1x2x3) is negative semi definite the origin is stable
V1 = (x1x2x3) is negative definite, the origin is asymptotically stable
V1 = (x1x2x3) is positive definite the origin is unstable
4 Numerical solution
Substituting (3.2.1) and (3.2.2) in (1.1)-(1.3), it becomes
kV 
kTz
dx 
=  − d1 + 1 *  x + µy − 1 * + nonlinearterms
dt 
1 + αV* 
1 + αV*
kV
kTz
dy
= 1 * x − (d 2 + µ ) y + 1 * + nonlinearterms
dt 1 + αV*
1 + αV*
dz
= k 2 y − Cz + nonlinearterms
dt
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
277
dx k1 ( x + T* )( z + V* )
=
dt
1 + α ( z0 + V* )
dy k1V* x ( x + T* )( z + V* ) −
=
dt
1 + α ( z0 + V* )
(4.1)
dz
= k2 y − cz
dt
In order to find the approximate solution to the model, power series solution is
used.
Let the solution to the system (1) be
x(t ) = x0 + a1t + a2t 2 +  + ant n
y (t ) = y0 + b1t + b2t 2 +  + bnt n
z (t ) = z0 + m1t + m2t 2 +  + mnt n
Let
x1 (t=
) x0 + a1t
(4.2)
y1 (t=
) y0 + b1t
z1 (t=
) z0 + m1t
2
2
=
x12 (t ) a=
b=
m1
1 , y1 (t )
1 , z1 (t )
(4.2.1)
Substituting (4.2) and (4.2.1) in (4.1), the system becomes
a1 =
b1 =
−  d1 (1 + α z0 + αV* ) + k1V*  x0 + µ y0 (1 + α z0 + αV* ) − k1T* z0
1 + α ( z0 + V* )
k1V* x0 − ( d 2 + µ )(1 + α z0 + αV* )  y0 + k1T* z0
1 + α ( z0 + V* )
m1 k2 y0 − Cz0
=
Then equation (4.2) can be written as
 −  d (1 + α z0 + αV* ) + k1V*  x0 + µ y0 (1 + α z0 + αV* ) − k1T* z0 
x1 ( t=
t
) x0 +   1

z
V
1
α
+
+
(
)
0
*


(4.2.2)
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Computation and series solution for mathematical model of HIV/AIDS
 k V x − ( d + µ )(1 + α z0 + αV* )  y0 + k1T* z0 
y1 ( t=
t
) y0 +  1 * 0  2

1 + α ( z0 + V* )


z1 ( t ) =+
z0 ( k2 y0 − Cz0 ) t
Let
 −  d1 (1 + α z0 + αV* ) + k1V*  x0 + µ y0 (1 + α z0 + αV* ) − k1T* z0 
x2 ( t ) =
x0 +  
 t + a2t 2


1 + α ( z0 + V* )


 k1V* x0 − ( d 2 + µ )(1 + α z0 + αV* )  y0 + k1T* z0 
(4.2.3)
y2 ( t ) =
y0 + 
 t + b2t 2


1
z
V
α
+
+
(
)
0
*


2
z2 ( t ) =+
z0 ( k2 y0 − Cz0 ) t + m2t
Perturb (4.2.3) and substitute into (1), we have
1 −  d1 (1 + α z0 + αV* ) + k1V* 
a2 = { 
2
1 + α ( z0 + V* )
−  d1 (1 + α z0 + αV* ) + k1V*  x0 + µ y0 (1 + α z0 + αV* ) − k1T* z0
×( 
)
1 + α ( z0 + V* )
+ µ(
k1V* x0 + ( d1 + µ )(1 + α z0 + αV* ) y0 + k1T* z0
k T ( k y − Cz0 )
)− 1 * 2 0
}
1 + α ( z0 + V* )
(1 + α z0 + αV* )
k1T*
1
b2 = {
2 (1 + α z0 + αV* )
(4.2.4)
−  d1 (1 + α z0 + αV* ) + k1V*  x0 + µ y0 (1 + α z0 + αV* ) − k1T* z0
)
×( 
1 + α ( z0 + V* )
− ( d1 + µ ) (
k1V* x0 − ( d1 + µ )(1 + α z0 + αV* ) y0 + k1T* z0
k T ( k y − Cz0 )
)+ 1 * 2 0
}
1 + α ( z0 + V* )
(1 + α z0 + αV* )
k V x − ( d1 + µ )(1 + α z0 + αV* ) y0 + k1T* z0
1
{k2 ( 1 * 0
) − C ( k2 y0 − Cz0 )}
2
1 + α ( z0 + V* )
Substituting (4.2.2) into (4.2.4)
m2
a2
b2
=
k1T*
1 −  d1 (1 + α z0 + αV* ) + k1V* 
(
a1 + µ b1 −
m)
2
1 + α ( z0 + V* )
(1 + α z0 + αV* ) 1
k1T*
k1T*
1
(
a1 − ( d1 + µ ) b1 +
m)
2 (1 + α z0 + αV* )
(1 + α z0 + αV* ) 1
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
m2
=
279
1
(k2b1 − Cm1 )
2
Let
x0 ≠ 0, y0 =
0, z0 =
0, T* =≠
T* 0, Ti* =
0, V* =
0.
Substituting case 1 in (4.2.2)
a1 = −d1 x0 , b1 = 0 , m1 = 0
d1
d2
a2 =
− 1 x0 , b2 =
(−d1 x0 ) =
0, m2 =
0.
2
2
Then the series become
x(t ) = x0 + x0 d1t + x0
d12t 2
d nt n
+  + x0 1
2
n
when x0 = 20 , d 2 = 0.01
x(t ) =20(1 − 0.01t +
0.012 2
t +)
2
when x0 = 10 , d 2 = 1
t2
x(t=
) 10(1 − t + +)
2
Case 2
Let x0 ≠ 0, y0 =
0, z0 =
0, T* =≠
T* 0, Ti* =
0, V* =
0
Substituting case 2 in (4.2.2)
a=
2
k1T0
1
(d1a1 + µ b1 −
m)
2
(1 + α z0 ) 1
k1T0
1
b2 = (− ( d1 + µ ) b1 +
m)
2
(1 + α z0 ) 1
m2
=
1
(k2b1 − Cm1 )
2
If x0 = 10 , y0 = 2 , z0 = 2 , T0 = 5 and considering other parameter k=
k=
3,
1
2
d=
d=
1 , and c = 1 .
1
2
We have the following solutions:
280
Computation and series solution for mathematical model of HIV/AIDS
Figure 1: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= 1/5 and α= 1/5
Figure 2: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= ½ and α= 1/5
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
281
Figure 3: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= 1 and α= 1/5
The graph of µ= 3/2
12
10
8
6
4
2
-4
0,5
0,475
0,45
0,425
0,4
0,375
0,35
0,325
0,3
0,25
0,275
0,2
0,225
0,15
0,175
0,1
0,125
0,075
0,05
-2
0,025
0
x
y
z
Figure 4: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= 3/2 and α= 1/5
282
Computation and series solution for mathematical model of HIV/AIDS
10
The graph of µ=2
8
6
4
0
-2
0,025
0,05
0,075
0,1
0,125
0,15
0,175
0,2
0,225
0,25
0,275
0,3
0,325
0,35
0,375
0,4
0,425
0,45
0,475
0,5
2
-4
x
y
z
Figure 5: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= 2 and α= 1/5
The graph of µ=3
10
9
8
7
6
5
4
x
3
2
0
y
0,025
0,05
0,075
0,1
0,125
0,15
0,175
0,2
0,225
0,25
0,275
0,3
0,325
0,35
0,375
0,4
0,425
0,45
0,475
0,5
1
z
Figure 6: Graph of x (CD4+T cells), y (infected cells), z (virus) against time at
µ= 3 and α= 1/5
D.O. Odulaja, L.M. Erinle-Ibrahim and A.C. Loyinmi
283
5 Conclusion
Figure 1 shows that the healthy CD4+T cells are all infected around t = 0.325,
when µ=0.2
Figure 2 shows that the healthy CD4+T cells are all infected around t = 0.3125,
when µ=0.5
Figure 3 shows that the healthy CD4+T cells are all infected around t = 0.3625,
when µ=1
Figure 4 shows that the healthy CD4+T cells are all infected around t = 0.3875,
when µ=1.5
Figure 5 shows that the healthy CD4+T cells are all infected around t = 0.4, when
µ=2
Figure 6 shows that the healthy CD4+T cells are all infected around t = 0.4875,
when µ=3
The above results show that as µ increases, the duration of time for all the
CD4+T cells to get infected also increases µ is the recovery rate of the CD4+T
cells. This implies that when the recovery rate µ is high, CD4+T cells take longer
time for all to get infected.
In this paper, we modified an existing HIV/AIDS model. We investigated the
characteristic equation and discussed the stability of equilibrium points by finding
the eigenvalues of the model that were previously considered.
We solved existing characteristic equations numerically using realistic values
for the parameters and we interpreted the graphs that resulted from the numerical
solution.
The stability criteria showed that HIV may not lead to full blown AIDS since
the healthy CD4+T cells may never die out completely.
284
Computation and series solution for mathematical model of HIV/AIDS
References
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global dynamics of a model for HIV infection of CD4+ T cells, JNAMP, 11,
(2007), 103-110.
[2] R.O. Ayeni, A.O. Popoola and J.K. Ogunmola, Some new results on affinity
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(2010), 74-79.
[3] W.R. Derrick and S. Grossman, Elementary Differential Equations, Addison
Wesley, Reading, 1976.
[4] L.B. Steven and A.S. Peter, Substance Abuse Treatment For Persons with
HIV/AIDS, U.S. Department of health and services, (2008), 27-29.
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model of HIV/AIDS, M.ed dissertation, Tai Solarin University of Education,
Ogun State, Nigeria, 2013.