What SEIR adds
The SIR model assumes an infected person can transmit immediately. SEIR introduces an exposed stage (E): infection has occurred, but transmission has not started. The progression is S → E → I → R: susceptible, exposed, infectious and recovered with permanent immunity.
Latency and incubation are different. The latent period ends when infectiousness begins; the incubation period ends when symptoms appear. Transmission can precede symptoms. Here I represents people who can transmit, whether symptomatic or not.
The population is closed and S + E + I + R = N is conserved. Initially S₀ = N − I₀, E(0) = R(0) = 0 and I(0) = I₀. The SIR reference starts from the same state to isolate the effect of latency. With I₀ = 0, no outbreak appears spontaneously.
Equations and mean durations
All three rates have units day⁻¹. β combines contacts and transmission probability, σ is the progression rate to infectiousness, and γ is the recovery rate. They are rates, not daily probabilities. The mean latent period is 1/σ and, with γ > 0, the mean infectious period is 1/γ.
Constant rates imply exponentially distributed residence times: individuals may spend different durations in each stage, rather than exactly the mean. This basic model is described in chapter 2 of Keeling and Rohani, Modeling Infectious Diseases in Humans and Animals.
σ ranges from 0.01 to 2 day⁻¹, corresponding to a mean latent period of 100 to 0.5 days. With σ = 0 nobody would ever leave E, so it is outside this model. With γ = 0, the model becomes the SEI limit without recovery; with β = γ = 0 and E(0) = 0, all compartments are constant.
What adding latency does not change
With γ > 0, R₀ does not depend on σ: all exposed individuals eventually enter I and have the same mean infectious duration. Re(0) and the herd immunity threshold HIT are unchanged too. The threshold assumes homogeneous mixing and perfect protection; it does not predict final outbreak size.
Adding the E and I equations gives d(E+I)/dt = β·S·I/N − γ·I. With β > 0, the fractions s = S/N, e = E/N and i = I/N satisfy the invariant:
Thus E + I is the same function of S as I in the SIR reference, traversed at a different speed. With an initial infection and Re(0) > 1, E + I peaks at S* = N/R₀ and equals the peak of I in SIR. S* does not mark the peak of I in SEIR. If Re(0) ≤ 1, both global maxima equal I₀.
For β, γ > 0 and I₀ > 0, final size is also the same: the final susceptible fraction satisfies ln(s∞/s₀) = −R₀(1 − s∞). These equalities apply to the complete outbreak; observed maxima and cumulative infections at a given day can differ.
What changes: growth, doubling and peak
Early on, with S approximately constant at S₀, the dominant exponential rate r satisfies:
For a growing outbreak, r is lower than β·s₀ − γ, the SIR rate without latency. Doubling takes longer. In the examples, the infectious peak is lower and later, while the peak of E + I keeps the same height. Increasing σ shortens latency and brings the trajectory closer to SIR.
r is not the instantaneous initial slope of I. Since E(0) = 0, dI/dt(0) = −γ·I₀: I initially falls when recovery and initial infection are present, even if E + I is growing. Exponential behavior emerges after that transient and before S changes substantially. Near the threshold, a rebound may never occur or may stay below I₀; the global maximum is then the initial value.
The intrinsic mean generation interval, from an individual’s infection to the infections they cause in the model, is Tg = 1/σ + 1/γ. It is 1/γ in SIR. This is neither the interval between symptoms nor an estimate of intervals observed throughout the outbreak. Without transmission or seeding, theoretical rates do not imply an observable growth phase.
Estimating reproduction from growth
With γ > 0 and s₀ > 0, this identity shows that the same positive r requires greater reproduction when latency is included. Using the SIR relation, Re = 1 + r/γ, underestimates it compared with this SEIR model. This differs from fixing β and γ in the simulator: in that comparison R₀ stays the same and r changes.
Wallinga and Lipsitch (2007) explain the dependence on the generation interval distribution. Wearing, Rohani and Keeling (2005) show why omitting latency or imposing unsuitable distributions can bias estimates and conclusions about control.
Observation time and the E + I < 1 criterion
N and I₀ are integer counts; the dynamics are continuous and may produce fractional individuals as a mathematical approximation. E + I < 1 marks the first downward crossing below one individual, not stochastic extinction or the clinical end of an epidemic.
A short observation window may end before the peaks. The tool then reports the observed maximum with a warning; that value may still be I₀ during the initial decline. Cumulative infections equal E + I + R at the end of observation and include I₀. More digits allow inspection of the calculation but do not increase epidemiological certainty.
Examples and parameter sources
The β/γ pairs are the illustrative SIR examples, retained for comparison. R₀ references provide context; they do not make these parameters a fit to a real population. Latencies come from specific studies:
- Original COVID-19: Xin and colleagues (2022) estimate a mean latent period of 5.5 days (95% CI: 5.1–5.9); σ = 0.18 gives 5.556 days. They use viral detection by PCR as a proxy for the onset of shedding, not a direct measurement of every transmission or a universal value for later variants.
- Influenza: Cori and colleagues (2012) estimate a 1.6-day latent period from viral excretion profiles. Their estimated infectious duration is 1 day; this example retains the illustrative 5 days from SIR, so it does not reproduce the paper’s complete fit. Carrat and colleagues (2008) describe a sharp rise in shedding between 0.5 and 1 day after inoculation; this is a different measure from the fitted mean latent period.
- Measles: Conlan and colleagues (2010) report 8 days of latency and 5 days of infectiousness from household studies. This example takes the latency but retains the illustrative 10 infectious days from SIR. Keeping R₀ = 15 with 5 days would require β = 3, outside this tool’s domain.
Assumptions and limits
The model assumes homogeneous mixing, constant rates, a closed population and permanent immunity. It has no age structure, births, deaths, reinfection, hospitalization, interventions or data fitting. Exponentially distributed residence times are a simplification: other distributions change the dynamics. This tool supports exploration of the model; it does not produce epidemiological forecasts.