# SIR Model

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

### R0

R0, or basic reproductive number, is the average number of people who get infected from one single case of a certain disease in "the zero state", which means nobody else in the group is immunized or infected. In Epidemiology, we use Re, or effective reproduction number, to denote the average number of people who get infected from one single case in a "given state" (for example, at a given time t).

But usually, just a single R0 could give us a picture about how infectious the disease is. Suppose one person in a party is the first case of a disease and its R0 is 2, then there will be about two people in the party got this disease. When they go back home, each of them will spread the disease to their two family members and so on.

$$\text{In theory, } \begin{cases} R_0<1, \text{it will finally die out without control} \\ R_0=1, \text{it will grow gradually} \\ R_0>1, \text{there will be an outbreak}\end{cases}$$

Use slider below to change R0 in the graph and observe the difference.

 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