Epistemic Status: Highly speculative. I threw this together quickly, and wrote this to document how I went about it. This is an attempt to create a first toy model, so others can error correct and improve, and upon which less-toy models can hopefully be built. You can see the spreadsheet with my work here. Please take this and run with it, and please don’t take this as more than trying stuff to see what’s wrong with it.
No one seems to be creating models of various scenarios in a way that feels remotely realistic, or even in a way that feels super simplified but that can be used as intuition pumps or baselines.
This post aims to fix that, or at least provide a first step.
At this point, we mostly know we’re f***ed, and that the new strain is at least ~40% more infectious, probably 50%+ more infectious, perhaps as high as 65%-70%.
Speeding up vaccinations seems to be the opposite of what we are doing, but seeing exactly what it would take to get us out of this mess seems like a worthwhile exercise. Time to start building toy models.
Completely Naive Model
To start off, I did the quickest thing at every step to see what happened.
Note that we’re measuring and predicting actual cases not positive tests. The observed and official numbers will always be several times lower than the real ones.
This model assumes:
- Covid Machine Learning’s estimate of old infections is correct.
- Covid Machine Learning’s estimates of current infections are roughly correct, about 600,000 per day.
- Based on sequencing results, we estimate 0.06% of new cases for cycle ending December 19.
- We assume every ‘infection cycle’ is five days, so if you are infected on day 0, you infect others on day 5.
- The initial strain had an effective R0=1 on December 19, when things began.
- The new strain is 50% more infectious than the old one.
- Immunity is total if you have been infected.
- Vaccinations don’t exist.
- Control systems don’t exist, people don’t adjust behavior at all.
- Heterogeneity doesn’t exist, people are all the same. Straight SIR.
That’s deeply silly, of course, but it’s not more deeply silly than models that have been offered and taken seriously in the past, despite being obvious nonsense. Assuming away vaccinations is the most obviously nonsensical part of that, but control systems and heterogeneity point in the other direction in many places, and our vaccination progress has been pathetic, so it makes sense to start off extra naive.
I do believe approximately in assumptions 1, 2, 3, 4, 5 and 7, so we have three things we’ll need to fix: Heterogeneity, control systems and vaccinations. And we’ll need to vary the infectiousness level of the new strain, considering at least the 50% and 65% cases.
We’ll also want to add deaths, especially since the control system likely depends on that number.
First let’s check out what you get without fixing any of the three issues.
You get this:
This is rather disastrous.
The safest day between now and the endgame is around Pi Day, March 14, at roughly 40% of current levels.
The most dangerous day is around June 1, at around 244% of current infection levels. If you are not infected by then, you’d be 380% more at risk than today, because of increased immunity in others. In addition, the hospital system would presumably be under at least extreme strain.
The number of infections reaches current levels twice. On the way back up, it happens on about April 20, and on the way back down, on about July 1. Note that July 1 is still much more dangerous than today for those still vulnerable.
We get to 10% of current levels around August 8, 10% of that around September 1.
The new strain is 1% of infections by January 20, 10% by February 20, 50% by March 20.
Approximately 63% of the population ends up infected.
We can contrast that with a 65% more infectious strain:
That is a true disaster. This is not the same scale as the previous graph. Check the Y-axis.
The safest day is now February 17 or so, most dangerous is April 24 and it’s really bad with over three million infected on that day alone. Hospitals would be facing five times the current case levels, and would be fully in triage mode at best.
We get back to current levels by around May 21, 10% of that by June 10, 10% of that around July 1.
The new strain is 1% of infections by January 15, 10% by February 9, 50% by March 2.
Approximately 75% of the population ends up infected.
We are not currently approximating deaths. If we did, since we are assuming zero vaccinations and heterogeneity, we’d choose a fixed IFR and then adjust for the collapse of the hospital system. It would not be pretty.
That’s all an intuition pump of why there’s a very large difference between 50% and 65% more infectious. It compounds quickly, and the crisis hits much faster and much harder. The second scenario’s peak crisis has twice as many simultaneous cases.
The most obvious missing knob is vaccinations.
We presumably add some number of people who are vaccinated, and thus effectively immune, on some accelerating function from week to week.
Given how things are going, it seems safe to treat being vaccinated and previously infected as mostly uncorrelated.
As of January 5, about three weeks after approval, 1.4% of the population (4.73 million doses) has been confirmed as vaccinated. That’s a really pathetic pace, and didn’t even require second doses. It seems safe to assume we’ll do better than that going forward, but how much better? That’s much harder to say. The limiting factor could quickly shift to the limited vaccine supply soon, so one approach is to assume that we administer most of the doses we have purchased. Biden has vowed 100 million shots in 100 days, so my ‘baseline scenario’ will assume we can do 250,000 vaccinations a day from December 14 until January 20, then 500,000 vaccinations a day starting January 21, and they are (for now) chosen at random.
Once someone is vaccinated, how immune are they and how fast? I will assume for now that everyone gets shot number two at the three week mark. My understanding is that you’re 80% immune by about day 10, and that with the booster we will say that jumps to 95% immune on day 30. We’ll say that you get no effect until day 10, people will be careful and be dealing with side effects at home, but also will have had to take some physical infection risk to get the shot, so let’s say those effects cancel.
To make my life easier without changing the math much, we’ll say that you get full 95% protection on day 15, and before that you have nothing, and simulate that by counting vaccinations as if they happen on day+15. It’s basically the same thing, and avoids having to keep track of who catches the virus before the vaccine can work.
There’s the model where such folk are X% immune to each potential infection, and there’s the model where such people are X% likely to be immune period and (100-X)% likely to be fully vulnerable. I’m going to model the second version, so all we have to do is multiply our vaccination numbers by 95%.
So each cycle, there will be an additional 2.5 million people vaccinated, which we will add to either the “Vaccinated and Previously Infected” or “Vaccinated and Never Infected,” and both categories will be fully safe.
Let’s see what that does.
Excellent news! It’s a much better picture if we’re looking at only 50% more infectiousness, despite being a rather pathetic pace of vaccinations.
For a 50% more infectious new virus, the worst remaining day is already behind us.
It takes over from the old strain at exactly the same pace (of course), but the peak is no longer terrifying, because there’s enough immunity to turn the tide quickly.
There is still a secondary peak later on, effectively prolonging our agony for those not yet vaccinated from May until September. That sucks, but it’s not a crisis situation.
We end up with 36.7% infected, versus 63% infected without the vaccine. Big improvement.
However, if you change from 50% more infectious to 65% more, we get a very different picture.
Now we have a terrifying peak at the end of April.
We end up with 51.5% infected. Much better than 74%, and a peak of 5,000,000 per cycle we still see what looks like hospitals very overwhelmed in May and June.
This still is a more reassuring picture than I had going on. At 50% additional infectiousness rather than 65%, we have a very good sporting chance to have the vaccine arrive almost exactly on time, and keep the final wave in check.
This is strong motivation to push hard on the vaccine side of the race. If we can accelerate faster than this, it will help a lot. Conversely, much slower than this, and things get much worse.
Now we need to add control systems. This will potentially help in some places, but also hurt us in others.
Adding Control Systems
It is known that the pandemic has involved powerful control systems of various sorts.
As people see infection rates, hospitalizations and deaths go up, they take less risk. As they see such things go down, they take more risks. Governments do the same, opening and closing various businesses, schools and other locations and activities to keep things in balance. For our purposes here, these forces all look similar. The question is how they would react in these new circumstances.
My model has the following gears:
- People are slowly getting what might be termed pandemic fatigue. Over time, they are less willing to put their lives on hold, and thus do riskier things, which has been enough to prevent us from winning via herd immunity so far. Magnitude of this is hard to say.
- People react to some combination of infections, deaths, hospitalizations, and the reactions of governments and other authorities. It’s not clear what is the central mix of these elements. Hospitalizations and deaths seem to play a large role, thus introducing lag. It’s not clear what would happen if deaths and infections diverged, such as if nursing home residents were vaccinated and thus protected, but it could get weird.
- People react to levels, not rates of change. Things rapidly getting worse or better doesn’t much matter to them. Mainly they care about how dangerous things seem to be. This is a lot of the evidence for pandemic fatigue – if we are stabilized at relatively high levels now versus before, despite substantial immunity and better knowledge, there must be a factor pushing in the other direction.
- Because people’s observations lag at least a week behind (for infections) and as much as a month or more behind (for deaths), and people then take time to adjust especially for the official systems, the control system will be slow to react to changes. This is much of what causes waves, peaks and valleys.
- In a full ‘back to normal’ scenario we’d be looking at R0 of somewhere between 2.5 and 4.5 before any immunity effects. In the past I’ve used 4, but often seen smaller numbers. Which means that given R0 is currently close to 1 with 20% infected (e.g. before immunity R0 ~ 1.25) people really, really aren’t taking this all that seriously in aggregate.
- The alternative hypothesis to #1 and #5 is that the virus has already mutated at least once to be more infectious, maybe several times, and even the old strain has baseline R0 higher than 4.
For now, we are assuming vaccinations are random, so they won’t impact the IFR. While that remains true, we can approximate the combination of all these factors by an average of the infection counts of the past several weeks, with infections lagging one cycle, deaths lagging four or five, and some reaction time, so let’s say we take the average of the last six cycles.
To start, let’s also ignore phase shifts like overloading hospitals, and ignore fatigue on the hopes that vaccines coming soon will cancel it out, although there’s an argument that in practice some people do the opposite.
As a first guess then, let’s take that average of the previous six cycles, the ratio of that to current levels, then raise that to some exponent. As sanity checks, I ask myself what I think people would be capable of doing from here if things got how much worse, and when they’d mostly return to normal, and I noted that I’d be highly surprised if such forces were sufficient to cause an additional peak after the new strain started declining again, with vaccinations continuing and a lot of immunity.
This settled me on an exponent of 0.25, which is the limit of where a final peak does not emerge with 65% additional infectiousness. You can see it almost happening on the chart but it stalls out.
Now what happens?
With 65% more infectiousness:
That’s a higher peak than before. That makes sense, because lag is your enemy here. By the time people realize things are getting bad again, they’ve let the situation get further out of control, and it happens about a month faster, in early April instead of early May. We also see an extended die-out period afterwards, which seems realistic. We end with 54% infected, slightly higher than without the control system.
With 50% more infectiousness instead:
Instead of the new peak being below the old peak, now it’s slightly above it, although it does not last long which gives less time for disastrous scenarios to accumulate. Final infected percent is around 49%, again somewhat higher.
Weird things happen with overeager control systems. I don’t expect this at all, but if you set the exponent fully to 1 and allow full pre-pandemic behavior to emerge, there’s even a ‘false dawn’ scenario, where it looks like it’s over, people go fully back to normal, and then it isn’t over, and there’s a final even bigger crisis, and then it happens again even bigger:
That’s because even with a lot of immunity, by assumption the new virus spreads really, really fast when everyone ignores it. Reducing to an old base R0 of 3 makes both later peaks stop at 1.2mm cases a day.
That doesn’t mean I think such things are likely, but it is worth noting that we could be this stupid. I can’t rule it out.
I also added a knob to make current R0 only move a percentage towards the target R0 to avoid dramatic shifts like the ones in that graph, if one wants to do that. It’s not clear the knob does much extra work.
The big tricky thing is that the control system largely depends on hospitalizations and deaths, and those depend on who gets sick. If we vaccinate the vulnerable first, hospitalizations and deaths will begin to lag behind case counts. That’s great in terms of patient outcomes, but has a dangerous side effect.
What does that do to the control system?
There are two major heterogeneity effects I think require incorporation before the model will seem complete: Selective vaccination, and selective infection.
Selective infection, as I see it, is mostly about different levels of risk taking, and also different levels of vulnerability, which presumably is correlated with superspreading. If you take twice as much risk, you are twice as likely to be infected each day. Depending on the distribution of risk taking, this can mean either not much, or a hell of a lot. If the superspreaders slash super-risk-takers are super out of control, they can get taken out quickly, and a little immunity can go a long way.
This seems very plausible to me. If anything, I continue not to believe that immunity isn’t doing a lot more than it is. Some people wear masks and hide, others disdain masks and go to crowded bars. Some get to work at home, some have to be essential frontline workers. This doesn’t seem like it should be complicated.
Me and many of those I know, both family and friends, are doing effective prevention, well above 90% below pre-pandemic activities, and were generally doing less of the risky things to begin with. Yet if people overall took more than 75% less risk than they started with, we would not have a pandemic before the new strain arrives.
It seems likely to me this effect is roughly fractal. If we say that one third of the people take two thirds of the risk (which seems crazy low) then I’d suspect that one third of that top third takes 4/9ths of the total risk, and so on, in both directions. Then each person is likely already infected proportional to how much risk they take.
We then also need to consider that those taking more risk are probably less likely to accept a vaccination, although some policies put them at the front of the line. No idea how that works out.
It all gets complicated, but we can probably treat this for now as one knob, by saying the first 50% of those who get infected will have taken on X% of the combined risk, and treating vaccinations as still roughly random with respect to risk taking (see the other adjustment for death rate concerns) since we have forces pulling in both directions, at least for now. Previously we’ve set this X to 50, which is too high, and thus we have been underestimating immunity from infections (assuming that reinfection remains super rare).
Then we have heterogeneity from vaccination. One aspect of this as noted before is how much risk such people take. For now, I’m willing to not worry about that, or set a knob and default it to no adjustment.
The more interesting question is what happens in terms of vaccinating the vulnerable. Right now, we are vaccinating on two fronts.
Nursing home residents, who are 1% of the population and 40%, are in the first wave of vaccinations everywhere. At some point, all areas add the elderly, but most are waiting.
Simultaneously, health care workers start the other angle of vaccination attack, followed by other essential workers, politically powerful or influential people, and those “most deserving” of allocation via politics and power. There are some old people in this group, but overall they are frontline and so tend to be younger than average, and less vulnerable than average, at least for those over 18.
Because risk grows so dramatically with age, doing some very old people early is more important than worrying about the other group being unusually young. Thus, we should expect to continue to lower the IFR further as more elderly get vaccinated.
On the flip side of that, IFR will go up if hospitals are overwhelmed. We don’t see evidence of that now, so it seems like hospitals can mostly handle current levels, but we are seeing definite signs of strain. At a minimum, patients with other conditions are suffering. So the behavioral effects should come into play more rapidly anywhere above current levels.
So presumably, we should add a column for IFR baseline, and one for IFR effective given infection levels and who has been vaccinated at a given stage, and adjust people’s reactions based partly on that, with a knob for how much weight they give hospitalizations/deaths versus infections.
To be conservative, I’ll assume an IFR of 0.45% to start, to make us line up with recent death numbers.
How much can we cut deaths? Let’s mostly focus on age:
Nursing homes have about a third of all Covid deaths, presumably almost all from people who are 70+, so we’ll want to note that we’re going there first, then assume we have two tracks, one for politically approved adults who have random risk, and one for old people in descending order.
Right now, the vast majority of doses in most places are going to the politically approved – there are many times more prioritized workers than there are residents of nursing homes. In the second phase, it plausibly reverses, as we hit a wall of who is plausibly better than the elderly. Things have been so crazy and random it’s hard to know.
About 6% of the population are health care workers, depending on your definition. If you expand that to other ‘essential’ workers you can get more or less as many extra political choices as you want. We’ll make a knob, but let’s assume for now that the first stage does all health care workers and those 70+, combined 14% of the population. Nursing homes going very early means that the early doses should be at least that effective. Then we have an even split between random people and those 60-69 or so, who are 20% of deaths (and the majority of remaining deaths) while being 7% of the population.
So the first 14% of the population protects against roughly 67% (two thirds) of deaths, then the second 14% protects against another 20%, which is about two-thirds of the remainder once again. Hospitalizations should be somewhat less extreme, but still not that dissimilar. After that, it’s likely that they open things up to everyone, but still exclude kids, so there’s a quarter of the population left out that basically never dies, and we sample from the rest.
Then we have to decide how much such factors matter to people, versus observing their own risk or case numbers, when deciding what to do. And also we have to consider that if people you know are vaccinated, you might then not care as much because you can’t infect them, and you sense less risk.
As a default let’s do an even split. Half of consideration is deaths. The other half is case counts, unadjusted for immunity because people don’t seem to make that adjustment.
Thus, at 14% vaccinated, relative death rates will be down by 62%, which reduces perception of risk in this model by 31%. At 28% vaccinated, relative death rates are down by 75%, reducing perception of risk by 37%. Then we don’t improve further.
In the interests of simplicity, let’s call that a decrease in risk perception of 33% and deaths by 66%, phased in linearly over the first 15% of the population vaccinated. Close enough.
Of course, it takes time to kick in, so we’ll give it 8 cycles (3 for the vaccine to work, 5 for people to actually die). Again, that’s an approximation but should be fine, as the effect comes in continuously anyway.
We’re not considering kids here at all, who are almost never hospitalized but also relatively rarely tested, I’m not sure how that adjustment works.
Before setting that up, the obvious prediction is that this will create much higher peaks in terms of number of infections under our control system assumptions, but still greatly reduce deaths.
Here’s what we get at max effect, when we have the control system only look at deaths, and with zero control system memory (so it doesn’t look at how safe we were being last week):
65% more infectious:
50% more infectious:
Those are substantially higher peaks than without the modification. 63% and 56% of people get infected respectively in these new scenarios by the end, again substantial boosts.
If the control system is 50% deaths/hospitalizations and 50% infections, you get an answer halfway between the two extreme scenarios, as you would expect.
While substantial, those are much smaller bad effects than the positive effect on saving lives.
(Reminder: So far there have been about 350k dead, and the next three weeks are already baked in.)
There’s a clear strong benefit to vaccinating old people early in terms of deaths. With the more infectious variant under plausible numbers, we end up with 663k dead, but with random vaccinations we would have had 844k dead.
Note that if we take the new strain out entirely, our model predicts a slow but steady improvement from here, with a combined 525k dead:
I think I’ll stop there, but give everyone a chance to look at the spreadsheet, spot errors, and make copies to try for yourself. You can use this link.
No doubt there remain errors of all kinds, both spreadsheet errors and conceptual mistakes, and things that weren’t incorporated, including things I know about (e.g. I’m intentionally not using the fractal nature of risk taking, which is hard to estimate but could be a very large advantage).
This is not meant to be an answer. This is meant to get juices flowing, and at least ask some of the questions and throw out concrete possible futures. Let’s work to improve it, and/or inspire other attempts.