1.3- Previous Epidemiological Models and Their Limitations
The first application of mathematical modelling to the spread of infectious disease was carried
out by Daniel Bernoulli in 1760
In order to apply this vast catalog of mathematical techniques to the study of computer virus spread, we view a single computing system as an individual in a population of similar individuals. Following the usual epidemiological approach, we neglect the details of infection inside a single system and consider an individual to be in one of a small number of discrete states, e.g. infected, uninfected, immune, etc. One might object to such a simplistic classification because some types of computer viruses, such as those that infect a large class of executable files in a system, can cause a system to become ``more'' infected over time -- thereby increasing the rate at which infection can be transmitted from that system. However, the time scale on which the internal infection occurs is generally much shorter than that on which the infection spreads to other systems, so such a simplification is quite reasonable. A further simplification which is characteristic of epidemiology is to abstract the details of viral transmission into a probability per unit time that a particular infected individual will infect a particular uninfected individual. Likewise, we abstract the details of detection and removal of a virus into a probability per unit time for an infected individual to be ``cured''. One could in principle derive the infection rates from the known details of the transmission process and the pattern of program sharing. If this information is unavailable (as it was for Bernoulli), it is often possible to simply measure the rates or infer them by fitting the observed course of an epidemic to a model. Most current epidemiological models are homogeneous, in the sense that an infected individual is equally likely to infect any of the susceptible individuals. Taken literally, this means that a man sneezing in Chicago is as likely to infect someone in New Delhi as he is someone else in Chicago. This approximation turns out to be adequate for diseases such as influenza, in which the disease can be transmitted via casual contact. However, its validity is generally conceded to be questionable for diseases in which each individual has a limited number of potentially infectious contacts. Program sharing is far from homogeneous, as one can readily establish by a bit of introspection. Most individuals exchange the majority of their programs with just a few other individuals, and never have any contact with the vast majority of the world's population. Another aspect of program sharing which must be taken into account in models is the fact that it can be strongly asymmetric. For example, the rate at which a retailer ships software greatly exceeds the rate at which a customer sends software to the retailer. Such asymmetry is occasionally important in the biological realm as well, particularly in the case of sexually transmitted diseases.
Recognized deficiencies of the assumption of homogeneous, symmetric interactions
have encouraged a variety of attempts to incorporate heterogeneity and asymmetry into biological
models. The spatial model is one method that has been used to account for local, symmetric
interactions
|
.
Although his work predated
the identification of the agent responsible for the transmission of smallpox by a century, he
formulated and solved a differential equation describing the dynamics of the infection which
is still of value in our day. The development of mathematical epidemiology was stalled by a
lack of understanding of the mechanism of infectious spread until the beginning of this
century
.
McKendrick developed the first
stochastic theory in 1926
,
and in the 1930's Kermack and McKendrick established the
extremely important threshold theorem
.
The model presented in this paper is general
enough to encompass both of these approaches, the original homogeneous model, and a variety of
other heterogeneous interaction models, some of which we shall explore in this paper.