March 3, 2023
The maths of the COVID-19 era. The pandemic told in four equations
The COVID-19 pandemic has brought a greater understanding of illness to the general public including the learning of specialist terminology such as “superspreader”, “strains”, and “lockdowns”.
Some of what we have learned was related to maths throughout both the transmission process and precautions to prevent infection. In this article, I will explain those processes through four different equations using my own calculations.
Calculus #1: The R
The transmission rate ( R ) is the ability of a disease to spread. It shows the number of people on average that an infected person will pass the virus to. If this value is below one (<1) then it will not be a concern as the disease will eventually stop spreading as not enough new people are being infected. If the number is over one (>1) however then the numbers will keep increasing unless other precautions are put into place to prevent it.
During May 2020, the R value within the UK had reached between 1 and 1.2 which caused a significant threat to the population. [For this article, R will be treated as constant despite the transmission rates of different COVID-19 variants]
Let’s see what happens when the R of a virus alters:
R=1 means that each infected person will spread the virus to one other person. This newly infected person would then only be able to spread the virus to one other person; meaning there would be three people infected. This would only increase by one each time so within ten steps the number of infected people would increase from one to ten.
R=2 means that each infected person will spread the virus to two other people. Those two newly infected people would then spread the virus to two more people each. This means that the number of people becoming infected would double each time therefore starting with one infected person would mean that after ten steps the number of infected people would increase from one to one thousand (1,023 to be exact).
A difference of R, as little as one, has dramatic effects on the number of those infected within a population which without intervention would continue to increase at a rapid rate thus resulting in increased numbers of deaths.
Calculus #2: The lockdown
The UK government took action using lockdowns and masks in response to high R values posing a threat to the general public. These measures reduce the R of the virus as the number of people that can be spreading the virus is significantly reduced. This is solved because infected people are unable to come into a close enough range for transmission with others due to lockdowns and facemasks which prevent the pathogens of the virus from spreading through breathing and coughing, lowering the R number. Pathogens are organisms that infect and replicate within human cells to cause disease.
R=1 or less doesn’t support the decision to enact lockdowns as the number of people infected would remain low enough that it does not significantly affect the population, and so it would be unnecessary for the government to take drastic action.
R=2, resulting in the use of lockdowns reduces the R number to a much more manageable level as its rate is slowed by a lack of transmittable access.
Assuming that all measures taken will reduce the R to 1.5 means that, statistically speaking, one person infects 1.5 other persons. I calculated that within ten steps, this would mean decreasing the number of people infected starting with one person from 1,023 to 113 (113.008359375).
There’s a price to these precautions which is again maths. Restrictions such as lockdowns are difficult to maintain as they put a significant strain on the economy as people are unable to go to work to earn money or to go and spend it outside their homes.
COVID-19 resulted in very high levels of public spending from the government in order to help support the economy under financial strain. The current estimates of the cost of the UK government measures announced the range of about £310 to £410 billion. The spending in 2020/21 (from the latest relevant data available online) was about £167 billion higher than what had been originally planned for prior to the pandemic. The extra money supplied was spent on public services like the NHS, support for businesses, and support for individuals.
Calculus #3: The vaccine
As a result of continued large scale infection, there was a need to develop and distribute an effective vaccine to the general public as soon as possible. This would act in a similar way to the previous restrictions since it would reduce the R value by decreasing the number of people who become affected. Some people were anxious to take the vaccine due to fears of possible health complications and its possible failure; however, these fears could be quickly eliminated with a greater understanding of the clinical testing and mathematics involved in the process.
Efficacy says how much the vaccine will cut the R rate.
The efficacy of a vaccine is measured in a controlled clinical trial in which the outcome of a group of vaccinated people is compared with that of a control group who received a placebo. A placebo is an inactive substance that looks like and is given the same way as the vaccine which has no effect on a person’s health. This measures the amount that the vaccine lowered the risk of getting sick in a population. The UK government has estimated that 2 doses of the COVID-19 vaccine were between 65% and 95% effective at preventing symptomatic disease with the Delta variant. This means that within a vaccinated population 65% to 95% fewer people will contract COVID-19 after coming into contact with it.
For one vaccinated person, efficacy is 95% meaning that the risk of becoming infected is lowered by 95% for that individual. So now, this person has a very slim chance of contracting the virus but is still surrounded by people who are ill and let’s assume that on 10 separate occasions, they are exposed to the virus. This would mean that the vaccinated person getting the virus at least once would have a probability of 0.4, which is still a threat to their safety.
Update on the current situation in the UK:
As of January 2023, the percentage of people testing positive of COVID-19 in England is 1.62%.
Calculus #4: The herd immunity
Furthermore, this protection is increased by herd immunity which is when enough of the population is vaccinated that a fewer number will come into contact with the virus and are therefore less likely to become infected. This is ultimately the goal of vaccinations as it results in the virus being reduced to a level that no longer poses a threat to the population.
The UK government was able to estimate how the likelihood of self-reporting long COVID-19 changed after being vaccinated using statistical models which take into account factors such as days since infection, age, and sex. These models involved a large sample of randomly selected private households within the UK with all participants being asked to provide swab samples at every follow-up visit. This means that this particular analysis applies to all people with different stages of COVID-19.
Efficacy = 90%
If no one is vaccinated, the virus would spread across the entire population since we would continue to be in contact with it and our immune systems would be unable to fight it.
If half the population is vaccinated against the virus, then the number of those experiencing symptoms will be decreased, however, it generally takes about 70-90% of a population to be immune for herd immunity to be effective.
If 90% of the population is vaccinated, then those who come into contact with the virus will be less likely to experience symptoms and pass it on. This means fewer people will come into contact with it and therefore the virus will eventually reach a low enough level that it has very little impact on the population as a whole.