A Three Dimensional Infection Age-Structured Mathematical Model Of Hiv/Aids Dynamics Incorporating Removed Class

A Three Dimensional Infection Age-Structured Mathematical Model Of Hiv/Aids Dynamics Incorporating Removed Class

Bawa, Musa

Department of Mathematics/Computer Science Ibrahim Badamasi Babangida University, Lapai, Nigeria.

Abstract

The research proposes a three dimensional infection age structured deterministic mathematical model of the HIV/AIDS disease dynamics and introduce requisite parameters on the effects of public campaign and drug administration aimed at slowing down the death of the victims. The population is divided into three compartments of the susceptible, removed and the infected members with age structure introduced in the infected class. The susceptible are virus free but are prone to infection through specific transmission pattern that is, coming in contact with infected body fluids such as blood, sexual fluids and breast milk. Members of the removed class are not prone to infection due to their adherence to warnings or change in behavior through public enlightenment. Members in the infected class are HIV virus carriers and are at various stages of infection. The three distinct classes led to a set of model equations with two ordinary differential equations and one partial differential equation with parameter values used to represent the consequential interactive characteristics of the population. The zero and non zero equilibrium states were analysed for stability or otherwise of the model and bounds obtained for sustenance of the population.

Immune Deficiency Syndrome (HIV / AIDS) disease dynamics which is a system of ordinary and partial differential equations. The population p(t) is divided into three classes of the susceptible s(t); this is the class in which members are virus free but are prone to infection as they interact with the infected class; the next class is the removed R(t). which is the class of those not susceptible to infection, possibly due to their adherence to warnings or changed behavior or enlightenment through public campaign; while the last class is the infected I(t); this is the class of those that have contracted the virus and are at various stages of infection.

The infected class I(t) is structured by the infection age with the density function

(t,a), where t is the time parameter and a is the infection age. There is a maximum infection age T at which a member of the infected class I(t) must quit the compartment via death; and so 0 a T. This not withstanding a member of the class could still die by natural causes at rate µ, the latter is also applicable to the susceptible class s(t) and the removed class R(t).

Members of the class S(t) move into the class R(t) due to change in behavior or / and as a result of effective public campaign at a rate .

The death rate via infection is given by

a = + tan a ; is the maximum death

1. Introduction

rate due to

2TK

infection while K is a control

The research work proposes a deterministic mathematical model of the Human Immunodeficiency Virus / Acquired

parameter which could be associated with the measure of slowing down the death of the infected, such as the effectiveness of the anti retroviral drugs which give victims

longer life span. A high value of K will imply high effectiveness of such measure and vice versa.

It is assumed that while the new births in S(t) are born therein, the off-springs of I(t) are divided between S(t) and I(t) in the proportions and (I – ) i.e. a proportion (I – ) of the offspring of I(t) are born with the virus.

The equilibrium states are then analysed for stability.The work complements and extends the work of Bawa

[2] by incorporating the removed class. According to Benyah [4], mathematical modeling is an evolving process, as new insight is gained the process begins again as additional factors are considered.

= the proportion of the off-spring of the infected which are virus free at birth 0 1

t = time

a = infection age

T = maximum infection age; i.e. when a = T, the infected member dies of the disease.

3. Equilibrium States

At the equilibrium state, let

= ; = ; = ; , = ()

(7)

0

0

= T ()

(8)

2. The Model Equations

The model equations are given by:

ds (t) = S t + R t + S t + I t

dt

S t I t (1)

dR (t) = s t R(t) (2)

dt

(, ) + (, ) + + a t, a = 0 (3)

0 = 0 = + (1 ) (9)

Substituting (7) and (8) into (1) and (3) give

+ + + = 0 (10)

= 0 (11)

a

d(a) + h a a = 0 (12)

a = + tan

2TK

(4) da

0

0

I t = T t, a da 0 a T

t, 0 = B t =

(5)

Where = + = + () (13)

From (12)

S t I t + I I t ; 0, a = a (6)

With the parameters given by:

= natural birth rate for the population

= 0 exp{ ()}

0

0

0

0

Let = exp{ ()}

Equation (14) becomes

(14)

(15)

µ = 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

= 0 () (16) From equation (8)

= (0) (17)

Where

efforts at slowing down the death of infected members like drug administration.

= ()

(18)

= rate of removal of the susceptible into the removed class; due to public campaign; i.e. measure of the effectiveness of the public campaign against infection

Using equation (9) in equation (17) gives

z = xz + 1 z (19)

We then solve equations (10), (11) and

(19) simultaneously, for (x, y, z) which gives the non-zero equilibrium state of the model.

From equation (19)

a + 1 = 1

Where p , q, r are constants

, = + () (27) With

0

0

= () (28)

So

= 1(1)

(20)

Substituting (24) to (28) into the model equations (1) to (3)

d + p t = + p t( 2+0) y + r t

dt

Substituting equation (20) in (11) give

+ p t + p t + z + q t

+ p t [z] (29)

y = (1(1) )

(21) Taking cogniza(2n1c)e of equation (10) and neglecting term of order 2 gives

Substituting equations (21) and (20) in

(10) give

p t = p t + r t p t p t +

2 2 q t xq t zp t

1 + 1 + 1 2 + 1

2 + 1 1 z = 0

z = +2 (1 1 )

1 1

0 = z p + r + (0

x)q (30)

Also from equation (2) and taking cognizance of equation (11)

+ = 0 (31)

z = + (1 1 )

1 1

Where

(22)

Also equation(22) (27) and (28) when substituted into (3) gives

a + a t + a +

1

t a

= : = exp ()

0 0

(23)

a t + h a a + a t = 0

Using equation (12), this reduces to

a t + t d(a) + h a a t = 0

da

The equilibrium states are the zero state (x, y, z) = (0, 0, 0) and the non-zero state given by the equations (20) to (22).

d(a) + h a + a = 0 32)

da

Solving the ordinary differential equation

(32) gives

4 .The Characteristic Equation

As in Akinwande [1], the equilibrium states are perturbed as follow:

= 0 exp{ ( + )

0

0

Integrating (33) over [0,T] gives

(33)

Let S t = + p t ; p t = p et (24) = 0 (2 4) { + }

0

I t = z + q t ; q t = q et (25)

0 0

(25)

Using equation (28), this reduces to

R t = y + r t ; r t = r et (26) (26)

= = 0 { + ()}

+ 0

0 + 1 1

= 0 40

0 0 0

= 0 () (34)

With

and the characteristics equation for the model is therefore given by

= [exp{ ( + ())}] (35)

0 0

+ + ) [

We now find (0) as follow From equation (9)

0 = + (1 )

From equation (27)

, 0 = + ()

, = 0 + = () (36) And = + 1 ()

Substitute (24) to (28) into (6) and use (9) and (36) to get

= + + + 2

+ 1 + (1 )

Compare this with (36) using (9) for 0

gives

0 + 0 = ()

+ 1 + 0

= + +

+ 1 + 1

× + 1 1

+

= 0 (41)

1. Stability of the Zero Equilibrium State.

   At the zero equilibrium state (x, y, z) = (0, 0, 0), the characteristics equation (41) takes the form;

   + + 1 0

   1 = 0 42

   With

   + + = 0 43

   1 1 = 0 44

   Equation (43) is a quadratic equation in

   with negative roots, when <µ.

   In order to investigate the nature of the root of (44) from equation (35)

   = +

   0 = + + 2

   + 1

   Neglecting terms of order 2 gives

   0 0

   0

   0

   =

   45

   0 = + + 1 37

   0 32

   = + + (1 ) () (38)

   + + 1 1

   = 0 (39)

   The co-efficients of , , in (30), (31) and (39) give the Jacobian determinant.

   Applying the result of Bellman and Cooke[3] to analyse the equation (44) for the stability or otherwise of the zero state. Let equation

   (44) take the form

   = 1 1 = 0 46

   = 46

   = 1 + 2 (47)

   the condition for Re<0 will then be given by the inequality.

   2 1

   2 1

   1 0 0 0 2 0 > 0 48

   Form (45)

   = +

   2 0 = 0 62

   1

   1

   0 = 0 63

   2

   2

   0 = 1 (64)

   The inequality (48) gives

   1 1 1 < 0 65

   0 0

   1 > 0 66

   And so

   = ()

   0

   = ()

   0

   = + 49

   The inequality (65) holds if

   1 1 < 0 67

   We define after simplifying and substituting, that

   = 1 +

   ()

   2

   +

   = 50

   0

   1

   2

   2

   exp

   ln cos 2

   And so

   = () 51

   0

   (68)

   Also

   0 = = ; 0 = 0

   0

   52

   < < 0 (69)

2. Stability of The Non-Zero

Equilibrium State.

= (53)

0

0 = 0 (54)

For the stability or non stability of the non-zero state, let the characteristics equation (41) takes the form

And

= (55)

0

0 = = 56

0

= 0 70

If we set = 70

= + 71

And from (35)

From the above equations: = + ()

1 = 1 1	57
= 1	(58)
= 1 ()	(59)
= 1	(60)
1 0 = 1 1	61

1 = 1 1	57
= 1	(58)
= 1 ()	(59)
= 1	(60)
1 0 = 1 1	61

0 0

= (72)

2 0

1 = ()

0

2

= () + ()

And so

= (73)

And

0 = 0

0 0 = ( + 1

= (74)

0

And so

+ + 1

(83)

,

0 = = ;

0

0 = 0 (75)

0 = + 1

+ 1

= (76)

0

0 = 0 (77)

2

+ (84)

We now need to obtain the condition for which the inequality.

= ()

0

And

78

0 0 0 0 > 0 (85)

Holds i.e

0 = = 79

0

:

= +

+ + (1 )

+ 1

+ + (80)

= + 1

+ + 1

2 + 1

+ + 1 )

(

(81)

= + 1

+ + 1 + (

2 )

+ 1 2

+ 1

+ (82)

And so

0 0 > 0 (86)

Since G(0) = 0. The expression F(0) 0 is cumbersome and therefore the nature of the sign is not clearly decipherable, hence we shall examine this condition in the neighbourhood of

= 0. 0 0 :

0 = + 1 1

= + (87)

So

0 = (88)

0 = + 1

2 (89)

Taking

( 2 ) < 0

2 ,

(0) > 0

if 2 + > + 1 (90)

Hence if J(K) <0, the non-zero state will be unstable, and if otherwise it will be stable at least locally.

Conclusion

We noted that for J(K)<0, K must be very low, this gives the condition for the stability of the origin and the instability of the non-zero state. On the other hand when J(K)>0, K must be correspondingly high, which leads to the instability of the origin and the possible stability of the non-zero state, locally.

A low level of K indicates high rate of death among the infected, hence the stability of the origin leading to a wiping out of the population. While a high level of K indicates longer life span for the infected and hence the instability of the origin and stability of the non zero state. This suggests a usefulness of the use of measures like the anti retroviral drugs to enhance the life- span of the infected.

From the inequality (90), weobtain a bound for , the measure of public campaign effectiveness, given by:

1 + 2

References

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

2. M. Bawa, A Two Dimensional Infection Age-Structured Mathematical Model of the Dynamics of HIV/AIDS, International Journal of Engineering Research & Technology, Vol. 2, Issue 2, 2013.

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

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

> 1 (1 ) (91)

And the inequalities

> 0 92

and

>

1 0 + 2

1 (1 ) (93)

give the critical values for K and respectively. This suggests that the public awareness campaign should target a success level not below the value J(kc) while the slowing down of infected victims death should not slide below . This may ensure the non-extinction of the population.