Existence Of Global Positive Solution Research Articles

Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model

In this paper, a stochastic HIV/AIDS epidemic model is presented to study the synthetic effect of ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments among MSM ( men who have sex with men). Firstly, we give the global stability of disease-free equilibrium and the endemic equilibrium in terms of basic reproduction number R0 for deterministic model. And then the existence of global positive solutions and the existence of unique ergodic stationary distribution under R0S>1 for stochastic model are given. Further, the long-time stochastic dynamic of the model is investigated, including the criteria on the extinction and persistence in mean for the stochastic model. Finally, we give some numerical simulations to illustrate our theoretical results, and the sensitive analysis shows that ART (antiretroviral therapy) and PrEP (pre-exposure prophylaxis) treatments can effectively control the spread of AIDS among MSM population.

The Stationary Distribution and Extinction of a Stochastic Five‐Dimensional COVID‐19 Model

This paper investigates a stochastic SEIAIR epidemic model with nonlinear incidence and contacting distance to describe the transmission dynamics of COVID‐19. Firstly, we show the global existence of positive solution of the system. Then, using the Lyapunov function method and theory of stochastic analysis, we set out the sufficient conditions of the existence and uniqueness of an ergodic stationary distribution to the stochastic model. Furthermore, we obtain the sufficient condition for extinction of the disease. Finally, we go through some numerical simulations to demonstrate the theoretical results. Our study extends and improves the results of previous studies.

Dynamic analysis and optimal control of a stochastic investor sentiment contagion model considering sentiments isolation with random parametric perturbations

Investor sentiment contagion has a profound influence on economic and social development. This paper explores the diverse influences of various investor sentiments in modern society on the economy and society. It also investigates the interference of various uncertain factors on investor sentiments in the modern economy and society. On this basis, the dual-system stochastic SPA2G2R model was constructed, incorporating positive and negative sentiments, as well as a supervision and isolation mechanism. The global existence of positive solutions was established, and sufficient conditions for the disappearance and steady distribution of investor sentiment were calculated. An optimal control strategy for the stochastic model was put forward, with numerical simulation supporting the theoretical analysis results. A comparison with parameter changes in the deterministic model was also conducted. The research reveals a competitive relationship between different investor sentiments. Enhancing societal guidance mechanisms promotes positive investor sentiment contagion. Timely control by the supervisory department effectively curbs the spread of investor sentiment. Additionally, white noise promotes investor sentiment contagion, suggesting effective regulation through control of noise intensity and disturbance parameters.

Generalized result on the global existence of positive solutions for a parabolic reaction diffusion model with a full diffusion matrix

"In this paper, we study the global existence in time of solutions for a parabolic reaction di usion model with a full matrix of di usion coe cients on a bounded domain. The technique used is based on compact semigroup methods and some estimates. Our objective is to show, under appropriate hypotheses, that the proposed model has a global solution with a large choice of nonlinearities."

The effect of Lévy noise and white noise on a Leslie-Gower predator-prey system with prey refuge

A stochastic Leslie-Gower predator-prey system with prey refuge is discussed, L$ \acute{e} $vy noise and white noise are adopted to simulate the random change of environment. The existence of global positive solution and the stochastic boundedness is proved, the threshold conditions of extinction and stable in the mean are given, the long time behavior of solution for the predator-prey system is also considered. In the end, some examples of numerical simulation are given to verify these conclusions. Compared to the deterministic system, the prey population and the predator population will be extinct, when the intensity of noise is large enough, however, it does not happen in the deterministic system, where the positive equilibrium is always globally stable. These results indicate that the stochastic change of environment has great influence on the population dynamic system.

Practical Exponential Stability of Impulsive Stochastic Food Chain System with Time-Varying Delays

This paper studies the practical exponential stability of an impulsive stochastic food chain system with time-varying delays (ISOFCSs). By constructing an auxiliary system equivalent to the original system and comparison theorem, the existence of global positive solutions to the suggested system is discussed. Moreover, we investigate the sufficient conditions for the exponential stability and practical exponential stability of the system, which is given by Razumikhin technique and the Lyapunov method. In addition, when Razumikhin’s condition holds, the exponential stability and practical exponential stability of species are independent of time delay. Finally, numerical simulation finds the validity of the method.

Deterministic and stochastic evolution of rumor propagation model with media coverage and class‐age‐dependent education

Considering the media coverage and age‐dependent education, a deterministic and a stochastic class‐age‐structured rumor propagation models are studied, respectively. First, the deterministic rumor propagation model is characterized by a coupled system of ordinary and partial differential equations. The positivity and boundness of solutions are proved, and the basic reproduction number is derived. Second, the stochastic rumor propagation model is formulated by stochastic differential equations. The existence of global positive solutions in model is discussed with the Itô's formula and stochastic Lyapunov function. Additionally, by utilizing comparison principle of stochastic differential equations and the strong law of large numbers, several sufficient conditions for extinction and persistence of the rumor are derived. Finally, numerical simulations are carried out for illustrating the results.

Dynamic behaviors of a stochastic new logistic model for microbial growth in foods

In this paper, a stochastic new logistic model is proposed, and dynamic behaviors of the model have been analyzed. The existence of global positive solution of the model is proved. In the persistence analysis of the model, the weak persistence and stochastic persistence of the model are proved. Some numerical simulations are presented to illustrate the dynamic properties of the model.

Dynamics of a Stochastic HIV Infection Model with Logistic Growth and CTLs Immune Response under Regime Switching

Considering the influences of uncertain factors on the reproduction of virus in vivo, a stochastic HIV model with CTLs’ immune response and logistic growth was developed to research the dynamics of HIV, where uncertain factors are white noise and telegraph noise. which are described by Brownian motion and Markovian switching, respectively. We show, firstly, the existence of global positive solutions of this model. Further, by constructing suitable stochastic Lyapunov functions with regime switching, some sufficient conditions for the existence and uniqueness of the stationary distribution and the conditions for extinction are obtained. Finally, the main results are explained by some numerical examples. Theoretical analysis and numerical simulation show that low-intensity white noise can maintain the persistence of the virus, and high intensity white noise can make the virus extinct after a period of time with multi-states.

Analysis of an eco‐epidemiological system with Lévy noise

An eco‐epidemiological system with Lévy noise and white noise is proposed. In this prey–predator system, assume that the predator population is infected by disease and divided into two classes: susceptible predator and infected predator. The existence of global positive solution is derived. Threshold conditions of extinction and persistence in the mean for prey and predator population are given by Itô's formula and comparison theorem for stochastic model. The long‐time behavior of this random system is also discussed by constructing suitable Lyapunov function, which shows the relationship between the solution of stochastic system and the equilibria of corresponding deterministic system. Finally, these results are verified by numerical simulation. Our results illustrate whether Lévy noise and white noise are both unfavorable to the population size of species.

Stationary distribution and long-time behavior of COVID-19 model with stochastic effect

The coronavirus disease (COVID-19) is a dangerous pandemic and it spreads to many people in most of the world. In this paper, we propose a COVID-19 model with the assumption that it is affected by randomness. For positivity, we prove the global existence of positive solution and the system exhibits extinction under certain parametric restrictions. Moreover, we establish the stability region for the stochastic model under the behavior of stationary distribution. The stationary distribution gives the guarantee of the appearance of infection in the population. Besides that, we find the reproduction ratio [Formula: see text] for prevail and disappear of infection within the human population. From the graphical representation, we have validated the threshold conditions that define in our theoretical findings.

Global existence and finite time blow-up for a stochastic non-local reaction-diffusion equation

In this paper, we investigate the existence and blow-up of positive solutions for a stochastic non-local reaction-diffusion equation. Firstly, we give the conditions for the existence of global positive solutions and estimate the probability of existence of non-trivial positive global solution. Secondly, by choosing some special initial data, we prove that stochastic non-local reaction-diffusion equation blows up in finite time in the point-wise sense with probability 1. The random upper bounds for blow-up times are obtained. Furthermore, by increasing the amplitude of the initial data, we can get a blow-up in any short time with a positive probability. Finally, we prove that the blow-up set is the whole region.

Stability on positive pseudo almost periodic solutions of HPDCNNs incorporating [formula omitted] operator

This article is involved with a class of high-order proportional delayed cellular neural networks incorporating D operator. To start with, we validate the boundedness and the global existence of positive solutions. Moreover, by exploiting differential inequality techniques and with the aid of Lyapunov function method, some testable conditions are obtained to assure the positiveness and global exponential stability on pseudo almost periodic solutions of the proposed models. And lastly, a numerical example is afforded to verify the effectiveness and feasibility of the theoretical findings.

Dynamical behavior of a stochastic SIQS epidemic model on scale-free networks

In order to study the impact of random environments during the spread of the disease, we propose a novel stochastic SIQS model on scale-free networks, which introduces stochastic perturbations to the infected rates. We first obtain the existence of global positive solutions. Moreover, by constructing appropriate stochastic Lyapunov functions, we prove sufficient conditions for extinction and persistence of the disease. Finally, we verify the analysis results through numerical simulations. In addition, the results of previous studies are also improved in our research.

Attraction–repulsion taxis mechanisms in a predator–prey model

We consider a predator–prey model where the predator population favors the prey through biased diffusion toward the prey density, while the prey population employs a chemical repulsive mechanism. This leads to a quasilinear parabolic system. We first establish the global existence of positive solutions. Thereafter we show the existence of nontrivial steady state solutions via bifurcation theory, then we discuss the stability of these branch solutions. Through numerical simulation we analyze the nature of patterns formed and interpret results in terms of the survival and distribution of the two populations.

Dynamic analysis of a stochastic four species food-chain model with harvesting and distributed delay

A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest in each species and the effect of time delay for interaction between species are considered, which makes the model more applicable in real situations. Specifically, we address the stochastic global dynamics behavior, including the existence of global positive solutions, stochastic ultimate boundedness, extinction with probability one, persistence in mean and global stability. The asymptotic stability in the probability distribution is obtained, and the criterion for the existence and non-existence of the optimal harvesting strategy is also derived. Furthermore, this paper can provide reference for the research of general n-species stochastic food-chain models.

Dynamic behavior of a stochastic SIRS model with two viruses

Abstract To study the effect of environmental noise on the spread of the disease, a stochastic Susceptible, Infective, Removed and Susceptible (SIRS) model with two viruses is introduced in this paper. Sufficient conditions for global existence of positive solution and stochastically asymptotic stability of disease-free equilibrium in the model are given. Then, it is shown that the positive solution is stochastically ultimately bounded and the moment average in time of the positive solution is bounded. Our results mean that the environmental noise suppresses the growth rate of the individuals and drives the disease to extinction under certain conditions. Finally, numerical simulations are given to illustrate our main results.

Evolutionary Dynamics of Stochastic SEIR Models with Migration and Human Awareness in Complex Networks

In this paper, a stochastic SEIR (Susceptible-Exposed-Infected-Removed) epidemic dynamic model with migration and human awareness in complex networks is constructed. The awareness is described by an exponential function. The existence of global positive solutions for the stochastic system in complex networks is obtained. The sufficient conditions are presented for the extinction and persistence of the disease. Under the conditions of disease persistence, the distance between the stochastic solution and the local disease equilibrium of the corresponding deterministic system is estimated in the time sense. Some numerical experiments are also presented to illustrate the theoretical results. Although the awareness introduced in the model cannot affect the extinction of the disease, the scale of the disease will eventually decrease as human awareness increases.

The stationary distribution in a class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion

A class of stochastic SIRS epidemic models with non-monotonic incidence and degenerate diffusion is investigated. By using the Lyapunov function method, the existence of global positive solutions and the ultimate boundedness with probability one are obtained. By using the Markov semigroups theory, Fokker–Planck equation and Khasminskiǐ functions, the existence of unique stationary distribution for the model is established. That is, when the stochastic basic reproduction number R0S>1 and some extra conditions are satisfied then the distribution density of any positive solutions of the model converges to a unique invariant density as t→+∞. Finally, the main conclusions and open problems are illustrated and verified by the numerical simulations.

Existence of Global Solutions for Stochastic Population Models with Time Delays

In this work, we study a class of stochastic multi-species predator-prey model with time delays. By constructing a appropriate Lyapunov functions, the existence of global positive solution for the models is proved, and some known results are generalized.