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Epidemiology and Infection\\
\\
A Multi-state Spatio-temporal Markov Model for Categorized Incidence of Meningitis in Sub-Saharan Africa\\
\\
L. Agier, M. Stanton, G. Soga, P. J. Diggle\\
\\
Supplementary Material 

\newpage

Supplementary Table S1: $SR$ values for competing models \\
\\
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\begin{tabular}{|l|cc|cc|}
	\cline{2-5}
	\hline
	Weight &amp; \multicolumn{2}{c|}{ Neighbours=all districts} &amp; \multicolumn{2}{c|}{Neighbours=adjacent districts}\\
	&amp; $N=1$ 	&amp; $N=2$	&amp; $N=1$	&amp; $N=2$\\
	\hline
	constant	&amp; *	&amp; *	&amp; -5399	&amp; -5408\\
	(number of neighbours)$^{-1}$	&amp; *	&amp; *	&amp; -5407	&amp; -5407\\
	(centroidsâ€™ distance)$^{-1}$	&amp; -5406	&amp; -5424	&amp; -5453	&amp; -5454\\
	population density of $d'$	&amp; *	&amp; *	&amp; -5639	&amp; -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.\\
\\
\begin{tabular}{|ll|cc|cc|}
	\hline
%	&amp; $i$	&amp; \multicolumn{2}{c}{$j=1$}	&amp; \multicolumn{2}{c}{$j=2$ \\
		&amp; &amp; estimate	&amp; standard error	&amp; estimate	&amp; standard error\\
	\hline
	$\alpha_{ij}$ 	&amp; 00	&amp; -6.24	&amp; 0.33	&amp; -8.69	&amp; 0.66\\
		&amp; 01	&amp; -3.35	&amp; 0.19	&amp; -5.49	&amp; 0.36\\
		&amp; 02	&amp; -2.88	&amp; 0.45	&amp; -5.15	&amp; 0.60\\
		&amp; 10	&amp; -2.84	&amp; 0.22	&amp; -4.76	&amp; 0.49\\
		&amp; 11	&amp; -1.93	&amp; 0.16	&amp; -4.36	&amp; 0.39\\
		&amp; 12	&amp; -1.03	&amp; 0.69	&amp; -2.94	&amp; 0.63\\
		&amp; 20	&amp; -4.30	&amp; 0.91	&amp; -3.49	&amp; 0.33\\
		&amp; 21	&amp; -0.87	&amp; 0.70	&amp; -2.16	&amp; 0.49\\
		&amp; 22	&amp; -2.13	&amp; 0.55	&amp; -2.20	&amp; 0.39\\
	\hline
	$\gamma_{ij}$	&amp; 00	&amp; 0.84	&amp; 0.08	&amp; 0.76	&amp; 0.19\\
		&amp; 01	&amp; 0.91	&amp; 0.18	&amp; 1.30	&amp; 0.35\\
		&amp; 02	&amp; 1.04	&amp; 0.36	&amp; 1.60	&amp; 0.51\\
		&amp; 10	&amp; 1.01	&amp; 0.18	&amp; 1.46	&amp; 0.29\\
		&amp; 11	&amp; 0.79	&amp; 0.19	&amp; 1.61	&amp; 0.29\\
		&amp; 12	&amp; 1.00	&amp; 0.39	&amp; 2.26	&amp; 0.42\\
		&amp; 20	&amp; 0.52	&amp; 0.50	&amp; 0.96	&amp; 0.39\\
		&amp; 21	&amp; 0.94	&amp; 0.42	&amp; 2.15	&amp; 0.44\\
		&amp; 22	&amp; -0.30	&amp; 0.34	&amp; 1.92	&amp; 0.28\\
	\hline
	$\delta_{ij}$ 	&amp; 00	&amp; 1.64	&amp; 0.11	&amp; 2.00	&amp; 0.28\\
		&amp; 01	&amp; 0.90	&amp; 0.34	&amp; 0.88	&amp; 0.64\\
		&amp; 02	&amp; 0.60	&amp; 0.73	&amp; 1.62	&amp; 1.05\\
		&amp; 10	&amp; 0.91	&amp; 0.28	&amp; 1.20	&amp; 0.44\\
		&amp; 11	&amp; 0.64	&amp; 0.34	&amp; 1.62	&amp; 0.54\\
		&amp; 12	&amp; -0.18	&amp; 0.88	&amp; 0.84	&amp; 1.02\\
		&amp; 20	&amp; 2.93	&amp; 1.10	&amp; 1.10	&amp; 0.62\\
		&amp; 21	&amp; -0.17	&amp; 0.93	&amp; 1.01	&amp; 0.94\\
		&amp; 22	&amp; 1.65	&amp; 0.77	&amp; 2.80	&amp; 0.61\\
	\hline
	$\zeta_{ij}$	&amp; 00	&amp; 0.49	&amp; 0.03	&amp; 0.63	&amp; 0.05\\
		&amp; 01	&amp; 0.26	&amp; 0.05	&amp; 0.36	&amp; 0.08\\
		&amp; 02	&amp; 0.20	&amp; 0.08	&amp; 0.39	&amp; 0.09\\
		&amp; 10	&amp; 0.23	&amp; 0.04	&amp; 0.37	&amp; 0.06\\
		&amp; 11	&amp; 0.20	&amp; 0.05	&amp; 0.34	&amp; 0.05\\
		&amp; 12	&amp; 0.13	&amp; 0.06	&amp; 0.29	&amp; 0.07\\
		&amp; 20	&amp; 0.07	&amp; 0.10	&amp; 0.35	&amp; 0.08\\
		&amp; 21	&amp; 0.28	&amp; 0.10	&amp; 0.45	&amp; 0.10\\
		&amp; 22	&amp; 0.10	&amp; 0.05	&amp; 0.26	&amp; 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
%	&amp; $d$	&amp; \multicolumn{2}{c|}{$j=1$}	&amp; \multicolumn{2}{c|}{$j=2$}\\
		&amp; &amp; estimate	&amp; standard error	&amp; estimate	&amp; standard error\\
	\hline
	$\beta_{dj}$	&amp; AGUIE	&amp; 1.40	&amp; 0.37	&amp; 1.97	&amp; 0.67\\
	&amp; ARLIT	&amp; 0.44	&amp; 0.46	&amp; 1.71	&amp; 0.79\\
	&amp; BILMA	&amp; 0.94	&amp; 0.40	&amp; 2.44	&amp; 0.69\\
	&amp; BIRNI.NKONNI	&amp; 1.69	&amp; 0.36	&amp; 1.90	&amp; 0.66\\
	&amp; BOBOYE	&amp; 1.10	&amp; 0.38	&amp; 1.59	&amp; 0.69\\
	&amp; BOUZA	&amp; 0.72	&amp; 0.37	&amp; 0.41	&amp; 0.70\\
	&amp; DAKORO	&amp; 0.42	&amp; 0.38	&amp; 0.20	&amp; 0.70\\
	&amp; DIFFA	&amp; 0.53	&amp; 0.45	&amp; 1.88	&amp; 0.74\\
	&amp; DOGON.DOUTCHI	&amp; 0.99	&amp; 0.37	&amp; 1.11	&amp; 0.67\\
	&amp; DOSSO	&amp; 0.87	&amp; 0.38	&amp; 1.12	&amp; 0.69\\
	&amp; FILINGUE	&amp; 0.21	&amp; 0.40	&amp; 0.21	&amp; 0.71\\
	&amp; GAYA	&amp; 1.02	&amp; 0.38	&amp; 1.59	&amp; 0.68\\
	&amp; GOURE	&amp; 0.61	&amp; 0.40	&amp; 0.89	&amp; 0.73\\
	&amp; GUIDAN.ROUMDJI	&amp; 0.77	&amp; 0.37	&amp; 0.77	&amp; 0.67\\
	&amp; ILLELA	&amp; 0.90	&amp; 0.38	&amp; 1.26	&amp; 0.67\\
	&amp; KEITA	&amp; 1.08	&amp; 0.37	&amp; 1.38	&amp; 0.68\\
	&amp; KOLLO	&amp; 0.41	&amp; 0.41	&amp; 0.28	&amp; 0.73\\
	&amp; LOGA	&amp; 1.09	&amp; 0.37	&amp; 1.43	&amp; 0.68\\
	&amp; MADAOUA	&amp; 1.41	&amp; 0.36	&amp; 1.99	&amp; 0.66\\
	&amp; MADAROUNFA	&amp; 0.87	&amp; 0.38	&amp; 1.62	&amp; 0.67\\
	&amp; MAGARIA	&amp; 1.29	&amp; 0.37	&amp; 1.67	&amp; 0.68\\
	&amp; MAINE.SOROA	&amp; -0.81	&amp; 0.67	&amp; 1.59	&amp; 0.78\\
	&amp; MARADI	&amp; 1.95	&amp; 0.36	&amp; 2.44	&amp; 0.66\\
	&amp; MATAMEYE	&amp; 1.37	&amp; 0.37	&amp; 1.86	&amp; 0.67\\
	&amp; MAYAHI	&amp; 0.71	&amp; 0.38	&amp; 1.49	&amp; 0.67\\
	&amp; MIRRIAH	&amp; 0.75	&amp; 0.38	&amp; 0.66	&amp; 0.69\\
	&amp; NGUIGMI	&amp; -1.10	&amp; 0.78	&amp; 0.86	&amp; 0.96\\
	&amp; NIAMEY	&amp; 1.09	&amp; 0.39	&amp; 1.62	&amp; 0.71\\
	&amp; OUALLAM	&amp; 0.39	&amp; 0.44	&amp; 0.93	&amp; 0.76\\
	&amp; SAY	&amp; 0.84	&amp; 0.39	&amp; 1.42	&amp; 0.69\\
	&amp; TAHOUA	&amp; 1.05	&amp; 0.38	&amp; 1.50	&amp; 0.68\\
	&amp; TANOUT	&amp; 0.96	&amp; 0.38	&amp; 1.49	&amp; 0.69\\
	&amp; TCHINTABARADEN	&amp; 0.96	&amp; 0.38	&amp; 1.59	&amp; 0.69\\
	&amp; TERA	&amp; 0.87	&amp; 0.40	&amp; 1.81	&amp; 0.69\\
	&amp; TESSAOUA	&amp; 1.16	&amp; 0.37	&amp; 1.05	&amp; 0.68\\
	&amp; TILLABERI	&amp; 1.12	&amp; 0.38	&amp; 1.07	&amp; 0.71\\
	&amp; ZINDER	&amp; 1.55	&amp; 0.37	&amp; 2.58	&amp; 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
 	$i$ 	&amp; \multicolumn{3}{c|}{Alert}	&amp; \multicolumn{3}{c|}{Epidemic}\\
	&amp; $f_{dij} (t)=0$	&amp; $f_{dij} (t)=1$	&amp; week	&amp;$f_{dij} (t)=0$	&amp;$f_{dij} (t)=1$	&amp; week\\
	\hline
	00	&amp; 2.8\%	&amp; 4.5\%	&amp; 9	&amp; 0.6\%	&amp; 1.0\%	&amp; 10\\
	01	&amp; 21.6\%	&amp; 25.8\%	&amp; 6	&amp; 5.8\%	&amp; 7.6\%	&amp; 5\\
	02	&amp; 26.7\%	&amp; 29.1\%	&amp; 4	&amp; 13.8\%	&amp; 18.2\%	&amp; 7\\
	10	&amp; 21.8\%	&amp; 20.3\%	&amp; 6	&amp; 76.3\%	&amp; 78.3\%	&amp; 6\\
	11	&amp; 38.5\%	&amp; 40.2\%	&amp; 6	&amp; 20.6\%	&amp; 24.8\%	&amp; 7\\
	12	&amp; 45.4\%	&amp; 43.8\%	&amp; 51	&amp; 43.8\%	&amp; 48.8\%	&amp; 3\\
	20	&amp; 30.5\%	&amp; 29.6\%	&amp; 12	&amp; 27.3\%	&amp; 34.4\%	&amp; 7\\
	21	&amp; 36.4\%	&amp; 34.3\%	&amp; 51	&amp; 61.6\%	&amp; 67.5\%	&amp; 4\\
	22	&amp; 22.1\%	&amp; 20.2\%	&amp; 15	&amp; 87.3\%	&amp; 89.5\%	&amp; 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.\\
\\
\begin{tabular}{|l|cccc|}
	\hline
	&amp; Latent	&amp; Alert	&amp; Epidemic	&amp; Annual number of cases\\
	\hline
	1986	&amp; 0.022	&amp; 0.016	&amp; 0.009	&amp; 394\\
	1987	&amp; 0.129	&amp; 0.121	&amp; 0.099	&amp; 3327\\
	1988	&amp; 0.073	&amp; 0.064	&amp; 0.025	&amp; 2361\\
	1989	&amp; 0.142	&amp; 0.129	&amp; 0.068	&amp; 3739\\
	1990	&amp; 0.069	&amp; 0.062	&amp; 0.030	&amp; 2252\\
	1991	&amp; 0.113	&amp; 0.119	&amp; 0.088	&amp; 4208\\
	1992	&amp; 0.265	&amp; 0.227	&amp; 0.232	&amp; 6947\\
	1993	&amp; 0.265	&amp; 0.236	&amp; 0.254	&amp; 11025\\
	1994	&amp; 0.242	&amp; 0.230	&amp; 0.232	&amp; 11838\\
	1995	&amp; 0.468	&amp; 0.580	&amp; 0.639	&amp; 43203\\
	1996	&amp; 0.403	&amp; 0.670	&amp; 0.578	&amp; 16745\\
	1997	&amp; 0.086	&amp; 0.080	&amp; 0.042	&amp; 4930\\
	1998	&amp; 0.066	&amp; 0.054	&amp; 0.016	&amp; 2328\\
	1999	&amp; 0.327	&amp; 0.221	&amp; 0.211	&amp; 5592\\
	2000	&amp; 0.274	&amp; 0.278	&amp; 0.293	&amp; 14633\\
	2001	&amp; 0.219	&amp; 0.196	&amp; 0.110	&amp; 8833\\
	2002	&amp; 0.150	&amp; 0.143	&amp; 0.075	&amp; 5785\\
	2003	&amp; 0.127	&amp; 0.131	&amp; 0.053	&amp; 9006\\
	2004	&amp; 0.134	&amp; 0.131	&amp; 0.040	&amp; 4153\\
	2005	&amp; 0.065	&amp; 0.055	&amp; 0.013	&amp; 1291\\
	2006	&amp; 0.051	&amp; 0.061	&amp; 0.053	&amp; 4465\\
	2007	&amp; 0.108	&amp; 0.091	&amp; 0.016	&amp; 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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\begin {center}
\includegraphics[width=5 in]{Agier_SuppFig1.eps}
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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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\includegraphics[width=5 in]{Agier_SuppFig2.eps}
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\end{document}

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