A Stochastic Model for Mycoplasma Pneumoniae Outbreak with Staged Progression

Mycoplasma pneumoniae (Mp) 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 Mp, 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.