Weekly cholera morbidity and mortality data for each city neighborhood and outlying communities were obtained from datasets compiled by contemporary physicians conducting active surveillance during the 1853 cholera epidemic in Copenhagen [22]. A corps of physicians traveled door-to-door to residences, hospitals (“sick-houses”), and “poor-houses” diagnosing and tabulating cases. Hospitalized cases were assigned to the neighborhood of residence, unless they were already in hospital prior to diagnosis, in which case they were geolocated to the hospital’s address. Six of the 13 neighborhoods were aggregated into two neighborhoods labeled “Combined upper” (Nørre and Klædebo) and “Combined lower” (Frimands, Strand, Snarens and Vester) due to low case counts and small geographic area. We excluded 621 (9%) cases that could not be geolocated, consisting of 122 (2%) cases from docked ships, and 499 (7%) from scattered outlying communities.

Cholera cases were defined as patients with rice-water diarrhea and evidence of severe dehydration [23], making the historical diagnostic criteria stricter than the current WHO suspected cholera case definition [24]. Population size for each neighborhood was interpolated between the 1850 and 1855 census assuming a linear growth model. Population density was estimated for each neighborhood by georeferencing the neighborhoods using a Geographic Information System (GIS) and calculating area. Hydraulic data was digitized from a contemporary map [21] showing the layout and direction of flow of all water-pipes supplying drinking water for the city (Fig 1A). We created a binary asymmetric transition matrix of hydraulic connectivity describing the flow of water between neighborhoods, and a binary symmetric matrix describing neighborhoods that share a border (as a proxy for sewage runoff and human connectivity) (S1 Table and S2 Table). All GIS work was done in QGIS [25].

To model the number of infectious people at each day, we fit a discrete time susceptible-infected-recovered (SIR) model to imputed daily case data We used the basic model structure from Azman et al. [17], with modifications as described below. In order to simulate daily case data (necessary because of the short generation interval for cholera), weekly cases were randomly reallocated to the seven days preceding the reported date. This reallocation was repeated 10 times to give 10 possible realizations of the epidemic. Each of the nine neighborhoods were considered as a discrete population. The model-predicted number of infected people in each neighborhood at each time-step was Poisson distributed where the mean was a function of the fraction susceptible in that neighborhood and a sum of the internal and external forces of infection.

We constructed a series of nested models in which the infection rate of susceptible individuals living in neighborhood i, or the force of infection, λ_i, was the sum of an internal force of infection, β_i, and an external force of infection from neighborhood j upon neighborhood i, α_j,i (Fig 2), such that: λi=βiIi+∑j≠iαj,iIj

Each model had differing assumptions about α and β. Starting from the simplest base-case model to the most complex saturated model, we allowed: (1) a single β and single α for all neighborhoods such that β_i = β and α_j,i = α, (2) an individual β_i for each neighborhood and a single α for all neighborhoods such that α_j,i = α, and (3) an individual β_i for each neighborhood and an asymmetric α_j,i such that α_j,i ≠ α_i,j (S1 Text).

In order to estimate the effect of the water supply on the epidemic, we used two methods. First, we fit a linear regression model using the log of the median cross-neighborhood transmission coefficients (α_j,i) from model 3 (saturated model) as the outcome and tested whether the hydraulic-transition and geographic-proximity variables were significant predictors of between-neighborhood transmission at a significance level of 0.05 (S1 Text, S1 Table and S2 Table). Second, we incorporated the hydraulic-connectivity and geographic-proximity matrices into model 2, producing models 2b and 2c, respectively. To assess the effect of hydraulic connectivity (model 2b), we allowed α_j,i to vary depending on whether the neighborhoods were connected via water pipes, such that: αj,i={α0ifnowaterconnectionj→iα0+α1ifwaterconnectionj→i

To incorporate geographic proximity (model 2c), we added an additional term if the neighborhoods shared a border. The resulting α_j,i can be described as: αj,i={α0ifnosharedborderorwaterconnectionj→iα0+α1ifnosharedborderbutwaterconnectionj→iα0+α1+α2ifsharedborderandwaterconnectionj→i

Parameter estimation was done using Markov chain Monte Carlo (MCMC) methods in JAGS 4.2.0 [26] in R 3.3.1 [27]. Each model was run once on each of the 10 possible realizations of the imputed daily epidemic data. We used four MCMC chains with a burn-in of 40,000 iterations and sampled the subsequent 60,000 iterations. Chain convergence was assessed with a potential scale reduction factor (R^) cutoff of 1.05 and a visual assessment of the trace plots. Posterior samples from all four chains and all 10 data realizations were pooled and summarized (S1 Text). Model preference was assessed using the average Watanabe-Akaike Information Criterion (WAIC); a difference of five in WAIC was considered significant, translating to a fold difference of e^5/2 = 12.2 in terms of posterior probability on the model state space [28]. WAIC was chosen due to recent developments in this field suggesting WAIC is better able to capture the tradeoffs between model complexity and fit in a Bayesian framework than the more commonly used DIC [29,30].

We validated our model by simulating the outbreak in each neighborhood. To do so, we selected a parameter set from the joint posterior distribution and then drew from the number of cases at the next time step from the appropriate Poisson distribution in each neighborhood. Point-wise prediction intervals were constructed by taking the 2.5 and 97.5 percentiles at each time-step. We then used the simulated data to refit the model parameters to test if the model could recapture the original parameter estimates.

To assess the heterogeneity of transmission efficiency by neighborhood, we calculated the internal reproductive number (R_int), the outflowing reproductive number (R_out), the inflowing reproductive number (R_in), and the total reproductive number (R_tot) within each neighborhood using the force of infection coefficients. The R_int, R_out, and R_tot can be interpreted as the number of cases a single infectious case produces within its own neighborhood, in all other neighborhoods, and in the whole city respectively, while the R_in represents the number of cases produced within one neighborhood by a single infectious case in the rest of the city.

The epidemic began on June 12 with two reported cases in the Nyboder neighborhood among individuals working on (or in close contact with) ships. The epidemic soon spread to other neighborhoods (Fig 3A and 3B, S1 Dataset). The epidemic was declared over on October 1, although four cases were reported subsequently in October. A total of 7,173 cases were reported, of which 5,953 (83%) were community acquired and 1,220 (17%) were acquired in hospitals or poor-houses. A total of 4,717 died, resulting in a case fatality ratio (CFR) of 66%. The CFR among hospital/poor-house-acquired infections was 77%, as compared to 63% in community-acquired infections. A total of 6,552 cases (91%) occurred within the city walls and were included in this analysis.

The outbreak was spatially heterogeneous over the city, despite its small geographic size. Attack rates within the city walls ranged from 2.1 to 9.6 per 100, and the CFR ranged from 76% to 54% (Table 1, Fig 1C and 1D). The neighborhood attack rates were not associated with the neighborhood’s population density as assessed by a linear regression (p = 0.99).

Model selection carried out using WAIC (Table 2) indicated the best model allowed for asymmetric and heterogeneous transmission between neighborhoods (model 3); this heterogeneity could not be explained by hydraulic and geographic connectivity (model 2b and 2c). Similarly, the linear regression on the median of the cross-neighborhood transmission coefficients (α_j,i) (Fig 4) from the fully saturated model (model 3) provided no support that a hydraulic connection between neighborhoods was significantly associated with the force of transmission between neighborhoods, even after controlling for geographic proximity (Table 3). The internal forces of transmission in each neighborhood were not strongly associated with the neighborhood’s population density (p = 0.11).

Using the fully saturated model (model 3), we simulated the outbreak in each neighborhood one day ahead, drawing a new parameter combination from the joint posterior distribution for each simulation (Fig 5). The R_int ranged from 0.2 (0.0–0.5) in Østerto to 1.7 (1.5–1.8) in St. Annæ Vester (Fig 6). One of the nine neighborhoods (St. Annæ Vester), had R_int >1, meaning it could maintain epidemics without infections from outside; in three other neighborhoods (St. Annæ Øster, Christianshavn and Nyboder), R_int was above 1 but the credible interval spanned 1.0, and in all remaining neighborhoods the R_int was below 1.0.

To validate the model, we re-estimated the model parameters from the simulated data for each of the 10 epidemic realizations. In seven of the 10 realizations, the 95% CI estimated from the simulated data overlapped 95% CI fitted from the original data for all 83 parameters. In the remaining three realizations overlap occurred in 82, 81 and 80 parameters respectively.