Ashleigh R. Tuite, MSc, MPH; Joseph Tien, PhD; Marisa Eisenberg, PhD; David J.D. Earn, PhD; Junling Ma, PhD; David N. Fisman, MD, MPH
Acknowledgment: The authors thank Dr. Nathaniel Hupert and his colleagues at the U.S. Centers for Disease Control and Prevention Preparedness Modeling Unit for guidance and useful discussions related to the 2010–2011 cholera epidemic in Haiti.
Grant Support: Ms. Tuite and Dr. Fisman are supported by a grant from the Institute of Population and Public Health of the Canadian Institutes of Health Research (PH2-#22228). Dr. Eisenberg is supported by the National Science Foundation (agreement 0635561). Dr. Earn receives support from the Natural Sciences and Engineering Research Council of Canada and the James S. McDonnell Foundation.
Potential Conflicts of Interest: Disclosures can be viewed at www.acponline.org/authors/icmje/ConflictOfInterestForms.do?msNum=M11-0096.
Reproducible Research Statement:Study protocol: Not applicable. Statistical code: Available from Ms. Tuite (e-mail, email@example.com) or Dr. Fisman (e-mail, firstname.lastname@example.org). Data set: Publically available from HealthMap (http://healthmap.org/en/), the MSPP of the Republic of Haiti (2), and the U.S. Centers for Disease Control and Prevention (15).
Requests for Single Reprints: David N. Fisman, MD, MPH, Dalla Lana School of Public Health, 155 College Street, Room 678, Toronto, Ontario M5T 3M7, Canada; e-mail, email@example.com.
Current Author Addresses: Ms. Tuite and Dr. Fisman: Dalla Lana School of Public Health, University of Toronto, 155 College Street, 6th Floor, Toronto, Ontario M5T 3M7, Canada.
Dr. Tien: Department of Mathematics, Ohio State University, Columbus, OH 43210.
Dr. Eisenberg: Mathematical Biosciences Institute, Ohio State University, Jennings Hall, 3rd Floor, 1735 Neil Avenue, Columbus, OH 43210.
Dr. Earn: Department of Mathematics and Statistics and the M.G. DeGroote Institute for Infectious Disease Research, McMaster University, 1280 Main Street West, Hamilton, Ontario L8S 4K1, Canada.
Dr. Ma: Department of Mathematics and Statistics, University of Victoria, Social Sciences and Mathematics Building A425, 3800 Finnerty Road, Victoria, British Columbia V8P 5C2, Canada.
Author Contributions: Conception and design: A.R. Tuite, J. Tien, M. Eisenberg, D.J.D. Earn, D.N. Fisman.
Analysis and interpretation of the data: A.R. Tuite, J. Tien, M. Eisenberg, D.J.D. Earn, J. Ma, D.N. Fisman.
Drafting of the article: A.R. Tuite, D.N. Fisman.
Critical revision of the article for important intellectual content: J. Tien, M. Eisenberg, D.J.D. Earn, D.N. Fisman.
Final approval of the article: D.J.D. Earn, D.N. Fisman.
Statistical expertise: D.N. Fisman.
Administrative, technical, or logistic support: J. Ma.
Collection and assembly of data: A.R. Tuite, J. Tien, M. Eisenberg, D.N. Fisman.
Tuite AR, Tien J, Eisenberg M, Earn DJ, Ma J, Fisman DN. Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions. Ann Intern Med. 2011;154:593-601. doi: 10.7326/0003-4819-154-9-201105030-00334
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Published: Ann Intern Med. 2011;154(9):593-601.
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 (R0, 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.
Haiti is in the midst of a cholera epidemic. Surveillance data to inform public health decision making are limited.
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.
The model does not include the effect of antibiotic treatment on transmission of cholera.
A publicly available tool to assist in managing the cholera epidemic in Haiti has been developed and can be further modified and refined.
Haiti, the poorest country in the Western Hemisphere (1), is in the midst of a cholera epidemic that has reportedly killed more than 4000 people and infected about 217 000 (as of 30 January 2011). Approximately one half of those infected have been hospitalized, and case-fatality rates in both community and hospital settings have been approximately 2% (2). The first laboratory-confirmed case was reported in the department of Artibonite (3–5), but cases have now been reported in all 10 administrative departments in the country. Control measures for the epidemic seem to be slowing transmission of the disease (2). The Pan American Health Organization has suggested that a vaccine campaign may be initiated (6–8), and low-cost approaches to the provision of cleaner water supplies have been advocated (3, 9).
Although epidemiologic surveillance constitutes an important component of the public health response to such an epidemic, publicly available surveillance data from Haiti have been relatively limited to date, whereas time series on hospitalizations and deaths by region are available (2). Epidemic data, combined with geographic and demographic information of a country's administrative regions, can be used to formulate and calibrate models of epidemics. These models can provide valuable insights into the nature of disease spread, help project the time course of the epidemic, and provide guidance on optimal control strategies (10–12).
Viboud and colleagues (13) and others (14) have demonstrated that spatial patterns of epidemic spread can be represented by using “gravity” models that characterize the interaction between geographic regions, with respect to epidemic spread as a function of population mass (13) and distance (13, 14). We used a gravity model to accurately predict the sequence and timing of regional cholera epidemics in Haiti by using publicly available data. We used a model based on the best available data and calibrated to reproduce the initial reported epidemic curve for Haiti and then evaluate the probable time course of Haiti's cholera epidemic in the absence of effective intervention and explore the potential effects of competing and complementary control strategies, including vaccine distribution and provision of clean water.
Information on the timing and magnitude of the cholera epidemic in Haiti was obtained from 3 complementary sources: the HealthMap system (http://healthmap.org/en/), the Ministère de la Santé Publique et de la Population (MSPP) of the Republic of Haiti (2), and a summary report by the U.S. Centers for Disease Control and Prevention (15). The HealthMap system is an automated surveillance platform that continually identifies, characterizes, and maps events of public health and medical importance, including outbreaks and epidemics, on the basis of Internet-based information sources (16). Data inputs for HealthMap include media sources, discussion groups, and government Web sites. As such, HealthMap may be more sensitive than traditional surveillance systems but may also be limited by unfavorable signal–noise ratios (16).
Because regional MSPP data had inconsistencies (for example, cumulative case counts reported at a given date often disagreed with cumulative case counts that were manual tabulated from serial reports), we calibrated a geographically explicit “meta-population” model (17) based on both national estimates of cumulative hospitalized case counts and dates of first reported case in each region.
We built a “susceptible–infectious–recovered” compartmental transmission model that characterized the population as susceptible to infection, infected and infectious to others, or recovered or otherwise “removed” from risk for further infection. We also added a water compartment to the model. The water compartment could be contaminated by infected and infectious persons and could in turn infect susceptible persons. Appendix Figure 1 is a schematic representation of the model. This modeling approach was described by Tien and Earn (18) for the simulation of diseases that can be transmitted through both person-to-person contact and contaminated water, such as cholera.
For each of the 10 departments in Haiti, the population is divided into “susceptible,” “infectious,” and “recovered.” Infection spreads through contact of susceptible with infectious persons both within a given department and in other departments, as well as through contamination of water sources. Additional model details are provided in the Appendix.
We constructed the models for the population of each of Haiti's 10 administrative departments. These populations were combined to form a meta-population model, in which disease could spread both within a given department (or “patch”) or between patches (implicitly reflecting the movement of people from one region to another) (19). The model was constructed by using the Berkeley Madonna software package (developed by Robert Macey and George Oster at the University of California at Berkeley, Berkeley, California). Complete model details and additional information on model parameterization are provided in the Appendix.
The basic reproductive number (R0) is an important index of epidemic potential for a communicable disease (10–12, 20). Existing estimates of R0 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).
Plausible ranges for the environmental survival of 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.
Because cumulative data on case counts for each department were inconsistent, we calibrated our model by modifying the dependence of transmission rate on distance (d−n) to best reproduce the ordering of case appearance by department (Appendix Table 1). We considered values of 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 R0, 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).
Appendix Table 1.
We performed 1000 stochastic simulations, with parameters drawn from uniform distributions for plausible ranges of values to develop projections of time to epidemic peak in each of Haiti's 10 departments.
Recent reports suggested the availability of approximately 1 million doses of a 50% effective vaccine (31) that can be used to control Haiti's cholera epidemic (7, 8), but the timing of their availability is unclear. Because the most widely available vaccine preparation is administered on a 2-dose schedule, with doses administered at least 1 week apart (32), this would constitute vaccination for 500 000 people in Haiti. We introduced vaccination into the model by moving persons from the susceptible compartment to the recovered (immune) compartment at a time representing completion of vaccination. We explored the projected effect of vaccination on total case counts under different assumptions about time to completion of vaccination by using 3 strategies of vaccine distribution: equal distribution to all 10 of Haiti's departments, distribution in proportion to the population, and distribution according to a model-based projection of optimal allocation of vaccine. Optimal allocation was estimated by iteratively minimizing total case counts subject to the constraint of 500 000 available doses, again by using Berkeley Madonna's optimize function. We also explored the relative effect of replacing vaccination with provision of clean water (9, 33) to the same number of people who could be vaccinated and the number of people who would need to receive clean water to have the same effect on epidemic spread as that achievable through vaccination.
We calibrated our base-case model to reproduce initial dynamics of cholera transmission in Haiti, in the absence of effective disease-control interventions. However, the rate of growth in case counts has slowed appreciably over time, which probably reflects the effects of disease-control efforts by governmental and nongovernmental agencies. We evaluated the probable effect of disease-control interventions by refitting our model to incorporate data from January 2011, with decreases in the effective reproductive number simulated by using a simple discount factor (f) (34), such that the reproductive number (R) at time t was taken to be Rt = R0/(1 + f)t.
We received no funding for this study.
Appendix Table 2 includes the best-fit parameter estimates based on 1000 stochastic simulations. The best-fit estimate for R0 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).
Appendix Table 2.
Close correlation is observed between model projections and reported dates of epidemic onset by region. MSPP = Ministère de la Santé Publique et de la Population.
The model was well-calibrated to reported cumulative hospitalizations. MSPP = Ministère de la Santé Publique et de la Population.
Figure 2 presents the projected timing of peaks in the epidemics, based on 1000 stochastic simulations with parameter values drawn from plausible ranges. Without effective intervention, the epidemic in the initially seeded region (Artibonite) was projected to peak at 100 to 150 days after initial seeding. Most other regional epidemics peaked at 200 to 250 days after initial seeding. Epidemics in the Sud, Grand'Anse, and Nippes departments were projected to peak last, nearly 1 year after the initial case in Artibonite. They continued at an incidence of more than 200 cases per 100 000 persons in December 2011 in these regions. An animated representation of the course of regional epidemics, by department, is available online (Supplement).
Timing is based on 1000 stochastic simulations, as described in the text. An increasing intensity of color denotes more simulations projecting that region's epidemic to peak at a given time. The epidemic in Artibonite, home of the initial outbreak, is projected to peak first; epidemics in Grand'Anse, Nippes, and Sud are projected to peak last.
We evaluated the effect of 3 competing strategies for use of 1 million doses of vaccine (sufficient for vaccination of approximately 500 000 persons): equal allocation to each of Haiti's 10 departments; allocation in proportion to the population; and allocation based on optimization, such that the total cases were reduced maximally. Unsurprisingly, for each strategy earlier completion of vaccination was associated with a greater reduction in case counts than was later completion of vaccination, although even vaccination before the onset of the epidemic did not reduce total case counts by more than 3.2% because of the limited quantity of vaccine available. The relative advantage of using an optimized distribution strategy, compared with vaccination based on proportional or equal allocation, increased the later that vaccination occurred, although the relative benefit of vaccination in all strategies diminished with time (Figure 3, 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).
Top. Optimized allocation was always more effective than either equal or proportional allocation of vaccine, although the difference between strategies increased with delay in vaccination. Bottom. Compared with allocation of clean water, vaccination was projected to reduce far more cases of cholera.
Appendix Table 3.
The combination strategy demonstrates a superadditive effect.
Optimal allocation of vaccine focused largely on Ouest department, because of the highly “connected” nature of this department with its large population (focused in Port-au-Prince) and central location, and on Centre department, which serves as a “crossroads” for cholera spread. However, because vaccination is increasingly delayed, larger shares of the vaccine were optimally reallocated to such regions as Nippes and Grand'Anse, which were projected to peak latest (Appendix Figure 5).
Optimized allocation of vaccine focuses largely on Ouest department, because of its large population, and Centre department, which serves as a “crossroads” for cholera spread. As vaccination is increasingly delayed, larger shares are allocated to peripheral regions where epidemics occurred later. Numbers in parentheses represent the ordering of outbreaks.
We calibrated our base-case model to reproduce initial dynamics of cholera transmission in Haiti in the absence of disease-control efforts interventions. Model fits obtained by using a longer time series, for hospitalizations to 14 January 2011, suggested a higher R0 (2.90), with a decrease in R0 by an average of 1.8% per day (Figure 4). By contrasting this epidemic with simulations using an R0 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).
A best-fit model (dotted line) incorporating an R0 of 2.90 and discounted at a rate of 1.8% per day reproduces reported cumulative hospitalizations well. Estimated values for the reproductive number by time are plotted. The reproductive number may have fallen below 1.0 in mid-December; <1.0 is no longer associated with exponential growth (11). The gap between case counts with the discounted model, and models in which the reproductive number (2.78 [base case] and 2.90) is reduced only through depletion of susceptible persons through infection provides an index of the effect of disease-control interventions in Haiti to date. MSPP = Ministère de la Santé Publique et de la Population; R0 = basic reproductive number.
Although cholera is preventable through treatment of sewage and provision of clean drinking water (as are other highly virulent enteric diseases), it continues to take a terrible toll worldwide (35–37). The ongoing cholera epidemic in Haiti is a stark reminder of the degree to which this infrastructure, taken for granted in high-income countries, remains unavailable in most of the world. The 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).
Mathematical models of infectious diseases, such as the one we describe here, represent an idealized or a simplified version of complex disease systems, but they may provide insights into the behavior of these systems and can serve as a tool for decision making under uncertainty (10–12). We constructed a relatively simple patch model representing Haiti, with each department influencing infection risk in persons who reside in other regions in a manner proportionate to the size of the departments in question and inversely proportional to the square-distance between them. We found that such a simple gravity model accurately reproduced both the initial magnitude of the cholera epidemic in Haiti and the timing of onset of regional epidemics. Gravity models have been used to project the movement of influenza in affluent countries, such as the United States and France, and have also been used to model the dynamics of measles and influenza in the United Kingdom in the prevaccination era (39, 40). To our knowledge, our study is the first application of a gravity model of epidemic spread in a low-income country.
On the basis of a model that is well-calibrated to the initial months of the cholera epidemic in Haiti, we can produce plausible projections on the future tempo of this epidemic in the absence of an effective intervention, but we can also quantify the effect of disease-control interventions in Haiti. Although any attempt to precisely predict the future with a mathematical model should be viewed with caution, we can qualitatively project with confidence that the cholera epidemic in Haiti is likely to continue for several months and, without effective intervention, has not yet peaked in the last-affected departments. However, deviation of the epidemic curve from initial model projections suggests that interventions on the ground are changing the dynamics of this disease and have probably already averted a substantial burden of morbidity and mortality.
Both vaccination and provision of clean water should mitigate this epidemic; however, our projections suggest that the limited number of vaccine doses available, combined with the substantial time to distribute the vaccine in a country with limited transportation infrastructure and health care resources, will make the effect of vaccination fairly modest. However, given the potential for thousands of additional cholera cases in Haiti and the high (>1%) case-fatality rate associated with cholera in Haiti, a reduction in total case counts of even a few percent will translate into a substantial number of lives saved. Furthermore, the transmissibility of cholera in Haiti (as assessed by the reproductive number) has probably been reduced substantially by ongoing disease-control efforts. The relative effect of a given level of vaccine coverage will be higher if the reproductive number is lower (11, 12), and as such, the effect of vaccination may be greater than projected in our analysis.
Using our model as a platform to evaluate optimal strategies for distributing the vaccine, we project that, regardless of vaccination timing, optimized distribution is more effective than equal distribution to all departments or distribution based on population. Although an optimized strategy favors provision of most vaccine doses to the populous central regions of Haiti, as the epidemic progresses more vaccine should be reallocated to regions of the country that are “weakly connected” and consequently have late-onset epidemics that have not peaked.
We also evaluated the relative effect of provision of clean, safe water to all people who might be reached by vaccination. Although universal access to clean water would probably have a major effect on cholera dynamics, clean water distributed to a relatively small subset of the population had a much smaller effect on case counts than did vaccination of the same number of people. Although this may seem counterintuitive, the result would be expected on the basis of the important role that susceptible persons play in perpetuating epidemics: People who have clean water are still vulnerable to infection through other routes, such as person-to-person transmission, whereas vaccinated persons are effectively removed both as potential cholera recipients and as sources of cholera for others.
As with any mathematical model, ours is a simplified representation of reality and is limited by the nongranular nature of the data available for model parameterization in this particular outbreak. Nonetheless, we believe that model credibility is strengthened by excellent calibration, with respect to both epidemic timing and national cumulative hospitalizations. Our treatment of vaccination as a “step function” is probably at odds with the practical realities of vaccine distribution, but nonetheless should provide qualitative insights into the relative value of vaccination strategies and approaches to targeting these limited resources. Furthermore, we have not considered antimicrobial drug use as a component of cholera-control interventions; limited data suggest that such antimicrobials as azithromycin could be used to decrease the duration of infection in patients with cholera (41). Our model does, however, provide a platform for the exploration of such an approach.
Our projections are the results of a model calibrated to 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.
In conclusion, a relatively simple patch dynamic transmission model of cholera that explicitly represents the “gravitational pull” of larger regions and shorter between-region distances in Haiti reproduces the dynamics of the current epidemic in Haiti. The model projects that the epidemic is likely to last well into 2011 and suggests that adaptive strategies for vaccination may provide a modest reduction in morbidity and mortality in the economically challenged country. Although the model's projections are not substitutes for empirical study of disease dynamics, we hope they may spur the international community to provide the additional logistic, economic, and political support that is needed to quell this epidemic.
We built a “susceptible–infectious–recovered” compartmental transmission model that characterized the population as susceptible to infection, infected and infectious to others, or recovered or otherwise “removed” from risk for further infection. We also added a water compartment to the model. The water compartment could be contaminated by infected and infectious persons and could in turn infect susceptible persons. Appendix Figure 1 presents a schematic representation of the model. This model approach was described by Tien and Earn (18) for the simulation of diseases that can be transmitted through both person-to-person contact and contaminated water, such as cholera.
We constructed the models for the population of each of Haiti's 10 administrative departments. These populations were combined to form a meta-population model, in which disease could spread both within a given department (or patch) or between patches (reflecting the movement of people from one region to another) (19). The model was constructed by using ordinary differential equations, such that for the ith patch (with i from 1 to 10):
Here, si, xi, and ri are the proportions of the population in the susceptible, infectious, and recovered (immune) or removed (through cholera-specific mortality) states, respectively, and wi is the concentration of 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.
The force of infection (λ) in the ith patch is:
βXi and βwi represent the transmission rate from infectious persons and water in the ith patch to susceptible persons in the ith patch, and [thgr ]ij represents the influence of infection prevalence in the jth patch on incidence in the ith patch, according to the relation:
Here, κ is a scalar multiplied by the gravity matrix representing between-patch coupling, pi and pj are population sizes, d is the distance between 2 patches, and n is a power that determines the strength of the dependence of transmission rate on distance.
The R0 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:
We assumed that cholera could be transmitted through either contaminated water or close contact but that waterborne transmission was a far more important method of transmission. Because both βI and µ are small relative to βw and γ:
We introduced vaccination into the model by moving persons from the susceptible compartment to the recovered (immune) compartment when vaccination was completed. We assumed that the vaccine was administered equally across the population in a given department (that is, we did not preferentially allocate vaccine to susceptible persons, but vaccination had no effect on infectious or recovered persons):
Here, vi is the probability of vaccination within a department.
We also explored the relative effect of replacing vaccination with provision of clean water to the same number of people who could be vaccinated and the number of people who would need to receive clean water to have the same effect on epidemic spread as that achievable through vaccination. We assumed that clean water was introduced at the same time as vaccination and continued for the model run (up to 2 years). Provision of clean water reduced the number of persons who were susceptible to infection through contaminated water, without changing their susceptibility to infection through person-to-person transmission, such that:
Here, ci 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.
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