## Abstract

Haiti is in the midst of a cholera epidemic. Surveillance data for formulating models of the epidemic are limited, but such models can aid understanding of epidemic processes and help define control strategies.

To predict, by using a mathematical model, the sequence and timing of regional cholera epidemics in Haiti and explore the potential effects of disease-control strategies.

Compartmental mathematical model allowing person-to-person and waterborne transmission of cholera. Within- and between-region epidemic spread was modeled, with the latter dependent on population sizes and distance between regional centroids (a “gravity” model).

Haiti, 2010 to 2011.

Haitian hospitalization data, 2009 census data, literature-derived parameter values, and model calibration.

Dates of epidemic onset and hospitalizations.

The plausible range for cholera's basic reproductive number (R_{0}, defined as the number of secondary cases per primary case in a susceptible population without intervention) was 2.06 to 2.78. The order and timing of regional cholera outbreaks predicted by the gravity model were closely correlated with empirical observations. Analysis of changes in disease dynamics over time suggests that public health interventions have substantially affected this epidemic. A limited vaccine supply provided late in the epidemic was projected to have a modest effect.

Assumptions were simplified, which was necessary for modeling. Projections are based on the initial dynamics of the epidemic, which may change.

Despite limited surveillance data from the cholera epidemic in Haiti, a model simulating between-region disease transmission according to population and distance closely reproduces reported disease patterns. This model is a tool that planners, policymakers, and medical personnel seeking to manage the epidemic could use immediately.

None.

### Editors' Notes

#### Context

Haiti is in the midst of a cholera epidemic. Surveillance data to inform public health decision making are limited.

#### Contribution

The authors constructed a mathematical model of epidemic dynamics that is based on both population and distance. The model's results closely match the contour of the epidemic to date. The model was used to project the probable effect of different approaches to allocation of vaccines and clean water on the course of the epidemic.

#### Caution

The model does not include the effect of antibiotic treatment on transmission of cholera.

#### Implication

A publicly available tool to assist in managing the cholera epidemic in Haiti has been developed and can be further modified and refined.

## Methods

### Data Sources

### Model

### Model Parameterization

_{0}) is an important index of epidemic potential for a communicable disease (10–12, 20). Existing estimates of R

_{0}for cholera vary widely (21–23). We obtained seed estimates from the published biomedical literature of the duration of infectiousness in patients with cholera and durability of

*Vibrio cholerae*in source water (18). We assumed that cholera could be transmitted through either contaminated water or close contact but that waterborne transmission (or consumption of food items contaminated with infective water) was a far more important method of transmission (24–26). Most documented cases of person-to-person transmission occur in households, usually because of infection in persons involved in food preparation (27, 28).

*V. cholerae*were derived from studies of bacterial survival in sediments (29). Population sizes were obtained from 2009 Haitian census data (30). We used centroids for each department's capital city to calculate straight-line distances between patches (Appendix Figure 2). To account for the shape of the country, we assumed that Haiti was divided into 2 zones when calculating distances: the lower zone included Grand'Anse, Sud, Nippes, and Sud-Est, and the upper zone included Nord-Ouest, Nord, Nord-Est, Artibonite, and Centre. Ouest was treated as a conduit between the 2 zones, such that the distance between 2 departments in different zones was calculated as the sum of the distance between the department in zone 1 and Ouest and between Ouest and the department in zone 2. For example, the distance between Grand'Anse and Centre was the sum of the distance between the capital of Grand'Anse and the capital of Ouest plus the distance between the capital of Ouest and the capital of Centre.

### Model Calibration

*d*) to best reproduce the ordering of case appearance by department (Appendix Table 1). We considered values of

^{−}^{n}*n*that ranged from 0.8 to 3.0. Goodness of ordering was evaluated by calculating Spearman correlation coefficients for observed versus expected order of first case appearance in the 9 non-Artibonite departments. Ordering was optimized for

*n*ranging from 2.0 to 2.4 (and was identical for

*n*in this range). For simplicity, we used an

*n*of 2.0. We subsequently optimized model fit through iterative adjustment of components of R

_{0}, as well as κ (the “gravitational constant”) and [xgr ] (the pathogen decay rate in water) to minimize the least-squares distance between overall model hospitalization estimates and those reported by MSPP, by using the optimize function of the Berkeley Madonna software package. We estimated credible intervals for model parameters by iteratively refitting our multipatch susceptible–infectious–water–recovered model, assuming that errors in case counts were Poisson-distributed (20).

### Optimization of Control Strategies

### Changing Dynamics of Infection Over Time

*f*) (34), such that the reproductive number (

*R*) at time

*t*was taken to be R

_{t}= R

_{0}/(1 +

*f*)

^{t}.

### Role of the Funding Source

## Results

### Model Parameterization and Calibration

_{0}for cholera was approximately 2.78 (plausible range, 2.06 to 2.78), which is within the range generated by our group and others (21–23). Using an inverse square-distance term in our gravity model combined with best-fit parameters, we found close correlation between model projections of ordering initial cases and initial case dates by department and between those reported by the MSPP and the U.S. Centers for Disease Control and Prevention (15) (Spearman correlation coefficient for ordering, 0.97 [

*P*< 0.001]; Spearman correlation coefficient for case dates, 0.92 [

*P*< 0.001]) (Figure 1). Cumulative hospitalized case counts projected by the model agreed closely with those reported by the MSPP (Appendix Figure 3).

### Estimated Time to Peak and Duration of Epidemic Under Initial Conditions

**Supplement**).

### Optimal Intervention Strategies

*top*). Nonetheless, the benefit associated with vaccination was far greater than that of allocation of clean water (Figure 3,

*bottom*). We estimated that 1.7 to 2.0 times as many people would need to be given access to clean water to achieve an equal reduction in cholera cases that can be achieved through optimal allocation of vaccine (Appendix Table 3). The combination of clean water and vaccination was projected to have a superadditive effect (projected reduction in cases resulting from both interventions was greater than the sum of the effects of individual interventions) (Appendix Figure 4).

### Changing Dynamics of Infection Over Time

_{0}(2.90), with a decrease in R

_{0}by an average of 1.8% per day (Figure 4). By contrasting this epidemic with simulations using an R

_{0}of 2.78 (as in the base case) or 2.90, we estimate that disease-control interventions have probably prevented thousands of cases of cholera (as of February 2011).

## Discussion

*V. cholerae*strain responsible for the current epidemic was presumably introduced by a person or persons visiting Haiti from South Asia (38). Regardless of the route of introduction, the limited availability of clean drinking water and sewage treatment in Haiti have permitted the rapid emergence of a cholera epidemic that is now active in all regions of the country. As of February 2011, more than 4000 cholera-related deaths have been recorded (2).

*initial*epidemic dynamics in the absence of an intervention; the animated representation of the course of regional epidemics (

**Supplement**) ignores control measures, which are likely to reduce the severity of regional epidemics (and may already be doing so, thanks to the hard work and dedication of government health workers and local and international relief workers). Our subsequent refitted model suggests that these efforts have resulted in a substantial decrease in the transmissibility of cholera in Haiti.

## References

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### Appendix

#### Model Details

*i*th patch (with

*i*from 1 to 10):

*s*,

_{i}*x*, and

_{i}*r*are the proportions of the population in the susceptible, infectious, and recovered (immune) or removed (through cholera-specific mortality) states, respectively, and

_{i}*w*is the concentration of

_{i}*V. cholerae*in local waters, rescaled by using the parameter [xgr ] to reflect the burden of infection in the population. Rescaling is described in detail by Tien and Earn (18). The parameter γ is the inverse of the mean duration of infectiousness (that is, the recovery rate in persons who survive and time to death in those who do not), and µ is the birth rate and noncholera mortality rate, which can effectively be ignored on the short time frame associated with this epidemic.

*i*th patch is:

*and β*

_{Xi}*represent the transmission rate from infectious persons and water in the*

_{wi}*i*th patch to susceptible persons in the

*i*th patch, and [thgr ]

*represents the influence of infection prevalence in the*

_{ij}*j*th patch on incidence in the

*i*th patch, according to the relation:

*p*and

_{i}*p*are population sizes,

_{j}*d*is the distance between 2 patches, and

*n*is a power that determines the strength of the dependence of transmission rate on distance.

#### Model Parameterization

_{0}for cholera can be expressed in terms of infectiousness of water sources and infectiousness of person-to-person contact, and it is inversely related to the rate of recovery from disease and the all-cause mortality rate in the population (18), such that for a single-patch model:

*and µ are small relative to β*

_{I}*and γ:*

_{w}#### Optimization of Control Strategies

##### Vaccination

*v*is the probability of vaccination within a department.

_{i}##### Provision of Clean Water

*c*is the probability of provision of clean water within a department. As with vaccination, there was no preferential provision of clean water to susceptible persons.

_{i}
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