A Theoretical Framework of HIV: Sigmoidal Vertical Transmission Function

Introduction Scientific and medical discoveries have made advances in understanding virology, immunology and pathogenesis of HIV infection. These advances have led to the design of effective prevention and treatment strategies of HIV infection in infants, children and adolescents. For instance, antiretroviral therapy (ART) during pregnancy, delivering and breast feeding can eliminate mother-to-child transmission. There are challenges associated with ART implementation especially in resource limited settings [1]. Timely initiation of antiretroviral therapy (ART) in HIV infected individuals has drastically reduced mortality and prolonged their lives [1]. Furthermore, improved survival and reduced HIV transmission risks by HIV infected individuals has led to decline in HIV incidence and increase HIV prevalence [1]. According to [2] an estimated that 34 million people were living with HIV at the end of 2011. Among these, 330000 were children less than 15 years, 2.5 million new HIV infected people. Vertical HIV transmission from mother-to-child accounts for more than 90% of paediatric AIDS [2-4]. Sub-Saharan Africa continues to be the world’s most affected region [5,6]. In order to prevent and control the spread of infectious diseases, mathematicians have embarked upon studies that investigate the role of treatment functions in epidemiological models. Studies of classical models have revealed that treatment functions are important in decreasing the spread of epidemiological diseases. Hampanda [7] applied theoretical frameworks to understand social barriers to prevention of mother-to-child transmission. Chibaya et al. [8] investigated the effect of vertical transmission and evolution of drug resistance on the dynamics of the disease. Baryarama et al. [3] formulated and analysed an HIV/AIDS model with complacency described by an AIDS population dependent function with inverse relation. It is upon this background that we consider a mathematical model that investigates the role of sigmoidal treatment functions in predicting the burden of vertical transmission. This paper acknowledges various challenges ranging from financial (unaffordable of drugs), organizational (insufficient facilities), (physical) limited access to treatment or resources to social (cultural, decision making processes). The paper is arranged as follows: section (2) describes model formulation, section (3) model analysis; section (4) numerical simulations and concluding remarks are done in section (5). The mathematical Model We consider an epidemic model which incorporates the effects of vertical transmission described by a sigmoidal function. We divide the population into five compartments, namely, Susceptible S, Asymptomatic infectives X, Symptomatic infectives Y, Treated infectives Z and AIDS A following the WHO stages of HIV infection. The susceptible population is replenished by births or recruitment at the rate and through 0 π


Introduction
Scientific and medical discoveries have made advances in understanding virology, immunology and pathogenesis of HIV infection. These advances have led to the design of effective prevention and treatment strategies of HIV infection in infants, children and adolescents. For instance, antiretroviral therapy (ART) during pregnancy, delivering and breast feeding can eliminate mother-to-child transmission. There are challenges associated with ART implementation especially in resource limited settings [1]. Timely initiation of antiretroviral therapy (ART) in HIV infected individuals has drastically reduced mortality and prolonged their lives [1]. Furthermore, improved survival and reduced HIV transmission risks by HIV infected individuals has led to decline in HIV incidence and increase HIV prevalence [1].
According to [2] an estimated that 34 million people were living with HIV at the end of 2011. Among these, 330000 were children less than 15 years, 2.5 million new HIV infected people. Vertical HIV transmission from mother-to-child accounts for more than 90% of paediatric AIDS [2][3][4]. Sub-Saharan Africa continues to be the world's most affected region [5,6].
In order to prevent and control the spread of infectious diseases, mathematicians have embarked upon studies that investigate the role of treatment functions in epidemiological models. Studies of classical models have revealed that treatment functions are important in decreasing the spread of epidemiological diseases. Hampanda [7] applied theoretical frameworks to understand social barriers to prevention of mother-to-child transmission. Chibaya et al. [8] investigated the effect of vertical transmission and evolution of drug resistance on the dynamics of the disease. Baryarama et al. [3] formulated and analysed an HIV/AIDS model with complacency described by an AIDS population dependent function with inverse relation.
It is upon this background that we consider a mathematical model that investigates the role of sigmoidal treatment functions in predicting the burden of vertical transmission. This paper acknowledges various challenges ranging from financial (unaffordable of drugs), organizational (insufficient facilities), (physical) limited access to treatment or resources to social (cultural, decision making processes). The paper is arranged as follows: section (2) describes model formulation, section (3) model analysis; section (4) numerical simulations and concluding remarks are done in section (5).

The mathematical Model
We consider an epidemic model which incorporates the effects of vertical transmission described by a sigmoidal function. We divide the population into five compartments, namely, Susceptible S, Asymptomatic infectives X, Symptomatic infectives Y, Treated infectives Z and AIDS A following the WHO stages of HIV infection. The susceptible population is replenished by births or recruitment at the rate and through Biostatistics and Biometrics Open Access Journal births of infection-free individuals at the rate ) ) ( where is the per capita birth rate and is modelled using a piecewise sigmoidal function, ) (t ω given by The above description leads to a system of differential equations 0 (1 ( )) , ) .
with initial conditions and the total population given by The force of infection is given by The parameter c measures the effective contacts and is the probability of transmission per contact. Adding all the equations of system (1), we obtain changes in the total population describe by Thus, the total population is limited by The distinction between this model and other models of HIV is that we allow for treatment of AIDS individuals who recover from the disease to treated class. We attempt to solve a non-autonomous system of equations to predict the epidemic outcome based on the cumulative infective cases

Invariant region
We analysis and discuss model (1) in a biological feasible region given by It is easy to show that the region Ω is positively invariant and it is sufficient to consider solutions in Ω . Hence, all solution of model (1)

Steady States Solution
To obtain the equilibrium points, we set the right hand side of the model (1) to zero and solve the system of non-linear equations The formula of the reproduction number due to vertical transmission 0 ( ), R t allows us to compute the incremental reproduction number 0 which measures or assesses the rate of reduction of new infections per unit of time. Knowledge of this quantity may help adjust some of the critical control factors to achieve desired results. Now, combining equations in system (6) and * * * , or It is worth noting that these solutions hold provided

Disease-free equilibrium
The solution (7) leads to the disease-free equilibrium given by and it exists for all values of Linearization of system (1) and construction of appropriate Lyapunov function [9] of the form We are able to establish the following theorem: Theorem 3.1: The disease-free equilibrium (DFE) (exists for all 0 R ) is locally and globally asymptotically stable 0 1. R < if and unstable if 0 1.

Endemic equilibrium
The result (8)  ] . x Noting that * N can be expressed as * *  (8),we obtain ( ) The result of the stability analysis of the endemic equilibrium point can be summarized by the following theorem. Theorem 3.2 The endemic equilibrium (EE) exists and it is locally asymptotically stable if 0 1 R > and unstable if 0 1 R < :

Numerical Simulation
To illustrate analytical results, we simulated system (

Conclusion
The study formulated and analyzed an HIV theoretical framework described by sigmoidal vertical transmission function with the aim of investigating the role of vertical transmission in predicting the outcome of an epidemic. The study shows that the disease clears from the population if 1 0 < R and persist if 1 0 > R : Furthermore, the study underscores the importance of estimation of critical factors influencing disease transmission dynamic. We believe our model formulation, presents a useful framework, which can be calibrated with available data to obtain insights in effects of vertical transmission in the presents of treatment therapy. The study may further aid determination of optimal time control for treatment inception to bring about elimination of vertical transmission.