
Previous Article
Coinfection in a stochastic model for bacteriophage systems
 DCDSB Home
 This Issue

Next Article
Cyclicity of $ (1,3) $switching FF type equilibria
Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems
Department of Mathematics, University of Georgia, Athens, GA 30602, USA 
A decomposition principle for nonlinear dynamic compartmental systems is introduced in the present paper. This theory is based on the mutually exclusive and exhaustive, analytical and dynamic, novel system and subsystem partitioning methodologies. A deterministic mathematical method is developed for the dynamic analysis of nonlinear compartmental systems based on the proposed theory. The dynamic method enables tracking the evolution of all initial stocks, external inputs, and arbitrary intercompartmental flows as well as the associated storages derived from these stocks, inputs, and flows individually and separately within the system. The transient and the dynamic direct, indirect, acyclic, cycling, and transfer ($\texttt{diact}$) flows and associated storages transmitted along a particular flow path or from one compartmentdirectly or indirectlyto 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.
References:
[1] 
D. Anderson, Compartmental Modeling and Tracer Kinetics, Lecture Notes in Biomathematics, 50. SpringerVerlag, Berlin, 1983. doi: 10.1007/9783642518614. Google Scholar 
[2] 
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part i, Nature, 280 (1979), 361367. doi: 10.1038/280361a0. Google Scholar 
[3] 
H. Coskun, Dynamic ecological system analysis, Heliyon, 5 (2019). Preprint: https://doi.org/10.31219/osf.io/35xkb. Google Scholar 
[4] 
H. Coskun, Dynamic ecological system measures, Results in Applied Mathematics, 2019. Preprint: https://doi.org/10.31219/osf.io/j2pd3. Google Scholar 
[5] 
H. Coskun, Static ecological system analysis, Theoretical Ecology, 2019, 136. Preprint: https://doi.org/10.31219/osf.io/zqxc5. Google Scholar 
[6] 
H. Coskun, Static ecological system measures, Theoretical Ecology, 2019, 126. Preprint: https://doi.org/10.31219/osf.io/g4xzt. Google Scholar 
[7] 
L. EdelsteinKeshet, Mathematical Models in Biology, Classics in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898719147. Google Scholar 
[8] 
B. Hannon, The structure of ecosystems, Journal of Theoretical Biology, 41 (1973), 535546. Google Scholar 
[9] 
P. W. Hippe, Environ analysis of linear compartmental systems: The dynamic, timeinvariant case, Ecological Modelling, 19 (1983), 126. Google Scholar 
[10] 
J. Jacquez and C. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 4379. doi: 10.1137/1035003. Google Scholar 
[11] 
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 115 (1927), 700–721, http://rspa.royalsocietypublishing.org/content/115/772/700. Google Scholar 
[12] 
W. W. Leontief, Quantitative input and output relations in the economic systems of the united states, The Review of Economic Statistics, 18 (1936), 105125. Google Scholar 
[13] 
W. W. Leontief, Inputoutput Economics, Oxford University Press on Demand, New York, 1986. Google Scholar 
[14] 
B. C. Patten, Systems approach to the concept of environment, Ohio Journal of Science, 78 (1978), 206222. Google Scholar 
[15] 
J. Szyrmer and R. E. Ulanowicz, Total flows in ecosystems, Ecological Modelling, 35 (1987), 123136. Google Scholar 
[16] 
L. J. Tilly, The structure and dynamics of Cone Spring, Ecological Monographs, 38 (1968), 169197. Google Scholar 
show all references
References:
[1] 
D. Anderson, Compartmental Modeling and Tracer Kinetics, Lecture Notes in Biomathematics, 50. SpringerVerlag, Berlin, 1983. doi: 10.1007/9783642518614. Google Scholar 
[2] 
R. M. Anderson and R. M. May, Population biology of infectious diseases: Part i, Nature, 280 (1979), 361367. doi: 10.1038/280361a0. Google Scholar 
[3] 
H. Coskun, Dynamic ecological system analysis, Heliyon, 5 (2019). Preprint: https://doi.org/10.31219/osf.io/35xkb. Google Scholar 
[4] 
H. Coskun, Dynamic ecological system measures, Results in Applied Mathematics, 2019. Preprint: https://doi.org/10.31219/osf.io/j2pd3. Google Scholar 
[5] 
H. Coskun, Static ecological system analysis, Theoretical Ecology, 2019, 136. Preprint: https://doi.org/10.31219/osf.io/zqxc5. Google Scholar 
[6] 
H. Coskun, Static ecological system measures, Theoretical Ecology, 2019, 126. Preprint: https://doi.org/10.31219/osf.io/g4xzt. Google Scholar 
[7] 
L. EdelsteinKeshet, Mathematical Models in Biology, Classics in Applied Mathematics, 46. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2005. doi: 10.1137/1.9780898719147. Google Scholar 
[8] 
B. Hannon, The structure of ecosystems, Journal of Theoretical Biology, 41 (1973), 535546. Google Scholar 
[9] 
P. W. Hippe, Environ analysis of linear compartmental systems: The dynamic, timeinvariant case, Ecological Modelling, 19 (1983), 126. Google Scholar 
[10] 
J. Jacquez and C. Simon, Qualitative theory of compartmental systems, SIAM Review, 35 (1993), 4379. doi: 10.1137/1035003. Google Scholar 
[11] 
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 115 (1927), 700–721, http://rspa.royalsocietypublishing.org/content/115/772/700. Google Scholar 
[12] 
W. W. Leontief, Quantitative input and output relations in the economic systems of the united states, The Review of Economic Statistics, 18 (1936), 105125. Google Scholar 
[13] 
W. W. Leontief, Inputoutput Economics, Oxford University Press on Demand, New York, 1986. Google Scholar 
[14] 
B. C. Patten, Systems approach to the concept of environment, Ohio Journal of Science, 78 (1978), 206222. Google Scholar 
[15] 
J. Szyrmer and R. E. Ulanowicz, Total flows in ecosystems, Ecological Modelling, 35 (1987), 123136. Google Scholar 
[16] 
L. J. Tilly, The structure and dynamics of Cone Spring, Ecological Monographs, 38 (1968), 169197. Google Scholar 
$\texttt{diact}$  flow distribution matrix  flows 
$\texttt{d}$  
$\texttt{i}$  
$\texttt{a}$  
$\texttt{c}$  
$\texttt{t}$ 
$\texttt{diact}$  flow distribution matrix  flows 
$\texttt{d}$  
$\texttt{i}$  
$\texttt{a}$  
$\texttt{c}$  
$\texttt{t}$ 
$\texttt{diact}$  flow and storage distribution matrices  flows  storages  
$\texttt{d}$  

$\texttt{i}$  
$\texttt{a}$  
$\texttt{c}$  
$\texttt{t}$ 
$\texttt{diact}$  flow and storage distribution matrices  flows  storages  
$\texttt{d}$  

$\texttt{i}$  
$\texttt{a}$  
$\texttt{c}$  
$\texttt{t}$ 
[1] 
Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391421. doi: 10.3934/jcd.2014.1.391 
[2] 
Alessia Marigo, Benedetto Piccoli. A model for biological dynamic networks. Networks & Heterogeneous Media, 2011, 6 (4) : 647663. doi: 10.3934/nhm.2011.6.647 
[3] 
Magdi S. Mahmoud. Output feedback overlapping control design of interconnected systems with input saturation. Numerical Algebra, Control & Optimization, 2016, 6 (2) : 127151. doi: 10.3934/naco.2016004 
[4] 
James P. Nelson, Mark J. Balas. Direct model reference adaptive control of linear systems with input/output delays. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 445462. doi: 10.3934/naco.2013.3.445 
[5] 
Xilin Fu, Zhang Chen. New discrete analogue of neural networks with nonlinear amplification function and its periodic dynamic analysis. Conference Publications, 2007, 2007 (Special) : 391398. doi: 10.3934/proc.2007.2007.391 
[6] 
Andrey Olypher, Jean Vaillant. On the properties of inputtooutput transformations in neuronal networks. Mathematical Biosciences & Engineering, 2016, 13 (3) : 579596. doi: 10.3934/mbe.2016009 
[7] 
Toufik Bakir, Bernard Bonnard, Jérémy Rouot. A case study of optimal inputoutput system with sampleddata control: Ding et al. force and fatigue muscular control model. Networks & Heterogeneous Media, 2019, 14 (1) : 79100. doi: 10.3934/nhm.2019005 
[8] 
Alexei Pokrovskii, Dmitrii Rachinskii. Effect of positive feedback on Devil's staircase inputoutput relationship. Discrete & Continuous Dynamical Systems  S, 2013, 6 (4) : 10951112. doi: 10.3934/dcdss.2013.6.1095 
[9] 
Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 7584. doi: 10.3934/cpaa.2004.3.75 
[10] 
Bogdan Sasu, A. L. Sasu. Inputoutput conditions for the asymptotic behavior of linear skewproduct flows and applications. Communications on Pure & Applied Analysis, 2006, 5 (3) : 551569. doi: 10.3934/cpaa.2006.5.551 
[11] 
Suoqin Jin, FangXiang Wu, Xiufen Zou. Domain control of nonlinear networked systems and applications to complex disease networks. Discrete & Continuous Dynamical Systems  B, 2017, 22 (6) : 21692206. doi: 10.3934/dcdsb.2017091 
[12] 
Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial & Management Optimization, 2011, 7 (2) : 497521. doi: 10.3934/jimo.2011.7.497 
[13] 
Mustaffa Alfatlawi, Vaibhav Srivastava. An incremental approach to online dynamic mode decomposition for timevarying systems with applications to EEG data modeling. Journal of Computational Dynamics, 2020, 7 (2) : 209241. doi: 10.3934/jcd.2020009 
[14] 
Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial & Management Optimization, 2008, 4 (1) : 115. doi: 10.3934/jimo.2008.4.1 
[15] 
TaiPing Liu, ShihHsien Yu. Hyperbolic conservation laws and dynamic systems. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 143145. doi: 10.3934/dcds.2000.6.143 
[16] 
Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonaldiffusion equation. Discrete & Continuous Dynamical Systems  S, 2016, 9 (1) : 109123. doi: 10.3934/dcdss.2016.9.109 
[17] 
Changjun Yu, Honglei Xu, Kok Lay Teo. Preface: Advances in theory and real world applications of control and dynamic optimization. Discrete & Continuous Dynamical Systems  S, 2020, 13 (6) : iiii. doi: 10.3934/dcdss.2020094 
[18] 
Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165191. doi: 10.3934/jcd.2015002 
[19] 
Hao Zhang, Scott T. M. Dawson, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta. Evaluating the accuracy of the dynamic mode decomposition. Journal of Computational Dynamics, 2020, 7 (1) : 3556. doi: 10.3934/jcd.2020002 
[20] 
Quoc T. Luu, Paul DuChateau. The relative biologic effectiveness versus linear energy transfer curve as an outputinput relation for linear cellular systems. Mathematical Biosciences & Engineering, 2009, 6 (3) : 591602. doi: 10.3934/mbe.2009.6.591 
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]