From rates to small-interval probabilities

A transition rate is an instantaneous event intensity. Over a sufficiently short interval, multiplying the current rate by the interval length gives the leading approximation to the event probability.

Return to the SIS scenario

At state \(I(t)=i\), the current infection and recovery rates are:

\[ b(i)=\beta\frac{(N-i)i}{N}, \qquad d(i)=\gamma i. \]

For \(N=1000\), \(i=10\), \(\beta=0.3\) and \(\gamma=0.1\):

\[ b(10)=2.97\text{ events/day}, \qquad d(10)=1.00\text{ event/day}. \]

We now ask what these rates imply during a short future interval of length \(h\) days, conditional on the current state.

Why multiply rate by time?

Rate
events per day

Short time
days

\[ \frac{\text{events}}{\text{day}} \times \text{day} = \text{events}. \]

For a one-event transition, this dimensionless leading quantity becomes the approximate probability of that event. Multiplication is therefore required both by the instantaneous-rate definition and by the units.

The precise small-interval statements

Conditional on \(I(t)=i\), as \(h\downarrow0\):

One infection

\[\Pr(i\to i+1)=b(i)h+o(h).\]

One recovery

\[\Pr(i\to i-1)=d(i)h+o(h).\]

No event

\[\Pr(i\to i)=1-[b(i)+d(i)]h+o(h).\]

The notation \(o(h)\), pronounced “little o of h,” represents terms that become negligible compared with \(h\) as the interval shrinks:

\[ \frac{o(h)}{h}\longrightarrow0 \qquad\text{as }h\downarrow0. \]

Why the no-event probability is the remainder

To first order in \(h\), the possible outcomes are one infection, one recovery or no event. Their probabilities must sum to 1:

\[ b(i)h+d(i)h+ \big(1-[b(i)+d(i)]h\big) =1. \]

The no-event probability falls as the current event rates or interval length increase.

What about two or more events?

One event

Its probability is of order \(h\): approximately proportional to the interval length.

Two or more events

Its probability is of smaller order, usually \(O(h^2)\), and therefore negligible relative to \(h\) as \(h\to0\).

\[ \Pr(\text{two or more events in }(t,t+h]) =o(h). \]

This is why only one-jump destinations appear in the infinitesimal CTMC description. Over a longer interval, several events may occur and the first-order formulas should not be treated as exact.

Numerical example

Use \(h=0.01\) day, approximately 14.4 minutes:

\[ \begin{aligned} \Pr(\text{infection})&\approx2.97(0.01)=0.0297,\\ \Pr(\text{recovery})&\approx1.00(0.01)=0.0100,\\ \Pr(\text{no event})&\approx1-3.97(0.01)=0.9603. \end{aligned} \]

Thus infection has approximately 2.97% probability, recovery 1%, and no event 96.03% during this short interval, conditional on the state remaining the current state until a jump.

Necessary validity condition

The first-order no-event approximation must not be negative:

\[ [b(i)+d(i)]h\le1. \]

If the current total rate is positive:

\[ h\le\frac{1}{b(i)+d(i)}. \]

This condition is necessary for valid first-order probabilities at the current state, but it is not sufficient for high accuracy. A value close to 1 leaves substantial possibility of multiple events over the interval, which the first-order approximation omits.

“Valid” and “sufficiently small” are different

Question	Meaning
Are the approximate probabilities valid?	They lie between 0 and 1 and sum to 1.
Is the interval sufficiently small?	Ignored higher-order and multiple-event effects are acceptably small for the intended use.

There is no universal cutoff such as 0.1 that guarantees adequate accuracy for every model. The required interval depends on the rates, state space and purpose of the calculation.

This does not turn the CTMC into a DTMC simulation

The short-interval formulas define the local behaviour of the continuous-time process. A Gillespie CTMC simulation does not advance through a chosen sequence of tiny fixed steps. Instead, it generates the next event time directly.

Lesson 3 derives the total rate and exponential waiting time used for that direct simulation.

Interactive Python laboratory

The code calculates first-order probabilities for several interval lengths, checks validity and plots how the approximation changes with \(h\). It does not draw random events.

Interactive PythonRates to short-interval probabilities

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

# Current SIS state and transition rates
N = 1000
i = 10
beta = 0.30
gamma = 0.10

infection_rate = beta * (N - i) * i / N
recovery_rate = gamma * i
total_rate = infection_rate + recovery_rate

if total_rate < 0:
    raise ValueError("Total rate cannot be negative.")

if total_rate == 0:
    maximum_valid_h = float("inf")
else:
    maximum_valid_h = 1 / total_rate

# Compare several short-interval choices
h_values = np.array([0.001, 0.01, 0.05, 0.10, 0.25, 0.30])

rows = []

for h in h_values:
    if h <= 0:
        raise ValueError("h must be positive.")

p_infection = infection_rate * h
    p_recovery = recovery_rate * h
    p_no_event = 1 - total_rate * h

valid = (
        p_infection >= 0
        and p_recovery >= 0
        and p_no_event >= -1e-12
    )

rows.append({
        "h (days)": h,
        "infection probability": p_infection,
        "recovery probability": p_recovery,
        "no-event probability": p_no_event,
        "total": p_infection + p_recovery + p_no_event,
        "valid first-order probabilities": valid
    })

probability_table = pd.DataFrame(rows)

print(f"Infection rate: {infection_rate:.3f} events/day")
print(f"Recovery rate: {recovery_rate:.3f} events/day")
print(f"Current total rate: {total_rate:.3f} events/day")
print(f"Maximum h for non-negative first-order remainder: "
      f"{maximum_valid_h:.4f} days")
print()
print(probability_table.round(5).to_string(index=False))

# Display the first-order formulas over a range of h
h_grid = np.linspace(0, 0.32, 250)
p_infection_grid = infection_rate * h_grid
p_recovery_grid = recovery_rate * h_grid
p_no_event_grid = 1 - total_rate * h_grid

plt.figure(figsize=(9, 5))
plt.plot(
    h_grid,
    p_infection_grid,
    label="Infection: b(i)h",
    color="#dc2626"
)
plt.plot(
    h_grid,
    p_recovery_grid,
    label="Recovery: d(i)h",
    color="#2563eb"
)
plt.plot(
    h_grid,
    p_no_event_grid,
    label="No event: 1 - [b(i)+d(i)]h",
    color="#16a34a"
)
plt.axvline(
    maximum_valid_h,
    color="black",
    linestyle="--",
    label="Validity boundary"
)
plt.axhline(0, color="black", linewidth=0.8)
plt.xlabel("Interval length h (days)")
plt.ylabel("First-order probability")
plt.title("Short-interval probability approximations")
plt.ylim(-0.3, 1.1)
plt.legend()
plt.grid(alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

Understand the new code

Code	Meaning
`maximum_valid_h = 1 / total_rate`	Finds the current-state interval boundary at which the first-order no-event remainder reaches zero.
`float("inf")`	Represents no finite upper boundary when the total event rate is zero.
`p_no_event >= -1e-12`	Allows only a tiny negative floating-point round-off value, not a genuinely negative probability.
`np.linspace(0, 0.32, 250)`	Creates 250 evenly spaced interval lengths for drawing the approximation curves.

Why `h <= 0` can be checked exactly

\(h\) is a directly supplied interval length. Zero and negative intervals are invalid inputs, so the code checks h <= 0 directly.

The tolerance appears only in a calculated floating-point probability, where arithmetic may produce a tiny round-off difference from the mathematical value zero.

Biological interpretation

At the current state, infection has the larger rate, so its short-interval probability is larger than the recovery probability. As \(h\) shrinks, both event probabilities shrink towards zero and no event becomes increasingly likely.

Increasing \(h\) makes the first-order event probabilities larger, but eventually the approximation becomes unsuitable because more than one event may occur within the interval. Direct CTMC simulation avoids choosing this artificial fixed step.

What this lesson has—and has not—done

You can now:

derive event probabilities to first order from transition rates;
calculate the no-event remainder;
interpret \(o(h)\) and neglected multiple-event probability;
check the necessary non-negativity condition; and
distinguish mathematical validity from approximation accuracy.

Not yet: Lesson 3 combines the competing rates and uses the exact exponential waiting-time distribution to generate when the next CTMC event occurs.