I need Math People to help me find out how long it takes 2 Last (Wo)Men on Earth to meet.

August 23, 2008 9:36 AM

How long should it reasonably take before 2 "Last Men on Earth" meet?

I'm writing a story about a post-apocalyptic mathematician, something to do with zombies. Any such character would naturally be inclined to work the following problem in such a situation. My difficulty lies in myself having very few mathematical skills (2 +2 is four, heh heh heh, me smart). So, Hive Mind, assist me!

Question: Assuming that there has been a global catastrophe of Hollywood proportions and most of the human race has died off (or become zombies, etc), and assuming that any lone survivors will, eventually, come to think of themselves as the "Last (Wo)Man on Earth" in the Richard Matheson vein, and assuming a slow rate of travel among these LMOE, how long will it take before 2 LMOE meet, and how many LMOE are there on Earth in total?

Assumption 1: Most of the human race has died off. Let's assume that 99.99% of the global human population is gone, just so we have something to work with.

Assumption 2: Lone survivors come to believe themselves as the Last Man on Earth. Let's assume that it takes 3 months of being alone to come to this assumption. This assumption is contingent on not seeing another -living- human being during this time, so we can expect city dwellers to have a margin of error here leading them to come to this conclusion later than their rural or islander counterparts, being as there should (at first) be more survivors in cities. However, they can't be considered LMOE unless they end of alone, so it may take some time for their fellow survivors to die off and thus the margin of error. If their fellow survivors never die off, they can't be considered LMOE. Let's say any group of survivors has a half-life of 1 week, and no group can begin with more than 8 survivors.

Assumption 3: A slow rate of travel among the LMOE. Let's say, for the sake of the problem, that half the LMOE are home-based and half are travel-based. Home-based LMOE stay within their little "fortified zone" of about 5 square miles (gathering supplies, killing zombies, and waiting out the storm going slowly insane). Travel-Based LMOE move slowly (at about 5 miles per day) due to travel hazards such as zombie hordes, traffic blocks, fires, etc. They're determined to find other survivors and thus keep themselves from madness.

Hidden Assumption 4: The game begins on Day 0. Day 0 is the day when the travel-based LMOE begin moving, the home-based LMOE start gathering supplies, and city-based groups of survivors begin dying off down to single LMOE.

How many days will it take before any 2 LMOE meet?


I realize I am not a mathematician (I am in fact, and English major and the opposite of a mathematician), and so any data that needs to be filled in, altered, or deleted should be in order for the problem to be solved.

Cheers.

posted by mr_book to science & nature (24 comments total)

Well, one factor that seems to be missing is any sort of broadcast or signal that the home-based LMOE would use to broadcast his(her) position. Shortwave radio can travel around the world, so, potentially this would greatly speed up the meeting.

Then again, if every LMOE really believes that he is the LMOE then there probably wouldn't be any motivation to establish such a beacon. Then again, every single treatment I've seen of LMOE (in film at least) has such a beacon in place, usually in the form of a radio transmission.

Otherwise LMOE could wander the earth pretty much randomly without ever coming into contact with each other - unless they were drawn to some geographic point for other reasons.

Sorry I can't help with the math... You should check out Quite Earth. Great LMOE treatment.
posted by wfrgms at 9:52 AM on August 23, 2008

Let's assume that no short-wave communication is possible, then.
posted by mr_book at 10:05 AM on August 23, 2008

You also need to specify whether they can meet one another "live" only through, well, random walks. I seem to recall that a common conceit in these stories is that there's a likely gathering place (like the clock in Grand Central) to which survivors would gravitate and then monitor. Even if not, note that they could leave notes/signs for each other. You would have to address whether the zombies could understand those. It would be neat if they did, somehow.
posted by Clyde Mnestra at 10:12 AM on August 23, 2008

0.01% of world population = 650,000 people.

How far apart can they be before we score a "hit"?

Lets say we're doing Day of the Triffids here - column of smoke by day, pillar of light by night. Then we can assume that, if you're in the same city, you'll find each other.

Population of New York = 19,300,000. 0.01% of 19,300,000 = 1930 people.
posted by Leon at 10:22 AM on August 23, 2008

At 99.99%, I don't think it would take very long. For example, in DC where I live, excluding the suburbs the population is ~588,000. If 99.99% don't make it, that still leaves about 60 survivors in an area of 68 square miles. For Manhattan, there are about 1,600,000 people, so you would expect about 160 survivors in 23 square miles. I think you might either need to make it 99.9999% mortality, or say that the zombies get the people in the highly populated areas because they are easier for the zombies to find as well. Of course, at 99.9999% only about 300 people would survive in the whole US.
posted by procrastination at 10:22 AM on August 23, 2008

This seems to presume the world is America.

If one is in America and one in [insert continent] the answer is never, unless they sail or fly somehow.
posted by A189Nut at 10:49 AM on August 23, 2008

Otherwise LMOE could wander the earth pretty much randomly without ever coming into contact with each other - unless they were drawn to some geographic point for other reasons.

I think LMOE will totally be drawn towards specific geographic locations: famous ones. I'm in Austin, and if I suspected I was alone after some disaster shit, I'd head to the Texas State Capitol building and spray paint a big-ass note there "ERIK WILL BE HERE EVERYDAY AT NOON" or some other crap that I stole from "I Am Legend". Then I'd go to the UT clock tower and spray paint a different message and a different time. If another LMOE in my town was too dumb to pick one of those buildings as a meet-up place, then I don't think they'd be much help to me.

If it turns out I'm the only LMOE in Austin who thought of either of those two places to meet up, and I really wanted to meet other people, I suppose I'd head to either that US Capitol building and just wait. If that didn't work, I'd head to NYC, and hang out by the Empire State Building.

I think every city (and country) has famous buildings that everyone knows about, and that people would gravitate towards them as meeting places. It's not like adventurous LMOEs are just going to wander around aimlessly.
posted by 23skidoo at 10:51 AM on August 23, 2008

Why would a traveling LMOE be traveling? If he or she believes that no one else is alive why would he or she be out looking for other people? Is it that they don't really think no one else lives, or is it that they just want something to do to keep busy, even if it is pointless?
posted by oddman at 11:15 AM on August 23, 2008

James Surowiecki writes about 23skidoo's technique in Wisdom of the Crowds. There have been experiments where people were told "you're supposed to meet someone in New York, but you don't know the time nor the place. Where would you be at which time?", and a good part of the people gave the same answer (noon at Central Station, I think).

So yes, the people wouldn't move randomly at all in a realistic scenario.
posted by dhoe at 11:18 AM on August 23, 2008 [2 favorites]

.01% survival would reduce the population to a level roughly in line with that of the mid-18th century. They didn't have shortwave radio then either. I think finding other humans would be just about as difficult in your apocalyptic wasteland as it was then.

How exactly would the population be decimated? If it were anything gradual (even just a few days) I'm sure that most of the population would never lose contact with society unless they choose to. As millions of people died in New York thousands of other people would gather together in increasingly large groups in a huge funnel directed at New York. (And Chicago, Tokyo, etc...)

How exactly would shortwave radio be restricted? The very idea that a person could converse with another person on the other side of the globe was an invention almost as powerful as the solution to the physical and technical puzzles that first facilitated it. Other radio frequencies passed through relays, wires, lasers, what happened to the satellites?, smoke signals, dixie cups... whatever. In this day and age, thousands of people in Chicago (assuming some gathering) won't tolerate being connected to thousands more in New York by the pony express.

I think 99.99% doesn't do exactly what you had in mind?
posted by stuart_s at 12:13 PM on August 23, 2008

kill as many people as necessary to make the problem work.
posted by mr_book at 12:29 PM on August 23, 2008 [2 favorites]

How exactly would shortwave radio be restricted?

We're talking mathematician here, shortwave is a hardware problem.

The smart mathematician would stay put. He or she would announce his or her presence with authority by blowing shit up and making a big fire in the sky. Also smoke.

Then he or she would start a search algorithm. The search vector itself would be very interesting. Can he or she somehow tag zombies like sperm whales and release them back into the wild? Somehow the tags could lead back to him or her? Maybe the tagged zombies could pass this condition on to zombies they meet along the way? That'd be awesome, exploit the fuckers instead of bashing their heads in.
posted by stubby phillips at 12:38 PM on August 23, 2008

A zombie virus with a message encrypted into it that the travelling zombie killers would understand.
posted by stubby phillips at 12:40 PM on August 23, 2008

If (a) enough people have died that someone in a place like Manhattan can believe themselves to be a LMOE, (b) they don't have technology to signal each other, and (c) they are travelling essentially randomly/arbitrarily (that is, not being drawn to a common destination), then I suspect they will never encounter one another. The earth is very big.
posted by winston at 12:57 PM on August 23, 2008

We could kill you, then we wouldn't have this problem.

Ehem. Sorry.

You could go the other way. A pretty serious hike for an unconditioned person is twenty miles in a day If a person standing on level ground can see / is visible from an area of a few acres, and your characters have to cover a certain amount of ground to forage for food and hide from the zombies, you can estimate how much land each character will survey in a day. Exactly how much, of course, depends on your characters and how observant they are. Now the number of people who survive depends on the story you want to tell: how far apart are they, and how long do they take to intersect.

Go take a walk around your neighborhood for a couple hours. Take a printed-out map of your neighborhood, so that your route fills up a page of paper. Mark on the map where you go, where you could see, hiding places you could have been seen from, woodsy areas that you couldn't see into at all. What if someone followed you on your walk an hour later. Would their path have crossed yours?

I expect that two people exploring an empty city might meet each other in weeks or months. I can imagine deciding to walk to a different city if I didn't see anyone for a few months, or a year.

Have you read Stephen King's Cell? Similar premise: everyone who uses a cell phone after The Traumatic Mystery becomes a mindless killing machine.
posted by fantabulous timewaster at 1:03 PM on August 23, 2008

Sounds like, to make your scenario probably you'd need something like The Change from the Emberverse universe. Either that or a premise change, because people WILL get a hold of one another if they can in short order.
posted by ZaneJ. at 1:04 PM on August 23, 2008

I feel like there's a very specific question in here. Kind of like the scene in Cast Away where Tom Hanks does some quick calculations to figure out the search area his rescuers would have to cover, and realizes: "that's twice the size of Texas".

Your problem only works if the LMOE is alone in a rural (or semi-rural) area. Like, perhaps a small town, 8 hours drive from anything else. A big city is the natural destination for survivors and a rational person looking for others would come up with beacons or something. But forget about that for now.

Let's assume your narrator is the only survivor in a region of 100,000 people; assuming similar death rates, perhaps there are 3000 people in the US or so. Suppose they are evenly distributed across the US land area (not realistic though - most of them would be in major cities). The US is about 3.5 million sq miles (land area) which means each person is alone is an area of about a thousand square miles. Hmm...

I could go through a calculation but there's no point. The result is hugely sensitive to how the populations are distributed, not to mention initial survivor estimates, and a whole lot of other factors. Your mathematician would realize such a calculation is meaningless with so much uncertainty. It could range from hours to years.
posted by PercussivePaul at 1:09 PM on August 23, 2008 [2 favorites]

Why would a traveling LMOE be traveling? If he or she believes that no one else is alive why would he or she be out looking for other people? Is it that they don't really think no one else lives, or is it that they just want something to do to keep busy, even if it is pointless?

There are other motivations to travel. Later-stage resource gathering, but also, potentially, fun. If I were the last person on the planet, I'd be going around from place to place breaking in and having lots of fun checking out people's possessions, finding cool instruments, books, people's underwear and whatever.
posted by wackybrit at 2:10 PM on August 23, 2008

Stuart_s said:

.01% survival would reduce the population to a level roughly in line with that of the mid-18th century.

This is not correct. If .01% survival is ~650,000, this would be lower than the estimated population in 10,000 BC. You must have calculated 10% survival rate.

As for the original question, it seems unanswerable with the given information. We would need to know what it is that is killing everyone off, how the "disease" spreads, the population densities of the particular areas these LMOE are in, and so on.

Assuming a completely equal rate of death spread over the planet, if the survivors are in Manhattan, they would find each other relatively quickly. If they are in Mongolia, it's plausible they would never find each other.

How would your fictional mathematician know that precisely 99.99% has died off? You as the author know this, but this variable is not known to him. He wouldn't bother trying to do this calculation.
posted by wigglin at 2:21 PM on August 23, 2008

I'd wait by the pig in Seattle's Pike Place Market.
posted by iconjack at 2:28 PM on August 23, 2008

wigglin - .01 *(6 billion) = 60 million. If we're America-centric, it's worth noting that 350 million -> 3.5 million.

iconjack - ah, but would you wait there if that's where the zombies (on some weird primal instinct) were going, too?

More generally, if everyone has reason to hide and not broadcast their existence, the survivors will be less likely to find one another. If I were a survivor in hiding, I would probably try to think of an isolated area in which hunter-gathering would be relatively easy. Probably not Appalachia, to avoid the Lovecraftian horrors known to inhabit those parts; but maybe someplace like the pacific northwest, where winters are relatively mild, population not terribly dense, and wild life is rampant. Nothing coastal; too many 'people' around, though the Lost Coast in Northern California might fit the bill.
posted by kaibutsu at 12:28 AM on August 24, 2008

kaibutsu: .01 * 6B is 1%. You need .0001 * 6B, which is two orders of magnitude smaller.

To me, the biggest variable here is Smart Zombies vs. Dumb Zombies. If the zombies are the groaning, knuckle-dragging type, than it seems reasonable to me that the vast majority of people would leave trail markings of where they went, signs and beacons miles in every direction, any sort of thing to comfort them that IF there is someone else, we won't just be two ships passing in the night.

Smart zombies which could follow these markings/beacons, would, obviously, make this a bad idea, and greatly diminish the odds of success.
posted by paisley henosis at 1:11 AM on August 24, 2008

kaibutsu:

Referring to the original question, I used .01%, not .01 in my response.
posted by wigglin at 4:45 AM on August 24, 2008

Metafilter: kill as many people as necessary to make the problem work.
posted by dersins at 5:07 PM on November 20, 2008