Computational Models:
From Mathematical Equations to Computational Experiments

The previous section presented a series of mathematical models that describe generalized contagion through systems of ordinary differential equations. Those models revealed how populations evolve under well-defined assumptions and provided the theoretical foundation for understanding the dynamics of fictional epidemiological events.

Mathematical models, however, represent populations through average quantities and assume that every individual is equally likely to interact with every other individual—a simplifying assumption commonly known as the well-mixed assumption.

While this assumption makes mathematical analysis possible, real populations seldom behave this way. Individuals occupy space, interact locally, form clusters, and experience chance encounters that influence the evolution of the entire system.

To investigate these spatial and stochastic effects, the mathematical formulations presented in the previous section were transformed into computational models based on multi-agent systems. The scientific hypotheses remain unchanged; only the representation changes—from systems of equations describing populations to autonomous agents whose interactions collectively reproduce population behavior.

Rather than representing an entire population by a set of continuous variables, a multi-agent system represents every individual as an autonomous computational agent.

Each agent occupies a location within a simulated environment and exists in one of the states defined by the mathematical framework:

• Susceptible

• Captured

• Assimilated

• Rescued

• Educated

• Defiant

The transition rates appearing in the differential equations become probabilistic interaction rules that determine how agents change state over time.

Instead of solving equations directly, the simulation repeatedly applies these local interaction rules to thousands of individual agents.

From these simple interactions, population-level behavior emerges naturally.

This shift illustrates one of the central ideas of computational science:

Complex collective behavior can emerge from the repeated application of simple local rules.

The computational implementation followed the same progressive philosophy used in constructing the mathematical models.

Beginning with the simplest two-state system (SAss), successive agent states were introduced one at a time to produce increasingly expressive computational models: 

SAss  --> SCA  --> SCAR --> SCARE --> SCARED

Each additional agent state introduced new interaction rules while preserving the structure of the previous models. The computational framework therefore mirrors the layered evolution of the mathematical framework presented earlier in the archive.

Once a computational world has been created, what scientific questions can we ask within it?

Unlike mathematical models, computational models allow researchers to investigate how spatial organization, stochastic interactions, and environmental assumptions influence the behavior of complex systems.

Rather than conducting a single simulation, this research employed a factorial experimental design in which three independent factors were systematically varied while observing their effects on generalized contagion dynamics.

Together, these factors define the computational world being investigated.

Factor 1: Agent-Based Model

The first factor determines the behavioral complexity of the simulated population.

Five progressively richer agent-based models were investigated:

• SAss

• SCA

• SCAR

• SCARE

• SCARED

Each successive model introduces one additional behavioral state while preserving the structure of its predecessor, mirroring the layered evolution of the mathematical framework.

Factor 2: Spatial Boundary Conditions

The second factor specifies how agents interact with the boundaries of the simulated environment.

Different boundary conditions alter encounter opportunities by changing how agents move and interact near the limits of the computational domain. Depending on the boundary condition, individuals may

• wrap around the domain (periodic)

• reflect from the boundary (reflective)

• remain confined within fixed limits

Boundary conditions influence encounter opportunities and therefore alter the dynamics of generalized contagion.

Factor 3: Initial Population Distribution

The third factor determines the initial spatial arrangement of agents before the simulation begins.

Different initial configurations include

• uniform

• random

• clustered

• Gaussian

Although the governing interaction rules remain identical, different initial conditions may produce substantially different emergent behaviors.

The Factorial Design

The complete computational experiment combines these three factors:

Agent-based Model × Boundary Condition × Initial Population Distribution --> Emergent Population Dynamics

Each simulation therefore represents a unique computational experiment defined by a particular combination of behavioral complexity, environmental structure, and initial population organization.

Before investigating spatial phenomena, the computational implementation was first validated against the mathematical models.

Under well-mixed assumptions, the agent-based simulations reproduce the deterministic behavior predicted by the systems of differential equations. This agreement demonstrates that the computational implementation faithfully represents the underlying mathematical formulations.

Validation establishes confidence that any differences observed under non-well-mixed conditions arise from the introduction of spatial organization and stochastic interactions rather than from errors in the computational implementation.

One of the first observations made by readers of simulation results is that the trajectories produced by agent-based models often appear noticeably different from the smooth curves generated by differential equations.

This difference is expected.

Systems of ordinary differential equations describe the average behavior of infinitely divisible populations under deterministic assumptions. Their solutions are therefore smooth and continuous.

In contrast, multi-agent simulations operate on discrete individuals whose interactions occur probabilistically.

Individual encounters happen at different times.

Some interactions occur by chance.

Others never occur at all.

Small random differences accumulate as the simulation progresses.

Consequently, simulation trajectories naturally exhibit small fluctuations around the average trend.

These fluctuations are not computational errors.

They are the expected signature of stochastic individual interactions.

As the number of agents increases or the results are averaged across many independent simulation runs, these fluctuations diminish and the aggregate behavior approaches the deterministic predictions of the mathematical model.

This agreement between the well-mixed agent-based simulations and the differential equation models provides an important validation of the computational implementation.

Once validated, the computational framework becomes a laboratory for investigating scientific questions that mathematical models alone cannot address.

Introducing explicit spatial organization allows researchers to examine how locality influences contagion dynamics.

Changing boundary conditions reveals how environmental geometry affects interactions.

Varying the initial distribution of agents explores the influence of historical circumstances on subsequent system evolution.

Together, these computational experiments extend the mathematical models beyond their classical assumptions and reveal behaviors that emerge only when individual interactions occur within space.

The computational framework presented here establishes the foundation for the simulation studies described in the following sections (upcoming).

Using the factorial experimental design introduced above, the subsequent pages examine how combinations of behavioral complexity, spatial organization, and initial conditions influence generalized contagion, revealing emergent phenomena that complement—and extend—the predictions of the mathematical models.