Our work is based on a deterministic model of HIV-1 dynamics, firstly appeared in [7], which takes into account the models developed by Perelson and his followers [3, 10, 13, 37]. These models are well presented and take specific biological reality into account. Our initial model has been extended here by adding two more variables, Cytotoxic T Lymphocytes (CTLs) cells and TB, to capture dynamics of co-infection of HIV and TB.

As standard, we describe the variations of the quantities of the modelled entities as a set of differential equations. We start considering a pool of immature CD4 T cells, represented by the variable

*U*, see equation (

1). These cells are continuously produced by the thymus at a rate

*N*
_{
U
}, and evolve into differentiated, uninfected T-cells,

*T* at a rate

*δ*
^{
U
}. Also, the TNF (

*F*) contributes to the clearance of naive T-cells, via

, and naive T cells are produced at fix rate

*k*
_{
Z
}due to CTLs (

*Z*) response. T cells (

*T*) are described by considering their different strains (

*T*
_{
i
}), which we do not detail here (see [

7]). Beyond being produced, they can become infected (

*I*
_{
k
}) by interacting with the virus strains

*V*
_{
k
}at rate

*β*
_{
k
}, or die at rate

*δ*
^{
T
}, (equation (

5)). Note that the infection parameter

*β* is not constant over time, but depends on the distribution of the viral strains R5 and X4 [

9]. Infected T cells are cleared out at a fixed rate,

*δ*
^{
I
}, and also due to the action of CTLs with rate

, (equation (

2)). Equation (

6) describes the budding of viruses, i.e. infected cells produce new viruses at rate

*π*, and the fact that virus particles may be nonviable or being cleared out at rate

*c* by immunoglobulin binding and subsequent engulfments by the macrophages. Next equation (

3) describes latent TB (

*B*) in the blood which propagates with the rate

*α* when T cells goes below a given threshold representing the efficacy of the immune system. The equation (

7) describes the TNF (

*F*) dependence on X4 strains and on the presence of TB in the blood and the fact that the efficacy of such a factor naturally decay in time. Finally, the response of Cytotoxic T Lymphocyte cells

*Z* is as in equation (

4): CTLs response depends on number of infected cells (I) and is cleared out at fixed rate

*b*. This model is general enough to be used as a framework for fitting real data and simulating superinfection and co-infection patterns.