## Wednesday, April 29, 2009

### Pandemics

Hmm... should I cancel my travel tomorrow to the Perimeter Institute? What exactly is the cost benefit analysis of flying to Toronto from Eugene (via Denver) right now to meet with possibly virus bearing economists and physicists from around the world? How much do those activities elevate my chances of catching H1N1, or, even worse, infecting my family?

I have no answers, although some years ago I wrote a paper on epidemics with collaborator A. Zee of KITP in Santa Barbara. That research was inspired by SARS, not H1N1, but it does have a certain timeless quality to it ;-) Astute readers will note that the bi-linear diagonalization in equation (6) resembles the transformation used on mass matrices in particle physics.

http://arxiv.org/abs/cond-mat/0306628

S. Hsu, A. Zee

We develop simple models for the global spread of infectious diseases, emphasizing human mobility via air travel and the variation of public health infrastructure from region to region. We derive formulas relating the total and peak number of infections in two countries to the rate of travel between them and their respective epidemiological parameters.

From the conclusions:

One interesting conclusion from our models is that typical international mobility – the probability per unit time of international travel for a given infected individual, estimated at mi→j ∼ 10^−5 per week – is still sufficiently small that a country with well-developed public health infrastructure (effectively, a negative eigenvalue λ) can resist an epidemic even when other more populous countries experience complete saturation. In the quasi-realistic simulation 1 (figures (1),(2)), of order 10^5 infections occur in country 2, even though the disease has swept completely through country 1. In reaching this conclusion, we kept the mobility parameter fixed during the outbreak, and did not assume any draconian quarantine on international travellers arriving in country 2. Such measures would reduce the number of infections in country 2 considerably. Of course, this conclusion assumes that the public health infrastructure in country 2 remains robust during the outbreak. In the nonlinear simulation 3 (figures (6), (7)), we see that a breakdown in the medical system can lead to grave consequences.

In the case of two countries, one of which is a “reservoir” with positive eigenvalue λ_i and the other with negative eigenvalue λ_j , a good rule of thumb arising from equation (11) is that the ratio of total number of infections in the two countries is given by the fraction of infected individuals who migrate in a timescale |λ_j|^−1 , which is the “halving” time for the epidemic in country j . In our simulation 1, this timescale is about two months, and the fractional mobility over that period is ∼ 10^−4, leading to 10^5 infections in country 2 if the entire 10^9 population of country 1 is infected. The maximum number of infections at any given time in the simulation is 5 10^3 (figure (2)). If the medical system of country 2 can treat this number of patients without breaking down (entering the nonlinear regime), it can prevent a larger outbreak.