Globally Asymptotically Stable Research Articles

Mathematical Model for the Dynamics of Paragonimiasis Disease in Human, Snail, and Crustacean Populations

This paper proposes a mathematical model for the dynamics of Paragonimiasis transmission in human, snail, and crustacean populations. We proved that the solutions to the differential equations are both positive and bounded. In addition, we calculated the equilibrium states of Paragonimiasis in both its free and present forms. We computed the basic reproduction number using the next-generation matrix. Furthermore, we used the Routh-Hurwitz criteria to determine local stability, which revealed that the system is locally asymptotically stable (LAS). Similarly, we assessed the global stability of the disease-free equilibrium and found that the system is globally asymptotically stable (GAS). To better understand the dynamics of Paragonimiasis transmission, we carried out sensitivity index analyses and computational simulations. Our data show varied indices of rise and decline as parameters are changed. Asymptptically stable, Dynamics, Endemic equilibrium, Reproduction number, Transmission 

Global stability of secondary DENV infection models with non-specific and strain-specific CTLs

Dengue virus (DENV) is a highly perilous virus that is transmitted to humans through mosquito bites and causes dengue fever. Consequently, extensive efforts are being made to develop effective treatments and vaccines. Mathematical modeling plays a significant role in comprehending the dynamics of DENV within a host in the presence of cytotoxic T lymphocytes (CTL) immune response. This study examines two models for secondary DENV infections that elucidate the dynamics of DENV under the influence of two types of CTL responses, namely non-specific and strain-specific responses. The first model encompasses five compartments, which consist of uninfected monocytes, infected monocytes, free DENV particles, non-specific CTLs, and strain-specific CTLs. In the second model, latently infected cells are introduced into the model. We posit that the CTL responsiveness is determined by a combination of self-regulating CTL response and a predator-prey-like CTL response. The model's solutions are verified to be nonnegativity and bounded and the model possesses two equilibrium states: the uninfected equilibrium EQ0 and the infected equilibrium EQ⁎. Furthermore, we calculate the basic reproduction number R0, which determines the existence and stability of the model's equilibria. We examine the global stability by constructing suitable Lyapunov functions. Our analysis reveals that if R0≤1, then EQ0 is globally asymptotically stable (G.A.S), and if R0>1, then EQ0 is unstable while EQ⁎ is G.A.S. To illustrate our findings analytically, we conduct numerical simulations for each model. Additionally, we perform sensitivity analysis to demonstrate how the parameter values of the proposed model impact R0 given a set of data. Finally, we discuss the implications of including the CTL immune response and latently infected cells in the secondary DENV infection model. Our study demonstrates that incorporating the CTL immune response and latently infected cells diminishes R0 and enhances the system's stability around EQ0.

Quantitative analysis of a fractional order of the $ SEI_{c}\, I_{\eta} VR $ epidemic model with vaccination strategy

<abstract><p>This work focused on studying the effect of vaccination rate $ \kappa $ on reducing the outbreak of infectious diseases, especially if the infected individuals do not have any symptoms. We employed the fractional order derivative in this study since it has a high degree of accuracy. Recently, a lot of scientists have been interested in fractional-order models. It is considered a modern direction in the mathematical modeling of epidemiology systems. Therefore, a fractional order of the SEIR epidemic model with two types of infected groups and vaccination strategy was formulated and investigated in this paper. The proposed model includes the following classes: susceptible $ \mathrm{S}(t) $, exposed $ \mathrm{E}(t) $, asymptomatic infected $ \mathrm{I_{c}}(t) $, symptomatic infected $ \mathrm{I_{\eta}}(t) $, vaccinated $ \mathrm{V}(t) $, and recovered $ \mathrm{R}(t) $. We began our study by creating the existence, non-negativity, and boundedness of the solutions of the proposed model. Moreover, we established the basic reproduction number $ \mathcal{R}_{0} $, that was used to examine the existence and stability of the equilibrium points for the presented model. By creating appropriate Lyapunov functions, we proved the global stability of the free-disease equilibrium point and endemic equilibrium point. We concluded that the free-disease equilibrium point is globally asymptotically stable (GAS) when $ \mathcal{R}_{0}\, \leq \, 1 $, while the endemic equilibrium point is GAS if $ \mathcal{R}_{0} > 1 $. Therefore, we indicated the increasing vaccination rate $ \kappa $ leads to reducing $ \mathcal{R}_0 $. These findings confirm the important role of vaccination rate $ \kappa $ in fighting the spread of infectious diseases. Moreover, the numerical simulations were introduced to validate theoretical results that are given in this work by applying the predictor-corrector PECE method of Adams-Bashforth-Moulton. Further more, the impact of the vaccination rate $ \kappa $ was explored numerically and we found that, as $ \kappa $ increases, the $ \mathcal{R}_{0} $ is decreased. This means the vaccine can be useful in reducing the spread of infectious diseases.</p></abstract>

Mathematical Modeling and Analysis of COVID-19 Infection: Application to the Kingdom of Saudi Arabia Data

To evaluate the probable effects of the COVID-19 outbreak on Saudi Arabia, a novel dynamical model is developed. Using the most recent instances of COVID-19 infection that have been reported in Saudi Arabia, we examine the roles of quarantine and hospitalization. The model’s mathematical outcomes are displayed. The model’s infection-free equilibrium is displayed, and asymptotically, it is determined to be both locally and globally stable. We demonstrate that the model is locally asymptotically stable (LAS) for the basic reproduction R 0 < 1 . The model is globally asymptotically stable (GAS) when R 0 ≤ 1 . To estimate the model parameters, recent COVID-19 instances in KSA that began between May 1, 2022, and August 4, 2022, are taken into account. We achieve the needed data fitting considering the approach of nonlinear least square, and we demonstrate that the predicted basic reproduction number is R 0 ≈ 1.2988 . Graphical representations of the calculated parameters and their effects on disease eradication are provided. The findings show that the most effective way to reduce the number of new instances of infection is to limit the contact of exposed, asymptomatic, symptomatic, and hospitalized people with vulnerable. The percentage of exposed people who are quarantined also plays a big part in lowering the number of infected cases.

Stability and bifurcation analysis of a Taenia saginata model with control measures

Bovine cysticercosis and human taeniasis are neglected diseases caused by the beef tapeworm Taenia saginata. These diseases affect both human and animal health, rural livestock producers’ livelihoods, and the economies of the nations. Bovine cysticercosis makes beef unfit for human consumption, thus decreasing the cattle market value. In this study, a mathematical model for the dynamics of human taeniasis and bovine cysticercosis is examined in the context of diseases’ control efforts. The analysis of the basic model shows that both disease free and endemic equilibria exist. The basic reproduction number R0 is computed by the next generation method. The disease free equilibrium is globally asymptotically stable (GAS) when R0<1 whereas the endemic equilibrium is GAS when R0>1. To determine parameters that drive the diseases, the normalized forward sensitivity index method is adopted. The findings demonstrate that human and animal recruitment rates, the probability of humans to contract taeniasis, the rate at which humans with taeniasis defecate in the environment and T. saginata eggs’ natural mortality rate influence the diseases’ dynamics. The effects of several interventions including vaccination of cattle, treatment of infected humans and cattle, proper beef cooking, enhanced hygiene and sanitation, and the use of chemicals to kill T.saginata eggs in the environment are evaluated. When such interventions are administered, the model exhibits forward bifurcation, and secondary infections reduce significantly with time. Therefore, to control the diseases we recommend that more efforts be directed to treat humans with taeniasis and proper beef cooking and meat inspection be encouraged.

Dynamic effects of asymptomatic infections on malaria transmission

Asymptomatic transmission is a way of spreading malaria from asymptomatic infected individuals to susceptible ones by mosquito bites. Asymptomatic malaria-infected individuals do not show clinical symptoms, but they can still transmit the disease. Motivated by mathematical models with asymptomatic infections, we propose a human–mosquito model for malaria transmission dynamics with asymptomatic infections and standard incidence rates. We calculate the basic reproduction number R0 of the model and then prove that the model exists a unique endemic equilibrium E∗ if and only if R0>1. By the Lyapunov function method, the global stability of equilibria of the model is obtained. It is shown that the disease-free equilibrium E0 is globally asymptotically stable (GAS) if and only if R0≤1, which implies that malaria will be extinct; and the endemic equilibrium E∗ is GAS if and only if R0>1, which suggests that malaria will persist. In addition, the effects of partial malaria-related parameters on the dynamic transmission of malaria are studied and analysed numerically. The results of numerical simulations show that asymptomatic infections can shift the epidemic dynamics of malaria and have a significant influence on the transmission of malaria.

The impact of the face mask on SARS-CoV-2 disease: Mathematical modeling with a case study

The mathematical modeling of new emerging infectious diseases gaining high attention from the researcher’s around the world. We formulate a mathematical model with the statistical data of coronavirus observed in Saudi Arabia from March 01, 2021 to September 30, 2021, is considered. Particularly, we focus on studying recent infected data in Saudi Arabia and obtaining reasonable fitting to the model by investigating its realistic parameters. We show the model-related results and their dynamic analysis. The stability results for the disease-free and endemic cases are shown. We show when R0<1, then the system at the disease-free case is locally asymptotically stable (LAS). When R0≤1, we show that the model is globally asymptotically stable (GAS) at the disease free case. We study the endemic equilibria and analyze its global dynamics whenever R0>1. We determine that the model gives the forward bifurcation. Moreover, we do data fitting to the model and then perform numerical experiments to justify the theoretical results. The sensitive parameters are used to study the behavior of the model and provide some illustrating results for the possible elimination of infection in the Kingdom of Saudi Arabia.

An improved method of global dynamics: Analyzing the COVID-19 model with time delays and exposed infection.

Considering the transmission characteristics of the coronavirus disease 2019 (COVID-19), there are certain time delays in the transition from susceptible individuals to exposed individuals after contact with exposed, symptomatically infected, and asymptomatically infected individuals. A COVID-19 model with time delays and exposed infection is developed and then the global dynamics of this model is investigated by an improved method; moreover, the numerical simulations are carried out. It is shown that the COVID-19-free equilibrium T0 is globally asymptotically stable (GAS) if and only if the control reproduction number Rc≤1, while T0 is unstable and the COVID-19 equilibrium T∗ is GAS if and only if Rc>1. The numerical results reveal that strengthening quarantine measures is helpful to control the COVID-19 epidemic in India. Furthermore, when Rc<1, the numbers of symptomatically infected, asymptomatically infected, and quarantined individuals eventually tend to the zero equilibrium state, and with the increase in the time delay, the three kinds of variables change faster and their peaks become larger; when Rc>1, the three kinds of variables eventually tend to the positive equilibrium state, which are oscillatory and the amplitudes of the oscillation enlarge as the value of time delay increases. The numerical results show that when Rc<1, the smaller the value of time delay, the smaller the final epidemic size. In short, the longer it takes time for susceptible individuals to transform exposed individuals, the harder COVID-19 will be controlled.

A Global Control Strategy to Remove Slow Delay Conditions for Delayed Systems with Complicated Polynomial Terms

Nonlinear systems constantly suffer from time-varying delays, which cover slow delays and nonslow delays. The existing results were used to impose slow delay conditions and were used to study the control problems, but few pieces of research studies have discussed the case of systems with nonslow delay. In this work, we remove the slow time-delay condition and consider the nonlinear delayed system with complicated polynomial terms. By proposing a dynamic gain method and constructing a new Lyapunov–Razumikhin (L-R) function, we successfully construct a stable controller, which guarantees that the plant is globally asymptotically stable (GAS). An example is utilized to verify the raised control scheme.

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.

MATHEMATICAL ANALYSIS OF TRANSMISSION DYNAMICS OF HIV/ AIDS IN RISK STRUCTURED POPULATION

A model for assessing the impact of key population at risk in spreading HIV/ AIDS is designed. The model is shown to have a globally asymptotically stable (GAS) disease-free equilibrium whenever the associated reproduction number is less than unity. It has a unique GAS endemic equilibrium whenever reproduction number exceeds unity, if there is no back and forth movement in susceptible classes. It is shown that if infected individuals with high risk abandon their risky behavioral practices, then the disease burden will reduce. Furthermore, if infected individuals with low risk do not adopt risky behaviors then the transmission of the disease will decrease. Partial Rank Correlation Coefficient (PRCC) indices are assessed using all the infection-related classes and basic reproduction number of the model. Uncertainty and sensitivity analysis results show that controlling the effective contact rate, disease-induced death rate in AIDS stage with low and high risk classes will help controlling the transmission of the disease.

Complex dynamics and impulsive control of a chemostat model under the ratio threshold policy

In this paper, we study the periodic solution and global stability of a chemostat model under impulsive control. First, we investigate the positivity and boundedness of the solution of the controlled system. Second, we find the periodic solution of the controlled system by employing the Poincare map and Brouwer’s fixed-point theorem. Furthermore, we obtain a sufficient condition which allows the existence of orbitally stable order-k periodic solutions (k=1,2) by using the comparison method and the vector field analysis. We find that the controlled system exists a unique positive equilibrium point that is globally asymptotically stable (GAS) under some conditions. Finally, we provide two numerical examples to verify the correctness of the theoretical results.

Global investigation for an "SIS" model for COVID-19 epidemic with asymptomatic infection.

In this paper, we analyse a dynamical system taking into account the asymptomatic infection and we consider optimal control strategies based on a regular network. We obtain basic mathematical results for the model without control. We compute the basic reproduction number (R) by using the method of the next generation matrix then we analyse the local stability and global stability of the equilibria (disease-free equilibrium (DFE) and endemic equilibrium (EE)). We prove that DFE is LAS (locally asymptotically stable) when R<1 and it is unstable when R>1. Further, the existence, the uniqueness and the stability of EE is carried out. We deduce that when R>1, EE exists and is unique and it is LAS. By using generalized Bendixson-Dulac theorem, we prove that DFE is GAS (globally asymptotically stable) if R<1 and that the unique endemic equilibrium is globally asymptotically stable when R>1. Later, by using Pontryagin's maximum principle, we propose several reasonable optimal control strategies to the control and the prevention of the disease. We mathematically formulate these strategies. The unique optimal solution was expressed using adjoint variables. A particular numerical scheme was applied to solve the control problem. Finally, several numerical simulations that validate the obtained results were presented.

Singular Perturbation: When the Perturbation Parameter Becomes a State-Dependent Function

This paper introduces a new class of singularly perturbed systems in which the small, but constant, perturbation coefficient in standard singular perturbation theory is replaced by a state-dependent function. This generalization is aimed at broadening the applicability of singular perturbation theory in practice. For this class of singularly perturbed systems, it is assumed that the boundary-layer subsystem is globally asymptotically stable (GAS) at the origin and the reduced subsystem is input-to-state stable (ISS) with respect to the state of the boundary-layer subsystem. Under a mild monotonicity condition, sufficient conditions on the perturbation functions are given under which the singularly perturbed system is GAS at the origin. ISS and nonlinear small-gain techniques are exploited in the stability analysis. The efficacy of the proposed theoretical result is validated via its applications to tackling integral control and feedback optimization problems.

Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate

&lt;abstract&gt;&lt;p&gt;A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number $ {R}_{0} $ is calculated. We focus on the global dynamics of the model with respect to $ {R}_{0} $. If and only if $ {R}_{0} &amp;gt; 1 $, the model exists a unique malaria-infected equilibrium $ E^{\ast} $, whereas it always possesses the malaria-free equilibrium $ E_{0} $. We first prove the local stability of the equilibria $ E_0 $ and $ E^{\ast} $ by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium $ {E}_{0} $ is globally asymptotically stable (GAS) when $ {R}_{0} &amp;lt; 1 $, globally attractive (GA) when $ {R}_{0} = 1 $ and unstable when $ {R}_{0} &amp;gt; 1 $; the equilibrium $ E^{\ast} $ is GAS if and only if $ {R}_{0} &amp;gt; 1 $. In particular, in order to obtain global attractivity of the equilibrium $ E^{\ast} $, we demonstrate the weak persistence of the system for $ {R}_{0} &amp;gt; 1 $. Our results imply that malaria will gradually disappear if $ {R}_{0}\leq1 $ and persistently exist if $ {R}_{0} &amp;gt; 1 $.&lt;/p&gt;&lt;/abstract&gt;

Fractional dynamics of the transmission phenomena of dengue infection with vaccination

&lt;p style='text-indent:20px;'&gt;Dengue infection brings unimaginable damage in low-income countries across the world to public health, social, and economical fields. Scientists rely on modeling to preferable understand the transmission phenomena of dengue in order to forecast some preventive measures and to give better data for the development of vaccines and medications. In present work, we construct a compartmental model for dengue infection via fractional derivatives to show the influence of memory. The suggested system's solution is investigated for existence as well as uniqueness with the help of fractional calculus fundamental characteristics. The model is analyzed and the reproduction number is obtained through Next-generation Matrix technique, moreover, the sensitivity of the reproduction number is determined using the partial rank correlation coefficient (PRCC) method. We have shown that if &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ R_0&amp;lt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the disease-free steady-state of the dengue model is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) in the absence of vaccine. In particular case, we have shown that the infection of dengue persists uniformly in the system for &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; greater than one. Furthermore, we quantitatively showed memory's impact on &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ R_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; by fluctuation of different factors. Our result predicted that the biting frequency, mosquito recruiting rate, and memory index are the most sensitive aspects in efficiently controlling new dengue illnesses, according to our findings. The effect of vaccination on the suggested system's threshold is also investigated. Novel numerical schemes are presented to conceptualize the time series of the system in different scenarios. The findings suggested that more accurate and precious results are obtained through non-integer derivative and that memory plays a beneficial role in the management of dengue disease. We demonstrated that manipulating the index of memory in the system may manage the system reproduction number and endemic level of dengue infection.&lt;/p&gt;

Data-Driven Stability Certificate of Interconnected Homogeneous Networks via ISS Properties

This letter is concerned with a compositional data-driven approach for stability certificate of interconnected homogeneous networks with (partially) unknown dynamics while providing 100% correctness guarantees (as opposed to probabilistic confidence). The proposed framework enjoys input-to-state stability (ISS) properties of subsystems described by ISS Lyapunov functions. In our data-driven scheme, we first reformulate the corresponding conditions of ISS Lyapunov functions as a robust optimization program (ROP). Due to appearing unknown dynamics of subsystems in the constraint of ROP, we propose a scenario optimization program (SOP) by collecting data from trajectories of each unknown subsystem. We solve SOP and construct an ISS Lyapunov function for each subsystem with unknown dynamics. We accordingly leverage a compositional technique based on -type small-gain reasoning and construct a Lyapunov function for an unknown interconnected network based on ISS Lyapunov functions of individual subsystems. We demonstrate the efficacy of our data-driven approach over a room temperature network containing 1000 rooms with unknown dynamics. Given collected data from each unknown room, we verify that the unknown interconnected network is globally asymptotically stable (GAS) with 100% correctness guarantee.

Global Stabilization for a Class of Stochastic Nonlinear Time-Delay Systems With Unknown Measurement Drifts and Its Application.

This article studies the control problem for a class of stochastic nonlinear time-delay systems with uncertain output functions. Under the appropriate assumptions, a stabilization controller is explicitly constructed by applying the adding a power integrator method. Then, using the Lyapunov-Krasovskii functionals to address time-delay, it is proven that the designed controller can guarantee the closed-loop system to be globally asymptotically stable (GAS) in probability. Finally, two simulations show that the control strategy is effective and can be applied to the actual system.

Global stability of a general HTLV-I infection model with Cytotoxic T-Lymphocyte immune response and mitotic transmission

Mathematical models have been considered as a robust tool to support biological and medical studies of human viral infections. The global stability of viral infection models remains an important and largely open research problem. Such results are necessary to evaluate treatment strategies for infections and to establish thresholds for treatment rates. Human T-lymphotropic virus class I (HTLV-I) is a retrovirus which infects the CD4+T cells and causes chronic and deadly diseases. In this article, we developed a general nonlinear system of ODEs which describes the within-host dynamics of HTLV-I under the effect Cytotoxic T-Lymphocytes (CTLs) immunity. The mitotic division of actively infected cells are modeled. We consider general nonlinear functions for the generation, proliferation and clearancerates for all types of cells. The incidence rate of infection is also modeled by a general nonlinear function. These general functions are assumed to satisfy a set of suitable conditions and include several forms presented in the literature. We determine a bounded domain for the system’s solutions. We prove the existence of the system’s equilibrium points and determine two threshold numbers, the basic reproductive number R0 and the CTL immunity stimulation number R1. We establish the global stability of all equilibrium points by constructing Lyapunov function and applying Lyapunov-LaSalle asymptotic stability theorem. Under certain conditions it is shown that ifR0≤1, then the infection-free equilibrium point is globally asymptotically stable (GAS) and the HTLV-I infection is cleared. If R1≤1<R0, then the infected equilibrium point without CTL immunity is GAS and the HTLV-I infection becomes chronic with no sustained CTL immune response. If R1>1, then the infected equilibrium point with CTL immunity is GAS and the infection becomes chronic with persistent CTL immune response. We present numerical simulations for the system by choosing special shapes of the general functions. The effect of Crowley-Martin functional response and mitotic division of actively infected cells on the HTLV-I progression are studied. Our results cover and improve several ones presented in the literature.

Global threshold analysis on a diffusive host–pathogen model with hyperinfectivity and nonlinear incidence functions

In this paper, we are concerned with the mathematical analysis of a host–pathogen model with diffusion, hyperinfectivity and nonlinear incidence. We define the basic reproduction number ℜ0 by the spectral radius of the next generation operator, and study the relation between ℜ0 and the principal eigenvalue of the problem linearized at the disease-free steady state (DFSS). Under some assumptions, we show the threshold property of ℜ0: if ℜ0<1, then the DFSS is globally asymptotically stable (GAS), whereas if ℜ0>1, then the system is uniformly persistent and a positive steady state (PSS) exists. Moreover, for the special case where all parameters are constants, we show that the PSS is GAS for ℜ0>1. Numerical simulation suggests that the spatial heterogeneity could enhance the intensity of epidemic, whereas the diffusion effect could reduce it.