Social networks and epidemics

Simple SIR models essentially assume that the infection rate of any  population is homogeneous, i.e. everyone stands an equal chance of getting infected. However this isn’t really true, because some people have very large social networks, while  the majority vary from  the the low tens, down to a handful of regular contacts. Politicians, business leaders and Scientists tend to have very large networks of regular contacts across the world, whereas most people have far smaller family and work related networks. The so-called super-spreaders are the first infected members of large social networks.

Social networks derived from news reports concerning the Iraq Insurgency in 2007 (my work). This just illustrates how large variations in network size can be.

At the start of an infectious disease outbreak the probability of someone getting infected if they are a member of a large network and then spreading it is very high. These are the super spreaders which force the initial reproduction rate R0 to very high values. It is no real surprise that Boris Johnson, Mat Hancock, Chris Whitty, Dominic Cummins and Neil Fergusson all got infected together as they formed part of the government’s  large COVID/SAGE  network. However as an outbreak progresses, the larger networks become increasingly already infected, so that remaining networks for new infections will therefore tend to get smaller and smaller. As a direct result of this gradual process, R will naturally begin to reduce,  even without imposing any extra social distancing policies. R is simply proportional to the size of network that the average infected person belongs to, so as mean network size diminishes so does R.

The Imperial model has a complex model of the distribution of UK population in cities, towns and villages and simulates travel and commuting. However as far as I can tell it  does not include social networks nor could it easily do so. There are just too many types of network for them to model – entertainment, businesses, councils, sports clubs, etc. All we can really do is to estimate how R might reduce naturally with time as the susceptible network size reduces.

My model uses a contact rate \beta which is the contact rate per day resulting in infection and 1/g which represents the average infectious period in days. Initially beta is 0.5 and 1/g = 5 but in order to simulate the decrease in social network size I reduce \beta by 0.003 per day. As a result R slowly reduces from 2.5 on March 1st to 1.0 on June 2nd. Note that this does not include any government led extra social distancing measures.

Here is a comparison  with this modified model with UK accumulated deaths

UK cumulative deaths compared to a slowing in R with average social network size

It is clear that the model doesn’t properly represents the shape of the UK data. It also seems clear that the first cases in UK also appeared probably 2 weeks before March 1st. One reason the curve can’t fit is because  the UK also imposed a lockdown on March 23rd which changed the dynamics. Next we look at Sweden which did not impose a lockdown.

Comparison of a reducing network model  with Sweden’s  accumulated deaths per million people.

The model is almost a perfect fit, so  I suspect that Sweden may well  be on the right track. If this intuitive decrease in  R due to ever decreasing  social networks of the susceptible really  is the driver, then by mid June the Swedish epidemic will be over through herd immunity.

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17 Responses to Social networks and epidemics

  1. mick wilson says:

    nice work, Clive. Do you intend to analyse other country tends eg Australia or new Zealand ?

  2. Cytokinin says:

    Until a few days ago, I would have agreed with your analysis, however whilst banging on about herd immunity, it was pointed out to me that it is probable that, as with flu, immunity is likely to last for only two, or three months. This means that the super spreaders will catch it again, and again… It also means that other people with smaller networks will also go through this cycle. It may be that the virus has a season, but so far this does not appear to be the case, in which case we may just have to contend with people dying until the virus mutates into something less lethal but more infectious. Whether we can make a vaccine that gives long term immunity is anyone’s guess.

    It looks like both of us will have to quarantine for fourteen days when we visit daughters in Australia.

  3. Sweden has implemented restrictions and social distancing measures. Work from home, reduced travel, restaurants with insufficient distancing closed, closed cinemas, no football matches or other sporting events, closed high schools and universities, no large gatherings etc.

    The Sweden myth is a myth. Number of new ICU-patients peaked first week av April. It was not because of herd immunity. It was because of social distancing measures.

  4. Andrew Carey says:

    “A study of 391 cases of COVID-19 and 1,286 of their contacts, in the Shenzhen region of China, found that 80 percent of cases were transmitted by just nine percent of carriers,”
    Ok, that’s quite a small study, and I’d be comfortable reading modelling attempts based on a rough 80:20 rule, but any decent model has to incorporate a three part decay rate – one for the gradual removal of the super spreaders who get it early, and one for enforced and voluntary behavioural changes. Imv of course.

    • Clive Best says:

      As far as I can tell Neil Ferguson’s model ignores this super-spreader effect. He models population distributions across cities and countryside, commuting etc. but essentially treats everyone the same.

  5. mesocyclone says:

    Nice work! I found this from your comment on Nic Lewis’ post on Judith Curry’s blog.

    I have no idea if the reduction factor you use is at all reasonable. I don’t even know how to tell, unless some studies have been done. But the idea certainly makes a lot of sense.

  6. Clive and everyone, is this a fair comment? Software always needs to be debugged – even the best programmers (maybe especially the best programmers in my experience!) leave bugs, which they have to eliminate by painstaking testing.

    A big problem with modelling complex systems, such as climates and epidemics, that there’s not much to test them against, especially when you try to make predictions of future behavior. One way to test a model is to see if it matches the data collected from the world AFTER the model is created. But your model can’t expect and isn’t claimed to be very accurate, so you need a lot of data to check it. But by the time you have a long data series new events have occured in the world, so your model is obviously out of date.

    I.e. you need more independent observations to check your model than there are parameters in the model, or you’re just pattern-matching.

    Catch-22 situation. So using models to predict the future is very tricky.

    Therefore models need to be very simple, and code eyeballed (checked by others?) as much as possible to try to make sure they are doing what they’re supposed to do.

    Clive’s models look pretty simple to me. Is it possible, even in principle, to go much further than them?

    Is it surprising that Neil Ferguson’s models fall apart – given how complex they are?

    Interested in your thoughts.

    • Clive Best says:

      My model is about 80 lines of Python. I can’t believe it contains any bugs, although the assumptions may be naive.

      Fergusson’s code is (was) 15,000 lines of C in a single file. As well as an SIR type model it also tries to simulate the interactions of people across the UK through work, play, schools, bars, etc.

      As far as I can tell we get the same answer for a 65 million population R=2.4 IFR(Infection Fatality Rate) = 1% -> 500,000 deaths

      In addition he tries to simulate the effect on R of different lockdown measures. However his model was originally written for Flu pandemics and treats people homogeneously., whereas in reality Covid-19 has a huge variation with the age of infected person. Most deaths occur in care homes and hospitals which themselves have now become centres of cross-infection. So R depends hugely on locality. It is misleading to quote a national value of R.

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