Interventions such as vaccination, antibiotic administration, and provision of clean water all can decrease the number of cholera cases. Vaccination reduces the number of fully susceptible persons, reduces infectiousness (ie, the rate of contamination of the water supply), and reduces the probability of becoming symptomatic when infected. Antibiotic administration shortens the duration of illness and perhaps reduces the concentration of vibrios excreted during illness. Access to clean water reduces consumption of vibrios. Each of these interventions will result in qualitative decreases in the extent of the epidemic. The benefits will be a combination of direct effects on those receiving the intervention, and indirect effects on those who benefit from reduced exposure because others received the intervention; in the case of vaccines, the latter effect is known as herd immunity.
The estimated direct impact of these interventions is often an input variable for transmission models; for example, these models assume that a certain proportion of the population is vaccinated and that the vaccine is effective in a particular fraction of the population (all-or-nothing efficacy) or reduces the infectiousness of each contact by a fixed fraction (leaky efficacy).32
Thus the role of the transmission models, over and above the assumptions about how interventions affect those who receive them, is to quantify the indirect effects of interventions – how much interventions can slow transmission and protect those who are not directly protected by the intervention. These quantitative results about the impact of interventions depend on the parameter values used in the model. In this sense (setting aside issues of model specification), the value of model-based predictions depends on the extent to which the predictions about indirect effects are robust to uncertainties about the value of input parameters.
Uncertainties in the values of input parameters can translate into massive uncertainties in the values of model predictions. We present an example in the (see eAppendix
] for details), and focus for this example on the impact of uncertainty in δ, the rate of removal of cholera from the water supply. In the Codeço model,15
the lifespan of cholera in the water supply is represented as 3 days (δ=1/3 days-1
). In other models, the lifespan of cholera in the water reservoir is set at 30 days,3,5,6,16
estimated at approximately 4.5 days 4,25
or fitted at approximately 41 days.7
Given that this term reflects the rates at which infectious vibrios become noninfectious due to death or physiologic change, one would expect the lifespan to be highly context-dependent and to vary based on the conditions of the water reservoir. Studies from the 1960s 33,34
examined cholera lifespan in a variety of water types (such as well-water and sea-water) and under a variety of conditions (including sun exposure and temperature variation). In these studies, cholera lifespan is reported from 4 to 80+ days depending on water source and condition.
Influence on R0 of lifespan of cholera in water reservoir
Variation in the assumed survival of cholera in water directly translates into variation in the distribution of assumed serial intervals for cholera transmission. This in turn changes estimates of R0
, the basic reproductive number, when these are obtained by fitting a model to the initial growth rate of the epidemic.35
When these estimates of R0
are in turn used to model interventions (by extending the model, after fitting to initial-growth data, into the future and considering the impact of interventions on transmission), the various values of R0
can give dramatically different predictions for the population-level effects of the interventions.
The proportion of a randomly mixing population that must be effectively vaccinated to prevent an epidemic from taking place is known as the critical vaccination threshold. This threshold is expressed as 1-1/R0
(effective vaccination means fraction vaccinated, or coverage, multiplied by vaccine efficacy 36
). For a model fitted to the early growth rate of the epidemic, varying the lifespan of infectious cholera vibrios in the aquatic reservoir (a parameter for which there are no data, but which is a key component to the serial interval) leads to very large changes in the inferred value of R0
and the corresponding critical vaccination threshold. Fitting the model with the lifespan of infectious vibrios set at 30 and then at 3 days changes the fitted R0
from 6 to 1.95, while the critical vaccination threshold decreases from 83% to 49%. Let us assume pre-epidemic vaccination of 70% of the population with a non-leaky vaccine that has 70% efficacy (in keeping with estimates for populations with less natural immunity than the endemic populations in which the vaccine was trialed37
). If R0
= 1.95, then pre-vaccination of a population would prevent an epidemic, whereas if R0
= 6, then nearly all unvaccinated persons will become infected. Thus, using parameter values found in the literature, the indirect benefits of vaccination (which is what the model is meant to quantify) range from almost complete protection of all unvaccinated persons to no protection. (See the eAppendix
] for further discussion of this issue.)
If one is willing to make strong assumptions, the problems of estimating R0
based on the initial growth rate and on assumed-duration parameters can be overcome in an idealized model by fitting to an epidemic curve with a known peak in cases.38
However, one must assume homogenously mixing and homogeneous population (which is implausible as we argue in the previous section); fixed reporting ratio throughout the epidemic, which is not the case39,40
; fixed asymptomatic to symptomatic ratio throughout the epidemic, for which we know of no supporting data; and a single-peaked epidemic, which has not been the case in multiple locations in both Haiti and Zimbabwe.29,41
Even if these assumptions were tenable, this approach can only be used once the epidemic has peaked and so cannot be employed at the start of an epidemic to guide interventions.
The claims made in this section are particular to the actual parameter values required for cholera models and the range of uncertainty that exists for them, in particular for the duration of infectiousness in contaminated water. Sensitivity analyses are necessary in all prediction models for infectious disease transmission, but here we argue, more specifically, that the uncertainty in just one parameter of cholera models can nearly eliminate the predictive power of these models. Within the range of possible values of this parameter, the qualitative predictions of the model range from substantial indirect vaccine effects to almost no indirect vaccine effects.