Deriving first-moment equations

A moment equation follows from stochastic event changes and rates; it is not obtained by averaging a deterministic equation informally.

A linear infection–removal process

Let \(I(t)\) change through

Event	Change	Rate in state \(i\)
New infectious individual	+1	\(\lambda i\)
Removal	−1	\(\mu i\)

This early-outbreak branching model ignores susceptible depletion so that the rates are linear.

Conditional expected change

Given \(I(t)=i\), over a short interval \(h\),

\[ \mathbb E[\Delta I\mid I(t)=i] = (+1)\lambda i h+(-1)\mu i h+o(h). \]

Divide by \(h\) and let \(h\) tend to zero:

\[ \frac{d}{dt}\mathbb E[I(t)] = \mathbb E[(\lambda-\mu)I(t)]. \]

Use linearity of expectation

\[ \frac{dm_1}{dt}=(\lambda-\mu)m_1, \qquad m_1(t)=\mathbb E[I(t)]. \]

Because the rate is linear in \(I\), the right-hand side contains only the first moment. The equation is closed.

Solution and interpretation

\[m_1(t)=I_0e^{(\lambda-\mu)t}.\]

If \(\lambda>\mu\), the expected infectious population grows; if \(\lambda<\mu\), it declines. Positive expected growth does not rule out extinction in individual stochastic paths.

Interactive Python laboratory

Interactive PythonFirst moment versus simulations

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

def branching_path_on_grid(rng, I0, birth, death, grid):
    I = I0
    time = 0.0
    event_times = [0.0]
    values = [I]

while I > 0 and time < grid[-1]:
        total_rate = (birth + death) * I
        wait = rng.exponential(1 / total_rate)
        if time + wait > grid[-1]:
            break
        time += wait
        I += 1 if rng.random() < birth / (birth + death) else -1
        event_times.append(time)
        values.append(I)

positions = np.searchsorted(
        event_times, grid, side="right"
    ) - 1
    return np.array(values)[positions]

I0 = 5
birth = 0.18
death = 0.12
grid = np.linspace(0, 20, 81)
M = 3000
rng = np.random.default_rng(250)

paths = np.array([
    branching_path_on_grid(rng, I0, birth, death, grid)
    for _ in range(M)
])

monte_carlo_mean = paths.mean(axis=0)
moment_solution = I0 * np.exp((birth - death) * grid)

selected = [0, 5, 10, 15, 20]
indices = [np.argmin(np.abs(grid - t)) for t in selected]
comparison = pd.DataFrame({
    "time": selected,
    "moment equation": moment_solution[indices],
    "simulation mean": monte_carlo_mean[indices]
})

print(comparison.round(4).to_string(index=False))

plt.figure(figsize=(9.5, 5.1))
plt.plot(grid, moment_solution, color="#174c34",
         linewidth=2.2, label="First-moment equation")
plt.plot(grid, monte_carlo_mean, color="#b42318",
         linestyle="--", label="Monte Carlo mean")
plt.xlabel("Time")
plt.ylabel("Expected infectious population")
plt.title("Derived first moment agrees with simulation")
plt.legend()
plt.grid(alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

What was—and was not—assumed

The equation describes the exact first moment of this linear birth–death model. It is not a deterministic approximation to one trajectory. Its derivation used the stochastic transition rates and expectation. Nonlinear epidemic rates will introduce higher moments; lesson 07 explains that problem.

What this lesson adds

You can now derive a first-moment differential equation from event changes and rates, use conditional expectation and linearity, solve the closed linear equation and compare it with Monte Carlo evidence.