Tuberculosis (TB) and human immunodeficiency virus (HIV) can be considered a deadly human syndemic. In this paper, we formulate a model for TB and HIV transmission dynamics. The model considers both TB and acquired immune deficiency syndrome (AIDS) treatment for individuals with only one of the two infectious diseases or both. The basic reproduction number and equilibrium points are determined and stability is analyzed. Through simulations, we show that TB treatment for individuals with only TB infection reduces the number of individuals that become coinfected with TB and HIV/AIDS and reduces the diseases (TB and AIDS) induced deaths. Analogously, the treatment of individuals with only AIDS also reduces the number of coinfected individuals. Further, TB treatment for coinfected individuals in the active and latent stage of TB disease implies a decrease of the number of individuals that passes from HIVpositive to AIDS.
Tuberculosis (TB) and human immunodeficiency virus/acquired immune deficiency syndrome (HIV/AIDS) are the leading causes of death from an infectious disease worldwide [
Following UNAIDS global report on AIDS epidemic 2013 [
Collaborative TB/HIV activities (including HIV testing, ART therapy, and TB preventive measures) are crucial for the reduction of TBHIV coinfected individuals. The World Health Organization (WHO) estimates that these collaborative activities prevented 1.3 million people from dying, from 2005 to 2012. However, significant challenges remain: the reduction of tuberculosis related deaths among people living with HIV has slowed in recent years; the ART therapy is not being delivered to TBHIV coinfected patients in the majority of the countries with the largest number of TB/HIV patients; the pace of treatment scaleup for TB/HIV patients has slowed; less than half of notified TB patients were tested for HIV in 2012; and only a small fraction of TB/HIVinfected individuals received TB preventive therapy [
The study of the joint dynamics of TB and HIV presents formidable mathematical challenges due to the fact that the models of transmission are quite distinct [
The paper is organized as follows. Section
The model subdivides the human population into 10 mutually exclusive compartments, namely, susceptible individuals (
Individuals leave the latent TB class
HIVinfected individuals (with no AIDS symptoms) progress to the AIDS class
HIVinfected individuals (preAIDS) coinfected with TBdisease, in the active stage
HIVinfected individuals (with AIDS symptoms), coinfected with TB, are treated for HIV, at a rate
The aforementioned assumptions result in the following system of differential equations that describes the transmission dynamics of TB and HIV disease:
Model for TBHIV/AIDS transmission with treatment.
Let
Adding all equations in model (
The region
Model (
the diseasefree equilibrium (no disease):
the HIVAIDS free equilibrium:
the TBfree equilibrium:
the syndemic equilibrium:
The following theorem states the stability of the equilibrium points.
The diseasefree equilibrium
Details of the proof of Theorem
Explicit expressions for the coinfection endemic equilibrium
For numerical simulations, we consider the following initial conditions for system (
Parameters of the TBHIV/AIDS model (
Symbol  Value  References  Symbol  Value  References 










[  

Variable 




Variable 







[  







[ 





[ 





[ 





[ 










In Table
Effect of

4.3  6  10  15  50 



0.99788  1.39239  2.32065  3.48097  11.60326 

0.00397  903.93492  2206.57268  2870.72755  3804.50589 
Effect of

0.051  0.055  0.07  0.09  0.99 



0.93669  1.01016  1.28566  1.65299  1.81829 

0.01708  135.73817  2516.54721  4472.84980  4930.48696 

0.00266  21.07182  390.59491  694.23361  765.26396 
In Figure
Stability of the diseasefree equilibrium (
Figure
Stability of the syndemic equilibrium
Consider
Impact of TB treatment on singleinfected individuals with disease induced death.
Impact of TB treatment on singleinfected individuals with no disease induced death.
Figure
Impact of AIDS treatment on singleinfected individuals with disease induced death.
Impact of AIDS treatment on singleinfected individuals with no disease induced death.
Impact of TB and AIDS treatment on coinfected individuals with no disease induced death.
The basic reproduction number represents the expected average number of new infections produced by a single infectious individual when in contact with a completely susceptible population [
In this Appendix, we provide details of the proof of Theorem
The Jacobian matrix of the system (
The diseasefree equilibrium is denoted by
Following [
In what follows we prove the local asymptotic stability of the endemic equilibrium
The Jacobian of the system (
The Jacobian
For the system (
For the sign of
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA, University of Aveiro) and