Novel moment closure approximations in stochastic epidemics

Abstract

Moment closure approximations are used to provide analytic approximations to nonlinear 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 thirdorder 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 novel closure approximation is developed in order to capture the behaviour of the stochastic SIS model at the critical point of persistence or extinction of the process. This mixture approximation, is a third-order approximation and comprises a probability distribution designed to capture the behaviour of the system conditioned on non-extinction (quasi-equilibrium) and a probability mass at 0 which represents the probability of extinction. Two 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 show: 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.

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

“…The stochastic logistic model is the stochastic birth-death analogous model of the well-known deterministic Verhulst-Pearl equations (Verhulst, 1838;Pearl and Reed, 1920;Pielou, 1977) and has been extensively used for modeling stochasticity in population biology Kiffe, 2002, 1996;Matis et al, 1998;Krishnarajah et al, 2005). For this continuous-time birth-death Markov process, the conditional probabilities of a unit increase and decrease, respectively, in an "infinitesimal" time interval (t,t + dt] are given by…”

“…Continuous-time birth-death Markov processes have been extensively used for modeling stochasticity in population biology Kiffe, 2002, 1996;Matis et al, 1998;Krishnarajah et al, 2005). The time evolution of these processes is typically described by a single equation for a probability function, where time and species populations appear as independent variables, called the Master or Kolmogorov equation (Bailey, 1964).…”

Stationarity in moment closure and quasi-stationarity of the SIS model

“…The ODE of the nth moment of infecteds, derived from the master equation, (see Krishnarajah et al [8]) is given by…”

“…In the cases where the distributions are skewed, the closure method used consisted in using cumulant truncation by setting the fourth cumulant equal to zero. Krishnarajah et al [8] explored the use of mixture distributions, based on a mixture of a point mass at zero representing extinction of the process, and log-normal and beta-binomial distributions to approximate the distribution conditional upon non-extinction. Clancy and Mendy [5] found that: to the quasi-stationary distribution of an SIS model is preferred; (ii) in the supercritical region, a beta-binomial distribution is preferred.…”

Fractional Bioeconomic Systems: Optimal Control Problems, Theory and Applications

“…First group operates in terms of stochastic control (Yong, 1999) and (Biagini et al, 2002), second one is based on converting the task (9) to non-random fractional optimal control (Jumarie, 2003). It is also possible to use system of moments equations instead of equation (5) as it was proposed in (Krishnarajaha et al, 2005) and (Lloyd, 2004). Unfortunately, there are some limitations, namely the redefinition of MSY for the model (5) and in a consequence finding an optimal harvest cannot be done by classical approaches (Bousquet et al, 2008) and numerical solution for stochastic control problems is highly complicated even for linear SDEs.…”