An SEIS deadly disease is introduced in an exponentially growing population. Without the treatment, three thresholds quantities R0, R1, R2 determine the dynamic of the epidemic and that of the population. With the treatment, three other threshold quantities R0T, R1T, R2T govern the dynamic of the epidemic and that of the population. We define the treatment effort and determine the critical treatment effort needed to prevent the disease to invade the population

SEIS model; threshold quantities; treatment effort; numerical simulations