Global stability for age-infection-structured human immunodeficiency virus model with heterogeneous transmission

In this paper, we analyze the global asymptotic behaviors of a mathematical susceptible-infected(SI) age-infection-structured human immunodeficiency virus(HIV) model with heterogeneous transmission. Mathematical analysis shows that the local and global dynamics are completely determined by the basic reproductive number R 0 . If R 0 < 1 , disease-free equilibrium is globally asymptotically stable. If R 0 > 1 , it shows that disease-free equilibrium is unstable and the unique endemic equilibrium is globally asymptotically stable. The proofs of global stability utilize Lyapunov functions. Besides, the numerical simulations are illustrated to support these theoretical results and sensitivity analysis of each parameter for R 0 is performed by the method of partial rank correlation coefficient(PRCC).

Asymptotic properties of the Lotka-Volterra competition and mutualism model under stochastic perturbations

Stochastically perturbed models, where the white noise type stochastic perturbations are proportional to the current system state, the most realistically describe real-life biosystems. However, such models essentially have no equilibrium states apart from one at the origin. This feature makes analysis of such models extremely difficult. Probably, the best result that can be found for such models is finding of accurate estimations of a region in the model phase space that serves as an attractor for model trajectories. In this paper, we consider a classical stochastically perturbed Lotka-Volterra model of competing or symbiotic populations, where the white noise type perturbations are proportional to the current system state. Using the direct Lyapunov method in a combination with a recently developed technique, we establish global asymptotic properties of this model. In order to do this, we, firstly, construct a Lyapunov function that is applicable to the both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method allows to find estimates for this region. We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization ('stabilization by noise'). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (that for this model occurs via the process that is known as the 'stochastic stabilization', or the 'stabilization by noise') are found applying the Khasminskii-type Lyapunov functions.

Walking position commanded NAO robot using nonlinear disturbance observer-based fixed-time terminal sliding mode

The walking stability of a humanoid robot is a fundamental problem due to the complex nonlinear dynamic model of the robot's legs. This work introduces the performance tracking control for the humanoid NAO robot by using a Nonlinear Disturbance Observer (NDO)-based Fixed-time Terminal Sliding Mode (FTSM). The influence of uncertain external disturbance is considered while implementing the control strategy to improve the walking motion of the NAO robot. An NDO is adapted to estimate the uncertainties and external disturbances. A novel FTSM surface is proposed to drive the tracking errors to zero in fixed-time. The designed NDO-based FTSM control law achieves robustness while reducing the chattering phenomenon. The Lyapunov's stability theory is used to establish the fixed-time stability of the sliding surface and system states under the proposed control method. To validate the performance of the proposed NDO-based FTSM control, a real-time experiment was conducted on a humanoid NAO robot to demonstrate the improved tracking performance in the presence of the uncertain perturbation effect. The effectiveness of the proposed controller design is validated on a flat, upward inclined surface, and compared to another controller.

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 EQ 0 and the infected equilibrium EQ ⁎ . Furthermore, we calculate the basic reproduction number R 0 , 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 R 0 ≤ 1 , then EQ 0 is globally asymptotically stable (G.A.S), and if R 0 > 1 , then EQ 0 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 R 0 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 R 0 and enhances the system's stability around EQ 0 .

Modeling of PID controlled 3DOF robotic manipulator using Lyapunov function for enhancing trajectory tracking and robustness exploiting Golden Jackal algorithm

In this study, a three degrees of freedom (3 DOF) rigid-link robotic manipulator (RLM) has been simulated by using the Simscape model and the mathematical model derived by Lagrange method. The robot arm has been regulated by an Optimized PID Controller to achieve better tracking performance and reasonable robustness against disturbances and payload uncertainty. To optimize the controller, a novel nature-inspired Golden Jackal Optimization (GJO) algorithm has been used due to its efficient exploration that increases the diversity of the released solutions and its exploitation schemes which enhance the best-explored solutions. The tuning process has utilized a Lyapunov stability function as the objective function (OF) and the efficacy of the proposed algorithm is evaluated through a comprehensive comparison with various state-of-the-art metaheuristic techniques such as Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Jellyfish Search Optimizer (JSO), Whale Optimization Algorithm (WOA), Arithmetic Optimization Algorithm (AOA) and Sine Cosine Algorithm (SCA). The assessment has been conducted on benchmark error-based functions, providing rigorous testing and validation of the algorithm's performance. Furthermore, the performance evaluation has focused on the system's robustness against disturbances, noise, and variations in the payload mass, particularly in the context of Pick and Place (PNP) industrial tasks. The results of simulation have demonstrated that the optimized system, employing the Lyapunov function, demonstrated superior performance in minimizing the objective function value compared to other benchmark functions.

Threshold dynamics of a stochastic mathematical model for Wolbachia infections

A stochastic mathematical model is proposed to study how environmental heterogeneity and the augmentation of mosquitoes with Wolbachia bacteria affect the outcomes of dengue disease. The existence and uniqueness of the positive solutions of the system are studied. Then the V-geometrically ergodicity and stochastic ultimate boundedness are investigated. Further, threshold conditions for successful population replacement are derived and the existence of a unique ergodic steady-state distribution of the system is explored. The results show that the ratio of infected to uninfected mosquitoes has a great influence on population replacement. Moreover, environmental noise plays a significant role in control of dengue fever.

A fractional-order mathematical model for malaria and COVID-19 co-infection dynamics

This study proposes a fractional-order mathematical model for malaria and COVID-19 co-infection using the Atangana-Baleanu Derivative. We explain the various stages of the diseases together in humans and mosquitoes, and we also establish the existence and uniqueness of the fractional order co-infection model solution using the fixed point theorem. We conduct the qualitative analysis along with an epidemic indicator, the basic reproduction number R0 of this model. We investigate the global stability at the disease and endemic free equilibrium of the malaria-only, COVID-19-only, and co-infection models. We run different simulations of the fractional-order co-infection model using a two-step Lagrange interpolation polynomial approximate method with the aid of the Maple software package. The results reveal that reducing the risk of malaria and COVID-19 by taking preventive measures will reduce the risk factor for getting COVID-19 after contracting malaria and will also reduce the risk factor for getting malaria after contracting COVID-19 even to the point of extinction.

A study on the transmission dynamics of COVID-19 considering the impact of asymptomatic infection

The COVID-19 epidemic has been spreading around the world for nearly three years, and asymptomatic infections have exacerbated the spread of the epidemic. To analyse and evaluate the role of asymptomatic infections in the spread of the epidemic, we establish an improved COVID-19 infectious disease dynamics model. We fit the epidemic data in the four time periods corresponding to the selected 614G, Alpha, Delta and Omicron variants and obtain the proportion of asymptomatic persons among the infected persons gradually increased and with the increase of the detection ratio, the cumulative number of cases has dropped significantly, but the decline in the proportion of asymptomatic infections is not obvious. Therefore, in view of the hidden transmission of asymptomatic infections, the cooperation between various epidemic prevention and control policies is required to effectively curb the spread of the epidemic.

Recurrent general type-2 fuzzy neural networks for nonlinear dynamic systems identification

This paper introduces a recurrent general type-2 Takagi-Sugeno-Kang fuzzy neural network (RGT2-TSKFNN) for the identification of nonlinear systems. In the proposed structure, the general type-2 fuzzy set (GT2FS) and a recurrent fuzzy neural network (RFNN) are combined to obviate the data uncertainties. The fuzzy firing strengths in the developed structure are returned to the network input as internal variables. In the proposed structure, GT2FS is utilized to characterize the antecedent parts while the consequent parts are performed using TSK type. The issues of constructing a RGT2-TSKFNN involve type reduction, structure learning as well as parameter learning. An efficient strategy is developed by utilizing alpha-cuts to decompose a GT2FS into several interval type-2 fuzzy sets (IT2FSs). In order to solve the computation time of the type-reduction issue, a direct defuzzification method is used instead of iterative nature of Karnik-Mendel​ (KM) algorithm. Type-2 fuzzy clustering and Lyapunov criteria are utilized for online structure learning as well as the antecedent and consequent parameters, respectively for reducing the number of rules and guaranteeing the stability of the proposed RGT2-TSKFNN. The reported comparative analysis of the simulation results is utilized to estimate the performance of the proposed RGT2-TSKFNN with respect to other popular type-2 FNNs (T2FNNs) methodologies.

SIRC epidemic model with cross-immunity and multiple time delays

Multi-strain diseases lead to the development of some degree of cross-immunity among people. In the present paper, we propose a multi-delayed SIRC epidemic model with incubation and immunity time delays. Here we aim to examine and investigate the effects of incubation delay [Formula: see text] and the impact of vaccine which provides partial/cross-immunity with immunity delay parameter ([Formula: see text]) on the disease dynamics. Also, we study the impact of the strength of cross-immunity [Formula: see text] on the disease prevalence. The positivity and boundedness of the solutions of the epidemic model have been established. Two different types of equilibrium points (disease-free and endemic) have been deduced. Expression for basic reproduction number has been derived. The stability conditions and Hopf-bifurcation about both the equilibrium points in the absence and presence of both delays have been discussed. The Lyapunov stability conditions about the endemic equilibrium point have been established. Numerical simulations have been performed to support our analytical results. We quantitatively demonstrate how oscillations and Hopf-bifurcation allow time delays to alter the dynamics of the system. The combined impacts of both the delays on disease prevalence has been studied. Through parameter sensitivity analysis, we observe that the infected population decreases with an increase in vaccination rate and the system starts to stabilize early with the increase in cross-immunity rate. Global sensitivity analysis for the basic reproduction number has been performed using Latin hypercube sampling and partial rank correlation coefficients techniques. The combined effect of vaccination rate with transmission rate and vaccination rate with re-infection probability (i.e. strength of cross-immunity) on [Formula: see text] have been discussed. Our research underlines the need to take cross-immunity and time delays into account in the epidemic model in order to better understand disease dynamics.

Dynamics of consumer-resource reaction-diffusion models: single and multiple consumer species

A consumer-resource reaction-diffusion model with a single consumer species was proposed and experimentally studied by Zhang et al.(Ecol Lett 20:1118-1128, 2017). Analytical study on its dynamics was further performed by He et al.(J Math Biol 78:1605-1636, 2019). In this work, we completely settle the conjecture proposed by He et al.(J Math Biol 78:1605-1636, 2019) about the global dynamics of the consumer-resource model for small yield rate. We then study a multi-species consumer-resource model where all the consumer species compete with each other through depression of the limited resources by consumption and there is no direct competition between them. We show that in this case, all consumer species persist uniformly, which implies that "competition exclusion" phenomenon will never happen. We also clarify its dynamics in both homogeneous and heterogeneous environments under various circumstances.

A two-galectin network establishes mesenchymal condensation phenotype in limb development

Previous work showed that Gal-1A and Gal-8, two proteins belonging to the galactoside-binding galectin family, are the earliest determinants of the patterning of the skeletal elements of embryonic chicken limbs, and further, that their experimentally determined interactions in the embryonic limb bud can be interpreted via a reaction-diffusion-adhesion (2GL: two galectin plus ligands) model. Here, we use an ordinary differential equation-based approach to analyze the intrinsic switching modality of the 2GL network and characterize the network behavior independent of the diffusive and adhesive arms of the patterning mechanism. We identify two states: where the concentrations of both the galectins are respectively, negligible, and very high. This bistable switch-like system arises via a saddle-node bifurcation from a monostable state. For the case of mass-action production terms, we provide an explicit Lyapunov function for the system, which shows that it has no periodic solutions. Our model therefore predicts that the galectin network may exist in low expression and high expression states separated in space or time, without any intermediate states. We test these predictions in experiments performed with high density cultures of chick limb mesenchymal cells and observe that cells inside precartilage protocondensations express Gal-1A at a much higher rate than those outside, for which it was negligible. The Gal-1A and -8-based patterning network is therefore sufficient to partition the mesenchymal cell population into two discrete cell states with different developmental (chondrogenic vs. non-chondrogenic) fates. When incorporated into an adhesion and diffusion-enabled framework this system can generate a spatially patterned limb skeleton.

Natural Parameter Conditions for Singular Perturbations of Chemical and Biochemical Reaction Networks

We consider reaction networks that admit a singular perturbation reduction in a certain parameter range. The focus of this paper is on deriving "small parameters" (briefly for small perturbation parameters), to gauge the accuracy of the reduction, in a manner that is consistent, amenable to computation and permits an interpretation in chemical or biochemical terms. Our work is based on local timescale estimates via ratios of the real parts of eigenvalues of the Jacobian near critical manifolds. This approach modifies the one introduced by Segel and Slemrod and is familiar from computational singular perturbation theory. While parameters derived by this method cannot provide universal quantitative estimates for the accuracy of a reduction, they represent a critical first step toward this end. Working directly with eigenvalues is generally unfeasible, and at best cumbersome. Therefore we focus on the coefficients of the characteristic polynomial to derive parameters, and relate them to timescales. Thus, we obtain distinguished parameters for systems of arbitrary dimension, with particular emphasis on reduction to dimension one. As a first application, we discuss the Michaelis-Menten reaction mechanism system in various settings, with new and perhaps surprising results. We proceed to investigate more complex enzyme catalyzed reaction mechanisms (uncompetitive, competitive inhibition and cooperativity) of dimension three, with reductions to dimension one and two. The distinguished parameters we derive for these three-dimensional systems are new. In fact, no rigorous derivation of small parameters seems to exist in the literature so far. Numerical simulations are included to illustrate the efficacy of the parameters obtained, but also to show that certain limitations must be observed.

Stability of a delayed SARS-CoV-2 reactivation model with logistic growth and adaptive immune response

This paper develops and analyzes a SARS-CoV-2 dynamics model with logistic growth of healthy epithelial cells, CTL immune and humoral (antibody) immune responses. The model is incorporated with four mixed (distributed/discrete) time delays, delay in the formation of latent infected epithelial cells, delay in the formation of active infected epithelial cells, delay in the activation of latent infected epithelial cells, and maturation delay of new SARS-CoV-2 particles. We establish that the model's solutions are non-negative and ultimately bounded. We deduce that the model has five steady states and their existence and stability are perfectly determined by four threshold parameters. We study the global stability of the model's steady states using Lyapunov method. The analytical results are enhanced by numerical simulations. The impact of intracellular time delays on the dynamical behavior of the SARS-CoV-2 is addressed. We noted that increasing the time delay period can suppress the viral replication and control the infection. This could be helpful to create new drugs that extend the delay time period.

Dynamical behavior of a stochastic SIQS model via isolation with regime-switching

A stochastic SIQS model via isolation with regime-switching is studied in this paper. The range of positive solution of the model is presented. Threshold to determine extinction and invariant measure is obtained by a new technique, which can be seen as the sufficient and almost necessary condition. Meantime, a value to judge the existence of stationary distribution is acquired by constructing the suitable hybrid Lyapunov function. Two values are proved to be consistent. Several examples are enumerated to test the theoretical results.

MATHEMATICAL MODELLING APPROACH OF THE STUDY OF EBOLA VIRUS DISEASE TRANSMISSION DYNAMICS IN A DEVELOPING COUNTRY

Background

Ebola Virus causes disease both in human and non-human primates especially in developing countries. In 2014 during its outbreak, it led to majority of deaths especially in some impoverished area of West Africa and its effect is still witnessed up till date.

Materials and Methods

We studied the spread of Ebola virus and obtained a system of equations comprising of eighteen equations which completely described the transmission of Ebola Virus in a population where control measures were incorporated and a major source of contacting the disease which is the traditional washing of dead bodies was also incorporated. We investigated the local stability of the disease-free equilibrium using the Jacobian Matrix approach and the disease- endemic stability using the center manifold theorem. We also investigated the global stability of the equilibrium points using the LaSalle's Invariant principle.

Results

The result showed that the disease-free and endemic equilibrium where both local and globally stable and that the system exhibits a forward bifurcation.

Conclusions

Numerical simulations were carried out and our graphs show that vaccine and condom use is best for susceptible population, quarantine is best for exposed population, isolation is best for infectious population and proper burial of the diseased dead is the best to avoid further disease spread in the population and have quicker and better recovery.

Control analysis of virotherapy chaotic system

In this work, we consider a chaotic system that plays a vital role in the treatment of cancer by injection of a virus externally. Due to the sensitivity of this disease, most of its treatments are highly risky. Therefore, we have designed control inputs using adaptive and passive control techniques for virotherapy. Both controllers are designed to bring global stability to the cancer system with the aid of a quadratic Lyapunov function. Furthermore, we use simulations to verify our controllers. Moreover, we show that our adaptive control technique gives better results in comparison.

Dynamic analysis of stochastic delay mutualistic system of leaf-cutter ants with stage structure and their fungus garden

In this paper, we propose a stochastic delay mutualistic model of leaf-cutter ants with stage structure and their fungus garden, in which we explore how the discrete delay and white noise affect the dynamic of the population system. The existence and uniqueness of global positive solution are proved, and the asymptotic behaviours of the stochastic model around the positive equilibrium point of the deterministic model are also investigated. Furthermore, the sufficient conditions for the persistence of the population are established. Finally, some numerical simulations are performed to show the effect of random environmental fluctuation on the model.

Optimal Control Strategy of an Online Game Addiction Model with Incomplete Recovery

Since the global COVID-19 pandemic in 2020, some people who have dropped out of online game have become re-addicted to it due to the order of stay-at-home, making the phenomenon of online game addiction even worse. Controlling the prevalence of online game addiction is of great significance to protect people's healthy life. For this purpose, a mathematical model of online game addiction with incomplete recovery and relapse is established. First, we analyze the basic properties of the model and obtain the expression of the basic reproduction number and all equilibria. By constructing suitable Lyapunov functions, the global asymptotical stability of the equilibria are proved. Then in the numerical simulation, we use the least squares estimation method to fit the real data of e-sports users in China from 2010 to 2020, and obtain the estimated value of all parameters. The approximation value of the basic reproduction number is obtained as R 0 ≈ 5.05 . The result reflects that the spread of game addiction in China is very serious. The stability of the equilibria are proved by using the estimated parameter values. Finally, the simulation results between with control and without control during 2020 to 2050 are compared, and the optimal control strategy is found by comparing the total infectious people. The results of optimal control suggest that if we increase our continuous attention to incompletely recovered people, we can prevent more people from becoming addicted to games. The findings in this paper reveal new mechanisms of game addiction transmission and demonstrate a more detailed and reliable control strategy.

Trajectory Modeling by Distributed Gaussian Processes in Multiagent Systems

This paper considers trajectory a modeling problem for a multi-agent system by using the Gaussian processes. The Gaussian process, as the typical data-driven method, is well suited to characterize the model uncertainties and perturbations in a complex environment. To address model uncertainties and noises disturbances, a distributed Gaussian process is proposed to characterize the system model by using local information exchange among neighboring agents, in which a number of agents cooperate without central coordination to estimate a common Gaussian process function based on local measurements and datum received from neighbors. In addition, both the continuous-time system model and the discrete-time system model are considered, in which we design a control Lyapunov function to learn the continuous-time model, and a distributed model predictive control-based approach is used to learn the discrete-time model. Furthermore, we apply a Kullback-Leibler average consensus fusion algorithm to fuse the local prediction results (mean and variance) of the desired Gaussian process. The performance of the proposed distributed Gaussian process is analyzed and is verified by two trajectory tracking examples.

Dynamic of a two-strain COVID-19 model with vaccination

COVID-19 is a respiratory illness caused by an ribonucleic acid (RNA) virus prone to mutations. In December 2020, variants with different characteristics that could affect transmissibility emerged around the world. To address this new dynamic of the disease, we formulate and analyze a mathematical model of a two-strain COVID-19 transmission dynamics with strain 1 vaccination. The model is theoretically analyzed and sufficient conditions for the stability of its equilibria are derived. In addition to the disease-free and endemic equilibria, the model also has single-strain 1 and strain 2 endemic equilibria. Using the center manifold theory, it is shown that the model does not exhibit the phenomenon of backward bifurcation, and global stability of the model equilibria are proved using various approaches. Simulations to support the model theoretical results are provided. We calculate the basic reproductive number R 1 and R 2 for both strains independently. Results indicate that - both strains will persist when R 1 > 1 and R 2 > 1 - Stain 2 could establish itself as the dominant strain if R 1 < 1 and R 2 > 1 , or when R 2 > R 1 > 1 . However, because of de novo herd immunity due to strain 1 vaccine efficacy and provided the initial stain 2 transmission threshold parameter R 2 is controlled to remain below unity, strain 2 will not establish itself/persist in the community.

Resilient control design for networked DC microgrids under time-constrained DoS attacks

This paper studies the resilient current controller design for the networked DC microgrid system with multiple constant power loads (CPLs) under a new type of time-constrained denial-of-service (DoS) attack. Different from the existing DoS attack models, which are often characterized by DoS frequency and DoS duration, this paper only considers the duration characteristics of the sporadic/aperiodic DoS attacks, and proposes a new type of time-constrained DoS attack model. Under the effects of such DoS attacks, a switching state feedback control law is constructed and a switching-like DC microgrid system model is then established. Furthermore, based on an attack-parameter-dependent time-varying Lyapunov function (TVLF) method, the exponential stability criterion of the resulting DC microgrid system under aperiodic DoS attacks is derived, and a new resilient controller design method is proposed. Finally, simulation studies are given to verify the effectiveness and merits of the proposed resilient control design scheme on achieving the desired control performance and attack resilience.

Derivation and Analysis of a Discrete Predator-Prey Model

We derive a discrete predator-prey model from first principles, assuming that the prey population grows to carrying capacity in the absence of predators and that the predator population requires prey in order to grow. The proposed derivation method exploits a technique known from economics that describes the relationship between continuous and discrete compounding of bonds. We extend standard phase plane analysis by introducing the next iterate root-curve associated with the nontrivial prey nullcline. Using this curve in combination with the nullclines and direction field, we show that the prey-only equilibrium is globally asymptotic stability if the prey consumption-energy rate of the predator is below a certain threshold that implies that the maximal rate of change of the predator is negative. We also use a Lyapunov function to provide an alternative proof. If the prey consumption-energy rate is above this threshold, and hence the maximal rate of change of the predator is positive, the discrete phase plane method introduced is used to show that the coexistence equilibrium exists and solutions oscillate around it. We provide the parameter values for which the coexistence equilibrium exists and determine when it is locally asymptotically stable and when it destabilizes by means of a supercritical Neimark-Sacker bifurcation. We bound the amplitude of the closed invariant curves born from the Neimark-Sacker bifurcation as a function of the model parameters.

Effectiveness of lock down to curtail the spread of corona virus: A mathematical model

In this paper, we have considered a mathematical model that deals with the effectiveness of the measures that may be helpful for reducing the spread of the COVID-19 virus in the society. Here we have illustrated the importance of lock down in controlling and maintaining the spread of the COVID-19 virus. The impact of the virus on the susceptible population has been considered in the model. Also, we have taken into account the susceptible population, which by taking preventive measures viz., by having strong immunity, maintaining social distancing, wearing PPE kits and masks etc., is able to reduce the possibility of getting infected from the virus. Local as well as global stability of the equilibrium points of the model have been studied using Lyapunov function and the geometrical approach techniques. Basic reproduction number has also been obtained by using the next generation matrix. To show the effectiveness of the model, different cases obtained by varying the parameters involved in the model have been considered. A comparison between the actual number of infected cases in India and that obtained by the proposed model, showing the effectiveness of the proposed model, has also been carried out.

Predefined-Time Stability/Synchronization of Coupled Memristive Neural Networks With Multi-Links and Application in Secure Communication

This paper explores the realization of a predefined-time synchronization problem for coupled memristive neural networks with multi-links (MCMNN) via nonlinear control. Several effective conditions are obtained to achieve the predefined-time synchronization of MCMNN based on the controller and Lyapunov function. Moreover, the settling time can be tunable based on a parameter designed by the controller, which is more flexible than fixed-time synchronization. Then based on the predefined-time stability criterion and the tunable settling time, we propose a secure communication scheme. This scheme can determine security of communication in the aspect of encrypting the plaintext signal with the participation of multi-links topology and coupled form. Meanwhile, the plaintext signals can be recovered well according to the given new predefined-time stability theorem. Finally, numerical simulations are given to verify the effectiveness of the obtained theoretical results and the feasibility of the secure communication scheme.

Stability and bifurcation analysis of an HIV-1 infection model with a general incidence and CTL immune response

In this paper, with eclipse stage in consideration, we propose an HIV-1 infection model with a general incidence rate and CTL immune response. We first study the existence and local stability of equilibria, which is characterized by the basic infection reproduction number R0 and the basic immunity reproduction number R1. The local stability analysis indicates the occurrence of transcritical bifurcations of equilibria. We confirm the bifurcations at the disease-free equilibrium and the infected immune-free equilibrium with transmission rate and the decay rate of CTLs as bifurcation parameters, respectively. Then we apply the approach of Lyapunov functions to establish the global stability of the equilibria, which is determined by the two basic reproduction numbers. These theoretical results are supported with numerical simulations. Moreover, we also identify the high sensitivity parameters by carrying out the sensitivity analysis of the two basic reproduction numbers to the model parameters.

Stochastic probical strategies in a delay virus infection model to combat COVID-19

In disease model systems, random noises and time delay factors play key role in interpreting disease dynamics to comprehend deeper insights into the course of dynamics. An endeavor to forecast intercellular behavioral dynamics of SARS-CoV-2 virus via Infection model with responsive host immune mechanisms forms the crux of this research study. Incorporation of time delay factor into infection transmission rates in noisy system epitomizes spectacular view on internal viral dynamics and stability properties are rigorously analyzed around equilibrium steady states to probe feasible strategies in mitigating rapid spread. Efforts to perceive inocular view on infection dynamics are not limited to theoretical frontiers but are substantiated with empirically simulated outcomes and visualized as graphical upshots. Discussions on numerical investigations emphasized shorter incubation periods and vaccination at pertinent time intervals to restrain massive spread and exhibit total immunity against SARS-CoV-2 infections.

Local Stability of McKean-Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies

A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.

Finite-time stabilization of a perturbed chaotic finance model

•

This article proposes a new robust nonlinear controller that stabilizes a chaotic finance system in a finite-time without cancellation of the spacecraft's nonlinear terms, it improves the efficiency of the closed-loop.

•

It accomplishes an oscillation-free faster convergence of the perturbed state variables to the desired steady-state.

•

The proposed controller is insensitive to the parameter uncertainties of the nonlinear terms and exogenous disturbances.

•

The paper performs a comparative study to verify the performance and efficiency of the proposed controller.

Introduction

Robust, stable financial systems significantly improve the growth of an economic system. The stabilization of financial systems poses the following challenges. The state variables’ trajectories (i) lie outside the basin of attraction, (ii) have high oscillations, and (iii) converge to the equilibrium state slowly.

Objectives

This paper aims to design a controller that develops a robust, stable financial closed-loop system to address the challenges above by (i) attracting all state variables to the origin, (ii) reducing the oscillations, and (iii) increasing the gradient of the convergence.

Methods

This paper proposes a detailed mathematical analysis of the steady-state stability, dissipative characteristics, the Lyapunov exponents, bifurcation phenomena, and Poincare maps of chaotic financial dynamic systems. The proposed controller does not cancel the nonlinear terms appearing in the closed-loop. This structure is robust to the smoothly varying system parameters and improves closed-loop efficiency. Further, the controller eradicates the effects of inevitable exogenous disturbances and accomplishes a faster, oscillation-free convergence of the perturbed state variables to the desired steady-state within a finite time. The Lyapunov stability analysis proves the closed-loop global stability. The paper also discusses finite-time stability analysis and describes the controller parameters’ effects on the convergence rates. Computer-based simulations endorse the theoretical findings, and the comparative study highlights the benefits.

Results

Theoretical analysis proofs and computer simulation results verify that the proposed controller compels the state trajectories, including trajectories outside the basin of attraction, to the origin within finite time without oscillations while being faster than the other controllers discussed in the comparative study section.

Conclusions

This article proposes a novel robust, nonlinear finite-time controller for the robust stabilization of the chaotic finance model. It provides an in-depth analysis based on the Lyapunov stability theory and computer simulation results to verify the robust convergence of the state variables to the origin.

Spatial waves and temporal oscillations in vertebrate limb development

The mesenchymal tissue of the developing vertebrate limb bud is an excitable medium that sustains both spatial and temporal periodic phenomena. The first of these is the outcome of general Turing-type reaction-diffusion dynamics that generate spatial standing waves of cell condensations. These condensations are transformed into the nodules and rods of the cartilaginous, and eventually (in most species) the bony, endoskeleton. In the second, temporal periodicity results from intracellular regulatory dynamics that generate oscillations in the expression of one or more gene whose products modulate the spatial patterning system. Here we review experimental evidence from the chicken embryo, interpreted by a set of mathematical and computational models, that the spatial wave-forming system is based on two glycan-binding proteins, galectin-1A and galectin-8 in interaction with each other and the cells that produce them, and that the temporal oscillation occurs in the expression of the transcriptional coregulator Hes1. The multicellular synchronization of the Hes1 oscillation across the limb bud serves to coordinate the biochemical states of the mesenchymal cells globally, thereby refining and sharpening the spatial pattern. Significantly, the wave-forming reaction-diffusion-based mechanism itself, unlike most Turing-type systems, does not contain an oscillatory core, and may have evolved to this condition as it came to incorporate the cell-matrix adhesion module that enabled its pattern-forming capability.

Dynamics of SIR model with vaccination and heterogeneous behavioral response of individuals modeled by the Preisach operator

We study global dynamics of an SIR model with vaccination, where we assume that individuals respond differently to dynamics of the epidemic. Their heterogeneous response is modeled by the Preisach hysteresis operator. We present a condition for the global stability of the infection-free equilibrium state. If this condition does not hold true, the model has a connected set of endemic equilibrium states characterized by different proportion of infected and immune individuals. In this case, we show that every trajectory converges either to an endemic equilibrium or to a periodic orbit. Under additional natural assumptions, the periodic attractor is excluded, and we guarantee the convergence of each trajectory to an endemic equilibrium state. The global stability analysis uses a family of Lyapunov functions corresponding to the family of branches of the hysteresis operator.

Stability analysis and simulation of the novel Corornavirus mathematical model via the Caputo fractional-order derivative: A case study of Algeria

The novel coronavirus infectious disease (or COVID-19) almost spread widely around the world and causes a huge panic in the human population. To explore the complex dynamics of this novel infection, several mathematical epidemic models have been adopted and simulated using the statistical data of COVID-19 in various regions. In this paper, we present a new nonlinear fractional order model in the Caputo sense to analyze and simulate the dynamics of this viral disease with a case study of Algeria. Initially, after the model formulation, we utilize the well-known least square approach to estimate the model parameters from the reported COVID-19 cases in Algeria for a selected period of time. We perform the existence and uniqueness of the model solution which are proved via the Picard-Lindelöf method. We further compute the basic reproduction numbers and equilibrium points, then we explore the local and global stability of both the disease-free equilibrium point and the endemic equilibrium point. Finally, numerical results and graphical simulation are given to demonstrate the impact of various model parameters and fractional order on the disease dynamics and control.

Stability of an HTLV-HIV coinfection model with multiple delays and CTL-mediated immunity

In the literature, several mathematical models have been formulated and developed to describe the within-host dynamics of either human immunodeficiency virus (HIV) or human T-lymphotropic virus type I (HTLV-I) monoinfections. In this paper, we formulate and analyze a novel within-host dynamics model of HTLV-HIV coinfection taking into consideration the response of cytotoxic T lymphocytes (CTLs). The uninfected CD 4 + T cells can be infected via HIV by two mechanisms, free-to-cell and infected-to-cell. On the other hand, the HTLV-I has two modes for transmission, (i) horizontal, via direct infected-to-cell touch, and (ii) vertical, by mitotic division of active HTLV-infected cells. It is well known that the intracellular time delays play an important role in within-host virus dynamics. In this work, we consider six types of distributed-time delays. We investigate the fundamental properties of solutions. Then, we calculate the steady states of the model in terms of threshold parameters. Moreover, we study the global stability of the steady states by using the Lyapunov method. We conduct numerical simulations to illustrate and support our theoretical results. In addition, we discuss the effect of multiple time delays on stability of the steady states of the system.

Modeling and analysis of a within-host HIV/HTLV-I co-infection

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the CD4 + T cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome (AIDS), while HTLV-I is the causative agent for adult T-cell leukemia (ATL) and HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP). Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. The present paper is concerned with the formulation and investigation of a new HIV/HTLV-I co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4 + T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell transmission. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact through the virological synapse, and (ii) vertical transmission through the mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and Lyapunov-LaSalle asymptotic stability theorem. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Mathematical modeling and analysis for controlling the spread of infectious diseases

In this work, we present and discuss the approaches, that are used for modeling and surveillance of dynamics of infectious diseases by considering the early stage asymptomatic and later stage symptomatic infections. We highlight the conceptual ideas and mathematical tools needed for such infectious disease modeling. We compute the basic reproduction number of the proposed model and investigate the qualitative behaviours of the infectious disease model such as, local and global stability of equilibria for the non-delayed as well as delayed system. At the end, we perform numerical simulations to validate the effectiveness of the derived results.

Mathematical analysis of COVID-19 via new mathematical model

We develop a new mathematical model by including the resistive class together with quarantine class and use it to investigate the transmission dynamics of the novel corona virus disease (COVID-19). Our developed model consists of four compartments, namely the susceptible class, S ( t ) , the healthy (resistive) class, H ( t ) , the infected class, I ( t ) and the quarantine class, Q ( t ) . We derive basic properties like, boundedness and positivity, of our proposed model in a biologically feasible region. To discuss the local as well as the global behaviour of the possible equilibria of the model, we compute the threshold quantity. The linearization and Lyapunov function theory are used to derive conditions for the stability analysis of the possible equilibrium states. We present numerical simulations to support our investigations. The simulations are compared with the available real data for Wuhan city in China, where the infection was initially originated.

A Diffusive Sveir Epidemic Model with Time Delay and General Incidence

In this paper, we consider a delayed diffusive SVEIR model with general incidence. We first establish the threshold dynamics of this model. Using a Nonstandard Finite Difference (NSFD) scheme, we then give the discretization of the continuous model. Applying Lyapunov functions, global stability of the equilibria are established. Numerical simulations are presented to validate the obtained results. The prolonged time delay can lead to the elimination of the infectiousness.

ELECTRONIC SUPPLEMENTARY MATERIAL

Supplementary material is available in the online version of this article at 10.1007/s10473-021-0421-9.

Fractional Order Model for the Role of Mild Cases in the Transmission of COVID-19

Most of the nations with deplorable health conditions lack rapid COVID-19diagnostic test due to limited testing kits and laboratories. The un-diagnosticmild cases (who show no critical sign and symptoms) play the role as a route that spread the infection unknowingly to healthy individuals. In this paper, we present a fractional order SIR model incorporating individual with mild cases as a compartment to become SMIR model. The existence of the solutions of the model is investigated by solving the fractional Gronwall's inequality using the Laplace transform approach. The equilibrium solutions (DFE & Endemic) are found to be locally asymptotically stable, and subsequently the basic reproduction number is obtained. Also the global stability analysis is carried out by constructing Lyapunov function. Lastly, numerical simulations that support analytic solution follow. It was also shown that when the rate of infection of the mild cases increases, there is equivalent increase in the overall population of infected individuals. Hence to curtail the spread of the disease there is need to take care of the Mild cases as well.

Modeling dynamics of cancer radiovirotherapy

Advances in genetic engineering have paved the way for a new therapy for cancer, which is called virotherapy. This treatment uses genetically engineered viruses which selectively infect, replicate in, and destroy cancer cells without damaging normal cells. Furthermore, current research and clinical trials have indicated that these viruses can be delivered as single agents or in combination with other therapies. In this paper, we propose systems of ordinary differential equations for modeling the dynamics of aggressive tumor growth under radiovirotherapy treatment. We divide the treatment period into two phases; consequently, we present two mathematical models. First, we formulate the virotherapy model as Phase I of the treatment. Then we extend the model to include radiotherapy in combination with virotherapy as Phase II of the treatment. Comprehensive qualitative analyses of both models are conducted. Furthermore, numerical experiments are performed in order to support the analytical results. An analysis of the parameters is also carried out to investigate their effects on the outcome of the treatment. Overall, the analytical results reveal that radiovirotherapy is more effective than, and a good alternative to, virotherapy, as it is capable of eradicating tumors completely.

Stability analysis of a neural field self-organizing map

We provide theoretical conditions guaranteeing that a self-organizing map efficiently develops representations of the input space. The study relies on a neural field model of spatiotemporal activity in area 3b of the primary somatosensory cortex. We rely on Lyapunov's theory for neural fields to derive theoretical conditions for stability. We verify the theoretical conditions by numerical experiments. The analysis highlights the key role played by the balance between excitation and inhibition of lateral synaptic coupling and the strength of synaptic gains in the formation and maintenance of self-organizing maps.

Analysis of a within-host HIV/HTLV-I co-infection model with immunity

Human immunodeficiency virus (HIV) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that attack the immune cells and impair their functions. Both HIV and HTLV-I can be transmitted between individuals through direct contact with certain body fluids from infected individuals. Therefore, a person can be co-infected with both viruses. HIV causes acquired immunodeficiency syndrome, while HTLV-I is the causative agent for adult T-cell leukemia and HTLV-I-associated myelopathy/tropical spastic paraparesis. Several mathematical models have been developed in the literature to describe the within-host dynamics of HIV and HTLV-I mono-infections. However, modeling a within-host dynamics of HIV/HTLV-I co-infection has not been involved. In the present paper, we are concerned to formulate and analyze a new HIV/HTLV co-infection model under the effect of Cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD4⁺T cells, silent HIV-infected cells, active HIV-infected cells, silent HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by two routes of transmission, virus-to-cell and cell-to-cell. On the other side, HTLV-I has two modes of transmission, (i) horizontal transmission via direct cell-to-cell contact, and (ii) vertical transmission through mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We define a set of threshold parameters which govern the existence and stability of all equilibria of the model. We explore the global asymptotic stability of all equilibria by utilizing Lyapunov function and LaSalle's invariance principle. We have presented numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we evaluate the effect of HTLV-I infection on the HIV dynamics and vice versa.

Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia

The Aim of this research is construct the SEIR model for COVID-19, Stability Analysis and numerical simulation of the SEIR model on the spread of COVID-19. The method used to construct the model is the SEIR model by considering vaccination and isolation factors as model parameters, the analysis of the model uses the generation matrix method to obtain the basic reproduction numbers and the global stability of the COVID-19 distribution model. Numerical simulation models use secondary data on the number of COVID-19 cases in Indonesia. The results obtained are the SEIR model for COVID-19; model analysis yields global stability from the spread of COVID-19; The results of the analysis also provide information if no vaccine, Indonesia is endemic COVID-19. Then the simulation results provide a prediction picture of the number of COVID-19 in Indonesia in the following days, the simulation results also show that the vaccine can accelerate COVID-19 healing and maximum isolation can slow the spread of COVID-19. The results obtained can be used as a reference for early prevention of the spread of COVID-19 in Indonesia.

Modeling the effects of contact tracing on COVID-19 transmission

In this paper, a mathematical model for COVID-19 that involves contact tracing is studied. The contact tracing-induced reproduction number R q and equilibrium for the model are determined and stabilities are examined. The global stabilities results are achieved by constructing Lyapunov functions. The contact tracing-induced reproduction number R q is compared with the basic reproduction number R 0 for the model in the absence of any intervention to assess the possible benefits of the contact tracing strategy.

Machine Learning for Modeling the Singular Multi-Pantograph Equations

In this study, a new approach to basis of intelligent systems and machine learning algorithms is introduced for solving singular multi-pantograph differential equations (SMDEs). For the first time, a type-2 fuzzy logic based approach is formulated to find an approximated solution. The rules of the suggested type-2 fuzzy logic system (T2-FLS) are optimized by the square root cubature Kalman filter (SCKF) such that the proposed fineness function to be minimized. Furthermore, the stability and boundedness of the estimation error is proved by novel approach on basis of Lyapunov theorem. The accuracy and robustness of the suggested algorithm is verified by several statistical examinations. It is shown that the suggested method results in an accurate solution with rapid convergence and a lower computational cost.

Universal Gorban's Entropies: Geometric Case Study

Recently, A.N. Gorban presented a rich family of universal Lyapunov functions for any linear or non-linear reaction network with detailed or complex balance. Two main elements of the construction algorithm are partial equilibria of reactions and convex envelopes of families of functions. These new functions aimed to resolve "the mystery" about the difference between the rich family of Lyapunov functions (f-divergences) for linear kinetics and a limited collection of Lyapunov functions for non-linear networks in thermodynamic conditions. The lack of examples did not allow to evaluate the difference between Gorban's entropies and the classical Boltzmann-Gibbs-Shannon entropy despite obvious difference in their construction. In this paper, Gorban's results are briefly reviewed, and these functions are analysed and compared for several mechanisms of chemical reactions. The level sets and dynamics along the kinetic trajectories are analysed. The most pronounced difference between the new and classical thermodynamic Lyapunov functions was found far from the partial equilibria, whereas when some fast elementary reactions became close to equilibrium then this difference decreased and vanished in partial equilibria.

Global stability of discrete virus dynamics models with humoural immunity and latency

This paper studies the global stability of discrete-time viral infection models with humoural immunity. We consider both latently and actively infected cells. We study also a model with general production and clearance rates of all compartments as well as general incidence rate of infection. We use nonstandard finite difference method to discretize the continuous-time models. The positivity and boundedness of solutions of the discrete models are established. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number [Formula: see text] and the humoural immune response activation number [Formula: see text]. The results signify that the infection dies out if [Formula: see text]. Moreover, the infection persists with inactive immune response if [Formula: see text] and with active immune response if [Formula: see text]. We illustrate our theoretical results by using numerical simulations.

Micrometer Backstepping Control System for Linear Motion Single Axis Robot Machine Drive

In order to cut down influence on the uncertainty disturbances of a linear motion single axis robot machine, such as the external load force, the cogging force, the column friction force, the Stribeck force, and the parameters variations, the micrometer backstepping control system, using an amended recurrent Gottlieb polynomials neural network and altered ant colony optimization (AACO) with the compensated controller, is put forward for a linear motion single axis robot machine drive system mounted on the linear-optical ruler with 1 um resolution. To achieve high-precision control performance, an adaptive law of the amended recurrent Gottlieb polynomials neural network based on the Lyapunov function is proposed to estimate the lumped uncertainty. Besides this, a novel error-estimated law of the compensated controller is also proposed to compensate for the estimated error between the lumped uncertainty and the amended recurrent Gottlieb polynomials neural network with the adaptive law. Meanwhile, the AACO is used to regulate two variable learning rates in the weights of the amended recurrent Gottlieb polynomials neural network to speed up the convergent speed. The main contributions of this paper are: (1) The digital signal processor (DSP)-based current-regulation pulse width modulation (PWM) control scheme being successfully applied to control the linear motion single axis robot machine drive system; (2) the micrometer backstepping control system using an amended recurrent Gottlieb polynomials neural network with the compensated controller being successfully derived according to the Lyapunov function to diminish the lumped uncertainty effect; (3) achieving high-precision control performance, where an adaptive law of the amended recurrent Gottlieb polynomials neural network based on the Lyapunov function is successfully applied to estimate the lumped uncertainty; (4) a novel error-estimated law of the compensated controller being successfully used to compensate for the estimated error; and (5) the AACO being successfully used to regulate two variable learning rates in the weights of the amended recurrent Gottlieb polynomials neural network to speed up the convergent speed. Finally, the effectiveness of the proposed control scheme is also verified by the experimental results.

Analysis of Malaria Control Measures' Effectiveness Using Multistage Vector Model

We analyze an epidemiological model to evaluate the effectiveness of multiple means of control in malaria-endemic areas. The mathematical model consists of a system of several ordinary differential equations and is based on a multi-compartment representation of the system. The model takes into account the multiple resting-questing stages undergone by adult female mosquitoes during the period in which they function as disease vectors. We compute the basic reproduction number [Formula: see text] and show that if [Formula: see text], the disease-free equilibrium is globally asymptotically stable (GAS) on the nonnegative orthant. If [Formula: see text], the system admits a unique endemic equilibrium (EE) that is GAS. We perform a sensitivity analysis of the dependence of [Formula: see text] and the EE on parameters related to control measures, such as killing effectiveness and bite prevention. Finally, we discuss the implications for a comprehensive, cost-effective strategy for malaria control.

Dynamic and game theory of infectious disease stigmas

Stigmas are a primal phenomena, ubiquitous in human societies past and present. Some evolutionary anthropologists have argued that stigmatization in response to disease is an adaptive behavior because stigmatization may help people and communities reduce the risks they face from infectious diseases and increase reproductive success. On the other hand, some cultural anthropologists and social critics argue that stigmatization has strong negative impacts on community health. One recent analysis resolved this conflict by hypothesizing that stigmas had individual and group-evolutionary benefits in the past but are now maladaptive because of intervening societal transitions. Here, we present a quantitative theory of infectious disease stigmatization. Using a four-compartment model of stigmatization against a chronic disease, we show a stigma ratio, being the ratio of net transmissions by stigmatized people to net transmissions by unstigmatized people, predicts the impact of stigmatization on lifetime infection risk. When stigmatized people are segregated from the rest of the population and there are no alternative interventions that reduce transmission, stigmatization can reduce prevalence and infection risk. When stigmas do not lead to segregation but do discourage behavior change and reduce access to medical interventions, stigmatization acts to increases the lifetime risk of infection in the community. We further show that fear of stigmas can create policy resistance to healthcare access. The societal consequences of fear are worse when effective medical treatment is available. We conclude that stigma's can be adaptive, but good healthcare and leaky ostracism can make stigmas against chronic infectious disease maladaptive, and that the deprecation of stigmas is a natural transition in the modern urban societies.

Mathematical analysis on deterministic and stochastic lake ecosystem models

In this paper, we propose and study the deterministic and stochastic lake ecosystem models to investigate the impact of terrestrial organic matter upon persistence of the plankton populations. By constructing Lyapunov function and using the LaSalle's Invariance Principle, we establish global properties of the deterministic model. The dynamical behavior of solutions fits well with some experimental results. It is concluded that the terrestrial organic matter plays an important role in influencing interactions between phytoplankton and zooplankton. Based on the fluctuations of lake ecosystem, we further develop a stochastically perturbed model. Theoretic analysis implies that the stochastic model exists a stationary distribution which is ergodic. The key point of our analysis is to enhance our knowledge of the factors governing the dynamics of plankton population models.