on options & epidemics

Daedalus, Spring 2008 by Ayres, Ian

The need to attend to volatility is important whenever options come into play. A number of years ago when I was teaching at Stanford, the university had a home mortgage program. The university would lend you half the purchase price of your house, if you give the university half the appreciation at the time of the sale. The program gave the university something akin to a call option on half your house. The university didn't have to bear any cost of home depreciation, but got half the upside if the housing value increased. I had a choice of buying a house in an unincorporated (and unzoned) new section of Mountain View or a relatively staid and seasoned development just south of the campus called College Terrace. Attending to volatility, you should be able to tell which house was more subsidized.

The mathematics of epidemiology developed independently from the mathematics of option pricing. But like call option prices, the force of an epidemic also rises with both the mean and variance of an underlying distribution. The force of an STD epidemic is, like an option, a kind of 'derivative,' in that it is derived from the mean and variance of the number of partners in a population.

It is immediately intuitive that an STD is more likely to spread when the average person in a population has a larger number of sexual partners, but the variance in number of sexual partners in a population also positively impacts an STD's expected replication rate. Epidemiologists have modeled the force of an epidemic in populations with heterogeneous sexual frequency to equal:

ρ^sub 0^ is the product of the transmission probability per partner (sometimes referred to as the 'efficiency' of transmission) and the average duration of the disease,

μ is the mean number of partners per unit time, and

σ^sup 2^ is the variance of the number of partners.

R^sub 0^ measures the 'reproductive rate' : the average number of secondary infections produced by a single index case in a population of susceptible persons. The dis ease rate is stable (or 'endemic') when the infector number (R^sub 0^) equals one; epidemic when greater than one ; and eventually zero (the disease will die out over time) when less than one.

The equation teaches us that for any given mean, increasing the variance in the number of partners will increase the epidemiological force of a disease. The intuition for the positive impact of variance is that populations with high variances in the number of sexual partners are likely to exhibit large connected networks of sexual nodes. The few members of the population with many sexual partners are likely to form connections with one another, as well as with members of the population who have few other sexual partners. Randomly infecting someone in a high variance network is therefore likely to spread the disease quickly, through these longer connecting chains. In a population with high variance, the few people with many sexual partners are the 'superspreaders' who tend to connect the rest of the population.