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8/15/2005
The Incidence of Epidemics, by Victor Niederhoffer:

Everybody has epidemics on his mind these days as we worry about infections introduced by bad people and we endeavor to eliminate all viruses, colds, and childhood diseases. The basic model for studying epidemics was developed by Kermack and McKendrick in 1927 and it's well covered in such books as Models in Biology by D. Brown. An excellent discussion of the slight extensions that have been made in the last 80 years is covered in The Mathematics of Disease by Matt Keeling. I also like Compartmental Models in Epidemiology in Wikipedia.

These models start with dividing the population into three compartments. Susceptibles, Infected and Recovered (immune). The susceptibles (S) get infected and that reduces the number of S for the next period. The Infected gets increased by the S that caught it and reduced by the number that recover. The key to the model is the relation between the infection rate, b and the recovery rate ( usually the Greek gamma) For example if there are 50 S and 50 I and 10% of the susceptibles get the disease then there will be 5 new infectives. But if the 50 old infectives take 10 days to gain recovery, at a constant rate, then 5 of the old infectives will no longer have it. The net result will be 0 new infectives, and the disease will be stable. The key variable that determines growth than is the relation of b to gamma. This is sometimes combined in the reproductive rate (R) which is the number of infectives produced by a single infective in a population. For high values of R the disease spreads like wild fire but quickly dies out with a proportion of susceptibles remaining at 1/r.

This simple model leads like most of the chaos and catastrophe and predator prey models to fantastic mathematical properties as the assumptions change about the relation of b to gamma and the initial population size changes. In most models, there are oscillations which get smaller and smaller, with the level of infections eventually reaching a constant. Various thresholds must be crossed for the epidemic to spread and there is a tipping point. The key assumption in all the models is how the rate of infections varies with the number of susceptibles and infectives in the population. The usual procedure is to assume that the number of infections varies linearly and directly with the product of S and I as this is used in physics for gas molecules. This leads to an approach to a maximum when the S and I are equal, but with the change in infectives reaching 0 at that point and increasing as the proportion of s to i approaches 0 or 1 according to the normal formulas for the product of s, and t-s which can be worked out starting with derivatives or the difference between two squares.

As to how this relates to markets, we're exposed to epidemics all the time. A rumor spreads about coming Fed or corporate action. A brokerage house infects some of its customers with an earnings estimate. A new system is sold to the masses , or an existing system is believed to work and is used by practitioners to establish a position until the number of new infectives dies out.

I like to work out some of these outbreaks of infections in markets with the following table

```        Period
1         2       3        4       5    ....           10
S
I
R
T
```

By varying the assumptions about the recovery rate, i.e. is the hazard rate increasing or decreasing as time elapses, and giving the recovered a chance to reenter the susceptible pool ( hope springs eternal for followers of systems), or those who adopt the latest nostra concerning bull or bear markets, growth versus value etc, many insights can be gained into the spread of these diseases.

Jim Sogi Responds

But The Chair wrote last week, "There is no evidence that it is possible to come up with a retrospective depiction of turning points that is inconsistent with randomness. Thus, ideas regarding sluggishness, selectivity, increasing highs and lows must be taken as untested and unproven."

The market opened the year at 1211.92 SPX after a 120 point rally in December, dropped 90 some points over the next 4 months, and in the 4 months since April is up 108 points. Can the tipping point be predicted non randomly?

Here is the famous story of John Snow who used spatial representations and statistical analysis to cure a great cholera epidemic in London.

"In 1854, when Cholera struck England once again, Snow was able to legitimate his argument that Cholera was spread through contaminated food or water. Snow, in investigating the epidemic, began plotting the location of deaths related to Cholera (see illustration). At the time, London was supplied its water by two water companies. One of these companies pulled its water out of the Thames River upstream of the main city while the second pulled its water from the river downstream from the city. A higher concentration of Cholera was found in the region of town supplied by the water company that drew its water form the downstream location. Water from this source could have been contaminated by the city's sewage. Furthermore, he found that in one particular location near the intersection of Cambridge and Broad Street, up to 500 deaths from Cholera occurred within 10 days.

After the panic-stricken officials followed Snow's advice to remove the handle of the Broad Street Pump that supplied the water to this neighborhood, the epidemic was contained. Through mapping the locations of deaths related to Cholera, Snow was able to pinpoint one of the major sources of causation of the disease and support his argument relating to the spread of Cholera.

Snow's classic study offers one of the most convincing arguments of the value of understanding and resolving a social problem through the use of spatial analysis."

John Snow: The London Cholera Epidemic of 1854 By Scott Crosier

Spacial analysis and statistical use of available data might describe and predict the tipping points. See here

Despite sage advice by my betters not to waste my time, I hypothesize that when bulls gets infected with bearitis they no longer trade at the ask, but trade at the bid, and at the tipping point, the market goes down. Using spatial analysis of the spread of bearitis, it should be possible to describe the tipping point.

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