# Epidemiological Parameters

### Common Variables and their Units

Using an SEIR model as an example lets discuss some common epidemiological parameters and what they mean. The figure shows the compartments for S=Susceptible, E=Exposed (but not yet infectious), I=Infectious (Shedding Virus), R=Recovered. The arrows show the transitions or people moving between the compartments in a specified time interval. Please also see the page Introduction to Compartment Models for more information (but please read this page first).

### What they mean

The table below shows the epidemiological parameters for the SEIR model shown above. Note that the UNITS or all or the parameters are inverse time.

Figure 1: SEIR Compartment Model

Each of the parameters are rate constants. The transmission rate β has a special role in that it appears inside the mass action term. This interaction term includes the product SI.

```If the compartments are all normalize so that the total population S+E+I+R = 1
then Susceptible individuals become exposed at a rate, βSI
For S+E+I+R = P, the rate of exposure (or the incidence) becomes βSI/P
```

### Common Questions

All of the parameters defined above (and as used in STEM) are rate constants so their values depend on the user specified time period.

#### Example 1: Mortality Rate

```The mortality rate μ represents the rate at which individuals die even with no disease.
For a constant population the mortality rate = the birth rate (or μ=μ*).
For humans, if the average life span is 50 years, and if the time period is 1 day, then
μ = (1/50 years) * (1 year/365.25 days) = (1/18262.5 days)
or
μ = 5.476 x 10-5[days-1].
```

#### Example 2: Infectious Mortality Rate

```The infectious mortality rate ξ represents the rate at which infected people die. It is not to be confused with
the net case fatality rate (CFR).

Consider, for example, a virulent disease that has a CFR of 0.5 (50% of people infected people die), a Period of Infection of
10 days and initially 100 people in the infected state. We can write down basic ordinary differential equation describing
how I changes from disease deaths:
I'(t) = -ξI(t)
Textbook algebra gives us:
I(t) = c*e(-ξt)
We now that I(0) = 100 so c=100. We also know that I(10)=50 (half the population has died). We get:
50 = 100*e(-ξ*10)
0.5 = e(-ξ*10)
ln(0.5) = -ξ*10
ξ = 0.0693
```
```In general, the infectious mortality rate ξ can be compute from CFR and γ by solving the equation:
CFR = e(-ξ/γ) or
ξ = -ln(CFR)*γ
```

#### The Difference between the Basic Reproductive Number and the Transmission Rate

A common question or confusion concerns the difference between the Basic Reproductive Number Ro and the Transmission Rate β. As discussed above, the transmission rate β, is the rate at which infectious cases cause secondary or new cases in a population, P, with S susceptible individuals. It is a rate constant and has units of inverse time (e.g., [days-1]).

The basic reproduction number , Ro, (sometimes called basic reproductive ratio) of an infection can be thought of as the number of cases one single case generates on average over the course of its infectious period, in a totaly susceptible (or otherwise uninfected) population. Ro is a dimensionless or unit-less parameter, and it is 'not a rate.

Epidemiologists want to compute or estimate Ro because;

```when
Ro < 1
the infection will die out in the long run. But if
and when
Ro > 1
the infection will spread in a population and can cause an epidemic.
```

The transmission rate β is an important control variable in an epidemiological model. The Basic Reproductive Number Ro can easily be calculated but it is a function of several epidemiological parameters. Conceptually (and by dimensional analysis) Ro is simply the ratio of the transmission rate (e.g. per day) divided by the recovery rate (e.g., per day).

In general Ro is the ratio of the transmission rate to the total rate at which individuals leave the infectious (I) compartment. So, if there is zero mortality,

```Ro = β/γ
```

with a non-zero mortality rate μ

```Ro = β/[γ + μ]
```

and with a net disease death rate ξ

```Ro = β/[γ + μ + ξ]
```

etc. Notice that in all three cases above the numerator and denominator have the same units so Ro is dimensionless. Obviously γ , μ , ξ and β must all be provided or specified using the same time units (e.g., [days-1].