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On the Swine Flu

April 27th, 2009 5 comments

So I've been MIA for the last month-and-a-half because of several exciting things that have happened to me, and I apologize for the hiatus... but it may be prolonged for a while still.  However, in the meantime, since a lot of my Mexican colleagues and students have been asking me about the swine flu, in somewhat of a panic, I thought I would try to put some numbers into some well-known SIR equations and see what insights can be obtained regarding the dynamics and proportions of the purported epidemic.  I believe this kind of analysis is routine in CDC in the US and in Secretaria de Salud in Mexico, although I'm not too sure how mathematically equipped the latter institution is in my home country. In much of what follows, I have to do some hand-waving because of the unavailability of accurate information in the web-o-sphere, or my inability to access true statistics.  Much of what I know I got from CNN and NYTIMES reports of today.  I find such fuzzy math unacceptable, personally, except to derive a notion of the magnitude of a problem, and so it would be a mistake to take the calculations as set fact or hard evidence.  Caveat in mind, I don't derive the model (it's up to the reader to find several excellent sites that might explain it), but instead delve to account for my assumptions and my results.

The SIR model (Susceptibles-Infecteds-Recovereds) uses coupled differential equations to analyze the progress of a disease in a closed population.  I focus on the population of Mexico City, assuming that the count of infected individuals nationally can be mostly found there.  Thus, out of Mexico City's 20 million people, most of the suspected 1600 currently-infecteds are in that city (all for simplicity).  Recovered individuals are the sum of those dead (149) plus those that did not die.  The coupled time-differential equations are: 

 \frac{dS}{dt} = -a S I

 \frac{dR}{dt} = b I

 \frac{dI}{dt} = a S I - b I

which basically says that the change in susceptible individuals is proportional to the amount infected and the amount currently susceptible, that the change in recovereds (removed from infection) is proportional to those who are already infected, and that the change in infecteds depends is the rate at which susceptibles get sick minus the rate at which infecteds get removed from the population.  The proportionality constants can be calculated with some cleverness (though not necessarily accuracy), like this:  

 a = \frac{-\frac{dS}{dt}}{S I}

and so we must guesstimate  \frac{dS}{dt} as well as  S and  I .  If there are currently 1600 individuals that are infected, linearly, 400 have been infected per day since this became news four days ago.  So my guesstimate is that the current rate  \frac{dS}{dt} = -400 individuals per day. The number of susceptibles is the population of Mexico City, so all 20 million, minus infecteds, about (I'm thinking naturally immune individuals are so few that the population number doesn't change much).  Finally, the number of current infecteds (Mexico-wide? focused in Mexico City) is 1600 according to the NYTIMES article I've been linking to.  This gives a value of  a = 1.25 \times 10^{-8} .  Calculating  b is a bit trickier.  The datum says that 149 people have died from the swine flu, but I don't think all people infected with the swine flu die.  I'm going to guesstimate that approximately 20% of the infecteds either die or recover (since apparently 10% of them die) in a day.  There's no reason for this except my hunch. So  b \approx .2 .

By the chain rule,  \frac{dI}{dt} = \frac{dI}{dS} \cdot \frac{dS}{dt} , and so  \frac{dI}{dS} = \frac{\frac{dI}{dt}}{\frac{dS}{dt}} .  With  I not zero, this means

 \frac{dI}{dS} = \frac{1.25 \times 10^{-8} S I - 0.2 I}{-1.25 \times 10^{-8} S I} = -1 + \frac{16,000,000}{S} .  The partial is zero at  S_* = 16,000,000 , and has initial conditions  S_0 = 19,998,400 and  I_0 = 1600 (twenty million total).

The value  S_* is called the threshold value, and it is less than the initial condition  S_0 \approx 20,000,000 .  This suggests an epidemic in fact occurs.

Luckily,  \frac{dI}{dS} can be solved in closed form, as

 I = -S + 1.6 \times 10^{7} ln(S) + C .

The value of  C is of course determined by the initial conditions, as  C = I_0 + S_0 - 1.6 \times 10^7 ln(S_0) \approx -2.489 \times 10^8 . Then 

 I = -S + 1.6 \times 10^7 ln (S) - 2.489 \times 10^8 .  

With this in mind, the maximum number of infecteds at a time occurs at  S_* = 16,000,000 and is

 I_* \approx 500,000 ,

or about half a million people, equivalent to about 2.5% of Mexico City's population.

If there is enough interest, I may calculate the time dynamics (how long the epidemic lasts, etc.) with numerical methods (as by Euler's method), unfortunately by hand since access to fast computers and cool software is limited to me at present.

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UPDATE May 5, 2009.

So it appears that the foundational numbers above were vastly overstated (since Mexico hadn't confirmed the particular strains of the alleged infecteds due to under-equipment): from the number of actual infecteds to the actual number of deaths related to the illness.  It now appears that the progression of the swine flu is a lot slower, and also that our derived coefficients are vastly different than originally thought.  Still, a happy exercise using the SIR equations.  I may yet post a new derivation that reflects reality more truly.