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HIV/AIDS is a public health problem especially in sub-Saharan Africa where majority of infections and deaths occur. Despite the large number of studies and efforts made in covering the data gap using mathematical models, little is known on how model estimates are confounded by the transmission variabilities that exist in stages of HIV progression. This work investigates the impact of including stages of HIV transmission in HIV/AIDS models. A deterministic HIV/AIDS model is developed and extended to include stages of HIV progression of infected individuals. Theoretical investigation of the models and numerical analyses indicate that the two models produce different estimates, with the model without stages producing lower estimates than the staged model. These results call for a careful consideration in evaluating the efficiency of HIV/AIDS models that are used to estimate and project the burden of HIV/AIDS disease.

HIV/AIDS epidemic continues to be the main killer disease in sub-Saharan Africa, with majority of infections and deaths occurring in adults. In 2013, the Joint United Nations Programme on HIV/AIDS (UNAIDS) and the World Health Organization (WHO) estimated that 35 million people were living with HIV in the world, 2.1 million were newly infected, and 1.5 million deaths occurred. Of these, 24.7 million lived in sub-Saharan Africa with 1.5 million new infections and 1.1 million AIDS deaths occurring in the same year [

HIV transmission is not uniform between countries and among different stages of the disease within an infected individual [

Viral load varies greatly between these stages. During the period of primary infection, viral load is typically high. The viral load drops as one enters the asymptomatic period, followed by a symptomatic/AIDS stage during which the viral load is extremely high. A community-based study in which couples are prospectively followed for 30 months to evaluate the risk of transmission in relation to viral load and other characteristics shows that the risk of infection increases as an infected person’s viral load increases [

The study of an epidemic, such as HIV, and its spread process in any community, is different to investigations in many other sciences. Data cannot be obtained through experiments in the population, but can only be obtained from surveys and results found in published or unpublished documents. These data are often not complete, may be inaccurate, and may vary with respect to methods used to collect them. Due to difficulties in obtaining HIV data, mathematical modelling and numerical simulation play an important role in analysing the behaviour of the epidemic, measuring its past, present, and future effect in a society.

Variations in the level of HIV transmission over time can have an impact on the estimates delivered from mathematical models. Existing models of transmission have either assumed a single group of infected individuals or have included stages of HIV progression to capture the transmission dynamics but have not studied how these two modelling approaches differ from each other. Lin et al. [

The HIV/AIDS model formulated here considers the total population,

The dynamics of the model are governed by the following system of differential equations:

where, r is the transmission rate and

In this case, the total populations varies. This can be due to

Because

which is positively invariant in the region:

Note that Equation (2) can be re-written as:

which integrates to:

indicating that when

This section derives stability conditions of the equilibrium points of the system in Equation (3).

Definition 2.1. Given a system of differential equations

We define a threshold factor:

from Equation (1) to represent the average number of secondary infections caused by one infective individual introduced into a completely susceptible population.

Theorem 2.2. For r > γ, the system in (3) always has a disease free equilibrium

Proof. From the second equation of (3), with the right hand side equal to zero at large t, then equilibrium points must satisfy:

or

and

Substituting (7) and (8) in (9) or in the first equation of (3) with the right hand side equal to zero, gives

Local stability of the equilibrium points is performed by introducing of small perturbations,

and substituting them in Equation (3). Because

The coefficients of the perturbations give the Jacobian matrix:

At the disease free equilibrium, the Jacobian matrix in (11) gives the following characteristic equation:

with eigenvalues

Substituting the endemic equilibrium point, the Jacobian matrix

where,

and

If

and the determinant,

Thus, by the Routh-Hurwitz criterion, all of the eigenvalues have negative real parts and the endemic equilibrium point is stable. On the other hand, if

Here, the model in (0) is extended to include stages of HIV progression. The infected group is divided into two subgroups: those in the primary stage of HIV infection,

From the above model, the total population at time t is given by:

The parameters

For stability analysis, the system in (17) is converted into proportions by letting

where

integrating to

The dynamic behaviour of the total population in this model is mainly governed by infected individuals who are in the asymptomatic stage of HIV infection,

Model (17) then becomes:

with

The above system have a positively invariant feasible region given by:

with all parameters positive.

The incidence of the disease is the proportion of new cases occurring in a population during a defined time interval. Using this model, incidence is given by:

where I is the incidence, and

The prevalence of the disease is defined as the proportion of infected individuals in a population. From the model, prevalence is simply infected individuals in

Using the Next-generation technique [

which is a linear combination of the threshold quantities of the infected individuals in the primary stage,

Because of variable population size, system (22) is complex and calculation of the endemic equilibrium points is difficult. The dynamic behaviour of the population size (Equation (21)) is considered.

Definition 3.3. As

Theorem 3.4. The system in (22) has a unique endemic equilibrium point if the threshold quantity

Proof. When the equilibrium is attained, the right hand side of system (22) goes to zero. Using the third equation in (22), we obtain:

Substituting (28) in the second equation of (22) while incorporating (23) gives:

where

Equation (29) gives

If

But we know that

If

If the system approaches a disease free equilibrium, then

But in this case,

If the system approaches the endemic equilibrium 3.2, then

where

From (34), we obtain

To analyze the stability of the equilibria, we establish a Jacobian matrix J and employ the Routh-Hurwitz technique to study the local stability of the equilibria. The Jacobian matrix of the system (22) is given by:

At the disease free equilibrium

From (36),

giving the characteristic equation:

The roots of (38) gives:

Clearly,

Linearizing the model around the endemic equilibrium point,

where

with

This section presents numerical simulation results of the models using parameter values described and presented in Section 4.1. We address the question whether it is necessary to incorporate stages of disease progression when modelling the spread of HIV/AIDS and seek to understand the effect of incorporating stages of HIV progression on the overall infection and spread of disease in the population as well as on estimation of future trend. In addition, we compare the two models by studying the effects of varying transmission rates on the models presented in this paper.

In sub-Saharan Africa, the average time lived by individuals is about 50 years. In this case, the natural mortality rate, ^{−1}. In the same region, the average birth rate, b is estimated to be 0.03. Studies have also estimated the waiting times in the first stage of HIV is 2 to 10 weeks while individuals in the asymptomatic stage spend about 10 to 15 years [

On the other hand, the first empirical data in sub-Saharan Africa communities show substantial variations in transmission among stages of HIV infection after sero-conversion [

The transmission rate, r as used in the simple model is estimated using the second equation in (3) at the steady state and given as:

where

From the model with stages,

where

As the rate at which individuals become infected is increased, then

Because prevalence is high in infected individuals, mortality becomes high (

In this paper, a simple model for HIV transmission has been formulated and extended to incorporate stages of HIV progression. Stability analysis of the models and numerical simulation examples has been performed to understand the impact of stages in estimates. The effect of varying the transmission rates r,

Our results indicate that when

The models produce different results. The model without stages produce estimates that are lower than the HIV estimates produced when stages are included. The nature of curves for y and

Although the models formulated are simple based on assumptions, and without fitting them to data, results show the importance of incorporating stages in models of HIV/AIDS. The results can not only be used to study how important stages of HIV infection are in the spread of HIV, but also they are helpful in evaluating the efficiency of HIV/AIDS models used in estimating and projecting the burden of HIV disease.

This work was developed from my MSc dissertation submitted to the University of Stellenbosch, South Africa with financial support from the African Institute for Mathematical sciences. The author acknowledges Fritz Hahne for valuable inputs.

Angelina MageniLutambi, (2015) Modelling the Impact of Stages of HIV Progression on Estimates. Advances in Infectious Diseases,05,101-113. doi: 10.4236/aid.2015.53012