With this information in conjunction with additional basic clinical information (ie, temperature, respiratory rate, oxygen saturation), we believe that a much needed improvement in the detection and case management of paediatric pneumonia is possible.
In that case, conditional probabilities are computed by solving a master equation (equation 2 in Additional file 1, "Basic information on Markov process", section 1.1).
Therefore, the dynamics is defined by transition rates inserted in a master equation (see Additional file 1, "Basic information on Markov process", section 1.1).
They can display a damped oscillating behavior, or none at all (see Additional file 1, "Basic information on Markov process", section 1.3).
According to theorems described in Additional file 1 ("Basic information on Markov process", theorems 6-8 and Corollary 3, section 1.3), a necessary condition for having damped oscillations is that the transition matrix has at least one non-real eigenvalue (see Additional file 1, "Basic information on Markov process", equation 4, section 1.1).
In the case of continuous time Markov process, instantaneous probabilities always converge to a stationary distribution (see Additional file 1, "Basic information on Markov process", corollary 2, section 1.2).
Therefore, a cell in a sub-population can only evolved in this sub-population (Additional file 1, "Basic information on Markov process", corollary 1, section 1.2 and from the definition of strongly connected component with no outgoing edges).
It can be shown (see Additional file 1, "Basic information on Markov process", corollary 2, section 1.2) that instantaneous probabilities of a continuous time Markov process converge to a stationary distribution.
A linear combination of stationary distributions is also a stationary distribution, since instantaneous probabilities are solutions of a master equation which is linear (see Additional file 1, "Basic information on Markov process", equation 2, section 1.1).
