# SIR Model

This chapter introduces SIR model, a model for spread of disease in Epidemiology, and some related terms.

### R0

When talking about the epidemic, prevalence will be one of the most common term in daily life. It is pretty easy to calculate the prevalence, which is the proportion of a particular population that founded to be affected by a medical condition at a given time. Even prevalence could be used to describe an epidemic, it might also describe some non-infectious disease such as heart disease, or even risk factors such as smoking.

$$\text{Prevalence} = \frac{\text{number with the disease}}{\text{population size}}$$

Incidence (or Incidence Risk), correspondingly, is the proportion of new cases in the population that is at risk of a medical condition (not affected yet) at a given time. Generally prevalence indicates how widespread of the disease, while incidence indicates the spreading in a given time period.

$$\text{Incidence} = \frac{\text{number of new cases of disease}}{\text{population size at risk}}$$

Click and drag hexagons to select new cases then press "Submit" to see the Prevalence and Incidence. (To make life easier, we ignore the possibilities of death and recovery here.)

 Prevalence Incidence 0 0

### Relationship Among Duration, Prevalence and Incidence Rate

Duration is the time period that a case stays in the prevalent case pool. Because duration will vary from person to person, we usually use average duration for a certain disease in a population instead.

Incidence Rate is the average incidence in a given period of time.

$$\text{Incidence Rate} = \frac{\text{incidence}}{\text{time period}}$$

Prevalence(P) depends on the Incidence Rate(IR) and Duration(D):

$$\frac{\text{P}}{\text{1-P}} \approx IR \times D$$

If P is small enough then 1-P is close to 1, we have:

$$P \approx IR \times D$$

You could think about a pool with inflow and outflow. If incidence rate is the rate of inflow and duration is the time period for water staying in the pool, then the water level will indicate the prevalence.

Suppose the total population is 100 and the visualization indicates final prevalence when incidence rate and duration rate is constant. You could drag bars below to adjust incidence rate and duration rate.

### Case Fatality Rate

Besides of the frequency of affected, death could also be a factor when study the epidemic. For example, Case Fatality Rate (CFR) is the proportion of people with the condition who die in a given period of time. Similar to Incidence, which is the percentage of affected in the at-risk pool, Case fatality rate is the percentage of death in the affected pool.

Intuitively, case fatality rate tells us how fatal a disease is. For example, the well-known Ebola virus has a case fatality rate from 83-90%. In the daily life, people usually describe the fatality of a given disease as "death rate", but in epidemiology death rate (or mortality rate) usually means the percentage of the people who die from a certain disease in a fixed population group such as nationality and race.

$$\text{CFR} = \frac{\text{Fatal Cases}}{\text{Detected Cases}}$$

You could click the figure on the right to control its status and check the current CFR in the population. Purple means detected while red means fatal.

Current Fatal Cases: 0

Current Detected Cases: 0

Current CFR: N/A