In an epidemic model, we consider a single population but consider every individual in the population as being classified according to their status with respect to a disease. In the classic model of a disease involving immunity, there are three categories in the population: individuals who are healthy but susceptible to become sick, those who are sick and therefore infectious, and those who once were sick but have recovered and retain immunity. These three variables are traditionally assigned the variables \(S\) (susceptible), \(I\) (infected), and \(R\) (recovered or removed).

The standard flow diagram for the SIR model tells the story of how an infection spreads through a population. Susceptible individuals become sick after interactions with infected individuals. When there are more infected individuals, it becomes more likely that a person will become infected. Once infected, individuals stay sick for some duration after which they recover. After they recover, they have developed an immunity for the illness and will not become sick again. In addition to this progression of the disease, individuals may die from any stage with some probability. New births in the population generally will be healthy unless our illness has vertical transmission of the disease.

A possible flow diagram for the SIR model is illustrated in the figure below. The three classes are represented by rectangles labeled by the appropriate state variable. Each rate (flow arrow) is labeled according to the process that it represents. A rate will generally always depend on the class that it leaves. To emphasize that the infection rate also depends on the number of infected individuals, we draw a link (dashed line) between the infected class and the infection flow arrow.

Simplifications of the model might come from eliminating some of the flow rates. For example, when studying a disease that progresses quickly relative to the population (such as influenza), we might ignore slow scale processes like birth and death. For a disease that instills permanent immunity, we might eliminate the loss of immunity rate. If a disease never gives immunity, then we might eliminate the recovered class altogether and have recovery send individuals directly back to the susceptible class. Some diseases require additional classes.

Ecosystems involve many different interacting species. Interactions might include competition, mutualism, parasitism and predation. In competition, two or more species rely on common resources such that the presence of one species reduces the availability of those resources for the other species. In mutualism, the presence of two interacting species together benefits each species relative to the populations living in isolation. In parasitism, one species takes resources from an individual of another species and may or may not lead to the death of the host but likely reduces the host's fitness. In predation, the predator kills and consumes individuals of the prey species.

The flow diagram for interactions between two species generally look the same, with the difference being whether interactions are beneficial or inhibitory. One way that the diagram can represent this is by using a different type of arrow for inhibitory effects. In the diagrams below, a regular arrow \(\rightarrow\) represents a beneficial interaction while a segment ending in a perpendicular cap \(\dashv\) represents an inhibitory interaction.

From a modeling perspective, we should note that the illustrated competition, mutualism and parasitism model diagrams are structurally equivalent. Only the nature of the interactions (beneficial or inhibitory) are different. Consequently, it may be possible to analyze related models for all three systems by considering different model parameters that might capture these differences.

Even within a single population, we might model the dynamics by a system that accounts for the number of individuals that are of different ages or stages of development. In the simplest cases, individuals steadily progress through a sequence of classifications (such as ages). The only way to leave the progression is to die. Often, only certain categories of individuals will be capable of reproduction (e.g., mature adults). The flow rates between classes generally reflect how long an individual stays in a class. Fast rates correspond to short dwell times. In some models, we may wish to allow individuals to revert to an earlier stage or to alternate between different mature states (e.g., fertile vs infertile).