About Epidemiological Parameters
Common Variables and their Units
Using an SEIR model as an example, let's 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
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
All of the parameters defined above (and as used in STEM) are rate constants so their values depend on the user specified time period.
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].
Infectious Mortality Rate (approximation)
The infectious mortality rate δ represents the rate at which infected people die. It is not to be confused with the net case fatality rate (CFR) which is the fraction of disease deaths PER case (not per day or unit time). CFR defines fraction of deaths over the entire period of infection
If the case fatality rate is very small (CFR << 1), and we defining the inverse of the period of infection as the recovery rate (γ) then we can simply approximate the infectious mortality rate as: δ ~ CFR * γ leaving other parameters the same. For example, suppose the case fatality rate for a new disease is 1% (so CFR = 0.01 [per case]). This means that 1% of people who get the new disease will die sometime during their illness. Suppose also that that the period of infection is 10 days (or the recovery rate γ = 0.1 days-1 ). Then the infectious mortality rate for this new disease is: δ ~ 0.01 * 0.1 = 0.001 per day.
Infectious Mortality Rate (more generally)
In general, for large mortality rates, one must also consider how the mortality rate affects the net period of infection. To understand the general solution, the figures below show the flow of individuals into and out of the infectious compartment as a "pipe model". For simplicity consider a steady state case where we always have a constant I ≡ 1. There is always then equal flow in and out of the compartment. In the figure below individuals in the I compartment leave by only one path (e.g., into an R compartment not shown). In this simple case the inverse of the period of infection is literally the "recovery rate". The flow out (and the flow in) is γ for I ≡ 1
Now let's look at a case where individuals may leave the Infectious compartment by two different paths (the flow is into two pipes). They may recover (flow γ*) or they may die from the disease (flow δ). If, for example, the rate of infectious mortality δ is not very small compared to γ, then clearly we need to make a correction to the recovery rate &gamma*. The steady state figure makes this clear. By definition we always have: γ ≡ 1/(Period of Infection)
so if we don't want the period of infection to change (i.e., if the flow out is still to add up to γ), then we need to adjust the recovery rate γ* A little algebra shows:
γ* = (1-CFR)γ
and the infectious mortality rate is:
δ = CFR γ
This satisfies the condition γ = δ + 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 δ
Ro = β/[γ + μ + δ] <equation 1>
etc. Notice that in all three cases above the numerator and denominator have the same units so Ro is dimensionless. Obviously γ , μ , δ and β must all be provided or specified using the same time units (e.g., [days-1].
As an example let's consider an outbreak of variola major smallpox. Suppose we had the following values for the epidemiological parameters.
- Period of infection = 20 days.
- This means γ = 0.05
- CFR = 1%. Then δ = 0.01*0.05 = 0.0005
- natural death rate (all other causes) μ = 1/50 years = 5.5x10-5 per day
- Transmission rate β = 0.3 per day
Then our denominator in equation 1 is γ + μ + δ = .05+.0005+.000055 = .050555 and we find Ro = β/[γ + μ + δ] = 0.3/.050555 = 5.9
of course given Ro (instead of β) one could instead compute transmission rate β per day by inverting equation 1.