# What is the relation between force of infection and infection rate?

There are different models to define the SIRD/SEIRD models of infection, for instance in one manual:

dS/dt = S - (α/N)SI


where S is the number of susceptible people, I the number of infected people, N the total population, and α the rate of infection. In another manual, the same compartment is defined by:

dS/dt = S - λS


where λ is the force of infection. Since the two formulae point to the same object (the decrease of susceptible people, dS/dt), it should be possible to merge them:

S - (α/N)SI = S - λS
(α/N)SI = λS
αI/N = λ


Is this equation correct?

Can either α or λ be calculated from epidemic curves?

For instance, it is also reported that

λ = βI


where β is the number of effective contacts per unit time. Could β be derived from epidemic curves?

Also, it is possible to derive the growth of infection Λ from an epidemic curve by deriving the slope of the regression line passing to the early, exponential cases of infection. How is Λ related to λ?

Is Λ the same as α?

• Is your 1st equation presented missing an 'S' on the last term? Jul 6, 2020 at 20:39
• Note: d[(ln(S)] / dt = 1- Lamba, which is a likely clue to answer your question by examining the ln of the exponential growth. Jul 6, 2020 at 20:59
• Sorry, there was indeed a missing S; corrected. Is Lambda Λ or λ? How do I relate d(ln(S))/dt to α? Thank you Jul 7, 2020 at 11:53

## 1 Answer

(This is growing too long to post as a comment...)

Well, if λ=βI, as you mention, then your two equations are exactly the same, with β=α/N, both α and β being constants. (Beware that N, the total population, includes the number of deaths, e.g. N = S + E + I + R + D, so it is a constant.).

The second equation's presentation is ambiguous, though, as it presents under a constant name (λ) something that is definitely not constant, as it depends on I. I suggest using only the first one, which is a lot clearer.

To estimate α or β, you have to use modelization and calibration.

The growth of infection, in the S(E)IRD model, is, in good logic, the derivative of I, dI/dt, whose value is (α/N)SI in SIRD, and εE in SEIRD (ε being a constant, the multiplicative inverse of the incubation period).

dI/dt is definitely not a constant either. Maybe you were looking for the initial growth of infection ?