[math-fun] An alternative math model for COVID
The 100-year-old differential equation model for infectious diseases -- the one with R/R0 -- isn't the only possibility. I'm going to suggest another one -- which I'm not claiming is necessarily correct, but which has some of the same characteristics as the R/R0 model. My model is more akin to 'predator-prey' models, only in this case, the predator is microscopic. The diffeq model suggests that 'herd immunity' is reached when the % of immune/already-had-its is large enough that there are too few uninfected people left to sustain a chain reaction. According to most estimates using this type of model, herd immunity % for COVID will be in the 70-80% range. But there is an alternative model where a kind of 'herd immunity' is reached at a far lower %. Consider a predator-prey model of lions and antelopes. The lions don't have to *win* the lion-v-antelope races; they just have to not come in *last* in order to eat. Suppose that a lion has a top speed of 20mph for 1/4 mile (I have no idea what a lion's actual speed is). All the antelopes have to do is to stay ahead of the lion for that period of time in order to survive. At time t=0, some % of the antelope herd can't achieve 20mph, and are bound to be eaten. But as time goes on, more and more of these 'slow' antelopes are eaten and no longer remain in the herd. At some point, *all* of these 'slow' antelopes will be gone, and since a famished lion isn't likely to get any faster, he will be forced to find another herd of antelopes to chase which might still contain some 'slow' antelopes. Note that in my model *everyone* who is exposed gets the disease -- i.e., R/R0 is exceedingly high, but only some % -- e.g., those with 'co-morbidities' -- will succumb. Note that while the herd immunity % in the diffeq model is dependent upon R/R0, the herd immunity % in my model is dependent on a 'critical velocity', and is independent of chain reaction dynamics. Thus, if 10% of the antelope population can't achieve 20mph, then the 'herd immunity' will be achieved at 10%. Yes, I have mixed up to some extent the difference between unexposed and immune, but in reality, most people only care if they survive, not whether they catch an asymptomatic case of the disease.
Henry, Would it be OK to say what you're talking about, please? Thanks, Dan
On Tuesday/1December/2020, at 9:21 AM, Henry Baker <hbaker1@pipeline.com> wrote:
The 100-year-old differential equation model for infectious diseases -- the one with R/R0 -- isn't the only possibility.
Henry — is your model the same as the basic (SIR) model in which some fraction of the population starts in the Recovered (and therefore immune) state? --Michael On Tue, Dec 1, 2020 at 3:29 PM Dan Asimov <asimov@msri.org> wrote:
Henry,
Would it be OK to say what you're talking about, please?
Thanks,
Dan
On Tuesday/1December/2020, at 9:21 AM, Henry Baker <hbaker1@pipeline.com> wrote:
The 100-year-old differential equation model for infectious diseases -- the one with R/R0 -- isn't the only possibility.
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-- Forewarned is worth an octopus in the bush.
Henry, that is a very interesting idea. It reminds a bit of molecular kinetics. I wonder if a statistical mechanics or thermodynamics approach (the latter not concerned so much with mechanism as with observables) might be useful? Social distancing lowers the temperature of society -- or maybe it lowers conductivity? Temperature also shows up as economic activity. Anyway, I think there is much to learn from more macro models. The SIR model and its variants does not readily describe this second wave with more cases but less severity. Predator-prey model helps but is maybe still too detailed to be straightforward to apply. I have tried a macro model with 5-year projection of excess deaths using CFR and IFR, to try to ignore details, and got two order-of conclusions: Maybe a 25 pct increase in overall mortality averaged over 2020-2024, and risk due to Covid is maybe 10x risk due to automobile. Anticipate invention of turn signals, rules of the road, seat belts, etc! Best, kr. On 01/12/2020, Henry Baker <hbaker1@pipeline.com> wrote:
The 100-year-old differential equation model for infectious diseases -- the one with R/R0 -- isn't the only possibility.
I'm going to suggest another one -- which I'm not claiming is necessarily correct, but which has some of the same characteristics as the R/R0 model.
My model is more akin to 'predator-prey' models, only in this case, the predator is microscopic.
The diffeq model suggests that 'herd immunity' is reached when the % of immune/already-had-its is large enough that there are too few uninfected people left to sustain a chain reaction.
According to most estimates using this type of model, herd immunity % for COVID will be in the 70-80% range.
But there is an alternative model where a kind of 'herd immunity' is reached at a far lower %.
Consider a predator-prey model of lions and antelopes. The lions don't have to *win* the lion-v-antelope races; they just have to not come in *last* in order to eat.
Suppose that a lion has a top speed of 20mph for 1/4 mile (I have no idea what a lion's actual speed is). All the antelopes have to do is to stay ahead of the lion for that period of time in order to survive.
At time t=0, some % of the antelope herd can't achieve 20mph, and are bound to be eaten. But as time goes on, more and more of these 'slow' antelopes are eaten and no longer remain in the herd. At some point, *all* of these 'slow' antelopes will be gone, and since a famished lion isn't likely to get any faster, he will be forced to find another herd of antelopes to chase which might still contain some 'slow' antelopes.
Note that in my model *everyone* who is exposed gets the disease -- i.e., R/R0 is exceedingly high, but only some % -- e.g., those with 'co-morbidities' -- will succumb.
Note that while the herd immunity % in the diffeq model is dependent upon R/R0, the herd immunity % in my model is dependent on a 'critical velocity', and is independent of chain reaction dynamics.
Thus, if 10% of the antelope population can't achieve 20mph, then the 'herd immunity' will be achieved at 10%.
Yes, I have mixed up to some extent the difference between unexposed and immune, but in reality, most people only care if they survive, not whether they catch an asymptomatic case of the disease.
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participants (4)
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Dan Asimov -
Henry Baker -
Ken Roberts -
Michael Kleber