Chapter 6 - Stochastic Dynamics

All the models considered thus far have been deterministic. This means they are essentially fixed “clockwork” systems; given the same starting conditions, exactly the same trajectory is always observed. Such a Newtonian view of the world does not apply to the dynamics of real pathogens. If it were possible to “re-run” a real-world epidemic, we would not expect to observe exactly the same people becoming infected at exactly the same times. Clearly, there is an important element of chance. Stochastic models are concerned with approximating or mimicking this random or probabilistic element. In general, the role played by chance will be most important whenever the number of infectious individuals is relatively small, which can be when the population size is small, when an infectious disease has just invaded, when control measures are successfully applied, or during the trough phase of an epidemic cycle. In such circumstances, it is especially important that stochasticity is taken into account and incorporated into models.
This chapter details three distinct methods of approximating the chance element in disease transmission and recovery: (1) introducing chance directly into the population variables, (2) by random parameter variation, and (3) individual-level, explicit modeling of the random events (Bartlett 1957). This third method is generally the most popular and will be the predominant focus of this chapter, illustrated by examples from the recent literature. All these examples have two elements in common. First, they predict different outcomes from the same initial conditions and, as such, multiple simulations are required to determine the expected range of behavior. Second, they all require the use of a random number-generating routine -- this can be thought of as a rather clever computational trick that can deliver an apparently random sequence of numbers in some prescribed range.