During a viral outbreak, the total number of infections follows what is known as a logistic function, an S-shaped curve characterised initially by exponential growth and then flattening out as either the outbreak is brought under control through external interventions such as quarantine, or the virus spreads through so much of the population that there are no new host possibilities. China’s Hubei province – in which the city of Wuhan was the initial source of the viral outbreak – illustrates an almost perfect logistic function. The virus runs rampant through the population (the exponential element of the function) before severe quarantine measures bring it under control.
Unlike pure exponential functions where the number of infected could continue to grow forever, logistic functions have a changing slope, calculated by dividing the total number of cases today by the total number of cases yesterday. This rate of change (let’s call it the ρ-value) is absolutely vital in understanding how outbreaks evolve and how severe they become. A ρ-value of exactly 1 means no new cases are being identified, i.e. the curve of total new cases flattens out, but anything above 1 means you are entering the dangerously exponential part of the function. A ρ-value of 1.5 implies a doubling of new cases roughly every two days.
As an example, using data from the World Health Organisation, in the last two weeks of January, Hubei had an average ρ-value of 1.421, pushing them solidly into an exponential outbreak. February brought an average ρ-value of 1.085 and the first two weeks of March have seen this ρ-value fall to around 1.002 – the outbreak is brought under control. These may seem like small changes, but they have massive real-world effects. Hubei currently sits with a total of around 67,786 confirmed cases. Had the ρ-value remained at 1.421 for just five more days before the virus was brought under control, Hubei would now sit with around 210,000 cases. This is why getting ahead of the virus is so absolutely vital. Bringing the ρ-value down should be an absolute imperative – hours literally count here.
Let’s extend these simple statistical inferences to South Africa by using some egregious forecasting techniques for the sake of illustration. On Sunday, President Cyril Ramaphosa confirmed that South Africa had 61 confirmed cases of Covid-19. A simple calculation shows the ρ-value average since Wednesday 11 May has been an extremely high 1.448. At this rate, the country will have over 12,000 cases by the end of March, over half a million by 9 April and over four million by 14 April, five days later.
These outcomes are, of course, nonsensical. The point to illustrate is that the actual number of cases is almost irrelevant: a country with thousands of cases and a low ρ-value is infinitely better placed than a country with a dozen cases and a high ρ-value.
This ρ-value can be thought of as the transmission rate (how quickly and easily the virus spreads), which is itself an aggregation of many other variables. Some of this is intrinsic to the virus; airborne viruses will naturally spread faster than viruses borne by body-fluid, but some of it can be manipulated through behaviour. Quarantining, social distancing and even handwashing are all ways to bring this ρ-value down. When President Ramaphosa spoke on Sunday, the travel restrictions, school closures and cancellations of large gatherings were all essentially attempts to keep this ρ-value suppressed.
Good forecasting looks at real-world lessons and fortunately, because South Africa is simply lagging the rest of the world in Covid-19 infections, we can lay out three more realistic scenarios for the country.
Scenario 1: Most optimistically, South Africa manages to follow in China’s footsteps almost exactly: quick identification of early cases, tracing and quarantining possible contacts and firm public measures to limit the spread of the virus all bring the ρ-value down to 1 fairly quickly, i.e. within 60 days as was the case in China. In this scenario, by early May new cases in South Africa would be largely under control with a total of around 32,000. At a current mortality rate of 3.75% (supposedly overstated internationally, but possibly realistic in a high HIV/TB South African context) total deaths would come to 1,120.
Scenario 2: In this still positive scenario, South Africa initially follows Italy – a good example for SA’s situation, because unlike China, Italy knew Covid-19 was a threat and had some time to prepare. The South African government implements strong quarantine measures and the population implements its own behavioural changes, i.e. handwashing. This scenario sees the outbreak run rapidly until the end of March but fortunately healthcare facilities are able to cope, and ρ-values come down quickly through April, mirroring those of South Korea, which reacted swiftly. There are very few new infections by mid-May – reflective of the roughly 60 days it took China to control the outbreak. In such a scenario, total cases end up around 700,000 people. Associated deaths would amount to 25,000.
Scenario 3: More pessimistically we acknowledge that the ability of our health system to respond as aggressively as those in China, Italy or South Korea is limited, and that the population in general is more at risk – living in close proximity, needing to use public transport, not having the luxury of being able to stay away from work and not having the infrastructure to facilitate frequent handwashing. In addition, certain elements (but not all) of the health system become overwhelmed. Now rapid Italian style ρ-values continue for an additional week into April, slowing South-Korean style ρ-values take an additional week to come about and Chinese style stability now only occurs at end-May. In such a scenario, confirmed cases rise to around 2.2 million – associated deaths at around 77,000.
These scenarios still rely on a large number of assumptions. For one, we do not yet have a good understanding on the length of time it takes to control the virus. The only country to have achieved this so far being China. For another, the currently infected South African population base caught the virus from travel, and probably has access to medical aid, general health resources and the ability to self-quarantine. If the virus is kept to only this base for as long as possible its effects will be more limited. If it spreads to the larger population base who do not have the same luxuries, the effects will naturally be more serious.
I also make only a limited attempt to incorporate economic effects into the scenarios. So far the health department appears to be admirably on top of the situation, but a wider outbreak will certainly punish an increasingly fragile public health system. In addition, the macroeconomic environment facing firms (and households) in South Africa was bleak even before the virus. A further contraction in demand will have ripple effects throughout the economy, on public finance and even socio-political direction for years to come.
These are topics for another day. Right now, if there is one thing I want to convey, it is that bringing down the Covid-19 ρ-value in South Africa should absolutely be an imperative. Think about your ρ-value every day, by washing your hands, increasing your social distancing, sneezing into your elbow, working from home or letting those who work for you stay home. Cumulative small changes can have a massive combined effect on the outcome of this tumultuous period.
For those who are interested, daily Covid-19 statistics are available for all countries from the World Health Organisation here. DM
Jeffrey Dinham is Senior Economist, Econometrix (Pty) Ltd
"The real problem of humanity is the following: we have paleolithic emotions; medieval institutions; and god-like technology" ~ Edward Wilson
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