A playground cordoned off due to the lockdown on March 26, 2020 in Auckland (Photo by Phil Walter/Getty Images)

Covid-19: The maths that explains why we’re locking down

Want to avoid Covid-19? Skip the high dose vitamin C, and take a moment to learn about the statistical modelling that helps us understand how diseases spread. Shaun Hendy explains.

Infectious diseases like Covid-19 spread from person to person. If, on average, every person who gets it goes on to infect two other people, then you have yourself an epidemic. Two becomes four, four becomes eight, which becomes 16, which becomes 32 …

This is exponential growth and it is exactly what we are seeing in places like the United States and Europe. In these countries, every person who gets Covid-19 is going on to infect two-and-a-half other people on average. The result is that the number of infected people in these countries is doubling every few days.

This is the first time that humans have encountered SARS-CoV-2, the virus that causes Covid-19. As far as we know none of us have acquired immunity to it yet, which means the virus can infect all of us. With no immunity, the exponential growth of infections could go on for months.

This is a scenario we have explored using some mathematics to simulate Susceptible-Infected-Recovered (SIR) models. A Te Pūnaha Matatini research team (Alex James, Mike Plank, Nic Steyn, and myself) have been working on building SIR-type models for Covid-19 for several weeks (there is a detailed summary of our work here and further details available on our website). We are one of several groups providing modelling of Covid-19 for the government.

The model shows what would happen if an outbreak took hold here. In a scenario where we did nothing about it, we found that the New Zealand health system would be swamped ten times over, leading to tens of thousands of deaths. This is very grim stuff, which is why our government is going to do its best to avoid this scenario.

In the first stage of the government’s response, people who came back from overseas with Covid-19 were isolated to prevent them from infecting others. The numbers of travellers testing positive has been rising rapidly as people fly back from countries where the virus is out of control. This is alarming, but we expect these numbers to start dropping in a few weeks now that we have tougher border restrictions in place.

Border controls were a good start, but it is possible that some people were missed, either because they didn’t develop symptoms or because they became infectious before their symptoms appeared. This means that we may have hidden transmission in the community taking place. For every person who shows up to hospital with severe symptoms of Covid-19, it is theoretically possible that there could be 100 others out there with it in the community.

Indeed, as testing has ramped up, the Ministry of Health has found several cases of Covid-19 that can’t be linked to overseas travel. These are the cases we are most concerned about, because our model shows that they could initiate run-away exponential growth in the disease. Without further action, we would be facing our worst case scenario.

Early in the crisis, the UK government talked about a ‘herd immunity’ strategy. They proposed letting the disease pass through their country’s population, while trying to quarantine those who are more likely to die from the disease (people over 70 or with other health conditions). Once those who got it recovered, their immune systems will likely have learned how to knock out SARS-CoV-2 when they next encounter it. With enough recovered people with immunity, the disease would no longer find new people to infect. The modelling shows that this strategy risks many lives, and the UK government abandoned it quickly after it was announced.

Instead, many countries have gone for lockdowns. By keeping people at home, restricting contact to family or housemates, this strategy greatly reduces the chances of healthy people getting the disease from an infected person. This seems to be slowing the spread down in Italy, although there is a lag between going to lockdown and any reductions in the numbers of new infections, because it can be up to two weeks before symptoms appear and people get tested.

New Zealand has had an opportunity to take a different approach. The government has raised our Alert Level to 4 from today, locking us down much earlier in the disease progression than has happened in other countries. With our model we looked at a scenario where the country was locked down for a year or more until a vaccine was available. It would work, reducing the number of deaths to a handful, but not many of us would want to spend a year stuck at home, unable to work or visit friends and family.

Luckily, we think we are at an early enough stage here that if we can slow the disease by locking down for a few weeks, we should be able to use contact tracing and testing to contain and then stamp out Covid-19 by isolating everyone who has it until they recover. From today our lockdown will begin to slow any community transmission down. If enough of us stay in our bubble, we will give the ministry’s contact tracing teams a chance to catch up with its spread. If we don’t stay at home then the lock down period will have to be even longer or we risk a runway epidemic.

Over the next four weeks we will be closely following the numbers of new infections detected and updating our model every day. In a few weeks’ time these models may start to tell us that we can drop the Alert Level in some parts of the country, perhaps if we restrict travel between regions. At the same time, we have teams looking at other ways to track the virus, using its changing genetic sequence for instance, as well as using big data. We are using maths to find the best ways to carry out contact tracing and how we can organise our hospitals should there be a flood of patients.

We will beat it, but it is going to take some maths, some time in our bubbles, as well as some kindness.


The Spinoff’s science content is made possible thanks to the support of The MacDiarmid Institute for Advanced Materials and Nanotechnology, a national institute devoted to scientific research.


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