What is exponential growth, and why is it relevant to epidemics? Kit Yates explains

Although the concept of exponential growth is not new in the public consciousness, a lot of misconceptions surround the idea. Exponential is often incorrectly used as a byword for rapid or large. It doesn’t have to be either of those. In fact, this is what is so deceptive about exponential growth. When it first starts it doesn’t look fast at all. 

When the delta variant of covid-19 first arrived in the UK, its week-on-week rise was small. On the 20 March 2021, just two cases were sequenced by the COVID-19 Genomic consortium (COG-UK). A week later it was 18—still small numbers, seemingly nothing to worry about. The next week it had jumped to 60 and the week after to 186, and then to 369. By the 1 May 2021 COG-UK sequenced 735 cases of the delta variant, accounting for 12% of all sequenced cases. Although these were just a fraction of the cases spreading in the community, the trend was clear. Sequenced cases were doubling every week. By that point it was too late to stop its spread. The delta variant now comprises 99% of all sequenced cases in the UK.

But what, precisely, is exponential growth and why is it relevant to epidemics? The mathematical definition says that a quantity that increases with a rate proportional to its current size will grow exponentially. This means that as the quantity increases so does the rate at which it grows. The more infected people there are, the more people they will infect, and the faster the cases will rise.

In March 2020, cases in the UK were thought to be doubling approximately every three and a half days. If this were true, locking down just a week earlier would have meant entering the lockdown with a quarter of the cases. Tens of thousands of lives would have been saved. This is the terrifying power of exponential growth.

When a disease spreads through a largely susceptible population, we can characterise its growth using the reproduction number, R. R tells us the number of new infections each infectious individual might expect to seed in the population during the course of their infectious period. If R is below one then cases will fall, but if R is above one, then cases will rise—exponentially. 

Sadly, this is the situation we find ourselves in again as covid spreads through the younger, largely unvaccinated population. For the UK, R, is currently estimated to be between 1.2-1.5. This suggests that the number of cases is growing by between 3% and 7% per day. 

That we will see a huge third wave is explicitly acknowledged by the government. Sajid Javid, the secretary of state for health and social care, expects to see 100,000 cases per day during the summer, despite not being able to put a figure on the resultant levels of hospital admissions. This has been rationalised by suggesting that the link between cases and hospitalisations has been broken. 

While vaccines have definitely weakened the link, they have not broken it. SAGE and SPI-M (its modelling subcommittee) suggested that from the start of April to early June the case-to-hospitalisation rate stayed approximately constant and that if this continued, exponential growth in cases would lead to exponential rises in hospitalisations. While it is likely that the link between hospitalisations and cases will continue to weaken as we vaccinate more, we can still expect to see rises in hospitalisation, which put pressure on an already stressed NHS. Indeed, we are seeing hospital admissions start to accelerate. Currently we are seeing over 500 new admissions per day with a week-on-week rise in admissions of over 55%. Hospitals in some regions are already having to cancel planned operations, including cancer care. Although deaths are at relatively low levels (compared to the thousands who were dying from COVID everyday in January) they are creeping upwards, growing 43% in the last week.

Of course, hospitalisation and death are not the only metrics against which to judge the impact of increased case numbers. Higher rates will disproportionately impact on younger unvaccinated people who have already lost so much to the ravages of the pandemic. Not only will our children suffer further educational disruption due to unmitigated transmission in schools, but we risk creating a generation impacted by chronic health problems. Deprived communities, who are disproportionately affected by covid, will continue to be disadvantaged by the government’s “Freedom Day” strategy. On top of this, allowing high levels of transmission in a partially vaccinated population increases the risk of vaccine resistant variants with the potential to undo much of the hard work carried out by the NHS through the vaccination programme.

The one mathematical consolation is that nothing can grow exponentially forever. Eventually, when enough people gain immunity (providing there are no immune evading variants, which is not a given) the virus will be denied the contacts between susceptible and infected people that it needs to continue to spread. But why we would try to achieve these high levels of immunity through natural infection, with all the attendant risks and none of the benefits of  allowing everyone for whom it is safe to be doubly vaccinated, is beyond most people’s comprehension.

As conditions change, we would expect the rate of growth to change. Schools breaking up for the summer will slow the rates of increase of cases as will increased immunity. On the other hand, further unlocking on 19 July 2021 risks accelerating the unmitigated spread of the virus, posing an ‘exponential’ threat to the health and wellbeing of a nation.

Kit Yates, senior lecturer, Department for Mathematical Sciences, University of Bath. Twitter: @Kit_yates_maths

Competing interests: KY is a member of Independent SAGE.