 Analytic perspective
 Open Access
Cholera transmission dynamic models for public health practitioners
 Isaac ChunHai Fung^{1, 2}Email author
https://doi.org/10.1186/17427622111
© Fung; licensee BioMed Central Ltd. 2014
 Received: 9 May 2013
 Accepted: 22 January 2014
 Published: 12 February 2014
Abstract
Great progress has been made in mathematical models of cholera transmission dynamics in recent years. However, little impact, if any, has been made by models upon public health decisionmaking and daytoday routine of epidemiologists. This paper provides a brief introduction to the basics of ordinary differential equation models of cholera transmission dynamics. We discuss a basic model adapted from Codeço (2001), and how it can be modified to incorporate different hypotheses, including the importance of asymptomatic or inapparent infections, and hyperinfectious V. cholerae and humantohuman transmission. We highlight three important challenges of cholera models: (1) model misspecification and parameter uncertainty, (2) modeling the impact of water, sanitation and hygiene interventions and (3) model structure. We use published models, especially those related to the 2010 Haitian outbreak as examples. We emphasize that the choice of models should be dictated by the research questions in mind. More collaboration is needed between policymakers, epidemiologists and modelers in public health.
Keywords
 Cholera
 Susceptible Population
 Basic Reproduction Number
 Model Misspecification
 Public Health Practitioner
Introduction
Since the 19th century, humans have experienced seven cholera pandemics. The seventh pandemic started in Indonesia in 1961 and continues to threaten vulnerable populations globally [1]. The cholera outbreak that began in October 2010 in Haiti, where cholera had been absent for a century, reminds us the importance of timely cholera prevention, treatment and control and the critical importance of water and sanitation infrastructure that has eliminated cholera from much of the developed world [2].
To better understand cholera epidemiology retrospectively and to predict the impact of interventions in the future, many researchers have begun using mathematical models as tools complementary to field epidemiology and statistical analysis. Mathematical models help us conceptualize the transmission dynamics in a quantitative way and allow us to test different hypotheses and understand their relative importance in silico. Important epidemiological observations and hypotheses for cholera have been modeled; examples include estimation of the basic reproduction number (R_{0}) [3, 4], seasonal variation in cholera incidence [5], inapparent cholera infections [6], hyperinfectivity of V. cholerae[7], humantohuman transmission [8], and the role of human mobility and river networks in transmission [5, 9]. Mathematical models also allow us to prospectively estimate the impact of various interventions, from treatment (oral rehydration therapy and antibiotics) to prevention (oral cholera vaccine (OCV), and water, sanitation and hygiene (WASH) interventions (e.g. [9–12])).
The purpose of this paper is to introduce cholera dynamic transmission models to public health practitioners, with an educational emphasis of conveying modeling concepts to students of these models. Models are simple, but not simplistic representations of the real world. They are used to capture the “essence” of a complex phenomenon. Models may help us better understand the relationship between different parts of the system. Some models may shed light on past epidemics while some may help us forecast the future. Here we define dynamic transmission models as models that explicitly simulate the transmission dynamics of infectious diseases in time. This paper will focus on the ordinary differential equation (ODE) models (populationbased continuoustime models as contrast to populationbased discretetime models using difference equations), while we will mention relevant agentbased models where appropriate (e.g. [9]).
Through a basic model, we will explain the major parameters and how interventions may change them. We will discuss how different assumptions and hypotheses can be accommodated by making changes to the model’s structure. Focus is given to the way different research questions dictate the model structure. Published models were chosen as illustrations and the list is not meant to be exhaustive. Priority is given to papers that model specifically the 2010 Haiti cholera epidemic. Instead of being a systematic review of all existing cholera models, my aim is to highlight three current major challenges of modeling efforts of cholera transmission dynamics: (1) parameter uncertainty and model misspecification; (2) interventions (especially, water, sanitation and hygiene); and (3) model structure. Spatial and climatic elements are also important features but they are beyond the scope of this paper (they are briefly discussed in the Additional file 1). For a detailed review of the recent cholera modeling literature, please refer to ref. [13].
Some basic concepts
First, let us review some basic concepts. In an ODE model of infectious diseases, we divide the population into a number of compartments. For example, in a SusceptibleInfectedRecovered (SIR) model, the population is divided into three compartments depending on their status of being susceptible to the infection (S), being infected and infectious (I), and having recovered from the infection (R). Individuals in each compartment were assumed to be homogeneously mixing with each other [14]. The ODEs of the model capture the change of the number of individuals in each compartment over continuous time. While ODE models have their own set of assumptions and limitations, they are commonly used in epidemiologic modeling because we can use a few equations to represent the transmission dynamics and create an easytounderstand model for public health practice.
The basic reproduction number, R_{0}, is usually defined as the number of individuals that an infected (and infectious) individual can infect when he or she is introduced into a completely susceptible population. For example, for a disease with R_{0} = 2, an infected individual on average infects two individuals in a totally susceptible population. The effective reproduction number, R or R_{E}, is defined as the number of individuals infected by a typical infectious individual when a fraction of the population is protected from infection through immunity, prophylaxis or nonpharmaceutical interventions [15]. For example, for a disease with R_{0} = 2, and if half of the population is immune to this disease, R_{E} = R_{0} * ½ = 1.
ODE models can be programmed in computers using different languages, software and platforms, for example, C, C++, Matlab, Mathematica, R, and Berkeley Madonna. For further details of these models, public health students of mathematical modeling may refer to general modeling texts, for example, Anderson and May [16], Cummings and Lessler [14], Keeling and Rohani [17], and Vynnycky and White [15].
The basic model
Following the example of Grad et al. [18], we adapt the model of Codeço [19] as our basic model through which we explain how the transmission dynamics of cholera is modeled mathematically.
Figure 1 presents a schematic of the basic model. The black boxes represent people: susceptible (S in equations in “The basic model” in Additional file 1); infectious (I); and recovered (R). The blue circle represents cholera bacterial concentration in the water reservoir (B).

Black arrows: Susceptible people become infected/infectious and they later recover and become immune.

Blue arrows: Infectious people contaminate the water supply with bacteria and the bacteria decay.

Red arrow: Susceptible people are exposed to contaminated water and may become infected.

Gray arrows: People are born into the susceptible population; they may die as a result of cholera infection or other reasons.
Please refer to the Additional file 1 for the equations and explanations of the variable and parameters. For models that simulate an outbreak within a short period of time (e.g. one year), one can ignore the dynamics of population growth (birth rate and death rate, gray arrows) and assume a constant population.
 1.
Infected individuals are infectious and contribute to bacteria shedding, which imply that asymptomatic individuals contribute as much bacteria to the water supply as symptomatic individuals.
 2.
Immunity obtained through infection lasts longer than the timeframe studied by the model (for example, 1 year).
These assumptions will be relaxed later as we modify the model structure to accommodate asymptomatic individuals and waning immunity.
In the following sections, we will discuss three current major challenges of modeling efforts of cholera transmission dynamics: (1) parameter uncertainty and model misspecification; (2) interventions (especially, water, sanitation and hygiene), and (3) model structure.
Model misspecification and parameter uncertainty
 1.
β: the “contact rate” between the susceptible population with contaminated water,
 2.
B: the level of contamination of the water supply (V. cholerae concentration), and
 3.
κ: the concentration of V. cholerae at which the infection rate is 50% of the maximum infection rate, that is β.
Model misspecification
The “contact rate” and the V. cholerae concentration are largely unknown in most contexts. As Grad et al. [18] have rightly pointed out, there are no simple methods that can convert results of experimental studies (for example, “a measured doseresponse relationship between number of vibrios ingested and the risk of infection” [18]) into the “contact rate” between susceptible individuals and bacteria in water (β), and the concentration of V. cholerae in the water reservoir that will make 50% of the susceptible population ill (κ) [18]. The rate at which susceptible individuals become infected is determined by many variables in reality, most of which cannot be easily measured. As the “contact rate” (β) can rarely be measured directly from experimental studies, it is usually estimated by fitting models to time series data. These problems are referred as model misspecification, where the item of interest is different from what the model actually models, e.g. empirical experiments provide dose data in terms of the number of bacteria, while the model needs the bacteria concentration data in the environmental water [18].
Parameter uncertainty
The per capita recovery rate is probably the most certain of all parameters in the model. It is approximately equal to the reciprocal of the duration of infection (1/γ), a parameter that more data are available. Cholera life span in water reservoir (1/δ) depends on the local environment. While it is largely unmeasured in many endemic or epidemic contexts, modelers can use historical experimental data from the literature and therefore this parameter is also relatively certain. The rate of water contamination by infectious people shedding V. cholerae into the water reservoir (ξ) depends on both bacteria shedding of the infected individuals (a biological quantity) and the level of sanitation in the environment (an environmental assessment). This is largely unknown in most contexts. These problems are that of parameter uncertainty.
Parameters assumed or fitted based on selected published mathematical models of cholera (partly adapted from Grad et al., 2012[18])
Symbol  Parameters  Range  Comments  Potential data from field epidemiology 

β  Rate of “contact” with reservoir water (days^{1})  10^{5} to 1  Difficult to convert empirical data into this “contact” rate.  Identity and location of drinking water sources; frequency of water usage and volume drawn from these sources 
1/γ  Duration of cholera infection (days)  2.9 to 14  The most certain among the 5 parameters  Clinical data 
1/δ  Cholera life span in water reservoir (days)  3 to 41  Usually not measured; depending on local environment (temperature, salinity), nature of the water source (running or static), cholera phage concentration. Historical experimental data available.  Water samples for microbiological experiments 
ξ  Rate of water contamination by humans, i.e. rate of increase in V. cholerae concentration in the water reservoir (cells * mL^{1} * person^{1} * day^{1})  0.01 to 10  Usually not measured; depending on infection severity, sanitation provision and water reservoir size.  Clinical data: frequency and volume of watery stool and especially concentration of vibrios in watery stool. 
κ  Concentration of cholera that yields 50% chance of infection (cells/mL)  10^{5} to 10^{6}  The dose–response curves depend on strain and biological context (e.g. gastric acidity). While empirical data provided data for doses (number of bacteria), the parameter measures in concentration.  Based on the volume of water intake per person per day and the vibrio concentration in the water samples, one can estimate the dose of vibrio intake per person per day 
Equally important is data collection from the field that informs model parameterization (see Table 1). For example, in a neighborhood affected by cholera, we can investigate the various sources of drinking water for a given household, their relative importance in terms of volume drawn or frequency used, and the concentration of cholera vibrios and their decay rate in water samples collected from these sources. Just as the human contact data for constructing the contact matrix between different age groups in a population is important for influenza transmission models [20], collecting water usage data from a community is important to the parameterization of people’s “contact” rate with contaminated water. Eisenberg, Robertson and Tien [21] recently suggested that if we can measure pathogen persistence time in environmental water sources (δ) or pathogen concentration in the water (B), we can better estimate the parameters of the waterborne transmission pathway.
Interventions
The second challenge is to model interventions correctly. Interventions can be represented in the model as a change in the value of a parameter, or a change in the model structure. I will first discuss treatment, and then OCV, followed by WASH interventions.
Treatment
The primary treatment for a cholera patient is oral rehydration treatment (ORT). It prevents dehydration and averts mortality [22]. Severe cases are given antibiotics to speed up their recovery and to reduce the amount of bacteria shed into the environment (see ref. [23], p.127). The effect of antibiotics treatment can be simulated in a model by increasing the recovery rate, γ, and by reducing the rate of water contamination by treated patients in terms of V. cholerae concentration in the water reservoir, ξ [11]. Another model simulated combined ORT with antibiotic treatment by decreasing cholerarelated death rate and increasing recovery rate [24]. Alternatively, patients under treatment can be represented by a distinct compartment [25]. In this case, there will be a rate at which infected patients receives treatment and the recovery rate of the treated patients respectively.
Vaccine and immunity
Not everyone vaccinated will be immune to infection. (For example, Shanchol confers 65% direct protection against cholera in a 5year followup period) [29]. Furthermore, there is an indirect effect through which unvaccinated individuals are protected in communities where some individuals are immunized. The concept of herd immunity refers to the fact that individuals immune to an infection will not transmit an infection since they are not infected in the first place. Therefore, by vaccinating individuals in a population, indirect protection is conferred to other members of the population who are not immunized. For nonimmunizing interventions like water, sanitation and hygiene interventions, a similar concept may apply and is sometimes known as herd protection or indirect protection.
Water, sanitation and hygiene interventions
Effect(s) on model parameters by water, sanitation and hygiene (wash) interventions
WASH interventions  Effect(s) on parameters 

Sanitation interventions and health promotion of their utilization  Reduce water contamination rate (ξ) 
Treatment of water at source (e.g. chlorination of piped water)  Increases the rate of bacteria removal from water (δ) 
Pointofuse water purification (via boiling, chlorination or filters)  Reduces the concentration of bacteria (B) of drinking water 
Using alternative source of drinking water  Reduces the “contact” rate between susceptible population with contaminated water (β) 
Reduction in transmission coefficient (“contact rate”) by water, sanitation and hygiene (WASH) interventions in selected published models of the Haiti epidemic
Model  WASH intervention that the model was supposed to simulate  Reduction in transmission coefficient (“contact” rate, β)  Empirical data sources for WASH interventions’ effectiveness or coverage 

Andrews and Basu [11]  Expansion of clean water provision  Exponential decline in β (1% decrease per week)  Estimated coverage of clean water since the outbreak’s beginning, from two progress reports by Red Cross and Oxfam respectively 
Bertuzzo et al. [12]  Sanitation: “a set of measures”, not explained in their paper  40% reduction for 1 month  None provided 
Chao et al. [9]  Educational campaign to promote improved hygiene and sanitation, that accompanies the vaccination campaign  10% or 30% (additional) reduction, in areas covered by vaccination campaign  None provided 
Tuite et al. [10]  Clean water provision, either to “the same number of people who could be vaccinated” or to “the number of people who would need to receive clean water to have the same effect on epidemic spread as that achievable through vaccination”  Reduction of waterborne transmission (but not humantohuman transmission) by a fraction that is the probability of provision of clean water within a Haitian department (equivalent to a province), for up to 2 years, beginning at the same time as vaccination program would do for the sake of comparison.  None provided. Implied assumption: 100% reduction of “contact” rate if covered by clean water provision. 
Probably the weakest link in modeling WASH interventions is the dearth of data that link the programmatic variables (e.g. implementation coverage) to the reduction of the transmission coefficient. For example: in one paper [24], while a value of 10 US dollars per the square of level of sanitation was provided, it would be in the interest of the readers to provide the means to convert such “level of sanitation” (i.e. the proportion of reduction in β) into any quantity of coverage of any sanitation projects in reality. Likewise, it will be beneficial to the readers if details can be provided as to the “set of measures” of sanitation that would lead to a 40% reduction in β over a period of one month in Haiti in another example [12]. Similarly, readers would benefit if a third example [9] could provide data to support their choice of 10% or 30% reduction in cholera exposure through a health education campaign of hygiene and sanitation that accompanies the vaccination campaign.
There are exceptions though. One model [11] simulated “the effect of a 1% per week reduction in the proportion of the population consuming contaminated water based on present estimates of clean water provision” in Haiti, by converting “the estimated proportion covered [by clean water provision] since the start of the cholera outbreak into a rate of [increasing] clean water provision”. Two progress reports published by Red Cross and Oxfam respectively were cited as references. Such a rate of increasing clean water provision, as a daily percent reduction of the rate of drinking contaminated water (β), led to an exponential decline of β [11]. This implies that as coverage of clean water provision increases in time (number of weeks, n), the “contact” rate with contaminated water (β) would reduce as: β*(1–0.01)^{n}. But it is difficult to tell how much more coverage increase per day is needed to achieve such an effect.
Another model [10] estimated the number of people who would need clean water provision to achieve the same effect as 500,000 people being vaccinated in Haiti. The implied assumption was that if clean water was provided, there would be a 100% reduction of waterborne transmission (but not humantohuman transmission). What mattered was coverage. (See section ‘Hyperinfectious bacteria and “humantohuman” transmission’ below.)
The WASH interventions that are chosen, and their effectiveness and coverage have a huge impact upon the results. Comparing a poorly defined WASH intervention with OCV could inadvertently misinform policymakers about which programs should be expanded.
While it is useful to illustrate ranges of possibilities, future studies should be designed to provide data to parameterize these models. Another example was a model that incorporated a separate compartment for people who received health education and therefore may be infected at a rate different from those who did not. It will be beneficial if empirical data can be provided to parameterize the rates of health education, of failure to comply with instructions of health education, and of infection rates of healtheducated individuals (all three parameters were “assumed”) [25]. Likewise, for the compartment for quarantine of healtheducated individuals who were exposed to cholera, their rate of quarantine after exposure and their rate of actually being infected, it will be beneficial if empirical data can be provided to parameterize them [25].
Model structure: additional components
The third challenge is to correctly build the model structure. There are debates in the literature as to the essential components of a model that successfully replicate observed cholera dynamics. These are tied to our understanding in biology and epidemiology as to the relative importance of certain features of the cholera life cycle or its epidemiology. The basic model can be modified to take these elements into account. In this section, we focus on two issues: (1) asymptomatic, or ‘inapparent’, infections, and (2) hyperinfectious bacteria and humantohuman transmission.
Asymptomatic infection
There was a debate with regard to the relative importance of asymptomatic infection to transmission dynamics [30]. As noted by Grad et al. [18], the basic model assumes that throughout an epidemic, there is a constant ratio of asymptomatic to symptomatic infections, and that the infectious dose “determines the likelihood of infection, but not the likelihood of being symptomatic” [18]. However, the volume of bacteria shedding is likely very different: A person with severe cholera shed a lot more stool than that shed by an asymptomatically infected person. Rate of diarrhea for severe cholera cases is as high as 500–1000 mL/h [31]. Severe cases may shed bacteria for one to two weeks while asymptomatic patients typically shed for one day [32]. It is also likely that the amount of viable Vibrio cholerae per gram of stool excreted by a symptomatically infected cholera patient is greater than (or equal to) the amount of viable vibrios per gram of stool excreted by an asymptomatically infected person. It is perhaps worth noting that for the most part, surveillance data only captures symptomatic infections.
The key to the debate in ref. [30] was how would we explain the rapid reduction of the effective reproduction number in the first few months of the Haitian outbreak. Underreporting of cases, including asymptomatic cases, should be taken into account when fitting modeling outputs to observed data (even if the model does not have a distinct compartment for asymptomatic cases). Nonetheless, the reduction in effective reproduction number during the first three months of the epidemic cannot be solely explained by the depletion of susceptible individuals through infection, as the surge in incidence in June and July 2011 (see Figure 1 of ref. [34]) would be difficult to explain. (For details of the debate and our comments, see Additional file 1).
Hyperinfectious bacteria and “humantohuman” transmission
The second issue is how important hyperinfectious V. cholerae are to the transmission process. A decade ago, Merrell et al. [35] discovered that freshly shed V. cholerae were much more infectious than those that were grown invitro. However, these hyperinfectious bacteria would lose their hyperinfectiousness once they were cultured in vitro in broth for 18 hours. Later, researchers demonstrated that mousepassaged V. cholerae also demonstrated similar hyperinfectious properties as those freshly shed by humans, but such properties would disappear after 24 hours in the in vitro environment [36]. It has also been demonstrated that growth in a biofilm induces a hyperinfectious phenotype of V. cholerae[37]. This was the basis of the hypothesis that freshly shed V. cholerae existed in a hyperinfectious state for less than one day and that they contributed to cholera transmission more than we previously expected. These implied that a socalled “humantohuman” transmission route played an important role than the environmental, “waterborne” route [38].
Parameters for hyperinfectious bacteria as found in selected published mathematical models (adapted from Grad et al., 2012[18])
To model “humantohuman” transmission is challenging in two aspects. Firstly, the relative magnitude of the transmission coefficient (“contact” rate) of “humantohuman” transmission to waterborne transmission is uncertain (See Table 4 and the Additional file 1). Secondly, to correctly capture the impact of interventions upon “humantohuman” transmission is not easy. Take for example, in Tuite et al.’s model [10], the relative reduction in total cases by “equal allocation of clean water” was much smaller than that by an “optimized allocation of vaccine”. The major reason was that Tuite et al. assumed that clean water provision stopped waterborne transmission but not “humantohuman” transmission. However, clean water provision may, in fact, reduce the “humantohuman” transmission. Cholera is transmitted via the oralfecal route. The hyperinfectious state only makes the necessary infectious dose (or the IC_{50}) much lower. Given that the “humantohuman” transmission is only a mathematical proxy of the impact of the hyperinfectious bacteria, clean water provision should have an impact on humantohuman transmission, even if it may not stop transmission completely.
Moving forward
Our research questions dictate our choice of models. For the purpose of public health practice and policymaking, we propose the following two directions for future development of cholera models.
The first direction is emergency preparedness and response for cholera outbreaks. During the early phase of the Haitian epidemic in 2010, the US Centers for Disease Control and Prevention (CDC) made use of Abrams et al.’s model [43] to inform policymakers (that model will be further discussed in the Additional file 1). In the future, we can crossvalidate models for both their model structure and parameters against various historical epidemiological datasets, and then use the validated models for outbreak response. In some outbreak scenarios, seasonality can be omitted from the model, as only a short time frame is needed. Elements of spatial heterogeneity can be included if relevant data are readily available. Modeling packages that use models of relatively few parameters and variables can be created and made readily available before the next outbreak. At the beginning of an outbreak when data are limited, field epidemiologists and policymakers (for example, Epidemic Intelligence Service officers and their superiors in the CDC) who are not trained in mathematical modeling can deploy such models to provide estimates of attack rates (cumulative incidence) and intervention effects in different scenarios. The model inputs will either be provided for by the model as default (obtained from historical data in the literature) or require users’ inputs (as estimated based on limited data at the beginning of an outbreak). To facilitate its use in developing countries, the use of software that requires expensive licenses can be avoided. Free software like R is a good alternative. Many public health practitioners find the availability of a userfriendly Graphical User Interface helpful. One example is to use Excel as the user’s interface to an executable file compiled from a C++ code, as in the influenza model Community Flu 2.0 that is available on CDC website [44].
The second direction is cholera control in endemic contexts. First, the elucidation of the drivers of, and their effects upon, seasonal patterns of cholera incidence, and the effect of population and hydraulic movements upon spatial heterogeneity of incidence, will help epidemiologists predict future outbreaks (some of the related models are briefly discussed in the Additional file 1). Second, the estimation of the longterm effects on cholera incidence and the return on investment of longterm infrastructure building and intervention programs will be valuable to policymakers. Complementary to this modeling effort, we will need to collect better data for intervention effectiveness (including indirect effect) and costs.
Conclusion
Dynamic transmission models of cholera have been developed very rapidly in recent years, especially after the 2010 Haitian outbreak. Many models have been published but few make any impact on decisionmakers and field epidemiologists. This paper provides an introduction to the basics of ordinary differential equation models of cholera transmission dynamics, in the hope that the usefulness of modeling in public health research and decisionmaking may be better appreciated. Field epidemiologists are crucial in the partnership with modelers as they provide actual data that help parameterize the models. Modeldriven data collection and datadriven model construction are equally important. Likewise, policy makers that are wellinformed with the assumptions and implications of mathematical models and the data that are used to parameterize them, will be able to use mathematical modeling studies to facilitate their decisionmaking. More collaboration between policy makers, epidemiologists and modelers is needed if we want to make progress in controlling cholera in Haiti and beyond.
Declarations
Acknowledgements
The author thanks Mr. Joseph Abrams, Dr. Bishwa Adhikari, Dr. David L. Fitter, Dr. Manoj Gambhir, Dr. John Glasser, Dr. Andrew J. Leidner, Dr. Martin I. Meltzer, Dr. Eric Mintz, Dr. Scott Santibanez, Dr. Zhisheng Shuai, Dr. Jordan Tappero and the anonymous reviewers for their comments on some of the early versions of this manuscript.
Disclaimer
The findings and conclusions expressed in this report do not necessarily represent the official position of the Centers for Disease Control and Prevention.
Authors’ Affiliations
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