Maths is present everywhere in our lives, subtly supporting us no matter what we’re doing: from each piece of technology we use, to optimising the production of the bananas we eat with our morning porridge. Despite playing such a big part in our lives, it’s rare that we see and hear about these formulas directly. That is, until the coronavirus pandemic.
From early March, we’ve been bombarded with countless statistics and graphs to explain the effect that Covid-19 has been having on the world; suddenly, simply watching the news each evening is a maths lesson in itself. While confusing, such mathematics has, thankfully, been extremely important in predicting and combating the virus and perhaps deserves more credit than it’s received. Here are two of the most crucial models and how they’ve impacted the fight against COVID-19.
Case Fatality Ratio
A standard measurement that estimates the proportion of deaths from the disease to the number of total cases diagnosed. While simple to define, it can often be challenging to be predict as it changes with time and is affected by many external factors. Instead, the delayed CFR (Case Fatality Ratio) is often used: CFRd(𝑡,𝜏𝑟𝑒𝑠)=𝐷(𝑡)𝑁(𝑡−𝜏𝑟𝑒𝑠)
where 𝐷(𝑡) represents the number of deaths at time t; 𝑁(𝑡−τres)specifies the number of diagnosed cases in the period (𝑡−𝜏𝑟𝑒𝑠); and 𝜏𝑟𝑒𝑠 denotes a corresponding time lag showing the duration from the day when the first symptoms occurred to the day of outcome.
Estimates from this value have improved as more data has been released, which has led to further measures for CFR; more specifically, for an individual vs the population. This has been key in determining the threat to specific demographics and how best to protect the public.
Basic Reproduction Number
The basic reproduction number 𝑅0 represents the average number of secondary cases an infected individual would transmit in a fully susceptible population. Until some kind of ‘herd immunity’ is achieved, 𝑅0 is equivalent to the well-known R-Value.
The concept is relatively simple. If each infected person infects two others, and those others two more each etc. then the number of cases will rise exponentially and so to slow and end infections R must fall below 1.
Determining this R value is not so straight forward, as to calculate the true real-time value requires data from the future – something we are clearly not yet capable of accessing. Fortunately, there are several mathematical models that use statistical tools, such as the Gamma distribution and Poisson regression, which are able to predict these values and then retrospectively analyse the findings against the real data in order to improve their calculations (a process known as machine learning).
The restrictions placed on everyday life is one of the biggest influences on the value of 𝑅0 and so naturally it features heavily in the government’s decision making and justification of the various levels of lockdown.