SIR Final Size Root

Solve the closed homogeneous SIR final-size equation with explicitly nonzero initial infectious fraction.

Description

Numerically solve the closed homogeneous SIR final-size equation for final susceptibility and infections occurring after the initial state.

Final epidemic size describes cumulative infection after an outbreak has run its course, rather than when its peak occurs. In a closed homogeneous SIR model, susceptible people move into infection and eventually leave the infectious compartment. A final-size relation links the remaining susceptible fraction to the initial state and reproduction number.^[1]

Initially infectious people contribute to future transmission but are already infected at the starting point. The additional infection fraction therefore equals the decline in susceptibility, excluding those initial infectious people. Any remaining initial population fraction is implicitly outside the susceptible and infectious compartments.

SIR Final Size Root solves the relation numerically. It requires a positive initial infectious fraction and returns two population fractions, without an epidemic timeline.

When to estimate cumulative infections in an SIR scenario

Use the tool for a fixed-parameter scenario with explicit initial compartments. It can help compare cumulative outcomes while distinguishing additional infections from the total that includes the initial infectious fraction.

Calculate final susceptibility under the supplied homogeneous SIR assumptions.
Compare changes in R0 or the initial susceptible fraction.
Check the accounting of initially infected people in a final-size estimate.

How to supply initial compartments and solve the final-size relation

Enter a positive Reproduction number for the SIR model.
Enter Initial susceptible fraction from 0 to 1 and Initial infectious fraction greater than 0 and at most 1.
Ensure the initial compartment fractions sum to at most one apart from a tiny numerical tolerance.
Run the tool and read Final susceptible fraction and New infection fraction as fractions of the original population.

s∞ = s0 × exp[−R0 × (s0 + i0 − s∞)]; newInfectionFraction = s0 − s∞

s0 and i0 are the initial fractions. The exponent includes initial infectious people plus subsequent infections.

The implementation brackets the root between zero and s0. It performs 150 bisection iterations on f(x) = x − s0 exp[−R0(s0+i0−x)], moving the upper bound when f(x) is positive and the lower bound otherwise. It returns the midpoint of the final bracket.

json

{
  "input": {
    "reproductionNumber": 1.1552453009332422,
    "initialSusceptibleFraction": 0.8,
    "initialInfectiousFraction": 0.2
  },
  "output": {
    "finalSusceptibleFraction": 0.4,
    "newInfectionFraction": 0.4
  }
}

Zero initial susceptibility is accepted and gives both outputs zero. Zero initial infectious fraction is rejected. The compartment-sum check allows a tolerance of 1e−12; invalid fractions or a larger excess produce an input error.

Limitations of a deterministic final-size root

The calculation assumes a closed homogeneous SIR scenario with fixed reproduction parameters and no return to susceptibility.
No stochastic fade-out, importation, time-varying intervention, contact structure, or uncertainty distribution is modeled.
New infection fraction excludes people already infectious initially. It should not be labeled total cumulative infection without that distinction.
A fixed bisection count does not provide a reported residual or convergence diagnostic, and the result gives no peak date or duration.

Alternative or Complementary threshold calculations

An instantaneous peak screen answers whether the infectious compartment would currently grow or decline. A herd-coverage calculation applies different assumptions about vaccine efficacy and coverage. Neither substitutes for the initial-condition accounting in this final-size relation.

References

Generality of the final size formula for an epidemic of a newly invading infectious disease — Bulletin of Mathematical Biology; J. Ma and D. J. D. Earn (2006)

Similar or alternative tools

SIR Infectious Peak Screen
Compare current susceptibility with the SIR zero-growth boundary without calculating final size.
Efficacy Adjusted Herd Coverage
Calculate a model immunity threshold and coverage requirement from R0 and a supplied efficacy fraction.