BOUNDS FOR THE EXPECTED VALUE OF ONE-STEP PROCESSES

Mean-field models are often used to approximate Markov processes with large state spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space {0,1,..., N} and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. While the mean-field model is a well known approximation, this lower bound is new, and unlike an asymptotic result, these bounds can be used for finite N. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.