Difference equations models

Difference equation (DE) models describe how the quantities of system components change over discrete time steps. You can think of them like tracking the balance of a bank account: each time step (e.g., a year), the new balance is calculated by adding the interest earned to the current balance. In biological terms, we might track how the concentration of a protein or the number of infected cells or people changes from one time point to the next.

Like Boolean models, difference equation models update the model components in steps — but with a key difference: in difference equation models, each step represents a quantitative amount of real time, such as 5 minutes, 1 hour, or 1 day.

In contrast to Boolean network models, difference equation models use quantitative variables. That means each component can take on a continuous range of values, such as the concentration of a molecule in solution or the proportion of a population that is infected with a particular disease.

Each model component is updated using a difference equation of the form:

x_i​ (t + Δt) =x_i ​(t) + f ( x (t) ) Δt

Here:

• x_i​ (t) is the value of component i at time t
  
• Δt is the length of the time step
  
• f ( x (t) ): the net rate of change, capturing the effect of production, degradation, interaction, or conversion processes

This means the value of component x_i at the next time point x_i​ (t + Δt) depends on its current value x_i (t), the change rate f ( x (t)) resulting from processes that produce and consume the component, and the length of the time interval Δt. If in a time step more is produced than consumed, the component’s value increases; if more is consumed than produced, it decreases. These processes can be more or less abstract, everything from synthesis and activation to infection and degradation, conversion and death. 

One limitation of this modeling approach is that it assumes changes occur only at fixed, evenly spaced time points. If the chosen time interval Δt is too large, the model may become unstable or biologically implausible — for instance, by producing negative concentrations. For many systems, particularly when you have experimental data collected at regular intervals, difference equation models strike a good balance between realism and simplicity.

One especially common and intuitive way to derive the equations for the change rate – particularly in biology – is through kinetic modelling, which we will introduce in the next section.