SEIR Latent Transition

Project exits from the SEIR exposed compartment under exponentially distributed latency with no new arrivals.

Description

Calculate expected exits and remaining population for an SEIR exposed compartment with exponential latency and no new arrivals.

In an SEIR model, susceptible, exposed, infectious, and removed compartments separate stages of an infection process. Exposed means infected but not yet infectious in this model. The latent period is the delay to infectiousness; it is different from an incubation period, which ends at symptom onset.^[1]

A simple exposed compartment assumes an exponentially distributed latent duration. Its reciprocal mean is a constant exit rate per day, and some members leave sooner or later than the mean. Starting with an exposed cohort and adding no new arrivals gives an exponential decline in the number still exposed.

SEIR Latent Transition returns that exit rate, expected exits during the supplied time step, and the remaining exposed population. Outputs are deterministic expectations and may be fractional even when the starting population is an integer.

When to project one exposed cohort

Use the tool to examine the exposed-to-infectious transition in a model step where the starting cohort and mean latency are supplied. It can help compare an exact exponential transition with a simple linear step approximation.

Calculate expected progression over a specified number of days.
Check that exits and remaining population account for the starting cohort.
Explore a mean latent duration without assuming every person exits at that exact time.

How to enter the cohort and calculate latent exits

Enter a nonnegative Exposed population, allowing a fractional model state.
Enter a positive Latent period days value as the mean exponential latency.
Enter nonnegative Step days for the transition interval.
Run the calculation and read Expected latent exits and Remaining exposed alongside the per-day rate.

σ = 1/latentPeriodDays; remainingExposed = E0 × exp(−σ × stepDays); expectedLatentExits = E0 − remainingExposed

The implementation uses the exponential expression directly rather than the Euler approximation E0 × σ × stepDays.

json

{
  "input": {
    "exposedPopulation": 100,
    "latentPeriodDays": 5,
    "stepDays": 1
  },
  "output": {
    "latentExitRatePerDay": 0.2,
    "expectedLatentExits": 18.12692469220182,
    "remainingExposed": 81.87307530779819
  }
}

Boundary	Result
Step days = 0	Zero exits and the full starting cohort remains.
Exposed population = 0	Both population outputs are zero; the rate is still calculated.
Latent period ≤ 0 or negative population/time	Input error.

Limitations of an isolated exponential transition

No new infections enter the exposed compartment during the step. A full SEIR update requires other flows and their timing.
Exponential latency assumes a constant exit hazard and does not represent a fixed delay or another latency distribution.
Expected exits are not integer realizations, individual transition times, or an uncertainty interval.
The tool does not evolve susceptible, infectious, or removed compartments, model symptoms, or estimate the latent period from data.

Alternative or Complementary timing and epidemic calculations

Keep population-model latency distinct from laboratory viral growth timing. An eclipse/latent timeline calculates differences between measured assay events, while an SIR final-size root estimates cumulative outcomes under another model representation.

References

Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection — Journal of Mathematical Biology; M. G. Roberts and J. A. P. Heesterbeek (2007)

Similar or alternative tools

Eclipse Latent Period Timeline
Calculate laboratory eclipse and release delays from supplied event hours, rather than an epidemiological exposed compartment.
SIR Final Size Root
Solve a fixed-parameter epidemic final-size relation without projecting individual latent exits.