Dynamical behavior and density function of a stochastic model of HPV infection and cervical cancer with a case study for Xinjiang,China

In this article, combining the transmission features of HPV infection and secondary cervical cancer in Xinjiang, China, a stochastic dynamical model for the HPV infection and secondary cervical cancer with environmental white noise is proposed. Firstly, the stochastic extinction of disease is investigated. A sufficient criterion for the asymptotic behavior of any positive solution of stochastic model revolving around the disease-free equilibrium of corresponding deterministic model is established. Secondly, a threshold criterion for the existence of unique ergodic stationary distribution is obtained by means of the auxiliary function. Furthermore, a new technique of partitioned matrix for the calculation of probability density function is proposed, the expression of a log-normal density function around the quasi-endemic equilibrium of stochastic model is calculated. Lastly, the best-fit parameter values in our model are identified by the MCMC algorithm on the basis of the cervical cancer data in Xinjiang province, China. The basic reproduction number is estimated as 1.3496 (95% CnI: (1.3458, 1.3716)). Then, to determine the key parameters of the model, the sensitivity analysis is explored. Some possible interventions and control measures are provided to reduce the HPV infection spread and cervical cancer in Xinjiang of China.     

Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network

A new SIRS epidemic model with infection age and relapse on a scale-free network is introduced. The basic reproduction number R₀ is defined. Asymptotic smoothness of solution and uniform persistence of system are proved. It is shown that the disease-free equilibrium is globally asymptotically stable by using Fluctuation Lemma if R₀ < 1 and the endemic equilibrium is globally asymptotically stable by constructing suitable Lyapunov functional if R₀ > 1. Effects of two immunization schemes are studied. Numerical simulations and sensitivity analysis are performed. Results manifest that infection age and degree of node play a significant role in controlling the emergence and spread of the epidemic disease.     

The Razumikhin approach on general decay stability for neutral stochastic functional differential equations

The topic of the paper is both the pth moment and almost sure stability on a general decay rate for neutral stochastic functional differential equations, by applying the Razumikhin approach. This concept is extended to neutral stochastic differential delay equations. The results obtained in the paper are more general and they may be specialized on the exponential, polynomial or logarithmic stability. Moreover, some neutral stochastic functional differential equations which are not pth moment or almost surely exponentially stable, could be stable with respect to a certain lower decay rate. In that sense, some nontrivial examples are presented to justify and illustrate the usefulness of the theory. More precisely, one can say anything about both the pth moment and almost sure exponential stability, although the solutions are pth moment and almost surely polynomially or logarithmically stable.     

Stabilization for multi-group coupled models with dispersal by feedback control based on discrete-time observations in diffusion part

In this paper, we study the almost surely stabilization for a class of unstable multi-group coupled models with dispersal (MGMs, in short) by feedback control design based on discrete-time observations in diffusion part. Specifically, we show that the corresponding stochastically controlled system is almost surely exponentially stable when the stochastic MGM is almost surely exponentially stable within a duration of time. Finally, an example is given to illustrate the obtained theoretical results.     

Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.     

Consensus of discrete-time multi-agent systems over packet dropouts channels

In this paper, we investigate the consensus problem for discrete linear multi-agent systems (MASs) with Markovian packet dropouts. Both identical and nonidentical packet dropouts are studied. For the discrete-time MASs under identical packet dropouts, we present the expectation of the total sojourn time of packet dropouts and successful message transmission, the switching number from packet dropouts to successful message transmission, and the awaken number of packet dropouts and successful message transmission. Based on these expectations, a linear consensus controller is designed through analyzing the transient properties of the Markov process such that the MASs can reach consensus almost surely for any initial distribution of packet dropouts. When it comes to the nonidentical packet dropouts where all the packets are independent and stochastic, a Markovian-lossy-channel based switching model (MLCBS model) is proposed. Based on the MLCBS model, we also propose an easy-to-implement linear consensus controller such that the MASs with nonidentical packet dropouts can achieve consensus almost surely. Finally, the theoretical results are illustrated by simulation examples.     

A new idea on almost sure permanence and uniform boundedness for a stochastic predator–prey model

The existence and uniqueness of stationary distribution and ergodic properties of a stochastic system are obtained. Especially, different from the existing methods, a new method is introduced to analyze almost sure permanence and uniform boundedness of the stochastic predator–prey model. This new idea is based on geometric structure of invariant set for a stochastic system. More specifically, we obtain our main conclusions by showing the invariant set for the stochastic population system lies in the interior of the first quadrant. It is interesting and surprising that the stochastic population model can guarantee a uniform boundedness almost surely. Some numerical simulations are carried out to support our results.     

经济系统中几个微分方程模型

建立了市场经济中供求关系的两类数学模型。根据商品数量的不同,对供给函数和需求函数的假设不同,建立了几个微分方程模型。研究了其中一个模型的Hopf分支问题,给出了均衡价格的局部稳定性条件和出现Hopf分支的条件     

Stability and Hopf bifurcation analysis of an eco-epidemiological model with delay

In this paper, an eco-epidemiological model with time delay is considered. The asymptotical stability of the three equilibria, the existence of stability switches about both the disease-free planar equilibrium and the positive equilibrium are investigated. It is found that Hopf bifurcation occurs when the delay τ passes through a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations at the positive equilibrium are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Host vector dynamics of a nonlinear pine wilt disease model in deterministic and stochastic environments

In this study, we develop a vector-host transmission model with general incidence rates for the dynamics of pine wilt disease in deterministic and stochastic environments. The existence and local asymptotic stability of equilibria are investigated in the deterministic case. We show the required conditions for the ergodic stationary distribution and extinction of the model in the stochastic case by constructing appropriate Lyapunov functions. Furthermore, by solving the corresponding Fokker-Planck equation, we obtain exact expressions of probability density function around the quasi-equilibrium of the stochastic model. Finally, we employ comprehensive numerical simulations to support our results and compare deterministic and stochastic situations.     

蓝细菌体系中外噪音维持的生物钟

采用数值模拟方法 ,研究了外部色噪音对蓝细菌生物钟体系的影响。结果表明,外噪音可以在体系的确定性稳态区诱导产生振荡和相干共振;在确定性振荡区,外噪音对振荡起到了破坏性作用,但是振荡的周期并没有随着外噪音的增加而明显变化,说明体系对外噪音具有鲁棒性;而在右侧分叉点附近的确定性振荡区,也发现了外噪音诱导相干共振现象。另外,研究了噪音相关时间的影响,发现其与噪音强度起相反的作用。     

Passivity-based resilient adaptive control for fuzzy stochastic delay systems with Markovian switching

The resilient adaptive controller design problem of a class of Itô-type Takagi–Sugeno (T–S) fuzzy stochastic systems with time-varying delay and Markovian switching is investigated. By utilizing improved matrix decoupling technique, passivity theory and stochastic Lyapunov–Krasovskii functional, LMIs-based sufficient conditions for the existence of resilient adaptive controller are provided such that the corresponding closed-loop system is almost surely asymptotically stable and robustly passive in the sense of expectation. The derived conditions can be easily solved with the help of LMI toolbox in Matlab. A simulation example is presented to illustrate the effectiveness of the proposed resilient adaptive control schemes.     

State feedback control for stochastic Markovian jump delay systems based on LaSalle-type theorem

This paper investigates the problem of stochastic stability and stabilization of stochastic Markovian jump delay systems (SMJDSs) based on LaSalle theorem. The time delays are assumed to be time-varying and numerous stochastic disturbances are considered. Attention is focused on the design of the mode-dependent state feedback controller for SMJDSs based on LaSalle theorem such that the closed-loop SMJDSs are almost surely asymptotically stable. The sufficient conditions for the solvability of the state feedback control problem are obtained in terms of linear matrix inequalities (LMIs). When the LMIs are feasible, the desired state feedback controller is also given. Two numerical examples including the vertical take-off and landing (VTOL) helicopter system are employed to demonstrate the effectiveness and usefulness of the method proposed in this paper     

Hierarchical Cucker–Smale flocking under random interactions with time-varying failure probabilities

This paper studies the hierarchical Cucker–Smale flocking model of sampled-data second-order discrete-time multi-agent systems under random interactions with time-varying failure probabilities. More precisely, each agent, at each sampling time point, can fail to see any of its superiors in the hierarchy. The random failures are not independent with varying failure rate probabilities. For this model with random interactions, we prove that the flocking would occur almost surely, i.e., agents’ velocities will converge almost surely to the velocity of the overall leader of the flock which moves with a varying velocity, the relative positions between agents and the overall leader converge almost surely. Finally, several numerical simulations are provided to illustrate the obtained results.     

Almost surely state estimation for multi-rate networked systems under random and malicious packet losses

This paper investigates the problem of a multi-rate networked system estimation with considering random and malicious packet losses. Three different rates are used: system sampling rate, measurement updating rate and estimation updating rate. Thus, the network energy can be saved. Since the plant and filtering are connected via network channel, the data packet losses unavoidably happen. In order to study the combination of the random and malicious packet losses, the probabilistic characterization for the link failures is applied, which leads to a stochastic estimation error system. The almost surely exponentially stability criteria is applied to guarantee this stochastic system stable. Finally, a cloud-aided vehicle suspension system is applied to verify the theoretical finding.     

Mathematical analysis of the global dynamics of an eco-epidemiological model with time delay

In this paper, an eco-epidemiological predator–prey model with time delay representing the gestation period of the predator is investigated. In the model, it is assumed that the predator population suffers a transmissible disease by contact. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcations at the disease-free equilibrium, the prey–infected predator equilibrium and the endemic-coexistence equilibrium are established. By means of Lyapunov functionals and LaSalle's invariance principle, sufficient conditions are obtained for the global asymptotic stability of the predator-extinction equilibrium, the disease-free equilibrium, the prey–infected predator equilibrium and the endemic-coexistence equilibrium of the model.     

Adaptive synchronization for delayed neural networks with stochastic perturbation

In this paper, an adaptive feedback controller is designed to achieve complete synchronization of unidirectionally coupled delayed neural networks with stochastic perturbation. LaSalle-type invariance principle for stochastic differential delay equations is employed to investigate the globally almost surely asymptotical stability of the error dynamical system. An example and numerical simulation are given to demonstrate the effectiveness of the theory results.