SIR Infectious Peak Screen
Screen the instantaneous infectious trend under a closed homogeneous SIR model; this is not calendar peak timing.
Description
Compare the susceptible fraction with the SIR growth boundary and report the instantaneous infectious trend under a homogeneous model.
The susceptible–infectious–removed (SIR) model groups a population by whether people can acquire infection, currently transmit it, or have left the infectious process. In its closed, homogeneous form, transmission and removal rates determine a reproduction number R0. As susceptibility falls, the effective reproduction number is R0 multiplied by the susceptible fraction.1
For a positive infectious compartment, the model predicts growth when that product exceeds one and decline when it is below one. Equality is the instantaneous zero-growth boundary. This threshold concerns the number currently infectious, rather than daily reported cases or a calendar date.
SIR Infectious Peak Screen returns the critical susceptible fraction, effective reproduction number, and a trend label. It evaluates a supplied state without simulating an epidemic or estimating R0 from observations.
When to check an SIR growth boundary
Use this screen while examining a homogeneous SIR scenario with an independently chosen reproduction number. It helps interpret a current susceptible fraction and distinguish a growth boundary from a peak-time forecast.
- Check which side of the model threshold a supplied state occupies.
- Compare susceptibility scenarios while holding R0 fixed.
- Identify when the calculated threshold lies outside the possible fraction range.
How to enter the state and evaluate its trend
- Enter Susceptible fraction from 0 to 1, such as 0.4 for 40%.
- Enter a positive Reproduction number representing the model R0; do not enter an already susceptibility-adjusted Rt.
- Run the screen and read the effective reproduction number and Infectious trend label.
criticalSusceptibleFraction = 1/R0; effectiveReproductionNumber = susceptibleFraction × R0 The implementation compares the product directly with one.
| Effective reproduction number | Infectious trend |
|---|---|
| Greater than 1 | growing |
| Less than 1 | declining |
| Exactly 1 | peak-boundary |
{
"input": {
"susceptibleFraction": 0.4,
"reproductionNumber": 2
},
"output": {
"criticalSusceptibleFraction": 0.5,
"effectiveReproductionNumber": 0.8,
"infectiousTrend": "declining"
}
}When R0 is below one, the critical fraction exceeds one and all allowed susceptible fractions give a declining label. Fractions outside [0,1], nonpositive R0, missing fields, and nonfinite inputs produce errors.
Limitations of an instantaneous SIR screen
- Closed population, homogeneous mixing, and the supplied model parameters are assumed. Changing contacts, importations, heterogeneous immunity, and reinfection are not represented.
- The tool does not accept the infectious compartment size. If it is zero, the actual SIR derivative is zero even though the tool still labels the supplied susceptibility product.
- No calendar peak, peak height, incidence curve, or uncertainty interval is calculated.
- The equality label uses exact numeric comparison, without a tolerance around one.
Alternative or Complementary epidemic-model calculations
Use a final-size calculation for cumulative infections under the corresponding fixed-parameter SIR assumptions. Estimating a reproduction number from growth requires a generation-interval model; it is a separate inference step rather than an output of this threshold screen.
References
-
The Mathematics of Infectious Diseases — SIAM Review; H. W. Hethcote (2000)
Similar or alternative tools
- SIR Final Size Root
Solve for final susceptibility and additional infection fraction with explicit initial susceptible and infectious fractions.
- Growth R0 Gamma Generation
Translate a growth rate into a reproduction number using a supplied gamma generation-interval model.