Four-Compartment Dynamical Model

• Four-compartment dynamical model is a framework that tracks four interacting subsystems via coupled differential equations, capturing nonlinear behaviors such as oscillations and chaos.
• Analytical methods like polynomial Lyapunov functionals and system decomposition ensure global existence and facilitate detailed bifurcation analysis.
• Applications in epidemiology, pharmacokinetics, and chemical kinetics illustrate its power in modeling complex systems through advanced simulation and data-driven techniques.

A four-compartment dynamical model is a framework in which the time evolution of a system is described by tracking the states of four interacting subsystems, or “compartments.” Each compartment is governed by its own dynamic equation, and the full system may include nonlinear interactions, memory effects, stochasticity, or spatial structure. This modeling paradigm is prevalent in fields such as chemical kinetics, epidemiology, pharmacology, biology, systems engineering, and complex network science. Four-compartment models can exhibit rich behaviors—such as oscillations, chaos, critical transitions, and intricate equilibrium structures—which require specialized analytical and computational tools for characterization.

1. Mathematical Structures and Representative Equations

A prototypical structure for a four-compartment dynamical model involves a system of coupled ordinary or partial differential equations. For example, in the spatially-extended reaction–diffusion Brusselator system, the state variables $u$, $v$, $w$, $z$ satisfy:

$$\begin{aligned} &u_t - a\Delta u = \alpha - (\beta + 1)u + u^2 v + D_1(w-u), \ &v_t - b\Delta v = \beta u - u^2 v + D_2(z-v), \ &w_t - c\Delta w = \alpha - (\beta + 1)w + w^2 z + D_3(u-w), \ &z_t - d\Delta z = \beta w - w^2 z + D_4(v-z), \ &u = v = w = z = 0 \text{ on } \partial \Omega. \end{aligned}$$

Such systems are distinguished by the presence of nonlinearities with non-constant sign, diffusion terms, and linear or nonlinear couplings among the compartments (Parshad et al., 2015). In epidemiology, a canonical example is the SCIRS model, where each compartment corresponds to a specific epidemiological state (e.g., Susceptible, Exposed, Infectious, Recovered), and transitions may involve memory-encoded or stochastic waiting times (Granger et al., 2022, Granger et al., 2023). In pharmacokinetics and pharmacodynamics (PK–PD), four compartments might represent drug distribution across spatial domains or cell states, with integer- or fractional-order kinetics (Daryakenari et al., 19 Sep 2024).

2. Analytical Methods: Existence, Long-Time Dynamics, and Decomposition

The analysis of four-compartment models necessitates tools adapted to the presence of nonlinear, sign-changing reaction terms and possible high-dimensional phase spaces. For the reaction–diffusion system, global existence of solutions is established via the construction of a polynomial Lyapunov functional $L_n(t)$, which enables the derivation of uniform-in-time a priori bounds even in the absence of classical sign conditions. The evolution of $L_n(t)$ is controlled by inequalities of the form

$L_n'(t) \leq C_5 L_n(t) + C_7 L_n(t)^{(n-1)/n},$

which, together with Poincaré inequality and integral estimates, rule out finite-time blow-up for both weak and strong solutions (Parshad et al., 2015).

For general nonlinear compartmental models, a comprehensive decomposition principle allows the tracking of substate and subflow dynamics originating from initial stocks, external inputs, and internal transfers. The governing equations are decomposed so that, for $n=4$, each compartment is separated into five subcompartments, and the corresponding flows are partitioned according to their source. This approach yields direct quantification of storage, throughflows, and the dynamic influence (direct, indirect, cyclic) of each compartment on the others (Coskun, 2018).

3. Memory, Stochasticity, and Advanced Model Features

Retarded or memory-dependent transition rates are a salient feature in modern four-compartment modeling, especially in epidemiology and chemical kinetics. For example, in SCIRS epidemic models with distributed delays, each compartment transition is characterized by a random waiting time drawn from a given PDF, leading to evolution equations of the form

$$\begin{aligned} \frac{ds}{dt} &= -\mathcal{A}(t) + \langle \mathcal{A}(t-t_C-t_I-t_R)\rangle, \ \frac{dc}{dt} &= \mathcal{A}(t) - \langle \mathcal{A}(t-t_C)\rangle, \ \frac{dj}{dt} &= \langle \mathcal{A}(t-t_C)\rangle - \langle \mathcal{A}(t-t_C-t_I)\rangle, \ \frac{dr}{dt} &= \langle \mathcal{A}(t-t_C-t_I)\rangle - \langle \mathcal{A}(t-t_C-t_I-t_R)\rangle, \end{aligned}$$

where $\langle\,\cdot\,\rangle$ denotes averaging over the waiting time distributions. This structure is recast via convolutions or, for fat-tailed PDFs, fractional derivatives (Granger et al., 2022, Granger et al., 2023). The stability, endemic equilibrium, and existence conditions for outbreaks are then linked analytically to the mean waiting times and the basic reproduction number $R_0$.

Recent advances have introduced stochastic simulation algorithms for compartment models with non-Markovian, fractional-order kinetics. These Monte Carlo methods combine Gillespie-type sampling with “clock tracking” for fractional waiting times (e.g., Mittag–Leffler), rigorously connecting the microstate dynamics to the mean-field fractional ODEs and revealing differences between deterministic and stochastic outcomes in small populations (Angstmann et al., 2023).

4. Compositionality, Model Reduction, and Dimensionality

Analyzing higher-dimensional models by direct state-space enumeration is generally intractable. Compositional methods and reduction techniques significantly mitigate this challenge:

• In the steady-state regime, coupled open dynamical systems’ equilibria are “compressed” into steady state matrices. Composition operations—serial (matrix multiplication), parallel (Kronecker product), and feedback (partial trace)—allow calculation of the global equilibrium structure for interconnected four-compartment systems by modular composition, thus avoiding exponential complexity (Spivak, 2015).
• For complex multi-compartment systems, effective dimension reduction can be achieved by mapping the master equation to a Doi–Peliti Hamiltonian, imposing conservation laws, and “freezing” all but a single order parameter. The reduced Hamiltonian $H^*(\eta,\theta)$ then governs the effective one-dimensional stochastic dynamics, enabling analysis of bifurcations, mean-field critical behaviors, and the structure of fluctuations (e.g., via associated Fokker–Planck or Langevin equations) (Visco et al., 17 Apr 2024). This approach provides efficient identification of stationary states, phase transitions (continuous or discontinuous), and stationary distributions in models originally defined in high dimensions.

5. Applications: Epidemics, Pharmacokinetics, and Beyond

Four-compartment models are applied across multiple scientific domains:

• Epidemiology: Extended SEIR and SCIRS models incorporate incubation, infection, recovery, and loss-of-immunity (resusceptibility) compartments. Delayed transition rates reflect realistic infection and immunity period distributions. Analytical criteria connect the disease-free and endemic states to $R_0 = \beta \langle t_I \rangle$; simulation studies using random walker models validate the theory and allow exploration of spatial effects, stochasticity, and oscillatory endemic states (Granger et al., 2022, Granger et al., 2023).
• Pharmacodynamics / Pharmacokinetics: In tumor modeling under chemotherapy, a four-compartment structure allows for fine-grained modeling of cell-state transitions, delay compartments, and tissue trapping. Fractional derivatives and time-varying parameters (e.g., for drug resistance or tolerance) can be incorporated. Physics-informed neural networks (PINNs) and fractional PINNs are used to infer hidden parameters, estimate time-dependent kinetics, and fit experimental data, providing insights into distributed delays, anomalous drug absorption, and adaptive responses (Daryakenari et al., 19 Sep 2024).
• Chemical Kinetics and Ecology: Generalizations of the Brusselator and related models to four compartments reveal chaotic attractors, high correlation dimensions (e.g., ~27.5), and finite fractal attractor dimension. Simulation-based attractor reconstruction (using Takens’ theorem and Grassberger–Procaccia algorithm) confirms the presence of complex spatiotemporal behavior, with numerical and theoretical bounds on the invariant set’s dimension (Parshad et al., 2015).

6. Numerical Techniques and Data-Driven Frameworks

Advanced data-driven and machine learning-based methods are increasingly being applied to four-compartment models:

• Dynamic Mode Decomposition (DMD): Coupled DMD approaches yield accurate low-dimensional reconstructions and preserve conservation laws in systems with strong inter-compartment coupling. Decoupled (compartment-wise) DMD can overlook conservation constraints and mutual influence, degrading predictive performance (Viguerie et al., 2021).
• Physics-Informed Neural Networks (PINNs/fPINNs): PINNs can embed ODE/PDE constraints directly into the loss, facilitating data assimilation and parameter inference even in the presence of fractional dynamics. The architecture typically involves multiple linear and nonlinear (feedback) layers, optimizing for both data fit and differential equation residual minimization. Scalability and generalization to larger datasets and more complex compartmental structures are supported (Ma, 2022, Daryakenari et al., 19 Sep 2024).

7. Attractors, Bifurcations, and Critical Behavior

Four-compartment models generically support rich phase-space phenomena, including chaotic attractors (with finite Hausdorff and fractal dimension), critical transitions, and bifurcations:

• Attractors: The existence, boundedness, and finite-dimensionality of global attractors can be established analytically by Lyapunov functionals and uniform Gronwall inequalities. The dimension of the attractor is bounded both from above (using trace estimates of the Fréchet derivative, Sobolev–Lieb–Thirring inequalities, and the geometry of the domain) and from below (by linearizing about steady states and analyzing the eigenvalue spectrum) (Parshad et al., 2015).
• Critical Transitions: Model reduction to an order parameter and the use of Hamiltonian formalism (e.g., via Cole–Hopf variable transformations and phase portrait analysis) permit detailed paper of phase transition structure (first-order vs. second-order), extinction events, and fluctuation-induced behavior in stochastic extensions (Visco et al., 17 Apr 2024).
• Comparison with Traditional Approaches: Classical compartment modeling, when generalized with fractional derivatives, retarded transitions, or compositional state space compression, shows improved descriptive adequacy for systems exhibiting anomalous transport, distributed delays, or emergent resistance/tolerance mechanisms—phenomena poorly captured by models with only Markovian transitions or fixed parameters.

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In summary, the four-compartment dynamical model provides a high-fidelity framework for modeling complex, interacting subsystems with nonlinear, stochastic, memory, and spatial features. Analytical, numerical, and machine learning-based methods collaboratively enable rigorous analysis of existence, attractor properties, critical transitions, and data-driven parameter estimation, with demonstrated applicability in fields ranging from chemical kinetics and ecology to epidemiology and pharmacodynamics (Parshad et al., 2015, Spivak, 2015, Coskun, 2018, Granger et al., 2022, Granger et al., 2023, Angstmann et al., 2023, Visco et al., 17 Apr 2024, Daryakenari et al., 19 Sep 2024).