STABILITY RESULTS OF A MATHEMATICAL MODEL FOR THE CONTROL OF HIV/AIDS WITH THE USE OF MALE AND FEMALE CONDOMS IN HETEROSEXUAL POPULATIONS

A compartmentalized deterministic mathematical model for the dynamics of HIV/AIDS under the use of male and female condoms has been formulated and studied qualitatively. Disease-free equilibria of the sub-models have been found to be locally and asymptotically stable. Stability results revealed threshold values for the proportions of susceptible and infected subpopulations that must use condom in order to achieve control, and possibly, eradication of HIV/AIDS in heterosexual populations. Condom use rate for the susceptible subpopulations has been found to be bounded above by the population’s birth rate, while that of the infected subpopulations is bounded below by a given threshold.


INTRODUCTION
Research has revealed a great deal of valuable medical, scientific and public health information about the human immunodeficiency virus (HIV), the causative agent for acquired immunodeficiency syndrome (AIDS), (Avert, 2008).HIV is spread by sexual contact with infected persons, by sharing hypodermic instruments (primarily for drug injection) with infected persons or through transfusion of infected blood (Avert, 2008).Babies born to HIV infected women may become infected before or during birth or breast feeding (CDC, 1999).Although the HIV prevalence is much lower in Nigeria than in other African countries such as South Africa and Zambia, the size of Nigeria's population (around 138 million) meant that by the end of 2007, there were an estimated 2,600,000 people infected with HIV, which is rather on the high side (UNAIDS, 2008).
Condoms are classified as medical devices and are regulated by the food and drug administrations across various countries of the world.There are many different types and brands of condoms available -however, only latex and polyurethane condoms provide highly effective barrier to HIV (CDC, 1993).For condom to provide maximum protection, they must be used consistently and correctly (CDC, 1988).
Once a sexually active HIV infected person is introduced into a heterosexual population, the virus begins to spread in the population due to interactions between susceptible and infective members of the population.The deployment of male and female condoms is aimed at reducing the chances of spreading and/or acquiring sexually transmitted diseases (STDs) including HIV.Condoms do not offer full protection against HIV and other STDs, but can significantly reduce the chances of transmitting and/or acquiring same (CDC, 1998).
About 80 per cent of HIV infections in Nigeria are transmitted through heterosexual contacts (Avert, 2008).The total number of condoms provided by international donors has been relatively low.Between 2000 and 2005, the average number of condoms distributed in Nigeria by donors was 5.9 per man, per year (UNFPA, 2005).A study in 2002 found that 75 percent of health service facilities that had been visited did not have any condoms or contraceptive supplies (Human Rights Watch, 2004).The number of female condoms sold in Nigeria has significantly increased, which indicates a greater awareness of sexual health issues.In 2003 only 25,000 female condoms had been sold, which increased to 375,000 in 2006375,000 in (UNFPA, 2007)).One major advantage of the female condom is that it does not rely upon the willingness of the man to use a condom himself (Family Health International, 2007).
Our main objective in this paper is to investigate the existence of the conditions for eradication of HIV in heterosexual populations under the use of male and female condoms.This is significant because the efforts channeled in the provision, promotion and counseling on the use of condom may amount to a waste if not optimally conducted.Two things are critical here; the subpopulations that should use condom, and the proportion of same subpopulations that must use the condom for effective results

THE MODEL EQUATIONS
Consider a heterosexual population with condom deployed in the various subpopulations.Assume that a proportion each of both infected and susceptible males and females use the condom (condoms are not 100 per cent efficacious).AIDS is an additional cause of dead for the infected persons.
Compartmentalizing the population into susceptible males not using the condom (S m ), susceptible females not using the condom (S f ), susceptible males using the condom (U m ), susceptible females using the condom (U f ), infected males not using the condom (I m ), infected females not using the condom (I f ), infected males using the condom (W m ), and infected females using the condom (W f ), and taking into consideration the epidemiological flow diagram in fig.2.1, which we put forward for the purpose of this research, we obtain the model equations given by equations (2.1) -(2.8).
We derive the model equations by considering the assumptions and the epidemiological flow diagram (fig.2.1).The incidence rates B m and B f are given as in Hseih (1996).
(2.11) dynamics incorporating the use of male and female condoms and * f β are the average number of sexual contacts for males, average number of sexual contacts for females, probability of transmission by infected males not using condom, probability of transmission by infected female not using the condom, probability of transmission by infected male using condom, and probability of transmission by infected female using the condom, respectively.Similarly, m ρ , f ρ , m δ , f δ represent the proportions of infected males, proportion of infected females, proportion of susceptible males and proportion of susceptible females using the condom, respectively.

MODEL EQUATIONS IN PROPORTIONS
Transforming the model equations into proportions has the advantage of reducing the total number of equations in the model, giving the equations in epidemiologically meaningful forms (for instance, the proportion of infected persons defines the prevalence of infection), (Avert, 2008).Consider the substitutions below, Using these substitutions in equations (2.1) -(2.8) and considering (2.9) to (2.12), we obtain the model equations in proportions as given by equations (3.1) -(3.6)

STABILITY ANALYSIS
We apply Hurwitz criterion (see David (1997) and Weisstein (2008)) to study the stability of the disease-free equilibrium (DFE) states of the various sub-models in this paper.

4.1
The sub-model with only infected males using the condom Here we analyze the stability of the DFE when only infected males use the condom.The sub-model in this case is obtained by setting in the general model.We therefore have the system of equations (4.1.1)for the sub-model.The conditions for stability have been stated and proved in the theorem that follows.
[ ] . The DFE for the system (4.1.1)is LAS if the inequality

Proof
The DFE, ( ) The Jacobian matrix evaluated at the DFE is given by The characteristic polynomial is given by STABILITY RESULTS OF A MATHEMATICAL MODEL FOR THE CONTROL OF HIV/AIDS ( ) [ For the characteristic polynomial to satisfy Hurwitz criterion (David, 1997), the inequality (4.1.2) must be satisfied It can be shown that the right-hand-side of (4.1.2) is positive if the inequality

4.2
The sub-model with only Infected Females using the condom Here we analyze the stability of the DFE when only infected females use the condom.The sub-model in this case is obtained by setting in the general model.We therefore have the system of equations (4.2.1) for the sub-model [ ]

Proof
The DFE, ( ) The Jacobian matrix evaluated at the DFE is given by The characteristic polynomial is given by For the eigenvalues to be all negative, the inequality (4.2.2) must be satisfied.
The Hurwitz criterion is satisfied by the characteristic polynomial provided the inequality (4.3.2) holds. ( ) The RHS of (4.3.2) is positive provided the inequality (4.3.3)holds

Sub-model with only susceptible females using condom
Here we analyze the stability of the DFE when only susceptible females use condom.The sub-model in this case is obtained by setting in the general model.We therefore have the system of equations (4.4.1) for the sub-model [ ] The Jacobian matrix evaluated at the DFE is The Hurwitz criterion is satisfied by the characteristic polynomial provided the inequality (4.4.2) holds.
Observe that the right-hand-side of the inequality (4.4.2) is positive if Therefore the DFE is locally and asymptotically stable.Thus the theorem is proved., and the sub-model is given by equations (4.5.1).

STABILITY RESULTS OF
, then the DFE is LAS.

Proof
The DFE is obtained as The Jacobian matrix evaluated at the DFE is as follows The Hurwitz criterion is satisfied by the characteristic polynomial provided the inequalities (4.5.2) and (4.5.3) hold., so that the model becomes The Jacobian matrix evaluated at the DFE is as follows

Re
The remaining roots of ) (λ p are the roots of the cubic equation ) )( ( ) ( The Hurwitz criterion is satisfied by the characteristic polynomial provided the inequalities (4.6.2) and (4.6.3)hold.Therefore the DFE of this sub-model is locally and asymptotically stable (LAS) Thus, the theorem is proved.

1S.
Musa, Department of Mathematics and Computer Science, Federal University of Technology, Yola M. O. Egwurube, Department of Mathematics and Computer Science, Federal University of Technology, Yola A. R. Kimbir, Department of Mathematics/Statistics/Computer Science, Federal University of Agriculture, Makurdi D. K. Igobi, Department of Mathematics and Computer Science, Federal University of Petroleum Resources, Effurun -Delta State

Figure 2 . 1 :
Figure 2.1: Epidemiological flow-diagram for HIV/AIDS with only susceptible males using condom Here we analyze the stability of the DFE when only susceptible males use condom.The sub-model in this case is obtained by setting model.We therefore have the system of equations (4.3.1) for the sub-model [ ] The Jacobian matrix evaluated at the DFE is with both Susceptible and Infected males using the condom Here we study the stability of the model in a situation where only males (a proportion each of susceptible and infected males) use the condom.In this case, with both Susceptible and Infected females using condom Here we study the stability of the model in a situation where only females (a proportion each of susceptible and infected females) use the condom.In this case, suspending the condom Here we study the dynamics of HIV when no condoms are employed in the entire population.sub-model as given by the system of equations (4.7.1).