Thread by @gro_tsen

Let me write a thread about the steady state of the SIRS epidemiological model and what it can tell us about what endemic covid might look like (how often we will catch it and whether we have any control over this), or at least, what the parameters involved are. 🧵⤵️ •1/43

(I had written a thread 🔽 about this at the beginning of the pandemic, in a different context, concerning other endemic coronaviruses, but I think it's now time to revisit it, and rather than refer to it, I'll redo it from scratch.) •2/43

https://twitter.com/gro_tsen/status/1254105664428834819

In the SIRS model, people go through three states cyclically: S = “susceptible”, i.e., NOT immune to infection; I = “infectious” (=infected); and R = “recovered”, which in this model should rather be taken to mean “currently immune”. Unlike in SIR, immunity DOES NOT last. •3/43

In other words, people will go cyclically through the three states S→I→R→S forever. (This is a very basic model and ignores, inter alia, deaths and birth in the population; but we can think of them as another form of the R→S transition.) •4/43

So we have three processes in this model:
‣ S+I → I+I (infection of susceptible),
‣ I → R (recovery of infected),
‣ R → S (loss of immunity).
Further, we assume first-order kinetics. We can call β,γ,δ the kinetic constants (see tweet 15) for these three processes: … •5/43

… this means that if s,i,r (btw 0 and 1) are the fraction of the population who are in states S,I,R, then the fraction infected per united time is β·i·s, the fraction who recover per unit time is γ·i, and the fraction who lose their immunity is δ·r. •6/43

So the dynamic SIRS model is given by the first-order differential equation system:
‣ i′ = β·i·s − γ·i
‣ r′ = γ·i − δ·r
‣ s′ = δ·r − β·i·s
… with s+i+r = 1 constantly. But I will NOT talk about this system in this form (too complicated! 😅). •7/43

Rather, what interests me is the “steady state” solution, viꝫ. the one where s,i,r are constant: people go through the various states, but their overall proportion remain steady in time. This is to model an endemic state where the disease is ever present. •8/43

Note that I am NOT claiming there won't be variations in the real-life endemic state. There will certainly be seasonal patterns. But imagine I'm taking year-long averages: I don't care about the wobbles, I care about the mean values. •9/43

Similarly, even though I much criticized SIR for ignoring heterogeneity in the population (‌

https://twitter.com/gro_tsen/status/1456197178985226241

‌), here the whole point is to ignore it because we are studying the population in aggregate, not the unpredictable dynamic but the steady state only. •10/43

Also note that while s,i,r can be interpreted to represent the fraction of the population that is in each state over a long-term average, they also represent the fraction of TIME that the average person spends in each state (🔽): … •11/43

https://twitter.com/gro_tsen/status/1450384063148216322

… so we care very much about the value of i, which represents not just “what fraction of the population will be infected with covid at a given moment of time” but also “what fraction of my life will I (average person) be infected with covid”. •12/43

Now the steady state is much easier to compute than arbitrary dynamics: just set s′=i′=r′=0 in the ODEs in tweet 7 above, and we get:
‣ β·i·s = γ·i = δ·r,
‣ still with s+i+r = 1,
— which is now just a system of algebraic equations that a high-school student might solve. •13/43

Sparing you the simple calculations, the steady-state solution (with i>0) is:
‣ s = γ/β,
‣ i = ((β−γ)δ)/(β(γ+δ)),
‣ r = ((β−γ)γ)/(β(γ+δ)).

Good! Enough pure math. Now what do all these values mean and what can be said about them? •14/43

First, what do the three kinetic constants mean? Well, γ is the (mean) rate or recovery, i.e., reciprocal of the time during which the average person is infectious. So in the case of covid, γ ~ 1/(2weeks). •15/43

As for δ, it's the inverse duration of immunity (where “immunity” here means “infection-preventing immunity”, i.e., sterilizing immunity, which is shorter than protective immunity against severe forms). Sadly, we don't know 1/δ, but it's probably measured in years. •16/43

Finally, β is the average number of people that an average infected person would infect per unit time if everyone around them were susceptible. Rather than study β, we generally study β/γ, which is the average number during the duration of infection: … •17/43

… this is known as the “basic reproduction number” and denoted R₀ (you've probably heard about it by now 😂), a most inconvenient notation because R already means “recovered”, so let me write 𝓡₀ instead [note: in some past threads I called this κ]. •18/43

Now one might think that 𝓡₀ would be known by now for covid, but actually it's a HUGE mess. What we CAN measure are effective dynamic reproduction numbers, and even that involves many subtleties (see this old thread 🔽), … •19/43

https://twitter.com/gro_tsen/status/1320327001258008578

… but recovering 𝓡₀ from this demands that we know what fraction are immune, because 𝓡₀ is for an epidemiologically naïve (i.e., 100% susceptible) population, and there is no longer any such thing. Even then, the dynamic 𝓡 and the steady-state 𝓡₀ might not agree. •20/43

And that's not even getting into the whole issue of variants which made the value of 𝓡₀ very much unknowable. Statements and computations about variant X being n% more infectious than variant Y are just bullsh💩t (see 🔽). •21/43

https://twitter.com/gro_tsen/status/1394085534826438658

So the bad news is: we know γ to some extent, but we don't know β nor δ. Maybe we can still say something about orders of magnitude, though! Call 𝓡₀ = β/γ (see above) and 𝓦 = δ/γ to get rid of time units: these are dimensionless parameters. •22/43

We can expect 𝓡₀=β/γ (which characterizes reproduction) to be somewhere between 2 and 20, and 𝓦= δ/γ (which characterizes waning of immunity) to be somewhere between 0.002 and 0.02. Yeah, these are HUGE margins, but at least note this: 𝓦≪1 while 𝓡₀>1. •23/43

Rewritten in terms of these parameters 𝓡₀=β/γ and 𝓦= δ/γ, the solution from tweet 14 is:
‣ s = 1/𝓡₀,
‣ i = (1−1/𝓡₀) · 𝓦/(1+𝓦),
‣ r = (1−1/𝓡₀)/(1+𝓦),
and since 1+𝓦 is very close to 1,
‣ s = 1/𝓡₀,
‣ i ≈ (1−1/𝓡₀) · 𝓦,
‣ r ≈ 1−1/𝓡₀.
•24/43

In the older thread referred to in tweet 2 above, I tried to estimate as best as I could, for the case of the already endemic coronavirus HCoV-OC43, the values of these parameters: •25/43

https://twitter.com/gro_tsen/status/1254105667528425473

I came up with 𝓡₀ ~ 4 and 𝓦 ~ 0.01 (with γ ~ 1/(15d)), or s ~ 25%, i ~ 0.8% and r ~ 75%. These are very rough estimates even for OC43 and may not apply to covid, but at least give us a ballpark value to anchor our expectations to when covid becomes endemic. •26/43

Note that these values are at least broadly compatible with those from the paper ncbi.nlm.nih.gov/pmc/articles/P… which found between 26% and 91% of seropositivity for OC43 antibodies, depending on the cutoff value used (tables 1 and 2): … •27/43

… their primary seropositivity cutoff value (giving 91% positive) is probably sensitive enough to detect all immune individuals in the “non infectable” sense, and their high cutoff value is presumably too high, so r would be somewhere in between 26% and 91% (😕). •28/43

Now here's a slightly surprising thing: for 𝓦 small, r and s depend essentially only on 𝓡₀. This may be counterintuitive: 𝓦 measures how immunity wanes, but the fraction r of people who are immune depends essentially on 𝓡₀, not really on 𝓦, … •29/43

… while conversely, the fraction i of people who are infected at a given time depends mostly on 𝓦 rather than 𝓡₀ which measures how the infection spreads. Essentially, that's because if 𝓦 is higher, people are infected more often, … •30/43

… so they spend more time infected, but they are therefore also made immune more frequently, which compensates the fact that they stay immune for shorter periods, and the total fraction of time that they are immune for doesn't vary (much). •31/43

The number of new infections per unit time is β·i·s = (1−1/𝓡₀) · 𝓦/(1+𝓦) · γ. With the “ballpark” values I gave in tweet 26, that's ~500 infections per day per million people. (Compare with ourworldindata.org/explorers/coro… — but note that most infections are not detected!) •32/43

Now let's consider the mean time between infections. It's equal to 1/δ (which is the mean time in state R) plus 1/(β·i) (which is the mean time in state S) or (𝓡₀+𝓦)/(𝓦·(𝓡₀−1)·γ) ≈ 𝓡₀/(𝓦·(𝓡₀−1)·γ). With the “ballpark” values of tweet 26, that's 5 years, … •33/43

… but these 5 years are broken down into 1/δ ~ 3¾ years of immunity plus 1/(β·i) ~ 1¼ years of no longer being immune. Here's why this is bad news if you want to avoid being infected by practicing some kind of social distancing instead of being an average person: … •34/43

… of these two terms, the one you have some control over is the 1/(β·i) term: if you cut your contacts by a factor A it should drop down to 1/(A·β·i) in your case; but the term 1/δ of biological origin will remain the same: … •35/43

… so in my example, if you have, say, 3 times fewer contacts than the average person, your mean time to infection rises from (3¾+1¼)=5 to (3¾+3×1¼)=7½ years, a gain of only 50% by practicing a rather severe form of social distancing! •36/43

Another point worth noting: if 𝓡₀ is large enough, then 1−1/𝓡₀ is close to 1, so the proportion of infected, i ≈ 𝓦 and the rate of new infections β·i·s ≈ 𝓦·γ do not depend on 𝓡. So having control measures in place is basically useless. •37/43

(For example, if you think 𝓡₀ is around 20, then by extremely strong measures you might reduce 𝓡 by a factor of 2, and this decreases 1−1/𝓡₀ by a measly 5%.) Again an example of people who argue for a very high 𝓡₀ shooting their own arguments in the foot! •38/43

OK, this is all using a very simplistic model in which immunity is an all-or-nothing phenomenon, and even in this very simplistic model, the two essential parameters 𝓡₀ and 𝓦 are very much unknown for how covid will behave (and not well known even for OC43). •39/43

It might be worth considering more sophisticated models, like SEIRS (which adds a time between infection and infectiousness) or one with gradual decline of immunity rather than all-or-nothing. I suspect no considerable difference would emerge. •40/43

⁃ the fraction of the population infected at a given time depends mostly on the ratio 𝓦 between infection and immunity durations; we can expect a fraction of a percent, but probably not much lower;
•41/43

⁃ the number of new cases might well be in the hundreds par day per million people (forever), and again this depends crucially on 𝓦 more than 𝓡₀;
⁃ the fraction of the population immune at a given time depends essentially on 𝓡₀, not really on 𝓦;
•42/43

⁃ increasing your time to infection by reducing your contacts is HARD, because it's mostly due to how long immunity lasts;
⁃ if 𝓡₀ is high, keeping control measures in place in the endemic state makes no sense (they might make sense BEFORE, to “flatten the curve”).
•43/43