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Predicting the pandemic: mathematical modelling tackles Covid-19

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The accuracy of the official coronavirus predictions depend, it seems, on mathematical models. How accurate are the models themselves?

500,000 people could die in the UK during the Covid-19 outbreak. Then it’s 250,000. Then 20,000. Or even fewer, if lots of us have already had the virus and we didn’t know about it. We could be under lockdown for three weeks, six months, 12 months or even two years.

The situation is changing every day and will continue to do so, which will mean more announcements and predictions and also a lot of speculating and hint-dropping.

Everyone wants to know what to expect next, what they might have to do next, and, most of all, whether they are safe now and how long we will all be at risk.

With so much speculation out there, what we need to know more than anything is this: can we rely on what we’re being told by the authorities? How accurate are the government announcements about Covid-19?

What our leaders are saying publicly about the outbreak is actually based on mathematical modelling systems provided by Imperial College London. In recent weeks, other researchers have applied their own models to the pandemic, as have experts elsewhere around the world.

There’s no vaccine for Covid-19 so, in general, modelling seeks to assess the likely impact of various non-pharmaceutical mitigation and suppression strategies on the spread of the virus and the number of deaths.

Aaron King, professor of ecology, evolutionary biology and complex systems at the University of Michigan, explains that these models use mathematics to make thinking precise and logic rigorous. “This enables analysts to work carefully from assumptions made to the conclusions they draw and translate data into key qualities necessary to understand and predict the outcome of epidemics under various policy scenarios,” he says.

The Imperial College researchers initially used data from the Covid-19 outbreak in Italy to inform their models and make recommendations about what best to do in the UK. However, as the pandemic spreads around the UK, more localised data will become available about the effectiveness of our government’s lockdown and social-distancing strategies and, from this, we’ll no doubt get more announcements and predictions.

Any infectious disease modelling is only as good as the data that’s fed into the computers, though. Covid-19 is a new virus, so modellers started with little data and have subsequently had to make assumptions about transmission rates, incubation periods, the time gap between when a person develops symptoms and becomes infectious, recovery rates and levels of immunity.

Professor James Wood, an infectious diseases expert from Cambridge University, explains that to understand how effective a model is means knowing what it’s actually for. “The Imperial model tries to deal with many things at once,” he says “Other models might focus on one specific thing, or one particular area; all of them help provide an overarching picture of what’s going on.”

King adds that the type of model used depends on the degree of detail that modellers wish to incorporate. “Early on with a novel pathogen we know very little, so typically start with basic elements of pathogen spread,” he says. “Once the pathogen arrives in a location it will spread through the population by a chain reaction process.”

‘A model cannot tell us with any certainty where the disease will explode next, which country or city. It might say that some places are more likely than others, but it can’t provide detail at the fine scale.’

Professor James Wood, infectious diseases expert, Cambridge University

King explains that four basic elements are used in most disease outbreak models. First is the number of people each infected individual will subsequently infect in the early stages of the outbreak, the R0, which modellers get an idea of by looking at the rate of exponential increase. For a while during the current outbreak, it was believed to be somewhere between 2.2 and 4. “This might look a relatively modest figure but it translates to a very high fraction of the population being infected,” King says.

The second element, the duration of the outbreak, is more difficult to predict, King believes. “Analysts gain more knowledge as they gain experience and observe different cases, but it’s still a variable quantity,” he says. “For instance, you don’t know how any one individual’s immune system will work on the virus.”

The case fatality ratio, the probability that an infected person dies, is the third basic element of disease outbreak modelling. This ratio is affected by the timing of the infections. “A sudden flood of severe infections will overwhelm any modern healthcare system; people who might have otherwise survived will die,” King says.

The fourth element is the asymptomatic ratio – the current number of unrecorded Covid-19 cases. Many people with mild Covid-19 symptoms don’t seek health care, don’t show up in official figures but are still capable of transmitting the disease and can also become immune to subsequent transmission. “Early strategies based on detecting infections are doomed to fail,” says King, “because we don’t know to what extent an unknown and seemingly large amount of transmission is going on under the radar.”

This is not necessarily bad news. An Oxford University study, published on 24 March, used mathematical modelling to suggest that far more people had been infected in the UK than previously thought, maybe more than half the population. In this scenario, the virus arrived much earlier than previously believed, by mid-January at the latest, and maybe even as early as December, spreading invisibly until the first severe cases were recorded at the end of February.

Using this model, only one in 1,000 people who caught the virus would need hospital treatment and, with large numbers of people already immune, the authorities could lift restrictions on the public much sooner than previously anticipated.

As with all the models, the headlines should not be taken literally as a statement of what is actually happening on the ground. When the Oxford researchers made their calculations, they had no better idea than their counterparts at Imperial about how many people had actually contracted the disease or were at risk of serious illness. Some experts criticised the study for encouraging people into wishful thinking, which the critics feared might encourage some to disobey government restrictions too early. Cambridge University’s Wood is more philosophical “If anything, the Oxford study showed the need for more rigorous serological testing,” he says.

Detection tests discover whether a person has the virus now and so tend to take place once the person has presented serious symptoms and has come for medical treatment. Models based on this data will inevitably provide a higher death rate as they are relating the number of deaths to the number of cases presented to the medical authorities.

A serological test looks for antibodies present in the blood that fight the virus and tell experts whether a person has had the virus at some point and recovered from it. With enough serological testing, experts then have more accurate data on how widespread the infection has really been. With this knowledge they can more accurately predict the infection rate, the duration of the outbreak and the case-fatality ratio. Western governments, including the UK, are in the process of buying and testing large numbers of serological kits.


Coronavirus curve graphs

Image credit: E&T


For current mathematical models to be accurate, analysts would also need data from the medical authorities that distinguish between deaths caused by the coronavirus and deaths that would have happened anyway. Often, this data is not known, not available or, at very least, the data is ambiguous.

It’s well known that a large majority of Covid-19 fatalities have been elderly people or people with underlying, often serious, medical conditions. Professor Neil Ferguson, from the Imperial College modelling team, has suggested that in the UK, as many as half to two-thirds of the deaths attributed to the virus could have happened without Covid-19. In the UK, 150,000 people die between January and March. Worldwide, around 56 million people die in a typical year.

Wood says that we mustn’t rely too heavily on predictive models as there are just too many uncertainties for even the most well thought out model to provide all the answers.

“Models can inform general policy and provide an over-arching picture, which reflects the analysis of the epidemic and its control, but there is still always the chance effect,” he says. “A model cannot tell us with any certainty where the disease will explode next, which country or city. It might say that some places are more likely than others, but it can’t provide detail at the fine scale.”

Wood also believes that people’s behaviour, how they follow advice, will be a more decisive factor in overcoming the current pandemic than models, or even government policy. He also believes that it’s vital that we quickly find ways of stopping animal-transmitted diseases from entering the human population in the first place.

The University of Michigan’s King thinks that governments and leaders can play a decisive and more effective role in managing an outbreak. “Earlier and more decisive policy responses during the initial stages of this outbreak would have made a difference,” he says. “As would the widespread availability of serological testing equipment from the outset.”

King also thinks that we need better international protocols to inform governments about what to do about the spread of a virus between countries through transport systems. Any set of protocols, he adds, should include global data collection and sharing.

“Often, a government’s first instinct is to cover things up, to see [an unfolding crisis] as a public relations issue rather than a public health problem,” King says. “We need to share information, experience and knowledge, not just suspicion and prejudice.”

On 30 March, the UK government announced that the UK’s R0 was now below one. Experts say that if an infection rate of below one is maintained, the Covid-19 virus will eventually die out. On the same day, however, Sir Patrick Vallance, the UK’s chief scientific adviser, said that we should pay less attention to day-to-day fluctuations and look over time to see what’s happening.

More data is needed. And, it seems, a lot more patience.


Flattening the curve

Researchers from Imperial College London have analysed the likely impact of multiple public health measures on slowing and suppressing the spread of coronavirus.

Much of the current UK government policy and advice on the pandemic stems from information from a team at Imperial College London, which has modelled the spread and impact of Covid-19.

Different strategies will have different outcomes: the aim of the research was to establish the effectiveness of various suppression strategy scenarios, especially considering the availability and requirements for intensive care unit (ICU) beds.

With no interventions at all, depicted by the black line (see graph above), the model suggests we would rapidly run out of available ICU beds. This number of beds – 8,000 – is shown in the red line and doesn’t account for any increase in capacity since the date of modelling.

The green line shows the effects of a suppression strategy that includes closing of schools and universities, case isolation and population-wide social distancing beginning in late March 2020. The orange line shows a strategy that includes household quarantine alongside case isolation and population-​wide social distancing. The blue shading shows the five-month period in which these interventions are assumed to remain in place.


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