 Open Access
 Total Downloads : 264
 Authors : Bawa, Musa
 Paper ID : IJERTV2IS2472
 Volume & Issue : Volume 02, Issue 02 (February 2013)
 Published (First Online): 28022013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Two Dimensional Infection AgeStructured Mathematical Model Of The Dynamics of HIV/AIDS
BAWA, Musa
Department of Mathematics/ Computer Science, Ibrahim Badamasi Babangida University, Lapai, Nigeria.
Abstract
The paper considers two compartmentalized mathematical model of HIV/AIDS disease dynamics of the susceptible and infected members with age structure introduced in the latter class. The susceptables are virus free but prone to infection through specific transmission pattern that is, coming in contact with infected body fluids such as blood, sexual fluids and breast milk. The total population is partitioned into two distinct classes giving rise to a set of model equations with one ordinary differential equation and one partial differential equation. Parameter values were used to represent the consequential interactive characteristics of the population. The equilibrium states and the corresponding characteristics equation were obtained. The Bellman and Cookes theorem is applied to analysed the equilibrium states of the model for stability and critical values of these parameters obtained. The result revealed that to sustain the population, the birth rate must be greater than the death rate among others.

Introduction
According to Benyah [3], mathematical modeling
processes. Burghes and Wood [4] opines that it could even be claimed that the spread of modern industrial civilization, for better or for worse, is partly a result of mans ability to solve the differential equations which govern so many of our industrial processes, be them chemical or engineering.
In this work, the population is partitioned into two compartments of the susceptible S(t), which is the class of members that are virus free but are prone to infection as they interact with the infected class.
The infected class I(t) consists of members that contracted the virus and are at various stages of infection. This class is structured by the infection age, with the density function (t, a) where t is the time and a is the infection age.
There is a maximum infection age T at which a member of the infected class must leave the compartment via death i.e. when a = T for 0 a T. However, a member of the class could still die by natural causes at a rate , which is also applicable to the susceptible class S(t).
Members of S(t) move into I(t) at a rate due to negative change in behavior. The gross death rate via infection is given by (a) = + tan a , is
is an evolving process, as new insight is gained the
process begins again as additional factors are
2TK
additional burden from infection while K is a co
ntrol
considered. The research work proposes a deterministic mathematical model of HIV/AIDS disease dynamics resulting into a system of ordinary and partial differential equations. Differential equations form very important mathematical tools used in producing models of physical and biological
It is assumed that while the new births in S(t) are born there in, the offsprings of I(t) are divided between S(t) and I(t) in the proportion and 1 respectively
parameter associated with the measure of slowing down the death of the infected member, such as the effectiveness of the anti retroviral drugs which give the victims longer life span. A high rate of K will imply high effectiveness of such measure and vice versa.
that is, 1 of the offsprings of I(t) are born with the virus.

The model equations
0
0
S1 = ()s(t) + I(t) s(t)I(t) (1) and I(t) = T (t, a)da, 0 a T (2)
(t, a) + (t, a) + (t +(a))(t, a) = 0 (3)
d(a) + (a)(a) = 0 (13)
d(a) = (a)da (14)
()
Integrating both sides
(a) = (0) exp { a (s) ds} (15)
t 0
Where (a) = + tan a
2TK
(4)
Let (a) = exp * a (s)ds} (16)
0
0
(t, 0) = B(t) = s(t)I(t) + (1) I(t) (5) and (0, a) = (a) (6)
S(0) = S0, I(0) = I0 (7)
With the parameters given by
= natural birth rate for the population;
= natural death rate for the population.
= rate of contracting the HIV virus.
(a) = gross death rate of the infected class.
= additional burden from infection.
K = measure of the effectiveness of efforts at slowing down the death of infected members.
That is,
(a) = (0)(a) (17)
and
0
0
y = (0) T (a)da = (0) (18) Using (11) and (18)
y = (y + (1) y) (19)
From (12) and (19).
= 1 (1 (1) ) (20)
Substituting (20) in (12)
() 1 (1(1) )
= the proportion of the offsprings of the infected
y =
[ (1(1) ) ](21)
which are virus free at birth 0 1.
T = maximum infection age i.e. when a = T the infected member dies of the disease.

Equilibrium states
At the equilibrium states, let
S(0) = , I(0) = y (8)
if (t, a) = (a) (9)
0
0
from (1.2), y = T (a) (10) from (5), (0) = (0) = y + (1)y (11) Substituting (9) to (11) into (1) and (3)
() + y y = 0 (12)
Hence, the zero equilibrium state, is (,y) = (0,0) and the nonzero equilibrium state is given by (20) and (21).

The characteristics equation
As in Akinwande [1], let the equilibrium state be perturbed as follows:
S(t) = +p(t), p(t) = p et (22)
I(t) = y + q(t); q( t) = q et (23)
Let (t. a) = (a) + (a) et (24)
0
0
With q = T(a)da (25) Substituting (22) to (25) into the model equations (1)
and (3)
)d d
(x + p et ) = () (+p et ) + (y+q et )
From (11), (0) = y + (1)y.
(+p et )(y+q et )
p et = () + () p et + y +q et
y q et yp et p q e2t
From equation (12) and neglecting terms of order 2, we have;
p et = () p et + q et q et yq et
or (y)p + () = 0 (26)
Substituting (24) into (3), gives
d((a) (a) et ] + d [(a)+(a) et ] + (a)[(a)+(a)et]
and (24) (t, a) = (a) + (a) et
But (t, 0) = (t) = (0) + (0) et (32)
From (5), (t) = s(t) + (1)I(t)
Substituting (22) to (25) into (5) and using (11)
and (32)
B(t) = ( + p et) (y + q et) + (1) (y +q et)
= y + p yet + q et + p q e2t + (1
)y + (1)q et
(33)
= 0
That is,
Compare this with (32) using (11) for (0) gives
y + (1)y + (0)et = y + p yet + q et
(a)et + d(a) + et d
(a) + (a)(a) + (a)(a)et =
+ p q e2t + (1)y + (1)q et
0
Since
neglecting terms of order 2
(0) = p y + q + (1)q (34)
d(a) + (a)(a) = 0
Then
(a)et + et d (a) + (a)(a)et = 0
d (a) +((a) +) (a) = 0 (27)
Solving the Ordinary Differented Equation (27), gives
d(a) = ((a)+ )da (28)
(a)
0
0
(a) = (0) exp { a((s) + )ds} (29) Integrating (29) over [0, T] gives
Substituting (0) in (30)
q = (yp + q + (1)q ) b() (35)
yp + [( + (1)) b()1)] q = 0 (36)
Using (26) and (36), we obtain the Jacobian determinant for the system with the eigen value
y
= 0 (37)
q = (0) T[ exp { a((s) + )s}] da
y ( + (1))b()1
0 0
or q = (0) b() (30)
Since q = (0) b(),where b() =
T exp { a((s) + s}] da (31)
and the characteristics equation is given by:
( y ) [( + (1))b()1] y(
0 0
) = 0 (38)
(0) is calculated as follows:

Stability of the zero equilibrium state
At the zero equilibrium state (x, y) = (0, 0),
the characteristic equation becomes: ( )[(1)b()1] = 0 (39)
That is,
either ( ) = 0 or (1)b()1 = 0 (40)
1 = (41)
This shows that 1 < 0, if <
The nature of the roots of the transcendental equation
(1)b()1 is now investigated.
0 0
0 0
Since b() = T exp { a( + (s)} da, which implies that
0
0
b() = T ea (a) da (42)
Using the approximation
0
0
b() = T(1a) (a) da =
Table 1. Stability analysis of the zero state.
K
T
D1(k)
Remar k
0.
3
0.
003
0.
4
1
0
0.3
1
0.1
2
0.00088
26
Instab ility
0.
4
0.
003
0.
4
1
0
0.3
1
0.1
2
0.00141
38
Instab ility
0.
5
0.
003
0.
4
1
0
0.3
1
0.1
2
0.00119
76
Instab ility
0.
6
0.
003
0.
4
1
0
0.3
1
0.1
2
0.00109
43
Instab ility
0.
7
0.
003
0.
4
1
0
0.3
1
0.1
2
0.00099
17
Instab ility
0.
8
0.00
3
0.
4
1
0
0.3
1
0.1
2
0.00081
81
Instab ility
0.
9
0.00
3
0.
4
1
0
0.3
1
0.1
2
0.00027
78
Instab ility
From table 1, D1 (k) > 0 when > , which implies the instability of the origin.

Stability analysis of the non zero state
At the non zero state
T (a) da T a(a) da (43)
1 1 [1 1 ]
0 0 (, y) = {
(1 (1 ) , } (48)
{ 1 1 }
0
0
= A where A = T a(a) da So, (1)b() 1 = 0 takes the form: (1)( A) 1 = 0 (44)
= (1) 1
(1)A
(45)
To analyse the non zero state for stability, we shall apply the result of Bellman and Cooke [2] to the characteristics equation (38) taking it in the form H() = 0
if we set = iw; and have H(iw) = F(w) + iG(w) (49)
Since b() = T exp { a( + (s)ds} da
0 0
So, sign = sign {(1 ) 1} (46)
b(iw) = T exp { a(iw + (s)} da
0
0
Let D1 (k) = (1) 1 (47)
So, the origin will be stable when D1 (k) < 0 and
= a e
iwa
0 0
(a)da (50)
T
Unstable when otherwise.
= 0 [(cos wa (sin wa)] (a) da = f(w) + ig(w) (51)
0
0
in f(w) = T (a) cos wa da (52)
0
0
and g(w) = T (a) sin wa da (53)
0
0
so , f(0) = T (a)da = (54) g(0) =0 (55)
0
0
Also , f(w) = T a(a) sin (wa) da
f(0) = 0 (56)
0
0
and, g(w) = T a(a) cos wa da
0
0
g(0) = T a(a)da = A (57)
Thus, H(iw) = ( y iw) [( + (1 )) b(iw) y ( ) (58)
H(iw) = ( y iw) [( + (1 )) f(w) + ig(w) 1] y ( )
= ( y iw) [( + (1 )) f(w) +
( + (1 ))ig(w) 1] y ( )
F(w) = ( y) ( + ( 1 )) f(w) + w( +
1 ))g(w)( y) y ( ) (59)
G(w) = ( y) ( + (1 )) g(w) w (
+(1 )) f(w) + w (60) G(0) = 0
F(0) = ( y) ( + (1 )) ( y)
y( ) (61) and
G1(0) = ( y) ( + (1 ))A ( + (1 )) + 1 (62)
F1(0) = 0
For stability or otherwise of the equilibrium state, we need to satisfy the condition for which the inequality F(0) G1(0) F1(0) G(0) > 0 holds.
The inequality then gives F(0) G1(0) > 0 Let D2(k) = F(0) G1(0)
Then, the nonzero state will be stable when D2(k) > 0
(63)
Table 2. Stability analysis of the nonzero state.
K 
T 
D2(k) 
Re mar k 

0 . 1 
0 . 2 
0. 01 
0. 01 5 
0. 00 1 
0 . 4 
1 
0.023285 41 
Sta ble 
0 . 2 
0 . 2 
0. 01 
0. 01 5 
0. 00 2 
0 . 4 
1 
0.023254 01 
Sta ble 
0 . 3 
0 . 2 
0. 01 
0. 01 5 
0. 00 3 
0 . 4 
1 
0.023553 26 
Sta ble 
0 . 4 
0 . 2 
0. 01 
0. 01 5 
0. 00 4 
0 . 4 
1 
0.023685 85 
Sta ble 
0 . 5 
0 . 2 
0. 01 
0. 01 5 
0. 00 5 
0 . 4 
1 
0.023817 56 
Sta ble 
0 . 6 
0 . 2 
0. 01 
0. 01 5 
0. 00 6 
0 . 4 
0.023948 39 
Sta ble 

0 . 7 
0 . 2 
0. 01 
0. 01 5 
0. 00 7 
0 . 4 
1 
0.024078 34 
Sta ble 
0 . 8 
0 . 2 
0. 01 
0. 01 5 
0. 00 8 
0 . 4 
1 1 
0.024207 41 
Sta ble 
0 . 9 
0 . 2 
0. 01 
0. 01 5 
0. 00 9 
0 . 4 
0.024335 60 
Sta ble 

1 
0 . 2 
0. 01 
0. 01 5 
0. 01 
0 . 4 
1 1 
0.024462 91 
Sta ble 
Fro table 2, D2(k)>0 which implies the stability of the non zero state.
Conclusion
The zero equilibrium state which is the state of population extinction will be stable when the birth rate is less than the death rate in addition to meeting the requirement of inequality D1(k)<0. The non zero state, which is the state of population sustenance will be stable if the inequality (63) is satisfied. So efforts must be geared toward meeting this non zero stability bound through public enlightenment.
References

N. I. Akinwande, On the Characteristics Equation of a Non Linear AgeStructured Population Model; ICTP, Trieste, Italy Preprint IC/99/153, 1999.

R. Bellman, and K. L. Cooke, Differencial Difference Equations; Academic Press, London, 1963.

F. Benyah, Introduction to Mathematical Modeling, 7th Regional College on Modelng, Simulation and Optimization, Cape Coast, Ghana, 2005.

D. N. Burghes, and A. D. Wood, Mathematical Models in the Social, Management and Life Sciences, Ellis Horwood Limited, New York, 1984.