SDE biological models
Stochastic differential equations extend ordinary differential-equation models by allowing the biological variables to fluctuate randomly through continuous time.
From a deterministic model to an SDE
Suppose a deterministic biological model is
\[\frac{dX}{dt}=f(X,t).\]In differential notation,
\[dX_t=f(X_t,t)dt.\]An SDE adds a stochastic contribution:
\[\boxed{dX_t=f(X_t,t)dt+g(X_t,t)dW_t}.\]| Part | Biological meaning |
|---|---|
| \(f(X_t,t)dt\) | systematic biological change: growth, infection, recovery, death, migration, etc. |
| \(g(X_t,t)dW_t\) | continuous random fluctuation around that systematic change |
Why not simply add random noise?
It is tempting to write
\[dX_t=f(X_t)dt+\sigma dW_t\]just to make the model stochastic. Sometimes that is a useful phenomenological model, but in mechanistic biology the form of the noise should ideally reflect the process generating the randomness.
Two common ways biological SDEs arise
| Approach | Idea |
|---|---|
| phenomenological noise | choose a diffusion term to represent unresolved environmental or measurement variability |
| diffusion approximation | derive drift and diffusion from underlying stochastic event rates such as births, deaths, infections and recoveries |
The second approach often gives the diffusion coefficient a direct mechanistic interpretation.
Example 1: stochastic population growth
The deterministic exponential-growth model is
\[\frac{dX}{dt}=rX.\]A simple stochastic version with multiplicative noise is
\[\boxed{dX_t=rX_tdt+\sigma X_tdW_t}.\]The drift \(rX_t\) gives systematic proportional growth. The diffusion \(\sigma X_t\) means the magnitude of random fluctuations increases with population size.
Why multiplicative noise can make biological sense
If the population is twice as large, there may be roughly twice as many opportunities for random births, deaths, interactions or environmental responses. A state-dependent diffusion term can represent that idea.
This differs from additive noise,
\[dX_t=rX_tdt+\sigma dW_t,\]where random fluctuations have the same absolute size regardless of the current population.
Same deterministic growth, different stochastic paths
Example 2: stochastic logistic growth
The deterministic logistic model is
\[\frac{dX}{dt}=rX\left(1-\frac{X}{K}\right).\]A stochastic extension might be
\[\boxed{dX_t=rX_t\left(1-\frac{X_t}{K}\right)dt+\sigma X_tdW_t}.\]Here the drift still contains density dependence and carrying capacity \(K\), while the diffusion represents proportional environmental variability.
Event-based derivation: the important idea
Now consider a population where the state changes through discrete events. Suppose state \(X=i\) has
\[\text{upward event rate}=b(i),\qquad\text{downward event rate}=d(i).\]Over a short interval \(\Delta t\), the expected change is approximately
\[E[\Delta X\mid X=i]\approx[b(i)-d(i)]\Delta t.\]The variance of the change is approximately
\[\operatorname{Var}(\Delta X\mid X=i)\approx[b(i)+d(i)]\Delta t.\]Why difference gives drift but sum gives variance
An upward event contributes \(+1\); a downward event contributes \(-1\). Their expected signed effects therefore subtract:
\[\boxed{\text{drift}=b(i)-d(i)}.\]But when variability is calculated, both event types contribute positive squared changes because
\[(+1)^2=(-1)^2=1.\]Therefore their variance contributions add:
\[\boxed{\text{variance rate}=b(i)+d(i)}.\]The corresponding diffusion approximation
The birth–death CTMC can therefore be approximated by
\[\boxed{dX_t=[b(X_t)-d(X_t)]dt+\sqrt{b(X_t)+d(X_t)}\,dW_t}.\]The deterministic drift comes from the net event rate. The diffusion comes from the total event variability.
Example 3: SIS epidemic
Let \(I(t)\) be the infectious population in an SIS model with total population \(N\). The event rates are
\[b(i)=\beta\frac{(N-i)i}{N}\]for infection and
\[d(i)=\gamma i\]for recovery.
The deterministic net change is
\[b(i)-d(i)=\beta\frac{(N-i)i}{N}-\gamma i.\]SIS diffusion approximation
Using the birth–death rule gives
\[\boxed{dI_t=\left[\beta\frac{(N-I_t)I_t}{N}-\gamma I_t\right]dt+\sqrt{\beta\frac{(N-I_t)I_t}{N}+\gamma I_t}\,dW_t}.\]The first bracket is exactly the deterministic SIS drift. The square-root term measures the stochastic variation generated by infection and recovery events.
Why this diffusion coefficient makes sense
If infection and recovery events are both frequent, there are many opportunities for random event-count fluctuations, so the diffusion magnitude is larger.
If both event rates approach zero, the event-driven stochastic variation also approaches zero.
CTMC and SDE describe randomness differently
| CTMC | SDE diffusion approximation |
|---|---|
| integer-valued states | continuous-valued states |
| individual events are explicit | many event fluctuations are approximated continuously |
| step-like trajectories | continuous rough trajectories |
| natural for small populations and extinction | often efficient for larger populations |
Visual comparison: deterministic, CTMC-like and SDE behaviour
Why an SDE can produce non-integer values
An SDE approximation may produce values such as
\[I(t)=12.7.\]That does not mean 0.7 of a person exists. It means the discrete event process has been approximated by a continuous stochastic variable.
This approximation is often more reasonable when counts are sufficiently large.
Boundary problems
A naïve SDE approximation can sometimes produce biologically impossible values such as negative populations.
Environmental noise versus demographic noise
| Type | Interpretation |
|---|---|
| demographic stochasticity | randomness from individual births, deaths, infections, recoveries and other discrete events |
| environmental stochasticity | random changes in external conditions affecting many individuals or rates simultaneously |
These sources can lead to different diffusion structures and should not automatically be represented by the same noise term.
Several biological variables
For a model with several compartments, write
\[\mathbf X_t=(X_{1,t},\ldots,X_{m,t})^T.\]A multidimensional SDE has the form
\[\boxed{d\mathbf X_t=\mathbf a(\mathbf X_t,t)dt+B(\mathbf X_t,t)d\mathbf W_t}.\]The drift vector \(\mathbf a\) describes systematic changes in all compartments. The diffusion matrix \(B\) describes the magnitudes and correlations of stochastic changes.
Why one Brownian motion is often not enough
Different biological events may create different directions of stochastic change. For an SIR epidemic, infection decreases \(S\) and increases \(I\), while recovery decreases \(I\) and increases \(R\).
Using separate Brownian drivers for distinct event mechanisms can preserve this structure and the induced correlations between compartments.
SIR event-vector view
Let
\[\mathbf X=(S,I,R)^T.\]An infection event has state-change vector
\[\boldsymbol\nu_1=(-1,+1,0)^T,\]while recovery has
\[\boldsymbol\nu_2=(0,-1,+1)^T.\]If their rates are \(a_1(\mathbf X)\) and \(a_2(\mathbf X)\), a diffusion approximation can be written schematically as
\[d\mathbf X=\left[\boldsymbol\nu_1a_1+\boldsymbol\nu_2a_2\right]dt+\boldsymbol\nu_1\sqrt{a_1}\,dW_1+\boldsymbol\nu_2\sqrt{a_2}\,dW_2.\]This shows how the stochastic terms can be derived directly from the underlying event mechanisms.
Why this event-vector formulation is useful
The same infection event simultaneously changes two compartments. Using its change vector in both the drift and diffusion automatically preserves that connection.
Euler–Maruyama simulation
For a one-dimensional biological SDE
\[dX_t=f(X_t,t)dt+g(X_t,t)dW_t,\]Euler–Maruyama uses
\[\boxed{X_{n+1}=X_n+f(X_n,t_n)\Delta t+g(X_n,t_n)\sqrt{\Delta t}\,Z_n},\]where
\[Z_n\sim N(0,1).\]This produces one numerical stochastic trajectory. Repeating the simulation produces a distribution of possible biological histories.
What can repeated SDE simulations tell us?
| Quantity | Biological question |
|---|---|
| mean trajectory | What is the average behaviour across possible histories? |
| variance | How uncertain is the state at time \(t\)? |
| peak distribution | How variable is the maximum epidemic or population size? |
| threshold probability | How often is a biologically important level crossed? |
SDEs are not automatically “better” than deterministic models
An SDE adds information about uncertainty, but it also adds assumptions and parameters. If the biological question concerns only average behaviour and random variability is negligible, a deterministic model may be sufficient.
If extinction, outbreak risk, threshold crossing or uncertainty matter, a stochastic model can provide information that a single deterministic trajectory cannot.
Choosing between CTMC and SDE
| Prefer CTMC when | Prefer SDE when |
|---|---|
| individual events matter | continuous approximation is acceptable |
| counts are small | counts are moderately large |
| exact extinction is important | continuous uncertainty is the main interest |
| event structure is central | computational efficiency or analytic approximation is useful |