Physicist Amok

Consider some population in its environment. The environment has some set of properties, and each member of the population likewise as some set of properties. Epidemiology concerns itself with the following problem: given some environment and population, predict the evolution over time of the properties of some individual placed in that environment, usually with respect to some disease or chemical. The issue is to select an appropriate set of properties for environment and population, and to construct empirically verifiable relationships among the values of these properties.

But first, what do we mean by a property? Hatsopoulos and Keenan have articulated a clearer definition than I can hope to:

A primitive property of a body is specified by describing an operation or test to which the body can be subjected. The value of the property at any time is the result of the operation performed at that time. Not all operations can be used in defining primitive properties of a body — only those whose result is uniquely related to the condition of the body at a particular time. Specifically, an operation defines a primitive property at time if the values obtained by successively repeating the operation on the body, beginning at and extending over decreasing intervals of time previous to , approach a limit as the interval approaches zero. (Hatsopoulous and Keenan, Principles of General Thermodynamics, 1981, p.7)

Then they go on to define properties as all functions of primitive properties. When we are concerned with populations, we have to extend this a little bit. We can measure an individual’s eye color, but what is a population’s eye color? For populations the relevant primitive properties generally take the form of probability distributions, and our measurement becomes some kind of sampling algorithm or time averaging. For the abstract object, the population, such a thing is a primitive property, but our actual measurement is on individuals.

Almost all epidemiological models use the rate of infectious doses being delivered to an individual, which we’ll call the infectivity . In general this is a property. Consider a model for tuberculosis in a hospital. Tuberculosis is carried by small droplets which are expelled into the air when an actively tuberculous patient coughs. We can construct a device which gathers droplets which come into contact with some area of its surface, and then checks for the presence of the bacillus in the droplets (for instance, by chemically extracting the DNA and amplifying genes specific to TB). Then we install this device in the hospital’s air handling units and record for some period of time. We can calibrate the device by sending a known quantity of bacilli containing droplets at it in the laboratory. In this case is a well posed, primitive property.

Unfortunately, usually shows up in combination with another factor, , the probability that an individual with specified properties will contract the disease in question when exposed to an infectious dose. is far more difficult to measure (in an ethical way), and probably depends on the properties associated with individuals.

To make matters worse, in many epidemiological problems, mechanical measurements of the infectivity aren’t available, much less of . If we are dealing with a relatively homogeneous population, a group of medical students in Denmark, say, then we can estimate the combination from the rate at which they are infected.

If we assume that infectious doses arrive independently, then the fraction of the population infected after some time is given by a Poisson distribution with parameter . We can’t tell how many of these people received multiple doses, so the only reliable number we get is those who are still healthy. The probability of zero infectious doses actually causing disease in time is given by . So if we have healthy individuals out of total, then . is a constant, so after some rearrangement we get . This method is due to RG Ferguson.