How should I model this infection spreading problem? Is this article's solution at all reproducible?

Question

While on the internet, I came across this quote in a Buzzfeed article, which granted, is probably not going to be the height of journalistic quality at all:

This is simple math. If one person sleeps with a pool of 10 people, as opposed to one person with a pool of 100 people, the former will have a higher risk [of contracting HIV].

This struck me as blatantly incorrect, because I assumed that:

1. The probability that someone has HIV is identical across populations before people start having sex.

2. Each person sleeps with each other person in his pool at most once. (This is naturally only possible after $n<10$ turns, where 1 turn is when the person pairs up with someone else and has sex.)

3. Only the person we are observing has sex with anyone. (This was more of an unconscious assumption, and I will ask about this later.)

4. Infected individuals also have sex with others. Moreover, the chances of obtaining one sexual partner is always identical to that of obtaining another.

Under these assumptions, it would seem that infection rates should be identical. (It occurs to me that I should be comparing how "quickly" the probability of having no infection decreases as the number of "turns" increases.)

Moreover, if I assume that (5) the person we are "observing" sleeps around more when there is a larger pool -- I did not specify to what extent, however, and this was more of an unintentional assumption like in (3) -- it would be obvious that because of (1), being in a larger pool increases your chances of getting infected: $\text{Pr}(\text{uninfected})^k$ decreases with $k$.

If I change (1) to "the proportion of people with HIV is equivalent at $k=0$" and drop (5), then it would still appear that one's chances of getting infected increase in the larger pool. (I was too lazy to write a proof, but I tested this with $k=2$ and with the proportion in both pools being 0.1, and on some intuitive level this makes sense to me.)

So far, not very good. It then occurred to me that (2) and (3) might not be entirely realistic assumptions, regardless of which version of (1) I use. So I thought of dropping (3), and assuming that (6) once someone sleeps with an infected individual, they too become infected.

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Unfortunately, this didn't appear to be "simple math" anymore: it occurred to me that the most straightforward way to work through this would possibly be to run a computer simulation. In any case, this leaves me with a few problems:

1. Is there a straightforward, "easy" way to come to the same conclusions the article author arrives at? (By easy, I mean something someone without too much statistics education might come up with -- I certainly couldn't come up with anything, and I've been through a few courses meant for people in technical fields while at university.) The only thing I could think of was if he assumed that the same number of infected individuals were in each pool, but this seems like an odd assumption to make to me.

2. Let's say we can bin his conclusion to the resolvability of the problem. If so, what tools would I need to arrive at a solution to this, if I take the route where I drop assumptions (2), (3), and (5), and if I use (6) as well? Can I simply solve this via some use of combinatorics -- assuming I use the revised version of (1), since the old version wouldn't make much sense that way -- or would something else be necessary?

Answer 1

I highly doubt that there's a closed form solution to this problem. The big problem with the closed form solution is that it will likely assume that something in the system is static. For instance, it gets much easier to model the situation if you accept your assumptions 2 and 3, but those assumptions really don't reflect reality.

Simulation is your best approach because you can set all of the variables (people) in motion at once. An overly simplified approach is in the code here. It assumes a population of 5,000 people, 0.3% of whom are infected. In the first run of the experiment, we assume each person selects a partner of the opposite sex at random ten times. If either partner is infected, the uninfected partner has a 10% chance of becoming infected. By the end of 10 turns, the total proportion of the population that is infected is 0.0066.

If you reset the experiment and have everyone pursue 100 partners, nearly everyone is infected by the end.

Keep in mind, this simulation inflates the spread of infection way too much. Mostly because it is assuming everyone in the population is having as many partners as anyone else, and that it occurs at random. That isn't a good assumption, and a more robust simulation would account for the majority of people having less than ten partners, and only a few really reaching the 100 mark.

But anyway, I think you'll find that the overall risk of an individual becoming infected increases with the number of partners primarily because the exposure pool (potentially infected people) is in constant motion