We provide a Galton--Watson model for the growth of a bacterial population in the presence of antibiotics. We assume that bacterial cells either die or duplicate, and the corresponding probabilities depend on the concentration of the antibiotic. Assuming that the mean offspring number is given by $m(c) = 2 / (1 + αc^β)$ for some $α, β$, where $c$ stands for the antibiotic concentration we obtain weakly consistent, asymptotically normal estimator both for $(α, β)$ and for the minimal inhibitory concentration (MIC), a relevant parameter in pharmacology. We apply our method to real data, where \emph{Chlamydia trachomatis} bacteria was treated by azithromycin and ciprofloxacin. For the measurements of \emph{Chlamydia} growth quantitative PCR technique was used. The 2-parameter model fits remarkably well to the biological data.