A possible error with mathematical ramifications

 Non-technical fact-check source

Dan Brown is the bestselling author of the novel The Da Vinci Code. His most recent bestseller, published in 2013, is Inferno. Like two of his earlier blockbusters it has been made into a movie. It stars Tom Hanks and Felicity Jones and is slated for release on October 28.

Today I want to talk about a curious aspect of the book Inferno, since it raises an interesting mathematical question.

Brown’s books are famous for their themes: cryptography, keys, symbols, codes, and conspiracy theories. The first four of these have a distinctive flavor of our field. Although we avoid the last in our work, it is easy to think of possible conspiracies that involve computational theory. How about these: certain groups already can factor large numbers, certain groups have real quantum computers, certain groups have trapdoors in cryptocurrencies, or …

The book has been out for awhile, but I only tried to read it the other day. It was tough to finish so I jumped to the end where the “secret” was exposed. Brown’s works have sold countless copies and yet have been attacked as being poorly written. He must be doing something very right. His prose may not be magical—whose is?—but his plots and the use of his themes usually makes for a terrific “cannot put down” book.

Well I put it down. But I must be the exception. If you haven’t read the book and wish to do so without “spoilers” then you can put down this column.

## In The Inferno

The Inferno is about the release of a powerful virus that changes the world. Before I go into the mathematical issues this virus raises I must point out that Brown’s work has often been criticized for making scientific errors and overstepping the bounds of “plausible suspension of disbelief.” I think it is a great honor—really—that so many posts and discussions are around mistakes that he has made. Clearly there is huge interest in his books.

Examples of such criticism of The Inferno have addressed the DNA science involved, the kind of virus used, the hows of genetic engineering and virus detection, and the population projections, some of which we get into below. There is also an entire book about Brown’s novel, Secrets of Inferno

However, none of these seems to address a simple point that we hadn’t found anywhere, until Ken noticed it raised here on the often-helpful FourmiLab site maintained by the popular science writer John Walker. It appears when you click “Show Spoilers” on that page, so again you may stop reading if you don’t wish to know.

## How The Virus Works—or Doesn’t?

How does the virus work? The goal of the virus is to stop population explosion.

The book hints that it is airborne, so we may assume that everyone in the world is infected by it—all women in particular. Brown says that 1/3 are made infertile. There are two ways to think about this statement. It depends on the exact definition of the mechanism causing infertility.

The first way is that when you get infected by the virus a coin is flipped and with probability 1/3 you are unable to have children. That is, when the virus attacks your original DNA there is a 1/3 chance the altered genes render you infertile. In the 2/3-case that the virus embeds in a way that does not cause infertility, that gets passed on to children and there is no further effect. In the 1/3-case that the alteration causes infertility, that property too gets passed on. Except, that is, for the issue in this famous quote:

Having Children Is Hereditary: If Your Parents Didn’t Have Any, Then You Probably Won’t Either.

Thus the effect “dies out” almost immediately; it would necessarily be just one-shot on the current generation.

The second way is that the virus allows the initial receiver to be fertile but has its effect when (female) children are born. In one third of cases the woman becomes infertile, and otherwise is able to have children when she grows up.

In this case the effect seems to work as claimed in the book. Children all get the virus and it keeps flipping coins forever. Walker still isn’t sure—we won’t reveal here the words he hides but you can find them. In any event, the point remains that this would become a much more complex virus. And Brown does not explain this point in his book—at least I am unsure if he even sees the necessary distinctions.

The other discussions focus on issues like how society would react to this reduction in fertility. Except for part of one we noted above, however, none seems to address the novel’s mathematical presumptions.

## The Math and Aftermath

The purpose of the virus is to reduce the growth rate in the world’s population. By how much is not clear in the book. The over-arching issue is that it is hard to find conditions under which the projection of the effect is stable.

For example, suppose we can divide time into discrete units of generations so that the world population of women after ${t}$ generations follows the exponential growth curve ${N = N_0 r^t}$. Ignoring the natural rate of infertility and male-female imbalance and other factors for simplicity, this envisions ${N}$ women having ${r}$ female children on average. The intent seems to be to replace this with ${2N/3}$ women having ${r}$ female children each, for ${N' = 2rN/3}$ in the next generation. This means multiplying ${N}$ by ${\frac{2}{3}r}$, so

$\displaystyle N'_t = N_0 \left(\frac{2}{3}r\right)^t$

becomes the new curve. The problem is that this tends to zero unless ${r \geq 3/2}$, whereas the estimates of ${r}$ that you can get from tables such as this are uniformly lower at least since 2000.

The point is that the blunt “1/3” factor of the virus is thinking only in such simplistic terms about “exponential growth”—yet in the same terms there is no region of stability. Either growth remains exponential or humanity crashes. Maybe the latter possibility is implicit in the dark allusions to Dante Alighieri’s Inferno that permeate the plot.

In reality, as our source points out, it would not take much for humanity to compensate. If a generation is 30 years and we are missing 33% of women, then what’s needed is for just over 3% of the remaining women to change their minds about not having a child in any given year. We don’t want to trivialize the effect of infertility, but there is much more to adaptability than the book’s tenet presumes.

## Open Problems

Have you read the book? What do you think about the math?

1. July 25, 2016 6:49 am

Interesting.

To me, all of this is moot as far as the plot goes, because all that is required is that the characters _believe_ in the outcome, whether it is true or even likely, or not.

And the analysis does not consider mutations, or cultural adaptions to the new reality.

July 25, 2016 2:13 pm

Two points:

First, cultural adaptation in response to a 1 in 3 infertility rate seems like a great way to achieve a stable population. Assume that with normal levels of fertility a high enough percentage of people are determined to reproduce (regardless of the total population) that world population grows exponentially but that, without adaptation, a 1 in 3 infertility rate would lead to population collapse.

Presumably, if 1 in 3 people were infertile incentives would kick in to encourage people to have more children. A desire to avoid the economic hardship that would accompany a drop in population (you can’t enjoy retirement if no one is around to make the goods you want to consume) society would incentivize having children. Those incentives would, however, cost money so would be adjusted downward if the population looked to be increasing.

In short by moving reproduction from a regime driven by a blind unmoderated individual desire to have children it would shift it to a societally moderated regime responsive to the total projected population.

Second, the assumption that it would be harder to engineer a virus which affected each later generation with a 1/3 chance of becoming infertile seems to assume the 1/3 factor is the result of some kind of internal modulation of a 100% effective means of rendering women infertile. If, instead, the virus has no modulation at all but it’s just that it’s only in 1/3 of the cases that the women is successfully rendered infertile it actually sounds easier. Of course there are tons of other problems with such a scheme.

3. July 26, 2016 7:25 am

No problem with your math, but with your literary crit. It isn’t “plausible suspension of disbelief”; it’s “willing suspension of disbelief” (Samuel Taylor Coleridge, Biographia Literaria). That’s an important difference. Something along the lines of “willing suspension of disbelief happens whenever fiction is read. Even in the most realistic fiction, we know those people don’t exist, we know those events didn’t actually happen. There are ghosts in ghost stories; we know there are no ghosts, but we enjoy them. There’s faster-than-light travel in a lot of science fiction, but we accept that, too–and that’s also ignoring a simple mathematical error: sqrt(1-v^2/c^2) is imaginary for v > c. And fantasy literature is, well, fantasy.

So: your critique of the math is interesting, and I’m not big on Dan Brown’s novels, so it’s no skin off my back, but not really to the point. Why should we expect Brown’s virology to work better than, say, Asimov’s relativity? Except that Asimov certainly understood the math, and Brown probably doesn’t. Plausible suspension of disbelief has the extraneous and poorly defined requirement that what we’re asked to believe in a fiction is “plausible.” That really isn’t relevant; only the act of imagination that allows us to enter the fiction in the first place. And the real issue is whether the writer’s imagination is powerful enough to make us believe, in spite of plausibility.

I recently wrote a Goodreads review of a novel about robots in Antarctica that apparently had vacuum tube logic. They were lubricated with some kind of machine oil with properties that were independent of temperature. Some things are hard to take, particularly if the story isn’t that good. So that’s the real question: is Brown’s imagination powerful enough that we overlook the math? Asimov’s imagination is that powerful (even though, re-reading Asimov 40+ years later, I’m surprised at how poor the writing is.) Is Brown’s?