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SARS COV-2 (Covid-19) has imposed a monumental socio-economic burden worldwide, and its impact still lingers. We propose a deterministic model to describe the transmission dynamics of Covid-19, emphasizing the effects of vaccination on the prevailing epidemic. The proposed model incorporates current information on Covid-19, such as reinfection, waning of immunity derived from the vaccine, and infectiousness of the pre-symptomatic individuals into the disease dynamics. Moreover, the model analysis reveals that it exhibits the phenomenon of backward bifurcation, thus suggesting that driving the model reproduction number below unity may not suffice to drive the epidemic toward extinction. The model is fitted to real-life data to estimate values for some of the unknown parameters. In addition, the model epidemic threshold and equilibria are determined while the criteria for the stability of each equilibrium solution are established using the Metzler approach. A sensitivity analysis of the model is performed based on the Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficients (PRCCs) approaches to illustrate the impact of the various model parameters and explore the dependency of control reproduction number on its constituents parameters, which invariably gives insight on what needs to be done to contain the pandemic effectively. The foregoing notwithstanding, the contour plots of the control reproduction number concerning some of the salient parameters indicate that increasing vaccination coverage and decreasing vaccine waning rate would remarkably reduce the value of the reproduction number below unity, thus facilitating the possible elimination of the disease from the population. Finally, the model is solved numerically and simulated for different scenarios of disease outbreaks with the findings discussed.

In this paper, the non-zero finite positive state consensus problem of leader-following multi-agent systems (MASs) connected over a directed network is investigated. The multi-agent network consists of homogeneous linear time-invariant (LTI) positive agents whose minimal positive state-space realization is assumed to be known. In this paper, a state-feedback hierarchical control protocol is proposed where the local controller synthesizes a singular, Lyapunov stable, and Metzler system matrix with a simple dominant eigenvalue at origin. The obtained consensus vector is the positive eigenvector associated with the zero eigenvalue of the synthesized system matrix. The controller gain matrices are obtained by solving a set of necessary and sufficient conditions derived in linear programing framework. A numerical example is given to elucidate the usefulness of the proposed results. [ABSTRACT FROM AUTHOR]

The positivity and cyclicity of descriptor linear electrical circuits with chain structure is considered. Two classes of descriptor linear electrical circuits are analyzed. Some new properties of these classes of electrical circuits are established. The results are extended to fractional descriptor linear electrical circuits. [ABSTRACT FROM AUTHOR]

This paper investigates the problem of positive stabilizability of single‐input LTI positive systems. Firstly, for a single‐input continuous‐time positive linear system, it has been derived that a necessary condition of the existence of a stablilizing linear time‐invariant controller is the number of nonnegative real poles not being greater than one. Inspired by that, the continuous‐time positive stabilizability of systems with unstable complex poles is studied in this paper, where the third‐order and higher‐order cases are considered. Secondly, an enhanced method to construct the static state‐feedback gain of a sampled positive system is presented, which takes account of the positivity preservation in the sampling interval. According to the method, the positive stabilizablity of the single‐input second‐order sampled positive system is analyzed. Some numerical examples are finally presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

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The cyclicity of the state matrices of positive linear electrical circuits with the chain structure is considered. Two classes of positive linear electrical circuits with the chain structure and cyclic Metzler state matrices are analyzed. Some new properties of these classes of positive electrical circuits are established. The results are extended to fractional linear electrical circuits. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we investigate the problem of finite-time stability (FTS) of linear non-autonomous systems with time-varying delays. By constructing an appropriated function, we derive some explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Finally, two examples are given to show the effectiveness of the main results.

Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz, and the distance between matrices is measured in l ∞ , l 1 , and in the max norm. We provide either explicit solutions or efficient algorithms for obtaining the closest (un)stable matrix. The procedure for finding the closest stable Metzler matrix is based on the recently introduced selective greedy spectral method for optimizing the Perron eigenvalue. Originally intended for non-negative matrices, here is generalized to Metzler matrices. The efficiency of the new algorithms is demonstrated in examples and numerical experiments for the dimension of up to 2000. Applications to dynamical systems, linear switching systems, and sign-matrices are considered. [ABSTRACT FROM AUTHOR]

This chapter proposes a novel interval observer for continuous-time linear systems with unknown disturbance. Based on the stability criterion of a Metzler matrix, the interval observer design problem is converted into a series of nonlinear inequalities. To attenuate the effect of unknown disturbance on estimation error, an interval observer design method based on constrained optimization is proposed. The proposed interval observer is able to estimate upper and lower bounds of the states in the general assumption that disturbance is unknown but bounded. Thus it is particularly suitable for fault detection for uncertain linear systems. Therefore, the proposed method is further used to generate dynamic thresholds to achieve fault detection. Finally, a flight control system is simulated to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

This paper is concerned with the positivity and stability of linear systems in continuous-time form with stochastic disturbances. Firstly, the definition of stochastic positivity in the sense of probability is introduced, which characterizes the minimum level for a stochastic linear system to be positive. Secondly, through rigorous analysis, sufficient criterion is derived under which the stochastic differential equation is stochastically positive in the sense of probability. Next, by resorting to the properties of Metzler matrices, mathematical expectation of the solution for the stochastic linear system is investigated to be always positive under some mild restrictions. Then, the exponentially p-moment stability is addressed for the stochastic linear positive system. Finally, two numerical examples are provided to demonstrate the applicability and effectiveness of the derived theoretical results. [ABSTRACT FROM AUTHOR]

To gain more insight into the molecular mechanism that regulates eukaryotic cell cycle, we propose a modification to an existing Mathematical model of mitosis-promoting factor control in Xenopus oocyte extract. To achieve this, we redirect how cell exits from mitosis using ubiquitin ligase enzyme APCP:Cdc20 which is acting as a modifier in cyclin degradation. We present a modified wiring map using combination of mass action kinetics and Michaelis-Menten kinetics to represent the biochemical processes. This results in a system of non-linear ordinary differential equations. The equilibria of the system were analyzed analytically. Using MPF as a bifurcation parameter, we analytically derived conditions under which bistability can occur in the MPF activation module. We find that for bistability to occur, it is necessary that the rate of phosphorylation of Wee1 must be greater than the rate of its dephosphorylation and that the concentration of MPF must lie in certain range of parameter values. Numerical simulations conducted show that the time taken to exit the cell cycle is shorter than those published in some reports and the sensitivity analysis results suggest that the rate of synthesis of cyclin is the most sensitive parameter in the dynamics of cell cycle. Through numerical experiments, we found new parameter sets that will give rise to bistability in the MPF activation module. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of finite-time stability (FTS) of linear non-autonomous systems with time-varying delays. By constructing an appropriated function, we derive some explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Finally, two examples are given to show the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The master equation plays an important role in many scientific fields including physics, chemistry, systems biology, physical finance and sociodynamics. We consider the master equation with periodic transition rates. This may represent an external periodic excitation like the 24 h solar day in biological systems or periodic traffic lights in a model of vehicular traffic. Using tools from systems and control theory, we prove that under mild technical conditions every solution of the master equation converges to a periodic solution with the same period as the rates. In other words, the master equation entrains (or phase locks) to periodic excitations. We describe two applications of our theoretical results to important models from statistical mechanics and epidemiology.

This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations.

In this paper, an interval sliding mode observer design method for uncertain systems is proposed. Uncertainty is assumed between a known minimum value and a maximum value. The observer is then constructed via a convex weighted sum of an upper estimator corresponding to the maximum value of the uncertainty and a lower estimator corresponding to the minimum value of the uncertainty. The weighting factor is calculated at each time, from the different measured outputs and the bounds of the interval of the estimator. [ABSTRACT FROM AUTHOR]

This paper considers an extension of diagonal stability for continuous fractional positive linear systems (FPLS) with the fractional order 0 < α < 1 . Based on diagonal stability of Metzler matrix, an extension that involves the combination of a diagonal positive definite matrix and a skew-symmetric anti-diagonal matrix related a Hurwitz and Metzler matrix is established. Combining this extension and the linear matrix inequalities (LMIs) criteria of stability for fractional order systems (FOS), a necessary and sufficient condition for extension diagonal stability of FPLS is presented. A state feedback controller is given, which ensures the stabilization and positivity of the closed-loop systems. Numerical examples are provided to demonstrate the effectiveness and applicability of the proposed methods. [ABSTRACT FROM AUTHOR]

To date, our knowledge of Rift Valley fever (RVF) disease spread and maintenance is still limited, as flooding, humid weather and presence of biting insects such as mosquitoes, have not completely explained RVF outbreaks. We propose a model that includes livestock, mosquitoes and ticks compartments structured according to their questing and feeding behaviour in order to study the possible role of ticks on the dynamics of RVF. To quantify disease transmission at the initial stage of the epidemic, we derive an explicit formula of the basic reproductive number, $$R_0$$ . Using the concept of Metzler matrix, we state necessary conditions for global asymptotic stability of the disease-free equilibrium. Results suggest that although host-ticks interactions may serve as disease reservoirs or disease amplifiers, the Aedes reproductive number should be kept under unity if disease post-epizootics activities are to be controlled. Results of both local and global sensitivity analysis of selected model parameters indicate that $$R_0$$ is more sensitive to the ticks attachment and detachment rates, probability of transmission from ticks to host and from host to ticks, length of infection in livestock and ticks death rate. Furthermore, when comparing the mean value of $$R_0$$ with that from previous studies which did not include ticks we found that our $$R_0$$ is very much larger resulting in an increase in the exponential phase of an outbreak. These findings suggest that if ticks are capable of transmitting the virus, they may be contributing to disease outbreaks and endemicity. [ABSTRACT FROM AUTHOR]

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey–Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number R 0 and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium—usually termed the endemic equilibrium (EE)—that exists if, and only if, R 0 > 1 . We also show that, if R 0 ≤ 1 , then the DFE equilibrium is globally asymptotically stable, while when R 0 > 1 , we have that the EE is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

Objectives: Mathematical modelling is of interest to study the dynamics of coronavirus disease 2019 (COVID-19), and models such as SEIR (Susceptible-Exposed-Infected-Recovered) have been considered. This article describes the development of a compartmental transmission network model - Susceptible-Exposed-Quarantine-Infectious-Infectious, undetected-Infectious, home-based care-Hospitalized-Vaccinated-Recovered-Dead - to simulate the dynamics of COVID-19 in order to account for specific measures put into place by the Government of Rwanda to prevent further spread of the disease., Methods: The compartments of this model are connected by parameters, some of which are known from the literature, and others are estimated from available data using the least squares method. For the stability of the model, equilibrium points were determined and the basic reproduction number R 0 was studied; R 0 is an indicator for contagiousness., Results: The model showed that secondary infections are generated from the exposed group, the asymptomatic group, the infected (symptomatic) group, the infected (undetected) group, the infected (home-based care) group and the hospitalized group. The formulated model was reliable and fit the data. Furthermore, the estimated R 0 of 2.16 shows that COVID-19 will persist without the application of control measures., Conclusions: This article presents results regarding predicted spread of COVID-19 in Rwanda., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2023 The Author(s).)

The authors address control system design based on an interval observer for non‐linear switched systems with non‐linear vector functions that are assumed to satisfy Lipschitz conditions. First, the observer gain satisfying a Metzler matrix can be solved by optimisations of linear matrix inequalities (LMIs). Second, an interval observer is designed to estimate the states of non‐linear switched systems with the average dwell time scheme, and sufficient conditions for state estimation are presented in terms of an LMI formulation. Third, based on an interval observer, state feedback matrices are designed to construct an asymptotically stabilising switching controller. Finally, a numerical example is provided to demonstrate the efficiency of the approach. [ABSTRACT FROM AUTHOR]

We present a new design for continuous–discrete observers for a large class of continuous time nonlinear time-varying systems with discrete time measurements. Using the notion of cooperative systems, we show that the solutions of the observers converge to the solutions of the original system, under conditions on the nonlinear terms and on the largest sampling interval. Our conditions are given by explicit expressions. [ABSTRACT FROM AUTHOR]

A new method is given to estimate an ultimate state bound on a time-varying linear system with delay and bounded disturbances by using some results on Metzler matrices. The effectiveness of the obtained results is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

Abstract: In this note, the problem of state bounding for linear time-varying systems with delay and bounded disturbances input is considered for the first time. By using a novel approach which does not involve the Lyapunov–Krasovskii functional method, new explicit delay-independent conditions are derived for the existence of a ball such that all the state trajectories of the system converge exponentially within it. A numerical example is given to illustrate the effectiveness of the obtained result. [Copyright &y& Elsevier]

Abstract: In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result. [Copyright &y& Elsevier]

Chattering is an undesirable phenomenon characterised by infinitely fast switching which may cause equipment damage in real systems. To avoid its occurrence, this study proposes a chattering‐free switching strategy for continuous‐time switched linear systems ensuring global asymptotical stability and a guaranteed cost level associated to the rms gain of a class of input to output signals. The switching function is designed considering a minimum dwell‐time constraint in order to avoid chattering and a maximum one to ensure robustness with respect to sampling jitters and implementation imperfections as, for instance, delays in the switching process. The conditions are based on Riccati–Metzler inequalities which take into account an equivalent discrete‐time switched linear system obtained from the continuous‐time one guided by a sampled switching rule without any kind of approximation. As a new result, for a subclass of Metzler matrices, necessary and sufficient conditions for the existence of a solution for the Riccati–Metzler inequalities are provided. Theoretical aspects are illustrated by some academical examples. [ABSTRACT FROM AUTHOR]

In this paper, the positive state consensus problem of a network of homogeneous LTI positive agents is presented. A hierarchical control protocol is applied to achieve finite positive state consensus over directed networks. The minimal state-space realization considered for the interconnected positive agents is not restricted to be (internally) positive state-space realization, which is in contrast to the existing literature. A set of sufficient conditions is derived in linear programming framework which is solved to find controller gain matrices involved in the consensus protocol. Also, in this paper, an observer-based hierarchical control protocol has been derived to achieve finite positive state consensus. Numerical examples are given to demonstrate the usefulness of the proposed results. [ABSTRACT FROM AUTHOR]

We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron-Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching. [ABSTRACT FROM AUTHOR]

The problem of the existence and determination of the set ofMetzler matrices for given stable polynomials is formulated and solved. Necessary and sufficient conditions are established for the existence of the set of Metzler matrices for given stable polynomials. A procedure for finding the set of Metzler matrices for given stable polynomials is proposed and illustrated with numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is that of investigating the asymptotic exponential cone of a single Metzler matrix, introduced in [23], and of defining and analysing the new concept of asymptotic exponential cone of a family of Metzler matrices (along a certain direction). These results will provide necessary and/or sufficient conditions for the solvability of an interesting algebraic problem that arises in the context of continuous-time positive switched systems and, specifically, in the investigation of the reachability property [21,22,25]. [Copyright &y& Elsevier]

Linear differential systems ẋ (t) = Ax(t) (A ϵ ℝnxn, x0 = x(0) ϵ ℝn, t ⩾ 0) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of A furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some a ⩾ 0, A+aI must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions x0 that result in nonnegative states x(t) in finite time are shown to form a convex cone that is related to the matrix exponential etA and its eventual nonnegativity. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the zero pattern properties and the asymptotic evolution of the trajectories of a continuous-time positive system are investigated. To this end, we need to introduce some new tools and to derive some new results, within the broad research area of nonnegative matrix theory, which enable use to explore the zero pattern and the elementary modes of the exponential of a Metzler matrix. Specifically, a normal form for the exponential of a Metzler matrix is provided, and the concept of echelon basis (consisting of eigenvectors and generalized eigenvectors of the Metzler matrix) is introduced. By making use of these two ingredients, a detailed result about the dominant mode of each single block appearing in the normal form of the exponential matrix is provided. This allows to obtain a “modal decomposition” of the exponential matrix, emphasizing the column dominant modes. As a result, the zero pattern as well as the asymptotic behavior of every free state evolution, depending on the zero pattern of the initial state, can be easily determined. [Copyright &y& Elsevier]

This paper investigates the stabilizability of linear systems with closed-loop positivity. A necessary and sufficient condition for the existence of desired state-feedback controllers guaranteeing the resultant closed-loop system to be asymptotically stable and positive is obtained. Both continuous- and discrete-time cases are considered, and all of the conditions are expressed as linear matrix inequalities which can be easily verified by using standard numerical software. Numerical examples are provided to illustrate the proposed conditions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper presents a formal description and analysis of an SIR (involving susceptible- infectious-recovered subpopulations) epidemic model in a patchy environment with vaccination controls being constant and proportional to the susceptible subpopulations. The patchy environment is due to the fact that there is a partial interchange of all the subpopulations considered in the model between the various patches what is modelled through the so-called travel matrices. It is assumed that the vaccination controls are administered at each community health centre of a particular patch while either the total information or a partial information of the total subpopulations, including the interchanging ones, is shared by all the set of health centres of the whole environment under study. In the case that not all the information of the subpopulations distributions at other patches are known by the health centre of each particular patch, the feedback vaccination rule would have a decentralized nature. The paper investigates the existence, allocation (depending on the vaccination control gains) and uniqueness of the disease-free equilibrium point as well as the existence of at least a stable endemic equilibrium point. Such a point coincides with the disease-free equilibrium point if the reproduction number is unity. The stability and instability of the disease-free equilibrium point are ensured under the values of the disease reproduction number guaranteeing, respectively, the un-attainability (the reproduction number being less than unity) and stability (the reproduction number being more than unity) of the endemic equilibrium point. The whole set of the potential endemic equilibrium points is characterized and a particular case is also described related to its uniqueness in the case when the patchy model reduces to a unique patch. Vaccination control laws including feedback are proposed which can take into account shared information between the various patches. It is not assumed that there are in the most general case, symmetry-type constrains on the population fluxes between the various patches or in the associated control gains parameterizations. [ABSTRACT FROM AUTHOR]

In this study, a deterministic model for the transmission dynamics of yellow fever (YF) in a human-mosquito setting in the presence of control measures is constructed and rigorously analyzed. In addition to horizontal transmissions, vertical transmission within mosquito population is incorporated. Analysis of the mosquito-only component of the model shows that the reduced model has a mosquito-extinction equilibrium, which is globally-asymptotically stable whenever the basic offspring number ( N 0 ) is less than unity. The vaccinated and type reproduction numbers of the full-model are computed. Condition for global-asymptotic stability of the disease-free equilibrium of the model when N 0 > 1 is presented. It is shown that, fractional dosing of YF vaccine does not meet YF vaccination requirements. Optimal control theory is applied to the model to characterize the controls parameters. Using Pontryagin's maximum principle and modified forward-backward sweep technique, the necessary conditions for existence of solutions to the optimal control problem is determined. Numerical simulations of the models to assess the effect of fractional vaccine dosing on the disease dynamics and global sensitivity analysis are presented., (© Springer Nature India Private Limited 2020.)