About the algorithm for calculating the final probabilities

Abstract

Abstract We consider a Markov process with continuous time and a countable number of states, known as the (general) Bartlett–Mac-Kendrick epidemic. Kolmogorov’s equation for the exponential (double) generating function of the final probabilities is a hyperbolic partial differential equation. Its solution requires the calculation of the final probabilities for trajectories specifying one of the boundary conditions. It is shown that the desired trajectory is described as the Dick paths, enumerated by Catalan numbers. For arbitrary trajectories of the final probabilities of the general epidemic process, a calculation algorithm based on the contex-free grammar is proposed. The enumerative problem is solved.