:: Scientific Models > Mathematical Models
The previous section introduced scientific models as complementary ways of investigating complex systems. This page explores the first of those perspectives by expressing the conceptual framework as systems of differential equations.
How do researchers transform observations about complex human interactions into mathematical models that can be analyzed, simulated, and ultimately used to better understand real-world systems?
Every scientific discipline develops its own language for describing the phenomena it seeks to understand. In computational science, mathematics serves as that language. Rather than replacing observation or experimentation, mathematical models complement them by expressing relationships with precision, allowing researchers to explore scenarios that would otherwise be difficult, impractical, expensive, or ethically impossible to investigate directly.
In this project, fictional epidemiological events were not modeled simply because they were entertaining stories. They were chosen because they provide controlled environments in which different forms of contagion—biological, social, behavioral, and informational—can be examined without the ethical and practical constraints associated with real-world systems. Once the essential interactions are identified, they can be represented mathematically, analyzed computationally, and interpreted scientifically.
The mathematical models presented in this archive therefore represent successive stages of abstraction. Each model was designed to answer a particular scientific question while remaining connected to a common conceptual framework. Together they illustrate how computational scientists gradually refine their understanding of a complex phenomenon by constructing a family of related models rather than searching for a single "perfect" model.
One of the defining characteristics of computational science is the ability to construct abstract representations of complex systems. Before writing equations, researchers first decide what should be represented, what assumptions are reasonable, and which details can be temporarily ignored without losing the essential behavior of the system.
The research presented in this archive followed precisely this process. Instead of immediately developing sophisticated mathematical equations, the investigation progressed through a sequence of increasingly refined abstractions (visually explained by the figure).
This progression illustrates an important principle of computational thinking: effective models are not copies of reality; they are purposeful simplifications that preserve the relationships necessary to answer a scientific question.
Although the project produced several mathematical models, they should not be viewed as isolated equations developed independently of one another. Instead, they form a family of related models that systematically explore different assumptions about contagion, recovery, rehabilitation, and long-term resilience.
The original proposal envisioned this family of models as part of a broader design space whose dimensions included:
the fictional phenomenon being represented (vampires, zombies, Borg, aswangs, and others);
the conceptual framework describing the possible states of individuals;
the mathematical formulation governing transitions between states;
the spatial representation of the population (zero-, one-, and two-dimensional);
the initial distribution of individuals;
and the boundary conditions governing movement through space.
Rather than producing a single simulation, the project was designed to investigate how different combinations of these factors influence the overall dynamics of the modeled population. In this sense, the research represents not merely the construction of mathematical models but the systematic exploration of a computational design space.
The progression of models developed throughout the project reflects an evolving scientific understanding of recoverable contagion. Each successive model was motivated by a new research question that could not be answered by its predecessor.
Scientific Question: How does a population behave before any fictional contagion is introduced?
The foundation of every epidemiological model is an understanding of the population in the absence of disease or contagion. Model 0 therefore establishes the baseline dynamics against which all subsequent fictional epidemics can be compared. It serves as the control model for the entire framework.
Continue to Model 0
Scientific Question: What happens when susceptible individuals become permanently assimilated after exposure?
The simplest contagion model introduces irreversible transformation from susceptible individuals to assimilated individuals. This model captures the fundamental mechanism of transmission while deliberately omitting intermediate stages of capture, rescue, or rehabilitation.
Continue to Model 1
Scientific Question: Can transmission be understood more accurately if capture and assimilation are modeled as separate processes?
Rather than assuming that transformation occurs immediately after contact, the SCA model distinguishes between being captured and becoming fully assimilated. This additional compartment allows the model to represent delayed progression and provides a richer description of the underlying interaction dynamics.
Continue to Model 2
Scientific Question: What changes when individuals can be rescued before permanent assimilation?
The introduction of rescue fundamentally changes the character of the epidemic. Unlike classical terminal epidemic models, SCAR introduces the possibility of intervention, allowing the system to explore how rescue efforts influence the long-term evolution of the population.
Continue to Model 3
Scientific Question: Is rescue alone sufficient, or must rescued individuals undergo rehabilitation before fully returning to society?
Real-world social problems rarely end at rescue. Individuals often require education, rehabilitation, counseling, or recovery before they can successfully reintegrate into the community. The SCARE model explicitly incorporates this transition, making the mathematical framework applicable to a broader class of recoverable social phenomena.
Continue to Model 4
Scientific Question: Can recovery produce long-term resilience against future contagion?
The final model extends the framework by introducing the Defiant compartment. Rather than merely recovering, rehabilitated individuals become resistant to future assimilation and may even contribute to protecting susceptible members of the population. This extension represents one of the project's principal conceptual contributions: recovery is modeled not as a return to the original state but as the emergence of a new state characterized by acquired resilience.
Continue to Model 5
To many readers, differential equations appear to be collections of unfamiliar symbols. Computational scientists, however, read them differently. Each equation tells the story of how a population changes through time, while each mathematical term corresponds to a specific interaction identified in the conceptual framework.
Every transition between compartments represents a scientific assumption about how individuals interact. Each parameter quantifies the strength of those interactions. Together, the equations provide a precise description of how local interactions give rise to population-level behavior.
The equations therefore represent more than mathematical notation—they are formal descriptions of scientific hypotheses.
Mathematics alone does not answer scientific questions. The systems of equations developed in this project become computational models only after they are translated into algorithms that computers can repeatedly evaluate.
Numerical methods approximate the continuous behavior described by differential equations, allowing thousands of alternative scenarios to be explored efficiently under different assumptions, parameter values, and spatial conditions. Computer simulation thus serves as the bridge between mathematical theory and scientific discovery.
Although this archive focuses primarily on the conceptual and mathematical foundations of the models, it also points toward the broader discipline of scientific computing, where mathematical formulations become software capable of supporting experimentation, visualization, and hypothesis generation.
Viewed individually, each mathematical model represents a specific contribution to the project. Viewed collectively, however, they reveal something more significant: the gradual evolution of a framework for modeling recoverable forms of contagion. Rather than extending classical epidemiological models solely by adding mathematical complexity, the research progressively incorporated concepts such as rescue, rehabilitation, education, and long-term resilience—features that make the framework applicable to many real-world social systems beyond infectious disease.
The models therefore document not only the mathematics developed during the project but also the evolution of the underlying scientific questions that motivated their creation.
Scientific models are never final. Every model is both a summary of current understanding and an invitation to ask better questions.
As you explore the models presented in this archive, consider the following:
Which assumptions would you retain, and which would you reconsider?
What other forms of contagion—biological, social, technological, or informational—might benefit from a similar modeling framework?
How might stochastic processes, network structures, adaptive behavior, or machine learning enrich these models?
Could entirely new compartments be introduced to represent phenomena not envisioned in the original project?
How would you translate these mathematical models into scalable scientific software for modern computational platforms?
The history preserved in this archive is therefore not only a record of completed research. It is also an invitation to future computational scientists to continue the conversation.
Mathematics becomes a scientific language when it transforms ideas into relationships that can be explored, questioned, and tested.
The mathematical models developed in this section establish the theoretical foundations of generalized contagion. Yet mathematics represents only one perspective. To investigate how these same principles unfold through individual interactions, spatial organization, and stochastic processes, the next section transforms these equations into computational models based on multi-agent systems.