  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
$\{0,1,\ldots,N\}$ 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. 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.
