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Epidemic Theory: Herd Immunity


Epidemics that strike without warning, killing and incapacitating people indiscriminately, are dramatic and terrifying natural phenomena, equaled only by floods, earthquakes, and fires in the devastation they can cause, and often exceeding them in the horror and fear they evoke. Ancient priests and physicians seized on any supernatural or natural explanation for such epidemics. They blamed the wrath of a vengeful god, evil spirits, or a convenient scapegoatwitches, the Jews, passing strangers. By the time of the Renaissance, people were blaming the climate and weather, with or without the conjunction of astrological signs. The miasma theory of disease was concordant with this view. The birth of bacteriology, the discovery of infectious pathogens, and the ascendancy of the germ theory of disease led to more rational analysis of the observed facts and development of more logical explanations. From the public health perspective, it is as important to discover what leads to the decline and disappearance of epidemics as to understand why and how they begin and continue.

William Farr (18071883) was the first to discern mathematical principles governing the behavior of epidemics. William Hamer, Ronald Ross, and other public health specialists in the early twentieth century developed refined mathematical models, factoring into their equations the variables involved in determining the interactions of disease agents, human hosts, and environmental conditions. Ross's models showed the interaction of mosquitoes, malaria parasites, and humans under varying conditions. Hamer modeled common infectious fevers of childhood. Modern concepts of epidemic theory evolved from these beginnings.


At its simplest, epidemic theory considers three variables: agent, host, and environment. Each of these has many components, howeverhost-agent interactions vary greatly, and variations in environmental conditions influence the interactions in innumerable ways. Epidemic theory is therefore inherently extremely complex, involving advanced stochastic mathematics. Epidemic theory has been verified by empirical observations, and by experimental epidemiology, in which infectious pathogens are introduced into colonies of mice or rats and the effects (disease and death outcomes) are observed. This enables epidemiologists to construct simple mathematical rules about the behavior of agents and hosts, while observations in the field provide data on variations in environmental conditions. Epidemic theory has also been used to study cancer, where the agent is a carcinogen, the host undergoes cellular and molecular (immunological) reactions, and the environment exposes the host to many influences, including other carcinogens. Social and behavioral scientists have applied epidemic theory to the study of the propagation and transmission of ideas, political beliefs, rumors, and behavioral epidemics. The HIV/AIDS (human immunodeficiency virus/acquired immunodeficiency syndrome) epidemic, new challenges in controlling tropical infectious diseases, and emerging infectious diseases from the 1980s onward have all aroused renewed interest in epidemic theory.

The Agent. Infectious pathogens vary in size and biological makeup from protein particles (prions) and ultramicroscopic viruses to multicellular tapeworms many meters long. They are spread by direct contact; person-to-person contact; droplet spread; through personal articles, clothing, utensils, and other belongings (known collectively as fomites); by way of a common vehicle such as water, food, milk, or contaminated air; by insect and other vectors; from animals that are their natural hosts, and from the inanimate environment. Survival of the agent is crucialif it cannot survive, it cannot invade and infect new hosts, and the epidemic ends. Many agents can flourish in some natural habitat without humans, many have alternate hosts, while a few are absolutely dependent on their human hosts for survival and propagation. Moreover, agents require a susceptible host. The probability that an infectious agent will encounter a susceptible host while the agent remains viable is a critically important variable in epidemic theory. In constructing mathematical models of epidemics, all possible variations in all these aspects of the agent's behavior in relation to that of the host and the environment must be taken into account.

The Environment. Some agents, and some insect vectors that carry infectious agents, can survive and/or transmit infection only within a narrow temperature range. Some agents coexist with other living things; for example, Vibrio cholerae flourishes in a symbiotic relationship with certain plankton species. The bacillus of tuberculosis thrives in dusty dark corners of crowded dwellings. Tetanus spores can survive almost indefinitely in soil. Staphylococcus and many other pathogens live on human skin. The variations are limitless. For any given pathogenic organism the range of tolerable environmental conditions may be wide or narrow. Any epidemic model of a specific disease must allow for these variations of the causative organism.

The Host. When an infectious agent invades a host, defensive immune responses are invoked to protect the host from harm. Immunity can also be conferred passivelyby maternal to infant transmission of antibodies across the placenta or in maternal milkor by vaccination or immunization. Immunity may be temporary, long-lasting, even permanent. One has to consider both individual hosts and the population as a whole, known in this context as the "herd."

Herd Immunity. The probability of an infectious agent encountering a susceptible host in which the agent can survive, propagate the infection, and sustain an epidemic depends on the proportion of susceptible hosts in the herd, or population. When an infectious agent is introduced into a population that has never previously encountered it, all are susceptible. As the epidemic passes through successive hosts, leaving them immune, progressively higher proportions of the population become immune. When a sufficiently high proportion of the population becomes immune to the infectious agent, the epidemic subsides and eventually ceases. The proportion required to reduce susceptibility to a level where an epidemic cannot be sustained depends on many variations in the properties of the herd, the environment, and the agent. A common cold virus introduced into a virgin herd infects everyone and confers transient immunity on everyone after recovery. If the herd is large, say 100,000 or more, common cold viruses may continue to circulate long enough for immunity to wear off, and the same individuals can be reinfected. Infection with the measles virus confers lifelong immunity. An epidemic begins only when there is a sufficiently large susceptible population. In the early twentieth century, industrial nations with high birth rates saw this happen about every other year.

Empirical observations confirm epidemic theory, showing that the probability of a diphtheria epidemic is reduced to near the vanishing point when 50 to 60 percent of the population have been rendered resistant either by previous infection or immunization. When a population has a level of lifelong immunity to a certain disease such that an epidemic of that disease cannot occur, the population is said to have herd immunity. Mathematical models can be constructed for many common and some rare infectious diseases, factoring in all the possible known variables to calculate the numbers and proportions required to achieve herd immunity. These are very useful for planning and evaluating control strategies.

In epidemics spread by person-to-person contact, a simple mathematical equation, the principle of mass action, expresses the incidence of new cases in relation to the number of current cases, the number remaining susceptible, and the proportion of total possible contacts between infectious cases and susceptible individuals that lead to infection. With common-source epidemics (e.g., waterborne or food-borne), and with vector-borne epidemics, the number of variable factors is much greater and the mathematics correspondingly more complex.

John M. Last

(see also: Epidemics; Epidemiologic Surveillance; Epidemiology; Farr, William; Pathogenic Organisms; Theories of Health and Illness )


Bailey, N. T. J. (1957). The Mathematical Theory of Epidemics. London: Griffin.

Fine, P. E. M. (1993). "Herd Immunity; History, Theory, Practice." Epidemiologic Reviews 15:265302.

Greenwood, M. (1935). Epidemics and Crowd Diseases; An Introduction to the Study of Epidemiology. London: Williams and Norgate.

Hamer, W. (1928). Epidemiology Old and New. London: Kegan Paul.

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