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Article Contents

# Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems

• * Corresponding author: Huseyin Coskun
• A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the mutually exclusive and exhaustive, analytical and dynamic, novel system and subsystem partitioning methodologies. A deterministic mathematical method is developed for the dynamic analysis of nonlinear compartmental systems based on the proposed theory. The dynamic method enables tracking the evolution of all initial stocks, external inputs, and arbitrary intercompartmental flows as well as the associated storages derived from these stocks, inputs, and flows individually and separately within the system. The transient and the dynamic direct, indirect, acyclic, cycling, and transfer ($\texttt{diact}$) flows and associated storages transmitted along a particular flow path or from one compartment-directly or indirectly-to any other are then analytically characterized, systematically classified, and mathematically formulated. Thus, the dynamic influence of one compartment, in terms of flow and storage transfer, directly or indirectly on any other compartment is ascertained. Consequently, new mathematical system analysis tools are formulated as quantitative system indicators. The proposed mathematical method is then applied to various models from literature to demonstrate its efficiency and wide applicability.

Mathematics Subject Classification: Primary: 34A34, 35A24, 37C60, 92C42; Secondary: 37N25, 37N40, 70G60, 91B74, 92B20, 92D30, 92D40, 93C15, 94A15.

 Citation:

• Figure 1.  Schematic representation of the dynamic subcompartmentalization in a three-compartment model system. Each subsystem is colored differently; the second subsystem ($k = 2$) is blue, for example. Only the subcompartments in the same subsystem ($x_{1_2}(t)$, $x_{2_2}(t)$, and $x_{3_2}(t)$ in the second subsystem, for example) interact with each other. Subsystem $k$ receives external input only at subcompartment ${k_k}$. The initial subsystem (gray) receives no external input. The dynamic flow decomposition is not represented in this figure. Compare this figure with Fig. 2, in which the subcompartmentalization and corresponding flow decomposition are illustrated for $x_1(t)$ only

Figure 2.  Schematic representation of the dynamic flow decomposition in a three-compartment model system. The figure illustrates the subcompartmentalization of compartment $1$ and the corresponding flow decomposition of $f_{j1}(t,\mathit{\boldsymbol{ x}})$. The figure also illustrates further decomposition of initial subcompartment $1_0$ and the corresponding initial subflow function, $f_{j_0 1_0}(t,\bf{x})$ (both dark gray)

Figure 3.  Schematic representation of the dynamic subsystem decomposition. The transient inflow and outflow rate functions, $f^w_{\ell_k j_k i_k}(t)$ and $f^w_{n_k \ell_k j_k}(t)$, at and associated transient substorage, $x^w_{n_k \ell_k j_k}(t)$, in subcompartment ${\ell_k}$ along subflow path $p^w_{n_k j_k} = i_k \mapsto j_k \to \ell_k \to n_k$

Figure 4.  Schematic representation of the simple and composite $\texttt{diact}$ flows. Solid arrows represent direct flows, and dashed arrows represent indirect flows through other compartments (not shown). The composite $\texttt{diact}$ flows (black) generated by outward throughflow $\hat{\tau}_{j}(t,\mathit{\boldsymbol{x}}) - \hat{\tau}_{j_0}(t,{\bf x})$ (i.e. derived from all external inputs): direct flow, $\tau^\texttt{d}_{i j}(t)$, indirect flow, $\tau^\texttt{i}_{ij}(t)$, acyclic flow, $\tau^\texttt{a}_{ij}(t) = \tau^\texttt{t}_{ij}(t) - \tau^\texttt{c}_{ij}(t)$, cycling flow, $\tau^\texttt{c}_{ij}(t)$, and transfer flow, $\tau^\texttt{t}_{ij}(t)$. The simple $\texttt{diact}$ flows (blue) generated by outward subthroughflow $\hat{\tau}_{i_i}(t,{\bf x})$ (i.e. derived from single external input $z_{i}(t)$): direct flow, ${\tau}^\texttt{d}_{j_i}(t) = {\tau}^\texttt{d}_{j_i i_i}(t)$, indirect flow, ${\tau}^\texttt{i}_{j_i}(t) = \tau^\texttt{i}_{j_i i_i}(t)$, acyclic flow, ${\tau}^\texttt{a}_{j_i}(t) = \tau^\texttt{a}_{j_i i_i}(t) = {\tau}^\texttt{t}_{j_i}(t) - {\tau}^\texttt{c}_{j_i}(t)$, cycling flow, ${\tau}^\texttt{c}_{j_i}(t) = \tau^\texttt{c}_{j_i i_i}(t)$, and transfer flow, $\tau^{\texttt{t}}_{j_i}(t) = \tilde{\tau}_{j_i}(t,{\bf x}) = \check{\tau}_{j_i}(t,{\bf x}) - z_{j_i}(t)$. Note that the cycling flows at the terminal (sub)compartment may include the segments of the direct and/or indirect flows at that (sub)compartment, if the cycling flows indirectly pass through the corresponding initial (sub)compartment (see Fig. 10). Therefore, the acyclic flows are composed of the segments of the direct and/or indirect flows

Figure 5.  Schematic representation of the model network. Arrows are labeled by the corresponding rate constants. Subflow paths $p^2_{0_1 1_1}$ and $p^3_{0_1 1_1}$ along which the transient external outputs are computed are red (subsystems are not shown) (Case study 3.1)

Figure 6.  Numerical results for the evolution of the initial populations, $\underline {\mathit{\boldsymbol{x}}} \left( t \right)$, and state variables, $\mathit{\boldsymbol{x}}(t)$. The populations generated by external inputs alone, $\bar{\mathit{\boldsymbol{x}}}(t)$, are presented in Fig. 7 (Case study 3.1)

Figure 7.  The graphical representations of the initial substate and substate functions ${\underline x _{{i_k}}}\left( t \right)$ and ${x}_{i_k}(t) = \bar{x}_{i}(t)$ for all $i,k$. The substates that are equal to zero are not labeled. (Case study 3.1)

Figure 8.  The graphical representations of the transient substorage functions at subcompartment $2_1$, $x^{1,m}_{3_1 2_1 1_1}(t)$, along path $p^1_{2_1 1_1}$, and the transient output rates $f^{2}_{0_1 2_1 1_1}(t)$ and $f^{3}_{0_1 2_1 1_1}(t)$ along paths $p^2_{0_1 1_1}$ and $p^3_{0_1 1_1}$, respectively (Case study 3.1)

Figure 9.  The graphical representation for the indirect storages from compartment $1$ to $3$ through $2$ within both the subsystems and initial subsystems, $x^\texttt{i}_{31}(t)$ and $x^\texttt{i}_{31}(t)$, respectively, and the residence times, $r_i(t)$ (Case study 3.1)

Figure 10.  Schematic representation for the complementary nature of the simple indirect and cycling flows within the $k^{th}$ subsystem. The composite direct subflow, $f_{k_k i_k}(t,{\bf x})$, is represented by solid arrow. This subflow also contributes to the simple cycling flow at subcompartment $k_k$. The simple indirect subflow, ${\tau}^\texttt{i}_{i_k k_k}(t)$, is represented by dashed arrow

Figure 11.  Schematic representation of the model network. Subflow path $p^1_{1_1}$, along which the cycling flow and storage functions are computed, is red (subsystems are not shown) (Case study E.1)

Figure 12.  The graphical representation of the substorage and inward subthroughflow matrices, $X(t)$ and $\check{T}(t)$, for time dependent input $z(t) = \left[ 3+\operatorname{sin}(t), 3+\operatorname{sin}(2 \, t) \right]^T$, and composite cycling flows and storages, $\tau^c_{i_0i_0}(t)+\tau^c_i(t)$ and $x^c_{i_0i_0}(t)+x^c_i(t)$ (Case study E.1)

Figure 13.  Schematic representation of the model network. (subsystems are not shown) (Case study E.2)

Table 1.  The dynamic $\texttt{diact}$ flow distribution and the simple and composite $\texttt{diact}$ (sub)flow matrices. The superscript ($^\texttt{*}$) in each equation represents any of the $\texttt{diact}$ symbols. For the sake of readability, the function arguments are dropped

 $\texttt{diact}$ flow distribution matrix flows $\texttt{d}$ $N^\texttt{d} = F \, \mathcal{T}^{-1}$ $\texttt{i}$ $N^\texttt{i}= {N}^\texttt{*} \, (\mathcal{T} - \mathcal{\hat T}_0)$ ${T^*} = {N^*}\left( {{\cal T} - {{\widehat {\cal T}}_0}} \right)$ $\texttt{a}$ ${N}^\texttt{a} = \tilde{T} \, \hat{\mathsf{T}}^{-1} - \tilde{\mathsf{T}} \, \hat{\mathsf{T}}^{-1} \, \hat{T} \, \hat{\mathsf{T}}^{-1}$ ${T}^\texttt{*}_\ell= {N}^\texttt{*} \, \mathcal{\hat T}_\ell$ $\texttt{c}$ ${N}^\texttt{c} = \tilde{\mathsf{T}} \, \hat{\mathsf{T}}^{-1} \, \hat{T} \, \hat{\mathsf{T}}^{-1}$ ${\tilde T^*} = {N^*}{\mkern 1mu} \mathsf{\hat T}$ $\texttt{t}$ $N^\texttt{t} = \tilde{T} \, \hat{\mathsf{T}}^{-1}$

Table 2.  The $\texttt{diact}$ flow and storage distribution and the simple and composite $\texttt{diact}$ (sub)flow and (sub)storage matrices. The superscript ($^\texttt{*}$) in each equation represents any of the $\texttt{diact}$ symbols

 $\texttt{diact}$ flow and storage distribution matrices flows storages $\texttt{d}$ $N^\texttt{d} = F \, \mathcal{T}^{-1}$ ${S}^\texttt{*} = \mathcal{R} \, {N}^\texttt{*}$ $\texttt{i}$ $N^\texttt{i} = (N-I) \, \mathcal{N}^{-1} - F \,\mathcal{T}^{-1}$ ${T}^\texttt{*} = {N}^\texttt{*} \, \mathcal{T}$ ${X}^\texttt{*} = {S}^\texttt{*} \, \mathcal{T}$ $\texttt{a}$ ${N}^\texttt{a} = (\mathcal{N}^{-1} \, N - I) \, \mathcal{N}^{-1}$ ${T}^\texttt{*}_\ell = {N}^\texttt{*} \, \mathcal{T}_\ell$ ${X}^\texttt{*}_\ell = {S}^\texttt{*} \, \mathcal{T}_\ell$ $\texttt{c}$ ${N}^\texttt{c} = (N - \mathcal{N}^{-1} N) \, \mathcal{N}^{-1}$ $\tilde{T}^\texttt{*} = {N}^\texttt{*} \, \mathsf{T}$ $\tilde{X}^\texttt{*} = {S}^\texttt{*} \, \mathsf{T}$ } $\texttt{t}$ $N^\texttt{t} = (N-I) \, \mathcal{N}^{-1}$
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