# Epidemiological Parameters

**About 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, βSIFor S+E+I+R = P, the rate of exposure (or theincidence) 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 rateξrepresents the rate at which infected people die. It isnotto be confused with the netcase 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 howIchanges from disease deaths:I'(t) = -ξI(t)Textbook algebra gives us:I(t) = c*eWe now that^{(-ξt)}I(0) = 100so c=100. We also know thatI(10)=50(half the population has died). We get:50 = 100*e^{(-ξ*10)}0.5 = e^{(-ξ*10)}ln(0.5) = -ξ*10ξ = 0.0693

In general, the infectious mortality rate ξ can be compute fromCFRandγby solving the equation:CFR = eor^{(-ξ/γ)}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 R_{o} 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* , R_{o}, (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. R_{o} is a dimensionless or unit-less parameter, and it is '*not a rate*.

Epidemiologists want to compute or estimate R_{o} because;

when R_{o}< 1 the infection will die out in the long run. But if and when R_{o}> 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 R_{o} can easily be calculated but it *is a function of several epidemiological parameters*. Conceptually (and by dimensional analysis) R_{o} is simply the ratio of the transmission rate (e.g. per day) divided by the recovery rate (e.g., per day).

In general R_{o} 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,

R_{o}= β/γ

with a non-zero mortality rate μ

R_{o}= β/[γ + μ]

and with a net disease death rate ξ

R_{o}= β/[γ + μ + ξ]

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