abstract: # Background
  .
  The recent emergence of hypervirulent subtypes of avian influenza has underlined the potentially devastating effects of pandemic influenza. Were such a virus to acquire the ability to spread efficiently between humans, control would almost certainly be hampered by limited vaccine supplies unless global spread could be substantially delayed. Moreover, the large increases that have occurred in international air travel might be expected to lead to more rapid global dissemination than in previous pandemics.
  .
  # Methods and Findings
  .
  To evaluate the potential of local control measures and travel restrictions to impede global dissemination, we developed stochastic models of the international spread of influenza based on extensions of coupled epidemic transmission models. These models have been shown to be capable of accurately forecasting local and global spread of epidemic and pandemic influenza. We show that under most scenarios restrictions on air travel are likely to be of surprisingly little value in delaying epidemics, unless almost all travel ceases very soon after epidemics are detected.
  .
  # Conclusions
  .
  Interventions to reduce local transmission of influenza are likely to be more effective at reducing the rate of global spread and less vulnerable to implementation delays than air travel restrictions. Nevertheless, under the most plausible scenarios, achievable delays are small compared with the time needed to accumulate substantial vaccine stocks.
author: Ben S Cooper; Richard J Pitman; W. John Edmunds; Nigel J Gay\* To whom correspondence should be addressed. E-mail: <email@example.com>[^1]
date: 2006-06
institute: **1**Statistics, Modelling, and Bioinformatics Department, Centre for Infections, Health Protection Agency, London, United Kingdom; Mexican National Institutes of HealthMexico
references:
title: Delaying the International Spread of Pandemic Influenza

# Introduction

The scale of threat posed by hypervirulent avian influenza subtypes \[1,2\], and the memory of the 20–100 million who died in the 1918 pandemic \[3,4\], warrant consideration of large-scale, concerted, and potentially highly disruptive control measures \[5\]. Were such a virus to acquire the ability to spread efficiently between humans, control would almost certainly be hampered by limited vaccine supplies \[6\]. Interventions able to substantially impede global spread, by providing time for vaccine stocks to accumulate, could have profound public health benefits.

Border controls and World Health Organization travel advisories formed central and sometimes controversial components of the control efforts during the severe acute respiratory syndrome (SARS) epidemic \[7,8\], and travel restriction is thought likely to occur during an influenza pandemic (although enforcement is currently considered by the World Health Organization to be impractical in most countries) \[9\]. In the absence of sufficient vaccine stocks, other control measures such as the use of antiviral agents could also be used \[10,11\]. Ideally, such measures would reduce the average number of secondary cases caused by each primary case (the effective reproduction number, *R* ~t~) to below one, making sustained transmission impossible. This happened during the SARS epidemic, where isolation, quarantine, and behaviour change were able to bring about control \[12\]. The much shorter serial interval for influenza makes the chances for early epidemic termination much lower \[13\]. The main value of interventions is more likely to be in reducing the incidence and slowing the rate of spread of the virus.

To evaluate the potential of travel restriction and local control measures to impede global dissemination we developed a stochastic (i.e., probabilistic) model of the international spread of influenza based on extensions of coupled deterministic epidemic transmission models \[14–19\]. This class of models has been shown to be capable of accurately forecasting local and global spread of epidemic and pandemic influenza \[14–19\] and accounting for the global distribution of other pathogens \[20,21\], but has not previously been used to assess the impact of travel restrictions or other control options for pandemic influenza.

# Methods

We used a metapopulation model that consists of a set of coupled dynamic epidemic transmission models (Figure 1). Each component model represents one city and tracks the progression of individuals through four classes: susceptible to infection (S); exposed to the virus but not yet infectious (E); infectious (I); and recovered and no longer susceptible (R). We assumed that infectiousness coincides with disease onset and that infectious cases do not travel.

Previous work has used deterministic approximations to study the evolution of this system \[14–19\]. With this approach the first case in each city (except the originating city) was assumed to occur only when the average number of incubating cases arriving from other cities exceeded one, an approximation that will artificially slow the rate of spread between cities. Our stochastic model, which has a similar underlying structure to its deterministic counterpart, avoids this distortion and uses probabilistic transitions to capture the inherent uncertainty in the course of the epidemic. This is more appropriate than the deterministic approaches previously adopted, because chance effects dominate in the early stages of the epidemic in each city and in the seeding of each city's epidemic. As well as providing greater realism, this approach allowed us to quantify the uncertainty in model predictions due to demographic stochasticity. Thus, rather than assuming that each city's epidemic starts at a determined time, we assumed that the initiation of an epidemic depends on the timing of a sequence of chance events: a person incubating the virus must board a plane; that person must infect others in the destination city; some of those others must cause further transmission, and so on. Each time the model is run, even with identical starting conditions and parameter values, a different answer is obtained. This stochastic model was used to estimate key parameters using data from the 1968–1969 (1968/9) influenza pandemic and to evaluate the impact of interventions using contemporary demographic and transport data. Mathematical details of the model are provided in Protocol S1.

Coupling between cities was estimated using data from the International Air Transport Association for 2002 that gives the number of seats on flights between 105 cities, including the 100 with the highest number of international scheduled passengers and all 52 used in the 1968/9 data. City sizes were taken from the United Nations urban agglomeration data (available at: <http://unstats.un.org/unsd/demographic/sconcerns/densurb/urban.aspx>). When fitting models to 1968/9 data, air transport data, sizes of urban agglomerations, and influenza data were taken from a previous study \[14,15\].

To select from variants of the basic model, we compared deterministic and stochastic model fits to data from the 1968/9 pandemic by choosing parameters to minimize the sum of squared deviations (SSQ) between times of observed and predicted peaks (this contrasts with previous work that aimed to forecast the pandemic, and therefore based parameter estimates only on data from the first affected city \[14\]). A major strength of our approach is that the observed time of an epidemic peak should be unaffected even by large between-country variation in influenza reporting rates. The predicted epidemic peak for a given city was defined as the day on which the highest number of infected people first developed symptoms. When a city had more than one observed peak in the 1968/9 data, only the first peak was used. We used the statistic SSQ(*m*)/(*n* − 2*m*) to compare model fits \[22\], where *m* is the number of fitted parameters (between one and four), and *n* the number of fitted data points (the number of cities with observed and predicted epidemic peak times). This formula selects models that are parsimonious and fit the data well. The full stochastic model was then used to estimate parameters for the best-fitting model by choosing parameters that minimised the mean SSQ from ten simulation runs for each combination of parameter values.

We evaluated models with sine wave, square wave, and no seasonal variation in transmission parameters for cities outside the tropics. In the sine wave formulation, the peak transmissibility occurred on the shortest day in each hemisphere, while in the square wave formulation, peak transmissibility lasted 6 mo, also centred on the winter solstice. We also considered model formulations where the transmission parameter in the tropics was taken as the maximum, minimum, and mean over 1 y of that outside the tropics. For some parameter values, the model predicted no epidemic peaks in some cities for which an epidemic peak was in fact recorded. When fitting the models we penalized these regions of parameter space by arbitrarily assigning a deviation between model and data of 500 d.

## Interventions

We used the stochastic model to consider the effects of (i) reducing local transmission (this simulates the effects of isolation, behaviour changes, antiviral use, or other measures that may reduce the average number of secondary cases produced by one primary case); and (ii) restricting travel to and from affected cities. We assessed the ability of these measures to delay epidemics in individual cities. We considered only major epidemics, which we defined as those peaking with at least one case per 10,000 people per day. We assumed that measures were introduced only after the first 100 symptomatic cases in each city except the originating city, for which 1,000 cases were required, although we also evaluated the sensitivity of the results to these assumptions.

We considered a number of other scenarios to assess the sensitivity of the results to the most important unknowns: patterns of seasonal variation in influenza transmission; variation in transmissibility between tropical and temperate regions; the proportion of individuals initially susceptible to the virus; the basic reproduction number, *R* ~0~ (defined as the mean number of secondary cases in a local and susceptible population caused by the introduction of one primary case); the distribution of the infectious period; the city in which the pandemic begins; and the date on which the virus first begins to spread.

# Results

Amongst the model variants considered, the best fit to data from the 1968/9 pandemic was achieved when transmissibility varied sinusoidally in temperate regions and was constant and equal to the north/south maximum in the tropics. We used this model to estimate key parameters using 1968/9 data, and to evaluate the impact of interventions. Models without seasonal forcing terms gave poor fits to data and could not account for the large differences in epidemic timing between cities in the north and south temperate regions. Models in which transmissibility in the tropics was set to the north/south mean also performed surprisingly poorly, with best-fit SSQs approximately three times greater than those obtained when transmissibility in the tropics was set to the north/south maximum (Figure 2A and 2B). Less surprisingly, models that assumed all cities were equally connected by air travel (but with the same total volume of air traffic) also performed poorly, with best-fit SSQs about twice as large as those from the models that used the air travel data. Previous work with the deterministic version of the model has assumed a square wave variation in transmissibility, assigning transmission outside the influenza season to be one-tenth of the value during the season \[14\]. We found the fit to data under this assumption to be substantially poorer compared with models in which maximum and minimum seasonal transmission parameters were both estimated.

An exploration of the parameter space for the best-fit model showed that, assuming 60% of the population to be initially susceptible (the approximate value estimated previously \[14\]), maximum *R* ~0~ values (*R* ~0,max~) ranging from about 2.5 to 3.5 gave the best fits to data, while minimum *R* ~0~ values (*R* ~0,min~) between about 0.5 and 1.5 had the most support (Figure 2C). The maximum *R* ~0~ value and the fraction initially susceptible could not be identified simultaneously: A high value of one implied a low value of the other (Figure 2D). However, the initial maximum *effective* reproduction number, *R* ~max~ (equal to the product of the two and giving the average number of secondary cases produced by one primary case in an actual population, accounting for immunity) was well defined, with only a narrow range of values between about 1.5 and 2.2 supported by the data. This result is consistent with other estimates from influenza pandemics \[14,23\]. We therefore took as our baseline scenario an *R* ~0,max~ value of 3 and an *R* ~0,min~ of 1.2, assumed 60% of the population to be initially susceptible, and used a model in which the *R* ~0~ value varied sinusoidally and peaked in midwinter, and in which the pandemic originated in Hong Kong on 1 June.

The model showed good agreement with data from the 1968/9 pandemic, with observed epidemic peaks almost always occurring at times when the model predicted a very high probability of influenza activity (Figure 3). Observed and predicted times of epidemic peaks differed, on average, by 31 d. There were, however, some anomalies: the first epidemic peaks occurred much later than might have been expected in London and Tokyo, and somewhat earlier than predicted in Manila and Madras.

Despite large variation in the timing of predicted epidemic peaks in individual cities between simulation runs, the overall course of the pandemic was quite predictable (Figure 3A), although there was markedly more between-run variability in the tropics and the south than in the north. The roughly ten-fold increase in air traffic since 1968 causes epidemics in most cities to peak between 1 and 2 mo earlier than they would have done in 1968 (in some southern hemisphere cities the epidemic peaks 1 y earlier) and substantially reduces variation between simulation runs (Figure 3B). The model reproduced another interesting aspect of influenza epidemiology: the tendency for peak periods of influenza activity in the tropics to shift with latitude, so that in the northern tropics they are closer to countries north of the tropics, while the southern tropics tend to be more closely aligned with countries south of the tropics \[24\]. This occurs despite the fact that the model has no explicit assumptions about seasonality for cities in the tropics; the behaviour arises only as a result of the strength of transport connections between different regions. It is also notable that the pandemic starts early enough to allow some probability of influenza activity in the south during the end of the flu season in 1968. Despite this, predicted epidemic peaks (the weeks with the greatest number of reported cases in each location) still occur in 1969 in the south.

When we used the model to evaluate interventions using contemporary air travel and demographic data, we found that travel restrictions to and from affected cities would slow epidemic spread, but unless almost all air travel from affected cities (i.e., greater than 99%) was suspended, the potential for delaying the pandemic was limited (Figures 4–6 and Table 1). Even when 99.9% of air traffic was suspended, most cities had a low probability of ultimately escaping the pandemic (Figure 4), and delays large enough to be of clinical significance (6 mo or more) were common only if interventions were made after the first few cases (Figure 5). Interventions that reduced transmission could typically lead to more pronounced delays (Figures 5 and 6 and Table 1), although only when *R* ~t~ was reduced to slightly above one were these sufficient to delay epidemics until the next influenza season. These findings were not highly sensitive to assumptions about initial susceptibility and transmissibility (Table 1).

Median Delays in Epidemic Peak for Diverse Interventions and Percentage of Cities Experiencing Major Epidemics

![](pmed.0030212.t001)

Decreasing the number initially susceptible (while holding *R* ~max~ constant) has two opposing effects (Table 1). First, within cities the time between seeding with influenza cases and the epidemic peak decreases. This is because the initial epidemic growth rate is unaffected, but each new case represents a greater proportional reduction in the susceptibles and causes a greater reduction in *R* ~t~ (the epidemic peaks when *R* ~t~ = 1). Conversely, between-city dynamics are slowed because there are fewer infectious people to spread the disease. Which effect dominates varies between cities; those affected at the start of the pandemic tend to experience peak activity earlier when there are fewer initial susceptibles; for the rest it usually occurs later. Although travel restriction always reduces the rate of spread between cities, under most scenarios so many people become infected that even near-total restriction has remarkably little effect. However, for a given *R* ~max~, the smaller the number of susceptibles the greater the impact of this intervention. For example, when 90% of the population are initially immune, the most extreme travel restrictions can be quite effective in preventing international spread. Conversely, reducing transmission has the greatest effect on impeding international spread when (for a given *R* ~max~) more people are susceptible. The large delays and reductions in the number of affected cities result from two effects acting in the same direction: The reduced *R* ~t~ slows the epidemic within each city (delaying epidemic peaks), and the reduced total number of cases reduces the rate of spread between cities. Larger reductions in transmission led, in extreme cases, to smaller delays in epidemic peaks (Table 1). This happened only when *R* ~t~ was reduced to below one, causing the epidemic decline to begin immediately; the peak therefore occurred at the time of the intervention, earlier than it would have done with a less effective intervention. Under such circumstances the time of the epidemic peak is not a good measure for fully evaluating local control measures.

Previous influenza modelling work has used both square and sine wave seasonal forcing terms \[14,25\]. We found that the outcomes of interventions were not highly sensitive to the precise assumptions made. The delays in the timing of epidemic peaks depended only to a limited extent on the city in which the pandemic started and to a somewhat greater extent on the date of release (Table 2), with larger delays more likely when the first cases occurred towards the end of the influenza season in the place of origin. Results were, however, highly sensitive to the timing of the intervention (Figure 5). Large delays in the timing of epidemic peaks and the prevention of epidemics in a large number of locations could be achieved with the most extreme interventions, but only when they were made sufficiently early. However, making the interventions after fewer than 1,000 cases in the place of origin had minimal additional benefit in slowing pandemic spread. Similarly, preemptive travel restrictions had no advantage over interventions made after one case in affected cities (Figure 5A and 5B).

Median Delays in Timing of Epidemic Peaks with Different Dates and Locations for the Start of the Pandemic

![](pmed.0030212.t002)

The course of infection with a future pandemic influenza virus might differ in important ways from our baseline assumptions, and could be quite unlike typical interpandemic influenza. We therefore assessed the robustness of our conclusions to the assumed latent and infectious periods. We found that assuming a greater degree of infectiousness early in the course of infection (reducing the serial interval from 4.2 to 2.6 d, as suggested by recent analysis of household influenza transmission data \[11,26\]) did not substantially alter the conclusions about the value of the interventions (Figure 6A–6C) compared with the baseline scenario (Figure 5C and 5D), although if this assumption was used when fitting the model to the 1968/9 data the estimated value of *R* ~max~ was reduced from about 1.8 to 1.5. Conclusions were also robust to moderate variation in the distribution of the latent period (Figure 6D–6F). If, however, the virus behaved more like the SARS coronavirus, with extended latent and infectious periods (Figure 6G–6I), a greatly delayed rate of global spread could be expected, giving more chance of delaying epidemics until the next influenza season. In this case, smaller reductions in travel and transmission can achieve clinically significant delays (6 mo or more) in epidemic take-off in many cities. Assuming reduced transmission in the tropics (Figure 6J–6L) also led to a substantial reduction in the rate of global dissemination. Under this scenario much smaller reductions in transmission would be sufficient to greatly reduce the chance of a pandemic; this happens because the lower transmission in the tropics (where the virus is assumed to originate) means that a further transmission reduction of just 21% would be sufficient to make sustained spread impossible in this region.

# Discussion

The relative ineffectiveness of travel restrictions for controlling pandemic influenza is a consequence of the rapid initial rate of growth of the epidemic in each city and the large number of people infected. For example, with a serial interval of 3 d, ignoring depletion of susceptibles, an *R* ~t~ of two would cause a 128-fold increase in new cases within 21 d (128 = 2^21/3^). This means that if travel from the first affected city was restricted to 1/128 of its former value on (and after) day 1, there would be approximately the same number of influenza cases leaving the city on day 21 + *t* as there would have been on day *t* had there been no intervention; even such an extreme intervention would therefore buy only about 3 wk. The highly connected nature of the air travel network prevents such minor delays between pairs of cities combining into substantial delays over the whole network.

Hufnagel et al. \[20\] used a related model to study the global spread of SARS. Although this model differed in important respects from the one used here (the implicit assumptions that air travel frequency varies with neither infection state nor country would not be tenable in the context of pandemic influenza), the conclusion that "remarkable success \[in SARS epidemic control\] is guaranteed if the largest cities are isolated in response to an outbreak" might, at first sight, be thought to apply equally to influenza. In fact, pandemic influenza is expected to have a much shorter serial interval than SARS, and delays in international spread that could be achieved by restricting almost all travel would be far more modest. Even if 99.9% of all travel could be stopped, epidemics in most cities would be delayed by no more than 4 mo. Moreover, the conclusion that a policy of isolating only the largest cities would guarantee success implicitly assumes that closing major airports would cause infected individuals who would have travelled through them to abandon their journeys rather than seek alternative routes, and that disease spread by routes other than air travel can be ignored without substantially altering the conclusions. This seems rather implausible, and for these reasons we think that the conclusions of Hufnagel and colleagues, while of undoubted theoretical interest, would be misleading if taken too literally.

Large and important uncertainties abound in influenza epidemiology: We do not know whether or not a significant proportion of transmission occurs before the onset of symptoms or whether subclinical infections are an important source of transmission, and we know very little about the determinants of seasonality \[24,27,28\]. In evaluating the potential to delay the spread of influenza by restricting travel and reducing transmission, we have systematically adopted optimistic assumptions, chosen to give the interventions the greatest chance of success. Thus we have assumed that seasonal effects are important (delaying the rate of spread outside the influenza season), and that asymptomatic cases do not contribute to transmission (minimizing the numbers capable of spreading the virus, and maximising the chance of detecting them); we have ignored travel that is not by air and not between major airports; and we have ignored the possibility of transmission during flights themselves. Despite these optimistic assumptions we found that even large and widely enforced travel restrictions would usually delay epidemic peaks by only a few days; to have a major impact, restrictions would have to be almost total and almost instantaneous. Only if a pandemic strain were considerably less transmissible, or had a considerably longer serial interval than influenza strains seen in the past, or if very few people were initially susceptible, would such measures be likely to have an important impact on the rate of pandemic spread. Local control measures able to reduce influenza transmission were found to have greater potential for reducing the rate of global spread (they could also substantially reduce the total number of cases, although an evaluation of this benefit is beyond the scope of this paper). Under most plausible scenarios, however, delays would still fall far short of those required to produce large quantities of vaccine unless they were implemented early and able to reduce *R* ~t~ to close to one. Elsewhere it has been shown that airport entry screening would be unlikely to detect more than 10% of passengers latently infected with influenza when boarding \[29\]. The results in this paper show that such an intervention would have a negligible impact on the course of a pandemic once it was underway.

The results also raise interesting questions about the importance of seasonality in influenza transmission. The evidence for strong seasonal effects in temperate regions found here with 1968/9 data is supported by a recent analysis of interpandemic influenza \[30\]. However, it is not clear how important such seasonal effects have been in previous pandemics, nor is it clear why a much better model fit should be obtained when transmission in the tropics is assumed to be the maximum (rather than the mean) of that in temperate regions. Indeed, a fuller understanding of the determinants of seasonal effects and their variation with latitude remains one of the outstanding problems of influenza epidemiology \[24,27,28\].

Recent models of pandemic influenza have accounted for household and social contact patterns \[10,11\]. While such details are needed for evaluating the possibility of containment at source, they would not be expected to affect the broad conclusions presented here. However, for a given *R* ~0~, assuming nonhomogeneous local mixing patterns would result in a somewhat reduced attack rate and rate of spread within each city, causing a slight decrease in the rate of global spread. For this reason, estimates of *R* ~max~ based on fitting models that assume homogeneous local mixing to pandemic data may underestimate the true value.

A new pandemic strain might not show the same pattern of seasonality as in 1968/9 and could potentially have greater transmissibility than strains seen previously. Both SARS and smallpox transmission can be greatly amplified by nosocomial spread \[31,32\]; a similar amplification effect could occur with an unusually virulent influenza virus that led to many hospitalisations. In these more pessimistic scenarios, even more heroic efforts would be required to have any chance of significantly delaying the virus's spread by restricting travel. The results here suggest that resources might be better directed at reducing transmission locally and at attempting to control outbreaks during the earliest stages of sustained human-to-human spread, when movement restrictions are likely to be a more valuable containment measure \[10,11\]

# Supporting Information

###### Detailed Description of the Model

(65 KB DOC)

Click here for additional data file.

## Editors' Summary

### Background.

Most people who get influenza (flu) recover quickly, although it can cause serious illness and death, most often in the elderly. Sometimes a new type of flu virus appears that is much more likely to kill. This happened, for example, in 1918, when a worldwide flu pandemic killed between 20 million and 100 million people. Recently, there have been concerns about a flu virus that affects birds, and often kills them. At present the virus does not pass easily from birds to humans, and it does not seem to pass from one human to another. However, the fear is that this virus might change and that human-to-human infection could then be possible. Should all this happen, the changed virus would be a major threat to human health. With current technology, it would take several months to produce enough vaccine for even a small proportion of the world's population. By that time, it would probably be too late; the virus would already have spread to most parts of the world. It is therefore important for health authorities to consider all the methods that might control the spread of the virus. With the increase in international travel that has taken place, the virus could spread more quickly than in previous worldwide pandemics. Restrictions on international travel might, therefore, be considered necessary, particularly travel by air.

### Why Was This Study Done?

It is important to estimate how useful restrictions on air travel might be in controlling the spread of a flu virus. Travel restrictions are usually unpopular and could themselves be harmful, and, if they are not effective, resources could be wasted on enforcing them.

### What Did the Researchers Do and Find?

This research involved mathematical modelling. In other words, complex calculations were done using information that is already available about how flu viruses spread, particularly information recorded during a worldwide flu outbreak in 1968–1969. Using this information, virtual experiments were carried out by simulating worldwide outbreaks on a computer. The researchers looked at how the virus might spread from one city to another and how travel restrictions might reduce the rate of spread. Their calculations allowed for such factors as the time of the year, the number of air passengers who might travel between the cities, and the fact that some people are more resistant to infection than others. From the use of their mathematical model, the researchers concluded that restrictions on air travel would achieve very little. This is probably because, compared with some other viruses, the flu virus is transmitted from one person to another very quickly and affects many people. Once a major outbreak was under way, banning flights from affected cities would be effective at significantly delaying worldwide spread only if almost all travel between cities could be stopped almost as soon as an outbreak was detected in each city. It would be more effective to take other measures that would control the spread of the virus locally. These measures could include use of vaccines and antiviral drugs if they were available and effective against the virus.

### Additional Information

Please access these Web sites via the online version of this summary at <http://dx.doi.org/10.1371/journal.pmed.0030212>.

•  Fact sheets are available about various aspects of flu from the Web site of the World Health Organization, which takes a global overview of the impact of the infection

Many health Web sites aimed at patients provide basic information about flu.

• US National Institute of Allergy and Infectious Diseases page about flu

• National Institute of Allergy and Infectious Diseases fact sheet about cold and flu symptoms

• US Centers for Disease Control and Prevention page about flu

• The Journal of the American Medical Association's patient page about influenza

• Page on flu from BBC Health

• Information about pandemic influenza from The Health Protection Agency

We thank Emilia Vynnycky and Maria Zambon (Health Protection Agency) for helpful discussions; the International Air Transport Association for supplying data; and Professor Neil Ferguson (Imperial College) for helpful suggestions that led to significant improvements in the manuscript.

**Author contributions.** BSC, WJE, and NJG conceived the study. RJP obtained and processed the travel data. BSC implemented and analysed the model. All authors contributed to writing the manuscript.

# References

## Abbreviations

IQR

:   interquartile range

SARS

:   severe acute respiratory syndrome

SSQ

:   sum of squared deviations

[^1]: **Competing Interests:** The authors have declared that no competing interests exist.


abstract: Impaired physical performance is a common but not obligate result of a low carbohydrate diet. Lessons from traditional Inuit culture indicate that time for adaptation, optimized sodium and potassium nutriture, and constraint of protein to 15–25 % of daily energy expenditure allow unimpaired endurance performance despite nutritional ketosis.
author: Stephen D Phinney
date: 2004
institute: 16108 Boothbay Court, Elk Grove, CA 95758, United States of America
references:
title: Ketogenic diets and physical performance

# Introduction

In the opinion of most physicians and nutrition scientists, carbohydrate must constitute a major component of one's daily energy intake if optimum physical performance is to be maintained \[1\]. This consensus view is based upon a long list of published studies performed over the last century that links muscle glycogen stores to high intensity exercise. It has also been reinforced by the clinical experience of many physicians, whose patients following low carbohydrate formula or food diets frequently complain of lightheadedness, weakness, and ease of fatigue.

During the time that this consensus view of the necessity of carbohydrate for vigorous exercise was forming, the last pure hunting cultures among the peoples of North America finally lost out in competition with expanding European cultural influences. Between 1850 and 1930, the routine consumption of carbohydrates spread north from the U.S. Plains States through central Canada, where the indigenous peoples had heretofore made at most seasonal use of this nutrient class. However the last of these groups to practice their traditional diet, the Inuit people of the Canadian and Alaskan Arctic regions, were luckily observed by modern scientists before their traditional dietary practices were substantially altered. The reports of these early scientists imply that the Inuit people were physically unhampered despite consuming a diet that was essentially free of identifiable carbohydrate.

Given this juxtaposition of clinical research results favoring carbohydrate against observed functional well-being in traditional cultures consuming none, it is an interesting challenge to understand how these opposing perspectives can be explained. This paper will review the observations of early explorer scientists among the Inuit, track the controversy that they stimulated among nutritionists in the last century, and utilize some of the forgotten lessons from the Inuit culture to explain how well-being and physical performance can be maintained in the absence of significant dietary carbohydrate.

# The origins of carbohydrate supremacy

Until the development of agriculture over last few millennia, our human ancestors' consumption of dietary carbohydrate was opportunistic. As some groups adapted to hunting and fishing for their sustenance, they were able to move into temperate and then arctic regions, where limited access to wild grain, nuts, and fruit dictated sustained dependence upon fat and protein as primary sources of dietary energy.

With the development of agriculture came the ability to grow and store grain, allowing societies to remain in a stable physical location, build permanent dwellings, and potentially stimulating the development of written language (those early stone tablets would have been difficult to transport from camp to camp on a dog sled). Starting from locations in the Middle East and Asia, cultures based upon agricultural wheat and rice spread over 5 millennia to dominate Europe, Africa, and the Americas.

With its ability to support a non-nomadic life style, greater population density, and permanent communities; there were clear advantages of agriculture-based societies over those based upon hunting and fishing, particularly as agricultural communities built the infrastructure to support trade and transport. Given its success in this competition of cultures (and by implication, the competition of their diets), it is an easy assumption that a grain-based diet is functionally superior to one based upon the meat and fish (fat and protein) of the hunting societies that they superseded.

As the science of nutrition developed in the early 20^th^ Century, numerous comparative studies were undertaken to assess differences between diets. Although there were some advocates of low carbohydrate diets (eg, the Banting diet of the 19^th^ Century, promoted for weight loss and diabetes control), the prevailing premise for these studies was that carbohydrate was a necessary nutrient for optimum human health and function. Among studies confirming this view, a classic was the 1939 study by two Danish scientists, Christensen and Hansen \[2\]. They did a crossover study of low carbohydrate, moderate carbohydrate, and high carbohydrate diets, each lasting one week. At the end of each diet, the subjects' endurance time to exhaustion on a stationary bicycle was assessed. Compared to the mean endurance time on the low carb diet of 81 minutes, the subjects were able to ride for 206 minutes after the high carb diet.

During the Second World War, another oft-cited study was performed, this time examining the practicality of pemmican (a mixture of dried meat and fat) as a light-weight emergency ration for soldiers. This experiment by Kark et al \[3\] involved abruptly switching soldiers in winter training in the Canadian Arctic from standard carbohydrate-containing rations to pemmican. This study only lasted 3 days, as the soldiers rapidly became unable to complete their assigned tasks, which included pulling loaded sleds 25-miles per day through deep snow.

With the resurgence of biomedical science in the 1960's came development of the percutaneous needle biopsy, facilitating assessment of intra-muscular fuel stores and metabolism. This led to the concept of muscle glycogen as the limiting fuel for high intensity exercise \[4\] and to the nutritional strategy of carbohydrate loading \[5\]. The clear consensus that developed from this research was that fat had limited utility as a fuel for vigorous exercise, and that humans are physically impaired if given a low carbohydrate diet.

# The hunter's counterpoint – practical observations on ketogenic diets

Although high-carbohydrate diets might be more effective in short-term tests of high-intensity exercise, there are multiple clues in the published literature that the debilitating effects of ketogenic diets are overstated. Not only is there the demographic evidence that whole populations of people lived for millennia as hunters, but there are many reports of Europeans crossing over to live within the cultures of these hunting societies without apparent impediment.

One of the earliest documented demonstrations of physical stamina during a ketogenic diet was the Schwatka 1878–80 expedition in search of the lost Royal Navy Franklin expedition. The Schwatka expedition, sponsored by the *New York Herald* and the American Geographical Society, departed from the west coast of Hudson's Bay in April of 1879 with 4 Caucasians, 3 families of Inuits, and 3 heavily laden dog sleds. Totaling 18 people, they started out with a month's supply of food (mostly walrus blubber) and a prodigious supply of ammunition for their hunting rifles. After covering over 3000 miles on foot over ice, snow and tundra, all 18 members of the original party plus their 44 dogs returned to Hudson's Bay in March of 1880. Once their initial provisions were depleted, the expedition's only source of additional food was hunting and fishing, as there were no other sources of supply along their route.

The leader of this expedition, Lt. Frederick Schwatka, was a graduate of both West Point and Bellevue Hospital Medical College. His summary of the expedition was published as a news article in the *New York Herald* in the Fall of 1880, but his written diary was lost for 85 years until its discovery and publication by the Marine Historical Association of Mystic CT in 1965 \[6\]. This fascinating 117-page saga describes how Schwatka, a frontiersman and U.S. Army surgeon, collaborated with his Inuit guides to accomplish a remarkable feat of physical endurance.

In one notation, Schwatka provides an interesting insight into his weaning from their initial supply of carbohydrate-containing food.

"*When first thrown wholly upon a diet of reindeer meat, it seems inadequate to properly nourish the system, and there is an apparent weakness and inability to perform severe exertive fatiguing journeys. But this soon passes away in the course of two or three weeks."*

This observation, written a century before the current author first came to grips with the issue of "keto-adaptation", offers an early clue to resolve the dichotomy between impaired performance with low carbohydrate diets in the laboratory and their lack of debilitating effects when taken among people practiced in their use. That Schwatka was not impaired by his prolonged experience eating meat and fat is evidenced by his diary entry for the period 12–14 March 1880, during which he and an Inuit companion walked the last 65 miles in less than 48 hours to make a scheduled rendezvous with a whaling ship and complete his journey home.

Twenty-six years later, a Harvard-trained anthropologist named Vilhjalmur Stefansson entered the Arctic with the purpose of studying the Inuit language and culture. Having been born in 1879 in Manitoba and grown up in North Dakota, it is unlikely that Stefansson was aware of the Schwatka expedition or its reported technique of extended dogsled travel while living by hunting. However when separated from his expedition and thus his source of supply over the winter of 1906–7, Stefansson was taken in by a group of Inuit on the Canadian Arctic coast. With the arrival of spring in June of 1907, he both spoke their language and had acquired their skill of living and traveling by dogsled on a hunter's diet.

For the next decade, Stefansson traveled extensively over the arctic mainland and among the islands to the north. During this period, he was away from the outposts of European settlement for periods of up to 18 months at a time, and in the remote regions of the Canadian Arctic he lived with groups of Inuit for whom he was the first European they had met.

Stefansson wrote extensively about these experiences in both the scientific literature and in books for the lay public \[7\]. One of the main themes of his writing was the adaptation of the Inuit culture to survive as nomadic groups in the arctic on a diet consisting solely of the products of hunting and fishing. Coming as it did in the same time period that the science of nutrition was blossoming with the discovery and characterization of vitamins (eg, the first vitamin to be chemically defined was thiamin by Funk in 1911), Stefansson's claim that one could live and function well on the products of just one food group caused tremendous controversy \[8\].

Subjected to great criticism and even scorn, Stefansson agreed to recreate the Inuit diet under scientific observation. Therefore, for the calendar year of 1929 he and a colleague from his arctic explorations ate a diet consisting of meat and fat for 12 months. This experiment, supervised by Dr. Eugene DuBois, was conducted at Bellevue Hospital in New York. For the first 3 months of this study, the two explorers were under constant observation to guarantee dietary compliance, after which they were allowed more freedom of movement but with frequent tests to document that they remained in ketosis. This study was reported in multiple peer-reviewed publications, the primary reports being published in the Journal of Biological Chemistry in 1930 \[9,10\], As noted by DuBois \[8\], the study results were essentially "negative", in that both subjects survived the 12 months in apparent good health, having no signs of scurvy (which was predicted to occur within the first 3 months) or other deficiency diseases.

It is interesting to note from the careful observations published from the Bellevue study that Stafansson ate relatively modestly of protein, deriving between 80–85% of his dietary energy from fat and only 15–20% from protein \[9\]. This was, and still remains, at odds with the popular conception that the Inuit ate a high protein diet, whereas in reality it appears to have been a high fat diet with a moderate intake of protein. In his writings, Stefansson notes that the Inuit were careful to limit their intake of lean meat, giving excess lean meat to their dogs and reserving the higher fat portions for human consumption \[11\].

It is also interesting to conjecture that the vigorous defense of his arctic observations by Stefansson may have led indirectly to the development of the carbohydrate loading hypothesis. Stefansson was a polarizing influence in the field of nutrition, and his advocacy of pemmican as an emergency ration for troops during the Second World War led directly to the Kark study quoted above, which in turn was a predecessor to many comparative dietary trials performed in Europe and the U.S. in later decades.

# Modern ketogenic diet performance studies

There was a resurgence of interest in very low calorie ketogenic diets for weight loss in the 1970's, followed closely by the complications (including sudden death) associated with the Liquid Protein diet popularized in 1976. However, the fatigue and apparent cardiac dysfunction caused by this collagen-based fad diet stood in stark contrast to the published experience of arctic explorers such as Schwatka and Stefansson. In addition, physicians who monitored patients following very low calorie diets observed wide variations between the exercise-tolerance of these individuals.

Given that the elegant research on the metabolism of total fasting by Dr. George Cahill and colleagues had demonstrated that full adaptation of nitrogen, fat, and carbohydrate metabolism required a number of weeks \[12\], it seemed reasonable to hypothesize that exercise tolerance would take more than a week to recover after removal of carbohydrate from the diet. This view was supported by the subsequent discovery of the prescient adaptation quote from Schwatka's diary \[6\] noted above.

To test this hypothesis, the current author (under the mentorship of Drs. Ethan Sims and Edward Horton at the University of Vermont) undertook a study of subjects given a very low calorie ketogenic diet for 6 weeks in a metabolic research ward \[13\]. The protein for this diet, along with a modicum of inherent fat, was provided by lean meat, fish, and poultry providing 1.2 grams of protein per kg of reference ("ideal") body weight daily. In addition, mindful that the natriuresis of fasting could reduce circulating blood volume and cause secondary renal potassium wasting, the subjects were prescribed 3 grams of supplemental sodium as bouillion and 25 mEq (1 g) of potassium as bicarbonate daily.

Treadmill performance testing of these subjects included determinations of peak aerobic power (VO~2~max) after a 2-week weight maintenance baseline diet, and again after 6 weeks of the ketogenic weight loss diet. Endurance time to exhaustion was quantitated at 75% of the baseline VO~2~max. This endurance test was repeated again after one week of weight loss and finally after 6 weeks of weight loss. Other than these tests, the subjects did no training exercise during their participation in this study. To compensate for the fact that the average subject had lost over10 kg, the final endurance treadmill test was performed with the subject carrying a backpack equivalent in weight to the amount lost.

The energy expenditure data (expressed as oxygen consumption) and exercise times across this 8-week inpatient study are shown in Table 1. That these subjects'peak aerobic power did not decline despite 6 weeks of a carbohydrate-free, severely hypocaloric diet implies that the protein and mineral contents of the diet were adequate to preserve functional tissue. As can be noted, endurance time to exhaustion was reduced after one week of the ketogenic diet, but it was significantly increased over the baseline value by the 6-week time point. However the interpretation of this endurance test is confounded by the fact that the oxygen cost (ie, energy cost) of the treadmill exercise had significantly decreased following the weight loss, and this occurred despite the subjects being made to carry a backpack loaded to bring them back to their initial exercise test weight.

Exercise parameters of Vermont study \[13\]

|                          | **Baseline** | **Week 1** | **Week 6** |
|--------------------------|--------------|------------|------------|
| **VO~2~max** (LPM)       | 2.49         | --         | 2.49       |
| **Exercise VO~2~**(LPM)  | 1.88\*       | 1.71       | 1.50\*     |
| **Endurance time** (min) | 168^+^       | 130        | 249^+^     |

LPM, liter per minute \*week 6 \< baseline, P \< 0.05 ^+^week 6 \> baseline, P \< 0.01

This question of improved efficiency notwithstanding, it is clear that our subjects experienced a delayed adaptation to the ketogenic diet, having reduced endurance performance after one week followed by a recovery to or above baseline in the period between one and six weeks. Given the reduced energy cost of the exercise despite the backpack, the extent of this adaptation cannot be determined from this study. To explain this improved exercise efficiency, we can speculate that humans are more efficient carrying weight in a modern backpack than under their skin as excess body fat. It is also possible that these untrained subjects became more comfortable with prolonged treadmill walking by their third test, and therefore improving their overall efficiency.

Given the uncertainties of this study caused by the subject's weight loss and potential for improved technique with multiple tests, the current author undertook a second study under the mentorship of Dr. Bruce Bistrian at MIT in Cambridge MA \[14,15\]. The diet employed in this followup study was patterned after that consumed by Stefansson during his year in the Bellevue study (and thus presumably close to that traditionally consumed by the Inuit) with the intention that the subjects would be in ketosis without weight loss.

This second study utilized competitive bicycle racers as subjects, confined to a metabolic ward for 5 weeks. In the first week, subjects ate a weight maintenance (eucaloric) diet providing 67% of non-protein energy as carbohydrate, during which time baseline performance studies were performed. This was followed by 4 weeks of a eucaloric ketogenic diet (EKD) providing 83% of energy as fat, 15% as protein, and less than 3% as carbohydrate. The meat, fish, and poultry that provided this diets protein, also provided 1.5 g/d of potassium and was prepared to contain 2 g/d of sodium. These inherent minerals were supplemented daily with an additional 1 g of potassium as bicarbonate, 3 grams of sodium as bouillon, 600 mg of calcium, 300 mg of magnesium, and a standard multivitamin.

The bicyclist subjects of this study noted a modest decline in their energy level while on training rides during the first week of the Inuit diet, after which subjective performance was reasonably restored except for their sprint capability, which remained constrained during the period of carbohydrate restriction. On average, subjects lost 0.7 kg in the first week of the EKD, after which their weight remained stable. Total body potassium (by ^40^K counting) revealed a 2% reduction in the first 2 weeks (commensurate with the muscle glycogen depletion documented by biopsy), after which it remained stable in the 4^th^ week of the EKD. These results are consistent with the observed reduction in body glycogen stores but otherwise excellent preservation of lean body mass during the EKD.

The results of physical performance testing are presented in Table 2. What is remarkable about these data is the lack of change in aerobic performance parameters across the 4-week adaptation period of the EKD. The endurance exercise test on the cycle ergometer was performed at 65% of VO~2~max, which translates in these highly trained athletes into a rate of energy expenditure of 960 kcal/hr. At this high level of energy expenditure, it is notable that the second test was performed at a mean respiratory quotient of 0.72, indicating that virtually all of the substrate for this high energy output was coming from fat. This is consistent with measures before and after exercise of muscle glycogen and blood glucose oxidation (data not shown), which revealed marked reductions in the use of these carbohydrate-derived substrates after adaptation to the EKD.

Exercise parameters of MIT EKD study \[15\]

|  | **VO~2~max** (LPM) | **Exercise VO~2~**(LPM) | **Exercise RQ** | **Endurance time** (min) |
|----|----|----|----|----|
| **Baseline** | 5.1 | 3.18 | 0.83\* | 147 |
| **EKD-4** | 5.0 | 3.21 | 0.72\* | 151 |

LPM, liter per minute \* P \< 0.01

Examining the results of these two ketogenic diet performance studies together indicates that both groups experienced a lag in performance across the first week or two of carbohydrate restriction, after which both peak aerobic power and sub-maximal (60–70% of VO~2~max) endurance performance were fully restored. In both studies, one with untrained subjects and the other with highly trained athletes who maintained their training throughout the study, there was no loss of VO~2~max despite the virtual absence of dietary carbohydrate for 4–6 weeks. This whole-body measure of oxidative metabolism could not be maintained unless there was excellent preservation of the full complement of functional tissues including skeletal muscle (and mitochondrial) mass, circulating red cell mass, and cardiopulmonary functions.

The possibility raised by the first study of improved endurance time after keto-adaptation was not substantiated by the second study employing highly trained athletes without the complicating variable of major weight loss. It is thus likely that the increased endurance time in the Vermont study was due to improved efficiency (ie, less hobbling from a backpack than from an equal weight of internal body fat) and/or improved acclimation to the endurance test procedure. Such acclimation would not be expected in the second study, as the highly trained bicycle racers were well conditioned to the stationary ergometer at the start of the study. It is also worth noting that the bicycle racers remained weight stable (excepting the half kilogram of reduced muscle glycogen) across the 4 weeks of the EKD, which was equi-caloric with the baseline diet. Although 4 weeks is a relatively short period to assess small differences in energy efficiency between diets, this observation implies that there was no great reduction in the efficiency of energy metabolism after keto-adaptation.

As a final note in this section, neither the Vermont study nor the MIT study has been refuted in the 2 decades since their publication. Understandably given the expense of human metabolic ward studies and the orthogonal conclusions of these two studies, neither study has been corroborated by a similar human study. However two subsequent animal studies examining physical performance after keto-adaptation have yielded results consistent with those presented above \[16,17\].

# Resolving the performance paradox

There are three factors that can help us explain the paradox presented by studies showing superior performance with high carbohydrate diets versus the present author's two studies noted above.

## Adaptation

The most obvious of these is the time allotted (or not) for keto-adaptation. In this context, the prescient observation of Schwatka (that adaptation to "a diet of reindeer meat" takes 2–3 weeks) says it all. None of the comparative low-carbohydrate versus high-carbohydrate studies done in support of the carbohydrate loading hypothesis sustained the low carbohydrate diet for more than 2 weeks \[5\], and most (including the classic report of Christensen and Hansen \[2\]) maintained their low-carbohydrate diets for 7 days or less.

There are to date no studies that carefully examine the optimum length of this keto-adapataion period, but it is clearly longer than one week and likely well advanced within 3–4 weeks. The process does not appear to happen any faster in highly trained athletes than in overweight or untrained individuals. This adaptation process also appears to require consistent adherence to carbohydrate restriction, as people who intermittently consume carbohydrates while attempting a ketogenic diet report subjectively reduced exercise tolerance.

## Sodium and potassium

The second factor differentiating the author's studies from many others is optimized mineral nutriture, which has benefits for both cardiovascular reserve in the short term and preservation of lean body mass and function over longer time periods. The Inuit people lived much of the year on coastal ice (which is partially desalinated sea water), and much of their food consisted of soup made with meat in a broth from this brackish source of water. When they went inland to hunt, they traditionally added caribou blood (also a rich source of sodium) to their soup. With these empirically derived techniques, the Inuit culture had adapted the available resources to optimize their intakes of both sodium and potassium.

When meat is baked, roasted, or broiled; or when it is boiled but the broth discarded, potassium initially present in the meat is lost, making it more difficult to maintain potassium balance in the absence of fruits and vegetables. Because our research subjects were accustomed to eating meat, fish, and poultry prepared as something other than soup, we chose to give them most of their sodium separately as bouillon and a modest additional supplement of potassium as potassium bicarbonate. With these supplements maintaining daily intakes for sodium at 3–5 g/d and total potassium at 2–3 g/d, our adult subjects were able to effectively maintain their circulatory reserve (ie, allowing vasodilatation during submaximal exercise) and effective nitrogen balance with functional tissue preservation.

An example of what happens when these mineral considerations are not heeded can be found in a study prominently published in 1980 \[18\]. This was a study designed to evaluate the relative value of "protein only" versus "protein plus carbohydrate" in the preservation of lean tissue during a weight loss diet. The protein only diet consisted solely of boiled turkey (taken without the broth), whereas the protein plus carbohydrate consisted of an equal number of calories provided as turkey plus grape juice. Monitored for 4 weeks in a metabolic ward, the subjects taking the protein plus carbohydrate did fairly well at maintaining lean body mass (measured by nitrogen balance), whereas those taking the protein only experienced a progressive loss of body nitrogen.

A clue to what was happening in this "Turkey Study" could be found in the potassium balance data provided in this report. Normally, nitrogen and potassium gains or losses are closely correlated, as they both are contained in lean tissue. Interestingly, the authors noted that the protein only diet subjects were losing nitrogen but gaining potassium. As noted in a rebuttal letter published soon after this report \[19\], this anomaly occurred because the authors assumed the potassium intake of their subjects based upon handbook values for raw turkey, not recognizing that half of this potassium was being discarded in the unconsumed broth. Deprived of this potassium (and also limited in their salt intake), these subjects were unable to benefit from the dietary protein provided and lost lean tissue. Also worthy of note, although this study was effectively refuted by a well-designed metabolic ward study published 3 years later \[20\], this "Turkey Study" continues to be quoted as an example of the limitations of low carbohydrate weight loss diets.

## Protein dose

The third dietary factor potentially affecting physical performance is adjusting protein intake to bring it within the optimum therapeutic window for human metabolism. The studies noted herein \[13-15,20\] demonstrate effective preservation of lean body mass and physical performance when protein is in the range of 1.2 – 1.7 g/kg reference body weight daily, provided in the context of adequate minerals. Picking the mid-range value of 1.5 g/kg-d, for adults with reference weights ranging from 60–80 kg, this translates into total daily protein intakes 90 to 120 g/d. This number is also consistent with the protein intake reported in the Bellevue study \[9\]. When expressed in the context of total daily energy expenditures of 2000–3000 kcal/d, about 15% of ones daily energy expenditure (or intake if the diet is eucaloric) needs to be provided as protein.

The effects of reducing daily protein intake to below 1.2 g/kg reference weight during a ketogenic diet include progressive loss of functional lean tissue and thus loss of physical performance, as demonstrated by Davis et al \[21\]. In this study, subjects given protein at 1.1 g/kg-d experienced a significant reduction in VO~2~max over a 3 month period on a ketogenic diet, whereas subjects given 1.5 g/kg-d maintained VO~2~max.

At the other end of the spectrum, higher protein intakes have the potential for negative side-effects if intake of this nutrient exceeds 25% of daily energy expenditure. One concern with higher levels of protein intake is the suppression of ketogenesis relative to an equi-caloric amount of fat (assuming that ketones are a beneficial adaptation to whole body fuel homeostasis). In addition, Stefansson describes a malady known by the Inuit as rabbit malaise \[8\]. This problem would occur in the early spring when very lean rabbits were the only available game, when people might be tempted to eat too much protein in the absence of an alternative source of dietary fat. The symptoms were reported to occur within a week, and included headache and lassitude. Such symptoms are not uncommon among people who casually undertake a "low carbohydrate, high protein" diet.

# Conclusions

Both observational and prospectively designed studies support the conclusion that submaximal endurance performance can be sustained despite the virtual exclusion of carbohydrate from the human diet. Clearly this result does not automatically follow the casual implementation of dietary carbohydrate restriction, however, as careful attention to time for keto-adaptation, mineral nutriture, and constraint of the daily protein dose is required. Contradictory results in the scientific literature can be explained by the lack of attention to these lessons learned (and for the most part now forgotten) by the cultures that traditionally lived by hunting. Therapeutic use of ketogenic diets should not require constraint of most forms of physical labor or recreational activity, with the one caveat that anaerobic (ie, weight lifting or sprint) performance is limited by the low muscle glycogen levels induced by a ketogenic diet, and this would strongly discourage its use under most conditions of competitive athletics.

# List of abbreviations

VO~2~max – maximum aerobic capacity

RQ – respiratory quotient

EKD – eucaloric ketogenic diet

# Competing interests

None declared.