Unlocking Complex Dynamics: How Biexponential Models Capture Dual Processes

Biexponential Model (Biexponential Model) is a class of mathematical models describing phenomena that evolve over time or other variables, characterized by the superposition of two exponential functions, allowing flexible description of systems with complex change processes.

Mathematical Expression

The basic form consists of the sum of two exponential functions:

y(t) = A 1 ·e k1·t + A 2 ·e k2·t , where t represents time, A 1 and A 2 are constants representing the initial contribution of each term, and k1, k2 are the respective exponential growth or decay rates.

The above equation is the standard form of the biexponential model. It describes how two processes with different rates combine to influence system change.

When k1 and k2 are negative, the model describes a dual‑exponential decay, commonly used for physical or biological decay processes such as drug metabolism, where a fast pathway eliminates a portion quickly and a slower pathway removes the remainder.

Assumptions

The overall system behavior is governed by two distinct driving processes: a fast process and a slow process.

Each process can be described by exponential decay or growth. The rate of change is proportional to the current state.

The two processes evolve independently without direct coupling.

The system output is the linear superposition of the two independent processes.

The figure below compares the graph of two single‑exponential functions with their sum, the biexponential function.

Application

In practice, data alone often cannot determine whether a single‑exponential or biexponential model fits better; mechanistic analysis is required to decide if the system consists of two superimposed processes.

Consider the spread of a new cultural trend. Two propagation pathways dominate:

Fast spread via social media and the internet, rapidly reaching a large audience.

Slow spread through word‑of‑mouth and offline activities, gradually permeating the society.

The fast pathway has a higher growth rate, while the slow pathway has a lower one. Adding the two yields a biexponential model that captures the overall diffusion of the culture.

Coupled Biexponential Model

In reality, the fast and slow processes may interact rather than remain completely independent. A coupled biexponential model introduces interaction terms, allowing the rate of one process to depend on the state of the other.

For example, the fast spread rate may be influenced by the existing slow‑spread audience, and vice versa. This can be expressed with differential equations that include coupling coefficients.

As an illustration, a movie’s buzz on social media (fast) quickly attracts viewers, while word‑of‑mouth (slow) gradually builds a stable audience; early rapid buzz creates a core group whose discussions further amplify social‑media diffusion.

Reference: Candia, C., Jara‑Figueroa, C., Rodriguez‑Sickert, C., Barabási, A.-L., & Hidalgo, C. A. (2018). The universal decay of collective memory and attention. Nature Human Behaviour . https://doi.org/10.1038/s41562-018-0474-5