The global stability and optimal control of the COVID-19 epidemic model

This paper considers stability analysis of a Susceptible-Exposed-Infected-Recovered-Virus(SEIRV)model with nonlinear incidence rates and indicates the severity and weakness of control factors for disease transmission.The Lyapunov function using Volterra-Lyapunov matrices makes it possible to study the global stability of the endemic equilibrium point.An optimal control strategy is proposed to prevent the spread of coronavirus,in addition to governmental intervention.The objective is to minimize together with the quantity of infected and exposed individuals while minimizing the total costs of treatment.A numerical study of the model is also carried out to investigate the analytical results.

GLOBAL STABILITY OF AN SIRS EPIDEMIC MODEL WITH DELAYS

In this article, an SIRS epidemic model spread by vectors （mosquitoes） which have an incubation time to become infectious is formulated. It is shown that a disease-free equilibrium point is globally stable if no endemic equilibrium point exists. Further, the endemic equilibrium point （if it exists） is globally stable with a respect ＂weak delay＂. Some known results are generalized.

GLOBAL STABILITY OF SIRS EPIDEMIC MODELS WITH A CLASS OF NONLINEAR INCIDENCE RATES AND DISTRIBUTED DELAYS

In this article, we establish the global asymptotic stability of a disease-free equilibrium and an endemic equilibrium of an SIRS epidemic model with a class of nonlin- ear incidence rates and distributed delays. By using strict monotonicity of the incidence function and constructing a Lyapunov functional, we obtain sufficient conditions under which the endemic equilibrium is globally asymptotically stable. When the nonlinear inci- dence rate is a saturated incidence rate, our result provides a new global stability condition for a small rate of immunity loss.

Global Stability Analysis of a Delayed SEIQR Epidemic Model with Quarantine and Latent

In this paper, we study a kind of the delayed SEIQR infectious disease model with the quarantine and latent, and get the threshold value which determines the global dynamics and the outcome of the disease. The model has a disease-free equilibrium which is unstable when the basic reproduction number is greater than unity. At the same time, it has a unique endemic equilibrium when the basic reproduction number is greater than unity. According to the mathematical dynamics analysis, we show that disease-free equilibrium and endemic equilibrium are locally asymptotically stable by using Hurwitz criterion and they are globally asymptotically stable by using suitable Lyapunov functions for any Besides, the SEIQR model with nonlinear incidence rate is studied, and the that the basic reproduction number is a unity can be found out. Finally, numerical simulations are performed to illustrate and verify the conclusions that will be useful for us to control the spread of infectious diseases. Meanwhile, the will effect changing trends of in system (1), which is obvious in simulations. Here, we take as an example to explain that.

Global Stability of An Eco-epidemiological Model with Beddington-DeAngelis Functional Response and Delay

In this paper,an eco-epidemiological model with Beddington-DeAngelis functional response and a time delay representing the gestation period of the predator is studied.By means of Lyapunov functionals and Laselle’s invariance principle,sufficient conditions are obtained for the global stability of the interior equilibrium and the disease-free equilibrium of the system,respectively.

Exponential stability of Takagi-Sugeno fuzzy systems with impulsive effects and small delays

This paper deals with the exponential stability of impulsive Takagi-Sugeno fuzzy systems with delay. Impulsive control and delayed fuzzy control are applied to the system, and the criterion on exponential stability expressed in terms of linear matrix inequalities （LMIs） is presented.

Stability analysis of a discrete biological model

In this paper, we investigate the equilibrium point, local and global behavior of the unique positive equilibrium point, and rate of convergence of positive solutions of following discrete biological model： Xn＋1=Xne^ln（α）（1-Yn）,Yn＋1=（γδ-1）Yn（1＋1/γδ-1-Yn/Xn）, where parameters α,γ,δ and the initial conditions Xo, ]Io are positive real numbers. Some numerical examples are given to verify theoretical results.

Stability Analysis for a Discrete SIR Epidemic Model with Delay and General Nonlinear Incidence Function

In this paper, we construct a backward difference scheme for a class of SIR epidemic model with general incidence f . The step sizeτ used in our discretization is one. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, the general incidence function f must satisfy certain assumptions, under which, we establish the global stability of endemic equilibrium when R0 >1. The global stability of diseases-free equilibrium is also established when R0 ≤1. In addition we present numerical results of the continuous and discrete model of the different class according to the value of basic reproduction number R0.

Global disturbance rejection of a class of nonlinear systems with unknown exosystems

An asymptotic rejection algorithm is proposed for a class of nonlinear systems that have not only additive nonlinear uncertainties but also unknown disturbances. The disturbances are generated from an unknown exosystem, and are assumed to be sinusoidal disturbances with unknown amplitude and frequency. By using the technique of backstepping and adaptive control, a nonlinear state feedback controller is designed. Under the proposed controller, the system＇s state variables asymptotically converge to zero, and the disturbances are rejected completely. The approach used is an integration of the robust stabilization technique, adaptive technique, and backstepping technique.

Tracking Control of Uncertain Nonlinear Systems With Unknown Constant Input Delay

A robust delay compensator has been developed for a class of uncertain nonlinear systems with an unknown constant input delay.The control law consists of feedback terms based on the integral of past control values and a novel filtered tracking error,capable of compensating for input delays.Suitable Lyapunov-Krasovskii functionals are used to prove global uniformly ultimately bounded(GUUB)tracking,provided certain sufficient gain conditions,dependent on the bound of the delay,are satisfied.Simulation results illustrate the performance and robustness of the controller for different values of input delay.

GLOBAL ANALYSIS OF SOME EPIDEMIC MODELS WITH GENERAL CONTACT RATE AND CONSTANT IMMIGRATION

An epidemic models of SIR type and SIRS type with general contact rate and constant immigration of each class were discussed by means of theory of limit system and suitable Liapunov functions. In the absence of input of infectious individuals, the threshold of existence of endemic equilibrium is found.For the disease-free equilibrium and the endemic equilibrium of corresponding SIR model, the sufficient and necessary conditions of global asymptotical stabilities are all obtained.For corresponding SIRS model, the sufficient conditions of global asymptotical stabilities of the disease-free equilibrium and the endemic equilibrium are obtained. In the existence of input of infectious individuals, the models have no disease-free equilibrium. For corresponding SIR model, the endemic equilibrium is globally asymptotically stable; for corresponding SIRS model, the sufficient conditions of global asymptotical stability of the endemic equilibrium are obtained.

On the model-based networked control for singularly perturbed systems

In this paper, the control of a two-time-scale plant, where the sensor is connected to a linear controller/ actuator via a network is addressed. The slow and fast systems of singularly perturbed systems are used to produce an estimate of the plant state behavior between transmission times, by which one can reduce the usage of the network. The approximate solutions of the whole systems are derived and it is shown that the whole systems via the network control are generally asymptotically stable as long as their slow and fast systems are both stable. These results are also extended to the case of network delay.

Global Transmission Dynamics of a Schistosomiasis Model and Its Optimal Control

Drug treatment, snail control, cercariae control, improved sanitation and health education are the effective strategies which are used to control the schistosomiasis. In this paper, we consider a deterministic model for schistosomiasis transmission dynamics in order to explore the role of the several control strategies. The global stability of a schistosomiasis infection model that involves mating structure including male schistosomes, female schistosomes, paired schistosomes and snails is studied by constructing appropriate Lyapunov functions. We derive the basic reproduction number R0 for the deterministic model, and establish that the global dynamics are completely determined by the values of R0. We show that the disease can be eradicated when R0?≤1;otherwise, the system is persistent. In the case where ?R0?>1, we prove the existence, uniqueness and global asymptotic stability of an endemic steady state. Sensitivity analysis and simulations are carried out in order to determine the relative importance of different control strategies for disease transmission and prevalence. Next, optimal control theory is applied to investigate the control strategies for eliminating schistosomiasis using time dependent controls. The characterization of the optimal control is carried out via the Pontryagins Maximum Principle. The simulation results demonstrate that the insecticide is important in the control of schistosomiasis.

Stability Analysis for Stochastic Delayed High-order Neural Networks

In this paper, the global asymptotic stability analysis problem is considered for a class of stochastic high-order neural networks with tin.delays. Based on a Lyapunov-Krasovskii functional and the stochastic stability analysis theory, several sufficient conditions are derived in order to guarantee the global asymptotic convergence of the equilibtium paint in the mean square. Investigation shows that the addressed stochastic highorder delayed neural networks are globally asymptotically stable in the mean square if there are solutions to some linear matrix inequalities （LMIs）. Hence, the global asymptotic stability of the studied stochastic high-order delayed neural networks can be easily checked by the Matlab LMI toolbox. A numerical example is given to demonstrate the usefulness of the proposed global stability criteria.

Global stability and Hopf bifurcation of an eco-epidemiological model with time delay

In this paper,an eco-epidemiological model with time delay representing the gestation period of the predator is investigated.In the model,it is assumed that the predator population suffers a transmissible disease and the infected predators may recover from the disease and become susceptible again.By analyzing corresponding characteristic equations,the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free and coexistence equilibria are established,respectively.By means of Lyapunov functionals and LaSalle’s invariance principle,sufficient condi tions are obtained for the global stability of the coexistence equilibrium,the disease-free equilibrium and the predator-extinct equilibrium of the system,respectively.

Global Stability of a Time-delayed Malaria Model with Standard Incidence Rate

A four-dimensional delay differential equations(DDEs)model of malaria with standard incidence rate is proposed.By utilizing the limiting system of the model and Lyapunov direct method,the global stability of equilibria of the model is obtained with respect to the basic reproduction number R_(0).Specifically,it shows that the disease-free equilibrium E^(0)is globally asymptotically stable(GAS)for R_(0)<1,and globally attractive(GA)for R_(0)=1,while the endemic equilibrium E^(*)is GAS and E^(0)is unstable for R_(0)>1.Especially,to obtain the global stability of the equilibrium E^(*)for R_(0)>1,the weak persistence of the model is proved by some analysis techniques.

A Dynamical Study of Modeling the Transmission of Typhoid Fever throughDelayed Strategies

This study analyzes the transmission of typhoid fever caused by Salmonella typhi using a mathematical model thathighlights the significance of delay in its effectiveness.Time delays can affect the nature of patterns and slow downthe emergence of patterns in infected population density.The analyzed model is expanded with the equilibriumanalysis,reproduction number,and stability analysis.This study aims to establish and explore the non-standardfinite difference(NSFD)scheme for the typhoid fever virus transmission model with a time delay.In addition,the forward Euler method and Runge-Kutta method of order 4(RK-4)are also applied in the present research.Some significant properties,such as convergence,positivity,boundedness,and consistency,are explored,and theproposed scheme preserves all the mentioned properties.The theoretical validation is conducted on how NSFDoutperforms other methods in emulating key aspects of the continuous model,such as positive solution,stability,and equilibrium about delay.Hence,the above analysis also shows some of the limitations of the conventional finitedifference methods,such as forward Euler and RK-4 in simulating such critical behaviors.This becomes moreapparent when using larger steps.This indicated that NSFD is beneficial in identifying the essential characteristicsof the continuous model with higher accuracy than the traditional approaches.

Global stability for a delayed multi-group SIRS epidemic model with cure rate and incomplete recovery rate

In this paper, by applying Lyapunov functional approach, we establish a sufficient condition on the global stability of a ＂delayed＂ multi-group SIRS epidemic model with cure rate and incomplete recovery rate which does not depend on the delays and is an extension of the ＂light drug model＂ studied in the recent paper [Muroya, Li and Kuniya, Complete global analysis of an SIRS epidemic model with graded cure rate and incomplete recovery rate, J. Math. Anal. Appl. 410 （2014） 719-732] to a multi-group model. Applying a Lyapunov functional on total population of each compartment, we offer new techniques for the delayed system, how to prove the permanence, the existence of the endemic equilibrium and the global stability of disease-free equilibrium for the reproduction number R0 ≤ 1 and endemic equilibrium forR0 ≥ 1.

Robust fuzzy control of Takagi-Sugeno fuzzy neural networks with discontinuous activation functions and time delays

The problem of global robust asymptotical stability for a class of Takagi-Sugeno fuzzy neural networks（TSFNN） with discontinuous activation functions and time delays is investigated by using Lyapunov stability theory.Based on linear matrix inequalities（LMIs）,we originally propose robust fuzzy control to guarantee the global robust asymptotical stability of TSFNNs.Compared with the existing literature,this paper removes the assumptions on the neuron activations such as Lipschitz conditions,bounded,monotonic increasing property or the right-limit value is bigger than the left one at the discontinuous point.Thus,the results are more general and wider.Finally,two numerical examples are given to show the effectiveness of the proposed stability results.

Global Exponential Stability of Variable Delay and Time－varying Measure Differential Systems with Impulse Effect

This work concerns the stability properties of impulsive solution for a class of variable delay and time-varying measure differential systems. By means of the techniques of constructing broken line function to overcome the difficulties in employing comparison principle and Lyapunov function with discontinuous derivative, some criteria of global exponential asymptotic stability for the system are established. An example is given to illustrate the applicability of results obtained.