Written by Rashid Mansoor

12 min read

Growth Curves and an Exponential Pandemic

  • Life Sciences

How do you stop the spread of a contagious pathogen throughout a heavily globalised world?

This is a question that is currently plaguing governments and organisations around the world as the exponential growth of Coronavirus continues. Nearly every country has implemented different means and measures in order to curb the spread or ‘flatten the curve’.

Despite some countries implementing what seem to be draconian measures, the spread of the virus is yet to still be fully contained in many areas of the world. Regardless of whether you look towards Sweden encouraging herd immunity, or India implementing a total lockdown – countries everywhere are trying to do what they think is right to help their populations survive.

An Empty Munich Subway – ‘The Great Empty’ by The New York Times

Having recently taken into account many of the articles that are listing pieces of information surrounding the spread of the virus, I was interested to look at some of the comparative data and how technology can be applied in such circumstances in order to enable better and more effective decision-making.

One item that has come up repeatedly is the comparable trend of cases and deaths from the disease between the United Kingdom and Italy.

Italy is arguably the country that has been worst affected by the disease so far. The Mediterranean country has had a severe number of cases and deaths in only a matter of weeks and, despite a total lockdown, Italy has had far more deaths attributable to coronavirus than China did at the same stage.

One popular argument for the main reason why the fatality rate is so high in Italy (an average of ~10%), in comparison to other nations (an average of ~4.5%), is because of the country’s ageing population and high amounts of ‘intergenerational contact’ – regular social contact between the old and young.

Researchers from the University of Oxford argued that Italy has one of the oldest populations in the world, with 23.3% of its population over the age of 65 – compared to 12% in China.

However, this only tells part of the story.

A common question that has come up time and time again is whether Italy started testing early enough. Researchers at the Sacco Hospital, Milan isolated a strain of the virus from an Italian patient in early March. This suggests that COVID-19 may have circulated the country, but primarily northern parts of the country, for weeks before it was detected.

This means that, when the reported quarantine of 16 million people in northern Italy was about to be implemented, a mass of the population fled to the southern part of the country to escape the lockdown – it makes the whole spread of the virus even harder to manage and track.

The Italian government has taken the harshest measures to curb the spread of the disease aside from China. Some of the measures include authorities banning any movement inside the country along with banning public gatherings and closing schools across the country.

So why are the comparisons being drawn between the two countries?

Picture 5

Italy & UK Cumulative Deaths – tracking against the cumulative number of deaths (Y-Axis) and the days from the first recorded death (X-Axis)

One reason is how closely the number of victims tracks between the two nations. The total number of deaths in Italy rose from 233 to 4,032 within two weeks, and as of the 21st March, there were 233 deaths from the virus in Britain and by the 31st this number had risen to 1789. As such, many have drawn the conclusion that the UK is to follow in Italy’s footsteps.

However, it’s not a perfect match.

The two data sets do not correlate exactly with UK deaths rising at a lower rate to Italy’s – ~30% per day to ~37% per day respectively. Likewise, despite Boris Johnson’s comparisons between the two countries many experts have emphasised the difficulty in drawing accurate conclusions about the virus by comparing what is known in other countries. It must also be noted that variations in health systems and demographics may well contribute to differences in the number of deaths.

So then, one could say the Italy-UK comparison is incorrect.

Perhaps, but unfortunately, research conducted by the Imperial College COVID-19 Response Team point towards some incredibly worrying numbers;

“...even if all patients were able to be treated, we predict there would still be in the order of 250,000 deaths in GB, and 1.1-1.2 million in the US.” – Imperial College COVID-19 Response Team

The important thing in a rapidly growing phenomenon such as an epidemic is the shape of the curve rather than absolute numbers. Looking at the general shape of the graph, the UK’s curve is still a worrying exponential.

For the UK’s forecast to be truly different from Italy the shape of the curve would need to show a flattening trend – a slower exponential curve is still exponentially and given a bit more time likely to saturate the population and overwhelm our best efforts to keep up with restorative measures.

There is currently not any good indication that this is happening by looking at daily rather than cumulative death rates.

Picture 4
Italy & UK Deaths by Day – tracking against the number of deaths-per-day (Y-Axis) and the days from the first recorded death (X-Axis)

Another thing to keep in mind is that the UK’s trend lines are relatively recent. We have about 2.5 weeks of data and the shape of the curve this early is more susceptible to the random variance that will get levelled out over the longer term.

So then, another question; why have other countries such as South Korea (1.3%), Singapore (0.3%), and Germany (0.4%) seen such low mortality rates in comparison?

South Korea, in particular, is an interesting case. With a population of ~51 million, their total deaths have followed a linear curve unlike Italy (~60 million) and the UK (~66 million) which both have comparable populations.

South Korea’s outbreak began like Italy’s, with a slow drip of cases soon turning into a dramatic escalation – at one point the country had an infection rate of 0.614% which is huge by comparison to other countries.

However, when you look at the infection rate today, South Korea has moved from an exponential to a largely linear trend. Data a week ago suggested the curve may turn logarithmic which usually indicates that an exponential trend has peaked or saturated and is beginning to level.

The linear trend it currently displays suggests that the trend may be picking up again, and in fact, the last ten or days shows an upward ascent to the curve moving towards super-linear growth. Or it may be that the strategies currently employed are sufficient to preserve the linear trend long-term and thus South Korea indeed is past the brunt of the pandemic.Picture 2

South Korea Cumulative Deaths – tracking against the number of deaths (Y-Axis) and the days from the first recorded death (X-Axis).

There are not any true exponentials in nature. The exponential trends we see, whether bacteria in a petri dish or Moore’s law, tend to be the early phase of a type of curve called a Sigmoid. When a growth process has hit saturation point the sigmoid curve moves from its exponential phase to a logarithmic phase. The linear trend seen in South Korea suggests that the cause of this is due to the methods of limiting the spread rather than due to oversaturation.

We will need to see linear or logarithmic trends in every country to know whether implemented methods are working.

Why is the growth rate in South Korea currently so slow? Well, the low fatality rate could be to do with the demographics who have contracted the disease. In South Korea, nearly a third of the cases have been people who are relatively young – between the ages of 20 and 29.

This, combined with a rigorous contact tracing programme and the quarantining of anyone the carrier has come into contact with has led to the sharp curbing of the spread of the disease.

A similar rationale has been suggested for Germany where their ‘super spreaders’ – those who picked up the disease early whilst skiing in Austria or Italy – were younger and healthier people, less likely to die from symptoms.

Looking at the numbers Germany, however, looks no better than the UK or Italy.

Their absolute numbers are lower but the shape of the curve is a very similar exponential. It has been suggested that Germany has far more infections and a far lower death rate, but as testing practices vary widely between countries, absolute infection rates are less reliable than deaths confirmed to be COVID-19 related.Picture 3

Cumulative Deaths & Deaths by Day Comparison – tracking against the cumulative / total number of deaths (Y-Axis) and the days from the first recorded death (X-Axis)

Germany is only four days behind the UK and is tracking astonishingly closely.

The really worrying trend, however, is not even Italy, but Spain. The gradient of the curve is well beyond anything we saw with Italy. Within a week, Spain looks set to outstrip Italy and become the epicentre of the pandemic – at least in Europe.

This is despite Spain enacting strict measures very early, imposing a shutdown on its entire 46 million population in mid-March.

Picture 1

Cumulative Deaths & Deaths by Day Comparison – tracking against the cumulative / total number of deaths (Y-Axis) and the days from the first recorded death (X-Axis)

So then, is the only way to stop the spread successfully to implement severe, country-wide quarantines?

In short, the answer is no but it depends on when it's implemented and it seems the traditional approaches to regular flu seasons will not be enough to curb the spread.

Professor Hugh Montgomery, a professor of medicine and the director of the University College London Institute for Human Health and Performance, recently told Channel 4’s Dispatches that the contagion rate of the coronavirus is far higher than that of the standard flu – 3 to 1.3-1.4 respectively.

By transmitting it to a further ten people at a rate of between 1.3-1.4, one person could be responsible for 14 separate cases of the flu – as those ten people pass it onto the next cohort of victims. By contrast, with coronavirus, that number skyrockets to 59,000 people.

A common technique is social distancing – measures people are encouraged to take to reduce social interaction between people. The UK Government has released more comprehensive guidance for individuals to follow, but in short, it follows three key recommendations;

  1. Only go outside for food, health reasons or work (where this absolutely cannot be done from home).
  2. Stay 2 metres (6ft) away from other people.
  3. Wash your hands as soon as you get home.

However, despite encouraging social distancing, populations in many major cities across the US and UK have been unwilling/unable to adhere to the measures. So without more stringent guidelines from governments, is social distancing just a fantasy?

Donald McNeil of The New York Times, recently explained how the virus could be stopped in a couple of weeks if everyone adhered to social distancing.

“Yeah, because within two weeks, the virus would die out on every surface that it was. People wouldn’t be interacting, so they wouldn’t transmit it. And everybody who has symptoms, the symptoms turn up in two weeks at the most. So you’d know who was sick. And even for the few asymptomatics, you’d be able to find them by doing tests. And so that’d be it. Epidemic over.” – Donald G. McNeil Jr, New York Times

Likewise, further research from the Imperial College COVID-19 Response Team that China “after very intense social distancing” have been able to curb the spread of the virus. And, by limiting the number of contact people have with each other, Public Health England has stated that it is possible to slow down the spread of the coronavirus.

So, we know which countries are implementing effective strategies and we also know that the most effective thing that we can do at this present moment is to adhere to the recommended social distancing measures.

Is there, however, anything that could be done differently in the future in order to stop the spread earlier?

Using mathematical models in epidemiology is nothing new. John Graunt was the first scientist who systematically tried to quantify causes of death back in 1662. However, to many, purely mathematical models that present numerical answers to these complex questions may appear as a black box – seemingly unrealistic, unhelpful, or confusing.

However, given the rise of new technologies over the past decade, the question arises whether there is a better way. A better way of taking the raw inputs and data points, and coming up with more accurate models faster.

A team of researchers led by London's Royal Veterinary College created epidemix, an interactive, online application allowing non-specialists to visualise disease transmission. At Hadean, we have applied our spatial simulation solution, Aether Engine, to design an initial simulation involving 100,000 entities in just three days.



Agent-based models have already been applied to map the spread of various diseases including the Ebola outbreak in West Africa in 2014, the seasonal spread of flu, and the Zika virus. All of these are with the distinct intention of helping policymakers make more informed decisions.

These tools were not viable or available to us fully, five or even ten years ago. However, due to the rise of cloud computing, we are able to harness more computing power than ever before which, at this current moment, we are not using fully.

The coronavirus is most certainly an unprecedented situation. Not since the outbreak of the Spanish Flu over 100 years ago has the world been collectively affected by such an epidemic.

There are, however, key differences around the world in regards to the spread and the effect of the virus caused by different social norms and demographics. In order to curb the spread and flatten the curve, it is integral governments implement and police the necessary social distancing measures.

However, at the moment this is purely reactive. In the future, policymakers and organisations alike need to use better modelling methodologies, using the tools we now possess in order to map the prospective outcomes of these diseases more effectively and earlier.

The impact of these new tools could mean saving thousands of lives.


We are delighted to announce that we are renewing our partnership with the Francis Crick Institute. The project will combine analysis of person-to-person interaction with insight into how COVID-19 transmits within an individual, providing a complete picture of the pathogen’s spread. Find out more here.