<p><i>Mycoplasma pneumoniae</i> (<i>Mp</i>) is one of the most common causes of community-acquired pneumonia in children. To uncover the effective interventions during an epidemic in crowded settings, we first develop a novel staged progression ordinary differential equation model for the transmission of <i>Mp</i>, incorporating the effects of isolation measures and correct diagnosis rate. The basic reproduction number is obtained by the next generation matrix approach. Based on the deterministic model, a continuous-time Markov chain (CTMC) model is formulated to account for demographic variability. An analytic estimate for the probability of a disease outbreak, as well as an explicit expression for the mean (variance) of the disease extinction time in the absence of an outbreak, is derived by a multi-type branching process approximation of the CTMC model. By fitting the model to real data from a primary school, we estimate some key parameters of our model. Numerical simulations indicate that: (i) if the effects of demographic variability are ignored, the time to extinction after an outbreak is likely to be significantly underestimated or overestimated, depending on the isolation proportion; (ii) the impact of disease transmission rate, isolation proportion, and correct diagnosis rate on the probability of a disease outbreak depends on the stage of infection in which an infected individual is first introduced; (iii) decreasing the transmission rate, increasing the isolation proportion, or improving the correct diagnosis rate can significantly reduce the mean final size after an outbreak; and (iv) improving the correct diagnosis rate can help reduce the number of severe pneumonia cases.</p>