As anticipated, the Coronavirus outbreak has definitively entered new phases in both the UK and the US, in the wake of ‘lockdown’ restrictions limiting non-essential production and consumption and encouraging populations to stay at home. These policies are centralised in the UK and State-wide in most of the US. As we can see from Chart 1 below, the log curve of cases and deaths is steadily flattening to the extent that we no longer have exponential growth of cases and deaths. It seems fairly clear that significant bending of the case curves took place after widespread social restrictions were imposed, and that the death curves followed with around a 10-day lag.
It’s too early to lift restrictions
It is utterly irresponsible to suggest, as many are doing (Trump supporters in the US and the usual Brexit suspects in the UK) that this means that we are in the clear and can now start to contemplate lifting restrictions on gathering socially, for work and on public transport. We do not have exponential growth, but despite all our efforts we still have steady linear growth in cases and deaths as indicated in Chart 2 below, and more clearly for deaths in Chart 3. Linear growth means that roughly constant numbers of people are acquiring the disease and are dying from it daily.
It is notable the extent to which the UK and US patterns of Covid-19 growth are now following each other, with similar rates of case growth and of the rise in deaths from the Coronavirus epidemic. (Chart 1)
Whilst both the UK and the US can be said to have been
slow in initiating forceful measures to deal with the Coronavirus epidemics in
their countries, the UK government under Prime Minister Boris Johnson has now,
albeit not always with the necessary clarity, announced shutdowns of most
social-mixing in Britain. Only ‘essential workers’ – a term yet to be
satisfactorily defined – should be leaving their homes except for shopping for
necessary goods, healthcare access or for suitably distancing exercise. There
are problems in obtaining adequate protective equipment for front-line health-workers,
but central government do seem to be making some effort to address this.
Unfortunately, in the US, the leadership vacuum in the
shape of Donald Trump is more interested in spreading misinformation and
bigging-up his own desultory (frankly negative) role in efforts to combat the virus
and its consequences. Any effective efforts to counsel isolation of those with
symptoms and general social-distancing, and to provide additional equipment and
space in anticipation of the inevitable rise in the number of cases needing
hospitalisation, have been taken by State governors, and mainly Democratic
ones. Worse still, Trump is now touting the idea that ‘the cure is worse than
the disease’ and that such restrictions as there are should be relaxed after another
two weeks to allow business (and from his point of view the stock market) to recover.
Apart from the fact that cases and deaths will almost certainly still be rising
at that time, the degree of complacency this signals is likely to be extremely
damaging to ongoing suppression efforts in the United States.
Some of the most important information about the Coronavirus (Covid-19) epidemic is to be found not from medical knowledge or in the lab but from basic mathematics. The key to understanding this behaviour is in the mathematics of exponential growth. What does this mean? There are two ways in which regular increases of anything can occur – either by constant addition – arithmetic growth – or by constant multiplication – exponential growth. We can illustrate the difference by starting from 1. If there is daily arithmetic growth of 2, then on the second day the total will be 1 + 2, so 3, on the third day the total will be 1 + 2 + 2, so 5, on the fourth day 1 + 2 + 2 + 2, so 7, and so on. If there is daily exponential growth of 2, then on the second day the total will be 1 × 2, so 2, on the third day 1 × 2 × 2, so 4, on the fourth day 1 × 2 × 2 × 2, so 8, and so on. The difference is in the sign – a plus sign in the case of arithmetic growth, a multiplication sign in the case of exponential growth. As is made clear by Chart 1 below, although the arithmetic growth gives higher totals initially, exponential growth very quickly afterwards leads to higher and rapidly increasing values.
Epidemics cause exponentially increasing numbers of
cases because for every person who is infected, that person can in turn infect
another. The number of people each infected person in turn infects every day multiplies
the number of cases. If we start off with one person who then infects one
other over 24 hours, and these two each infect another over the following 24
hours, and all four infected each in turn infect one other the next day, and so
on, then we have the daily exponential growth of 2 we described above. This
might be quite an extreme epidemic, but in any case where the number of new
infections is increasing each day, the growth will be exponential, rather than