Drift and diffusion
A stochastic differential equation separates change into two parts: a systematic part and a random part.
\[\boxed{dX_t=a(X_t,t)\,dt+b(X_t,t)\,dW_t}.\]First: what is \(X_t\)?
\(X_t\) is the quantity being modelled at time \(t\). In biology it might represent a population size, concentration, proportion, or a continuous approximation to the number of infectious individuals.
The SDE describes how \(X_t\) changes through time.
The two parts of an SDE
| Term | Name | Meaning |
|---|---|---|
| \(a(X_t,t)dt\) | drift | systematic local change |
| \(b(X_t,t)dW_t\) | diffusion | random local fluctuation |
It is useful to think of them separately before combining them.
Drift: the systematic direction
Suppose first that there is no random term:
\[dX_t=a\,dt.\]If \(a>0\), the process tends upward. If \(a<0\), it tends downward. If \(a=0\), there is no deterministic tendency at that instant.
Diffusion: the random fluctuation
Now suppose there is no drift:
\[dX_t=b\,dW_t.\]The Brownian increment has mean zero, so diffusion does not itself prescribe an upward or downward direction. Instead, \(b\) controls the scale of the random movement.
Over a finite small interval,
\[\Delta W=\sqrt{\Delta t}\,Z,\qquad Z\sim N(0,1),\]so the random contribution is
\[\boxed{b\sqrt{\Delta t}\,Z}.\]What does the diffusion coefficient \(b\) do?
If \(|b|\) is small, random fluctuations are relatively small. If \(|b|\) is large, they are larger. For the simple constant-coefficient case, the random contribution over \(\Delta t\) has variance
\[\boxed{b^2\Delta t}.\]Thus \(b^2\) is the local variance rate in this simple setting.
Same drift, different diffusion
The dashed straight line therefore shows the underlying deterministic tendency. It is not another stochastic trajectory.
Putting drift and diffusion together
Over a small interval \(\Delta t\),
\[\Delta X\approx a(X_t,t)\Delta t+b(X_t,t)\Delta W.\]Using the Brownian-increment rule gives
\[\boxed{\Delta X\approx a(X_t,t)\Delta t+b(X_t,t)\sqrt{\Delta t}\,Z}.\]This is the most useful way to understand one numerical SDE step.
One step, piece by piece
Suppose at the current state
\[a=0.4,\qquad b=0.3,\qquad\Delta t=0.01,\qquad Z=-1.2.\]The drift contribution is
\[a\Delta t=0.4(0.01)=0.004.\]The Brownian increment is
\[\Delta W=\sqrt{0.01}(-1.2)=-0.12.\]The diffusion contribution is
\[b\Delta W=0.3(-0.12)=-0.036.\]Therefore
\[\Delta X\approx0.004-0.036=-0.032.\]Drift is not the same as the realised change
This distinction is fundamental. If the drift is positive, the process can still move downward over a particular short interval because of diffusion.
Similarly, negative drift does not prevent occasional upward movements.
\[\text{drift}=\text{systematic tendency},\qquad\text{realised change}=\text{drift contribution}+\text{random contribution}.\]Why the two terms scale differently
Over a small interval, the drift contribution has characteristic size
\[a\Delta t,\]while the diffusion contribution has characteristic size
\[b\sqrt{\Delta t}.\]For very small \(\Delta t\), \(\sqrt{\Delta t}\) is much larger than \(\Delta t\). This is why an SDE path remains locally rough even when it has a clear long-term drift.
Many paths reveal the drift
For the constant-coefficient model
\[dX_t=a\,dt+b\,dW_t,\]starting from \(X_0\), the solution is
\[X_t=X_0+at+bW_t.\]Since \(E[W_t]=0\),
\[\boxed{E[X_t]=X_0+at}.\]This shows precisely how the drift determines the mean trajectory in this simple model.
State-dependent drift
In realistic models, drift need not be constant. For example,
\[a(X_t)=rX_t\left(1-\frac{X_t}{K}\right)\]describes logistic deterministic growth. The direction and magnitude of drift then depend on the current state.
State-dependent diffusion
Diffusion may also depend on the state. For example,
\[b(X_t)=\sigma X_t.\]Then the random fluctuation becomes larger when \(X_t\) is larger:
\[\sigma X_t\,dW_t.\]This is called multiplicative noise. When \(b\) is constant, the model instead has additive noise.
Biological interpretation must come from the model
The functions \(a\) and \(b\) are not merely mathematical decorations. They should reflect how the biological system changes and where its uncertainty comes from.
For an epidemic model, the drift may reproduce the deterministic infection and recovery balance, while diffusion may approximate fluctuations arising from random infection and recovery events.
Drift and diffusion in several variables
Biological models often contain several compartments, such as \(S,E,I,R\). Then the state is a vector and the SDE can be written schematically as
\[d\mathbf X_t=\mathbf a(\mathbf X_t,t)dt+B(\mathbf X_t,t)d\mathbf W_t.\]The drift becomes a vector and diffusion becomes a matrix. This allows different random event mechanisms to affect several compartments simultaneously and can preserve correlations between their changes.
Connection with Euler–Maruyama
The Euler–Maruyama method repeatedly applies the small-step rule
\[\boxed{X_{n+1}=X_n+a(X_n,t_n)\Delta t+b(X_n,t_n)\sqrt{\Delta t}\,Z_n},\]where each \(Z_n\sim N(0,1)\) is an independent standard normal draw.
So the ideas developed here—drift, diffusion and Brownian increments—lead directly to numerical simulation of SDEs.