(S)

Susceptible: Health and non immune individuals

(V)

Vaccinated: immunized individuals

(S_vih)

AIDS patient: people infected with HIV

(E)

Earlier latent: people infected but without clinic signs of disease

(L)

Later latent: People who remains without clinic signs

( $\alpha$ )

Vaccination coverage

${\eta }_S$

Recruitment rate of susceptible

${\pi }_1={\eta }_{S}S$

Number of susceptible rescued at birth

${\pi }_2={\eta }_{V}V$

Number of vaccinated rescued at birth

${\pi }_3={\eta }_{vih}{S}_{vih}$

Number of VIH/AIDS patients rescued at birth

(I)

People infected with Tuberculosis and are infectious

( $\epsilon$ )

AIDS-HIV prevalence

( $\omega$ )

Progression rate to latent earlier stage

( $\beta SI$ )

Incidence rate: number of new infected cases

( $\sigma =1-T_{\text{CD4}}$ )

Immune deficiency rate

( ${\theta }_2=1+\sigma$ )

Cost of immune deficiency rate

( ${\theta }_1$ )

Vaccine efficacity

$d_X\ge \mathrm{0,}{d}_E\ge \mathrm{0,}{d}_E\ge \mathrm{0,}{d}_T\ge 0$

Removal rate, include death due to the TB

( $\beta$ )

Effective contact rate

${\mu }_{vih}$

HIV specific death rate

p

Fraction of individuals who progress directly in TB active stage

$\nu$

Progression rate to active disease

$1-p$

Progression fraction to the late latent stage