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 decision-making and day-to-day 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

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 [

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 _{0}) [

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 (population-based continuous-time models as contrast to population-based discrete-time models using difference equations), while we will mention relevant agent-based models where appropriate (e.g. [

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

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 Susceptible-Infected-Recovered (S-I-R) 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 [

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 non-pharmaceutical interventions [_{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 [

Following the example of Grad et al. [

Figure

• 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

Here are the key assumptions:

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.

The first challenge is model misspecification and parameter uncertainty, that was highlighted by Grad et al. [

1. β: the “contact rate” between the susceptible population with contaminated water,

2. B: the level of contamination of the water supply (

3. κ: the concentration of

The “contact rate” and the

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

Therefore, in the cholera mathematical model literature, the values of the parameters used vary greatly as seen in Table

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 ^{-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

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.

The primary treatment for a cholera patient is oral rehydration treatment (ORT). It prevents dehydration and averts mortality [

People recovered from cholera develop immunity that protects them from being infected again for several years [

Vaccine and waning immunity (Model 1).

Not everyone vaccinated will be immune to infection. (For example, Shanchol confers 65% direct protection against cholera in a 5-year follow-up period) [

A dynamic model that explicitly simulates the transmission mechanism can take these factors into account, if we slightly modify the model structure as in Figure

Vaccine and waning immunity (Model 2).

Provision of clean water, sanitation and personal hygiene are all important interventions that can stop cholera transmission. In transmission dynamic models, one can simulate the effects of these interventions by changing the values of one or more parameters. Sanitation interventions, from latrines to flush toilets, reduce water contamination from human feces by separating them from the drinking water supply (reducing contamination rate, ξ). Chlorination of piped water removes bacteria from the water (increasing the removal rate of bacteria, δ). Point-of-use purification via boiling, chlorination, or filters, reduces the bacterial concentration in drinking water (reducing B). Interventions that promote alternative sources of drinking water reduce “contact” between susceptible populations and contaminated water (reducing β, as in [

Water, sanitation and hygiene interventions.

Effect(s) on model parameters by water, sanitation and hygiene (wash) interventions

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) |

Point-of-use 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

Andrews and Basu [ | 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. [ | Sanitation: “a set of measures”, not explained in their paper | 40% reduction for 1 month | None provided |

Chao et al. [ | 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. [ | 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 human-to-human transmission) by a fraction that is the probability of provision of clean water within a Haitian | 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 [

There are exceptions though. One model [^{n}. But it is difficult to tell how much more coverage increase per day is needed to achieve such an effect.

Another model [

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 policy-makers 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 health-educated individuals (all three parameters were “assumed”) [

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 human-to-human transmission.

There was a debate with regard to the relative importance of asymptomatic infection to transmission dynamics [

Bertuzzo et al. [

The key to the debate in ref. [

The second issue is how important hyperinfectious

Some modelers argue that these hyperinfectious bacteria hold the key to our understanding of cholera transmission dynamics (e.g. refs. [

Multiplier for infectiousness of freshly shed vibrio (hyperinfectious state) | 50 | 700 | 100 | [ |

Duration of hyperinfective state (hours) | 24 | 5 | 24 | [ |

For further discussion on these parameter values, please refer to the Additional file

“Human-to-human” infection incorporated into the basic model.

To model “human-to-human” transmission is challenging in two aspects. Firstly, the relative magnitude of the transmission coefficient (“contact” rate) of “human-to-human” transmission to waterborne transmission is uncertain (See Table _{50}) much lower. Given that the “human-to-human” transmission is only a mathematical proxy of the impact of the hyperinfectious bacteria, clean water provision should have an impact on human-to-human transmission, even if it may not stop transmission completely.

Our research questions dictate our choice of models. For the purpose of public health practice and policy-making, 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 [

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

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 decision-makers 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 decision-making may be better appreciated. Field epidemiologists are crucial in the partnership with modelers as they provide actual data that help parameterize the models. Model-driven data collection and data-driven model construction are equally important. Likewise, policy makers that are well-informed 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 decision-making. More collaboration between policy makers, epidemiologists and modelers is needed if we want to make progress in controlling cholera in Haiti and beyond.

The authors declare that they have no competing interests.

Online Supplementary Materials.

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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.

The findings and conclusions expressed in this report do not necessarily represent the official position of the Centers for Disease Control and Prevention.