Krishnarajah, I., Cook, A., Marion, G. and Gibson, G.
Bulletin of Mathematical Biology 67, 855-873.
log-normal approximation, beta-binomial approximation, normal approximation, non-normal, mixture approximation, moment-closure, epidemics, SIS, SI, R.Solani
||Moment closure approximations are used to provide analytic approximations to
non-linear stochastic models. They often provide insights into model behaviour and
help validate simulation results. However, existing closure schemes typically fail in
situations where the population distribution is highly skewed or extinctions occur.
In this study we address these problems by introducing novel second- and third-
order moment closure approximations which we apply to the stochastic SI and
SIS models. In the case of the SI model, which has a highly skewed distribution of
infection, we develop a second-order approximation based on the beta-binomial.
In addition, a closure approximation based on mixture distribution is developed in order to capture the
behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass
at 0 which represents the probability of extinction. Two third-order versions of this mixture
approximation are considered in which the log-normal and the beta-binomial
are used to model the quasi-equilibrium distribution. Comparison with simulation
results shows: 1) the beta-binomial approximation is flexible in shape and matches
the skewness predicted by simulation as shown by the stochastic SI model and 2)
mixture approximations are able to predict transient and extinction behaviour as
shown by the stochastic SIS model in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likelihood function and carry out point and interval parameter estimation.