Thread by @joshuasweitz

This thread is on #Covid19, heterogeneity, herd immunity, and the roots of SIR models; why mathematical choices we often take for granted have profound effects on interpreting unfolding epidemics.

@arxiv manuscript: arxiv.org/abs/2005.04704
Code: github.com/aapeterson/pow…

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Key take-away: our mathematical analysis of the *joint* dynamics of heterogeneity and infection reveals that the force of infection can reduce to a simple form: I x S x S (or variants thereof) rather than I x S.

This nonlinear change in epidemic models may have significant consequences to long-term predictions and lead to super-slowing down of epidemics (including reduced herd immunity thresholds).

To begin: the basic "SIR" (susceptible-infectious-recovered) models of epidemiology treat everyone identically. That is, the rate of new infections is proportional to the product of I and S densities (or fractions).

However, in an outbreak, some people will be more likely to be infected than others, because of where/how they work, live, or socialize, or even due to intrinsic differences (e.g., immune system state, age, etc.).

As a result, some simulation models have revealed that removing "highly susceptible" individuals from the pool of those that can get sick does two things: (i) reduces the number of susceptibles; (ii) reduces the vulnerability of those uninfected.

Prior work shows that variability in susceptibility may substantially reduce the herd immunity threshold from that expected naively from the SIR model:

@TheAtlantic:
theatlantic.com/health/archive…

Gomes et al.
doi.org/10.1101/2020.0…

Britton et al.
science.sciencemag.org/content/early/…

However, prior work has utilized complex models that have been mathematically challenging. We derive a new way to account for variability with a single heterogeneity parameter - showing that the force of infection reduces to I x S x S (or variants thereof).

To begin, let's imagine that the susceptibility of individuals looks like this.

(There is good reason to assume an exponential distribution based on a "maximum entropy" argument; however, we can also note this looks like a "super-spreader" curve.)

Since infection scales with susceptibility, then the rate of those newly infected is not the mean, but twice the mean:

In this case, the average newly infected person was *twice* as susceptible as the uninfected pool as a whole!

This means that not just the *number* of susceptible people decreases as the disease progresses, but also the *average susceptibility* of those uninfected.

We can show that the basic rate term in the Kermack-McKendrick model changes from 1st- to 2nd-order when we account for change, i.e., the force of the equation changes from ~I x S to ~ I x S x S. This fundamentally alters the long-term dynamics of the outbreak.

The strength in terms of R0 will not be changed, nor will the initial speed, but the final-size relationship will be systematically over-estimated w/o accounting for heterogeneity.

But what if susceptibilities are not exponentially distributed? For example, some have suggested that they follow the "80:20" rule, where the most susceptible 20% carry 80% of the risk.

We can account for this with a different distribution (a "gamma" distribution) and find that the rate changes to 5th order! (For the math nerds, the gamma is the "eigendistribution" of the contagion process.)

This provides a direct link between the variability of susceptibility and the power-law order in the SIR equations. No additional compartments needed, just a modified order for the S term!

We compared these different assumptions of variability for published data on the H1N1 outbreak of 2009, and found that including this in the model improves match with final outbreak size as measured with serology:

Importantly, this lets us account for variability with a single parameter based on the shape of the distribution, as opposed to previous treatments which are by nature much more complicated (and therefore opaque).

What does this mean for the COVID outbreak? Potentially it means that final sizes *may* be substantially less than predicted from classic SIR-based inferences (e.g., SEIR or similar):

We caution that for respiratory diseases, and Covid-19 in particular, there are many unknowns and even a reduced final size would lead to large-scale illness, hospitalization, and fatalities (see @bansallab, @nataliexdean & @CT_Bergstrom for discussions):

https://twitter.com/bansallab/status/1259970542624550913

But this may add be important in assessing locales that have been hard hit to assess whether new infections are reduced by susceptible depletion and, potentially the depletion of highly susceptible individuals.

In a broader sense, our findings formalize the notion that accounting for susceptibility leads to a dynamic slowing down of the epidemic process, beyond that from susceptible depletion alone, connecting heterogeneity with 'super-slowing'.

Tweets from a team of engineers, quant biologists, physicists, and information theorists:

Christopher Rose, Andrew J. Medford, C. Franklin Goldsmith, Tejs Vegge, myself, Andrew A. Peterson

Manuscript: arxiv.org/abs/2005.04704
Code: github.com/aapeterson/pow…

Feedback welcome.