A Likelihood Approach for Real-Time Calibration of Stochastic Compartmental Epidemic Models

<div><p>Stochastic transmission dynamic models are especially useful for studying the early emergence of novel pathogens given the importance of chance events when the number of infectious individuals is small. However, methods for parameter estimation and prediction for these types of stochastic models remain limited. In this manuscript, we describe a calibration and prediction framework for stochastic compartmental transmission models of epidemics. The proposed method, Multiple Shooting for Stochastic systems (MSS), applies a linear noise approximation to describe the size of the fluctuations, and uses each new surveillance observation to update the belief about the true epidemic state. Using simulated outbreaks of a novel viral pathogen, we evaluate the accuracy of MSS for real-time parameter estimation and prediction during epidemics. We assume that weekly counts for the number of new diagnosed cases are available and serve as an imperfect proxy of incidence. We show that MSS produces accurate estimates of key epidemic parameters (i.e. mean duration of infectiousness, <i>R</i>₀, and <i>R</i>_eff) and can provide an accurate estimate of the unobserved number of infectious individuals during the course of an epidemic. MSS also allows for accurate prediction of the number and timing of future hospitalizations and the overall attack rate. We compare the performance of MSS to three state-of-the-art benchmark methods: 1) a likelihood approximation with an assumption of independent Poisson observations; 2) a particle filtering method; and 3) an ensemble Kalman filter method. We find that MSS significantly outperforms each of these three benchmark methods in the majority of epidemic scenarios tested. In summary, MSS is a promising method that may improve on current approaches for calibration and prediction using stochastic models of epidemics.</p></div

An algorithm for real-time calibration of stochastic compartmental epidemic models.

<p>An algorithm for real-time calibration of stochastic compartmental epidemic models.</p

Sequence of observations during an epidemic.

<p>Sequence of observations during an epidemic.</p

Identifying the calibration method with statistically dominant performance for predictions at 0.05 significance level.

<p>Red, if MSS is statistically better than all benchmark methods. White, if all methods fail to demonstrate statistically dominant performance. Blue if I.Poi (benchmark A), Green if Particle Filter (benchmark B) and Black if the Ensemble Kalman filter (benchmark C) statistically outperform the MSS method. Multiple colors can occur if more than one method is significantly better than MSS.</p

A model for the outbreak of a novel viral pathogen.

<p>A model for the outbreak of a novel viral pathogen.</p

(A) Weekly clinical meningitis cases in Burkina Faso reported between 2002 and 2015 (data were made available from the Ministry of Health, Burkina Faso). (B) Percentage of confirmed meningitis cases that are associated to <i>Neisseria meningitis</i> serogroup A, <i>N</i>. <i>meningitidis</i> non-A serogroups (including C, W, and X), and other pathogens (including <i>Streptococcus pneumoniae</i> and <i>Haemophilus influenzae</i> type b) from 2002–2015.

<p>These estimates are obtained from WHO [<a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.ref031" target="_blank">31</a>] (for 2002), WHO Enhanced Meningitis Bulletin (for 2003–2005), Burkina Faso <i>Maladies Potentiel Épidémie</i> (MPE) surveillance data (for 2006 and 2012–2015), and Novak et al. [<a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.ref011" target="_blank">11</a>] (for 2007–2011) (see <a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.s002" target="_blank">S1 Data</a>). Men-A, meningitis serogroup A; Non Men-A, meningitis serogroups other than A.</p

The proposed model matches the key characteristics of meningitis epidemics in Burkina Faso observed between 2002 and 2015.

<p>(A) Age distribution of probable meningococcal meningitis in Burkina Faso from 2007–2011 [<a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.ref011" target="_blank">11</a>] versus the age distribution of cases generated by the model. (B) Estimated meningococcal carriage prevalence in different age groups from carriage survey studies in the African meningitis belt [<a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.ref034" target="_blank">34</a>] versus the age-specific average carriage prevalence obtained from the model. (C–D) Average and standard deviation of weekly clinical meningitis cases observed from 2002–2015 versus those produced by the model. (E) Cosine of the angle (<b><i>θ</i></b>) between the vectors of Fourier amplitude for observed and simulated meningitis time series (cosine of 1 indicates total match in periodicity and cosine of 0 indicates no overlap between the significant periods of two time series; see <a href="http://www.plosmedicine.org/article/info:doi/10.1371/journal.pmed.1002495#pmed.1002495.s001" target="_blank">S1 Text</a> for additional details). (F) Observed (Data) and simulated (Model) number of districts in each year between 2002 and 2015 in which the threshold of 10 meningitis cases per 100,000 population was exceeded. Cos, cosine; StDev, standard deviation.</p

Expected number of vaccines used per year (over a 30-year simulation period) for scenarios with and without strain replacement.

<p>Error bars represents 95% projection intervals (error bars that are shorter than the width of symbols are not shown). PMC, polyvalent meningococcal conjugate; PMP, polyvalent meningococcal polysaccharide.</p

Vaccination assumptions.

<p>Vaccination assumptions.</p

Cost-effectiveness of vaccination strategies, assuming a vaccine cost of US$4 per dose for PMP and PMC vaccines.

<p>Cost-effectiveness of vaccination strategies, assuming a vaccine cost of US$4 per dose for PMP and PMC vaccines.</p