Sequential Bayesian inference in hidden Markov stochastic kinetic models with application to detection and response to seasonal epidemics

Abstract

We study sequential Bayesian inference in continuous-time stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is modeling of seasonal infectious disease outbreaks represented through a compartmental epidemic model. We demonstrate inference in such models with several numerical illustrations and also discuss predictive analysis of epidemic countermeasures using sequential Bayes estimates.

Computational Method for Epidemic Detection in Multiple populations

“…Under full observations the detection problem (4) would be trivial, since one can directly track I (2) t and declare an outbreak as soon as there any infecteds in the second pool. However, realistically I (2) is not observed. Some of the reasons include mis-diagnoses among infecteds, patients not seeking care, false positives, mis-reporting or lack of reporting of epidemiological data, etc.…”

“…Remark 1. Note that detection costs are intrinsically defined in terms of the count of infecteds in Pool 2, I (2) , which is assumed to be unavailable to the policy-maker. Below we will operationalize (2) and (3) by taking conditional expectation with respect to information that is available, see (18)- (17).…”

“…ideally one takes τ = θ. However, this is not possible if only partial information is available about X, specifically about I (2) t . When τ and θ are different, C Delay penalizes the event {τ > θ}, and C FA penalizes {τ < θ}.…”

Efficient sampling of conditioned Markov jump processes

“…Markov jump processes (MJPs) can be used to model a wide range of discrete-valued, continuoustime processes. Our focus here is on the MJP representation of a reaction network, which has been ubiquitously applied in areas such as epidemiology (Fuchs, 2013;Lin and Ludkovski, 2013;McKinley et al, 2014), population ecology (Matis et al, 2007;Boys et al, 2008) and systems biology (Wilkinson, 2009(Wilkinson, , 2018Sherlock et al, 2014). Whilst exact, forward simulation of this class of MJP is straightforward (Gillespie, 1977), the reverse problem of performing fully Bayesian inference for the parameters governing the MJP given partial and/or noisy observations is made challenging by the intractability of the observed data likelihood.…”

Efficient $$\hbox {SMC}^2$$ SMC 2 schemes for stochastic kinetic models

“…Application areas include (but are not limited to) systems biology (Golightly and Wilkinson 2005;Wilkinson 2012), predator-prey interaction (Ferm et al 2008;Boys et al 2008) and epidemiology (Lin and Ludkovski 2013;McKinley et al 2014). Here, we focus on the MJP representation of a stochastic kinetic model (SKM), whereby transitions of species in a reaction network are described probabilistically via an instantaneous reaction rate or hazard, which depends on the current system state and a set of rate constants, with the latter typically the object of inference.…”