Formula used
The model tracks four compartments: susceptible (S), exposed (E), infectious (I), and recovered/removed (R). The governing equations use rates per day:
dE/dt = β(t)·S·I/N − σ·E
dI/dt = σ·E − γ·I
dR/dt = γ·I + v·S
Here β(t) can include an intervention window and optional seasonality. The peak time is the simulated time where I(t) reaches its maximum.
How to use this calculator
- Enter population size and initial compartment values (E0, I0, R0).
- Set rates: β for transmission, σ for incubation, and γ for recovery.
- Choose duration and a time step. Use RK4 for better peak accuracy.
- Optionally enable an intervention window, seasonality, or vaccination flow.
- Click Calculate Peak Time to view results above the form.
- Use the export buttons to download CSV or PDF for reporting.
Example data table
| Parameter | Value | Meaning |
|---|---|---|
| β | 0.35 1/day | Transmission strength per contact-day |
| σ | 0.1923 1/day | Incubation progression (≈ 5.2 days) |
| γ | 0.1429 1/day | Recovery rate (≈ 7.0 days) |
| E0 | 50 | Initial exposed individuals |
| I0 | 25 | Initial infectious individuals |
| Δt | 0.1 days | Numerical resolution for integration |
SEIR peak timing insights
1) Why peak time matters
Peak time indicates when the infectious compartment reaches its maximum, often aligning with the greatest demand on testing, isolation capacity, and clinical resources. In many practical scenarios, shifting the peak by even 7–14 days can reduce congestion and improve response efficiency.
2) Typical parameter scales
Rates are entered per day. Incubation is commonly 3–7 days, so σ often falls near 0.14–0.33. Infectious duration frequently ranges 4–10 days, giving γ near 0.10–0.25. Transmission β varies widely with behavior and mixing; values between 0.15 and 0.60 are often explored for scenario analysis.
3) Linking β and R estimates
The calculator reports an approximate basic reproduction estimate R0 ≈ β/γ and an initial effective reproduction Re ≈ (β/γ)·(S0/N). When Re is above 1, infections tend to grow; as S decreases, Re drops and the infectious peak becomes inevitable.
4) Incubation delays the visible peak
Because exposed individuals must progress to infectious at rate σ, peaks in E and I can be separated in time. Lower σ (longer incubation) typically delays the infectious peak and can broaden the curve, even if β and γ are unchanged.
5) Recovery sets the downturn speed
Higher γ shortens infectious duration and usually lowers peak I and advances the decline. For fixed β, increasing γ reduces β/γ, often pulling Re closer to 1 and moving the peak earlier with a smaller maximum.
6) Interventions reshape the curve
Use the intervention window to reduce β by a chosen percentage. A 30–60% reduction applied before rapid growth can delay the peak and reduce its height. If applied late, the peak time may shift only slightly because S has already decreased.
7) Seasonality and vaccination effects
Seasonality modulates β(t) with a sinusoid, which can create secondary waves when conditions improve. Vaccination in this tool is modeled as a constant flow from S to R, steadily lowering susceptible availability and reducing Re, often delaying or suppressing the peak.
8) Numerical quality and step size
Peak timing depends on integration accuracy. RK4 generally provides smoother trajectories than Euler at the same step size. For daily-scale rates, Δt between 0.05 and 0.25 days often balances speed and stability; smaller steps are recommended when strong interventions switch on sharply.
FAQs
1) What does “peak infectious time” mean?
It is the simulated day when I(t) reaches its maximum value. This is the point where the model expects the largest number of currently infectious individuals.
2) Why can the “new exposures/day” peak differ from the I(t) peak?
New exposures depend on S(t) and I(t) at that moment, while I(t) is influenced by incubation and recovery. The exposure peak can occur earlier than the infectious peak.
3) How do I choose σ and γ from periods?
Use σ ≈ 1/(incubation days) and γ ≈ 1/(infectious days). For example, 5.2-day incubation gives σ≈0.192, and 7-day infectious period gives γ≈0.143.
4) What step size is safe?
For most settings, Δt=0.1 days works well with RK4. If you use sharp interventions or very high β, reduce Δt to 0.05 or 0.02 days to stabilize peak timing.
5) What does the intervention window change?
It reduces β(t) by a percentage between the start and end times, representing distancing, masking, or mobility limits. Earlier and stronger reductions usually delay the peak and lower it.
6) How is vaccination modeled here?
Vaccination is a constant-rate flow from S to R (S→R), reducing susceptibility over time. It does not explicitly model dose schedules, waning immunity, or breakthrough infections.
7) Can I use this for forecasting real outbreaks?
Use it for scenario exploration and sensitivity, not precise forecasts. Real outcomes depend on heterogeneous contact patterns, reporting delays, behavior changes, and parameter uncertainty that a simple SEIR model may not capture.