Three-State Compartmental Model

• Three-state compartmental models are mathematical frameworks that partition systems into three distinct states with specified transition rates and nonlinear interactions.
• They capture complex dynamics such as traveling waves, bistability, and shock solutions, with applications across epidemiology, nuclear medicine, and social sciences.
• Analytical and numerical methods validate these models, enabling robust parameter inference and practical applications in diverse real-world systems.

A three-state compartmental model is a mathematical representation whereby a system, process, or population is dynamically partitioned into three distinct compartments or states, each governed by their own dynamics and coupled via transition rates. This framework is broadly utilized in epidemiology, biophysics, nuclear medicine, population dynamics, and social sciences to capture key mechanisms such as transport, lifecycle progression, or status changes that cannot be adequately described by simpler two-state models. The quintessential feature is explicit modeling of flows and transformations among three categories—typically two mobile/progressive states and one sedentary, absorbing, or intermediate state—with the possibility of elaborate transition networks, nonlinear feedbacks, and interaction terms.

1. Model Structures and Mathematical Formulation

Three-state compartmental models universally define three state variables, typically denoted as populations or concentrations $X_1(t)$, $X_2(t)$, $X_3(t)$, evolving according to coupled ordinary or partial differential equations. A canonical generic system reads:

$$\begin{aligned} \dot{X}_1 &= f_1(X_1, X_2, X_3, t) \ \dot{X}_2 &= f_2(X_1, X_2, X_3, t) \ \dot{X}_3 &= f_3(X_1, X_2, X_3, t) \end{aligned}$$

where the $f_i$ encode transitions due to endogenous kinetics, exogenous input, loss, and possibly nonlinear interactions.

The archetypal three-state "run-and-tumble + cell-cycle" PDE for Caulobacter crescentus is structured as:

$$\partial_t\begin{pmatrix}n_+ \ n_0 \ n_- \end{pmatrix} = D\partial_x^2 \begin{pmatrix}n_+ \ n_0 \ n_- \end{pmatrix} - \mathcal{V}\partial_x \begin{pmatrix}n_+ \ n_0 \ n_- \end{pmatrix} + \mathcal{M} \begin{pmatrix}n_+ \ n_0 \ n_- \end{pmatrix}$$

with $n_+$, $n_-$, $n_0$ representing densities of right-moving, left-moving, and settled cells (Breoni et al., 2022).

In nuclear medicine, a tracer-kinetics 3-compartmental ODE takes the canonical form:

$$\begin{cases} \dot{C}_1 = - (k_{e1} + k_{21}) C_1 + k_{12} C_2 + k_{1e} C_{1e}(t) \ \dot{C}_2 = k_{21} C_1 - (k_{12} + k_{32}) C_2 + k_{23} C_3 \ \dot{C}_3 = k_{32} C_2 - k_{23} C_3 \end{cases}$$

(Delbary et al., 2016)

Key variants include PDEs (with age-time structure for epidemiology (Brinks, 2014)), stochastic master equations (gene switching (Rubinstein et al., 2022)), and algebraic reductions with integrals of motion for demographic models (Postnikov, 2015).

2. Transition Networks and Biological Interpretations

Model structure is dictated by domain-specific transitions:

• Cell motility/cell cycle models: Right-movers and left-movers interconvert (tumble; $\lambda_e$), each can irreversibly settle into a sedentary state ($\lambda_s$); settled cells divide, repopulating the motile classes ($\lambda_d$); motile cells are subject to removal ($\mu$) (Breoni et al., 2022).
• Medical tracer models: Input flows from plasma to tissue compartments, with reversible and irreversible exchanges, excretion, and experimentally measurable output as weighted compartment sums (Delbary et al., 2016, Velten et al., 2023).
• Gene expression: "Inactive," "poised," "active" states transition sequentially; mRNA is produced only in the active state, degraded elsewhere (Rubinstein et al., 2022).
• Demography: Population, economic surplus, and literacy interact nonlinearly, with two integrals of motion (Postnikov, 2015).
• Public health (illness-death): Healthy, ill, and dead compartments, with incidence, remission, and state-dependent mortality rates (Brinks, 2014).
• Social trust: "Trusters," "skeptics," and "doubters" interact via bounded confidence, with peer influence, birth, and death (Meylahn et al., 2024).
• Transport models: Empty/strongly/weakly bound states in motor-mediated transport, including exclusion and internal biochemical kinetics (Zhang, 2011).

This diversity reflects the flexibility of the three-state paradigm, underpinning rich phenomena such as cell-wave formation, traffic jams, stochastic bursts, and bistability.

3. Analytical Solutions, Invariants, and Identifiability

Three-state models frequently admit analytical reduction via integrals of motion or generating function techniques:

• Demography: Two first integrals relate literacy and surplus to population; system reduces to a single nonlinear ODE, which interpolates between Malthusian, Kremer-type, and logistic (Thoularis–Wallace, Gompertz) growth (Postnikov, 2015).
• Tracer kinetics: Identifiability is proven via coprimality and resultant of transfer function polynomials in Laplace domain; uniqueness of physiological rate estimation depends on measurable output composition, with global injectivity for blood fraction $V\geq 1/2$ (Delbary et al., 2016).
• Gene expression: Steady-state distributions obtained exactly as ${}_2F_2$ generalized hypergeometric functions, yielding explicit formulas for all moments (Rubinstein et al., 2022).
• Public health prevalence: A single PDE links prevalence to all transmission and mortality rates, enabling direct incidence estimation from cross-sectional data (Brinks, 2014).

These results guarantee both structural and practical identifiability under parameter and measurement constraints, and facilitate robust parameter inference via classical and Bayesian computational techniques (Robinson et al., 2023).

4. Dynamical Phenomena: Waves, Shocks, Bifurcations, and Stationarity

Three-state models exhibit nontrivial dynamical regimes:

• Super-ballistic scaling: For cell-cycle-motility models, the settled-cell mean-square displacement scales as $t^3$ at early times, reflecting compounded “divide → swim → settle” kinetics (Breoni et al., 2022).
• Traveling waves/band formation: Nonlinear chemotactic attraction and repulsion drive wave-like aggregation patterns, with stability boundaries computable via linear analysis and numerical PDE integration (Breoni et al., 2022).
• Traffic jams (shock solutions): In intracellular transport, mean-field equations admit domain-wall solutions separating low- and high-density phases, with positions and heights controlled by biochemical rates and boundary conditions (Zhang, 2011).
• Bistability and bifurcations: Nonlinear phase diagrams with saddle-node bifurcations arise in lead pharmacokinetics and public health models, leading to history-dependent multi-equilibria (Radulescu et al., 2019, Brinks, 2014, Meylahn et al., 2024).
• Stationary states and extinction: Social-trust and crime dynamics display multiple stationary points whose stability depends on persuasion (μ), contagion (α), and removal (ν) rates—a non-monotonic relationship exists between convincing power and group survival (Meylahn et al., 2024, Ramponi et al., 23 May 2025).

Many systems are reducible analytically, but stochastic variants (Gillespie simulation) and diffusion approximations reveal rare events (extinction, bursts) inaccessible to the ODE mean-field limit (Meylahn et al., 2024).

5. Empirical Applications and Domain-Specific Examples

Three-state compartmental models have been validated and deployed in numerous domains:

Area	Model Roles	Key Features/Results
Microbiology	Swarmers/stalked phases of CC	Stationary ratios, $t^3$ scaling, waves, nonlinear stability (Breoni et al., 2022)
Nuclear medicine	Plasma/EES/hepatocyte tracer kinetics	Identifiability, data-fitting, biomarker extraction (Delbary et al., 2016, Velten et al., 2023)
Epidemiology	Healthy/ill/dead states	Prevalence PDE, incidence estimation, simulations (Brinks, 2014)
Gene regulation	Inactive/poised/active gene	Exact stationary distribution (${_2F_2}$), moment formulas (Rubinstein et al., 2022)
Demography	Population/surplus/literacy	Integrals of motion, unified ODE, growth phase transitions (Postnikov, 2015)
Pharmacokinetics	Blood/bone/soft tissue compartments	Bistability, historic dependence, nonlinear feedback (Radulescu et al., 2019)
Social Science	Trust/skeptic/doubter interactions	ODE and stochastic extinction, non-monotonicity (Meylahn et al., 2024)
Crime Modeling	Susceptible/incarcerated/educated	$R_0$ threshold, stability, empirical fitting (Ramponi et al., 23 May 2025)
Intracellular transport	Empty/strong/weak motor binding	Shock structure, phase diagrams, exclusion effects (Zhang, 2011)

6. Extensions, Limitations, and Generalizations

Three-state models are extensible in multiple directions:

• Higher-order compartmentalizations for more detailed lifecycle or kinetic steps.
• PDEs with space, age, or other structure (spatiotemporal or cohort models).
• Non-Markovian extensions via expanded state space, memory terms, or duration dependence.
• Covariate stratification for multipopulation models.
• Model-agnostic Bayesian and frequentist estimation frameworks supporting time-varying parameters, observational noise, and state-space filtering (Robinson et al., 2023).
• Limitations include dependence on Markov assumption, closed population requirement, identifiability restriction due to observable composition, and computational cost for stochastic inference.

These limitations are mitigated by analytic reductions, simulation studies, robust parameter identification results, and empirical validation across domains.

7. Impact and Significance

The three-state compartmental model provides a minimal yet versatile approximation of complex biological, physiological, and social systems. It enables researchers to:

• Capture intertwined nonlinear kinetics and transition sequences not representable in two-state models.
• Extract physically or epidemiologically meaningful parameters analytically or computationally.
• Predict long-term dynamics, including transitions, waves, metastable regimes, and stochastic phenomena.
• Calibrate and validate real-world interventions or innovations based on rigorous mathematical infrastructure, ensuring interpretability of fitted parameters and model outputs.

Its application has elucidated unexpected phenomena (e.g., super-ballistic spreading, bistability, traveling waves, extinction risks), provided quantifiable levers for policy and clinical intervention, and unified widely disparate modeling traditions under a common mechanistic framework.