In both the neighbourhood and mean-field models, the mean susceptible density S falls

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1 Supplementary Information Epidemic cholera spreads like wildfire Manojit Roy 1,2, Richard D. Zinck 1+, Menno J. Bouma 3, Mercedes Pascual 1,2 Supplementary note on model behaviour near 2 nd order criticality: In both the neighbourhood and mean-field models, the mean susceptible density S falls sharply from S = 1 across the critical point σ = 1 (Supplementary Fig. S1a). For the neighbourhood model, though, S quickly settles on to the SOC density S 0.4, whereas the mean-field density drops to a much lower value S 0.16 because of the much larger outbreaks generated in the latter model. The probability of a spanning cluster of connected susceptible cells, which determines the system-wide connecitivity of susceptible cells and therefore the spatial spread of infection and ultimately epidemic size, also drops sharply from 1 to 0 across the critical point for both models (Supplementary Fig. S1b). The coefficient of variation (CV) in outbreak size, on the other hand, exhibits a sharp peak at the critical point that rapidly dwindles away on either side (Supplementary Fig. S1c), more so for the meanfield model in the super-critical regime (σ > 1) because fat-tailed distributions in the neighbourhood model for all σ (Fig. 3) contribute to large size fluctuations and thus keep CV at a higher level. These properties at σ = 1 are characteristics of a 2 nd order phase transition 1, especially the divergence of size fluctuations. Their rapid increase suggests that intervention measures devised to gradually push the system towards the elimination boundary σ(r 0 ) 1 may have the unintended consequence of triggering explosive outbreaks as the target nears. A pre-emptive strategy such as mass vaccination that lowers the susceptible pool and thus rapidly and drastically reduces R 0 below 1 in between epidemics, would effectively reduce the susceptible population and take the system to the subcritical regime without experiencing the dynamics close to this boundary. The non-monotonic pattern of variation in the power-law exponent α with changes in σ in the neighbourhood model (Fig. 3, bottom panel) suggests the existence of a possible indicator of the approach to a tipping point 13 near the elimination boundary. As σ is lowered from 1, 1

2 at about σ 1.3 the slope falls below the critical slope of α = 1.5, and continues until it reaches a value α 1.7 at σ 1.04 (the tipping point), at which point it reverses course and rises towards the critical slope at σ = 1. Thus, the location of the first crossing over the critical slope, at σ 1.3, could provide such an indicator. Supplementary References: 1 Ma, S. K. Modern theory of cricial phenomena. (Benjamin, London, 1976). 2

3 Supplementary figure and table captions: Figure S1: The critical transition at σ = 1 in the neighbourhood (red line) and meanfield (blue line) models are compared. a, Average susceptible density S falls sharply from S = 1 across the critical point σ = 1 for both models. b, The susceptible spanning probability P drops from 1 to 0 across the critical point. c, The coefficient of variation (CV) of outbreak sizes exhibits a sharp peak at the critical point for both models. At the super-critical regime, CV for the neighbourhood model remains higher than that for the mean-field model. Figure S2: The size distributions for the neighbourhood model do not depend on neighbourhood size. a-f, Comparison of size distributions for the neighbourhood model with 4- and 8-neighbourhoods (red and blue circles respectively), at σ = 1, 1.04, 1.1, 1.5, 10 and, shows that the neighbourhood size has negligible impact on the shape of the distribution. Figure S3: Geographic maps are shown for the data locations. The top row shows the 5 groupings of countries in Africa chosen to aggregate the case data regionally (Methods). The middle row shows the 5 groupings of districts in the state of Punjab (former British India) chosen to aggregate the mortality data (Methods). The bottom row shows the individual districts of the state of Assam (India). These maps were produced by modifying with open source programs (inkscape and gimp) the respective shape files available in Wikipedia ( for Africa and for Assam, both under the Creative Commons License and for Punjab, an image in the public domain). Figure S4: Likelihood profiling for sigma shows a well defined lower bound but a flat confidence interval to the right, and therefore, a poorly defined upper bound and maximum. A representative profile likelihood is shown here for Sibsagar district in Assam (India), which gives a 95% C.I. of (2.2, ) (see Methods). Figure S5: The length of the fade-out intervals also shows a fat-tailed distribution. A representative plot of the frequency distribution n(t) of inter-epidemic intervals T is shown here for Region 1 in Punjab. 3

4 Figure S6: Size distribution in the sub-critical regime has an exponentially decreasing shape. The exponential size distribution at σ = 0.85 (sub-critical, in blue) shows a characteristic convex shape on the log-log scale, and is compared here to the linear shape for the power-law distribution at the critical point σ = 1 (in red). Figure S7: The size distribution does not depend on the area or type of event aggregation. The original distribution plot for the Region 1 of Punjab (in red) is compared with a representative plot obtained by aggregating five random districts Attock, Mooltan, Sheikhupura, Kangra and Simla (in blue), and also with a representative plot for the single district of Jhelum (in green). All three plots exhibit a similar distribution, indicating the absence of a significant bias due to the aggregation procedure. Table S1: Maximum likelihood estimates (MLE) and their 95% C.I. are shown for the parameter σ. The MLE values of σ for all locations (column 2) are significantly higher than 1. Because the maximum likelihood function varies little and remains flat to the right of these estimates, their confidence intervals (column 3) are broad on the right hand side. This means that only the lower bounds can be estimated meaningfully (shown in fig. 2). These lower bounds are of relevance to determine whether the dynamics are in the super-critical regime and bounded away from the critical point. Columns 4 and 5 show the estimated value and 95% C.I. of the other parameter θ. (The C.I. upper bound is limited by the largest value of θ used for the model simulation, which is 1500). 4

5 Supplementary Table S1 Data location MLE σ C.I. (σ) MLE θ C.I. (θ) Sibsagar (Assam) 100 (2.2, ) 1000 (700,>1500) Darrang (Assam) 2.33 (1.5, ) 1000 (500,>1500) Goalpara (Assam) 50 (4.8, ) 700 (420,1200) Lakhimpur (Assam) 50 (3.3, ) 680 (295,>1500) Nowogong (Assam) 50 (4.4, ) 480 (295,1030) All (Assam) 100 (10.5, ) 980 (720,>1500) Group1 (Africa) 100 (23.4, ) 900 (753,>1500) Group2 (Africa) 100 (14, ) 500 (309,670) Group3 (Africa) 100 (13.8, ) 1000 (800,>1500) Group4 (Africa) 100 (64, ) 620 (420,890) Group5 (Africa) 100 (18, ) 900 (710,>1500) All (Africa) 100 (50, ) 620 (780,>1500) Region1 (Punjab) 100 (8, ) 120 (96,209) Region2 (Punjab) 100 (10.8, ) 90 (66,130) Region3 (Punjab) 100 (19, ) 160 (124,206) Region4 (Punjab) 100 (14.1, ) 180 (136,235) Region5 (Punjab) 100 (11.2, ) 160 (108,232) All (Punjab) 100 (35, ) 160 (132,173) 5

6 Supplementary Figure S1 a b S σ Neighbourhood Mean field P(span) c σ CV σ 6

7 Supplementary Figure S2 a b c n(s) σ = 1 4 neighbour 8 neighbour d σ = 1.04 e σ = 1.1 f n(s) σ = 1.5 σ = size s size s σ = (SOC) size s 7

8 Supplementary Figure S3 Africa (Group 1) Africa (Group 2) Africa (Group 3) Africa (Group 4) Africa (Group 5) Punjab (Region 1) Punjab (Region 2) Punjab (Region 3) Punjab (Region 4) Punjab (Region 5) Assam (Sibsagar) Assam (Nowogong) Assam (Lakhimpur) Assam (Goalpara) Assam (Darrang) 8

9 Supplementary Figure S4 log likelihood σ 9

10 Supplementary Figure S5 frequency n(t) 5e 05 5e 04 5e 03 5e fade out interval size T 10

11 Supplementary Figure S6 n(s) σ = 0.85 σ = size s 11

12 Supplementary Figure S7 n(s) 1e 06 1e 04 1e 02 Region 1 (Punjab) Random Grouping Single District (Jhelum) s 12