ABSTRACT
Deterministic systems involving ordinary differential equations or delay differential equations, though widely used in practice (in part because of the ease of their use in simulation studies), have proven less descriptive when applied to problems with small population size. To address this issue, stochastic systems involving continuous time Markov chain (CTMC) models or CTMC models with delays are often used when dealing with low species count. The goals of this chapter are two-fold: one is to give an introduction on
how to simulate a CTMC model or a CTMC model with delays, and the other is to demonstrate how to construct a corresponding deterministic model for a stochastic one, and how to construct a corresponding stochastic model for a deterministic one. The methodology illustrated here is highly relevant to current researchers in the biological and physical sciences. As investigations and models of disease progression become more complex and as interest in initial phases (i.e., HIV acute infection, initial disease outbreaks) of infections or epidemics increases, the recognition becomes widespread that many of these efforts require small population number models for which ordinary differential equations (ODEs) or delay differential equations are unsuitable. These efforts will involve CTMC models (or CTMC models with delay) that have as limits (as population size increases) ordinary differential equations (or delay differential equations). As the interest in initial stages of infection grows, so also will the need grow for efficient simulation with these small population number models. We give a careful presentation of computational issues arising in simulations with such models. The following notation is used throughout the remainder of this chapter:
tstop denotes the final time, Z n is the set of n-dimensional column vectors
with integer components, and ei ∈ Zn is the ith unit column vector (that is, the ith entry of ei is 1 and all the other entries are zeros), i = 1, 2, . . . , n.
