Drift, diffusion and biological meaning

Every SDE step combines a predictable local tendency with a random fluctuation. These two contributions have different mathematical and biological roles.

The general SDE

\[dX(t)=\underbrace{f(X(t))dt}_{\text{drift change}}+\underbrace{g(X(t))dW(t)}_{\text{diffusion change}}.\]

The drift coefficient \(f\) gives the local expected rate of change. The diffusion coefficient \(g\) controls the size of random fluctuations; it is not itself the random number.

Biological derivation from infection and recovery

For an SIS infectious count, let infection and recovery event rates be \(a_1\) and \(a_2\). Infection changes \(I\) by +1 and recovery by −1.

\[f(I)=a_1(I)-a_2(I),\qquad g(I)=\sqrt{a_1(I)+a_2(I)}.\]

The rates subtract in the drift because the events move \(I\) in opposite directions. Their variances add in the diffusion because both random event streams contribute uncertainty.

Direction and magnitude are different

Quantity	Can be negative?	Interpretation
\(f(I)\)	Yes	Expected direction: positive upward, negative downward.
\(g(I)\)	Usually chosen non-negative	Scale of fluctuations.
\(g(I)dW\)	Yes	Realised random change can be positive or negative.

Units check

If \(I\) is measured in people and time in days, \(f\) has units people/day. Because \(dW\) has units \(\sqrt{\text{day}}\), \(g\) has units people/\(\sqrt{\text{day}}\). Both complete terms have units people.

Interactive Python laboratory

At one epidemic state, the drift contribution is fixed while repeated diffusion contributions vary around zero.

Interactive PythonDrift and diffusion at one SIS state

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

N = 1000
I = 100
S = N - I
beta = 0.30
gamma = 0.10
dt = 0.10
repetitions = 10000
seed = 9
rng = np.random.default_rng(seed)

infection_rate = beta * S * I / N
recovery_rate = gamma * I

drift = infection_rate - recovery_rate
diffusion = np.sqrt(
    infection_rate + recovery_rate
)

drift_change = drift * dt
dW = np.sqrt(dt) * rng.normal(
    0, 1, size=repetitions
)
diffusion_changes = diffusion * dW
total_changes = drift_change + diffusion_changes

summary = pd.DataFrame({
    "quantity": [
        "Drift coefficient f(I)",
        "Diffusion coefficient g(I)",
        "Fixed drift change f(I)dt",
        "Mean diffusion change",
        "SD diffusion change",
        "Mean total SDE change"
    ],
    "value": [
        drift,
        diffusion,
        drift_change,
        diffusion_changes.mean(),
        diffusion_changes.std(ddof=1),
        total_changes.mean()
    ]
})

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

plt.figure(figsize=(9, 5))
plt.hist(
    total_changes, bins=40,
    color="#79ae90", edgecolor="white"
)
plt.axvline(
    drift_change, color="#b42318",
    linestyle="--", linewidth=2,
    label=f"Drift change = {drift_change:.2f}"
)
plt.xlabel("SDE change over one interval")
plt.ylabel("Number of repetitions")
plt.title("Diffusion spreads changes around the drift")
plt.legend()
plt.grid(axis="y", alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

Conditions and limitations

The square-root expression requires non-negative event rates.
At an absorbing state where all rates vanish, both drift and diffusion vanish.
A diffusion approximation treats the state as continuous and can produce non-integer values.
Near small populations or boundaries, a CTMC may represent discrete extinction more faithfully.

What this lesson adds

You can now identify drift as expected direction, diffusion as fluctuation scale, derive both from opposing biological event rates, and check their units. The next lesson explains the Brownian increment that supplies the random sign and magnitude.