A Two Dimensional Infection Age-Structured Mathematical Model Of The Dynamics of HIV/AIDS

A Two Dimensional Infection Age-Structured 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.

1. 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 off-springs 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 off-springs of I(t) are born with the virus.

2. 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 off-springs 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.

3. 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 non-zero equilibrium state is given by (20) and (21).

4. 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:

5. 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.

6. 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 non-zero state will be stable when D2(k) > 0

(63)

Table 2. Stability analysis of the non-zero state.

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

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

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

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