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This study investigates the qualitative properties and numerical dynamics of a stochastic epidemiological model incorporating unreported cases with a general contact susceptible function, shedding light on the intricate dynamics of infectious diseases such as COVID-19. In the qualitative analysis, we rigorously examine the mathematical properties of the model, including the existence and positivity of solutions, and identify a critical threshold parameter, R s , pivotal in determining the long-term behavior of the system. Notably, our analysis reveals that the stochastic noise significantly influences the dynamics, leading to distinct outcomes: if R s exceeds unity, solutions converge exponentially to a unique invariant probability distribution, whereas values below one result in the extinction of infectious diseases at an exponential rate. In the numerical study, we delve into comprehensive simulations to validate our theoretical findings and explore the behavior of the model under various scenarios. Synthetic data simulations provide illustrative examples, showcasing both disease extinction and persistence phenomena. Furthermore, we investigate the impact of the susceptible contact function, g(S), on disease dynamics, and propose a selection method for optimizing this function based on real-world COVID-19 data from the UK. By integrating rigorous mathematical analysis with empirical data-driven insights, our study offers valuable contributions to understanding the complex dynamics of infectious diseases. [ABSTRACT FROM AUTHOR]

In this paper, a novel scheme is presented to estimate the uncertainty and disturbance of the spacecraft system by a compensatory vector calculated online from the information of measured angular velocities. The compensatory vector is used in the constrained attitude control law to compensate for the perturbations of the spacecraft system. The attitude controller considering the input limitations of reaction wheels is developed by solving an optimization problem through the Karush-Kuhn-Tucker (KKT) theorem. By using noisy angular velocities in the observer design, the stochastic stability of the closed loop system under the constrained multivariable controller is demonstrated. In the results, at first, the open-loop performance of the uncertainty and disturbance observer is evaluated through computer simulations. Subsequently, the performance of the proposed controller in both constrained and unconstrained versions is evaluated. Finally, the proposed controller performance in compensating for uncertainties and disturbances is compared with the adaptive backstepping controller reported in the literature. The comparative results of two controllers show the superior performance for the proposed constrained controller in the presence of disturbances, uncertainties and input constraints. Furthermore, the proposed control system could attenuate the effect of measurement noises of angular velocities by weighting the compensatory vector in the observer design. [ABSTRACT FROM AUTHOR]

The stochastic responses and the asymptotic stability with probability 1 of vibro-impact systems with friction under wideband noise excitation were investigated. The Zhuravlev non-smooth transformation and the stochastic averaging method were extended to obtain the steady-state probability density functions of the system. The accuracy of the method was verified through comparison of the theoretical results with those from the Monte Carlo simulations. The effects of the friction force and the vibro-impact restitution coefficient on the system responses were studied. Furthermore, The Lyapunov exponent of the linearized averaged It8 equation was derived and the stability of the trivial solution was determined with the Lyapunov exponent. The results show that, changing the frictional coefficient and the vibro-impact restitution coefficient could adjust the system stochastic stability [ABSTRACT FROM AUTHOR]

Capturing complicated dynamics and understanding the underlying controlling ecological processes is one of the major ecological issues. The Allee effect is an essential component in ecology, and considering it can have a substantial impact on system dynamics. In the present investigation, we analysed a prey-predator scenario in which the predator is a generalist since it feeds on prey populations and the Allee phenomenon impacts the prey population's growth. The influence of the Allee effect on the changing nature of the system is investigated. The stability and boundedness of the model's equilibria are extensively investigated. We found that including the Allee effect enhances the system's local and global behaviours through a detailed bifurcation analysis. The chaotic nature of the system is strongly impacted by the Allee effect, particularly once a specific threshold value is reached. In the study of bifurcation analysis, we looked into bifurcations such the presence of transcritical bifurcation and Hopf-bifurcation to chaos. We added stochastic perturbation to this problem by including random fluctuations in the sensitive parameters. Finally, we analysed the system's mean-square stochastic stability towards the internal equilibrium. As a result, it is discovered that the Allee effect and stochastic perturbation considerably influence the behaviour of the prey-predator system.

This paper introduces a novel resource-efficient control structure for remote path-following control of autonomous vehicles based on a comprehensive combination of Kalman filtering, non-uniform dual-rate sampling, periodic event-triggered communication, and prediction-based and packet-based control techniques. An essential component of the control solution is a non-uniform dual-rate extended Kalman filter (NUDREKF), which includes an h-step ahead prediction stage. The prediction error of the NUDREKF is ensured to be exponentially mean-square bounded. The algorithmic implementation of the filter is straightforward and triggered by periodic event conditions. The main goal of the approach is to achieve efficient usage of resources in a wireless networked control system (WNCS), while maintaining satisfactory path-following behavior for the vehicle (a holonomic Mecanum-wheeled robot). The proposal is additionally capable of coping with typical drawbacks of WNCS such as time-varying delays, and packet dropouts and disorder. A Simscape Multibody simulation application reveals reductions of up to 93% in resource usage compared to a nominal time-triggered control solution. The simulation results are experimentally validated in the holonomic Mecanum-wheeled robotic platform. • A novel, comprehensive periodic event-triggered networked control solution is proposed. • High reduction of resource usage (of up to 93%) can be reached while maintaining acceptable control performance. • A non-uniform dual-rate extended Kalman filter (NUDREKF) is essential in the control structure. • Some conditions are provided such that the prediction error of the NUDREKF is exponentially mean square bounded. • Real path-following control tests for a holonomic Mecanum-wheeled robot validate the control solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with stochastic stability and stochastic bifurcation of the FitzhugNagumo model with multiplicative white noise. We employ largest Lyapunov exponent and singular boundary theory to investigate local and global stochastic stability at the equilibrium point. In the rest, the solution of averaging the Ito diffusion equation and extreme point of steady-state probability density function provide sufficient conditions that the stochastic system undergoes pitchfork and phenomenological bifurcations. These theoretical results of the stochastic neuroscience model are confirmed by some numerical simulations and stochastic trajectories. Finally, we compare this approach with Rulkov approach and explain how pitchfork and phenomenological bifurcations describe spiking limit cycles and stability of neuron’s resting state. [ABSTRACT FROM AUTHOR]

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple H 2 ⊂ H 1 ⊂ H 0 to obtain a bound on the difference between the original solution and the perturbed solution. [ABSTRACT FROM AUTHOR]

We study a 2-player stochastic differential game of lobbying. Players invest in lobbying activities to alter the legislation in her own benefit. The payoffs are quadratic and uncertainty is driven by a Wiener process. We consider the Nash symmetric game where players face the same cost and extract symmetric payoffs, and we solve for Markov Perfect Equilibria (MPE) in the class of affine functions. First, we prove a general sufficient (catching up) optimality condition for two-player stochastic games with uncertainty driven by Wiener processes. Second, we prove that the number and nature of MPE depend on the extent of uncertainty (i.e. the variance of the Wiener processes). In particular, we prove that while a symmetric MPE always exists, two asymmetric MPE emerge if and only if uncertainty is large enough. Third, we study the stochastic stability of all the equilibria. We notably find, that the state converges to a stationary invariant distribution under asymmetric MPE. Fourth, we study the implications for rent dissipation asymptotically and compare the outcomes of symmetric vs asymmetric MPE in this respect, ultimately enhancing again the role of uncertainty. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a novel stochastic prey-predator model under random small immigration. Mainly, we prove boundedness for the solution of the model using probabilistic and analytic types of inequalities. Furthermore, possible conditions on the immigration for achieving stochastic square stability are obtained. The immigration of both prey and predator is assumed to be either constant and stochastically perturbed or proportional to the population and stochastically perturbed. In all cases, we arrived at the fact that stability can only be achieved if the immigration is small enough. We also show that as random immigration increases, the dynamic becomes destabilized and could lead to chaos. Lastly, we perform a computational analysis in order to verify the obtained theoretical results.

Abstract In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results.

Abstract Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

This paper studies a simple process of demand adjustment in cooperative games. In the process, a randomly chosen player makes the highest possible demand subject to the demands of other coalition members being satisfied. This process converges to the aspiration set; in convex games, this implies convergence to the core. We further introduce perturbations into the process, where players sometimes make a higher demand than feasible. These perturbations make the set of separating aspirations, i.e., demand vectors in which no player is indispensable in order for other players to achieve their demands, the one most resistant to mutations. We fully analyze this process for 3-player games. We further look at weighted majority games with two types of players. In these games, if the coalition of all small players is winning, the process converges to the unique separating aspiration; otherwise, there are many separating aspirations and the process reaches a neighbourhood of a separating aspiration. [ABSTRACT FROM AUTHOR]

To profit from price oscillations, investors frequently use threshold-type strategies where changes in the portfolio position are triggered by some indicators reaching prescribed levels. In this paper we investigate threshold-type strategies in the context of ergodic control. We make the first steps towards their optimization by proving ergodic properties of related functionals. Assuming Markovian price increments satisfying a minorization condition and (one-sided) boundedness we show, in particular, that for given thresholds, the distribution of the gains converges in the long run. We also extend recent results on the stability of overshoots of random walks from the i.i.d. increment case to Markovian increments, under suitable conditions. [ABSTRACT FROM AUTHOR]

We present an evolutionary model which allows us to study the impact relationship-specific investment has on bargaining. Agents are matched to play an investment and bargaining game. During bargaining, agents have an outside option to form a new relationship, but in exercising this option loses their current investment. We find that the stochastically stable post-investment bargaining convention is dependent on the cost of investment. In particular, the larger the cost of investment, the lower is the share of gross surplus that is received. This stands in contrast with previous studies. In addition, we find that there is under-investment. We disentangle the forces which lead to these two results. [ABSTRACT FROM AUTHOR]

In the present paper, we study stochastic stability and stochastic boundedness for the stochastic differential equation (SDE) with multi-delay of third order. The derived results extend and improve some earlier results in the relevant literature, which are related to the qualitative properties of solutions to third-order delay differential equations (DDEs) and SDEs with multi-delay. Two examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

In this article, a distributed diffusion Kalman filtering algorithm with event‐triggered communication (DDKF‐E) is studied for discrete‐time nonlinear systems. According to the event‐triggered communication protocol, the data between sensors and estimators are transmitted only when the predefined conditions are satisfied. Considering the characteristic of event‐triggered method and truncated error by linearization, an upper bound of the estimation error covariance matrix is obtained by using the variance‐constrained method. The Kalman gain is designed to minimize the upper bound and then two Riccati equations are obtained. Furthermore, the stochastic stability theory is used to prove the stability of DDKF‐E, and it is derived that the estimation error of DDKF‐E is exponentially bounded in mean square. Finally, numerical simulations validate the effectiveness of the DDKF‐E algorithm. [ABSTRACT FROM AUTHOR]

This paper provides some sufficient conditions for the existence and uniqueness and the stochastic stability of the global solution for nonlinear neutral stochastic functional differential equations. When the drift term and the diffusion term satisfy a locally Lipschitz condition, and the Lyapunov monotonicity condition has a sign-changed time-varying coefficient, the existence and uniqueness of the global solution for such equations will be studied by using the Lyapunov--Krasovskii function approach and the theory of stochastic analysis. The stability in pth-moment, the asymptotical stability in pth-moment, and the exponential stability in pth-moment will be investigated. Different characterizations for these three kinds of stochastic stability in moment will be established, which are presented with respect to integration conditions. These results have seldom been reported in the existing literature. The almost surely exponential stability for the global solution of such equations is also discussed. Some discussions and comparisons are provided. Two examples are given to check the effectiveness of the theoretical results obtained. [ABSTRACT FROM AUTHOR]

Considering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system. [ABSTRACT FROM AUTHOR]

I characterize the outcomes of a class of model-free reinforcement learning algorithms, such as stateless Q-learning, in a prisoner's dilemma. The behavior is studied in the limit as players stop experimenting after sufficiently exploring their options. A closed form relationship between the learning rate and game payoffs reveals whether the players will learn to cooperate or defect. The findings have implications for algorithmic collusion and also apply to asymmetric learners with different experimentation rules. [ABSTRACT FROM AUTHOR]

This paper investigates the σ-error pth $ (p>0) $ (p > 0) -moment stability and synthesis problems for a class of discrete-time semi-Markov jump singular systems (DTSMJSSs). In view of the Lyapunov functions approach, we first derive the σ-error pth $ (p>0) $ (p > 0) -moment stable behaviour of interested systems. Based on semi-Markov kernel approach and the techniques of eliminating power of matrices (EPMs), we then obtain several LMI conditions for σ-error mean square stability. Subsequently, due to the time-varying singular E-matrix in DTSMJSSs, two different forms of controller gain are exploited in order to design stabilising state-feedback strategies. In particular, computational analysis and comparison of the proposed control schemes are provided, which shows the numerical complexity of the proposed control schemes is vastly lower than that of the existing literature. Finally, an example has been performed to verify the obtained analytical results. [ABSTRACT FROM AUTHOR]

This paper aims to develop a stochastic perturbation into SEIR (Susceptible-Exposed-Infected-Removed) epidemic model including a saturated estimated incidence. A set of stochastic differential equations is used to study its behavior, with the assumption that each population's exposure to environmental unpredictability is represented by noise terms. This kind of randomness is considerably more reasonable and realistic in the proposed model. The current study has been viewed as strengthening the body of literature because there is less research on the dynamics of this kind of model. We discussed the structure of all equilibriums' existence and the dynamical behavior of all the steady states. The fundamental replication number for the proposed method was used to discuss the stability of every equilibrium point; if R0 < 1, the infected free equilibrium is resilient, and if R0 < 1, the endemic equilibrium is resilient. The system's value is primarily described by its ambient stochasticity, which takes the form of Gaussian white noise. Additionally, the suggested model can offer helpful data for comprehending, forecasting, and controlling the spread of various epidemics globally. Numerical simulations are run for a hypothetical set of parameter values to back up our analytical conclusions. [ABSTRACT FROM AUTHOR]

We propose an approach to neural network stochastic differential games of mean-field type and its corresponding stochastic stability analysis by means of adversarial training (aka adversarial attacks). This is a class of data-driven differential games where the distribution of the variables such as the system states and the decision-makers' strategies (control inputs) is incorporated into the problem. This work casts the cooperative/noncooperative game terminology into the deep learning framework where we talk about cooperative and noncooperative neural network computations that involve learning capabilities and neural network architectures. We suggest a method to computationally validate the feasibility of the approximated solutions via neural networks and evaluate the stochastic stability of the associated closed-loop system (state feedback Nash). Moreover, we enhance the stochastic stability by enlarging the training set with adversarial initial states to obtain a more robust neural network for a particular decision-maker. Finally, a worked-out example based on the linear-quadratic mean-field type game (LQ-MTG) that illustrates our methodology is presented. [ABSTRACT FROM AUTHOR]

Understanding population migration is essential for controlling highly infectious diseases. This paper studies the global dynamics of an infectious disease epidemic model incorporating population migration and a stochastic transmission rate. Our findings demonstrate that in deterministic and stochastic environments, the models exhibit global Lyapunov stability near the disease-free equilibrium point, determined by a threshold parameter. Furthermore, we analyze the effect of migration on infectious diseases. We discover that the number of infections and the peak value of the infection curve increase with a higher level of population migration. These results are supported by numerical illustrations that hold epidemiological relevance. Additionally, the disease-free equilibrium of the associated time delay model is linearly asymptotically stable, and the endemic equilibrium exhibits more bifurcation for larger time delay values.

The article devises an H∞$$ {H}_{\infty } $$‐norm theory for the CSVIU (control and state variations increase uncertainty) class of stochastic systems. This system model appeals to stochastic control problems to express the state evolution of a possibly nonlinear dynamic system restraint to poor modeling. It introduces the H∞$$ {H}_{\infty } $$ control with infinity energy disturbance signals, contrary to usual H∞$$ {H}_{\infty } $$ stochastic formulations, which mimic deterministic models dealing with finite energy disturbances. The reason is to portray the persistent perturbations due to the environment more naturally. In this regard, it develops a refined connection between a notion of stability and the system's power finiteness. It delves into the control solution employing the relations between H∞$$ {H}_{\infty } $$ optimization and differential games, connecting the worst‐case stability analysis of CSVIU systems with a perturbed Lyapunov type of equation. The norm characterization relies on the optimal cost induced by the min–max control strategy. The solvability of a generalized game‐type Riccati equation couples to the rise of a pure saddle point, yielding the global solution of the CSVIU dynamic game. In the way to the optimal stabilizing compensator, the article finds it based on linear feedback stabilizing gain from solving the Riccati equation together with a spectral radius test of an ensuing matrix. The optimal disturbance compensator produces inaction regions in the sense that, for sufficiently minor deviations from the model, the optimal action is constant or null in the face of the uncertainty involved. Finally, an explicit solution approximation developed by a Monte Carlo method appears, and a numerical example illustrates the synthesis. [ABSTRACT FROM AUTHOR]

This paper considers the problem of estimating unknown sparse time-varying signals for stochastic dynamic systems. To deal with the challenges of extensive sparsity, we resort to the compressed sensing method and propose a compressed Kalman filter (KF) algorithm. Our algorithm first compresses the original high-dimensional sparse regression vector via the sensing matrix and then obtains a KF estimate in the compressed low-dimensional space. Subsequently, the original high-dimensional sparse signals can be well recovered by a reconstruction technique. To ensure stability and establish upper bounds on the estimation errors, we introduce a compressed excitation condition without imposing independence or stationarity on the system signal, and therefore suitable for feedback systems. We further present the performance of the compressed KF algorithm. Specifically, we show that the mean square compressed tracking error matrix can be approximately calculated by a linear deterministic difference matrix equation, which can be readily evaluated, analyzed, and optimized. Finally, a numerical example demonstrates that our algorithm outperforms the standard uncompressed KF algorithm and other compressed algorithms for estimating high-dimensional sparse signals. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

This article studies the control problem of Markov jump systems subject to actuator saturation and partial mode information. The asynchronous phenomenon between the controller and the plant is addressed by introducing a hidden Markov model. By means of convex hull method, some sufficient conditions are obtained to guarantee the stochastic stability of the saturated Markov jump systems. Finally, a numerical example is given to illustrate the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

Power system instability causes many local or large-scale power outage accidents. To maintain sustainable development, a new power system construction aimed at maximizing new energy consumption is being put on the agenda. However, with a large increase in stochastic disturbance factors (SDFs), the system gradually shows strong stochasticity, and the stability presents greater complexity. It is necessary to analyze the impact on the system based on different processing methods of SDFs to maintain system stability. This paper delves into the impact of SDFs on system stability by analyzing and summarizing both stochastic variables and processes. Initially, the SDFs in the power system are meticulously analyzed and categorized. When the SDFs are treated as stochastic variables, the probabilistic stability is classified and evaluated based on a probability analysis method, which includes the probabilistic small-disturbance stability, the probabilistic transient stability, and the probabilistic voltage stability. When the SDFs are treated as stochastic processes, the stochastic stability is classified and evaluated by using a stochastic analysis method, including the stochastic small-disturbance stability, the stochastic transient stability, and the stochastic voltage stability. Finally, the research perspectives on SDFs and system stability are discussed and prospected. [ABSTRACT FROM AUTHOR]

In this paper, we study a stochastic SIS epidemic model with distributed delays. The positiveness of the solutions is established. We obtain sufficient conditions for the extinction of the disease through the study of stochastic stability of the disease-free equilibrium and stability of the same equilibrium in the mean. Compared to many works on the deterministic and stochastic SIS models and their stability, the distributed delays involved in the model offer new conditions with much more boundedness on the rate of losing immunity. The disease is extinct for small and large enough values of the intensity of noise and regardless of the initial history functions and the magnitude of the basic reproductive number R 0. [ABSTRACT FROM AUTHOR]

We consider the following network model motivated, in particular, by blockchains and peer-to-peer live streaming. Data packet flows arrive at the network nodes and need to be disseminated to all other nodes. Packets are relayed through the network via links of finite capacity. A packet leaves the network when it is disseminated to all nodes. Our focus is on two communication disciplines, which determine the order in which packets are transmitted over each link, namely Random-Useful (RU) and Oldest-Useful (OU). We show that RU has the maximum stability region in a general network. For the OU we demonstrate that, somewhat surprisingly, it does not in general have the maximum stability region. We prove that OU does achieve maximum stability in the important special case of a symmetric network, given by the full graph with equal capacities on all links and equal arrival rates at all nodes. We also give other stability results, and compare different disciplines' performances in a symmetric system via simulation. Finally, we study the cumulative delays experienced by a packet as it propagates through the symmetric system, specifically the delay asymptotic behavior as N → ∞ . We put forward some conjectures about this behavior, supported by heuristic arguments and simulation experiments. [ABSTRACT FROM AUTHOR]

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented. [ABSTRACT FROM AUTHOR]

This study introduces a novel stochastic variant for the Susceptible-Infected-Recovered-Susceptible (SIRS) system, focusing on perturbations involving the immunity decay rate. We determine a critical threshold value of the reproduction number, denoted as R 0 , which plays a pivotal role in understanding the system dynamics. Through rigorous mathematical derivations, we have shown that a unique solution exists for the system under consideration. Additionally, we leverage the powerful analytical tool of a stochastic Lyapunov function to evaluate the extinction and persistence of the infection, providing valuable insights into the system behavior under different conditions. Our analysis reveals that if R 0 < 1 , the disease will eventually vanish from the population, whereas if R 0 > 1 , an outbreak will ensue. To reinforce our findings, we provide computer simulations as supplementary evidence. [ABSTRACT FROM AUTHOR]

This paper concerns the 2D stochastic Ginzburg–Landau–Newell equations with a degenerate random forcing. We study the relationship between stationary distributions which correspond to the original and perturbed systems and then prove the stability of the stationary distribution. This suggests that the complexity of stochastic systems is likely to be robust. The main difficulty of the proof lies in estimating the expectation of exponential moments and controlling nonlinear terms while working on the evolution triple H2⊂H1⊂H0 to obtain a bound on the difference between the original solution and the perturbed solution.

In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented.

This paper investigates the stochastic stability of discrete time positive Markov jump nonlinear systems (PMJNS). Some definitions on stochastic stability for discrete time PMJNS are introduced first. Then, using the multiply max-separable Lyapunov function method, some stochastic stability criterions of discrete time PMJNS are provided, and some corresponding criterions are also provided for discrete time positive Markov jump linear systems (PMJLS). Different from previous conclusions that require subsystems to be stable or marginally stable, the obtained results allow some subsystems to be unstable. Based on the proposed criterions, the stochastic stability behavior of discrete time positive Markov jump systems can be obtained just from the algebraic properties of the system function and the probability characteristics of the Markov chain. To illustrate the main results, two simulation examples are provided at the end. [ABSTRACT FROM AUTHOR]

The present work addresses the problem of energy shaping for stochastic port-Hamiltonian systems. Energy shaping is a powerful technique that allows one to systematically find feedback laws to shape the Hamiltonian of a controlled system so that, under a general passivity condition, it converges to a desired configuration. Energy shaping has been recently generalized to consider stochastic port-Hamiltonian systems. Nonetheless, the resulting theory presents several limitations so that relevant examples, such as the additive noise case, are immediately ruled out from the possible use of energy shaping. In the current paper we continue the investigation of the properties of a weak notion of passivity for a stochastic system and derive a weak notion of convergence for the controlled system. Such weak notion of passivity is strictly related to the existence and uniqueness of an invariant measure for the system so that the theory developed has a purely probabilistic flavor. We will show how all the relevant results of energy shaping can be recover under the proposed weak setting. [ABSTRACT FROM AUTHOR]

This study extends existing formulae of the fixation probabilities of strategies for symmetric games and bimatrix games in finite populations and derives a counterpart for trimatrix games. This allows us to describe the stochastic evolutionary game dynamics when three players are assigned different roles and therefore are not interchangeable. Following previous studies, we also derived two types of stochastic stability conditions based on the obtained fixation probabilities; "strong stochastic stability," which requires that for any initial frequencies of strategies, the fixation probability of a combination of specific strategies is higher than that under neutrality and those of any other combinations are lower than neutrality; and "stochastic stability," which only requires that the fixation probability of a specific strategy combination be higher than that under neutrality for any initial frequencies of strategies. Thus, for the former, we obtain a clear correspondence with bimatrix games, but not necessarily for the latter. The results of applying our findings to triadic conflicts (the Impartial person and mediator game and the Fish in troubled waters game), the volunteer's dilemma, and coordination games are also reported. [ABSTRACT FROM AUTHOR]

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In this article, we study the dynamics of Norovirus infection by developing a stochastic epidemic model having Levy noise. The study shows that Levy noise and informative interventions have more influence on the said dynamics. Firstly, we show that the model has a unique global positive solution. After this, we formulated a stochastic threshold value as a necessary condition for the extinction and persistence in the mean of the proposed epidemic model. Finally, numerical simulation are drawn to verify our obtained results of the dynamics. The perturbed stochastic approach may affect the dynamical properties of the model and large noises will be greatly significant to minimize its transmission. [ABSTRACT FROM AUTHOR]

Path integral control integrated with backstepping control is proposed to address the practical regulation problem, wherein the system dynamics are represented as stochastic differential equations. Path integral control requires the sampling of uncontrolled trajectories to calculate the optimal control input. However, the probability of generating a low-cost trajectory from uncontrolled dynamics is low. This implies that the path integral control requires an excessive number of trajectory samples to approximate the optimal control input appropriately. Therefore, we propose an integrated method of backstepping and path integral control to provide a systematic approach for sampling stabilized trajectories that are close to the optimal one. This proposed method requires a relatively small number of samples than that of the path integral control and uses the terminal set to further reduce the computational load. In simulation studies, the proposed method is applied to a single-input single-output example and a continuous stirred-tank reactor for demonstration. The results show the advantages of integrating the backstepping control and the path integral control. [ABSTRACT FROM AUTHOR]

We study the threshold dynamics of a stochastic SAIRS‐type model with vaccination, where the role of asymptomatic and symptomatic infectious individuals is explicitly considered in the epidemic dynamics. In the model, the values of the disease transmission rate may switch between different levels under the effect of a semi‐Markov process. We provide sufficient conditions ensuring the almost surely epidemic extinction and persistence in time mean. In the case of disease persistence, we investigate the omega‐limit set of the system and give sufficient conditions for the existence and uniqueness of an invariant probability measure. [ABSTRACT FROM AUTHOR]

• Consider event-triggered extended dissipativity control for uncertain semi-Markov switching systems with disturbances. • Develop a mode- and disturbance-dependent switched event-triggered mechanism. • Propose a criterion for stochastic stability and extended dissipativity. • Present a co-design of the feedback-gain and event-triggered matrices in the framework of linear matrix inequalities. This paper is concerned with event-triggered extended dissipativity stabilization of uncertain semi-Markov switching systems with external disturbances. The purpose is to ensure the stochastic stability and extended dissipativity of the semi-Markov switching systems via using event-triggered control. A mode- and disturbance-dependent switched event-triggered mechanism is devised to determine the triggered moments, which can further reduce the number of triggered events compared to some existing switched event-triggered mechanisms. On the basis of the proposed mechanism, a time- and mode-dependent piecewise-defined Lyapunov functional is constructed to establish a criterion for the stochastic stability and extended dissipativity. Subsequently, a co-design scheme for the needed feedback-gain and event-triggered matrices is presented using some equivalent transformations of matrix inequalities. Lastly, a DC motor model and a robotic arm model are utilized to illustrate the validity of the designed controller. [ABSTRACT FROM AUTHOR]