Epidemiology is the subject that studies the patterns of health and illness and associated factors at the population level. The word “epidemiology” is derived from the Greek terms epi, which means “upon,” demos, which means “ people,” and logos, which means “study.” This etymology implies that the subject of epidemiology applies only to human populations. The role of father of epidemiology is often assigned to the Greek physician Hippocrates (460–377 b.c.e.), who described the connection between disease and environment [83]. The term “epidemiology” appears to have first been used to describe the study of epidemics in 1802 by the Spanish physician de Villalba in Epidemiologia Espanola [30]. Until the twentieth century, epidemiological studies were mostly concerned with infectious diseases. Nowadays, the leading causes of deaths worldwide are diseases such as stroke and coronary heart disease [132], positioning diseases that do not transmit from one person to another as a central concern of epidemiology. Among infectious diseases, those that dominate worldwide as a cause of death include lower respiratory infections (such as pneumonia) and HIV. In this book, we will be concerned with mathematical modeling of infectious diseases.

An infectious disease is a clinically evident illness resulting from the presence of a pathogenic microbial agent. The microbial agent causing the disease can be bacterial, viral, fungal, parasitic, or it can be toxic proteins, called prions. Infectious diseases caused by bacteria include tuberculosis and pneumonia; viral diseases include HIV and influenza; the most widespread fungal diseases are dermatomycoses; parasitic infections are caused by macroparasites such as protozoa, helminths, trematodes, and cestodes. An example of a prion-caused disease is Creutzfeldt–Jakob disease. Communicable diseases are infectious diseases that can be transmitted from one infectious person to another, directly or indirectly. Often, we do not make a distinction between infectious diseases and communicable diseases, since many of the infectious diseases are in fact communicable diseases. However, there are diseases that are infectious but not communicable. Tetanus is an example of such a disease. Transmittable diseases are infectious diseases that can be transmitted from one person to another through unnatural routes. For instance, Creutzfeldt–Jakob disease can be passed from one patient to another through surgical instruments or transplants. Nonetheless, the distinction between infectious diseases, communicable diseases, and transmittable diseases is subtle, and infectious diseases are often called communicable diseases or transmittable diseases because of their potential to be transmitted from one person to another.

Transmission of infectious diseases may occur through a variety of pathways. According to the means of transmission, infectious diseases are classified as follows:

For modeling purposes, we distinguish four types of transmission: direct, when the causative pathogen is transmitted from one person to another; vector-transmitted, when the causative agent is transmitted from a vector to a human; environmental transmission, when a human becomes infected through contact with a pathogen present in the environment; and vertical, when the pathogen is transmitted from mother to child at birth. Person-to-person and airborne diseases are usually modeled as directly transmitted when transmission occurs through contact between one person and another. What constitutes a “contact” sufficient for transmission in these diseases depends on the specific disease. In sexually transmitted diseases, sexual contact is necessary, while airborne diseases, which are often modeled as directly transmitted, require a certain degree of physical proximity without the necessity of touching. Modeling vector-borne transmission requires the inclusion of the dynamics of the vector in addition to the dynamics of the infected individuals. Environmentally transmitted diseases are typically modeled by separately modeling the dynamics of the virus in the environment and the transmission that occurs on contact between an individual and the free pathogen.

A pathogen reservoir is an ecological niche in which a pathogen lives and multiplies. Such a reservoir plays a significant role in the spread of the pathogen. According to their reservoir, the microbial agents are classified as human, animal, and environmental. Human pathogens circulate mostly among humans, and humans play a role in their transmission. Animal pathogens have vertebrate animals as a reservoir, and circulate primarily among animals. Epidemiologically, this is significant, because many such pathogens adapt to infect humans through animal-to-human transmission. Infections that spread from vertebrate animals to humans are called zoonoses. Environmental pathogens multiply primarily in the environment (typically water and soil) and spread from there to animal and human populations.

Many infectious diseases have more than one pathway of transmission. For instance, HIV is primarily transmitted through sexual contact, but it can also be transmitted by blood transfusion or needle sharing. Furthermore, HIV can be transmitted vertically at birth from an infected mother to her child. Avian influenza H5N1 is primarily transmitted through direct contact with infected poultry and rarely directly from human to human. However, significant evidence now exists that H5N1 can persist in the environment, and the environmental route of transmission is gaining more importance.

There are a number of concepts in epidemiology strictly related to infectious diseases. These concepts play an important role in the construction of mathematical models by adding various features to the model. Some of the most widely used concepts are listed below. Others will be introduced as new models are discussed.