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24: Language and Entropy (Information Theory in Language)
699 Plays7 years ago
Information theory was founded in 1948 by Claude Shannon, and is a way of both qualitatively and quantitatively describing the limits and processes involved in communication. Roughly speaking, when two entities communicate, they have a message, a medium, confusion, encoding, and decoding; and when two entities communicate, they transfer information between them. The amount of information that is possible to be transmitted can be increased or decreased by manipulating any of the aforementioned variables. One of the practical, and original, applications of information theory is to models of language. So what is entropy? How can we say language has it? And what structures within language with respect to information theory reveal deep insights about the nature of language itself?
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Transcript
Introduction and Personal Story
00:00:00
Speaker
The spectre of disease causes untold mayhem, anguish, and desolation. The extent to which the spectre has yielded its power, however, has been massively curtailed in the past century. To understand how this has been accomplished, we must understand the science and mathematics of epidemiology.
00:00:15
Speaker
Epidemiology is the field of study related to how disease unfolds in a population. So how has epidemiology improved our lives? What have we learned from it? And what can we do to learn more from it? All this and more on this episode of Breaking Math. Episode 25, Pandemic Panic. I'm Jonathan. And I'm Gabriel.
00:00:42
Speaker
And welcome to Breaking Math. You can find us on Patreon at patreon.com, such as Breaking Math Podcast. Also, we are on Twitter at BreakingMathPod. And we're on Facebook, Gmail as Breaking Math Podcast. And we have our website, BreakingMathPodcast.com. So what are we going to be talking about today? Today is all about epidemiology. And today we have a very interesting story about why we chose this episode. Jonathan, want to tell us a bit about the inspiration for this episode?
00:01:11
Speaker
Sure, I got very sick in late January, early February, so I had to go to the hospital for a couple weeks. And one of the things that I noticed about the hospital right away was everything worked like clockwork. And that, of course, is organization.
00:01:28
Speaker
They have reasons for operating like clockwork. That's not an observation in and of itself, but it led me to observe when it comes to infection, everybody was doing things like washing their hands. And it made me think, why are we doing this? I mean, I know why we're doing this, but in a system where things breed, like the Lotka-Volterra model that we talked about on episode three,
00:01:51
Speaker
Where you have rabbits and foxes The foxes eat the rabbits the foxes starve all that kind of stuff you can listen to it you have a lot of complicated math that goes along with that and that one thing led to another and That led to this episode on epidemiology
Impact of Modern Medicine and Historical Practices
00:02:07
Speaker
It's always very interesting when something very personal can be used as inspiration for this episode or for this podcast. We've always talked about how mathematics permeates all aspects of our lives. So I kind of like how you can use things like your hospital stay and talk about the mathematics of hospitals and how we improve our daily lives.
00:02:28
Speaker
Oh, yeah. And as far as an improvement to my daily life goes and your daily life and almost every probably everybody who is listening because they have an Internet connection has been impacted by this is just we don't die all the time anymore. And the rate of infectious disease reached its more or less its current point back in the fifties and from a very high rate. And my
00:02:54
Speaker
Dad back in the mid 40s got yellow fever and that was a common thing to happen back then because people just died left and right of Childhood illnesses all the time because vices are brutal and children are weak. Mm-hmm Wow
00:03:09
Speaker
Now, you had mentioned earlier about modern medicine. Would you say, in some sense, that modern medicine goes back to ancient Greece with Hippocrates? I would say that modern medicine, I would say that in as much as modernity can be traced back to modern Greece, yes. Hippocrates did some pretty neat things. One of the most important things we did was assume that disease has a natural cause instead of blaming people for their diseases.
00:03:39
Speaker
like was common back in the day. Not only did he do that but he experimented and then experimentation got better of course in the Renaissance era and then the Enlightenment era and then the Modern era and here we are today.
00:03:55
Speaker
Yeah, and one of the things that actually really fascinates me is both culture as well as science and empiricism and mathematics. Because even to this day, we have very strong elements of culture that permeate our thinking. I'm talking as a whole. I'm not just talking about, say, in Western medicine.
00:04:17
Speaker
But people seem to be really emotionally tied to ideas in medicine that are not mathematical. You know, I'm talking about like, oh gosh. Well, one example of that is just in our value of pride itself. Surgeons still, by and large, refuse to use a checklist, even though math and statistics shows us that using a checklist reduces the amount of errors by a lot. They think they're above that.
00:04:43
Speaker
And it's still hard for hospitals to get doctors to wash their hands when they should. That's crazy. Wow. So pride is an element. That absolutely makes sense. So we sometimes are still our own worst enemy or in our own evolution. We still haven't conquered all of our issues. Anyways, this episode, not to go too far on a tangent, this episode, of course, is to be firmly about mathematics and medicine.
Interactive Segment: 'Asking Math'
00:05:08
Speaker
Oh yeah, mathematics. Well, not necessarily medicine, but mathematics and disease. Pac said we're going to talk a lot about what medicine can do, but, um, I mean, we'll see. And we have a new segment on the show. That's right. This new segment, uh, we are currently calling asking math where, uh, essentially, uh, how, how would you describe it, Jonathan? You have a question about math or mathy things, send it to us and we'll see if we could answer it. If we don't, then we'll ignore you.
00:05:36
Speaker
No, actually, I look forward to this. This is an opportunity to interact with you as a listener. And also, this keeps us on our toes. This gives us a chance to really dig in and do our own research. We're going to learn a lot with this. Now, what's interesting is, I don't know how many of you all spend a lot of time researching on internet forums, but people can sometimes be right. Sometimes they can be partially right, and sometimes they can get really flamed when they are partially right. I understand that that can happen. I'd like to think that, for me personally,
00:06:06
Speaker
I will be very open about it when I've been corrected.
Foundations of Epidemiology: John Grant's Contributions
00:06:09
Speaker
It's just exciting for me because it's a chance to really delve into mathematical questions for our benefit and for your benefit. And it's also a way to get to know our audience better. You guys have done a good thing for us by giving us a platform and helping us get the message about certain mathematical concepts out to the world and in return we'd like to know more about you and your interests.
00:06:30
Speaker
Yeah, so please send us your breaking math questions. And currently, I think email is probably the best way you can also send it to us on Facebook.
00:06:38
Speaker
Yeah, and again, that email is breakingmathpodcast.com and that Facebook address is facebook.com slash breakingmathpodcast. Today I'm breaking math. We're going to talk a little bit at first with the history of the study of disease and epidemiology. We're then going to get into modeling epidemics. We're going to talk a few examples of those models and how to use them.
00:07:04
Speaker
So the first person we're gonna talk about his name was John Grunt and he was a haberdasher That is a really fun word to say I did not know what that meant before about 10 minutes ago for those who are not familiar Jonathan Don't explain what a haberdasher is a haberdasher in the United States is someone who is a purveyor of men's clothing and in England is a purveyor of sewing goods so
00:07:27
Speaker
back in the 1600s during which John Grant was alive. Maybe it was a mashup of either one. I didn't want to delve too far into the haberdasher angle this guy's like, it's not the main part of his life. Sorry, I go on tangents a lot. Okay. That'd be the last we hear of the haberdasher part. Aberdasher.
00:07:47
Speaker
But he was also an author. The plague broke out in the 1660s. And because of the plague, there's a lot of renewed interest in public health. He was an author of a book that talked about how to do that. But he also did some concrete research. His book was never put into effect. And he created the first life table.
00:08:12
Speaker
Wow, John Gaunt, haberdasher, author, mathematician. And what is the life table? Life table. So this is very interesting. This actually is a table where you take a population of people and you talk about how many of them are expected to die in a given time interval. Yeah. And now actually during the research, we pulled up one of his life tables. Oh, yeah, we did. And it's pretty shocking.
00:08:39
Speaker
For our listeners, why don't we read to you some of the more salient parts of this life table? So right off the bat, the infant mortality rate was really high, obviously, in the 1600s, and 36% of the population would die by age six. Right, out of this little table here, it says right here, of 100, as you just said, 36 die within the first six years. And then it says in the next 10 years, 24 more die, and then after that- So that's 50 already. 50% of the population dies by age 16.
00:09:08
Speaker
Yeah. Yeah. And then the next 10 years you have 15 more die. We're at 65% of the population dying by age 24. And the next third one by age 34, you have nine more deaths. Yeah. Then six, four, three, two, one. And a lot of people think that we didn't live past 30. We totally lived past 30. It's just that if you didn't live past age six, you probably, I mean, if you live past age six, you're probably going to make it pretty far.
00:09:34
Speaker
But yeah, but that's the reason why the life expectancy is low and actually is lowered in the United States right now because of obesity and the opiate epidemic. So this is dark stuff that we're dealing with. At the end of the day, we're dealing with a grim specter.
00:09:53
Speaker
Yeah, totally. In fact, I was just talking to Jonathan during the planning of this episode, and we had talked about this guy here, John Grant, and we're wondering if perhaps he was depressed with all the mathematics and tallying of death. Well, what I thought of is just from what I've read, people who study actuarial science, they're actually whole websites devoted to actuarial science.
00:10:18
Speaker
and on mortuary science, and there's a lot of crossover in tone. And I do believe that the same type of people who bury and take care of our deceased are the same type of people who study our deceased mathematically. And isn't there a case where in terms of the perception, in terms of societal perceptions of death, following a very significant plague, there is an increase in executions? Oh yeah, that's right. Yeah, after any plague,
John Snow and Cholera Mapping
00:10:48
Speaker
There's a big spike in executions. And because the theory is when death is so prevalent, the feeling of death is devalued for the living. Wow, that's fascinating. So no matter how you slice it, death certainly is part of life and is very, very worthy of study.
00:11:07
Speaker
Calculating who survives and who dies runs into statistics, and that's a big theme in epidemiology, is statistical models. So let's say in London there's an outbreak at a specific hospital of swine flu or something worse than swine flu, and we want to see what's going to happen. We create what's called a mathematical model.
00:11:30
Speaker
And it doesn't have, you know, several million variables. There's not a variable for each person. There's not a variable for each germ. You simplify and you simplify using a mathematical instinct, but also you use statistics to guide you. And statistics was created from gambling and death. Gambling and death. Interesting how you pair those two together.
00:11:55
Speaker
Yeah. So if you don't like, uh, you know, studying stats right now, then just, you have an excuse cause you don't like gambling or death. It's worth noting, uh, these life tables are, uh, were and are still used for insurance policies.
00:12:09
Speaker
Yeah, if you take out a life insurance policy, it costs more if you start that when you're older. Um, because there's, you know, there's a bigger chance you're going to die sooner. So the insurance company has less time to make money to pay off and, you know, less time crooks. But yeah, as we've mentioned, uh, the seeds of epidemiology, John Grunt's research and John Grunt's book.
00:12:32
Speaker
Um, which is about public health have been tied since the very beginning. Um, and with our next person, we'll see that too. Yeah. So moving down a couple, uh, about a century and a half, two centuries, we have Jon Snow. Where have I heard, where have I heard that name before? I think he's on the game of thrones. Yeah, exactly. He was an actual real guy. Want to tell us a little bit about Mr. Jon Snow.
00:12:57
Speaker
Oh, yeah. He is really the father of epidemiology or, you know, he's currently considered to be the father of epidemiology because he in 1854 graphed deaths on a map. That's all he did. But it was a massive step in the right direction.
00:13:18
Speaker
Broad Street, part of Soho in London, had an outbreak in 1854 of cholera, during which 616 people died. And if you recall, Louis Pasteur wouldn't do his research for like another 20 years or so.
00:13:33
Speaker
miasma versus germ theory was still up in the air. And this was another notch in the right direction of germ theory because he proved that cholera is spread through water. Because if you look at the dots on the map, and you can find this on the Wikipedia article about John Snow, they're all clustered around sources of water.
00:13:50
Speaker
Interesting. Yes, that's a really good way of studying a phenomena that may not be well understood at a given time. It's just seeing where things happen. Now that's not to be confused with the fact that correlation does not always equal causation, but in this case it was certainly very successful.
00:14:08
Speaker
Oh, yeah. I mean, with all science, what we're doing is we have a model in our head of how the universe works. We take the model of how the actual universe works, which is just the universe. We compare the two and we reevaluate and it's amazing how well that works. It's kind of bizarre sometimes. You know, and also for our listeners who may not be aware of this, can you talk a little bit about what miasma is?
00:14:32
Speaker
Miasma is air that smells bad. Well, that miasmic is used to describe air that smells bad. But people thought that caused disease. If you've ever seen a plague doctor mask,
Innovations in Epidemiological Studies
00:14:43
Speaker
it looks like a leather bird's mask. And those are filled with potpourri and used by plague doctors because they thought that good smelling air would cancel out bad smelling air. Wow.
00:14:53
Speaker
Wow, it's so fascinating how far we've gone. And of course, I always like to say this, we are not at the epitome of knowledge that we know of. There's so much happening now that I'm sure in 100 years we'll look back and think, wow, why do they believe that?
00:15:09
Speaker
Yeah, absolutely. There's always things that we need to improve upon. And it's like, it's like, I think with knowledge, a lot of times it's like trying to drain a lake. So the first thing you could do is get giant buckets, but then there's not, then the bucket gets unwieldy. So you use a spoon and spoon gets unwieldy. So you take some paper towels and you pat the bottom of the lake dry. So the lake gets drier and drier, but it's approaching something. And I feel like that's what we do with science.
00:15:38
Speaker
Yeah, absolutely. So just before we move on, can we talk more about the effect that John Snow and his research had on treating the disease for London as a whole?
00:15:52
Speaker
Well, it put an importance on identifying sources of dirty water. His recommendations for what to do, however, were not put in place until after his death, and by then a lot of the stuff was already suggested by other sources. So unfortunately, because like we've said before, England in the past had an issue with modernization.
00:16:14
Speaker
And they did things like keep bank records on notched sticks. And I can't remember the other thing that we talked about on a previous episode, but in this case, and in the case of John Grant, did not put in, put their recommendations into effect for many years. So now we're going to get into the late modern era. Yeah. So I think one of the first people who we are going to talk about for this section is a certain Dr. Janet Lane Clapon.
00:16:41
Speaker
Dr. Lane Clapon was remarkable for her time. She had a PhD and an MD, what's called a double doctor. And she pioneered the use of cohort studies in epidemiology. And remember, this is only about 50, 60 years after John Snow did his research. So it was still pretty new. I mean, this is the time that Einstein was doing this stuff, the time that World War I was about to break out.
00:17:09
Speaker
So it was quite a while ago. And cohort studies are when you do a cross-sectional study at time intervals. And a cross-sectional study is when you do a study of a population, a cross-section of the population at a certain, at a discrete point in time. So is this in any way related to how control groups are used in research?
00:17:35
Speaker
It is. And the relation between cohort studies and control groups is a little bit out of the scope of this episode, unfortunately, and we will do an episode on statistics. We promise you stats fans that. And the reason why this is so useful in epidemiology is because if you think about a virus or bacteria, what it does is it spreads. And if it spreads, then you want to know how it's changing because if
00:18:01
Speaker
Its main action is something that happens over time. You have to study what happens over time. And it was an obvious step like any genius step is, but revolutionary like any genius step is. Sure, let's talk about some of her cohort studies and what effect they had on society.
00:18:20
Speaker
Oh yeah, so there's debate at the time whether or not cows or breast milk was better for babies. So she divided the babies into two groups, or cohorts, cohort studies, and fed one with cow's milk and one with breast milk.
00:18:36
Speaker
and the ones who were fed breast milk got heavier and she used the statistical methods that she pioneered and this is the reason why she used a cohort study and the cohort study is more expensive than other types of studies sometimes but it's much more reliable in certain ways and she was able to establish that these results were unlikely to occur because of a fluke or because of confounding
Disease Vectors and Control Measures
00:19:02
Speaker
such as something other than the type of milk being responsible for the difference in weight. I'm sure there's plenty of other examples of confounding that ruins studies, but yeah, you made the point pretty well I think.
00:19:16
Speaker
Now we're going to talk a little bit about disease vectors. And a disease vector is just something that carries disease into an organism. So in this case, we clearly are not talking about a vector in the sense of something that has a direction and a magnitude. All we're talking about is a method of delivery vehicle. This is not the mathematical definition of a vector.
00:19:38
Speaker
No, yeah, a vector simply means carrier or transporter. You can see how it applies to arrows and you can see how it applies to disease. It's a very vague word. So what's the official definition of vector? The official definition of a vector is something that carries a disease into an organism. It's the delivery thing.
00:19:58
Speaker
So let's say it could be, for example, mosquitoes could be a vector or the vector could be something like a flea or a vector could be simple as snot. Oh yeah. Even mucus when you sneeze at work, that's a vector.
00:20:14
Speaker
And this is kind of a nebulous definition because it's the quote-unquote intermediate one. And where do you draw the line of intermediate? Is the lake a vector because mosquitoes breed in lakes? Traditionally, no. So we really have to understand that a vector
00:20:34
Speaker
in it, almost like the mathematical version of a vector is something that can be simplified very strongly without removing any of its useful information. So you can model it as a number, basically. And I'm sorry, who is the researcher who introduced the idea of disease vectors?
00:20:51
Speaker
The researcher who really did a lot of research on vectors with respect to malaria was a British man called Ronald Ross. Everybody's been British so far, except for I'm not sure if Janet Lane Clapin was. A concrete example of how he discovered a vector is he let these mosquitoes that were uninfected suck on this guy Hussein Khan's blood.
00:21:15
Speaker
who is in Sukharabad in India and inside these mosquitoes he found these round cells that were not part of the mosquito cells because they didn't look anything like animals like like insect cells and he was so excited by his discovery that he
00:21:35
Speaker
wrote this poem. This day, relenting God hath placed within my hand a wondrous thing and God be praised. At his command, seeking his secret deeds with tears and toiling breath, I find thy cunning seeds. O million murdering death, I know this little thing a myriad men will save. O death, where is thy sting? Thy victory, O grave.
00:21:56
Speaker
And this is an example where you don't need test tubes and stuff to really, I mean, you need test tubes, I guess, to find the vector inside of the mosquito or whatever. But what I mean is use a malaria net to get rid of, to decrease the spread of malaria by a lot.
00:22:13
Speaker
Malaria, like many of you know, is the primary historical killer of humans. And this just shows how, again, this is the third time we've mentioned it, but how important epidemiology really is. And I don't think it's political to say that when we responded to terrorism by decreasing the amount of effort that we put into eradicating polio in Afghanistan,
00:22:42
Speaker
that we're doing the world a grave disservice. So although disease vectors themselves are not mathematical, we certainly do get very mathematical in recent models of epidemics.
Understanding Disease Spread: R0 and Models
00:22:59
Speaker
So in this next segment we're going to talk about some mathematical models of epidemics. And the first thing that we are going to talk about is what's called a basic reproduction number. This is something that was found by a gentleman by the name of George McDonald in 1952 actually.
00:23:15
Speaker
And it's a really simple concept, and you'll see why in just a second. The basic reproduction number of a virus or bacteria or whatever is the number of new cases that a case generates on average. So if you're sick, how many people get sick because of you on average? And if less than one person gets sick on average, then the disease will die out.
00:23:36
Speaker
if more than one person gets sick on average, then the disease will live on and eventually take over the entire population. And the truth is a little bit between these two, but for fast epidemics, this number, the basic reproduction number, and we're going to call it R0 from now on, because that's what it's called in the literature, is very useful.
00:23:57
Speaker
So let's think of an example here then. We've got something like tuberculosis. So if tuberculosis can either cultivate slowly or quickly via different processes, then its total reproduction number is the sum of the base reproduction numbers. That is to say R0 is equal to the fast R0 added to the slow R0.
00:24:23
Speaker
And it really is not that surprising because if you have different ways that a disease can proliferate, then it's going to try all of them. So the number of people who you're going to infect is the sum of all of those. And that's the basic logic behind it. I know that is not one-to-one correlation. And the example of tuberculosis, how does it spread slowly and how does it spread quickly?
00:24:43
Speaker
I think it has to do with the way that the disease progresses. So if the disease progresses slowly, you calculate that number and we'll show you how to calculate the base reproduction number in just a second. And if it goes quickly, then it's a different one and it's just different expressions of the same disease. And so you might wonder why these aren't weighted.
00:25:06
Speaker
by like something, so why isn't it like the average of the fast and the slow one? And the reason is simply because the number of total tuberculosis cases is already a sum of the fast and slow methods that it's progressing, so we don't need to wait it again.
00:25:22
Speaker
We said that in the example of tuberculosis, R0 is equal to R0 fast plus R0 slow. Well, let's talk about what R0 actually is. There's an equation where R0 is described as beta over gamma, where beta is the contact rate. And in the case of disease, a contact rate would be what?
00:25:40
Speaker
So the beta is the contact rate, obviously, which is the rate at which a disease creates new diseases and gamma, which is
00:25:55
Speaker
uh, the infectious period is, um, which is how long a disease, uh, is infected, infectious for. So is it infectious for two weeks, like measles, um, is infectious for like years, like in the case of HIV. Um, that's what that is. Cool. Okay. And, um, to give you some examples of basic reproduction numbers, Ebola, which is known for burning itself out real fast, like a toddler or something.
00:26:24
Speaker
has an R naught of two, but measles has an R naught of 15. And measles is known for spreading extremely quickly. And Ebola doesn't have a lower one because if it's below one, like we said, it's non-viable. So as this is introduced, oh.
00:26:40
Speaker
So the takeaway of this is that we have a number and if we can make that number less than one, then we kill off the disease like we do in smallpox. And there's ways that we can deal with viruses. I mean, like what?
00:26:55
Speaker
Well, three ways that we have our quarantine, where one who is infected is isolated from other people as to not spread the disease. Also sanitation, which of course is just cleanliness. And then finally, we've got vaccines. And you'll notice that all three of these affect the beta number, the rate at which a disease can be.
00:27:17
Speaker
can spread because it's with sanitation you like you have less bacteria sitting around so there's less of that with quarantine the bacteria have nowhere to go and with vaccines the bacteria has have nothing to do or they just get eaten up by white blood cells or I don't know if that's how viruses are eaten in the body but yeah
00:27:42
Speaker
Now one model that's pretty widely used because it's pretty simple but can be used in conjunction with other models to create very complex and nuanced models is called the SIR model and that stands for susceptible, infected, and recovered.
00:27:59
Speaker
This is a differential equation, isn't that right? Yeah, and we're going to talk about what that means in just a second, but just so you're not lost, susceptible are people who haven't gotten the disease yet and they're not immune to it. Infected are the people who are infected, and recovered are the people who are recovered. It's pretty self-explanatory now that I think about it.
00:28:20
Speaker
And we talked about differential equations before, but this model that we have here, the SIR model, might be a very good way of actually explaining it if you had the background that we just gave you.
00:28:33
Speaker
for, uh, um, epidemiology. So, um, we have S I N R, which are the numbers of people who are, uh, you know, susceptible infected and recovered. And then we have S prime, I prime and our prime. And we're going to call these and prime just mean rate of change. So we're going to use the word rate of change from now on. And what's a rate of change?
00:28:55
Speaker
Uh, it's the rate with which something changes. Sorry. It is, it is pretty basic and basically what this is, is if I go a certain amount of distance, a certain amount of time, that's a rate of change.
00:29:10
Speaker
And S prime, I prime, and R prime are technically what are called the infinitesimal rates of change. So this is for a car what you'd see on your speedometer. That's your infinitesimal rate of change. That's with respect to time. Or increase in distance over unit of time. Increase in speed over unit of time would be acceleration.
00:29:30
Speaker
So now obviously we have susceptible individuals, right? And we got infected individuals. So what is the relationship between susceptible individuals and susceptible infected and recovered individuals? So this is interesting. So as we said, susceptible is S and infected individuals is I. We know that S or susceptible individuals become I or infected individuals. Basically S is going to go down as I increases.
00:29:59
Speaker
Also, infected individuals we know will eventually, or some of them at a given rate, will become recovered individuals. So I becomes R, I goes down as R increases.
00:30:11
Speaker
And you'll notice that S can only go down, R can only go up, but I can go either down or up. So if there's more people getting infected than R recovering, then I prime, the rate of change in I, is positive, otherwise it's negative or zero. Every day a fraction of infected individuals
00:30:31
Speaker
turn into recovered individuals. So let's say a disease is like measles and it runs for two weeks. So what percentage of the population is going to be turned into recovered individuals every day if the disease lasts two weeks? So it would be one fourteenth then.
00:30:46
Speaker
Yeah, one day out of 14 days. And by the way, recovered here also means dead. So if you die from measles, you're technically recovered from measles, which is funny. Yeah. To expand that, the rate at which susceptible people turns into infected people is dependent on both the susceptible population and the infected population.
00:31:11
Speaker
since this is a statistical average of interactions between people. The way that you think about this is let's say I have a thousand people who are infected and a thousand people who are susceptible. If the interactions are perfectly random, if for every person in the susceptible group there's a thousand people,
00:31:31
Speaker
in the infected group, so there's a million different interactions that could happen with equal probability in this model. And for those of you who keep track of the math, these are the two equations, r prime equals bi, where b is the rate of change of r with respect to i, and s prime is equal to negative a times s times i. And we're gonna keep the equation here because this is a simple version. Normally you do have to keep into account the number of the whole population.
00:31:59
Speaker
This leaves I prime or the rate of change of infected people is equal to A times S times I minus B times I. Yeah, so we have three different equations that govern the system. And we put together a little program that just kind of shows you
00:32:18
Speaker
how sensitive these variables are. So we're gonna run it with, and we're gonna use a slightly different equation, which you could find on the Facebook page. So let's say, all right, so what is the initial un-sick population, the initial susceptible population?
00:32:40
Speaker
So the initial un-sick population, let's do a nice large sum. How about 15,000? All right. 15,000 people who are not sick yet. How many people are sick? Oh, how about let's start small. I want to say 33. 33 people. And let's say no people have recovered yet. What's the chance that this disease spreads between a random contact between two people? How about half a percent? So if it's in decimal zero, zero five.
00:33:11
Speaker
And how long does this disease incubate for? Oh, how about eight days? Eight days. And by incubate, I meant how long is it susceptible? How long is it infectious for? And we're going to run that simulation over the course of a year.
00:33:34
Speaker
After a year, it looks like 373 people have gotten sick total, but the number of people who are actually sick has gone down to seven. So this disease is going to die off.
00:33:45
Speaker
Oh, okay. Yay. But look what happens when you change it from half a percent to five percent. Okay. Oh, wait, not much. Sorry. Well, this disease just isn't really that detrimental. Yeah. So let's see if half of the people. Okay. Now that everybody dies. Okay. Okay. Sorry. We just had to tweak the numbers a little bit. No. So, but that's that's one of the numbers that you could
00:34:15
Speaker
to, uh, tweak with, and sometimes you change it very slightly and it makes a huge difference, but let's, uh, see what changing the rate. So let's say it's a non-infectious for eight days. Let's say it's infectious for a whole month, 31 days. It's about the same honestly. Yeah. Wow. Goodness. So I wonder what our magic numbers are here then.
00:34:38
Speaker
Yeah. It seems like a lot of the ways that you slice this, uh, yeah. Um, it is not a very deadly disease that we designed. So let's boost the number of people that you start out with to a half a third of a thousand. Okay. Very good.
00:34:55
Speaker
After a year it infects a bunch of people but the number of sick people has gone down and after three years it's gone down further but only to 61. After 10 years
00:35:15
Speaker
About a third of the population has gone and they're still sick people. So the disease dies off painfully slowly with these parameters. But let's see what happens when we make it a little bit less infectious. So instead of 2.5%, let's make it 0.25%.
00:35:33
Speaker
So in that case, um, uh, about, uh, about only a fifth of the population has been, uh, infected by this, but it's going down stubbornly, stubbornly slowly. Oh, wow. Again, these are just arbitrary numbers that we just made up on the spot. So it'd be interesting to run this simulation a few times and to figure out what are the key numbers here.
00:35:57
Speaker
Now if we bumped up the part probability that an infection is one out of 20 times, then...
00:36:05
Speaker
about two thirds of the population gets sick over time. And if you give it unrestrained growth, instead of 15,000, 150,000, you have still two thirds of the population. Even though we started with the exact same amount. In fact, if we decrease the number of people from 333 to 33, however, when we decrease the number of people from 333 to 33,
00:36:32
Speaker
Instead of 100,000 people dying or getting sick, you only have about 5,000 out of the whole 150,000. So that's a very touching number right there. So reducing the amount of sick people by quarantining them is very effective in controlling disease. And this is just something that you study when you study disease.
Epidemiology and Memetics
00:36:53
Speaker
One of the things that's interesting is how we can apply the same mathematics that describe viruses as other things, like things in culture that go viral. As you know in this podcast, we have a fascination with language and culture as well as mathematics, and one of the aspects of language and culture that we can talk about are memes.
00:37:12
Speaker
And I mean, we're talking specifically in this case about the memes that you get from Facebook that have the pictures and text on them for one simple reason, to simplify the model and make it so that inoculation can be, not inoculation,
00:37:31
Speaker
so that being resistant to a virus can be a thing. So we're equating memes and viruses here. So what do memes gets? They get boring. Can you think of a meme for like two years ago or something? Oh gosh, you know, I'm sure there's a graveyard of all of the old memes where memes go to die, you know, like the I don't know if you all recall this one, but the Irma gird girl.
00:37:55
Speaker
Oh, yeah, that was so long ago. Actually, it wasn't even that long ago. Yeah. But yeah, so check it out. I'm going to go to Google and I'm going to type in old memes and see what comes up. So I'm typing old memes, cutting edge mathematical research here. I'm breaking math. Yes. Well, there's the 100 greatest memes of all time. You can look that up and let's see. Let's take a jock down memory lane here. OK.
00:38:22
Speaker
Oh yeah, like this guy, like the Dos Equis guy. You know, is he still around? Is he still? I haven't seen that in a meme in forever. So you get the point. Memes get boring over time. You've seen a meme, it's funny for a little bit, then it doesn't get boring. So now we're going to define S-I-N-R in terms of memes. So what's S?
00:38:43
Speaker
Susceptible population, so obviously this is anybody and of course if you've got something hot like a cute cat, you know, I think we are all susceptible to that because deep in our brains we know we all think cats are cute. Yeah, and so S is the people who haven't seen the meme. Next we have infected individuals, and so those are the people who have a risk of transmitting the meme.
00:39:03
Speaker
Uh, so those are people who have seen the meme and, um, aren't bored of it yet, right? Sure. Sure. Yeah. So that means they're, they are probably, and again, in this case, obviously the meme doesn't affect them aside from bringing some enjoyment into their otherwise mundane and lonely life. So, so they've seen the meme and they are, are motivated to press like and or share.
00:39:27
Speaker
Yeah, or they're the type of people who could press likers or not even. So if your grandma has seen a meme and she's gone within the period of time where I mean, could be funny, maybe she just doesn't find it funny. Like some people are just naturally immune to certain viruses. Maybe, you know, there's we go on and on with the parallels here. Oh, yeah, absolutely. And then are the people recovered from the meme and I'm calling you recovering because you do have to recover from the meme.
00:39:52
Speaker
Yeah. So, so that's at the point where you're no longer going to like and share. Yeah. You're going to hate it. You're going to hate it. Sort of like songs as well that get really old, no longer enjoying them. Yeah. And actually, yeah, for those of you in the audience who aren't, you know, under 35, uh, just replace meme with song and everything we said makes sense.
00:40:13
Speaker
And so A, we talked about A and B before in the equations. So A is the probability that if someone shares a meme with you, that you'll remember that meme. And B is the inverse of the lifespan of the meme. So how long the meme is intrinsically funny for before it gets old.
00:40:33
Speaker
With this, you could track the spread of social media and make predictions about social media. And in fact, people do do this all over all. It's big money right now.
Advanced Epidemiological Models
00:40:45
Speaker
So like with anything in math and in science, but especially in math, one discovery can be applied to so many different things.
00:40:53
Speaker
What's also fascinating is I find memes are also, there's some degree of a barrier in memes with respect to age. I took note of this back when I was a kindergarten teacher. I had a lot of kindergartners come in after recess and tell me very excitedly that they had discovered Indian clay.
00:41:12
Speaker
Now I realize that that name may not be very sensitive. I'm saying I'm sharing this as a fact of what they told me. That was a term that when I was in kindergarten I had used and at the time I thought that's what it was actually called. But it made me surprised that that term is still being passed around in the kindergarten age level.
00:41:33
Speaker
Oh yeah, there is definitely a barrier and it could be incorporated by the model technically, like by it actually, it is incorporated by the model technically by the model's own standards. So if, uh, if, like, if like most people over like 35, like I said, I'm going to share memes or people under 35 aren't gonna, you know,
00:41:56
Speaker
then transmission rates are just gonna be different. But as you know, as you've seen in personal experience, by just sharing memes, you have an insight into the way that epidemiology works, and it doesn't work like that. And with things like sexually transmitted diseases, there's a huge limitation on age ranges and things like that, and just so many different factors.
00:42:20
Speaker
So, good point that you brought up there. Interesting. Wow. And there's so much with our conscious experience that can be described via virology. This seems like it could be a continuing conversation even into future episodes.
00:42:36
Speaker
As many of you can see, there's limitations to SIR theory, but it's not the end of the world. And like any model, it doesn't give you full information about the world and doesn't fully represent the world.
00:42:53
Speaker
so graph theory is used a lot in studying epidemiology but so somebody who's a friend of the show did some work on this actually and it turns out that there's all kinds of really cool linear algebra and differential equations together related stuff that you could do with graph theory graph theory meaning bunch of nodes connected by arrows
00:43:21
Speaker
You could also do spatial differential equations and anything that could be modeled mathematically is going to be modeled by the world and is going to interact with diseases so you could throw that into the theory too.
Conclusion and Listener Engagement
00:43:36
Speaker
Epidemiology will be around as long as communicable diseases around, whether it be physical, mental, or otherwise systemic to the tradition that is human culture. We've made great strides in the last century, but this field is still ripe for new discoveries and creativity. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. You can find us on Patreon at patreon.com slash BreakingMathPodcast, if you want to buy us a sandwich. We are also on Twitter at BreakingMathPod.
00:44:02
Speaker
Facebook at facebook.com slash breaking math podcast and Gmail at breaking math podcast at gmail.com. You can check out our website that has a lot of different the widgets that you look at and we promise we'll update it soon. Yes, that's true. Guilty as charged. Also, please send us your math related questions. We'd love to start a question and answer segment that we'd love to get as cutting edge and as deep as possible.
00:44:28
Speaker
Yeah, and for those of you who ordered the poster, if you haven't received them by the time this episode is out, they're in the mail.