Metzler Matrix Research Articles

State Control Design of Ostensible Metzler Linear Systems with Unsigned Input Parameters

This paper deals with the design of a complete state control for input unsigned, rank deficient matrix parameters of a linear system with system dynamics defined by ostensible structures of Metzler matrices. The proposed solution is based on the principle of diagonal stabilization of positive systems and uses a stabilizing additional component over the decomposition of the Metzler matrix in solving the incomplete internal positivity of such linear system structures. The novelty of the proposed approach is the unified representation of the parametric constraints of the Metzler matrix and the structurally constrained system inputs using linear matrix inequalities, which guarantees that the closed-loop system will be asymptotically stable. Despite the complexity of the constraint conditions on this class of linear continuous systems, the design conditions are formulated using sharp linear matrix inequalities only. A detailed design process is presented using a system-linearized mathematical model to verify the superiority and practicality of the proposed method.

Global stability analysis and modelling onchocerciasis transmission dynamics with control measures

ABSTRACT Background: Onchocerciasis infection is one of the neglected tropical diseases targeted for eradication by 2030. The disease is usually transmitted to humans through the bites of black flies. These black flies mostly breed near well-oxygenated fast-running water bodies. The disease is common in mostly remote agricultural villages near rivers and streams. Objective: In this study, a deterministic model describing the infection dynamics of human onchocerciasis disease with control measures is presented. Methods: We derived the model’s reproductive number and used a stability theorem of a Metzler matrix to show that disease-free equilibrium is both locally and globally asymptotically stable whenever the reproductive number is less than one. Parameter contribution was conducted using sensitivity analysis. The model endemic equation is shown to be a cubic polynomial in the presence of infected immigrants and a quadratic form in their absence. Results: When the inflow of infected immigrants is null, the model endemic equation may admit a unique equilibrium if the reproductive number is greater than one, or admits multiple endemic equilibria if the reproductive number is less than unity. We carried out a sensitivity analysis to identify the significant parameters that contribute to onchocerciasis spread. Conclusion: Onchocerciasis disease can be eradicated if the importation of infected immigrants is properly monitored. The integration of the One Health concept in the public health system is key in tackling the emergence and spread of diseases.

Modeling the Inflow of Exposed and Infected Migrants on the Dynamics of Malaria

Malaria is currently a life-threatening vector borne disease which is endemic in most of the developing and underdeveloped countries associated with poor health care systems. In this study, a host-vector mathematical model that takes into account the inflow of human migrants who have been exposed or infected with malaria is formulated and analysed. The reproduction number of the mosquito vector population is derived and used as a threshold quantity for determining the existence of the model trivial and realistic steady states. The Routh-Hurwitz criterion and some stability theorems of Metzler matrices are used to show that the realistic disease free equilibrium is both locally and globally asymptotically stable whenever the disease reproductive number is less than one. We derived an equation for the model endemic condition and used Descartes Rule of Sign Change to established the conditions for the model to admit one or three endemic equilibrium state(s). It is further shown that in the absence of inflow of exposed or infected migrants, the model admits a globally asymptotically unique endemic equilibrium when R0>1 and two endemic equilibria when R0<1. Our local sensitivity analysis revealed that the adults mosquito removal and biting rates were respectively the most significant contributing parameters to the spread of malaria. The numerical simulations results suggested that the exposed and infected immigrants have no significant impact on the dynamical behaviour of the model population sub-classes.

Unstable cores are the source of instability in chemical reaction networks

In biochemical networks, complex dynamical features such as superlinear growth and oscillations are classically considered a consequence of autocatalysis. For the large class of parameter-rich kinetic models, which includes generalized mass action kinetics and Michaelis–Menten kinetics, we show that certain submatrices of the stoichiometric matrix, so-called unstable cores, are sufficient for a reaction network to admit instability and potentially give rise to such complex dynamical behaviour. The determinant of the submatrix distinguishes unstable-positive feedbacks, with a single real-positive eigenvalue, and unstable-negative feedbacks without real-positive eigenvalues. Autocatalytic cores turn out to be exactly the unstable-positive feedbacks that are Metzler matrices. Thus there are sources of dynamical instability in chemical networks that are unrelated to autocatalysis. We use such intuition to design non-autocatalytic biochemical networks with superlinear growth and oscillations.

Mathematical analysis of in-host Ebola virus infection dynamics model with sensitivity analysis

Ebola virus (EBOV) causes a haemorrhagic and lethal Ebola disease disastrous to human beings, which is transmitted by contact of body fluids of infected animals and humans. Presently, there are no therapies for the disease. In this paper, a mathematical model is proposed to investigate the in-vivo dynamics of EBOV infection with sensitivity analysis. A system of five non-linear ordinary differential equations constitutes the model, from which the basic reproduction number, R0 is calculated using the next generation matrix method. The parameter R0 is employed to analyze global stability of disease-free and endemic equilibria. Using the Metzler matrix operator, the results indicate that the disease-free equilibrium point is globally asymptotically stable provided that R0 < 1, which implies that the disease disappears from the host after some period of time. With Lyapunov Stability Theory and LaSalle Invariant Principle, the results indicate that the endemic equilibrium point is globally asymptotically stable provided that R0 > 1, which implies that the disease persists in the host. Sensitivity analysis of the basic reproduction number pertaining to the model parameters is achieved using forward normalized sensitivity index method. The results indicate that the parameters for infection rate, production rate of uninfected target cells and virus replication rate are positively sensitive. On the other hand, the parameters for natural death rate of target cells and natural death rate of the virus are negatively sensitive, implying that the basic reproduction number decreases as the parameters increase and vice versa. Besides, it is shown that the parameter for the infection rate is the most sensitive one while the parameter for the virus reproduction rate is the least sensitive one. Numerical simulations are used to validate the analytical results. The results suggest implementation of deliberate control measures to eradicate EBOV disease by considering sites in the model to which the most sensitive parameters are affiliated.

On exponential stability of switched functional differential equations with average dwell-time

We present a novel approach to the exponential stability of switched functional differential equations. Our approach does not involve Lyapunov functions. It is simple and based upon spectral properties of Metzler matrices, a comparison principle and the average dwell-time technique. Consequently, some new explicit criteria for the exponential stability of switched functional differential equations are derived. Some illustrative examples are given.

Ostensible Metzler Linear Uncertain Systems: Goals, LMI Synthesis, Constraints and Quadratic Stability

This paper deals with the design problem for a class of linear continuous systems with dynamics prescribed by the system matrix of an ostensible Metzler structure. The novelty of the proposed solution lies in the diagonal stabilization of the system, which uses the idea of decomposition of the ostensible Metzler matrix, preserving the incomplete positivity of the system during the synthesis. The proposed approach creates a unified framework that covers compactness of interval system parameter representation, Metzler parametric constraints, and quadratic stability. Combining these extensions, all of the conditions and constraints are expressed as linear matrix inequalities. Implications of the results, both for design and for research directions that follow from the proposed method, are discussed at the end of the paper. The efficiency of the method is illustrated by a numerical example.

Metzler asymptotic stability of initial time linear time-varying real-order systems

Determining the asymptotics of many time-varying systems associated with real orders remains a challenging issue in qualitative asymptotic theory. This paper introduces a new concept of Metzler asymptotic stability to linear time-varying real-order systems. The design class is considered in standard and block matrix forms whenever the initial time is defined on the real axis. The non-negativity of such systems is discussed by introducing a time-dependent Metzler matrix. New theoretical conditions are developed that guarantee the asymptotic stability of such systems. It is shown that, to ensure the Metzler asymptotic stability of such systems; it suffices to construct a suitable linear non-negative asymptotic stable system where the coefficient matrix should be Metzler. Examples illustrate the potential practical and promising applicability of the introduced results.

A Convex Formulation for Robust Control of Bilinear Positive Systems With Evolutionary Dynamics

In this article, optimal control of bilinear positive systems with uncertain evolutionary dynamics is studied. The first objective is to determine optimal control signals such that the effect of the disturbance signals on the outputs of the nominal dynamics is minimized in terms of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm performance index. Properties of the positive systems, Metzler matrices, and the exploitable structure of the problem are utilized to show that the underlying problem is representable as a standard convex optimization problem. Next, the optimal strategy is examined in the presence of polytopic uncertainties. In this case, the optimization problem is a semi-infinite min-max problem. It is shown that under some assumptions, the underlying problem is equivalent to a finite min-max problem, i.e., it suffices to check only the vertices of the polytopic set for <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal{H}_2$</tex-math> </inline-formula> norm minimization. The proposed approach for polytopic bilinear positive systems is non-iterative, does not involve parameter tuning or nonsmooth optimization algorithms, and can be solved using off-the-shelf solvers. The effectiveness of the proposed approach is investigated for the optimal combination therapy of human immunodeficiency virus (HIV).

Metzler/Zeta Correspondence

We present a formula for determinants on the Metzler matrices of digraphs. Furthermore, we introduce a walk-type zeta function with respect to this Metzler matrix of a finite torus graph and express its limit formula by using the integral expression.

When is a Matrix of Dimension Three Similar to a Metzler Matrix? Application to Interval Observer Design

A simple, necessary, and sufficient condition ensuring that a real matrix of dimension three is similar to a Metzler matrix is exhibited. When this condition is satisfied, a construction of the transfer matrix is given. This construction is used to design an interval observer for a family of continuous-time systems. An example is provided with interval observer design for the so-called love dynamics in the case of limit cycles

Novel global exponential stability results for a class of two-coupled-hub nonlinear genetic regulatory networks with time-varying delays

This work studies global exponential stability analysis for a class of nonlinear genetic regulatory networks (GRNs) with two-coupled-hub structure and time-varying delays. It is important to study the influence of hub genes in GRNs problem, due to there is the complexity arising from the interactions among hub gene nodes. In this article, the existence of GRN’s nonnegative equilibriums is presented and a sufficient condition such that all nonnegative equilibriums are positive can be further proved. In addition, novel global exponential stability (GES) criteria of coupled nonlinear GRN is established based on a Metzler matrix method. Furthermore, under the proposed delay-dependent and -independent GES sufficient conditions, it proves the considered GRNs has a unique non-negative GES equilibrium point. Compared with the existing GES criteria, it is worth stressing that the proposed GES conditions are easily verified by computing relative matrices eigenvalues and induced norms. Finally, the validity of the theoretical results is verified by three simulation examples.

Feedback stabilization and robustness analysis using bounds on fundamental matrices

We prove a new robust stabilization theorem for time-varying systems that have sampled measurements and time-varying disturbances, and we provide new observer designs, using novel bounds for fundamental matrices for systems with disturbances. Our main tools use properties of Metzler matrices and positive systems.

Modelling the impact of health care providers in transmission dynamics of COVID-19

In this paper, a mathematical model is proposed and analysed to assess the impacts of health care providers in transmission dynamics of COVID-19. The stability theory of differential equations is used to examine a mathematical model. The results of both local and global stability of disease-free equilibrium points were determined by using Routh–Hurwitz criteria and Metzler matrix method which verified that was locally and globally asymptotically stable. Also, the endemic equilibrium point was determined by the Lyapunov function which showed that E∗ was globally asymptotically stable under strict conditions. The findings revealed that non-diagnosed and undetected health care providers seems to contribute to high spread of COVID-19 in a community. Also, it illustrates that an increase in the number of non-diagnostic testing rates of health care providers may result in high infection rates in the community and contaminations of hospitals’ equipment. Therefore, the particular study recommend that there is a necessity of applying early diagnostic testing to curtail the COVID-19 transmission in the health care providers’ community and reduce contaminations of hospital’s equipment.

Modeling nosocomial infection of COVID-19 transmission dynamics

COVID-19 epidemic has posed an unprecedented threat to global public health. The disease has alarmed the healthcare system with the harm of nosocomial infection. Nosocomial spread of COVID-19 has been discovered and reported globally in different healthcare facilities. Asymptomatic patients and super-spreaders are sough to be among of the source of these infections. Thus, this study contributes to the subject by formulating a SEIHR mathematical model to gain the insight into nosocomial infection for COVID-19 transmission dynamics. The role of personal protective equipment θ is studied in the proposed model. Benefiting the next generation matrix method, R0 was computed. Routh–Hurwitz criterion and stable Metzler matrix theory revealed that COVID-19-free equilibrium point is locally and globally asymptotically stable whenever R0<1. Lyapunov function depicted that the endemic equilibrium point is globally asymptotically stable when R0>1. Further, the dynamics behavior of R0 was explored when varying θ. In the absence of θ, the value of R0 was 8.4584 which implies the expansion of the disease. When θ is introduced in the model, R0 was 0.4229, indicating the decrease of the disease in the community. Numerical solutions were simulated by using Runge–Kutta fourth-order method. Global sensitivity analysis is performed to present the most significant parameter. The numerical results illustrated mathematically that personal protective equipment can minimizes nosocomial infections of COVID-19.

Finite-time event-triggered control for switched affine systems with time-varying delay under state-dependent switching

Finite-time event-triggered control problem for switched affine systems with time-varying delay is investigated in this paper. A dynamical event-triggered scheme with multiple parameter matrices is proposed to avoid the waste of communication resources, which is more general than some existing event-triggered schemes with a common parameter matrix. By applying the proposed event-triggered scheme, a state-dependent switching signal and a state feedback controller are designed. By utilising an appropriate Lyapunov–Krasovskii functional and Metzler matrices, sufficient conditions for the existence of controller gains are firstly proposed such that the closed-loop system is finite-time bounded with finite-time disturbance attenuation performance. It is also proved that there is a lower bound between two adjacent triggers, i.e. Zeno phenomenon can be avoided. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed method.

On the similarity to nonnegative and Metzler Hessenberg forms

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

Positivity and uniform exponential stability for Volterra integro-dynamical systems on time scales

The main objective of this study is two-fold: First, we provide necessary and sufficient conditions for the positivity of Volterra integro-dynamical systems on time scales by means of Metzler matrices. Secondly, we show that positivity of the system implies uniform exponential stability of the trivial solution under sufficient conditions.

Convergence in nonautonomous linear differential equations with Kirchhoff coefficients

Abstract A class of nonautonomous linear ordinary differential equations is considered. The coefficient matrices are Metzler matrices with zero column sums such that their directed graphs have a common directed spanning tree. It is shown that if the off-diagonal elements of the coefficients are uniformly positive along the common directed spanning tree, then under mild additional assumptions the convergence of the Perron vectors of the coefficient matrices implies that all solutions tend to a finite limit at infinity. The value of the limit can be expressed in terms of the initial data.

Positive Consensus in Fractional-Order Interval Networked Systems

This brief investigates the positive consensus of fractional-order interval networked systems with the order α meeting 0 <; α ≤ 1 and 1 <; α <; 2, respectively. Based on the theories of fractional stability and properties of Metzler matrix, by virtue of some rigorous theoretical analyses, the distributed control protocol is proposed and sufficient conditions of positive consensus for fractional-order interval networked systems with 0 <; α ≤ 1 are given, which have no connection with the order. What's more, the results can be extended to conditions without using the information of Laplacian matrix with respect to networked topology. Furthermore, sufficient conditions for the systems with 1 <; α <; 2 to reach positive consensus are presented in terms of linear matrix inequality. Finally, simulation examples are given to verify the effective of results.