The model proposed can be described by a system of 9 differential equations. The period of time considered is divided in intervals of 1 year, and the equations were written considering a range of change during i years.

The system of equations is the following:

  dS/dt = - b (S I/N) + l - m S - l V - S Vx
  dSbcg/dt = - b (Sbcg I/N) - mSbcg + l V + S Vx
  dE/dt = b (S I/N) - k1 E - m E - r1 E
  dEbcg/dt = b (S I/N) - k2 Ebcg - m Ebcg - r2 Ebcg
  dEp/dt = r1 - k3 Ep - m Ep
  dEbcgp/dt = r2 - k4 Ebcgp - m Ebcgp
  dT/dt = r3 I- k5 T - m T
dI/dt = k1 E + k2 Ebcg + k3 Ep + k4 Ebcgp + k5 T - d I - r3 I
 
The differential equation for the whole population is:
 dN/dt = l - m N - d I
 
where
S: number of Suceptibles
Sbcg: number of Suceptable individuals with bcg
I: number of infectious individuals
E: number of latent individuals
Ebcg: number of latent individuals with bcg
T: number of treated individuals
Ep: number of latent individuals with prophilaxis
Ebcgp: number of latent individuals with bcg and prophilaxis i
N: total population number
m: mortality rate in all classes
d: TB-ralated mortality rate
l: birth rate
b: rate of infection per contact
V: rate of vaccination in newly born in interval i
Vx: rate of vaccination in a massive campaign
r1: rate of prophylaxis applied to individuals in class E during interval i
r2: rate of prophylaxis applied to individuals in class Ebcg during interval i
r3: rate of treatment to infectious individuals during interval i;
k1: progression rate to active TB in category E
k2: progression rate to active TB in category Ebcg
k3: progression rate to active TB in category Ep
k4: progression rate to active TB in category Ebcgp
k5: progression rate to active TB in category T