We're in an influenza epidemic, according to Center for Disease Control director Dr. Julie Gerberding. In an interview with CNN, she gave a few facts.

1. On average, 10-20 percent of Americans can expect to get the flu in any given year. The vast majority get well.
2. Flu victims should "self-triage" before seeking care, to avoid overwhelming medical facilities. Breathing trouble, fever that lasts more than four days, a blue tinge to the skin, lethargy or irritability, and seizures are signs that medical attention is needed.
3. Flu symptoms that resolve and return more forcefully a few days later could indicate a complicating bacterial infection, which is a reason to see a clinician.
4. People over 65, pregnant women or those with an underlying medical condition such as diabetes or heart disease should see a doctor early and not wait.

Item 2 suggests that the CDC is well versed with mathematical modeling.

Various epidemic-based movies and TV shows hint at mathematical modeling. In these, someone asks "What will happen if the virus gets out?" while standing in front of a big map. After a dramatic pause, lots of red dots start appearing on the map. Parameters -- "What happens if the airports are tightly controlled?" -- are never discussed in these Hollywood offerings. They are probably unaware of the data available at Gridded Population of the World. They may not have even read an issue of Morbidity and Mortality Weekly Report.

Figure 1. Imaginary epidemic, week 1. Outbreak near Dallas TX.

The most devastating epidemic of modern times was the 1918 influenza pandemic. At a time when the world population was 1.8 billion, the flu incapacitated 1 billion people and killed 25 million people in the space of 8 weeks. Most of the people who died were aged 20-34. For comparison, 55 million people died during the 6 years (Sep 1939 - Sep 1945) of World War II. During World War I (Jun 1914 - Nov 1918), 8.5 million people died. In fact, at a particularly crucial moment in WWI, German Chancellor Ludendorff caught the flu. Desiring to wake up fresh, he took a sleeping tonic, but accidentally overdosed. While he lay in a coma, Austria and Turkey surrendered. The war ended shortly thereafter. Ironically, President Woodrow Wilson and General John Pershing also caught the flu. More details are available in a documentary: The American Experience: Influenza 1918.

Figure 2. Imaginary epidemic, week 2. Virus spread by airports.

More recently, chickens have been stricken. In April 1983, a variety of the influenza virus killed 20 million chickens in Pennsylvania. The virus proved lethal because of a change in one animo acid. On 21 Dec 2003, Reuters reported the flu-related death of a half million chickens in South Korea. Mild flu epidemics happen every year. Whenever a large population of non-immunes exists, epidemics happen. In 1875, The King of the Fiji Islands returned from a diplomatic trip, infected with measles. Out of a population of 150 thousand, 40 thousand died of measles. Here are a few words on the 1918 flu:

The total mortality of the 1918 epidemic was 0.5 percent of the population. In a few places the mortality was much higher. In Samoa 25 percent of the people died. The Eskimos in Alaska suffered terribly; some villages were wiped out and others lost their entire adult population. In Nome, 176 out of 300 Eskimos died. The disease caused havoc in India where and estimated five million people died. [Beveridge]

The spread of flu has a strong mathematical basis. Serfling's model ( (Y=average mortality + trend + 52 week cycle + 26 week cycle + Error) or (Y = a + b₁t + b₂cos(p t / 26) + b₃sin(p t / 26) + b₄cos(p t / 26) + b₅cos(p t / 26) + E ) ), for instance, is used to estimate levels of influenza. The Kermack-McKendrick model (Susceptibles --rIS--> Infectives --qI--> Quarantines) can also be used to model the spread of an epidemic. The pictures I give here involve a more complicated model, which I am not satisfied with. I wanted to see how population density affects the spread, with factors like airports, roads, and hospital overload thrown in.

Figure 3. Imaginary epidemic, week 3. Continued outbreaks.

Let me discuss hospital overload. When a population is suddenly stricken, uninfected people must work much harder to maintain the normal quality of available services. In particular, medical services can be overwhelmed to uselessness, prompting a sharp increase in the mortality rate. At the peak of a 1918-type model, 80 million Americans would all be sick at once, and five percent of them would need some sort of medical attention. In the model, the effectiveness of hospitals during the worst week has a large influence on the severity of the epidemic. Hence, it makes sense that the CDC has started recommending self-triage in the current epidemic - just in case.

Figure 4. Imaginary epidemic, week 4.

The importance of controlling airports was dramatically demonstrated during the containment of the SARS coronavirus. In the pictures here, airports greatly exacerbate the spread of a modeled virus. During the SARS outbreak, such things as quarantine inspectors and health alert cards diligently contained the spread of the virus.

Figure 5. Imaginary epidemic, week 5.

The model illustrated here can be improved in many ways. That's part of the fun of mathematical modeling - improving the model and adding parameters.

1. In the model, only regions with a population density over 80,000 have an airport. That's a bad assumption.
2. Road traffic isn't accurately depicted.
3. Removals currently aren't done at all. Diagrams with red and blue areas would be an improvement.
4. CDC and WHO have shown their mettle, and would likely implement many effective countermeasures if something like 1918 got started. CDC intervention could be modeled.
5. The spread of this imaginary virus differs according to population density. This could be done better.
6. Immunization shots and antiviral drugs are apparently not plentiful enough to withstand a pandemic. An aspect could be added that shows what happens when drug supplies run out.
7. In a massive epidemic, a primary killer is not knowing how to care for the sick [Vaughn]. A humanity-driven media, which gave useful medical information to the public, might be modeled. Alternately, a ratings-driven media could be modeled.

If you make improvements to this model on a population grid, let me know. As an example of something to avoid with mathematical modeling, consider a public announcement made by 1918's Surgeon General Vaughan "If the epidemic continues its mathematical rate of acceleration, civilization could easily disappear from the face of the earth within a few weeks." That doesn't seem helpful.

References:

Beveridge, W. Influenza, The Last Great Plague. Watson Publishing Intl, 1977.

Billings, M. The 1918 Influenza Pandemic. http://www.stanford.edu/group/virus/uda/

Callahan, J. "The Spread of a Contagious Illness" http://www.math.smith.edu/~callahan/ili/pde.html

Chowder, K and Kenner, R. The American Experience: Influenza 1918. http://www.pbs.org/wgbh/amex/influenza/index.html

CDC.Morbidity and Mortality Weekly Report. http://www.cdc.gov/mmwr/index.html

CNN. "CDC Head Calls Flu Outbreak Epidemic." http://www.cnn.com/2003/HEALTH/12/19/sprj.flu03.epidemic/index.html

Guðnadóttir, S. The Spanish Influenza 1918. http://www.vortex.is/~sigrun/

Gridded Population of the World, http://sedac.ciesin.columbia.edu/plue/gpw/index.html?main.html&2

Lynch, E. "The Flu of 1918." The Pennsylvania Gazette. November 1998. http://www.upenn.edu/gazette/1198/lynch.html

Reuters. "South Korean Bird Flu Outbreak Spreads"http://www.reuters.co.uk/newsArticle.jhtml;?type=scienceNews&storyID=4027307

Vaughn, W. Influenza, An Epidemilogic Study. Baltimore,1921.

Mathematica Code:

The population-based epidemic model used here is available at http://library.wolfram.com/infocenter/MathSource/4906/.

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Comments are welcome. Please send comments to Ed Pegg Jr. at ed@mathpuzzle.com.

Ed Pegg Jr. is the webmaster for mathpuzzle.com. He works at Wolfram Research, Inc. as the administrator of the Mathematica Information Center.