## Covid-19 highlights the pros and cons of mathematical models

Use this forum to discuss the philosophy of science. Philosophy of science deals with the assumptions, foundations, and implications of science.
Steve3007
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### Covid-19 highlights the pros and cons of mathematical models

Look up "herd immunity" anywhere and the chances are you'll see the equation:

1 - 1 / R0

R0 is the average number of people infected by 1 person in a population where nobody has immunity. The output of the equation is the proportion of the population who need to be immune in order for the R number for that population to come down to 1. This is called a "threshold" because it is the point at which the number of infected people stops growing.

But as with any mathematical model, the derivation of this equation contains simplifications compared to what actually happens in the real world. Among them are:

It assumes a "well mixed" population. i.e. the people are evenly distributed, each person is equally likely to come into contact with others and the immune are evenly distributed through the vulnerable.

It assumes that the only mechanism by which the virus transmission is reduced is that the presence of immune people reduces the density (number per unit area) of vulnerable people and therefore reduces the distance and incidence of contact between them and the infected.

But despite, or perhaps because of, this simplicity, the model is popular with politicians in press conferences. Do you think this model is too simplistic? Could the equation be modified to take account of other factors?
Count Lucanor
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve3007 wrote: February 23rd, 2021, 1:40 pm
But despite, or perhaps because of, this simplicity, the model is popular with politicians in press conferences. Do you think this model is too simplistic? Could the equation be modified to take account of other factors?
Most likely the same approach used by economists, the favored sons of politicians. Mathematical models in Economy have been a complete failure, precisely because they depart from abstract assumptions that have very little to do with real life.
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

Count Lucanor wrote:Most likely the same approach used by economists, the favored sons of politicians. Mathematical models in Economy have been a complete failure, precisely because they depart from abstract assumptions that have very little to do with real life.
I suspect that's too pessimistic. The defining feature of a model is that it represents some but not all of reality. So if a model was literally a complete failure there would be no sense in which it could be called a model. I think every theory we create is a model. The key to creating models is to work out which aspects of reality can safely be neglected so the model is manageably simple but also tolerably accurate.

The interesting point for me here is the fact that the herd immunity model is one that has been in the public eye a lot because of Covid-19.
Pattern-chaser
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve3007 wrote: February 23rd, 2021, 1:40 pm But despite, or perhaps because of, this simplicity, the model is popular with politicians in press conferences. Do you think this model is too simplistic? Could the equation be modified to take account of other factors?

To explain stuff to people, simplicity and clarity are pretty important. So we need models, examples and metaphors. But these teaching aids (if I may call them that) must be balanced too. Too simple (simplistic) and we misinform and mislead. Too complicated and we confuse our audience. We have all seen (in other topics) how concepts such as herd immunity can be misapprehended.

So surely the equation could be modified, and made more accurate, but will that contribute to greater understanding? That is the critical question, I think.
Pattern-chaser

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Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

Pattern-chaser wrote:To explain stuff to people, simplicity and clarity are pretty important. So we need models, examples and metaphors. But these teaching aids (if I may call them that) must be balanced too. Too simple (simplistic) and we misinform and mislead. Too complicated and we confuse our audience.
Fair point.
We have all seen (in other topics) how concepts such as herd immunity can be misapprehended.
I think that's a special case. I think RJG's unwillingness to see the absurdity in his idea that immune people vacuum up the virus isn't simply a misapprehension. It seems to me to be a conscious decision to believe what he wants to believe regardless. This is evident in the fact that even when his belief leads him to state clear absurdities, such as that the density of a system of objects is unrelated to the average distance between the objects, he can't abandon it.

That kind of behaviour, in my experience, is more common here than it is in the world in general, because the nature of this forum is such that it contains an unusually large number of people with a strongly entrenched pre-existing belief that they came here with the specific purposes of defending.
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

Pattern-chaser wrote:So surely the equation could be modified, and made more accurate, but will that contribute to greater understanding? That is the critical question, I think.
If an equation (and associated model) like this is widely used to explain the basic principles of a process, such as herd immunity, then yes that is a critical question. In that case even that simple equation is probably not needed. All that's needed is a qualitative description of the basic principles of social distancing and herd immunity: that reducing the ability of the virus to find new vulnerable people by increasing the distance and decreasing the number of encounters between the vulnerable and the infected is what is key. I think the vast majority of people get that.
Count Lucanor
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve3007 wrote: February 24th, 2021, 5:13 am
Count Lucanor wrote:Most likely the same approach used by economists, the favored sons of politicians. Mathematical models in Economy have been a complete failure, precisely because they depart from abstract assumptions that have very little to do with real life.
I suspect that's too pessimistic. The defining feature of a model is that it represents some but not all of reality. So if a model was literally a complete failure there would be no sense in which it could be called a model. I think every theory we create is a model. The key to creating models is to work out which aspects of reality can safely be neglected so the model is manageably simple but also tolerably accurate.

The interesting point for me here is the fact that the herd immunity model is one that has been in the public eye a lot because of Covid-19.
I never said there was any problem with models per se, I mentioned specifically mathematical models in human affairs.
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

Count Lucanor wrote:I never said there was any problem with models per se, I mentioned specifically mathematical models in human affairs.
OK. Although I'd say that mathematics is just the language in which the model is expressed. It seems odd to think that the choice of language has a significant effect on the efficacy of the model. But I guess it does have a psychological effect. People often object to, as they see it, being "reduced to statistics". But if a model originally expressed in English is expressed in French they don't think they've been "reduced to French".
Count Lucanor
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve3007 wrote: February 25th, 2021, 7:57 am
Count Lucanor wrote:I never said there was any problem with models per se, I mentioned specifically mathematical models in human affairs.
OK. Although I'd say that mathematics is just the language in which the model is expressed. It seems odd to think that the choice of language has a significant effect on the efficacy of the model. But I guess it does have a psychological effect. People often object to, as they see it, being "reduced to statistics". But if a model originally expressed in English is expressed in French they don't think they've been "reduced to French".
I can see the analogy working between different spoken languages (i.e. English vs French), but not between two different fields of knowledge. Mathematics has a language, it uses expressions and requires a syntactic structure, but it is not merely a language to convey semantic meanings. The key difference between mathematical models and non-mathematical models applied to human affairs is that the first ones treat their objects of study as closed systems with statistical regularities, predetermined by natural forces that can be reduced to mathematical formulas.
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

Count Lucanor wrote:The key difference between mathematical models and non-mathematical models applied to human affairs is that the first ones treat their objects of study as closed systems with statistical regularities, predetermined by natural forces that can be reduced to mathematical formulas.
I disagree with this.
RJG
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve3007 wrote:It assumes that the only mechanism by which the virus transmission is reduced is that the presence of immune people reduces the density (number per unit area) of vulnerable people and therefore reduces the distance and incidence of contact between them and the infected.
Steve, this is an error. You are falsely assuming that the mechanism of "herd immunity" is the same mechanism of "social distancing". It is not. These are two separate and distinct mechanisms.

Let's assume the R0 value is 2.5. Using the equation T=1-(1/R0), this means that we could attain protection for ALL 100 people if we vaccinate 60 people in a herd of 100 people, within a given environment (e.g. a large room).

So now according to your assumption of how the mechanism works, we could, instead of vaccinating these 60 people, just simply remove them from the room, and we would still get the same protective effect, ...right?

In other words, according to your understanding, whether we vaccinate 60 people, or remove 60 people, the result is the same, ...right?
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

RJG wrote:Steve, this is an error. You are falsely assuming that the mechanism of "herd immunity" is the same mechanism of "social distancing". It is not. These are two separate and distinct mechanisms.
You are wrong.

As demonstrated previously:

viewtopic.php?p=377615#p377615

herd immunity, as described by 1 - 1 / R0 works by the same mechanism as social distancing. It works by reducing the number of encounters between vulnerable/infected people, by increasing the distance between them. See also the Wikipedia article:

https://en.wikipedia.org/wiki/Herd_immunity
Wikipedia article wrote:The greater the proportion of immune individuals in a community, the smaller the probability that non-immune individuals will come into contact with an infectious individual.
As the article says there, it is the reduced probability of encounters between non-immune and infectious individuals that is important. As previously said, it's nothing to do with immune people sucking up the virus. If it was then that would be clear in the derivation of the equation.
RJG wrote:Let's assume the R0 value is 2.5. Using the equation T=1-(1/R0), this means that we could attain protection for ALL 100 people if we vaccinate 60 people in a herd of 100 people, within a given environment (e.g. a large room).
You've misunderstood or simply mis-stated the meaning of the threshold value which is the output of that equation. It does not mean "protection for ALL 100 people". It means that at that threshold, the R value, for a given R0 value, is 1. That means that on average, at that threshold, each infected person infects 1 other person. i.e. the incidence of infected people neither grows nor shrinks at the threshold.
So now according to your assumption of how the mechanism works, we could, instead of vaccinating these 60 people, just simply remove them from the room, and we would still get the same protective effect, ...right?
If you could remove immune people from a population and leave everything else the same then yes, the protective effect is the same. That is then called social distancing. As we're seeing right now, it combats the disease but at great expense to the economy etc. We've been through all this.
In other words, according to your understanding, whether we vaccinate 60 people, or remove 60 people, the result is the same, ...right?
If the removal of the vaccinated/immune people didn't cause the remaining people to move closer together than they would otherwise have been, yes the result simply in terms of combatting the virus is the same. We're seeing that result right now.

RJG
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### Re: Covid-19 highlights the pros and cons of mathematical models

RJG wrote:So now according to your assumption of how the mechanism works, we could, instead of vaccinating these 60 people, just simply remove them from the room, and we would still get the same protective effect, ...right?

In other words, according to your understanding, whether we vaccinate 60 people, or remove 60 people, the result is the same, ...right?
Steve3007 wrote:If you could remove immune people from a population and leave everything else the same then yes…
This proves that this (your assumption) is NOT the mechanism of herd immunity.

For if we remove these people from the herd then we would have a saturation value of 0% immune people (to 100% vulnerable people). But yet, according to science, we need a saturation of 60% immune people to reach herd immunity. And yet you claim we can reach herd immunity with 0% immune people within a herd.

If the mechanism of herd immunity is what you say it is, then we could end covid right now!

All we would have to do is just ship 60% of the people within a herd (a given community, town, or state, etc) away from the herd and then VOILA! ...the 40% remaining would then instantly and magically be "immune" to the covid virus that still existed in their environment. We then could bring back the remaining 60% in two separate groups (so as to not dip below the threshold value) and EVERYONE would be protected! Repeat this process community-by-community across the world, and covid would be gone forever.

Part of the confusion is thinking (or more accurately, being indoctrinated to believe) that the virus transmits from person-to-person. It does not. The virus transmits from person-to-environment and environment-to-person. Removing people away from an environment, or herd, does not remove the virus in the environment.

The protection mechanism of herd immunity is not from increasing vector distances (social distancing), but is by STOPPING the virus transmission dead in its tracks. The virus stops and cannot pass or transmit past an immune person. Any virus that encounters an immune person is a dead virus; it is STOPPED dead in its tracks. Without an immune person there, there would be nothing to STOP the transmission. "Increasing the viral distance" is not the same, nor as effective in stopping the transmission as STOPPING the virus outright by an immune person.

Herd immunity works by "saturating" a population with immune people who "stop" the further transmission of the virus. Whereas, social distancing works by "distancing" vulnerable people away from infected people.

TWO different mechanisms.
Last edited by RJG on February 26th, 2021, 9:38 am, edited 1 time in total.
Steve3007
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### Re: Covid-19 highlights the pros and cons of mathematical models

RJG, we've been through all of this over and over again. I've tried to get you to see sense and failed. I've tried to get you to read the derivation of the equation for herd immunity and failed. I tried to get you to at least admit the errors in your most absurd statements, and failed. I then asked you if we were going to be going round the same stuff again if we discussed it again, and you had that post removed by a moderator.

You've said yourself more than once that, in your view, trying the same thing again and expecting different results is madness. So why would I bother?
RJG
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### Re: Covid-19 highlights the pros and cons of mathematical models

Steve, did you even read what I wrote? Did you not even try to comprehend my words?

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