Application of Sowunmi’s Lemma on the Stability Analysis of a Two Dimensional Infection Age – Structured Mathematical Model of HIV/AIDS

Application of Sowunmi’s Lemma on the Stability Analysis of a Two Dimensional Infection Age – Structured Mathematical Model of HIV/AIDS

BAWA, Musa

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

Abstract–The Sowunmis lemma is used to analyse the stability or otherwise of the zero equilibrium state of a two dimensional infection age structured mathematical model of the dynamics of HIV/AIDS. Two compartments of the susceptible and infected members were considered. 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 system gave rise to a set of model equations with one ordinary differential equation and one partial differential equation. The zero equilibrium state and the corresponding characteristics equation were obtained. The latter which arises from the perturbation of the equilibrium state of the system is of transcendental form, from which the lemma applies. The result revealed that to sustain the population, the birth rate must be greater than the death rate among others. This is consistent with the work of Bawa[7 ].

1. INTRODUCTION

   Mathematical modeling can be defined as the process of creating a mathematical representation of some phenomena in order to gain a better understanding of them. It is therefore an abstraction of reality into the world of mathematics. Any phenomenon which has the ability to grow or decay over time can be represented by a mathematical model and then solved analytically where feasible or in several cases tools of advanced calculus and functional analysis are employed to study and interpret the dynamics. According to Benyah [9], mathematical modeling is an evolving process, as new insight is gained the process begins again as additional factors are considered.

   The research work considers the work of Bawa[7],and applied the Sawunmis lemma to analyse the stability of the that is, 1 of the off-springs of I(t) are born with the virus.

   zero equilibrium state of the aged structured model which resulted in a system of ordinary and partial differential equations

   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 additional

   2TK

   burden from infection while K is a control parameter

   associated with the measure of slowing down the death of the infected member, which give the victims longer life span. A high rate 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 there in, the off-springs of I(t) are divided between S(t) and I(t) in the proportion and 1 respectively

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

   t

2. THE MODEL EQUATIONS

   Where (a) = + tan a

   2TK

   (4)

   0

   S1 = ()s(t) + I(t) s(t)I(t) (1) and I(t) = T (t, a)da, 0 a T (2)

   (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.

   = 1 (1 (1) ) (20)

   Substituting (20) in (12)

   () 1 (1(1) )

   y =

   [ (1(1) )]

   (21)

   (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.

   = the proportion of the off-springs of the infected 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)

   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 [3], 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

   With q = T(a)da (25)

   Substituting (22) to (25) into the model equations (1) and (3)

   if (t, a) = (a) (9)

   )d d

   (x + p et ) = () (+p et ) + (y+q et )

   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)

   d(a)

   (+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

   + (a)(a) = 0 (13)

   (y)p + () = 0

   (26)

   d(a) = (a)da (14)

   ( )

   Integrating both sides

   Substituting (24) into (3), gives

   d((a) (a) et ] + d [(a)+(a) et ] + (a)[(a)+(a)et] = 0

   0

   (a) = (0) exp { a (s) ds} (15)

   Let (a) = exp { a (s)ds} (16)

   That is,

   0 (a)et + d(a) + et d

   (a) + (a)(a) + (a)(a)et = 0

   That is,

   (a) = (0)(a) (17)

   Since

   and

   0

   y = (0) T (a)da = (0) (18) Using (11) and (18)

   y = (y + (1) y) (19)

   From (12) and (19).

   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

   (a) = (0) exp { a((s) + )ds} (29)

   Integrating (29) over [0, T] gives

   q = (0) T[ exp { a((s) + )ds}] da

   y ( + (1))b()1

   and the characteristics equation is given by:

   ( y ) [( + (1))b()1] y( ) = 0 (38)

   0 0

5. SOWUNMIS LEMMA

   or q = (0) b() (30)

   0

   Since q = (0) b(),where b() = T[exp

   0

   { a((s) + }ds] da (31)

   Suppose the characteristics equation arising from the perturbation of the equilibrium state of a dynamical system is of the transcendental form

   (0) is calculated as follows: From (11), (0) = y + (1)y.

   and (24) (t, a) = (a) + (a) et

   But (t, 0) = (t) = (0) + (0) et (32)

   From (5), (t) = s(t) + (1)I(t)

   et w(t)dt 1

   0

   Where is the eigen value and function of t. Let

   g(u) eut w(t) dt

   0

   (39)

   w(t) is some continuous

   p>(40)

   Substituting (22) to (25) into (5) and using (11) and (32)

   Be the real part of the expression obtained from the left

   B(t) = ( + p et) (y + q et) + (1) (y +q et)

   hand side of (39) by setting

   u iv , then a necessary

   = y + p yet + q et + p q e2t + (1

   condition for the local asymptotic stability of the equilibrium state of the system is given by

   )y + (1)q et

   (33)

   g(0) 1

   (41)

   Compare this with (32) using (11) for (0) gives

   The equilibrium or steady state is unstable if

   y + (1)y + (0)et = y + p yet + q et

   + p q e2t + (1)y + (1)q et neglecting terms of order 2

   g(0) 1

   Akinwande[5]. Proof:

   (42)

   (0) = p y + q + (1)q (34)

   Substituting (0) in (30)

   Let

   u iv , then (39) gives the pair of equations

   ut

   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

   0 e

   And

   0 e

   ut

   w(t) cos vtdt 1

   w(t) sin vtdt 0

   (43)

   (44)

   y

   = 0 (37)

   From stability theories [1], [2], [4], [6], [8], [9] and [10], a necessary and sufficient condition for local asymptotic stability is for the real part of the eigen value to be in the negative half plane, that is

   Re u 0

   From the real part of (43), we derive the function given by

   g(u)

   1 0

   And

   (52)

   g(u) eut w(t) dt

   0

   (45)

   g0 1 b0 1

   (53)

   Which is an exponentially monotone decreasing function of

   u. Further more, every solution of (45) is less or equal to the solution of (43). The solution Uc will be negative if

   Should hold simultaneously But

   b T exp a s dsda

   0 w(t) dt 1

   (46)

   0 0

   And positive

   So that

   b0 T exp a sdsda

   0 w(t) dt 1

   (47)

   0 0

6. STABILITY ANALYSIS OF THE ZERO EQUILIBRIUM STATE

Inequality (53) becomes

1 T exp a sdsda 1

The characteristic equation takes the form ( )[(1)b()1] = 0 (48)

Let

0 0

Which gives

D 1

T exp a sdsda

( ) = 0 or (1)b()1 = 0 (49)

1 = (50) That is 1 < 0, if <

1

Then

D1 1

0 0

(54)

We now apply the sawunmis lemma [ ], on the transcendental equation in (49). The lemma requires that if the transcendental equation can be expressed in the form

Hence, the constraints on the parameters which can guarantee stability are the inequality (54) together with

.

g() 1,

where is the eigen value.

CONCLUSION

Then the steady state of the system is unstable if

g(0) 1

And a necessary condition for the local asymptotic stability of the equilibrium state is given by

g(0) 1

From (49)

The zero equilibrium state will be stable when the birth rate is less than the death rate in addition to meeting the requirement of inequality (54). This is consistent with the work of Bawa[7].

REFERENCES

1. N. I. Akinwande, Local Stability Analysis of Equilibrium State of Mathematical Model of Yellow Fever Epidemics; J. Nig. Math.

   Soc. Vol.14, 1995

2. N. I. Akinwande, A Mathematical Model of Yellow Fever Epidemics; Afrika Matematika Serie 3, Vol. 6, 1996.

3. N. I. Akinwande, On the Characteristics Equation of a Non Linear

   1 b 1

   This gives

   g 1 b

   (51)

   Age-Structured Population Model; ICTP, Trieste, Italy Preprint IC/99/153, 1999.

4. N. I. Akinwande, Application of Sawunmis Proposition on the Characteristic Equation of a Dynamical System; Advances in Mathematics, Proceedings of a Memorial Conference in Honour of Late Professor C. O. A Sowunmi, Department of Mathematics, University of Ibadan, Ibadan, Nigeria. Vol. 1, 2009.

5. N. I. Akinwande, COA Sowunmis Lemma- A Result on the

   For stability of the zero equilibrium state, the inequalities

   Stability Analysis of the Equilibrium States of Mathematical

   Models of Population Dynamics, Darkol Press And Publishers, Lagos, 2o13.

6. T. M. Apostol, Mathematical Analysis; A Modern Approach to Advance Calculus; Addison-Wesley, London. 1964.

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

8. E. Beltrami, Mathematics for Dynamics Modeling; Academic Press Inc. London, 1989.

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

10. M. E. Gurtin & R. C. Maccamy, Non-Linear Age_Dependent Population Dynamics; Arch. Rat. Mech> Anal. 54, 1974.

11. C. O. A. Sowunmi, On a Set of Sufficient Conditions for the Exponential Asymptotic Stability of Equilibrium States of a Female Dominant Model ; J. Nig. Math. Soc. Vol. 6, 1987.