Thread by @phl43

Many have noted the cyclical nature of the pandemic, with waves that come and go seemingly at random, but the reasons for this remain mysterious. In this post, I argue that population structure could explain this and several other puzzling phenomena ⬇️ cspicenter.org/blog/waronscie…

I start by describing what an epidemic is supposed to look like according to standard epidemiological models. As long as people have the same number of contacts, incidence rises until herd immunity is reached, at which point it starts falling and eventually the epidemic dies.

If people's behavior changes and they reduce their number of contacts, because of a lockdown or because they do so voluntarily, incidence can fall before herd immunity is reached, but it starts rising again as soon as normal behavior resumes.

However, if we look at real data, the epidemic doesn't behave like that at all. In particular, incidence often starts falling long before the herd immunity threshold is reached. Sometimes, as in Sweden last spring, this can be explained by the fact that people's behavior changes.

But in many other cases, such as Florida last summer, incidence falls before the herd immunity threshold predicted by standard epidemiological models is reached even though, as far as we can tell based on mobility data, there was no clear change in behavior.

More generally, the effective reproduction number frequently undergoes large fluctuations that cannot be explained or very imperfectly by changes in behavior, such as what happened in France last fall.

I present a few examples to illustrate the point, but as I explain in the post, more systematic analyses confirm that, after the first wave, mobility data predict the effective reproduction number very poorly, so this observation isn't due to cherry-picking.

Now, mobility data are no doubt a very imperfect proxy of the relevant behavioral variables, so it's likely that with better data on behavior we'd find a stronger correlation with the effective reproduction number.

This prediction is vindicated to some extent by this very interesting, and unjustly ignored, paper recently published by @StenRuediger and his colleagues that used GPS data from cell phones in Germany to predict the effective reproduction number. pnas.org/content/118/31…

However, as I argue in the post, I think it's very difficult to deny that the effective reproduction number can undergo large fluctuations even in the absence of significant behavioral changes, which is hard to understand.

Of course, there are other factors that influence transmission (such as meteorological variables), but I argue in the post that they are not sufficient to explain the large fluctuations of the effective reproduction number we observe in the absence of behavioral changes.

Since SARS-CoV-2 is a respiratory virus that is transmitted by contact, transmission should ultimately depend on people's behavior, this is very puzzling. So how can we explain those fluctuations of the effective reproduction number without denying this basic fact?

What I propose in the post is that we can square this circule by taking into account population structure and how it can affect transmission even in the absence of behavioral changes.

Indeed, standard epidemiological models, of the sort that are used to make projections and study the impact of non-pharmaceutical interventions, assume that the population is homogeneous mixing or something close to it.

What this means is that models assume that someone who is infectious has the same probability of infecting everybody in the population or, since models used in applied work often divide the population into age groups, the same probability of infecting everyone in their age group.

Of course, this is totally unrealistic, since in practice if I'm infectious the probability that I'll infect most people in the population or even in my age group is effectively zero, because I don't even have any interaction with them and therefore couldn't possibly infect them.

In practice, the virus doesn't spread in a homogeneous population, but on a network based on people's patterns of interaction with each other. The topology of that network determines what paths the virus can take to spread on the population and not all paths are equally likely.

Now, suppose that this network can be divided into subnetworks that are internally well-connected, but only loosely connected to each other.

In network science, a network that has this property is said to have "community structure", which many real networks are observed to have. For instance, here is a network based on friendship relationships among a few thousand people on Facebook, which has this kind of structure.

If the population has that kind of structure, when one of the subnetworks is seeded, the virus starts spreading in that subnetworks until herd immunity is reached locally, at which point incidence goes down unless the virus manages to reach another subnetwork from there.

Thus, as different parts of the network get seeded and the virus spreads in them, we'd observe exactly the kind of waves that come and go before aggregate herd immunity is reached even in the absence of behavioral changes that we see in real data.

This makes sense intuitively, but can we observe this kind of behavior in simulations? In order to check, I created a model that randomly generate a network with that kind of structure and simulate the spread of the virus on that network.

However, because it's computationally very demanding to simulate the spread of a virus on a large network of individuals, I assumed that parts of the networks were internally so well-connected that they could be idealized as homogenous mixing populations.

Thus, instead of simulating the spread of the virus on a network of individuals, I simulate the spread on a network of homogeneous mixing populations that has community structure. Here is a graph that shows the network generated by the model for one of my simulations.

At the level of each subpopulation in the network, the model is a standard epidemiological model that assume homogeneous mixing, but people who are infected in one subpopulation can "travel" to another along the edges of the network and infect people over there.

(I put "travel" in scare quotes because people in different subpopulations may nevertheless be neighbors. What matters is who they interact with, not physical proximity, though obviously they are related. I discuss this point in more detail in the post.)

As you can see, the network is divided into subnetworks that are internally well-connected, but loosely connected to each other. Moreover, each edge is associated with a probability of "travel" along that edge, which is much greater for edges that stay within the same subnetwork.

For this simulation, I assumed a probability of "travel" of 5% along the edges that stay within the same subnetwork, but only 1 in 10,000 for edges that lead to a subpopulation in another subnetwork. There are more than 10,000 subpopulations, for a total population of ~5 million.

Here is a chart that shows the result of the simulation when I let the virus spread on that network. As you can see, the effective reproduction number undergoes wild fluctuations and the population experiences several waves at the aggregate level.

However, at the level of each subpopulation, the basic reproduction number was assumed to remain constant! Thus, this shows that, when the population has that kind of structure, the effective reproduction number can undergo large fluctuations even without any behavioral changes.

In order to make the process more intuitive, I created this animation showing how the virus spreads across subpopulations, which are represented by rectangles whose area is proportional to their size inside larger rectangles that represent the subnetworks to which they belong.

Unsurprisingly, if we increase the connectivity between subnetworks enough, the model behaves in a way that is more similar to what happens in a homogeneous mixing population.

For instance, if I use the same method to randomly generate a network but multiply the average number of edges between subnetworks by 10 and the probability of "travel" associated to those edges by 100, I obtain this epidemic.

Simulations on networks with community structure can produce all sort of epidemics, not just epidemics with large, sharply defined waves as above, but also epidemics that exhibit long plateaus with ups and downs. Just as we see in real data.

Thus, by relaxing the assumption of homogeneous population mixing and simulating the spread of the virus on a network with community structure, we can get the sort of behavior that we observe in the real world even with a constant basic reproduction number in each subpopulation.

But of course it doesn't mean that real populations have that kind of structure! Ultimately, since we don't have the sort of data we'd need to test it directly, this hypothesis remains speculative and will remain so until we collect that sort of data.

While there is no doubt that real populations are not homogeneous mixing, the hypothesis that they have the sort of community structure I assumed in my simulations is stronger than this claim and it's not something that we can simply take for granted.

In the post, I discuss some reasons to doubt it, but I ultimately conclude that they are less convincing than it may seem at first sight, so you should read it for a more thorough discussion of how realistic the model is.

Also, while I assumed for my simulations that people’s behavior didn’t change and that the basic reproduction number remained constant at the level of each subpopulation, I don't believe that it's true.

I made this unrealistic assumption to show that, even in the total absence of behavioral changes, the effective reproduction number can undergo large fluctuations due to population structure, but a more realistic model would take into account behavioral changes, the weather, etc.

I also assumed that the network doesn't change over time, but in reality behavioral changes — whether voluntary or imposed by government interventions — would not only affect the basic reproduction number but also change the properties of the network.

Thus, not only is the hypothesis that behavioral changes affect transmission not incompatible with the hypothesis that population structure does, but in part behavioral changes probably affect transmission through the effect it has on population structure.

Moreover, as I argue in the post, in addition to why the effective reproduction number sometimes undergoes large fluctuations in the absence of behavioral changes, population structure can also explain other puzzling phenomena.

For instance, many people have been surprised that countries where the prevalence of immunity is very high at the aggregate level have nevertheless experienced large outbreaks recently, which is usually ascribed to imperfect protection against infection and waning immunity.

But while I'm sure that imperfect protection and waning are part of the story, population structure may be another part, since a high prevalence of immunity in the population as a whole can hide much lower levels in some social networks.

Another phenomenon that is very hard to explain within the traditional modeling framework is the fact that both Alpha's and Delta's transmission advantages have been extremely variables over time and place.

I previously conjectured that population structure could explain this phenomenon, since different variants can spread in different parts of the network, where the prevalence of immunity is not the same and changes over time. cspicenter.org/blog/waronscie…

In this post, I confirm this intuition with simulations. I seed the population with one variant during the first part of the simulation and with another during the rest of the simulation, then I compute the second variant's transmission advantage over the first.

In the simulation, I assumed that both variants were equally transmissible (i. e. have the same basic reproduction number in every subpopulation), yet as you can see the second variant still had a significant but highly variable (across time and region) transmission advantage!

As I argued in my post on Delta's transmissibility advantage, the point is not that Delta and Alpha before that don't have a transmissibility advantage over previously established variants, but that population structure can make the estimates in the literature very misleading.

This is a more general problem: using the data simulated by my model, I show that, in the presence of complex population structure, several methods used in the scientific literature to estimate the effects of non-pharmaceutical interventions (NPIs) are totally unreliable.

Intuitively, this is easy to understand, for those methods usually assume a homogeneous mixing population. In such a population, putting aside factors like the weather, the effective reproduction number can only change due to behavioral changes and the accumulation of immunity.

But as we have seen, in the presence of population structure, the effective reproduction number can undergo large fluctuations even in the absence of behavioral changes and while immunity is still low, so those methods can ascribe to NPIs the effect of population structure.

I confirm this for 2 classes of models widely used in the literature to estimate the impact of NPIs by feeding to those models the data corresponding to the first wave in my first simulation above and falsely telling them lockdowns were in place during part of the simulation.

The first model I use is very similar to that used in Flaxman et al. (2020), which is probably the most widely cited study on the impact of NPIs. nature.com/articles/s4158…

As I already showed before, this model would be completely unreliable even if real populations actually were homogeneous mixing, but in this post I show that it's even worse in the presence of complex population structure. necpluribusimpar.net/lockdowns-scie…

I give the model data for the first 200 days of the simulation and tell it that a lockdown was in place for 75 days starting from t = 105. This makes sense because governments often decide to order lockdowns when incidence reaches a dangerously high level.

In fact, there was no lockdown in the simulation, but since the model assumes a homogeneous mixing population, it can only ascribe the fall of the effective reproduction number to this non-existent lockdown and therefore concludes that it reduced transmission by more than 40%!

I do the same thing with a model similar to that used in Chernozhukov et al. (2021), which belongs to a different class of models, but also implicitly assumes a homogeneous mixing population. sciencedirect.com/science/articl…

I already criticized this paper a few months ago, and already pointed out in passing that it rested on the homogeneous mixing assumption, but here I use simulated data to show how much population structure can bias the results. cspicenter.org/blog/waronscie…

In order to do that, I randomly distribute subpopulations into 15 regions (see post for details), give the data to the model and falsely tell it a lockdown was in place for 60 days in each region, after randomly drawing a starting date between t = 100 and t = 140 in each region.

I repeat this procedure 1,000 times and ask the model to estimate the effect of lockdowns on the daily growth rate of infections each time, which it does by looking at the correlation between the presence of a lockdown and the growth rate of infections.

Again, there was no lockdown in the simulation, so the real effect is zero. However, as you can see on this plot showing the distribution of the estimates found by the model, the model concludes that lockdowns reduced the daily growth of infections by 7% to 10%.

Thus, what those examples illustrate is that, in the presence of complex population structure, the methods used in the scientific literature to estimate the effects of non-pharmaceutical interventions (among other things) are *totally* unreliable!

Thus, even if the hypothesis that real populations have the sort of structure posited by my theory turned out to be false, the mere fact that it *might* be true should be concerning, because it would mean that a lot of things we have been assuming about the pandemic are wrong.

In general, we have been obsessing over quantities such as the effective reproduction number and the herd immunity threshold, but in the presence of complex population structure those are largely meaningless at the aggregate level.

Again, this post remains speculative, but I think it's good speculation. Indeed, not only is the hypothesis that real populations have that sort of structure not crazy, but if true it would mean that we have been thinking about the pandemic wrong.

It highlights a more general point, which is that data can only be interpreted with a model, so if the model is wrong the conclusions we draw from the data will also be wrong. Here I wanted to stress that assuming homogeneous mixing could result in very serious mistakes.

There is not doubt that real populations aren't homogeneous, so the only question is how exactly they are structured. I hope this post will encourage more people to think about how population structure might affect transmission and that data will be collected to test this theory.

P. S. Oops, I just realized that I had totally forgotten to upload the code for this post on GitHub, but it's now up and the piece has been updated with a link to the repository, which I also put here in case you want to have a look. github.com/phl43/populati…

ADDENDUM: Many people have brought up the debate about heterogeneity and the herd immunity threshold from last year in connection to my post, so I've added a note at the end of the post to clarify the relationship between that debate and what I'm talking about in this post.