Birth–death processes
A birth–death process is one of the simplest and most useful continuous-time Markov chains for biological populations. The state is an integer count, and each event changes that count by exactly one.
What is the state?
Let
\[X(t)=\text{number of individuals of interest at time }t.\]The possible states are typically
\[0,1,2,3,\ldots\]or, for a population with an upper limit \(N\),
\[0,1,2,\ldots,N.\]If the current state is \(X(t)=i\), the process can make two basic types of jump:
\[i\longrightarrow i+1\qquad\text{or}\qquad i\longrightarrow i-1.\]Why is this useful in biology?
Many biological quantities change one event at a time. A cell divides, an organism dies, an infection occurs, or an infectious individual recovers. Each event can increase or decrease the count by one.
| Modelled quantity | Birth-type event \(+1\) | Death-type event \(-1\) |
|---|---|---|
| animal population | birth | death |
| cell population | cell division | cell death |
| infectious population in SIS | new infection | recovery |
| molecule count | production | degradation |
The transition structure
Let \(b(i)\) be the birth rate and \(d(i)\) the death rate when the current state is \(i\). Then
\[i\longrightarrow i+1\quad\text{at rate }b(i),\]\[i\longrightarrow i-1\quad\text{at rate }d(i).\]This nearest-neighbour structure is what makes a birth–death process special: from an interior state \(i\), the next jump can only be to a neighbouring state.
Rates are not probabilities
The quantities \(b(i)\) and \(d(i)\) are rates per unit time. They are not themselves probabilities.
Over a sufficiently short interval \(\Delta t\), they give the approximations
\[P(i\to i+1)\approx b(i)\Delta t,\]\[P(i\to i-1)\approx d(i)\Delta t,\]\[P(i\to i)\approx1-[b(i)+d(i)]\Delta t.\]Why is there a probability of staying at \(i\)?
Continuous time does not mean that an event is occurring at every instant. The process normally spends some random time waiting in its current state.
During a short interval, either a birth occurs, a death occurs, or no event occurs. Therefore the no-event probability is approximately what remains after subtracting the two event probabilities from 1.
\[1-b(i)\Delta t-d(i)\Delta t=1-[b(i)+d(i)]\Delta t.\]Total event rate
Either a birth or a death can end the current waiting period. The total rate of leaving state \(i\) is therefore
\[\boxed{a(i)=b(i)+d(i)}.\]The waiting time \(T\) until the next event satisfies
\[\boxed{T\sim\operatorname{Exp}(a(i))}.\]Its expected value is
\[E[T]=\frac{1}{b(i)+d(i)}.\]A larger total rate means that events tend to occur more frequently and the average waiting time is shorter.
Which event happens next?
Once an event occurs, the relative sizes of the two rates determine its type:
\[\boxed{P(\text{birth next})=\frac{b(i)}{b(i)+d(i)}},\]\[\boxed{P(\text{death next})=\frac{d(i)}{b(i)+d(i)}}.\]These probabilities add to 1.
A numerical example
Suppose that at state \(i=10\),
\[b(10)=2.7\text{ per day},\qquad d(10)=1.0\text{ per day}.\]The total rate is
\[a(10)=2.7+1.0=3.7\text{ per day}.\]Hence the mean waiting time is
\[E[T]=\frac1{3.7}\approx0.270\text{ days}.\]When the next event occurs,
\[P(\text{birth next})=\frac{2.7}{3.7}\approx0.730,\]\[P(\text{death next})=\frac{1.0}{3.7}\approx0.270.\]So the next state is 11 with probability about 0.730 or 9 with probability about 0.270.
Birth and death rates can depend on population size
The rates need not be constant. In biological models they often depend on the current state:
\[b(i),\qquad d(i).\]This allows the model to represent density dependence, competition, limited susceptible populations and many other biological mechanisms.
Simple linear birth–death model
Suppose each individual independently gives birth at rate \(\lambda\) and dies at rate \(\mu\). With \(i\) individuals, the total rates are
\[b(i)=\lambda i,\qquad d(i)=\mu i.\]The factor \(i\) appears because any of the \(i\) individuals can generate the event.
What do \(\lambda\) and \(b(i)\) mean?
| Quantity | Meaning |
|---|---|
| \(\lambda\) | birth rate per individual |
| \(\lambda i=b(i)\) | total birth-event rate for the whole population when its size is \(i\) |
| \(\mu\) | death rate per individual |
| \(\mu i=d(i)\) | total death-event rate for the whole population when its size is \(i\) |
This distinction between an individual rate and a total event rate is fundamental in stochastic population modelling.
Expected direction of change
At state \(i\), births push the process upward and deaths push it downward. The local expected rate of change is therefore
\[\boxed{b(i)-d(i)}.\]For the linear model,
\[b(i)-d(i)=(\lambda-\mu)i.\]| Rates | Average tendency |
|---|---|
| \(\lambda>\mu\) | population tends to grow |
| \(\lambda=\mu\) | no net deterministic drift, although random fluctuations remain |
| \(\lambda<\mu\) | population tends to decline |
The SIS epidemic as a birth–death process
A particularly important example is the stochastic SIS epidemic. Let the state be the number of infectious individuals \(I(t)=i\).
A new infection increases \(i\) by one, so it acts as a birth-type event:
\[i\to i+1.\]A recovery decreases \(i\) by one, so it acts as a death-type event:
\[i\to i-1.\]Using frequency-dependent transmission, the rates can be written
\[\boxed{b(i)=\beta\frac{(N-i)i}{N}},\qquad\boxed{d(i)=\gamma i}.\]Here \(N-i\) is the susceptible population.
Why the SIS birth rate is nonlinear
In the SIS model,
\[b(i)=\beta\frac{(N-i)i}{N}.\]The rate depends on both infectious individuals \(i\) and susceptible individuals \(N-i\). When there are few infectious individuals, there are few sources of infection. When almost everyone is infectious, there are few susceptible individuals left to infect.
Therefore the infection rate is small near both ends and larger at intermediate states.
Where the rates are equal
For the parameters in the graph, the non-zero point where birth and death rates are equal satisfies
\[\beta\frac{(N-i)i}{N}=\gamma i.\]For \(i>0\), cancelling \(i\) gives
\[\beta\frac{N-i}{N}=\gamma.\]Hence
\[i=N\left(1-\frac{\gamma}{\beta}\right).\]With \(N=100\), \(\beta=0.3\) and \(\gamma=0.1\),
\[i\approx66.7.\]Below this level the infection birth rate exceeds the recovery rate; above it the recovery rate exceeds the infection rate. This describes the local average tendency, not a barrier that individual stochastic trajectories cannot cross.
Boundary states
At \(i=0\), a transition to \(-1\) is impossible. In an epidemic without imported infection,
\[b(0)=d(0)=0,\]so state 0 is absorbing.
If the maximum state is \(N\), then at \(i=N\) an upward transition is impossible. For the SIS model this occurs naturally because
\[b(N)=\beta\frac{(N-N)N}{N}=0.\]Extinction
If state 0 is absorbing, a major question is whether the process eventually reaches it.
For an epidemic, reaching \(I(t)=0\) means extinction of infection. For a biological population, it means extinction of the modelled population.
Stochastic models are especially valuable here because extinction can occur even when the corresponding deterministic model predicts positive growth.
Birth–death process versus deterministic population model
A deterministic model might use
\[\frac{dx}{dt}=b(x)-d(x)\]to describe the average direction of change. A birth–death process keeps the upward and downward events separate and random.
| Deterministic model | Birth–death process |
|---|---|
| uses net change \(b-d\) | keeps birth and death events separate |
| one trajectory from fixed initial conditions | many possible trajectories |
| population may be treated continuously | state is usually an integer count |
| cannot directly give extinction probability | can quantify extinction and other event probabilities |
How one trajectory is generated
At current state \(i\):
This is the same event-based logic used by the Gillespie algorithm.
Why birth–death processes matter
Birth–death processes provide a bridge between biological mechanisms and stochastic mathematics. Simple event rules become transition rates; the rates determine random waiting times and state changes; and the resulting process can be used to study population variability, extinction, persistence and outbreak probabilities.
They also provide one of the clearest settings for learning generator matrices, Kolmogorov equations and stochastic simulation.
What comes next?
Birth–death processes describe how a population changes through individual random events. Another fundamental process focuses specifically on counting how many events occur through time. This leads to the Poisson process.