Why biological randomness matters
Biological systems contain events whose exact occurrence cannot usually be predicted in advance. We may know how likely an infection, birth, death, recovery or mutation is, without knowing exactly when it will occur or which individual it will happen to.
Begin with one simple biological event
Suppose one infectious person meets a susceptible person. Transmission is possible, but it is not certain.
A deterministic description tends to represent many such uncertain events through an average rate. A stochastic description represents the uncertainty explicitly.
Rate does not tell us the exact next event
Suppose an infectious individual has a recovery rate
\[\gamma=0.2\text{ day}^{-1}.\]This rate describes the tendency to recover. It does not mean that this particular person must recover after exactly 5 days.
The value
\[\frac{1}{\gamma}=5\text{ days}\]is the mean infectious duration under the usual constant-rate assumption. One person may recover earlier and another later.
Probability over a short time interval
If an event occurs at rate \(r\), then over a sufficiently short interval \(\Delta t\),
\[\boxed{P(\text{event during }\Delta t)\approx r\Delta t}.\]For example, if the recovery rate is \(0.2\) per day and \(\Delta t=0.1\) day,
\[P(\text{recovery during the next }0.1\text{ day})\approx0.2(0.1)=0.02.\]So there is approximately a 2% probability of recovery during that short interval.
This simple connection between rates and probabilities will become fundamental when we construct continuous-time Markov chains.
What is random in a biological model?
Different models may treat different quantities as random.
| Biological process | What may be random? |
|---|---|
| epidemic | who becomes infected, when infection occurs, when recovery occurs |
| population dynamics | birth and death events |
| genetics | mutation, inheritance and genetic drift |
| cell biology | molecular reaction and gene-expression events |
| ecology | individual survival, reproduction, movement and encounters |
Deterministic model: one trajectory
A deterministic model specifies how the state changes from its current value. Once the initial conditions and parameters are fixed, running the model again gives the same trajectory.
This trajectory can be extremely useful. It describes the behaviour implied by the model without random event-to-event variation.
Stochastic model: many possible trajectories
Now imagine starting the same epidemic several times with exactly the same initial state and the same parameter values. If infection and recovery events occur randomly, the epidemics need not follow identical paths.
One trajectory produces a substantial outbreak. Another produces a smaller outbreak. A third dies out quickly. None of these differences requires us to change the parameters: they can arise from the random ordering and timing of events.
A concrete epidemic example
Suppose an epidemic begins with one infectious person. During the early stage, that person might:
| Possible sequence | Outcome |
|---|---|
| recover before infecting anyone | the outbreak ends immediately |
| infect one person and then recover | infection may continue, but extinction is still possible |
| infect several people before recovery | a larger outbreak becomes more likely |
A deterministic model averages over these possibilities. A stochastic model allows each possible event sequence to occur with an appropriate probability.
Why not just use the deterministic average?
This is one of the most important questions in stochastic modelling. Suppose many stochastic epidemic simulations are performed and their average is similar to a deterministic epidemic curve. Why bother with the stochastic model?
Because the average does not tell us how widely the individual outcomes vary.
The average can hide outcomes that are scientifically or practically crucial: early extinction, unusually large outbreaks, very high hospital demand, or rare events.
Expected outcome and possible outcome are different ideas
The expected value describes an average across the probability distribution. A single stochastic trajectory is one possible realisation of the process.
Therefore
\[\text{expected trajectory}\neq\text{the trajectory that must occur}.\]Indeed, the expected value need not itself be a possible individual outcome in a discrete biological system.
When does randomness matter most?
Small populations
One event represents a large fraction of the population, so random changes can have a large effect.
The beginning of an epidemic
With only a few infectious individuals, chance can determine whether infection disappears or establishes itself.
Near extinction
When only a few infected individuals remain, random recovery or death events can eliminate infection even when a deterministic model approaches zero only gradually.
Rare but important events
Mutation, emergence, spillover and unusually large transmission events may be uncommon but biologically important.
Thresholds and capacity limits
When a decision depends on whether a quantity crosses a threshold, the probability of crossing it can matter more than the average value.
Randomness is not the same as ignorance
Two different sources of uncertainty should be distinguished.
| Type | Meaning |
|---|---|
| Process randomness | events are represented as random even when model parameters are known |
| Parameter uncertainty | we do not know the true values of quantities such as transmission or recovery parameters exactly |
For example, a model can use a known recovery rate and still generate random recovery times. Conversely, uncertainty about the recovery rate itself is a different problem.
What does a stochastic model give us?
Instead of only one predicted curve, a stochastic model can provide quantities such as
\[P(\text{early extinction}),\] \[P(\text{large outbreak}),\] \[P(\text{hospital demand exceeds capacity}),\]as well as expected values, variances, distributions and ranges of possible outcomes.
Where this section is going
The rest of this topic develops the mathematics needed to describe this randomness precisely. We will move from random variables to stochastic processes, then to Markov chains, event rates, waiting times, transition and generator matrices, Gillespie simulation, Monte Carlo methods, and finally extinction and outbreak probabilities.