Making predictions about something as unpredictable as flu is foolhardy. I rarely (if ever) do it, but I’m going to do it now. I am predicting a bad and early H1N1 swine flu season in the northern hemisphere next fall and winter. The reasons for this departure from our usual custom is a paper in Nature from 2007 I just re-read. It’s entitled “Seasonal dynamics of recurrent epidemics” by Stone, Olinkyl and Huppert from Tel Aviv University and it appeared in vol. 446, pp 533-536, 2007 (doi:10.1038/nature05638; html version here, if you have a subscription). The paper isn’t specifically about flu and it is a mathematical modeling paper that presents an analytical condition for whether there will be a full epidemic or a minor “skip” in a seasonally forced SIR model. It’s mainly concerned with childhood diseases like measles, not flu. But it seemed to us it was of immediate relevance to swine flu. We’re going to try to explain why we were so struck by this paper (and if you want to know more about mathematical modeling for flu, you might try our 17 part series that takes a single paper and explains it, paragraph by paragraph and equation by equation at a level suitable for most educated readers; you won’t need it for this post, though).
It is an adage in the field of mathematical modeling that “all models are wrong, but some models are useful” (attributed to George Box). We think this is a model that is useful because it provides insights into what could be happening with flu. Not the whole story, but an important part of the story. Even relatively simple models for disease dynamics — for example, ones incorporating crude assumptions like the probability of transmission is proportional to the rate of contact between infected and susceptible people — can behave in very strange and counter-intuitive ways, especially when you include seasonal forcing in the equations. In simple terms, seasonal forcing says that some periodic environmental factor that varies throughout the year — opening of schools or absolute humidity or temperature — is altering the transmissibility of the virus. Many systems have their own inherent periodic behavior that don’t require outside forces. A weight on a spring bouncing up and down or a child on a swing (i.e., a pendulum) are two examples familiar to any college student taking physics or differential equations. The rate at which the child swings back and forth after an initial push is characteristic of the swing apparatus and only depends on the length of the chains attaching it to the top bar (yes, I am neglecting damping and non-linearities from larger angles; so sue me). The reason a pendulum clock can keep time is because the period of swing is regular and fixed. Now, after the child is swinging back and forth in a regular, periodic way characteristic of the swing set-up, do some periodic forcing. What this means is that you start to give the swing an additional push but not at intervals corresponding to its inherent periodicity but some other period completely independent of the swing’s inherent period. There are a huge number of behaviors of the swing that you can produce (including chaotic behavior) by different kinds of external forcing (pushing at odd intervals, say at the bottom of the path or as it is on the way up to you as you stand behind), and the mathematical analysis of inherently simple periodic systems with external forcing can quickly become intractable. But the authors of the Nature paper sidestep some of this analysis to see if there are some broad underlying regularities that might be useful:
Theoretical studies have shown that seasonal forcing can be responsible for inducing similar complex population dynamics such as higher-order cycles, resonances and deterministic chaos. These complex responses can easily mask any simple underlying mechanistic processes that might otherwise help in forecasting future epidemics. The modelling framework used here helps uncover, and gives new insights into, these processes. (Stone et al., Nature [cites omitted])
One of the innovations in this paper is that instead of trying to predict a specific outbreak, the authors concentrate on post-epidemic dynamics. In other words, they are looking at pairs or triples of outbreaks, not single outbreaks. It has been known for a long time that seasonal childhood diseases have really bad years, sometimes several in a row, and then suddenly a “skip” or year with a minor peak (the peak is there but its a mini-peak). Sometimes there will be several skips, then another bad year. Flu does the same thing. Last year was bad but the three previous seasons were “mild” but the one before that was also bad. Stone et al. tried to find a simple explanation for this behavior with the help of a mathematical model for seasonally forced infectious disease dynamics where there are just three kinds of people: those who are Susceptible, those who are Infected and those who are Recovered (or dead). It’s called an SIR model for the three categories. Our series on modeling antivirals looks at a model very similar to this in concept.
One of the main results in this paper is a mathematical formula for a threshold of the proportion of susceptibles in a population required at the outset of a season for it to be “bad” (have a high peak). If the number of people susceptible when the outbreak starts is above that threshold, then there is an epidemic that year. If it is below the threshold, there is a “skip.” The exact formula isn’t important for us, or even so much what goes into it (things like the size of the forcing function, the rate of entry into the population of new susceptibles, etc.). What is interesting is the insights it gives into what’s going on.
On one level the analysis just seems to confirm conventional epidemiologic wisdom: if there is a really bad year, most of the susceptibles are “used up” and the following year there aren’t enough people left who aren’t immune for the virus to get going. The virus confronts “herd immunity” produced by the previous bad year. But that doesn’t always happen and the analysis shows why. It’s not just how big the previous year’s outbreak is, but its timing within the flu season (which they refer to as early phase or late phase). Instead of paraphrasing it, I’m going to quote directly from their paper. The two symbols used are S0 and Sc. The first is the minimum number of susceptibles left in the wake of the last year’s outbreak, while the second is their threshold criterion. If S0 exceeds Sc then there will be a bad year:
First, consider the case in which there are only two main seasons each year, a ‘high’ season (high disease transmission) and a ‘low’ season (low disease transmission). Suppose an infected individual is introduced into a population of susceptibles during the high season. It makes a crucial difference whether the individual enters the population relatively early or late.
First, consider the scenario in which the infected individual is introduced early in the high season and proceeds to initiate an epidemic. This gives plentiful time for the development of a full-scale epidemic. These large protracted epidemics eventually die out, exhausting the susceptible pool (S0) in the process. If S0 < Sc, there are too few susceptibles to fuel an epidemic in the following year. Second, in contrast, should an infected individual enter the susceptible population very late in the high season, there may be little time available for the build up of a large-scale outbreak. Being late, the epidemic is more likely to be affected as the season changes from high to low. The smaller contact rate associated with the low season can act to curtail the epidemic, and cut it short. As a result, a large susceptible pool S0 remains. Should S0 > Sc, the number of susceptibles will be enough to trigger an outbreak in the following year.
Stone et al. present very credible evidence that when the peak of the previous year occurs late in the season and is cut short by the “off season” factors (whatever they might be), then the following year is bad, because the aborted late appearance (aborted by the lack of seasonal forcing) leaves enough susceptibles to exceed the threshold for a bad year. While the insights here came from looking at the behavior of a mathematical model, the reasoning is fairly robust to its assumptions. I think you can see where I am going with this. I am not predicting a bad year because number crunching in a mathematical model said it must be so. On the contrary, through the use of the model, we are able to see some implicit logic about what goes on in systems like this.
Here’s how I see it applying to swine flu. This started late in the flu season. We’re not sure when, exactly, but probably in March sometime. Because there was no natural immunity in the population and in other respects the virus transmitted with the facility of seasonal flu, it could spread pretty fast and widely before whatever factors involved in flu’s seasonal forcing lowered transmission to the point it started to subside. It’s true it is not subsiding everywhere but it is subsiding in many places in the north. However it is not the fact it is subsiding but the reasons why it is subsiding that are important. If it is starting to wane because it had burned itself out by using up the susceptibles, that would suggest next year wouldn’t be so bad. But in fact, while there was a lot of flu around, most people didn’t get it. If it is subsiding it is probably because whatever is involved in the seasonal forcing of flu (and we don’t really know what that is) has started to cut it short before the “tinder” of susceptibles was used up. Everyone expected this to happen when the summer came and the fact it didn’t happen right away was a surprise. It suggests this virus is quite transmissible and combined with the lack of immunity could overcome the extra push to transmissibility the seasonal forcing gives it. But it looks to be subsiding now. When the forcing starts again in the fall all the makings will be there for an early and big flu season if the threshold for it is exceeded. I feel pretty confident there are plenty of susceptibles around for the virus. True, I don’t know how many are needed because I don’t know what the threshold is. But I’m betting it’s not too high. Meanwhile the vaccine won’t be available to decrease the susceptibles before the virus can pick up a head of steam.
I would dearly like to be wrong about this and making any prediction about flu is undoubtedly stupid. Doing so on the basis of a mathematical model may be even more foolhardy. But sometimes you just go on scientific hunches, and my hunch is that Stone et al. have this pegged right, even if they didn’t intend it for this flu. We’ll just have to see. But meanwhile, I’d keep your seat belts fastened because I see evidence of turbulence ahead.