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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 \\
\\
%\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.\\
\\
\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.\\
\\
\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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