On the Swine Flu (a Year-and-a-Half Later)

This week I have been revisiting the swine flu scare of April 2009.  The Mexican government then was particularly anxious: it suspended regular classes, and many kids were just thrilled that they found themselves suddenly on vacation.  Some schools even advised kids that if they felt ill, they should excuse themselves from attending classes for at least a week.  Many took the slightest symptom as an excuse to take a short vacation... to the beach, no less.

A year ago in April I attempted to simulate, a grosso modo, not the time progress of the disease, but yes some of its generalities.  I estimated, for example, that if the swine flu were concentrated in Mexico City, with its sizable population of 20 million, at the outbreak rate of infection and, by clever deduction of the SIR differential equation constants of proportionality  a (and a wild guess of  b ), the total of infected individuals at one time would come to about 500,000, or 2.5% of the city's population.

One thing that struck me from Mexico's Secretaria de Salud infomercials at the time was that they graphed an exponential progression of the disease (which, after government intervention, supposedly decreased exponentially again). It was unrealistic and, whether the authorities minded or not, misinforming.

With the SIR model I recently calculated what could have been the time progress of the disease: I used the method that I developed in my previous post to facilitate the calculation, after I realized that my (Markovian) matrix formulation was equivalent to Euler's method: expanding out the matrix one obtains, indeed, Euler's estimation formulas (which we thought of, from the point-of-view of a Markov chain, as the one-step forward transformation of a vector pulled through or propagated by the Markov matrix).  Interestingly, then, one can say that a Markov transition matrix really what it does is use Euler's method to estimate the next step.  This is kind of an intriguing insight to me, because I never understood a transition matrix in this way.

The matrix formulation therefore also acquires an error as Euler's method does: picking a point away form the initial condition, if the number of steps used to get there is  n , the error (the difference between the approximate value and the true value) is approximately proportional to  1/n .

The matrix formulation, then, required that the differential equations of the SIR model be recast not in absolute terms but in relative proportions, as Duke University's website proposes from the beginning.  I didn't do this originally, and so my calculation of the constant  a must be modified.  I said that  dS/dt = -400 .  In terms of  s(t) , which is  S(t) / N where  N is the total population,  ds/dt = -400/20,000,000 .  The constant of proportionality  a , then, can be calculated as  a = \frac{-ds/dt}{s \cdot i} \approx 0.25 if, from initial conditions,  s = \frac{19,998,400}{20,000,000} and  i = \frac{1,600}{20,000,000} .  My calculation of the constant of proportionality  b remains unmodified, it was a guesstimate based in part on CDC records of disease recovery, if I remember correctly.  Recall that this  b , from the POV of the Markov chain matrix, is the transition probability of going from infected state to recovered state.  The initial conditions  S_0 = 19,998,400 ,  I_0 = 1,600 and  R_0 = 0 need to be recast in terms of proportions, too, as  s_0, i_0 above and  r_0 = 0 .  Strictly speaking, we had data at the time that deaths were about 149 people (which I attempted to use to estimate the constant  b ).  The initial condition vector would have to be modified to reflect this forgotten piece of information, but the truth is that the difference in trajectory is negligible, so, to remain consistent with the analysis of a year ago, I will leave  s and  r as they are.

Here's the timed graph I came up with.  It states that the time at which the number of infecteds is a maximum (about 500,000 as per the post of April of 2009; new numerical estimate using the matrix transform: 440,000) is approximately 145 days after the outbreak (April 27ish): maybe around September 19, 2009.  It also states that, at the end of the disease, roughly 8,000,000  people recover (or die).  This I know was not the case (not even thousands of people died, e.g.), largely in part because of medication and excellent medical care and, admittedly, competent government response (but it puts things in perspective, doesn't it? No wonder it was a "scare.").

Swine Flu SIR April 27, 2009

  1. Larry Baker
    July 11th, 2017 at 15:14 | #1

    I, too, was interested in the relationship between SIR modeling via differential equations vs. via Markov chains, and your two posts make the relationship very clear. It's quite interesting that your generalized transition matrices implement Euler's method! Thanks.

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