Data Assimilation for the Lorenz (1963) Model using Ensemble and Extended Kalman Filter

<p>Presentation prepared for the Biogeodynamics and Earth System Sciences Summer School (BESS), 23rd June 2011</p> <p> </p

SIRBV model.

<p>Schematic representation of the SIRB model at node <i>i</i> with the three additional compartments for vaccinated individuals: vaccinated susceptible individuals (<i>V</i>^<i>S</i>), vaccinated infected (<i>V</i>^<i>I</i>), and vaccinated recovered (<i>V</i>^<i>R</i>). Compartments <i>I</i> and <i>V</i>^<i>I</i> contribute to the local bacteria concentration (<i>B</i>). Compartment <i>S</i> is exposed to a force of infection <i>F</i>_<i>i</i> and enter either compartment <i>I</i> with rate <i>σF</i>_<i>i</i> or <i>R</i> with rate (1 − <i>σ</i>)<i>F</i>_<i>i</i>, respectively. These probabilities are rescaled of a factor 1 − <i>η</i> for the vaccinated susceptibles.</p

Forecasted incidence.

<p>Box-plot (percentiles 0.05, 0.25, 0.5, 0.75, 0.95) of the cumulative incidence (total number of new cases divided by the population size) forecasted in each commune of Grande Anse after the vaccination period (from November 12 to December 31, weeks 45-53) during the first projection. Results are obtained with (green) and without (red) the deployment of the OCV campaign in the communes highlighted in yellow. The cholera projections are driven by an ensemble of forecasted rainfall (S1), the maximum forecasted rainfall (S2), and the actual rainfall (S3). The circle represents the measured incidence during the same period, while asterisks are a constant projection based on the cases reported from the occurrence of Matthew up to the projection date (from October 4 to October 29).</p

Sensitivity of the ensemble RMSE to the OCV coverage and the vaccine efficacy during the second projection using measured rainfall (scenario S3).

<p>The minimum mean RMSE is obtained for coverage <i>ξ</i> = 0.6 and efficacy <i>η</i> = 1.0 (RMSE = 19.46), however the structure of the error is almost symmetric (RMSE = 19.49 for <i>ξ</i> = 1.0 and efficacy <i>η</i> = 0.6) indicating identifiability issues between coverage and efficacy. A slightly higher error, RMSE = 19.98, is associated to the second projection (<i>ξ</i> = 0.9 and efficacy <i>η</i> = 0.87).</p

First cholera projection.

<p>From September 3 to October 29, 2016: comparison between the weekly reported cases (white circles) and the model results forced by the DA procedure in the departments of Grande Anse, Sud, Ouest, and over all Haiti. The median of the ensemble is the blue line, while the blue area represents the ensemble 0.05-0.95 confidence interval. From October 29 to December 31, 2016: comparison between the weekly reported cases (crosses) and the model forecast (the ensemble median is the dashed line, the area represents the ensemble 0.05-0.95 confidence interval). Results in green are obtained considering the OCV campaign with vaccine efficacy set to <i>η</i> = 87% and coverage <i>ξ</i> = 90%. Results without OCV campaign are in red. The yellow shaded area shows the timing of the OCV campaign. Panels in the first row depict the average measured precipitation (blue) and the forecasted precipitation (red) under scenario S1 (left), S2 (center), and S3 (right).</p

Reported cases after Matthew.

<p>Maps of the reported cholera cases and the associated incidence during October 2016 at the communal level with the detailed weekly dynamics for the departments most affected after Matthew (Grande Anse (a), Sud (b) and Ouest (c)), and for the whole country (panel d).</p

Forecasted incidence.

<p>Same as <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1006127#pcbi.1006127.g003" target="_blank">Fig 3</a> for the communes in Sud.</p