The rules of contagion

Dave Strugnell | October 2020

In this bizarre year of 2020, and after several months spent trying to understand and model the coronavirus pandemic and its health and mortality impacts, it somehow seemed appropriate to find myself reading a book entitled The Rules of Contagion. Author Adam Kucharski has built his career on getting to grips with outbreaks of deadly, contagious diseases such as Ebola and Zika, and has a clear and firm grip on their dynamics that he outlines lucidly in the pages of Contagion. But, as he also vividly illustrates, it is not just diseases that can be studied through the lens of infection.

Kucharski opens with a fascinating history lesson, highlighting how the mathematical modelling of the spread of infectious diseases forever changed our understanding of these diseases and enabled far more effective responses to them. Of course, having an increasingly clearer biological sense of their causes didn’t do any harm: disease outbreaks tend to be episodic and to rise to peaks before falling to troughs, perhaps in several waves, and in mediaeval times, those attempting to make sense of the dynamics without that biological understanding were forced into convoluted conjectures by way of explanation. The most compelling theory available was one of astrological influence (indeed, the word influenza has its roots in the Italian for ‘influence’).

The early development of the now-common mathematical approach had its origins in the curiosity of one Ronald Ross, a surgeon who found himself posted by the British Empire to Bangalore in the early 1880s, and pitched headlong into drawn-out battles with its mosquitoes. Being a man of empirical science, he tested his theory that the excessive size of the mosquito population at his bungalow was related to the barrel of stagnant water outside his window by tipping the barrel over; history does not record how many incorrect theories were debunked prior to this, but by essentially removing the insects’ breeding ground, Ross found himself both relieved of the swarms that had previously greeted him every evening, and saddled with a newfound interest in mosquitoes which was to be lifelong. That interest came in useful a decade later when, while on sabbatical back in England, he came into contact with Patrick Manson, a leading researcher into tropical parasitic diseases, in particular malaria (interesting sidebar: this disease’s name too comes from the Italian and from superstitious mediaeval cause attribution, ‘mala aria’ meaning, literally, bad air). By the last decade of the nineteenth century, the cause of malaria was known to be a parasite named Plasmodium, but the medical fraternity was still clueless as to how the parasite was spread from host to host. Manson had just begun to speculate that mosquitoes might be carriers, and that was of course enough to pique Ross’s interest in the theory.

Drinking water was the first candidate to be tested, in a manner which would be unlikely to pass through any vigilant research ethics committee today: Ross had mosquitoes suck blood from a patient infected with malaria and then lay eggs in water, which he then had three volunteers drink; they were paid for their participation in the experiment, though surely not enough. And none came down with malaria, thus invalidating the drinking-water theory. Ross was quick, however, to move to what transpired to be the correct theory: that mosquito bites were responsible for the spread, which he confirmed successfully through experimentation on birds. This singular discovery led to Ross being awarded the 1902 Nobel Prize for medicine, the second ever awarded (though Manson had contributed richly to the work through his correspondence with Ross, he was never credited and came to the news of the Nobel award in the same way as most his countrymen, through the newspaper). After a discovery on which some scientists may have been tempted to rest their laurels, Ross continued pressing with his research, turning his attention now to the key question of how the transmission of malaria could be reduced or terminated. And here is where mathematics first came into the study of infection.

Suppose, Ross reasoned, that a village of 1,000 people had one person infected with malaria. Transmission to another villager would require, in the first place, a mosquito to bite the infected individual; he estimated that 1 in 4 would manage to bite someone, and therefore only one in 4,000 would manage to bite the right candidate. Only 1 in 3 of those, by his estimates, would survive long enough to produce the parasite (a process which takes time), and then again, only 1 in 4 of those surviving would manage to bite a human, thereby completing the infectious transmission. So, ultimately, it would take 48,000 mosquitoes for one successful new infection. More mosquitoes, or more infected people, would speed up this process; counterbalancing this, however, was the recovery of infected patients, who would be infectious no longer after a period. His visionary insight was that if the pace of recoveries exceeded the pace of new infections, eventually the disease would simply die out. It would therefore not be necessary to eliminate all mosquitoes, only to control their numbers below some mathematically-solvable critical point, to eventually drive the level of disease down to zero. These were the beginnings of the SIR (Susceptible-Infectious-Recovered) model, the simplest of compartmental models that any budding epidemiologist today will learn very early on in her studies, first formalised by William Kermack and Anderson McKendrick at Edinburgh’s Royal College of Physicians Laboratory in the 1920s.

The key insight of the SIR model is that the rate of new infections is a function of the pace of spread- the reproductive number- as well as the relative sizes of the infected and susceptible populations. Early on in an epidemic, most of the population is susceptible but there are very few infected people to spread the disease, so the pace of spread is low. But as it progresses, the size of the infected group grows and the pace picks up; there comes a point, however, when the number of recoveries starts to exceed the number of new infections, because of the declining size of the susceptible population, and at this point the disease is doomed to ultimate extinction, although many more people will be infected before it dies out. This point is the so-called herd immunity level. Simple epidemiological models which fail to account for heterogeneity in transmission of and susceptibility to Sars-Cov-2 would imply that between 50% and 70% of the population would need to be infected to reach herd immunity, barring a vaccine; arguments have also been mounted that infection levels as low as 20% may be sufficient to reach this threshold, taking account of large swathes of the population who are presumed to have some form of immunity from or substantial protection against infection, perhaps due to prior exposure to other coronaviruses and resultant T-cell cross-reactive protection. The bottom line is this: we don’t yet know where the herd immunity threshold lies for the pandemic we’re currently facing, and if anyone tells you that they know the answer to this, they should probably be resolutely ignored. (If I’ve learned anything in the humbling experience of trying to understand Covid-19, it’s that anybody who is bold enough to state that they’ve got it all figured out is clearly not paying conscious attention to the complexity, and is therefore not worth listening to.)

In his book The Prevention of Malaria, Ross indicated clearly that he had wider ambitions than the study of infections, and was interested more broadly in what he referred to as “happenings”, which could be either independent (e.g. accidents or divorces, where the “happening” of that occurrence to one person doesn’t materially affect the probability of it happening to another person), and dependent, where there is some degree of contagion between individuals. The shape of people affected over time follows a very different pattern in each case: for independent happenings, a concave curve, and for dependent happenings, the now-familiar S-curve with accelerating “infection” early on (the convex portion of the curve) as the infected population grows faster than the susceptible population, followed by slower, concave growth once the proportions are reversed. And Ross’s generalisation gave the hint that it was not just infectious diseases that could be modelled using this approach: the spread of literally anything that depended on transmission from person to person permitted exactly the same analytical framework.

In the early 1960s, Everett Rogers, a sociologist by background, applied the Ross framework for the first time to the transmission of new products and ideas, leapfrogging the term “viral” out of the narrow confines of the medical lexicon once and for all. As more and more people take to a new idea, it spreads like wildfire before reaching a point where the growth naturally levels off as the population still susceptible to the idea shrinks below a critical point. Kucharski applies this thinking to the 2008 financial crisis (funny how both of the most significant economic disasters of the 21st century to date are amenable to analysis in the same terms). The spread of collateralised debt obligations (CDOs), those clever but now-infamous packages of mortgage debts, spread contagiously through the US and global economies in a very similar way to a physical virus, the novelty and apparent high-value/low-risk combination serving to propagate their spread with great speed. The problem? Underpinning the low-risk assumption was the security of the physical assets and the conviction, based on a couple of decades of experience, that house prices rise inexorably. That assumption met the realities of economic supply and demand head-on in an immoveable object sort of way from 2007: mortgage risks are not independent, but rather correlated through a common link to underlying economic conditions, and the inevitable effect of many simultaneous fire-sales is a softening of the property market that weakens the security of the backing assets. Banks, at least those few who weren’t bailed out, learned this reality the hard way.

There are also some surprising aspects of life that lend themselves to analysis through the lens of contagion. One such is violent crime. Epidemiologist Gary Slutkin noticed similarities between maps of urban murders in the US and those depicting cholera outbreaks in Bangladesh, an observational feat that must surely have required either immense good fortune or paying attention to a degree which most of us (well, me, anyway) would find unachievable. The mechanisms? Revenge, normalisation of murder, propagation of a more generally violent culture… a host of factors, but the fact that one murder makes more murders more likely seems to be undeniable. This led Dr Slutkin to start the organisation Cure Violence, which aims to effectively cut the reproduction number of violent crime by despatching teams of “violence interruptors” to high-risk areas to attempt, through talking to members of affected communities, to prevent the reactions that perpetuate the violence. In one year of applying this practice in Chicago’s notorious West Garfield Park neighbourhood, gun violence incidents were reduced by a massive two-thirds.

And, of course, no book on contagion would be complete without a consideration of how things go viral in cyberspace. The same analytical framework lends itself to tweets and Facebook messages: the reproduction number reflecting the number of susceptible followers likely to retweet or like will help to predict the ultimate number who will see the post. But the role of influencers, to my surprise, turns out to be rather more muted than those who style themselves thus would have us believe. Consensus on what makes a post go viral remains elusive. So much for my plans of achieving global renown through one devastatingly brilliant post.

All in all, Kucharski provides an immensely interesting, considered and well-constructed read, particularly apposite in these weird pandemic times. It was particularly pleasant to get a historical perspective on some of the tools that we’ve been using in getting to grips with the intricacies and complexities of Covid-19, and to have the outlook we’ve brought to bear- informed by the best-quality academic literature available, open to a variety of possibilities, updated in the light of emerging evidence, but informing action in the face of unavoidable uncertainty- affirmed throughout, perhaps most neatly in this quote from the epidemiologist Alice Stewart:

“The trick is to get the best guess of the thickness of the ice when crossing a lake… The art of the game is to get the correct judgement of the weight of the evidence, knowing that your judgement is subject to change under the pressure of new observations.” 

¹Quite often, the argument that herd immunity is reached at low levels comes along with an unstated right-wing political agenda that ought to be interrogated before taking too seriously any scientific claims made.