Modelling the Effect of Screening and Treatment on Transmission of HIV/AIDS Infection in a Population

This paper examines the effect of screening and treatment on the transmission of HIV/AIDS infection in a population. A nonlinear mathematical model for the problem is proposed and analysed qualitatively using the stability theory of the differential equations. The effective reproduction number of the normalised model system (3) was obtained by using the next generation operator method. The results show that the disease free equilibrium is locally stable by using Routh Hurwitz criteria at threshold parameter less than unity and unstable at threshold parameter greater than unity. Globally, the disease free equilibrium is not stable due existence of forward bifurcation at threshold parameter equal to unity. Using Lyapunov method, endemic equilibrium is globally stable under certain conditions. However, analysis shows that, screening of unaware HIV infectives and treatment of screened HIV infectives have the effect of reducing the transmission of the disease. Numerical simulation of the model is implemented to investigate the sensitivity of certain key parameters on the voluntary screening and treatment of the screened infectives and full blown AIDS victims.


Introduction
The acquired immunodeficiency syndrome (AIDS) emerged in 1981 and has become an alarming sexuality transmitted disease throughout the world. HIV has shown a high degree of prevalence in populations all over the world [1]. It is common to young and adults. Individuals aged 15years and above are the most susceptible group of acquiring infection [2]. This is because they are sexually active and thus capable of reproduction. Routine screening of unaware infectives have now become an integral part of programs in low and middle income countries. People can get HIV tests at a health clinic, at special HIV voluntary counselling and testing (VCT) sites (UNAIDS, 2002). Screening for HIV facilitates immune system monitoring and early management of side infections and sexually transmitted infections (STI's) which can greatly improve long term health. It also facilitates referral to social and peer support. HIV positive test may help in changing behaviours that may help in transmitting infections. It also enables infectives to be aware and immediately take advantage of antiretroviral therapy (ART) that will help to manage infections and delay the onset of HIV symptoms. Improvement to access to HIV testing and counselling and antiretroviral therapy could significantly reduce infection rates ( [3], [4]). Estimates developed through epidemiological modelling suggest that HIV related mortality can be reduced by 20% between 2010 and 2015 if the guidelines for early treatment are broadly implemented The link between infectious diseases and screening must be understood in relation to infectives on the spread of HIV infections. From the initial models of [5,6,7], various refinements have been added into modelling frameworks, and specific issues have been addressed by [8,9,10,11].
So far few studies have been developed to analyse mathematically the impact of the screening of unaware infectives on the spread of HIV infections in a homogeneous population. However, none of these studies had considered the aspect of screening and treatment on the transmission of HIV/AIDS infections. It is well known that treatment of screened infectives may also play a major role in the transmission dynamics of the disease in a homogeneous population. [1] presented a theoretical framework for transmission of HIV/AIDS with screening of unaware infectives. [12] established and analyzed a mathematical model of the effect of screening the HIV infection in a homogeneous population with infective immigrants. However, in all the above studies, none of them incorporated the treatment of screened infectives and full blown AIDS patients. In this paper, it is therefore intended to analyze a Transmission of HIV/AIDS Infection in a Population model which will incorporate screening and treatment on the transmission of HIV/AIDS infection in a population. This is an extension of the work by [5] and the work by [12], by including the aspect of treatment of screened infectives in a homogeneous population in order to put more knowledge on the long and short term behaviour of the dynamics of the disease transmission and predict whether the disease will disappear (curbing down) or will persist. Thus we study and analyze a non linear mathematical model of the effect of screening and treatment on transmission of HIV/AIDS infection in a population. The model incorporates the assumption that all three invectives (Unaware infectives, Screened infectives and Treated infectives) move to full blown AIDS at different rates.

Model Formulation
A non linear mathematical model is proposed and analysed to study the effect of screening and treatment on transmission of HIV/AIDS infection in a population. The proposed model subdivides the population of interest into five sub population compartments depending on the HIV status of individuals. In modelling the dynamics, the population is divided into five subclasses: Susceptibles In formulating the model, the following assumptions are taken into consideration: (i) The rate of transmission is direct proportional to the susceptibles population and also to the ratio between the members of infected population to the total population.
(ii) Only screened infectives and full blown AIDS can be treated with ARV therapy (i.e. can move to treated class) at different rates 1 γ and 2 γ respectively. (iii) Unaware infectives, screened infectives and treated class will move to full blown AIDS at different rates 1 δ , 2 δ and σ respectively where 2 1 σ δ δ < < . (iv) Unaware infectives can only move to screened class and full blown AIDS and unaware infective can be screened at a rate 'θ '.
(v) Unaware infectives, screened infectives and treated class can infect susceptibles class at different rates 1 β , 2 β and 3 β respectively where 3  Taking into account the above considerations, we then have the following schematic flow diagram: The model is thus governed by the following system of non linear ordinary differential equations: The total population at time t is given by,

N t S t I t I t T t A t
We note that in the absence of the disease, the total population size N is stationary for , λ µ = declines for λ µ < and grows exponentially for λ µ > . So we shall assume that mortality rate µ , will be a function of state variables. For convenience, we analyse our model in terms of proportions of quantities instead of actual populations. This can be done by scaling the population of each class by the total populations. We make the transformation , in the classes , I T , and A respectively.
Therefore the normalised model system of equations (3) is both mathematically and epidemiologically well posed.

Model Analysis
The normalized model system of equations (3) will be analysed qualitatively to get insight into it is dynamical features which will give a better understanding of the effects of screening and treatment on the transmission of HIV/AIDS infection in a population. Threshold which governs elimination or persistence of HIV/AIDS will be determined and studied.

Disease Free Equilibrium (DFE)
The disease free equilibrium denoted by ' 0 E 'of the normalised model system (3) was obtained as

The Effective reproduction number e R
The effective reproduction number, e R of the normalised model system (3) was obtained by using the next generation operator method and is given by

Numerical Sensitivity Analysis Transmission of HIV/AIDS Infection in a Population
In determining how best to reduce human mortality and morbidity due to AIDS, the sensitivity indices of the effective reproductive number ' e R 'to the parameters in the model was calculated using approach of [13]. These indices tell us how crucial each parameter is to disease transmission and prevalence and discover parameters that have a high impact on e R and should be targeted by intervention strategies. The normalized forward sensitivity index of a variable to a parameter is a ratio of the relative change in the variable to the relative change in the parameter. When a variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.  Definition 1: The normalised forward sensitivity index of a variable 'p' that depends differentiable on a parameter 'q' is defined as: As we have an explicit formula for e R in equation (6)

Local Stability of Disease-free Equlibrium
To determine the local stability of disease free equilibrium, the variational matrix 0 M of the normalised model system Where ( ) The condition (8) is sufficient to satisfy all the equations So it is clear that for 1 e R < which corresponds to (9) This situation can also be realised easily when we try to assess the contribution of 1 , From the equations (11)-(13) above, it is clear that are the ones contributing the most on the transmission of the HIV/AIDS infection followed by screened infectives ' 2 i 'which keeps the disease endemic (i.e.  (1). However when the disease remain endemic the disease induced deaths reduce the equilibrium population size from 1 n = to * s .

Determination of Forward or Backward Bifurcation
From equation (16), it follows that there is no backward bifurcation since the value of 0 K = , hence no multiple equilibria. Therefore existence of unique endemic equilibrium which is locally stable for 1 e R > and unstable if 1 e R < was explored by a forward bifurcation diagram obtained when a graph of proportion of unaware infective ' 1 i 'against reproduction number ' e R ' was sketched as shown below:  (17). The local asymptotic stability of endemic equilibrium is analyzed by using the Centre Manifold theory [14] and shows that it is stable under certain conditions. The normalised model system (3) shows that it will exhibit a backward bifurcation which occurs at 1 e R = under certain condition otherwise it will exhibit a forward bifurcation at 1 e R = as shown in figure 2 and is locally stable.

Global Stability of Endemic Equilibrium
By direct calculation of the derivative of V along the solutions of (3) we have

Model with Screening and Treatment When Treated HIV Infectives do not Transmit the Infection in a
Population ( )   By analysing the four epidemiological situations discussed above, it may be concluded that in the presence of screening and treatment, if both the screened and treated HIV infectives decide to take preventive measures and do not transmit the infection, the disease will tend to the endemic state if both the screening and treatment rates are small. If the annual screening and treatment rates are very high (say 99% of the initial population), the disease may tend to disease free equilibrium point. However if screened and treated HIV infectives also contribute to the transmission of the disease, then even if both screening and treatment rates are very high, the disease is set up among the population as the system continues towards asymptotically stable endemic equilibrium point. Analysis also shows that the endemicity of the disease is reduced by screening of unaware HIV infectives and treatment of the screened HIV infectives in the population.  . It is seen from these figures that for any starting initial value, the solution curves tend to the equilibrium * E . Therefore we conclude that the normalised model system (3) is globally stable about this endemic equilibrium point * E for the estimated parameters above. Figure .4.2 below show the variation of the proportion of total population in all classes It is seen that the proportion of susceptible population decreases with time and then reaches its equilibrium position. This is due to treatment with ARV. Therefore infection becomes less endemic in the population. Initially proportion of unaware infectives increases but due to the increase in screening and treatment, the rates θ and 1 γ respectively, decreases then reaches its equilibrium position. This ultimately leads to the decrease of the proportion of AIDS infectives.

Proportion of Susceptibles Prorportion of Screened infectives
In      Figure 4.4 (iv) above. Thus as the screening rate becomes zero, the unaware infectives will continue maintaining sexual relationships in the community leading to the increase in the proportion of AIDS infective population.                 It can be observed that as screened and treated HIV infectives choose to participate in sexual interactions without exposing themselves, the proportion of unaware HIV infective population increases which in turn increases the proportion of AIDS patients population. When screened and treated HIV infectives do not take part in sexual interactions, means that   It is seen that as the movement rates from infective classes increases, proportions of all infective classes populations decreases (Figures 4.8 (ii), (iii) and (iv)) which in turn increases the proportion of AIDS patients population ( Figure.4.9) and also increases the proportion of susceptible population (Figure 4.8 (i)). It is observed from the figures that as α increases the proportion of AIDS patients population decreases and it is also observed that as the recruitment rate increases, the proportion of AIDS patient population increases.

Discussion and Conclusions
A non linear mathematical model has been proposed and analyzed to study the effect of screening of unaware HIV infectives and treatment of screened HIV infectives on the transmission of HIV/AIDS infection in a population. The disease free and endemic equilibria were obtained and their stabilities investigated. The model showed that the disease free equilibrium is locally stable by using Routh Hurwitz criteria at threshold parameter less than unity and unstable at threshold parameter greater than unity, but globally the disease free equilibrium is not stable due existence of forward bifurcation at threshold parameter equal to unity. Also the model analysis showed the existence of unique endemic equilibrium that is locally stable under certain conditions when the threshold parameter exceeds unity due to existence of forward bifurcation at threshold parameter equal to unity. Using Lyapunov method, endemic equilibrium is globally stable under certain conditions. A sensitivity analysis shows that ' 1 β ' is the most sensitive parameter on e R and the least is ' σ A numerical study of the model was carried out to see the effect of key parameters on the transmission of HIV/AIDS infections. The analysis shows that the screening of unaware HIV infectives and treatment of screened HIV infectives have the effect of reducing the transmission of the disease. It is observed that when the screened infectives and treated infectives do not participate in the transmission of the infection, the AIDS population is significantly reduced in comparison to the case where there is no screening and treatment. It is found that the disease becomes more endemic in the absence of screening and treatment and consequently the AIDS population increase. This has a consequence of people not knowing that they are infected and still practice sexual relationships without taking precautions. Screening makes people aware of their infection and reduce their viral load by using ARV

Time(years)
Proportion of Aids patients λ=0.1 λ=0.5 λ=1 Transmission of HIV/AIDS Infection in a Population treatment and consequently AIDS patients population decreases. This is due to the fact that on being aware of their infection and using ARV, people either abstain to interact sexually or use preventive measures like using condoms and by changing their behaviour which results in the decline of AIDS epidemic.
Based on the results of the study, we conclude that the most effective way to reduce the transmission of HIV and AIDS epidemic infection is to educate people to go for voluntary HIV screening and become aware of their HIV status and if an individual finds himself/herself HIV positive, then he/she has to be willing to undergo ARV treatment and therapy so as to reduce the viral load hence prolonging life. Furthermore people should be educated to be aware of the consequences of practising unsafe sex and other preventive measures against the infection. If the population shows positive attitude towards voluntary screening, treatment with ARV and preventive procedures, then the transmission of the disease can be controlled. Therefore the national HIV/AIDS control programs for all developing countries should increase the education programs on the importance of voluntary HIV screening, ARV treatment and preventive procedures to the community at all social classes especially lower classes and high risk groups so that the transmission of the disease can be controlled. Finally, more HIV/AIDS centres for voluntary screening and ARV treatment should be established across each country to ensure that more people have access to the facilities hence reducing the transmission of HIV and therefore reduce the AIDS epidemic.