Table 1.

Mathematical and computational techniques for modeling immune processes.

Technique	Description	Comments
ODE	Ordinary differential equations: describe the rate of change with respect to one other variable (e.g., population change over time, t)	Commonly used technique that can be used to quantify changes in population size over time
PDE	Partial differential equations: describe rate of change of a function of more than one variable with respect to one of those variables (e.g., motion through space x, y, and z as a function of time t)	Often used to describe changes occurring over both time and multiple spatial dimensions
Monte Carlo	Statistical random sampling method where outcomes are determined at random from input probability distribution functions	Stochastic technique to model deterministic processes, very frequently integrated within ABM, CPM, and other stochastic modeling approaches
Petri nets	Graph-based model describing network of events or “transitions” that occur depending on given conditions or “places,” a stochastic methodology	Computationally efficient can be effectively defined using SBML2. Capturing explicit spatial representation can be difficult
ABMs	Agent-based models are composed of individual entities specified as agents, which exist independently in a well-defined state: a set of attributes at a specific point in, e.g., time and space, with state transitions governed by a rule-set, often described in terms of finite state machines and other diagrammatic constructs using the Unified Modeling Language	There are a number of methodologies to generate ABMs. There are tools with user interfaces for constructing simpler lattice-based ABMS or “unconstrained” models manually coded as software in languages such as Java and C++
(Extended) cellular Potts modeling	A lattice-based modeling technique for simulating the collective behavior of cells. A cell is defined as a set of pixels within a lattice (sharing a “spin state”) and is updated pixel-by-pixel according to a mathematical function, which incorporates cell volume and surface/adhesion energies	Similar to an ABM but relies on effective energy functions (the Hamiltonian) to describe cellular adhesion, signaling, motility, and other physical phenomena
Hybridized models	Bringing together a range of different techniques generally within the context of an ABM or CPM, incorporating differential equations and a variety of other mathematical and computational techniques to effectively capture phenomena occurring over different spatiotemporal scales (e.g., intracellular activity)	Can take advantage of different modeling techniques, particularly applicable where there are multiple processes occurring in different scales of time and space