In this work, we set up a new discrete predator-prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator-prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results.

Functional connectivity, representing a statistical coupling relationship between different brain regions or electrodes, is an influential concept in clinical medicine and cognitive neuroscience. Electroencephalography-derived functional connectivity (EEG-FC) provides relevant characteristic information about individual differences in cognitive tasks and personality traits. However, it remains unclear whether these individual-dependent EEG-FCs remain relatively permanent across long-term sessions. This manuscript utilizes machine learning algorithms to explore the individual specificity and permanence of resting-state EEG connectivity patterns. We performed six recordings at different intervals during a six-month period to examine the variation and permanence of resting-state EEG-FC over a long period. The results indicated that the EEG-FC networks are quite subject-specific with a high-precision identification accuracy of greater than 90%. Meanwhile, the individual specificity remained stable and only varied slightly after six months. Furthermore, the specificity is mainly derived from the internal connectivity of the frontal lobe. Our work demonstrates the existence of specific and permanent EEG-FC patterns in the brain, providing potential information for biometric applications.

OBJECTIVE: As we all know, mathematical models provide very important information for the study of the human immunodeficiency virus type. Mathematical model of human immunodeficiency virus type-1 (HIV-1) infection with contact rate represented by Crowley-Martin function response is taken into account. The aims of this novel study is to checkthe local and global stability of the disease and also prevent the outbreak from the community.
METHODS: The mathematical model as well as optimal system of nonlinear differential equations are tackled numerically by Runge-Kutta fourth-order method. For global stability we use Lyapunov-LaSalle invariance principle and for the description of optimal control, Pontryagin\'s maximum principle is used.
RESULTS: Graphical results are depicted and examined with different parameters values versus the basic reproductive number R0 and also the plots with and without control. The density of infected cells continued to increase without treatment, but the concentration of these cells decreased after treatment. The intensity of the pathogenic virus before and after the optimal treatment. This indicates a sharp drop in the rate of pathogenic viruses after treatment. It prevents the production of viruses by preventing cell infection and minimizing side effects.
CONCLUSIONS: We analysed the model by defining the basic reproductive number, showing the boundedness, positivity and permanence of the solution, and proving the local and global stability of the infection-free state. We show that the threshold quantity R0 < 1, the elimination of HIV-1 infection from the T cell population, is eradicated; while for the threshold quantity R0 > 1, HIV-1 infection remains in the host. When the threshold quantity R0 > 1, then it shows that the steady-state of chronic disease is globally stable. Optimal control strategies are developed with the optimal control pair for the description of optimal control. To reduce the density of infected cells and viruses as well as maximize the density of healthy cells is determined by the objective functional.

An epidemic model with two parallel infectious stages and time delays and pulse vaccination is proposed. We introduce four thresholds and further obtain the conditions that the disease will be extinct or not. Corollaries show that under condition that θ > max { θ ∗ 1 , θ ∗ 2 } the disease will fade out, and if θ < min { θ 1 ∗ , θ 2 ∗ } , the disease will be endemic. Our results indicate that a larger pulse vaccination rate will lead to the eradication of a disease. Furthermore, two thresholds ℜ ∗ 1 and ℜ ∗ 2 show that the diversity of the contagiousness affects the basic properties of these models. In addition, numerical results indicate that the probability for an infected individual to enter different infective compartments greatly affects two infective compartments.

To understand the effect of impulsive vaccination and restricting infected individuals boarding transports on disease spread, we establish an SIR model with impulsive vaccination, impulsive dispersal and restricting infected individuals boarding transports. This SIR epidemic model for two regions, which are connected by transportation of non-infected individuals, portrays the evolvement of diseases. We prove that all solutions of the investigated system are uniformly ultimately bounded. We also prove that there exists globally asymptotically stable infection-free boundary periodic solution. The condition for permanence is discussed. It is concluded that the approach of impulsive vaccination and restricting infected individuals boarding transports provides reliable tactic basis for preventing disease spread.

In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number R 0 ≤ 1 , then disease-free equilibrium is globally attractive, and if R 0 > 1 , then the disease is permanent. When the model degenerates into SEIR model, it is proved that if R 0 > 1 , then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when R 0 > 1 , the endemic equilibrium of general SEIRS models is also globally attractive.

In this paper, the propagation of a nonlinear delay SIR epidemic using the double epidemic hypothesis is modeled. In the model, a system of impulsive functional differential equations is studied and the sufficient conditions for the global attractivity of the semi-trivial periodic solution are drawn. By use of new computational techniques for impulsive differential equations with delay, we prove that the system is permanent under appropriate conditions. The results show that time delay, pulse vaccination, and nonlinear incidence have significant effects on the dynamics behaviors of the model. The conditions for the control of the infection caused by viruses A and B are given.

To study the interaction of parasitoids and their insect hosts in laboratory environment, we propose a mathematical model incorporating impulsive resource inputs, stage-structure, maturation times and negative binomial distribution of parasitoid attacks. According to the adaptability of the insect host to the environment, we obtain conditions under which the system is uniformly permanent in two cases, which guarantee that the host and its parasitoid can coexist. By applying fixed point theory, we show existence of the positive periodic solution where the host and its parasitoid can coexist, and also obtain the conditions that ensure the existence of the parasitoid-extinction periodic solution. Our numerical analysis confirms and extends our theoretical results. The simulations show that when the total amount of resource is fixed, a smaller amount of recourse inputs with a shorter period of impulsive delivery results in smaller oscillation amplitude in the insect host population. However, the development of parasitoid population is not affected by the resource management strategy. It is also demonstrated that larger maturation times, either the host\'s or the parasitoid\'s, lead to the decline of the parasitoid population. But larger parasitoid\'s maturation time does accelerate the host\'s population growth. These are helpful for us to acquire a deeper knowledge of the host-parasitoid interaction in laboratory environment.

The aim of this paper is to investigate the dynamical behaviour of a class of three species Lotka-Volterra competitive-competitive-cooperative models with feedback controls and time delays. By developing a new analysis technique, we obtain some sufficient conditions that ensure these models have the dynamical property of permanence. We also give some sufficient conditions that guarantee the global attractivity of positive solutions for this system by constructing a new suitable Lyapunov function. Finally, we give some numerical simulations to illustrate our results in this paper.

In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.