Events and transition rates in continuous time

A continuous-time Markov chain changes state through discrete biological events, but those events occur at random times rather than on a fixed time grid.

DTMC time versus CTMC time

DTMC

The process is examined at fixed times \(0,\Delta t,2\Delta t,\ldots\). Transition probabilities describe what happens during each fixed step.

CTMC

The process may jump at any continuous time \(t\ge0\). The state remains constant between randomly timed events.

A CTMC trajectory therefore consists of horizontal waiting periods separated by instantaneous integer-valued jumps.

Scenario: an SIS infection

Let \(I(t)=i\) be the number of infectious people in a closed population \(N\). Then \(S(t)=N-i\).

Infection event

\[i\longrightarrow i+1.\]

One susceptible person becomes infectious.

Recovery event

\[i\longrightarrow i-1.\]

One infectious person returns to susceptibility.

No other one-jump destination is allowed in the basic SIS CTMC.

What is a transition rate?

A transition rate measures the instantaneous tendency for a particular event to occur while the process is in its current state.

For the SIS model:

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

\(b(i)\) is the infection rate and \(d(i)\) is the recovery rate at state \(i\). They are also called transition intensities or hazards.

Units are essential

If time is measured in days, \(\beta\) and \(\gamma\) have units per day. Because population counts are treated as counts, \(b(i)\) and \(d(i)\) have units of events per day.

A rate is not a probability. A rate may exceed 1. For example, an infection rate of 2.97 events per day does not mean a probability of 297%. It describes instantaneous event intensity at the current state.

Lesson 2 will connect a rate to probability over a sufficiently short interval. This page does not yet choose an interval or simulate an event time.

Why rates depend on the current state

The infection rate needs both susceptible and infectious people:

\[ b(i)=\beta\frac{S I}{N} =\beta\frac{(N-i)i}{N}. \]

Few infectious people give few opportunities for transmission.
Few susceptible people also limit transmission.
The infection rate is largest at an intermediate state.
The recovery rate \(d(i)=\gamma i\) increases linearly with the infectious count.

The parameters may be constant while the rates change, because the population state changes.

Formal instantaneous definition

For different states \(x\) and \(y\), the transition rate is:

\[ q(x,y) = \lim_{h\downarrow0} \frac{ \Pr(X(t+h)=y\mid X(t)=x) }{h}. \]

Intuitively, the probability of the jump is examined over an increasingly short positive interval \(h\), then divided by the interval length. The limiting quantity is probability per unit time.

This definition describes an instantaneous tendency. It does not claim that \(q(x,y)\) itself is a probability.

Boundary states

State \(i=0\)

\[b(0)=d(0)=0.\]

No infectious person remains. In the closed SIS model, state 0 is absorbing.

State \(i=N\)

\[b(N)=0,\qquad d(N)=\gamma N.\]

No susceptible person remains, so infection is impossible, but recovery remains possible.

When \(\beta>0\), \(\gamma>0\) and \(1\le i\le N-1\), both infection and recovery rates are positive.

Conditions for biologically valid rates

\(N\) must be a positive integer.
\(i\) must be an integer satisfying \(0\le i\le N\).
\(\beta\ge0\) and \(\gamma\ge0\).
Every allowed event must lead to a state inside the state space.
An impossible boundary event must have rate zero.

Interactive Python laboratory

The code calculates the event rates at one state, checks their conditions and plots how infection and recovery rates change across the complete SIS state space.

Interactive PythonSIS transition rates

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

def sis_rates(i, N, beta, gamma):
    susceptible = N - i
    infection_rate = beta * susceptible * i / N
    recovery_rate = gamma * i
    return infection_rate, recovery_rate

# Biological state and parameters
N = 1000
i = 10
beta = 0.30
gamma = 0.10

if not isinstance(N, int) or N <= 0:
    raise ValueError("N must be a positive integer.")

if not isinstance(i, int) or i < 0 or i > N:
    raise ValueError("i must be an integer between 0 and N.")

if beta < 0 or gamma < 0:
    raise ValueError("beta and gamma cannot be negative.")

infection_rate, recovery_rate = sis_rates(
    i, N, beta, gamma
)

rate_table = pd.DataFrame({
    "event": ["infection", "recovery"],
    "state change": ["i to i + 1", "i to i - 1"],
    "rate": [infection_rate, recovery_rate],
    "units": ["events per day", "events per day"]
})

print(f"Current state: I = {i}, S = {N - i}")
print(rate_table.round(4).to_string(index=False))
print()
print("These numbers are rates, not probabilities.")

# Rates at every possible state
states = np.arange(N + 1)
infection_rates = beta * (N - states) * states / N
recovery_rates = gamma * states

maximum_state = int(infection_rates.argmax())
maximum_infection_rate = infection_rates[maximum_state]

print()
print(f"Maximum infection rate: "
      f"{maximum_infection_rate:.2f} events per day")
print(f"It occurs at I = {maximum_state}")

plt.figure(figsize=(9, 4.9))
plt.plot(
    states,
    infection_rates,
    label="Infection rate b(i)",
    color="#dc2626"
)
plt.plot(
    states,
    recovery_rates,
    label="Recovery rate d(i)",
    color="#2563eb"
)
plt.scatter(
    [i],
    [infection_rate],
    color="#7c3aed",
    s=60,
    label=f"Current infection rate at I={i}"
)
plt.xlabel("Current infectious count i")
plt.ylabel("Transition rate (events per day)")
plt.title("SIS transition rates depend on the current state")
plt.legend()
plt.grid(alpha=0.25)
plt.tight_layout()

Output

Run the code to see the result.

Understand the new code

Code	Meaning
`def sis_rates(...)`	Packages the two state-dependent rate formulas into one reusable function.
`isinstance(N, int)`	Checks that \(N\) was supplied as a Python integer.
`np.arange(N + 1)`	Creates all integer states from 0 through \(N\), including both boundaries.
`.argmax()`	Returns the position at which the infection-rate array is largest.

Biological interpretation

At \(I=10\) and \(S=990\), the infection rate is 2.97 events per day and the recovery rate is 1 event per day. The current state therefore has a stronger instantaneous tendency towards infection than recovery.

This comparison does not yet tell us the probability of either event during a stated interval, the time of the next event, or which event will occur. Those are separate CTMC calculations developed in Lessons 2–5.

What this lesson has—and has not—done

You can now:

distinguish continuous-time jumps from fixed DTMC steps;
define biological events and their state changes;
calculate state-dependent transition rates with correct units;
distinguish a rate from a probability; and
identify boundary rates and absorbing states.

Not yet: Lesson 2 converts rates into small-interval probabilities. Exponential waiting times, total rates and random event selection remain for Lessons 3–5.