Math Gets Around » Math Goes Pop!

Finding Love with a Modified Drake’s Equation

Some time ago, I wrote an article on the optimal way to select a mate, assuming you know how many eligible partners exist, and that once you’ve dated someone, you can’t go back and date them again (sorry, Drew Barrymore and that dude from the Apple commercials). This is less romantically known as the secretary problem. Let me briefly recall the problem and its solution: suppose you have n candidates, from which you want to pick the best one. This applies to a variety of situations, from hiring a secretary to finding a girlfriend to apartment hunting. In either case, the outcome is the same: you should look at roughly the first n/e of them (yes, that e), and then select the first one after those n/e which is better than all that you have seen so far. While this strategy won’t guarantee you get the best choice, it will give you the best choice around 37% of the time.

The major problem with this model is that in many situations, the value of n is unknown. There are ways to circumvent this problem, which I will not discuss here. Instead, in the context of finding a mate, I offer the following method to calculate the number of partners you could reasonably expect to find in your area. This method recently gained some attention when Peter Backus, a Ph.D. candidate from the University of Warwick wrote a paper titled “Why I Don’t Have a Girlfriend.”

The basic technique involves modifying the Drake Equation, an equation used to estimate the number of potential extraterrestrial civilizations in our galaxy. For those who have never been introduced to this equation, it asserts the following:

N = R · f_p · n_e · f_l · f_i · f_c · L.

According to Wikipedia, these variables represent the following quantities:

R = the average rate of star formation per year in our galaxy,
f_p = the fraction of those stars that have planets,
n_e = the average number of planets that can potentially support life per star that has planets,
f_ℓ = the fraction of the above that actually go on to develop life at some point,
f_i = the fraction of the above that actually go on to develop intelligent life,
f_c = the fraction of civilizations that develop a technology that releases detectable signs of their existence into space,
L = the length of time such civilizations release detectable signals into space.

Given estimates for all of these parameters, one could then estimate the number of civilizations in our galaxy. Since we don’t know the values of any of these parameters, however, this is more of a thought experiment than anything else.

Nevertheless, the idea can be easily modified to try and find the number of eligible mates in a given area. Peter Backus’ approach is fairly specific to him, but he links to a more general approach discussed here, in which the following equation is presented:

n = P · f_t · f_o · f_c · A · R

In this case, the parameters are given by

P = the population. This could be the population of your university, your city, or your country, depending on how ambitious you are.
f_t = the fraction of that population which you would want to mate with, in broad terms. If you’re a straight male, this would be the fraction of females. If you’re a gay male, it would be the fraction of males, and so on.
f_o = the fraction of the population you’d want to mate with which wants to mate with you. For example, if you’re a straight male who wants to mate with females, this will compensate for the fact that some females will be lesbian and therefore unwilling to mate with you.
f_c = author Raymond Francis labels this the out fraction, and describes it as the answer to the question “Of the people in your target gender and orientation, how many of them are open enough about their sexuality to engage in a relationship of the sort you’re hoping for?” If you are straight, this value is likely 1, or very close to it. If not, things can be a little bit fuzzier.
A = the fraction of those remaining who fall within your desired age range. This is, of course, personal to you – if you’d like a socially acceptable age range, you could follow the “half your age plus seven” rule.
R = factors for any remaining filters you wish. Do you want your partner to have a certain level of education, or a certain income? Do you need a non-smoker, or demand a euphonium player? Here’s where you can fold that into the mix.

Yes, Wikipedia has a graph illustrating the half your age plus seven rule. Amazing.

With all these parameters accounted for, N will give you the number of potential mates in your area.

Let’s take this equation for a spin, shall we? Suppose you are a straight male living in Los Angeles, and looking for a girl to date in Los Angeles. According to Wikipedia, the estimated population of LA as of 2008 was 3,833,995. Of those, let’s say that 51% are female, and of the females, let’s posit that 90% are straight or bisexual. f_c should be high in this case – to be conservative, let’s put it at 95%.

To estimate the age filter, one can obtain some data from this site. Suppose you are a 25 year old man – then, absent any personal preference, the socially acceptable age range of women for you to date is between 19.5 and 36. According to census data, in 2000 there were 3,694,820 people in Los Angeles, and of them, 974,004 were between the ages of 20 and 34. Additionally, there were 251,632 people between the ages of 15 and 19, and 584,036 people between the ages of 35 and 44. If we make the assumption that ages are roughly uniformly distributed within these brackets, this gives us an additionlal 141,970 people either between the ages of 19.5 and 20, or between the ages of 35 and 36. Combining this gives a total of 1,115,974 people between the ages of 19.5 and 36 in Los Angeles in 2000, or roughly 30% of the population. Let’s use this for our value A.

Assuming you have no other restrictions (i.e. taking R = 1), this gives us n = 3,833,995 · .51 · .9 · .95 · .3 = 501,544. That’s a lot of ladies out there for the taking. Of course, taking R = 1 is probably unrealistic. It’s unlikely you want to date women who are married, for example, and everyone has their own personal taste that will decrease the pool even further. Once you’ve calculated your personal value for R, however, you then know how many eligible mates will be in your area. Given that, you’ll know how large n/e is, and then you’ll know how many people you should date before you think about settling down.

Although Peter Backus has received a fair amount of buzz for the short paper he has written on this idea, he readily admits that he is not the first to think of applying Drake’s equation in this situation. I’ve discussed mostly Raymond Francis’ approach here, but Backus has links to many other people that have discussed the idea on his website. In particular, here’s an exchange from CBS’s The Big Bang Theory (don’t worry, there’s a laugh track so you know when things are supposed to be funny).

In summary, not only can the Drake Equation be used to consider the existence of extraterrestrial life, it can also be used to consider potential mates right here on Earth. The next step, of course, is obvious: we must combine these two equations to calculate the number of potential extraterrestrial mates. Undoubtedly the number will be small, but one should never underestimate the power of love.

Interstellar love is a wonder to behold.

Math Gets Around: Breakfast

I admire the food blog Serious Eats because, as we’ve seen before, it’s not afraid to get a little mathematical. This month they have upped the ante with a post on the delicious object now known as the Mobius strip bagel.

Named for the classical geometric object of the same name, the Mobius strip bagel (and its cousin, the Mobius strip donut) give an elegant mathematical spin on ordinary edibles. In addition to the aesthetic value, the Mobius strip bagel also has the advantage of added surface area, meaning that one can pile on even more cream cheese before stuffing one’s face.

Mathematician George Hart has step-by-step instructions for the transformation from torus to Mobius strip here. I have yet to try this technique myself, but I can think of no better way to celebrate the holidays than by transforming breakfast food into mathematically themed breakfast food.

Comic (but not Comical) Mathematics

A few months ago, my girlfriend and I were persuaded to subscribe to the LA Times by a very nice man at a nearby Ralph’s store who offered us $20 in groceries for the exchange. “Just try it out,” he insisted, “because you can always cancel and we’ll simply pro-rate the cost based on how long you were a subscriber.”

Fantastic, we thought. Given the current uncertainty surrounding the future of the newspaper industry, subscribing made us feel like responsible citizens – like giving blood, but with fewer personal questions beforehand.

Unfortunately, once the newspaper began to arrive, we had to face the fact that we never read it. I think I skimmed through it a couple days that first week, but after that the papers would go from our doorstep to the recycling bin. Try as we might, we simply couldn’t fit a morning newspaper routine into our lifestyle. And so, with a heavy heart, we canceled our subscription.

To be honest, I haven’t really noticed a difference. All the news I need I can easily access on the internet. The only part I do sometimes miss is rifling through the comics. Sure, you can always go to a website like comics.com and see what your favorite syndicated cartoon characters are up to, but I still prefer the layout a newspaper provides, with dozens of comics squeezed together, allowing you to read them all rapid-fire without having to wade through a list of comics and be bothered by loading times.

My father remedies this by e-mailing me comics directly. Every day I know that I can look forward to the latest adventures of Liō, but this daily standard is sometimes intermingled with other comics. For example, just the other day my father sent me the following math-themed comic:

This is the work of a man named Dave Coverly, who apparently has been named Cartoonist of the Year by the National Cartoonists Society. More of his work can be found here. Some of his cartoons are quite good. Unfortunately, the one above is not.

The above cartoon has the potential to work on multiple levels, but fails to do so. Certainly anyone can read the comic and get the joke (ha ha, the description on the sign involves math, it’s funny because the books in the bin are math books!), but the sentence on the sign doesn’t make any sense. Forget that the mathematical expression is complicated – replace it by a number, such as 1.8. Then the sign reads “Math Book Sale: Buy 2, Get 1 at 1.8!”

Huh? What is this supposed to mean? If Coverly had written 1.8 instead of (3/4 x 2B) ÷ (7π/100), I would hope that an editor would’ve said something like “Hey man, that doesn’t make any sense, why not try for a joke that isn’t so sloppy?” Instead, we get basically the same punchline, but masked by a more convoluted expression. No doubt the reader is meant to simply see that the sign involves math, chuckle inwardly, and turn the page.

It may sound like I’m just trying to pick a fight, but I don’t think this kind of thing would fly with another subject. For example, suppose the comic had a Spanish theme rather than a Math theme. In such a case, Coverly’s comic might look something like this:

This cartoon, like the last one, works on one level: it’s a Spanish book sale, and the sign describes the sale with some Spanish. It’s not hard to get the joke.

However, anyone with a basic understanding of Spanish would probably find this comic more confusing than funny, because the Spanish phrase doesn’t make sense. The phrase certainly doesn’t explain the sale, but it’s also not particularly funny in it’s own right. Why wouldn’t the sign say “half price” in Spanish? Alternatively, one could write a joke in Spanish – this would be lost on people who don’t speak Spanish, but they would still be able to appreciate the basic joke, namely that the Spanish book sale sign is partially written in Spanish.

The situation with math should be no different than the situation with Spanish, but apparently a comic like this is allowed to slide by. Perhaps our population’s mathematical illiteracy is worse than our Spanish illiteracy. Even so, I think it’s fair to call out this comic for simply being lazy. It’s not hard to get the same joke across, but enhance it so that the sign actually makes sense. Here is just one way Coverly could’ve enriched his original idea:

I feel better already.

Of course, one could argue that Parade Magazine is not the place one should go for a daily dose of mathematical humor. And indeed, you’d be correct. Of the nationally syndicated strips, Foxtrot is the only one that consistently uses math for a punchline, but such examples are few and far between. As always, for math themed comics, one is best advised to search here.

Math Gets Around: Preventing the Zombie Apocalypse

If pop culture has taught us anything, it is that in the event of a zombie outbreak, we are royally screwed. When faced with an onslaught of classical zombies (of the type first made famous by Romero’s 1968 film Night of the Living Dead), films have shown again and again that we are no match for hordes of cannibalistic undead. With the more recent interpretation of zombies that are faster and smarter, our hopes for survival have diminished even further.

Despite overwhelming odds, however, it is not in our nature to simply roll over in the face of adversity. While the body count is usually high in films chronicling the eventual war between the living and the dead, in most cases there are a few who survive to continue the fight after the credits roll.

But how realistic is this depiction? How prepared are we to defend ourselves from being eaten alive by our deceased ancestors? And what strategies will give us the best chance of survival? You’ll be happy to know that mathematics can answer some of these questions.

Hopefully we are more prepared than Homer Simpson.
Students from the mathematics departments at Carleton University and the University of Ottawa have produced several mathematical models to predict what will happen in the event of a zombie outbreak, and how our response to such an outbreak may affect its outcome. The students, led by Professor Robert J. Smith? (this is not a question, he simply insists on this piece of punctuation after his name) used the theory of differential equations to see what would happen in the event that the dead rise from their graves in search of fresh meat.

As would be expected from a paper with such important pop-culture consequences, this research has already garnered a fair amount of attention. The Globe and Mail ran an article last Friday, as did Wired (and with a much cooler picture, I might add). However, if you really want to get to the heart of the matter, here’s a link to the original paper.

Their model eschews the post-28 Days Later interpretation of zombies, focusing instead on the lumbering, thoughtless monsters that have been the stuff of childhood nightmares for decades. One could argue, however, that the results presented in their work would be even grimmer were we to allow zombies the benefit of intelligence and the ability to run a 6:00 mile.

The underlying ideas are similar to those of the SIR model for the spread of infectious disease, which has been discussed elsewhere on this corner of the internet. One significant difference here, of course, is that when dealing with a zombie infestation, dead people may not necessarily stay dead. This adds complications to the theory, but also allows for a richer analysis.

It’s unclear whether or not the zombies in this model know how to dance.

In the end, their research gives us the following insights into the nature of zombie warfare:

With one exception, human-zombie coexistence is not possible. In order to save humanity from a zombie infestation, we must kill every last one of them.
Often times there is a period of latency from the time a person is bitten until the time they turn into a zombie. Whether or not such latency exists will not have an effect on whether or not the zombie hordes will overwhelm us – the only thing that changes is how long it will take for them to do so.
Quarantining is not an effective way to try and stop a zombie outbreak. All it will do is prolong our extinction.
The only model where coexistence is possible is a model in which a cure exists for zombification. Unfortunately (or fortunately, depending on your preferences), in such a world the zombie population would greatly outnumber the human population. In other words, we would survive, but not by much.
The only way to ensure our survival is to attack with decisive force as frequently as we are able. In particular, the model ignores the impact of birth and death rates, which is fine over a brief period of time, but becomes more significant as the fight continues, since more bodies means more potential zombies. In other words, the longer the fight goes on, the less likely we are to emerge from it.

For the self professed zombie expert, these findings may not be all that surprising. For the rest of us, however, it just goes to show you how far a little mathematics can take you, even when exploring the realms of the highly improbable. You may scoff at this article now, but when World War Z arrives, you’ll be thankful that someone took the time to conduct this preliminary research.

(Hat tip to Patrick for the link to the Globe & Mail article. With a colder climate that no doubt helps to preserve dead bodies, it’s no wonder that Canadians are blazing a trail with this research – they will be on the front lines when the time comes.)

Math Gets Around: On Birthdays and Trading Cards

Today marks the 1 year anniversary of Math Goes Pop! I started on somewhat of a whim after reading an article about compulsory Algebra I education for all California 8th graders (although what with our finances down the toilet, who knows what the current status is here). When I started writing I wasn’t sure there was enough content out there to sustain a blog with this one’s focus. Once I started digging, though, I found that the rabbit hole went quite deep, and so here I am a year later with plenty left to talk about (the recent obsession with pointless math holidays certainly has helped with my output).

Given the date, it seems fitting to begin by mentioning the birthday problem. This is a standard problem given in any introductory probability course, but many people find the result counter intuitive at first.

The birthday problem asks a simple question: if you have a room full of people, how many people do you need so that there’s a 50% probability that at least two of them share the same birthday? In the simplest setting, we make a few assumptions: for example, we usually ignore the presence of leap years, and we assume that each day of the year is equally likely to be someone’s birthday.

With these assumptions, solving the problem is not difficult. The probability that at least 2 people in a group of n share a birthday is 1 minus the probability that all n people have different birthdays. In other words, the probability is

1 – (1 – 1/365)(1 – 2/365)…(1 – (n-1)/365).

This is because the second person has a 1 – 1/365 = 364/365 chance of having a birthday different from the first person, the third person has a 363/365 chance of having a birthday different from the first two people, and so on.

So, to answer the question, we just need to find the smallest n such that the expression above is greater than 1/2. The punchline is that n only needs to be 23 or more in order for this inequality to hold. In other words, in any room with at least 23 people, there’s a better than 50% chance that two of them have the same birthday.

For many people, this number seems too low – after all, you may have been in several groups of 23 or more and never or rarely met someone with your birthday. The problem with this argument is that it addresses a different question. The reason why we only need 23 people to have a 50% chance of finding a common birthday is that we are placing no restriction on the date of the shared birthday. Requiring that someone share the same birthday as you fixes the date, and this changes the problem.

If instead you want to know how many people you need in a room so that there’s a 50% chance one of them will have the same birthday as you, this probability is given by the new expression

1 – (364/365)ⁿ.

To make this greater than 1/2, we need to take a much larger n: n = 253, to be precise.

As I said, this is a well-known problem in probability theory. A few months ago, however, I was asked about a variant of the birthday problem by my friend Gabe over at Motivated Grammar. Rather than looking for identical birthdays, what if you look for different birthdays? More specifically, how many people do you need in a room to guarantee with, say, a 90% certainty, that every day of the year is someone’s birthday?

Of course, you could always pick poorly so that everyone in the room has the same birthday, but the odds of this happening are quite low. In fact, this problem is more commonly known by another name: the coupon collector’s problem.

For this problem, suppose you are clipping coupons from a newspaper (any newspaper except for USA Today). There are n different coupons you can collect, but each newspaper only has 1 coupon, and you can’t see which coupon the newspaper has until you’ve bought the newspaper. In this context, the question becomes: what’s the probability that you’ll need to buy at least newspapers to collect n coupons?

If you prefer, you can think about this problem in terms of trading cards as well. Each pack of cards is akin to buying a set of newspapers, and you want to know the probability that you’ll need to buy at least a certain number of packs in order to collect all the cards. From baseball players to Pokemon, this same problem governs the distribution (assuming that all cards in the back are equally likely to be in the pack, which may not always be the case).

What’s known about this problem? Well, as I said above, even if you buy hundreds of thousands of cards, or stuff millions of people in a room, there’s no guarantee that you’ll collect every card or every date. However, on average, the number of cards you’ll need to go through to complete a set of size n is about n*logn.

In terms of birthdays, this says that if you want to collect every date, on average you’ll need to pool together around 2,153 people. Why such a large number? It’s not unreasonable to expect something like this – when you first begin collecting people, it won’t be hard to get people with different birthdays. However, as your numbers increase, you’ll get a new birthday less and less frequently. Finding that last birthday could prove to be quite elusive.

The same analysis works for trading cards. Trying to complete your collection of series 2 Teenage Mutant Ninja Turtles Animated Series trading cards? Well, my friend, with 88 cards total and 5 cards per pack, you can expect to buy around 79 packs of cards. Perhaps this box set would be a better investment.

Mondo to the max, indeed.

While the expected values are easy to calculate, it may be that you need to greatly exceed the expected value in order to complete your collection. However, one can use Markov inequalities to get bounds on the probabilities. For example, there’s a 90% chance that you’ll be able to hit every birthday if you take no more than about 21,535 people. To bump those odds up to 95%, take 43,069 people.

So, for parents whose children who have gotta catch ‘em all, you can use these methods to get a rough estimate for how much that completion will cost you. And if you’re trying to get a room full of people together so that every day of the year is someone’s birthday, I’d strongly suggest not picking people at random. What an awkward party that would be.

Math Gets Around in the Big City

Math has gotten a bit of a visibility boost recently, in the form of posts by professor Steven Strogatz at the New York Times blog. For three weeks, starting at the end of May, Professor Strogatz filled in for usual blogger Olivia Judson, and during that time he used the platform to write some highly readable musings that show the presence of mathematics in unlikely places, and touch on some of the directions math is headed in the 21st century.

Let me highlight the first post, titled “Math and the City.” Professor Strogatz begins this article by describing Zipf’s law, an observation attributed to linguist George Zipf regarding the distribution of words in a language (for a linguistic motivation, you can check the Wikipedia article on Zipf’s law).

One of these things is not like the other.

In the context of cities, the law states the following: in a given country, if you rank the cities within that country by population, the largest city should be about twice as large as the second largest city, about three times as large as the third largest city, and so on. In other words, a city’s population is inversely proportional to its rank.

Using the power of the internet, it’s not too hard to find current population data to try and verify this observation. Here’s a table illustrating Zipf’s law for U.S. cities (I pulled the population data from here):

City	Estimated Population (July 2007)	Zipf Law Ratio	Ranking
New York, NY	8,274,527	1	1
LA, CA	3,834,340	2.158	2
Chicago, IL	2,836,658	2.917	3
Houston, TX	2,208,180	3.747	4
Phoenix, AZ	1,552,259	5.331	5
Philadelphia, PA	1,449,634	5.708	6
San Antonio, TX	1,328,984	6.226	7
San Diego, CA	1,266,731	6.532	8
Dallas, TX	1,240,499	6.670	9
San Jose, CA	939,899	8.804	10

Those with a visual bent can also take a look at a graph of the data:

Professor Strogatz doesn’t provide heuristics for why Zipf’s law should be true, but he does observe that this phenomenon has been around for more than a century, and can be observed in countries throughout the world (with varying degrees of agreement). He then goes on to discuss more recent mathematical observations regarding urban development, including the observation that cities, as with many other things, enjoy economies of scale. For example:

If one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size… the bigger a city is, the fewer gas stations it has per person…

The same pattern holds for other measures of infrastructure. Whether you measure miles of roadway or length of electrical cables, you find that all of these also decrease, per person, as city size increases.

In other words, the distribution of infrastructure is not quite the same as the distribution of the population – as population grows, so too does infrastructure, but it can grow more slowly. Further discussion can be found in the article.

Of course, there are many questions here. First of all, a little digging will show you that this trend is stronger in some countries rather than others. Why is this? Also, why must we break down our analysis by country to look at these trends? Why don’t we see this pattern emerge if we simply rank cities independent of country?

Moreover, this Zipfian trend depends of course on one’s definition of the word “city.” If one extends the notion to municipal areas, the trends become less clear. So, how should we define what it means to be a city?

As I learned from a recent article on The Daily Dish, Tim Gulden of George Mason University has recently tried to answer some of these questions. By using nighttime satellite data, Professor Gulden and his coauthors were able to more rigorously and consistently measure city sizes globally, and were able to use these measurements to compare rankings for population, economic activity, and patented innovations.

In their paper (the abstract of which can be found here), the authors give evidence supporting a Zipf-type distribution for all three statistics, and use this data to argue against the idea that globalization is making the world “flatter,” i.e. more equidistributed with regards to things like population or economic activity. Instead, they argue that the world of the future will feature global ranks that follow more of a Zipf distribution, and that one reason why this hasn’t happened already is because it currently can be difficult to migrate between the barriers of different countries.

For more math in the spotlight, I’d encourage you to read Dr. Strogatz’s other posts (here and here). Happy reading!

Math Gets Around, and So Does Disease (Both Real and Virtual)

Most of us are familiar with the story of Chicken Little, the young chicken turn Disney sellout who one day has a major panic attack because she (or he, depending on the version you’re told) believes that the sky is falling.

No doubt this fable has conditioned many of us to be wary of chickens that try to warn us of impending crises. But given the recent media frenzy surrounding swine flu, perhaps we should turn our attention away from the chicken, concerns over avian flu notwithstanding, and focus a bit more on the humble pig.

There is some debate on this issue: while everyone seems to be in agreement that the swine flu outbreak is, so far, milder than many had anticipated, health officials have cautioned that we may not yet be out of the proverbial woods (or pigpen, as the case may be). At the same time, however, one can just as easily find articles that argue that maybe this whole thing has been overhyped.

Cute family film, or a foreshadowing of the coming apocalypse?

This divergence of opinion shows that we still have a great deal to learn about how to deal with potential pandemics. An improvement in our understanding will, among other things, necessitate a better understanding of how disease spreads within human populations, as well as a better understanding of how humans respond to health crises. Since we can’t really release diseases into the population to observe what will happen, the only thing we can do to guide our response is look at historical data, or try to model what would happen in the event of an outbreak. In either case, mathematics can provide us with valuable insight.

There are a few decent survey articles online that discuss how math can be used to model the spread of disease. For some discussion that involves only basic Algebra, the Wikipedia article on Mathematical modeling of infectious disease isn’t a bad place to start. Another good survey article by Professor Matt Keeling (now of the University of Warwick) can be found here.

The simplest (and crudest) way to model a disease would be to assume some sort of exponential growth: for example, person A gets a disease, and passes it on to 3 friends, who each pass it on to 3 friends, who each pass it on to 3 friends, and so on. This is the sort of model that people love to point to when trying to put fear into the hearts of the masses. For instance, if we postulate that swine flu spreads in approximately this fashion, with each infected person infecting another 3 people every day, it would take only three weeks for the entire world to become infected.

Of course, we know that such a conclusion is unrealistic – this means that this is not a very good model. To get a decent model for the spread of disease, we need to take into account certain parameters of the population and of the disease in question. For example, what are the birth and death rates in the population? How contagious is the disease, and once someone is sick, for how long can they infect other people?

One such model that keeps track of these and other parameters is dubbed the SIR model, so named because it partitions the population into three groups of people: Susceptible, Infectious, and Recovered. The Susceptible group has not caught the disease in question, although they are able to catch it if they come into contact with it. If a person catches the disease, that person moves into the infectious category – that person is now able to infect anyone he or she comes into contact with. After a certain period of time, the person will recover, and move to the recovered class. Once recovered, we assume the person cannot become infected by the same disease again.

Using differential equations, one can model the long term behavior of diseases under the SIR assumptions. The result is that the proportion of infected people follows a damped oscillatory pattern with time – in other words, when the disease is introduced, outbreaks are common, and result in an infections to a larger proportion of the population. As time progresses however, outbreaks become less common, and the disease stabilizes to an endemic state where a certain fixed proportion of the population is infected at all times.

SIR model over time. Image taken from the article cited above.

Once things have stabilized, the proportion of susceptible individuals is given by 1/R₀, where R₀ is the the average number of other individuals each infected individual will infect in a population that has no immunity to the disease (called the Basic reproduction number). In other words, the more infectious a disease, the smaller the susceptible population will be in time. This meshes well with our intuitive understanding of how diseases spread.

It’s worth noting that the values of R₀ have been estimated for various diseases. For example, we believe that the R₀ for HIV/AIDS lies somewhere between 2 and 5, while measles is between 12 and 18. This speaks to the fact that it’s much harder to infect someone with HIV than it is to infect them with measles.

This is by no means meant to be an exhaustive discussion of the ways in which mathematicians model disease. And, as with any model, the SIR model has its limitations. There are generalizations of the SIR model – interested parties can read about some of them here. However, in general, finding an appropriate model is one of the difficulties in trying to understand the spread of disease.

Constructing a model isn’t the only way we can use math to better understand the spread of disease. We can also analyze historical data from diseases that have already occurred to try and find patterns.

Some may find a treasure trove of data from the 1918 flu epidemic. Others may search for answers by looking at data from Ebola outbreaks. Moreover, some have found patterns by studying data from diseases that don’t even exist!

Well, that’s not entirely accurate. The disease does exist, but only in the far away land of Azeroth, where orcs and humans battle for baubles and trinkets, form guilds, and go on quests. It’s true – World of Warcraft has given us useful data regarding how people respond during an epidemic.

Whatever disease she has, I’m pretty sure you don’t want it.

Here’s the story, courtesy of this article on the phenomenon:

In 2005, the game’s designers at Blizzard Entertainment decided that some players’ characters had become too powerful, so they created a virus — called “Corrupted Blood” — that would make the game more challenging for the most powerful players.

Turns out that even in the virtual world, things don’t always turn out as planned. The virus quickly infected any nearby character, regardless of its relative strength.

The programmers imposed a mass quarantine…yet many players ignored the quarantine, spreading the virus. Eventually, more than four million of the game’s six million players worldwide were infected, and millions “died.”

The silver lining in this cloud of virtual death was found by researchers studying the spread of disease, including Rutgers professor Nina Fefferman. The article continues by saying that “Fefferman and a colleague studied the plague as it spread…[focusing] on how a pandemic would spread and affect society and the economy. Since then she’s been called upon by health agencies all over the world to consult on her findings, including the U.S. Department of Homeland Security and the U.S. army.”

Certainly the traditional avenues for analyzing pandemic data are open as well, but it is surprising to think that interesting conclusions can be drawn from the reaction to virtual diseases. This also allows WoW players to counter the argument that online games are a waste of time by asserting that they are, in fact, helping to prevent mass extinction at the hands of a new strain of disease.

And for that, World of Warcraft players, you have my heartfelt thanks.

Math Gets Around: The Entomology of Civil Engineering

In the continuing saga of animals that are better than you at math, it now appears that ants are much better than most of us at optimization. Granted, they may not be able to think abstractly, but in concrete terms, they far surpass us with a particular type of optimization: the efficiency of traffic flow.

As anyone who has gone to a picnic will tell you, ants do a very good job of creating traffic streams – their foot traffic moves steadily, and without the major pileups to which my fellow residents of Los Angeles have become so accustomed. One could argue that the wide expanse of park area is proportionately much larger for the humble ant than what most motorists have to live with, but even so, the march of the ant colony often appears quite regimented, even with space enough to make a wider path. How is it that ants can control their traffic so well?

This article from the Wired Science blog discuss how ants succeed where we fail. At the heart of the matter is a study from the University of Sydney on leafcutter ants. In order to give the ants a better sense of what it’s like trying to navigate through a congested urban landscape, scientists restricted the ants to naturally narrow pathways, such as the ends of tree branches, in order to better understand how these ants organize their traffic in cramped spaces.

With their superior understanding of traffic flows, could ants one day dominate the world? Some scientists say “Yes!”

The findings clash with most people’s behavior on the freeway:

In the latest findings, published in the February issue of the Journal of Experimental Biology, [entomologist Audrey] Dussutour’s team found that ants leaving the colony automatically gave right-of-way to those returning with food. Of the returning ants, some were empty-mandibled — but rather than passing their leaf-carrying, slow-moving brethren, they gathered in clusters and moved behind them.

Rather than try to outpace their slower moving brethren, those without loads to carry simply kept pace with the slower ants. This is at direct odds with what most people do on the roads – who wants to drive stuck behind a bus? Based on our own behavior, we may question the wisdom of the leaf cutter ant’s process.

As is often the case, however, nature knows best. By not trying to barrel ahead of the slower moving ants, the ants without any baggage saved time on average. Not by a paltry amount, either – the study estimates “that patience reduced the average delay experienced by an individual ant crossing a crowded three-meter bridge from 64 to 32 seconds.” That’s a 50% reduction in commute time!

One plausible explanation for the difference between our behavior and the ant’s behavior is that we are looking at different optimization problems. People in general are trying to minimize their own individual travel times, and the other cars on the road aren’t given much consideration. With (apparently) smaller egos, the problem in the ant’s case is to make the whole traffic network run as smoothly as possible, so food can be brought in quickly, and energy isn’t wasted in traffic jams.

The study helps give weight to the maxim that patience is a virtue. Haste while driving carries with it certain risks, risks that on average far outweigh the benefits that come from not trying to outpace others on the road.

It’s doubtful that this study will do much to change human behavior, but understanding efficient traffic flow algorithms certainly has its applications, from urban planning to the engineering of self-driving cars. Perhaps people would be more patient if they weren’t the ones doing the driving.

Unfortunately, the days of the self driving car are not yet upon us, so until that day arrives, we must be content with what we have. So, dear reader, take a cue from the noble ant, and slow it down when you’re on the road – over time, it may save you time.

A glimpse into cities of the future?

Math Gets Around: Holiday Treats

At this time of year, many people push their studies to the side in favor of roasted animals and pie. However, the activities of enlarging your waistline and mastering some mathematics need not be mutually exclusive. For evidence of this claim, I need only turn your attention to the culmination of thousands of years of human evolution: the Pecan Pie-cosahedron.

Pecans + math = crazy delicious.

This masterful work of craftsmanship was created by an individual known by the pseudonym of turkey tek over at instructables.com. The pie is so named because it has the shape of an icosahedron, arguably the most beautiful of the five¹ Platonic Solids (so named because of the Greek philosopher, not because the solids are just good friends). Even better, this isn’t turkey tek’s first foray into mathematically inspired baked goods: also on display is the formiddable Giant Fractal Pecan Pie.

Yes, even pie can be educational.
Such seminal work naturally gives rise to the question: In what other ways can one combine holiday sweets with mathematics? After some careful deliberation, I humbly submit the following ideas.

1) Gingerbread Geodesic Domes.

The gingerbread man leads a tragic life. His sole purpose is to be put on display in the hope that he will be purchased and devoured. Often his life is ended through the horrifying mutilation of his appendages, all in the name of holiday cheer.

It seems reasonable, then, to let the gingerbread man get the most out of his short and sweet existence by accomodating him with a relatively luxurious lifestyle. This is the primary role of the gingerbread house.

While there are many people who pride themselves in their ability to make a fantastic and functional gingerbread house, it is a sad fact that these days, with people looking for quick solutions, boxed gingerbread houses are becoming more and more common. The pieces of the house are all included – all that’s left for the builder is to attach them, so that the process becomes akin to building an edible lego set. Unfortunately, this leads to a certain degree of uniformity in gingerbread dwellings, not to mention questionable quality of the building materials itself.

The least we can do for our small edible acquaintances, it seems to me, is to have enough respect for them that we take the time to build them a unique home, made from fresh gingerbread. With this goal in mind, I believe the gingerbread geodesic dome would be greatly appreciated by gingerbread man and woman alike. Not only would it provide them with a comfortable place to spend their limited time on this earth, but it’s also useful should you wish to provide your gingerbread populace with facilities such as planetariums.

For those interested in learning more about gingerbread geodesic domes, here is an article from Mother Earth News that should provide you with enough information to get started.

This gingerbread family is living large.

2) Sugar Cookie Math Ornaments.

Apparently, in some families it is tradition to decorate the Christmas tree with edible ornaments. The ornaments are made from thin sugar cookies, which are often cut into holiday themed shapes.

However, I see no reason why the holiday motif should be allowed to overshadow equally important themes. Mathematics is always in season, so why not make some cookies that will allow your children to express themselves through mathematics? Let your children decorate the tree with their favorite equality, or let them turn the tree into a center for edible math problems, by allowing them to use the ornaments to pose questions they can then solve! With sugar cookie math ornaments, you can exploit the natural harmony that exists between math and the holidays.

Surprisingly, there’s not much in the way of math inspired cookie cutters, so you may have to do some custom work. You can find some number shaped cookie cutters, and it’s not hard to find Roman alphabet cutters as well. Greek letters, unfortunately, remain elusive.

3) The Dreidel.

This one is easy – the game of dreidel, as with other games of chance, naturally lends itself to many discussions of mathematics. In fact, there are even theorems in math that pertain to dreidels. For example, in 2004, mathematicians Thomas Robinson and Sujith Vijay from Rutgers University proved that the length of a game of dreidel is bounded by a constant times n² spins, where n is the number of tokens at play in the game.

If you’re looking for a holiday game that incorporates mathematics, you certainly can’t go wrong with a friendly game of dreidel.

You see, no matter what your plans for the holidays, it’s easy to incorporate mathematics in with your celebrations. Everyone knows that a party can only be made more lively with the inclusion of some math, so whether it’s one of these ideas, or one of your own, don’t be shy about putting giving math an important role this season.

1. For those of you who find it curious that there are only 5 Platonice solids (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron), here is one elementary proof of this fact.

Math Gets Around: Politics, Part 3

As promised, in this thrilling final installment to the relationship between math and voting (the first two parts can be found here and here), we will look at what many people see as the holy grail of voting systems: Range voting.

The concept of range voting is simple. Given a set of candidates, in a range voting system you simply put a score next to each name that reflects how strongly you support that candidate. Of course, this is quite different from our current voting system, where we only get to vote for one candidate, but more importantly, it differs significantly from other voting systems where you are just asked to rank candidates in order of preference, because a ranking gives no information about the degree to which your support varies from candidate to candidate.

For example, if Anna, Bob, and Charlie are all running for President, you and I may both prefer Anna to Bob, and Bob to Charlie. However, I may LOVE Anna and HATE Charlie, while you may be relatively indifferent, with only a slight preference for one over the other. In a ranked voting system, both our preferences would be recorded as A > B > C. However, using range voting, our preferences may look something more like: A = 99, B = 50, C = 0 for me, and A = 51, B = 50, C = 49 for you.

This example shows that range voting allows us to capture more information about voter preferences than the other voting systems discussed. Therefore, one might heuristically expect that because it captures more information, it leads to better results, i.e. a more accurate representation of the will of the people. The question, of course, is whether this is actually the case.

The answer may depend on your definition of “better,” but by most measures the range voting system comes out on top of the others. One important feature of this system is that it is not subject to the constraints of Arrow’s Impossibility Theorem (discussed in Part 1). In other words, range voting has basically every property you would like a voting system to have. Range voting doesn’t contradict Arrow’s theorem because Arrow’s theorem deals only with voting systems that only rank preferences.

There are many other benefits to range voting as well, and these benefits are no doubt well known by any range voting advocate. The website Rangevoting.org gives a list of reasons why range voting is so great – I won’t list them all here, but I will highlight some of the more interesting ones.

Range voting encourages honest voting, rather than strategic voting. There is never an incentive for you to score a candidate you support more lower than a candidate you support less.
Range voting allows for a larger range of political parties to flourish. Because you are not restricted to one vote, people from third parties can feel free to support their candidate without fear of “wasting their vote.” This is also good for independents who may not feel particularly strongly about any major party candidate.
(Perhaps the cutest result) Range voting maximizes the number of “pleasantly surprised” voters, i.e. the number of voters for whom the winner of the election is better (scored higher) than they thought it would be.

As with any other idea, though, range voting is not without its share of criticism. However, these criticisms pale in comparison to the critiques that can be made about our current voting system. The main critique with range voting has to do with strategic voters, and comes in two forms:

Why doesn’t this just degenerate into the system we already have? For example, if supporters for one candidate feel strongly enough, they will simply give that candidate the highest score and every other candidate a 0. Doesn’t this benefit dishonest voters, and hurt candidates whose supporters are honest and may not give their candidate the full score, or score everyone else with a 0?

This criticism is a little suspect, because while there are certainly people who may vote in this way, it’s certainly hard to believe that everyone will, or that even a disproportionate number of supporters of one candidate will. It’s more likely that roughly the number of people for each major candidate will feel strongly enough to vote in this way, so that in the end it should all balance out. There certainly are extremes of political belief, but this is true of both the left and the right, with a wider swath of moderates somewhere in the middle.

Doesn’t the system inflate support for third party candidates? For example, people will be more likely to throw support to a candidate they believe has no chance of winning – this will amplify support for lesser known candidates, and dampen support for well known candidates.

This seems plausible. One way to combat this is to require that a candidate receive a minimum number of scores in order to be viable. For example, we could say that in order for a candidate to be declared the winner, at least 10% of the population must have voted for the candidate. The percentage should be high enough to be significant, but not so high that it’s possible it couldn’t be obtained if enough voters strategically abstain from voting for a particular candidate.

However, it’s still far more likely that a major candidate with a large base of support will win over an independent candidate with a small base of fervent supporters. Overall, range voting will certainly reflect preferences better than the current system, so it’s hard to argue that this is much of a valid criticism when compared to the current system, where your support is for all intents and purposes meaningless unless it is for a major party candidate.

In summary, from a mathematical point of view, there really is no argument: range voting certainly trounces are current voting system, and it looks like it beats everything else as well. The question then becomes: why don’t we use it? I’m not sure there’s a good answer.

Range voting is not used by any democratic nation, but examples of it can be found all over the internet. For a concrete example you are probably familiar with, you need look no further than the Internet Movie Database. On this site, you are free to vote on any movie you like, giving it a rating between 1 and 10. These votes are then compiled into IMDB’s rankings of the top 250 movies of all time, which can be found here. What’s even better, the votes are compiled using a true Bayesian estimate, the formula for which can be found at the bottom of the page. If you have any doubts about the validity of range voting, you need only go view this list and see all the awesome movies on it to conclude that indeed, this system has it going on.

Of course, you may find that this list of movies is terrible, but in this case, don’t worry. It doesn’t mean that range voting doesn’t work, it just means you have bad taste in movies.

In conclusion, our voting system is horribly broken. There are solutions out there, but getting from where we currently are to a new voting system is a problem that goes beyond the scope of a pop culture math blog. For now, we’ll have to deal with things as is, and of course, that includes voting tomorrow, November 4th. So make sure you go out and do it. We can fight the larger fight of voting systems another day.

Math Goes Pop!

Ruminations on the Intersection Between Mathematics and Popular Culture