Dr Savi Maharaj, Lecturer in Computing Science, Faculty of Natural Sciences, University of Stirling

Dr Savi Maharaj discuses the role that social distancing plays in responding to epidemics, using studies using simulations and games. The virtual lecture is available to watch online, and a transcript is provided below.

Thank you for joining to listen to my talk, I am Savi Maharaj from Computing Science and Mathematics at the University of Stirling, and I’m going to tell you work that I started doing about 12 years ago and then left behind to move on to other things. I now slightly regret this decision because the topic has suddenly become extremely relevant to current events. Now everybody knows what social distancing is about.

I warn you that my slides today have been put together from a number of different talks, so you will see slight inconsistencies in naming and numbering – I hope that this won’t be a problem. My talk is about using computational techniques to study the use of social distancing to control epidemics of infectious disease. This is joint work. At first I worked with Adam Kleczkowski, who is now at the University of Strathclyde, and we developed and studied a computational agent-based model of social distancing. Our main interest was to look at the cost benefit trade-offs involved in controlling epidemics by social distancing, where the cost is the economic damage that has caused by social distancing, and the benefit is the reduction of the number of people who become ill. Our results were published in our 2012 paper, which is referenced here. Most of my talk will be based on that people.

We then did some work with some psychologists in which we turned our epidemic simulation into a computer game so that we could do experiments to see how much social distancing people chose to do when we put them inside a simulated epidemic. I’ll say a little bit about these results, which are published in our 2015 paper. I’ll end with a discussion of the limitations of the work and some thoughts about the real economics and benefits of social distancing as it applies to the scenario that we’re facing today.

So there’s no need nowadays to explain what social distancing is, or physical distancing as we are being advised to call it these days. It’s an ancient practice (there are stories about it being used during the bubonic plague) that is used to control the spread of epidemics. Our questions were, is it effective? Or rather, in what situations and when applied and in what way can it effective? And also, is it cost effective, or is it the case that the economic cost incurred by stopping people from going to work or going shopping, causing businesses to close, etc., is it the case that these outweigh the benefit of reducing the spread of the epidemic? This is what we set out to answer in our model.

Modelling

Now, how does one represent social distancing behaviour in the model? Traditional methods of modelling epidemics use differential equations and they rely on something called a mean field assumption, which basically means that the whole population mixes freely. There’s no spatial variation in who can contact whom. We can make these models to an extent more fine-grained by introducing more compartments represented as additional equations to represent populations at different locations, but that approach gets very complex. If you want it to be high resolution, you need quite a lot of compartments.

Another approach and the one that we’re going to adopt is to use a network model in which we talk about individuals who are connected by links. This has been used in a number of studies to model social distancing at a more fine-grained individual level, and that’s what we’re going to use. At the time we did the work, the existing individual models did not consider the economic aspect of social distancing, so this is what we set out to do. There was some other work that did look at the economic aspect but did not consider a network model.

So how does our model work? We have individuals, these little circles here, who are located on a network which shows how they can contact others. So focussing on this individual here, in the middle, each individual has two neighbourhoods: the infection or contact neighbourhood (the smaller circle in this in the picture) which this represents the people with whom they can make contact every day and transmit infection, and an awareness neighbourhood (the grey circle, which represents the source of information used in making decisions about social distancing). At each time, step, each person becomes aware of the level of infection within their awareness neighbourhood and they can expand or contract their contact to neighbourhood in response. This is the social distancing response, the way that we model it. So we see that the person here on the left has got only two infected cases in their awareness neighbourhood, so they’ve left their contact neighbourhood relatively large, whereas this person on the right has got quite a large number of infected cases within the awareness neighbourhood, so they’ve contracted their contact reaches quite a bit.

For the epidemic dynamics we base it on the traditional SIR are or Susceptible, Infected, Removed model and we add some tweaks that incorporate social distancing behaviour. So each individual starts off as being susceptible to the disease and then at every time step they make contact with the other individuals within their infection neighbourhoods, some of whom might be infected. A second parameter (P) represents the infectiousness of the disease and is basically the probability that a single contact between a susceptible and an infected individual will transmit the infection. If that happens, the individual becomes infected and at every time step up, an infected individual can recover with probability, (another parameter, Q).  

Social distancing is modelled by modifying the radius of the infection neighbourhood or the ‘contact neighbourhood’ in response to the number of infected cases in the awareness neighbourhood. There’s an important parameter over here which only to describe in the next slide, and this is a parameter that governs the strength of the social distancing response. How strongly some individuals respond is modelled by a parameter we call the risk attitude or Alpha, which is described by these equations here, which are not going to go into in detail. The important thing to know is that very small values of Alpha represent a highly cautious response, where the individual closes down contexts immediately when they’re when there are just a small number of infected cases within the awareness neighbourhood. Large values of Alpha represent a very relaxed response, where there is very little social distancing taking place.

This is what a simulation run looks like. The red individuals are the infected ones. Green are susceptible. Grey is removed or recovered, and yellow represents cautious susceptibilities who are practicing social distancing. In this particular scenario, we can see that the cautious people are forming a sort of band, separating the infected pieces from the rest of the population. The infected cases will eventually recover and the band of cautious social distancers will prevent it from reaching the wider population. So this is a case where social distancing is going to protect quite a large proportion of the population.

Economic Cost/Benefit

How do we move that model the economic impact, or the economic benefit and cost? We use a very simple model in which we count the contacts between pairs of individuals at every time step of the simulation and add them all up during the duration of the epidemic. So, the net benefit of social distancing is calculated as a weighted sum of the gain due to reducing the total number of individuals who get ill during the epidemic and subtracting from that the number of contacts lost due to social distancing. The weighting that we use is not based on any actual economic values, which is one of the limitations of our model. Here are some details about the implementation of the model and the number of simulation runs we did and so on.

So now for some results. So this is a typical result. Here we are varying only the infectiousness (P) of the disease, keeping all the other parameters constant with the values that you see here. So we’re comparing the case of no control with the case of social distancing applied with a risk attitude or alpha parameter having the value 0.11. In terms of the values used in our simulations, that represents a sort of moderate social distancing response. So, we see that what happens is that for the values of P, the infectiousness from about .05 to a bit more than point 3, social distancing has an effect. It reduces the final number of people who become ill. R infinity here on the y axis is the final number of infected cases and we see that for these particular values of P, there is a strong reduction in the number of infected cases.

Of course, social distancing also has the effect of reducing the number of contacts, so this is what we’re looking at here. This graph has a very interesting shape. We see that for the less infectious diseases, the loss of contacts is low because the disease dies out quickly. So this reduces the length of time that social distancing has to be done. For highly infectious diseases, the loss of contacts is higher, but still relatively low. This is because the level of social distancing used in this scenario was too weak to stop the highly infectious disease from rapidly infecting most of the population, and once that happened and the infected people recovered, they were able to resume normal social contacts. So the overall locus of contact is relatively low.

What’s interesting is this worst-case scenario, which with these parameters is around P equals 0.4. What’s happening here is that the number of lost contacts is highest. The reason for this will become apparent in the next slide where we look at the effect social distancing on the duration of the epidemic. So the duration is how long it takes for the final infected case to recover. We see that for diseases in that middle area with P around 0.4, moderate social distancing is causing the epidemic to last a lot longer than without social distancing. So this causes more loss of contacts over the whole epidemic because the social distancing behaviour is continued for a much longer time.

So bringing everything together and looking at the overall benefit – that is, counting both the reduced infection and the lost contacts, we see that we can classify the effect of social distancing into four types, depending on how infectious disease is over here with diseases that are not very infectious, the epidemic dies off quickly anyway, and social distancing has no effect. In the second zone, the social distancing efficiently stops the epidemic quickly, so the loss of contacts is only for a short time. This is the best scenario with the highest benefit. In the next zone, however, social distancing is too weak to stop the spread, and it only has the effect of prolonging the epidemic, causing greater loss of social contacts because social distancing is practised for longer. This is our worst scenario. Finally, in zone D we have a disease that is so infectious, it quickly sweeps through the population. So the epidemic ends relatively quickly, but social distancing is ineffective. So lots of people get ill, but the loss of contacts is not quite as much.

Varying the social distancing response

So now let’s look at what happens if we vary the strength of the social distancing response. In the previous slides we looked at a moderate response, now we are varying it. We see a very strong response where – we basically panic. We always cut social contact of theirs, even the least hint of infection. This is always going to stop the epidemic. No matter how infectious the disease is, and this always has maximum benefit. By contrast, a very relaxed response will not stop the epidemic, but also will not slow it very much so that the epidemic will spread quickly, will be over quickly and normal social contact can resume. This is a bit like the herd immunity strategy that we’ve heard discussed in relation to coronavirus. Now, notice that the result is that by the end of the epidemic, all or nearly all of the population will have been infected. Obviously this will be fine if it’s a very mild disease with very low fatality, but it could be very serious if it’s a disease that kills a lot of people. The worst scenario in our simulation here is the halfway response where we have a control that is too wishy washy – it does not suppress the epidemic, it only slows it, and causes huge economic damage, lots of loss of social contact for very little benefit. Now, one thing that you can’t see from this slide, however, is that even this weak social distancing can have other benefits in terms of reducing the peak impact of the epidemic. That is the largest number of people who are ill at any point in time. We didn’t consider this in our 2012 paper, but we did in the 2015 people, which is what I’m going to show you next.

2015 study – How do people really behave in pandemics?

So in our next study, we worked with psychologists to try to find out what social distancing behaviour choices real people might make. We made a game in which we asked people to play the role of an individual in an epidemic who gets the information daily about the number of infected cases in the neighbourhood and must make choices about how to change the size of their contact to neighbourhood. Making contacts and then simply a money but exposes them to the risk of infection, so the player has to navigate this trade-off.

What we found is that the observed a risk attitude (on the x axis) among participants weaker than what was needed for optimal control. So they were cautious, but not as cautious as would be the optimal. The figures show the outcomes from running simulations using the risk attitude that we saw with our participants. So the blue line is the observed risk attitude. The red horizontal line is showing the outcome with no control, and the green is the outcome with social distancing applied using that observed risk attitude. There’s only a small number in a small reduction in the total number who become infected. However, looking at the social contacts, we see that there is quite a large reduction in the number of social contacts. So this isn’t looking too good. We also see that the social distancing lengthens the duration of the epidemic.  However, crucially in this study, we also looked at the peak impact rate, that is the maximum proportion of the population who are ill at any point in time. We saw that although the control isn’t strong enough to stop the epidemic, it does have the benefit of significantly reducing the peak impact rate, and this does have the benefit of keeping the number of infected cases at any point in time at a manageable level so that health care systems are not overwhelmed. These days, this is what we’re calling flattening the curve.

So now, just to sum up, there are a number of limitations in our model. It would be great to work closely with public health experts and economists to make it a more realistic model. Some of the limitations are that we didn’t parametrized the model to reflect any real disease. We basically looked at a full range of parameters and I’ve shown you examples that gave us some interesting graphs. The economic model is very crude. It doesn’t account for different kinds of social contact that might have different benefits. Our model treats ‘removed individuals’ as recovered and immune rather than dead, which is obviously a much greater cost if the individuals die rather than recovering. We also considered voluntary social distancing in response to local conditions rather than the kind of enforced uniform lockdown across the whole of society that we’re seeing today. However, despite these limitations, we can see that some of our conclusions are consistent with other studies. Basically, for social distancing to work really effectively, it needs to be done very strongly. Even if it isn’t done that strongly, it can still have the effect of flattening the curve and reducing the peak impact.

Closing thoughts

To end, I want to share just a couple of interesting headlines I saw recently that suggest a very different analysis of the economic benefits of social distancing. These really gave me pause when I thought of these in relation to the study that that I had done. The first headline is about a study that counts the benefit of saving lives, something we did not do in our model because our individuals recovered from illness rather than dying. This study found that the economic benefits of social distancing as it’s applied today in the US for the COVID-19 pandemic are really considerable, to say the least. The second study takes a historical perspective and it looks at different US cities during the 1918 Spanish flu pandemic, and what it found is that the cities that applied social distancing rigorously and for longer periods actually experienced greater economic growth after the epidemic was over. This could be because they experienced fewer deaths, so they lost a smaller proportion of the young working population. I think this is quite an optimistic picture. There are some very scary headlines I’ve also seen saying that the current coronavirus lockdown is going to plunge the world into a deep depression. I think the picture is not quite so clear-cut as it might seem, and there is reason to hope that to the overall economic impact will be beneficial. So I’ll stop there. Thank you very much for listening.

Dr Savi Maharaj, University of Stirling, May 2020