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Epidemiology and Infection\\
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A Multi-state Spatio-temporal Markov Model for Categorized Incidence of Meningitis in Sub-Saharan Africa\\
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L. Agier, M. Stanton, G. Soga, P. J. Diggle\\
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Supplementary Material
\newpage
Supplementary Table S1: $SR$ values for competing models \\
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%\begin{tabular}{{\left}p{4cm}|*{2}{\centering}p{4cm}|*{2}{\centering}p{4cm}|} %\arraybackslash
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\begin{tabular}{|l|cc|cc|}
\cline{2-5}
\hline
Weight & \multicolumn{2}{c|}{ Neighbours=all districts} & \multicolumn{2}{c|}{Neighbours=adjacent districts}\\
& $N=1$ & $N=2$ & $N=1$ & $N=2$\\
\hline
constant & * & * & -5399 & -5408\\
(number of neighbours)$^{-1}$ & * & * & -5407 & -5407\\
(centroids’ distance)$^{-1}$ & -5406 & -5424 & -5453 & -5454\\
population density of $d'$ & * & * & -5639 & -5620\\
\hline
\end{tabular}
* covariate of constant values over all districts
\newpage
Supplementary Table S2: Maximum likelihood parameter estimates and standard errors for $\alpha_{ij},\gamma_{ij}, \delta_{ij}$ and $\zeta_{ij}$ estimates.\\
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\begin{tabular}{|ll|cc|cc|}
\hline
% & $i$ & \multicolumn{2}{c}{$j=1$} & \multicolumn{2}{c}{$j=2$ \\
& & estimate & standard error & estimate & standard error\\
\hline
$\alpha_{ij}$ & 00 & -6.24 & 0.33 & -8.69 & 0.66\\
& 01 & -3.35 & 0.19 & -5.49 & 0.36\\
& 02 & -2.88 & 0.45 & -5.15 & 0.60\\
& 10 & -2.84 & 0.22 & -4.76 & 0.49\\
& 11 & -1.93 & 0.16 & -4.36 & 0.39\\
& 12 & -1.03 & 0.69 & -2.94 & 0.63\\
& 20 & -4.30 & 0.91 & -3.49 & 0.33\\
& 21 & -0.87 & 0.70 & -2.16 & 0.49\\
& 22 & -2.13 & 0.55 & -2.20 & 0.39\\
\hline
$\gamma_{ij}$ & 00 & 0.84 & 0.08 & 0.76 & 0.19\\
& 01 & 0.91 & 0.18 & 1.30 & 0.35\\
& 02 & 1.04 & 0.36 & 1.60 & 0.51\\
& 10 & 1.01 & 0.18 & 1.46 & 0.29\\
& 11 & 0.79 & 0.19 & 1.61 & 0.29\\
& 12 & 1.00 & 0.39 & 2.26 & 0.42\\
& 20 & 0.52 & 0.50 & 0.96 & 0.39\\
& 21 & 0.94 & 0.42 & 2.15 & 0.44\\
& 22 & -0.30 & 0.34 & 1.92 & 0.28\\
\hline
$\delta_{ij}$ & 00 & 1.64 & 0.11 & 2.00 & 0.28\\
& 01 & 0.90 & 0.34 & 0.88 & 0.64\\
& 02 & 0.60 & 0.73 & 1.62 & 1.05\\
& 10 & 0.91 & 0.28 & 1.20 & 0.44\\
& 11 & 0.64 & 0.34 & 1.62 & 0.54\\
& 12 & -0.18 & 0.88 & 0.84 & 1.02\\
& 20 & 2.93 & 1.10 & 1.10 & 0.62\\
& 21 & -0.17 & 0.93 & 1.01 & 0.94\\
& 22 & 1.65 & 0.77 & 2.80 & 0.61\\
\hline
$\zeta_{ij}$ & 00 & 0.49 & 0.03 & 0.63 & 0.05\\
& 01 & 0.26 & 0.05 & 0.36 & 0.08\\
& 02 & 0.20 & 0.08 & 0.39 & 0.09\\
& 10 & 0.23 & 0.04 & 0.37 & 0.06\\
& 11 & 0.20 & 0.05 & 0.34 & 0.05\\
& 12 & 0.13 & 0.06 & 0.29 & 0.07\\
& 20 & 0.07 & 0.10 & 0.35 & 0.08\\
& 21 & 0.28 & 0.10 & 0.45 & 0.10\\
& 22 & 0.10 & 0.05 & 0.26 & 0.05\\
\hline
\end{tabular}
\newpage
Supplementary Table S3: Maximum likelihood parameter estimates and standard errors for $\beta_{dj}$ estimates.\\
\\
\begin{tabular}{|ll|cc|cc|}
\hline
% & $d$ & \multicolumn{2}{c|}{$j=1$} & \multicolumn{2}{c|}{$j=2$}\\
& & estimate & standard error & estimate & standard error\\
\hline
$\beta_{dj}$ & AGUIE & 1.40 & 0.37 & 1.97 & 0.67\\
& ARLIT & 0.44 & 0.46 & 1.71 & 0.79\\
& BILMA & 0.94 & 0.40 & 2.44 & 0.69\\
& BIRNI.NKONNI & 1.69 & 0.36 & 1.90 & 0.66\\
& BOBOYE & 1.10 & 0.38 & 1.59 & 0.69\\
& BOUZA & 0.72 & 0.37 & 0.41 & 0.70\\
& DAKORO & 0.42 & 0.38 & 0.20 & 0.70\\
& DIFFA & 0.53 & 0.45 & 1.88 & 0.74\\
& DOGON.DOUTCHI & 0.99 & 0.37 & 1.11 & 0.67\\
& DOSSO & 0.87 & 0.38 & 1.12 & 0.69\\
& FILINGUE & 0.21 & 0.40 & 0.21 & 0.71\\
& GAYA & 1.02 & 0.38 & 1.59 & 0.68\\
& GOURE & 0.61 & 0.40 & 0.89 & 0.73\\
& GUIDAN.ROUMDJI & 0.77 & 0.37 & 0.77 & 0.67\\
& ILLELA & 0.90 & 0.38 & 1.26 & 0.67\\
& KEITA & 1.08 & 0.37 & 1.38 & 0.68\\
& KOLLO & 0.41 & 0.41 & 0.28 & 0.73\\
& LOGA & 1.09 & 0.37 & 1.43 & 0.68\\
& MADAOUA & 1.41 & 0.36 & 1.99 & 0.66\\
& MADAROUNFA & 0.87 & 0.38 & 1.62 & 0.67\\
& MAGARIA & 1.29 & 0.37 & 1.67 & 0.68\\
& MAINE.SOROA & -0.81 & 0.67 & 1.59 & 0.78\\
& MARADI & 1.95 & 0.36 & 2.44 & 0.66\\
& MATAMEYE & 1.37 & 0.37 & 1.86 & 0.67\\
& MAYAHI & 0.71 & 0.38 & 1.49 & 0.67\\
& MIRRIAH & 0.75 & 0.38 & 0.66 & 0.69\\
& NGUIGMI & -1.10 & 0.78 & 0.86 & 0.96\\
& NIAMEY & 1.09 & 0.39 & 1.62 & 0.71\\
& OUALLAM & 0.39 & 0.44 & 0.93 & 0.76\\
& SAY & 0.84 & 0.39 & 1.42 & 0.69\\
& TAHOUA & 1.05 & 0.38 & 1.50 & 0.68\\
& TANOUT & 0.96 & 0.38 & 1.49 & 0.69\\
& TCHINTABARADEN & 0.96 & 0.38 & 1.59 & 0.69\\
& TERA & 0.87 & 0.40 & 1.81 & 0.69\\
& TESSAOUA & 1.16 & 0.37 & 1.05 & 0.68\\
& TILLABERI & 1.12 & 0.38 & 1.07 & 0.71\\
& ZINDER & 1.55 & 0.37 & 2.58 & 0.67\\
\hline
\end{tabular}
\newpage
Supplementary Table S4: Highest transition probabilities to switch to the alert and epidemic threshold for any given $i$ value (for an average district-specific intercept), for a spatial covariate $f_{dij} (t)$ equal to 0 and equal to 1. The week this highest transition probability is reached is specified.\\
\\
\begin{tabular}{|l|ccc|ccc|}
\hline
$i$ & \multicolumn{3}{c|}{Alert} & \multicolumn{3}{c|}{Epidemic}\\
& $f_{dij} (t)=0$ & $f_{dij} (t)=1$ & week &$f_{dij} (t)=0$ &$f_{dij} (t)=1$ & week\\
\hline
00 & 2.8\% & 4.5\% & 9 & 0.6\% & 1.0\% & 10\\
01 & 21.6\% & 25.8\% & 6 & 5.8\% & 7.6\% & 5\\
02 & 26.7\% & 29.1\% & 4 & 13.8\% & 18.2\% & 7\\
10 & 21.8\% & 20.3\% & 6 & 76.3\% & 78.3\% & 6\\
11 & 38.5\% & 40.2\% & 6 & 20.6\% & 24.8\% & 7\\
12 & 45.4\% & 43.8\% & 51 & 43.8\% & 48.8\% & 3\\
20 & 30.5\% & 29.6\% & 12 & 27.3\% & 34.4\% & 7\\
21 & 36.4\% & 34.3\% & 51 & 61.6\% & 67.5\% & 4\\
22 & 22.1\% & 20.2\% & 15 & 87.3\% & 89.5\% & 8\\
\hline
\end{tabular}
\newpage
Supplementary Table S5: Absolute difference between the fitted values and the cross-validates values for the probabilities of entering the latent state, alert, and epidemic state, averaged over weeks and districts within each year (given in \%), and annual number of cases.\\
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\begin{tabular}{|l|cccc|}
\hline
& Latent & Alert & Epidemic & Annual number of cases\\
\hline
1986 & 0.022 & 0.016 & 0.009 & 394\\
1987 & 0.129 & 0.121 & 0.099 & 3327\\
1988 & 0.073 & 0.064 & 0.025 & 2361\\
1989 & 0.142 & 0.129 & 0.068 & 3739\\
1990 & 0.069 & 0.062 & 0.030 & 2252\\
1991 & 0.113 & 0.119 & 0.088 & 4208\\
1992 & 0.265 & 0.227 & 0.232 & 6947\\
1993 & 0.265 & 0.236 & 0.254 & 11025\\
1994 & 0.242 & 0.230 & 0.232 & 11838\\
1995 & 0.468 & 0.580 & 0.639 & 43203\\
1996 & 0.403 & 0.670 & 0.578 & 16745\\
1997 & 0.086 & 0.080 & 0.042 & 4930\\
1998 & 0.066 & 0.054 & 0.016 & 2328\\
1999 & 0.327 & 0.221 & 0.211 & 5592\\
2000 & 0.274 & 0.278 & 0.293 & 14633\\
2001 & 0.219 & 0.196 & 0.110 & 8833\\
2002 & 0.150 & 0.143 & 0.075 & 5785\\
2003 & 0.127 & 0.131 & 0.053 & 9006\\
2004 & 0.134 & 0.131 & 0.040 & 4153\\
2005 & 0.065 & 0.055 & 0.013 & 1291\\
2006 & 0.051 & 0.061 & 0.053 & 4465\\
2007 & 0.108 & 0.091 & 0.016 & 809\\
\hline
\end{tabular}
\newpage
Supplementary Figure S1: Scoring rule R disaggregated by district (A), mean annual incidence per 100000 population computed over the 1986-2007 period by district (B) and district-level population density (C).\\
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\newpage
Supplementary Figure 2: Fitted values (y axis) are compared to cross-validated values (x axis) for the probabilities of entering the latent state (A), alert (B), and epidemic state (C). Colors code for meningitis-years. The green and pink dots represent the years 1996 and 1999, respectively.\\
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\end{document}