Calculator
Choose a method, enter values, then compute λ.
Example data table
Use this example to sanity-check expected scales.
| Scenario | Method | Inputs | Computed λ | Risk over one unit |
|---|---|---|---|---|
| Community outbreak | β · I / N | β=0.30, I=25, N=1000 | 0.00750000 | 0.00747293 |
| High-density contact | β · I | β=0.00040, I=25 | 0.01000000 | 0.00995017 |
| Incidence density | cases/(S·Δt) | cases=12, S=850, Δt=1 | 0.01411765 | 0.01401846 |
| Cumulative risk | -ln(1-p)/T | p=0.10, T=1 | 0.10536052 | 0.10000000 |
Formulas used
- Frequency-dependent transmission: λ = β · I / N
- Density-dependent transmission: λ = β · I
- Incidence-based hazard approximation: λ ≈ cases / (S · Δt)
- From cumulative risk: p = 1 − exp(−λT) ⇒ λ = −ln(1−p) / T
- Risk over one unit: Risk(1) = 1 − exp(−λ)
Interpretation: λ is the instantaneous infection pressure experienced by a susceptible individual under the chosen assumptions.
How to use this calculator
- Select the estimation method that fits your study design.
- Enter the required inputs for that method.
- Click Compute Force of Infection to see results.
- Review λ, risk, and survival for one time unit.
- Use CSV/PDF buttons to export the results table.
FAQs
1) What does “force of infection” mean?
It is the instantaneous per-susceptible rate of becoming infected. In many models, it acts like a hazard and links contact patterns to infection risk over time.
2) Which method should I choose?
Use β·I/N for frequency-dependent contact, often used in human mixing. Use β·I for density-dependent settings. Use incidence-based when you trust observed cases and susceptible time at risk.
3) What unit is λ reported in?
λ inherits your time unit. If Δt is in days, λ is per day. For the cumulative method, λ is per T unit. Keep your inputs consistent for interpretable outputs.
4) How is risk computed from λ?
Assuming λ is constant over the interval, risk over one unit is 1 − exp(−λ). This is the standard hazard-to-risk conversion used in survival and epidemic models.
5) Can λ change over time?
Yes. Real epidemics have time-varying λ driven by behavior, immunity, seasonality, and interventions. This tool estimates a single interval value; repeat calculations across intervals to study changes.
6) What are common pitfalls?
Mixing time units, using inconsistent population definitions, and double-counting cases are common issues. Also ensure I and N reflect the same population and time window as β.
7) How should I interpret λ versus incidence?
Incidence counts events in a period; λ is a rate per susceptible at an instant. When hazards are roughly constant, incidence density approximates λ, especially for short intervals.
8) Does the uncertainty option give an exact confidence interval?
No. It uses a simple normal approximation for incidence-based inputs. For formal inference, use exact Poisson or likelihood-based methods with clear assumptions and report your modeling choices.