Outbreak size, duration, peak size and peak time

Probability classifies whether an event occurred. Outcome analysis keeps the numerical result from every complete trajectory, revealing how large, long and intense epidemics can be.

Define each trajectory-level outcome

Final outbreak size

\[Z=N-S(T).\]

Total number ever infected, including people infectious initially.

Duration

\[T=\inf\{t:I(t)=0\}.\]

Time from the initial state until the final recovery.

Peak size

\[I_{\max}=\max_{0\le t\le T}I(t).\]

Largest number simultaneously infectious.

Peak time

\[T_{\mathrm{peak}}=\min\{t:I(t)=I_{\max}\}.\]

First time the maximum infectious count is reached.

Why definitions need small details

State whether initial infectious people are included in final size; this lesson includes them.
A maximum may be reached more than once; this lesson records its first occurrence.
Duration requires the full continuous-time clock and cannot be obtained from event types alone.
All simulations must finish at extinction before their final sizes are compared.

Mixtures can hide biological structure

With few initial infections, outcomes often split into early fade-outs and major epidemics. A single overall mean may fall between these groups and describe neither group well.

Report both when useful: unconditional summaries describe all possible starts; conditional summaries such as “given final size at least 20” describe the severity of trajectories classified as major outbreaks.

Interactive Python laboratory

Run 1,000 complete SIR epidemics. Each simulation returns exactly one row containing its four outcomes. The program then compares all runs with major-outbreak runs and explores dependence between size and peak.

Interactive PythonCTMC outbreak outcomes

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

def simulate_sir_outcomes(rng, N, I0, beta, gamma):
    S = N - I0
    I = I0
    R = 0
    time = 0.0

peak_I = I
    peak_time = 0.0

while I > 0:
        infection_rate = beta * S * I / N
        recovery_rate = gamma * I
        total_rate = infection_rate + recovery_rate

waiting_time = rng.exponential(1 / total_rate)
        time += waiting_time

if rng.random() < infection_rate / total_rate:
            S -= 1
            I += 1
        else:
            I -= 1
            R += 1

# Strict > records the first time the maximum is reached
        if I > peak_I:
            peak_I = I
            peak_time = time

return {
        "final size": N - S,
        "duration": time,
        "peak I": peak_I,
        "peak time": peak_time
    }

# Model and Monte Carlo settings
N = 200
I0 = 3
beta = 0.25
gamma = 0.10
number_of_runs = 1000
major_threshold = 20
seed = 88

rng = np.random.default_rng(seed)
rows = []

for run in range(number_of_runs):
    outcome = simulate_sir_outcomes(
        rng, N, I0, beta, gamma
    )
    outcome["run"] = run + 1
    rows.append(outcome)

outcomes = pd.DataFrame(rows)
outcomes["major outbreak"] = (
    outcomes["final size"] >= major_threshold
)

# Put the run number first
outcomes = outcomes[[
    "run", "final size", "duration",
    "peak I", "peak time", "major outbreak"
]]

major_outcomes = outcomes[outcomes["major outbreak"]]

summary = pd.DataFrame({
    "all runs": outcomes[
        ["final size", "duration", "peak I", "peak time"]
    ].median(),
    "major outbreaks": major_outcomes[
        ["final size", "duration", "peak I", "peak time"]
    ].median()
})
summary.index.name = "median outcome"

quantiles = outcomes[
    ["final size", "duration", "peak I", "peak time"]
].quantile([0.05, 0.50, 0.95])
quantiles.index = ["5th percentile", "median", "95th percentile"]

correlation = outcomes["final size"].corr(
    outcomes["peak I"]
)

print("First 10 trajectory outcomes")
print(outcomes.head(10).round(3).to_string(index=False))
print()
print("Unconditional outcome quantiles")
print(quantiles.round(3).to_string())
print()
print("Median: all runs versus major outbreaks")
print(summary.round(3).to_string())
print()
print(
    f"Major outbreaks: {len(major_outcomes)} "
    f"of {number_of_runs}"
)
print(
    "Correlation between final size and peak I: "
    f"{correlation:.3f}"
)

# Figure 1: distributions of all four outcomes
fig, axes = plt.subplots(2, 2, figsize=(10, 7.5))

columns = ["final size", "duration", "peak I", "peak time"]
titles = [
    "Final outbreak size",
    "Epidemic duration",
    "Peak infectious count",
    "Time of infectious peak"
]
xlabels = ["People", "Days", "People", "Days"]

for axis, column, title, xlabel in zip(
    axes.flat, columns, titles, xlabels
):
    axis.hist(outcomes[column], bins=25,
              color="#79ae90", edgecolor="white")
    axis.set_title(title)
    axis.set_xlabel(xlabel)
    axis.set_ylabel("Number of runs")
    axis.grid(axis="y", alpha=0.2)

fig.suptitle("Outcome distributions across complete CTMC epidemics")
fig.tight_layout()

# Figure 2: dependence between two outcomes
plt.figure(figsize=(8.2, 5.3))
colours = np.where(
    outcomes["major outbreak"], "#b42318", "#39815e"
)
plt.scatter(
    outcomes["final size"], outcomes["peak I"],
    c=colours, alpha=0.45, s=20
)
plt.axvline(major_threshold, color="black", linestyle="--",
            label=f"Major threshold = {major_threshold}")
plt.xlabel("Final outbreak size")
plt.ylabel("Peak infectious count")
plt.title("Large final sizes tend to accompany larger peaks")
plt.legend()
plt.grid(alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

Understand the outcome programming

Code	Purpose
`if I > peak_I`	Updates the peak only for a strictly larger value, preserving the first peak time.
`return {...}`	Returns one labelled outcome record from one trajectory.
`pd.DataFrame(rows)`	Creates a table with one row per simulated epidemic.
`.quantile([0.05, 0.50, 0.95])`	Reports the middle and tails without assuming a normal distribution.
`.corr(...)`	Measures linear association; it does not prove causation.

Interpret outcomes together

Final size measures total burden, while peak size measures maximum simultaneous prevalence and is more directly related to pressure on limited services. Duration measures how long transmission persists. Peak time affects how much preparation time may be available.

These summaries are related but not interchangeable. Two trajectories can have similar final sizes but different peaks or durations.

Avoid misleading summaries

Do not report only the mean when the distribution separates into fade-outs and large epidemics.
Do not condition on major outbreaks without saying so.
Do not treat a 5th–95th percentile range as a confidence interval for the mean.
Do not calculate duration from a simulation stopped at an arbitrary horizon.

What this lesson adds

You can now extract well-defined outcomes from each completed trajectory, construct an outcome table, report empirical quantiles, distinguish unconditional and conditional summaries, and examine dependence between epidemic burden and peak pressure. The final CTMC lesson concentrates on choosing and interpreting plots and tables correctly.