Thread by @gro_tsen

Some more numerical examples illustrating various possible outcomes of an epidemic with two viral variants — and, ultimately, how impossible to predict these things are. Let me explain what these graphs show. ⤵️ •1/20

Both images illustrate the behavior of an epidemic with two different pathogen forms: one starting with basic reproduction number 1.0 (the “ancestral form”), another starting with basic reproduction number 1.4 (the “variant”). Infection by either induces immunity to both. •2/20

In each scenario we start with the same initial conditions: 0.75% of the population is infected, and ✳︎of✳︎ those, 1% are by the variant, the rest by the ancestral strain. •3/20

We start with a wholly susceptible population, or, if you prefer, we ignore whatever proportion of the population might have been made immune prior to time t=0 and only consider those still susceptible at this point. •4/20

(This makes these data fairly plausible for various European countries at the point where the variants started circulating. I'm not claiming I'm modelling anything specific, just that the orders of magnitude are reasonable.) •5/20

In each scenario (I'll get to the difference between the two a few tweets down), four graphs are plotted. Abscissa is always time, using the same scale (recovery intervals, i.e., the mean time it takes for an infected individual to recover). •6/20

The upper-left graph shows the proportion of the population which is susceptible (green), infected (red) and recovered=immune (blue) as function of time; plus the proportion ✳︎of✳︎ infected which are by the variant (dashed purple). •7/20

The lower-left graph is precisely the same, but in log scale instead. The upper-right graph is the proportion infected but in a different (linear) scale, and with breakdown: by ancestral form (orange), variant (purple) and total (red). •8/20

Finally, the lower-right graph shows the current effective reproduction number: the overall reproduction number is in black, and the ones for each pathogen form are in lighter color. The R=1 line is shown for convenience. •9/20

So now how do the two scenarios differ? Well, the first one (left set of four graphs in first tweet, reproduced below) is a standard SIR model (trivially adapted to two pathogen forms). In other words, it assumes a perfectly homogeneous population. •10/20

The second scenario (right set of four graphs in first tweet, reproduced below) is a SIR model with heterogeneous susceptibility following an exponential distribution. I.e., instead of everyone being equally susceptible to the virus, … •11/20

… we now assume that (for biological and/or social) reasons, there are variations in susceptibility, and that these follow an exponential distribution with the same expected value. In my opinion this is a more plausible a priori hypothesis than equal susceptibility! •12/20

More generally it's easy to model the case of a Γ-distributed susceptibility with an arbitrary shape parameter. See the paper arxiv.org/abs/2008.00098 by Montalbán, Corder and Gomes (@mgmgomes1) for more. I think “homog.” and “exp heterog.” are natural cases to consider. •13/20

Note that the evolution of the proportion of cases infected by the variant is almost identical in both of my scenarios: the variant increases in proportion, following a logistic-like curve, and at t=25 it represents ≳98% of cases. •14/20

Yet the overall epidemic differs considerably in both cases. In the homogeneous scenario the final attack rate is 45%, but only 23% in the exp-distributed susceptibility scenario. And the peak fraction infected differs even more (2.3% in one case, 0.8% in the other). •15/20

So essentially, unless we have good data on individual variations of susceptibility to covid (both biological and social), ✱which of course we do not✱, it's entirely impossible to predict how the variants will behave, … •16/20

… ✳︎even✳︎ if we have very reliable data on the reproduction number of both forms and even of there are no temporal variations on people's behavior or other changes in infectiousness (which are fairly unreasonable assumptions of their own). •17/20

So while it's fun to play with such mathematical models, and while they can greatly inform us on the sort of things which CAN happen, they're quite incapable of predicting what WILL happen, and those who claim to do so are charlatans. See also: •18/20

https://twitter.com/gro_tsen/status/1249738327143501824

Some more numerical experiments illustrating the sort of behaviors we can get from SIR or SIR-like models (using similar graphs and code) are in the thread below. •19/20

https://twitter.com/gro_tsen/status/1362142052146937859

Finally, here is the Sage code I used to compute the models discussed in this thread: gist.github.com/Gro-Tsen/4eacb… — running it as is produces the first graph, and for the second, just change “hetcoef” to 2.0 (it is 1+1/k where k is the Γ-distribution shape parameter). •20/20