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We consider delayed chemical reaction networks with generalized kinetics of product form and show that complex balancing implies that within each positive stoichiometric compatibility class there is a unique positive equilibrium that is locally asymptotically stable relative to its class. The main tools of the proofs are respectively a version of the well-known classical logarithmic Lyapunov function applied to kinetic systems and its generalization to the delayed case as a Lyapunov-Krasovskii functional. Finally, we demonstrate our results through illustrative examples.

Incremental stability is assessed for the Unidirectional Flow Model, which describes the flow of a conserved quantity in a cascade of compartments. The model is governed by a cooperative system of nonlinear ordinary differential equations with a right-hand side that depends on an input function. Suitable conditions are given on the input function such that the solutions to the model converge to each other at an exponential rate. Milder conditions, compared with the latter, are given such that the solutions to the model converge to each other asymptotically. These conditions are related to inflow-connected and outflow-connected compartmental systems and are easy to check. Computational means to estimate the exponential rate are given and tested against numerical solutions of the model. A natural application of incremental stability is state estimation, demonstrated in the numerical experiments, put into a traffic flow context.

In this paper, we estimate epidemiological data of the COVID-19 pandemic in Hungary using only the daily number of hospitalized patients, and applying well-known techniques from systems and control theory. We use a previously published and validated compartmental model for the description of epidemic spread. Exploiting the fact that an important subsystem of the model is linear, first we compute the number of latent infected persons in time. Then an estimate can be given for the number of people in other compartments. From these data, it is possible to track the time dependent reproduction numbers via a recursive least squares estimate. The credibility of the obtained results is discussed using available data from the literature., Comment: 11 pages, 12 figures

A computation-oriented representation of uncertain kinetic systems is introduced and analysed in this paper. It is assumed that the monomial coefficients of the ODEs belong to a polytopic set, which defines a set of dynamical systems for an uncertain model. An optimization-based computation model is proposed for the structural analysis of uncertain models. It is shown that the so-called dense realization containing the maximum number of reactions (directed edges) is computable in polynomial time, and it forms a super-structure among all the possible reaction graphs corresponding to an uncertain kinetic model, assuming a fixed set of complexes. The set of core reactions present in all reaction graphs of an uncertain model is also studied. Most importantly, an algorithm is proposed to compute all possible reaction graph structures for an uncertain kinetic model., Comment: 23 pages, 5 figures

In this letter we introduce a class of delayed kinetic systems derived from mass action type reaction network models. We define the time delayed positive stoichiometric compatibility classes and the notion of complex balanced time delayed kinetic systems. We prove the uniqueness of equilibrium solutions within the time delayed positive stoichiometric compatibility classes for such models. In our main result we prove the semistability of the equilibrium solutions for complex balanced systems with arbitrary time delays using an appropriate Lyapunov-Krasovskii functional and LaSalle's invariance principle. As a consequence, we obtain that every positive complex balanced equilibrium solution is locally asymptotically stable relative to its positive stoichiometric compatibility class., Comment: 12 pages, 2 figures

In this paper an algorithm is given to determine all possible structurally different linearly conjugate realizations of a given kinetic polynomial system. The solution is based on the iterative search for constrained dense realizations using linear programming. Since there might exist exponentially many different reaction graph structures, we cannot expect to have a polynomial-time algorithm, but we can organize the computation in such a way that polynomial time is elapsed between displaying any two consecutive realizations. The correctness of the algorithm is proved, and possibilities of a parallel implementation are discussed. The operation of the method is shown on two illustrative examples., Comment: 19 pages, 7 figures

This paper studies the relations among system parameters, uniqueness, and stability of equilibria, for kinetic systems given in the form of polynomial ODEs. Such models are commonly used to describe the dynamics of nonnegative systems, with a wide range of application fields such as chemistry, systems biology, process modeling or even transportation systems. Using a flux-based description of kinetic models, a canonical representation of the set of all possible feasible equilibria is developed. The characterization is made in terms of strictly stable compartmental matrices to define the so-called family of solutions. Feasibility is imposed by a set of constraints, which are linear on a log-transformed space of complexes, and relate to the kernel of a matrix, the columns of which span the stoichiometric subspace. One particularly interesting representation of these constraints can be expressed in terms of a class of monotonous decreasing functions. This allows connections to be established with classical results in CRNT that relate to the existence and uniqueness of equilibria along positive stoichiometric compatibility classes. In particular, monotonicity can be employed to identify regions in the set of possible reaction rate coefficients leading to complex balancing, and to conclude uniqueness of equilibria for a class of positive deficiency networks. The latter result might support constructing an alternative proof of the well-known deficiency one theorem. The developed notions and results are illustrated through examples.

Mass-action kinetics is frequently used in systems biology to model the behaviour of interacting chemical species. Many important dynamical properties are known to hold for such systems if they are weakly reversible and have a low deficiency. In particular, the Deficiency Zero and Deficiency One Theorems guarantee strong regularity with regards to the number and stability of positive equilibrium states. It is also known that chemical reaction networks with disparate reaction structure can exhibit the same qualitative dynamics. The theory of linear conjugacy encapsulates the cases where this relationship is captured by a linear transformation. In this paper, we propose a mixed-integer linear programming algorithm capable of determining weakly reversible reaction networks with a minimal deficiency which are linearly conjugate to a given reaction network., Comment: 26 pages, 3 figures

In the first part of this paper, we propose new optimization-based methods for the computation of preferred (dense, sparse, reversible, detailed and complex balanced) linearly conjugate reaction network structures with mass action dynamics. The developed methods are extensions of previously published results on dynamically equivalent reaction networks and are based on mixed-integer linear programming. As related theoretical contributions we show that (i) dense linearly conjugate networks define a unique super-structure for any positive diagonal state transformation if the set of chemical complexes is given, and (ii) the existence of linearly conjugate detailed balanced and complex balanced networks do not depend on the selection of equilibrium points. In the second part of the paper it is shown that determining dynamically equivalent realizations to a network that is structurally fixed but parametrically not can also be written and solved as a mixed-integer linear programming problem. Several examples illustrate the presented computation methods., Comment: 29 pages, 1 figure

A numerically effective procedure for determining weakly reversible chemical reaction networks that are linearly conjugate to a known reaction network is proposed in this paper. The method is based on translating the structural and algebraic characteristics of weak reversibility to logical statements and solving the obtained set of linear (in)equalities in the framework of mixed integer linear programming. The unknowns in the problem are the reaction rate coefficients and the parameters of the linear conjugacy transformation. The efficacy of the approach is shown through numerical examples., Comment: 19 pages

An algorithm is given in this paper for the computation of dynamically equivalent weakly reversible realizations with the maximal number of reactions, for chemical reaction networks (CRNs) with mass action kinetics. The original problem statement can be traced back at least 30 years ago. The algorithm uses standard linear and mixed integer linear programming, and it is based on elementary graph theory and important former results on the dense realizations of CRNs. The proposed method is also capable of determining if no dynamically equivalent weakly reversible structure exists for a given reaction network with a previously fixed complex set., Comment: 18 pages, 9 figures

An algorithm is given in this paper for the computation of dynamically equivalent weakly reversible realizations with the maximal number of reactions, for chemical reaction networks (CRNs) with mass action kinetics., Comment: This paper has been withdrawn by the author because an important case was not taken into consideration in the algorithm description

Reversibility, weak reversibility and deficiency, detailed and complex balancing are generally not "encoded" in the kinetic differential equations but they are realization properties that may imply local or even global asymptotic stability of the underlying reaction kinetic system when further conditions are also fulfilled. In this paper, efficient numerical procedures are given for finding complex balanced or detailed balanced realizations of mass action type chemical reaction networks or kinetic dynamical systems in the framework of linear programming. The procedures are illustrated on numerical examples., Comment: submitted to J. Math. Chem

This paper presents new results about the optimization based generation of chemical reaction networks (CRNs) of higher deficiency. Firstly, it is shown that the graph structure of the realization containing the maximal number of reactions is unique if the set of possible complexes is fixed. Secondly, a mixed integer programming based numerical procedure is given for computing a realization containing the minimal/maximal number of complexes. Moreover, the linear inequalities corresponding to full reversibility of the CRN realization are also described. The theoretical results are illustrated on meaningful examples., Comment: 23 pages, 4 figures