Fred Brauer
January 31, 2005
 

Almost since the beginning of recorded history epidemics have developed suddenly in populations and then ended just as suddenly, leaving part of the population untouched. The Book of Exodus describes the plagues that Moses brought down upon Egypt , and there are many other biblical descriptions of epidemic outbreaks. The Black Death (thought to be bubonic plague) spread from Asia through Europe in several waves beginning in 1346, causing the death of one-third of the population of Europe between 1346 and 1350 and recurring regularly in Europe for more than 300 years, notably as the Great Plague in London of 1665-1666. Recurring invasions of cholera killed millions in India in the nineteenth century. The influenza epidemic of 1918-19 killed at least 20 million people overall, more than half a million in the United States . More recently, the Ebola virus, the SARS epidemic of 2002-3 have caused worldwide concern. Even more recently several strains of avian flu have forced the killing of millions of birds and caused worries about spread to humans.

An early triumph of mathematical epidemiology was the formulation in 1927 of a simple model by a public health physician, W. O. Kermack, and a biochemist, A. G. McKendrick. This model predicted that when an infective entered a population from outside, disease could develop suddenly and then disappear just as suddenly without infecting the entire community, behaviour similar to that observed in countless epidemics, namely that diseases develop suddenly and then disappear just as suddenly without infecting the entire community.

Kermack and McKendrick divided the population being studied into three classes with S denoting the number of individuals susceptible to the disease, that is, not (yet) infected, I the number of infected individuals, assumed infectious and able to spread the disease by contact with susceptibles, and R the number of individuals infected and then removed from the possibility of being re-infected or of spreading infection. Removal is carried out through isolation from the rest of the population, immunization against infection, recovery from the disease with immunity against reinfection, or through death caused by the disease. These characterizations of removed members are quite different from an epidemiological perspective and of course also from a human point of view, but are equivalent from a modelling point of view that takes into account only the state of an individual with respect to the disease.

The Kermack-McKendrick epidemic model makes very simple assumptions about the rates of disease transmission and removal. One of the assumptions is that the disease is transmitted from one individual of a population to another by direct contact. Thus it is not applicable to diseases which are transmitted by a vector , that is, diseases which are transmitted back and forth between two populations such as mosquitoes and birds as in West Nile virus. However, the ideas that go into the formulation of the Kermack-McKendrick model are also useful for the formulation of more complicated epidemic models.

The model contains only two parameters (the values of which are to be determined from observed data) and could be applied to many diseases transmitted by direct contact. While a more detailed model might be a better description of a specific disease, it would require more parameters. Since data are often incomplete and inaccurate because of under-reporting and mis-diagnosis at the beginning of an epidemic, a simple model may give better predictions.

When these assumptions are translated into mathematical statements of the transition rates between classes the result is a pair of equations, called differential equations, for the rates of change of the numbers of susceptible and infective members of the population. These equations can be analyzed by relatively simple mathematical methods, and the analysis predicts that an epidemic will pass through the population without infecting the entire population. It also reveals that there is a threshold phenomenon. If the average number of new infections caused by a single infective entering a susceptible population, called the basic reproduction number , is less than 1 no epidemic develops while if the basic reproduction number exceeds 1 there will be an epidemic.

If there is no vaccine or treatment available for a disease there have been essentially only two methods for trying to control the epidemic. One method has been isolation of diagnosed infectives both for treatment and to prevent them from passing on the infection. The second method has been quarantine of people who have had contact with infectives, to monitor whether they become infective themselves so that they can be isolated quickly, and also to reduce the risk that they will pass on the infection. The rates of isolation and quarantine may be varied, depending on decisions about the amount of effort to invest in these strategies. If and when a vaccine is developed, models might indicate whether it is more urgent to concentrate on vaccination or isolation and what vaccination strategy may be expected to be most effective.

Soon after the beginning of the SARS epidemic of 2002-3 scientists began looking at the formulation of models to describe this epidemic. Usually, this work was carried out by groups of scientists, including mathematicians, statisticians, epidemiologists, and microbiologists. For example, a network was set up by MITACS (Mathematics of Information Technology and Complex Systems) including scientists from Health Canada and universities across Canada . Some early observations were that the SARS epidemic was an example of a general class of models, not a completely separate situation, and that it was possible to formulate more general and more inclusive models of Kermack-McKendrick type. For this reason, the MITACS group, which was originally set up to study SARS, became a group studying the modelling of infectious diseases in general. Models have been developed which incorporate the additional features described above and have been used to try to answer such questions as whether control of an epidemic should concentrate on isolation of diagnosed infectives or quarantine of contacts of infectives, or a combination of both. For example, it appears that the speed with which measures are implemented are of critical importance in reducing the spread of the epidemic. It also appears that in an epidemic which can be controlled by isolation the additional gains from quarantine are slight.

A real epidemic differs considerably from an idealized model. In addition to the aspects neglected to keep a model sufficiently simple to be amenable to mathematical analysis, the initial stages of an epidemic are not at all as pictured in a model of Kermack-McKendrick type. A crucial hypothesis in the Kermack-McKendrick model is that the mixing of the population is homogeneous. In real epidemics, it usually turns out that most infectives do not transmit the infection to others but a few “superspreaders” may pass on the infection to many others. The relatively new field of network science, which has exploded since 1990, building on the mathematics of random graphs from the 1950’s and some sociological and psychological studies of the 1960’s, gives a way to take into account the contact patterns among people and give better insights into the development of an epidemic. This aspect of disease transmission modelling is being studied and promises to improve our understanding of how epidemics spread and how better to control an epidemic.

It is rarely possible to compare possible control strategies during an actual epidemic. For this reason, mathematical modelling of epidemics is a promising tool for comparison of possible strategies. Of course, it is important to remember that a model is only a model and will not be an accurate description of all aspects of an epidemic. However, examination of models may allow tentative conclusions even before the nature of a new disease is understood. Although mathematics has not yet cured any diseases (except possibly math anxiety), it may help in controlling future epidemics.
 

(Note: Another article on this topic by Fred Brauer appeared in Pi in the Sky in their November 2004 issue entitled What does Mathematics have to do with SARS at http://www.pims.math.ca/pi/)