We formulate a deterministic system of ordinary differential equations to quantify HAART treatment levels for patients coinfected with HIV and Kaposi's Sarcoma in a high HIV prevalence setting. A qualitative stability analysis of the equilibrium states is carried out and we find that the diseasefree equilibrium is globally attracting whenever the reproductive number
Kaposi's Sarcoma is a cancer that occurs mostly in humans with suppressed immune systems [
In competent immune systems, acquisition of HHV8 does not guarantee the development of KS; in fact, most individuals with a strong immune response could remain latently infected with HHV8 throughout their lifetime [
Recent research findings [
From what has been described above, we formulate a mathematical model which includes the following classes: a susceptible class,
The structure of the paper is as follows. In Section
AIDS is the progressive stage at which an HIVinfected individual loses competency of his immune system and becomes prone to opportunistic infections. For this reason, HIV coinfection models have found a lot of space in HIV epidemiology studies. HIV and tuberculosis coinfection have been studied by Cohen et al. [
We begin with a human population of susceptibles which is free of both KS and HIV denoted by
For simplicity of the model, we assume that all individuals in the class
HIVonly and coinfected individuals are assumed to progress to the asymptomatic preAIDS classes (
We assume that all infected individuals in the classes
The force of infection
The model flow diagram depicting this biological system is illustrated in Figure
Model flow diagram.
The above assumptions and formulations lead us to this nonlinear system of differential equations:
All parameters for the model system (
We denote by
In this section, we prove the following lemma.
The system (
Without loss of generality we may assume that
Suppose that the assertion
Then
From the differential inequality (
For the state variables in our model, we always take
From Lemma
Thus, in the region
The model (
From (
This number
If
for
The fixed point
Consider
Then
Since
The biological interpretation of Lemma
In this section following the approach in [
Let
Substituting the values of
From the solution
The model (
Having proved the existence of the endemic equilibrium point we now investigate its stability using the Centre Manifold Theory [
Without loss of generality, it is assumed that 0 is an equilibrium for system (
Assume the following.
Matrix
Let
The local dynamics of system (
We make the following change of variables:
The Jacobian matrix of system (
The linearised system of the transformed equations (
The left eigenvector of
For the model system (
From (
For the sign of
From (
By Theorem 11 in [
Since
To analyze the coinfection dynamics of KS and HIV, we investigate the KSinduced reproductive number in (
If we define
If we define the partial reproductive number without treatment to be
If
(a) 3D plot of
From Figures
Figures
Using the R programming environment, we ran numerical simulations of the model. The following data were input as initial conditions:
Parameter values used in the numerical simulations of model system (
Parameter values and their estimates.
Parameter  Symbol  Value  Source 

HIV rate of transmission 

0.4801*  Baggaley et al. [ 
Boily et al. [  
Death due to AIDS 

0.333  Malunguza et al. [ 
Mukandavire et al. [  
Death due to AIDS with KS 

0.067*  Malunguza et al. [ 
Mukandavire et al. [  
Rate of KS acquisition among HIVinfected cohort 

0.001  Assumed 
Rate of KS acquisition among preAIDS cohort 

0.002  Assumed 
Rate of KS acquisition among AIDS cohort 

0.003  Assumed 
Relative HIV infectiousness of an HIVinfected individual 

0.8  Assumed 
Relative HIV infectiousness of a coinfected individual 

1.1  Assumed 
Relative HIV infectiousness of a preAIDS individual 

1.2  Assumed 
Relative HIV infectiousness of an AIDS individual 

1.3  Assumed 
Recruitment rate of sexually mature individuals 

800*  Barley et al. [ 
Malunguza et al. [  
Mukandavire et al. [  
Natural mortality rate 

0.02  Mukandavire et al. [ 
Treatment rate of infected and coinfected cohorts 

Varies  Assumed 
Treatment rate of preAIDS and preAIDS coinfected cohorts 

Varies  Assumed 
Natural progression to preAIDS 

0.01  Assumed 
Rate of acute KS development in coinfected cohort 

0.0001  Assumed 
Rate of acute KS development in preAIDS coinfected cohort 

0.0002  Assumed 
Rate of acute KS development in AIDS coinfected cohort 

0.0003  Assumed 
Natural progression to AIDS 

0.1  Mukandavire et al. [ 
Natural progression to AIDS after treatment 

0.1  Assumed 
Our value for the rate of HIV transmission is the average of the minimum (0.011) and maximum values (0.95) for the same parameter in Baggaley et al. [
Figure
Epidemic curves for our model with the following treatment scenarios: (a) no treatment (
Figure
Our model makes no attempt to consider the cost associated with providing the kind of treatment we have discussed. However, it is noteworthy to consider that the gains associated with providing treatment to any more than
In numerous HIVpositive cohorts, susceptibility to KS development is incredibly high. Recent findings that antiretroviral treatment for HIV clinical symptoms can reverse KS in most patients suggest that the only obstacle in preventing KSrelated complications (or even deaths) is one's ability to access the treatment itself. This is especially the case in subSaharan Africa, where KS prevalence is higher than anywhere else in the world, yet access to proper treatment remains low. We have shown that providing treatment to just
In the future, we hope to extend this model to consider all modes of HHV8 transmission. One way to do this is by introducing separate classes of HHV8 infectious individuals. This will increase the overall reach of the model and hopefully provide a positive influence on policy making in areas affected by KS. For now, supporting KS education and awareness needs to become as important as providing universal access to antiretroviral treatment for HIV patients. Most southern African countries already have programs in place to encourage HIV screening; this is the perfect time to also test for HHV8 and provide KS counseling. Regardless of how long the HIV/AIDS epidemic lasts in Africa, there is no reason people should have to fight KS as well.
This work was done under SAMSA Masamu Program with support from the National Science Foundation under Grant no. DMS1050259. Lungu acknowledges the support of the University of Botswana, Massaro would like to acknowledge partial support from the NSF SCALEIT graduate fellowship program at the University of Tennessee, and Malunguza acknowledges the support of the National University of Science and Technology.