Conditions For Extinction Research Articles

Dynamic analysis of an HIV model with CTL immune response and logarithmic Ornstein–Uhlenbeck process

In this paper, we present a stochastic HIV model, including logistic growth, CTL immune response and logarithmic mean–reverting Ornstein–Uhlenbeck process. Firstly, we certify that there exists a unique global positive solution by rigorous mathematical analysis. Then, by constructing a suitable Lyapunov function and using the strong law of large numbers and Fatou’s lemma, we obtain the existence of the stationary distribution when stochastic reproduction number is greater than one. Moreover, we derive an approximate expression for the probability density function of the stochastic model around the infected equilibrium of its corresponding deterministic model. In addition, a critical condition for virus extinction is established. Finally, through numerical simulations, we investigate the effects of noise intensity and the logistic growth on model behavior, and we also explore the influence of key parameters on virus persistence and extinction.

A spatiotemporal model for the effects of toxicants on the competitive dynamics of aquatic species

In this paper, we develop a reaction–diffusion model with negative toxicant–taxis that incorporates spatiotemporally inhomogeneous toxicant input to investigate the impact of toxicants on the competitive dynamics of two species in a polluted aquatic environment. Here the negative toxicant–taxis models the evasive movement of avoiding toxicants by species. We establish the global well-posedness of the model, analyze the existence and stability of spatially homogeneous steady states, and derive sufficient conditions for species extinction and coexistence. Through linear stability analysis, we identify sufficient conditions on model parameters that destabilize spatially homogeneous steady states under spatiotemporally uniform toxicant input. Numerical experiments reveal the influence of key toxicant-related factors (input rate, taxis intensity, and diffusivity) on competition outcomes and species distributions. Notably, strong negative toxicant–taxis can induce spatial aggregation and segregation patterns between the species and the toxicant under uniform toxicant input. Our findings suggest that toxicant–taxis may promote population persistence and coexistence, particularly when the toxicant input is not uniform in space and time and the toxicant does not diffuse fast (i.e. weak diffusivity). However, strong toxicant diffusion can diminish the impact of taxis, adversely affecting population persistence and species coexistence.

On Models of Population Evolution of Three Interacting Species

In this paper, we first analyzed several basic population dynamics models interpreting the relationships between three species. These are the May-Leonard model with three competitors, some prey-predator models of three-species and a prey-predator model with a super-predator. Subsequently, in our work, we proposed a new three-species model consisting of a prey, a predator and a superpredator, including some important assumptions such as competition, self-defense and infected prey. We examined the various equilibrium points of proposed model, and determined the conditions for extinction and survival of species in the long term. Finally, we performed numerical illustrations using Maltlab software to corroborate the theoretical results.

A comparison of stochastic and deterministic dynamics of tuberculosis model

Tuberculosis is a deadly infectious disease leading to a major health concern in Southern India while there are major obstacles to controlling its spread. Here, we present a mathematical model with five compartments. We categorize the infected compartment into two subcategories: latent TB-infected individuals and active TB-infected individuals. By introducing white noise in a deterministic system, we formulate a stochastic system. We investigate the model in deterministic and stochastic framework, followed by data calibration using TB infection data from Kanyakumari District in South India from 2019 to 2023. In the deterministic model, we derive the disease-free and endemic equilibrium points, compute the basic reproduction number, and examine their stability. Moreover, we perform sensitivity analysis to evaluate how variations in model parameters affect TB prevalence. In addition, we study the uniqueness of solutions of stochastic model. After that we derive the conditions for disease extinction and stochastic permanence, and execute extensive simulations to capture the variability and randomness in TB transmission dynamics. This study signifies that stochastic dynamics is richer than deterministic one in the way of mitigation TB transmission.

Stationary distribution and extinction of an HCV transmission model with protection awareness and environmental fluctuations

We propose a stochastic HCV model with protection awareness and exposed-acute-chronic phases in this study. First of all, we show that the stochastic HCV model admits a unique global positive solution for any given positive initial values. Then, we verify that HCV model has a unique stationary distribution under the sufficient criterion R0s>1, which indicates that HCV transmission undergoes the persistence in the long term. Furthermore, we derive the sufficient conditions for HCV extinction under the condition R0e<1. As a consequence, we derive the relationships among the stochastic persistence index R0s, the stochastic extinction index R0e and the threshold (the basic reproduction number R0) of the model without fluctuations. The condition R0>R0s>1 reveals that the existence of the white noises triggers the less stochastic persistence index. While, the condition R0<R0e<1 reveals that, when the intensities of the white noises are enhanced, the value R0e triggers the stochastic extinction.

Dynamics of a multigroup stochastic SIQR epidemic model

Abstract In this paper, we consider a multigroup stochastic SIQR epidemic model with varying total population size. After proving the existence and uniqueness of the global solution to the system, we developed sufficient conditions for the existence of an stationary ergodic distribution of the positive solutions. Then we gave sufficient conditions for extinction of the diseases which is based on the basic reproduction number in its corresponding deterministic system.

Extinction and stationary distribution of a novel SIRS epidemic model with general incidence rate and Ornstein–Uhlenbeck process

We propose, in this paper, a novel stochastic SIRS epidemic model to characterize the effect of uncertainty on the distribution of infectious disease, where the general incidence rate and Ornstein–Uhlenbeck process are also introduced to describe the complexity of disease transmission. First, the existence and uniqueness of the global nonnegative solution of our model is obtained, which is the basis for the discussion of the dynamical behavior of the model. And then, we derive a sufficient condition for exponential extinction of infectious diseases. Furthermore, through constructing a Lyapunov function and using Fatou’s lemma, we obtain a sufficient criterion for the existence and ergodicity of a stationary distribution, which implies the persistence of the disease. In addition, the specific form of the density function of the model near the quasiendemic equilibrium is proposed by solving the corresponding Fokker–Planck equation and using some relevant algebraic equation theory. Finally, we explain the above theoretical results through some numerical simulations.

A new analytical study of prey–predator dynamical systems involving the effects of Hide-and-Escape and predation skill augmentation

The aim of this work is to provide a comprehensive analysis of the dynamics within an ecosystem characterized by interactions between prey and predators, through a combination of theoretical analysis and numerical simulation. The study investigates the impact of two pivotal effects: the “Hide-and-Escape” behavior exhibited by prey and the “Predation Skill Augmentation” strategy adopted by predators. Theoretical analysis identifies three equilibrium points, all of which show positive characteristics. Furthermore, local stability conditions are determined for each of these equilibrium points. The derivation of global stability conditions is also presented, and their validity and importance as local stability conditions for the coexistence point are demonstrated. In addition, Kolmogorov conditions for coexistence and extinction are verified, as well as local bifurcation analysis. For the validity of the results, a numerical simulation of the system using MATLAB is conducted.

Persistence and extinction of a reaction–diffusion vegetation-water system with time–space white noise

This study considers an ecosystem subjected to stochastic fluctuations, and presents a stochastic reaction–diffusion vegetation-water model with time–space noise. Using a truncation and stop time method, this paper gives several properties important to the mild solution such as existence uniqueness, positivity, and continuity with respect to initial values. Sufficient conditions for persistence and extinction of vegetation are derived. Theoretical results are demonstrated using numerical simulations.

Effects of sexual and vertical transmission on Zika virus dynamics under environmental fluctuations

Zika is a mosquito-transmitted viral disease which may spread directly by the vector or sexual transmission. Zika virus may persist in semen and urine for a long time after disappearance from the blood, those persons are known as the convalescent humans. It can also be transmitted vertically among mosquitoes. In this paper, we have considered an eight-compartment Zika model to study the effect of all the said aspects on the virus dynamics in deterministic as well as stochastic environment. In analytic part, we have computed basic reproduction number and discussed stability of different equilibria. We have shown the proposed model undergoes through transcritical bifurcation when the reproduction number is unity and validate the model with real infection data of Dominican Republic in 2016. To study the model in stochastic environments, the additive noise is taken into consideration which is formulated considering the standard Brownian white noises proportional to each class. We have obtained the condition for disease extinction and persistence in mean. All theoretical findings are justified by numerical simulations. Lastly, the paper is ended with some conclusions.

Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations.

A Stochastic Semi-Parametric SEIR Model with Infectivity in an Incubation Period

This paper introduces stochastic disturbances into a semi-parametric SEIR model with infectivity in an incubation period. The model combines the randomness of disease transmission and the nonlinearity of transmission rate, providing a flexible framework for more accurate description of the process of infectious disease transmission. On the basis of the discussion of the deterministic model, the stochastic semi-parametric SEIR model is studied. Firstly, we use Lyapunov analysis to prove the existence and uniqueness of global positive solutions for the model. Secondly, the conditions for disease extinction are established, and appropriate stochastic Lyapunov functions are constructed to discuss the asymptotic behavior of the model’s solution at the disease-free equilibrium point of the deterministic model. Finally, the specific transmission functions are enumerated, and the accuracy of the results are demonstrated through numerical simulations.

The effects of additional food and environmental stochasticity on the asymptotic properties of a nutrient–phytoplankton model

In this paper, we investigate the deterministic and stochastic features of an additional food provided nutrient–phytoplankton model, and discuss the effects of additional food and environmental stochasticity (noise) on its asymptotic properties. A detailed global dynamics analysis of the deterministic model is firstly provided, and then the critical conditions for persistence and extinction of phytoplankton is found. For the stochastic model, we establish some sufficient conditions for the exponential extinction of phytoplankton and the existence of stationary distribution. By solving the Fokker–Planck equation, the explicit expression of local probability density of the stochastic model is further derived. With the help of probability density, we obtain the mean, variance, skewness and kurtosis of the biomass of phytoplankton, and numerically find the additional food and stochasticity have complex effects on these statistical properties. Both theoretical and numerical findings indicate that the additional food source is favorable for the growth of phytoplankton, whereas the stochasticity is adverse to its growth. Moreover, the additional food source and stochasticity play important roles in the formation of phytoplankton bloom.

Persistence and extinction for stochastic HBV epidemic model with treatment cure rate

With the current struggles of the world nowadays with several epidemics, modeling the dynamics of diseaseoutbreaks has become much more important than any time before. In this context, the present paper aimsat studying a stochastic hepatitis B virus epidemic model with treatment cure rate. Our model consists ofthree epidemic compartments describing the interaction between the susceptible, the infected and the recovered individuals; an SIR model where the infected individuals transmit the infection to the susceptible ones with a transmission rate perturbed by white noise. Our paper begins by establishing that our hepatitis B stochastic model has unique global solution. It moves then to giving sufficient conditions for the stochastic extinction and persistence of the hepatitis B disease. Finally, our paper provides some numerical results to support the analytical study, showing numerically that the treatment cure rate facilitates the extinction of the hepatitis B disease among the population.

Stationary distribution of a double epidemic stochastic model driven by saturated incidence rates

The double epidemic study is one of the critical studies in recent times as humankind experiences various simultaneous spreads of diseases. Reproduction numbers are important, which helps to derive sufficient conditions for extinction, persistence and co-persistence of diseases. This work aims to establish the stationary distribution of a stochastic double epidemic model comprised of SIR and SIRS transmission hypothesis through saturated incidence rate, with the aid of Markov semigroup theory. By constructing a suitable Lyapunov functional, the required result is sufficient by conditions on reproduction numbers.

On the stochastic threshold of the COVID-19 epidemic model incorporating jump perturbations

This work delves into the intricate realm of epidemic modeling under the influence of unpredictable surroundings. By harnessing the power of white noise and Lévy noise, we construct a robust framework to capture the behavioral characteristics of the COVID-19 epidemic amidst erratic changes in the external environment. To enhance our comprehension of the intricate dynamics of the coronavirus, we conducted an investigation using a stochastic SIQS epidemic model that incorporates a dedicated compartment to represent populations under quarantine. Thanks to stochastic modeling techniques, we account for the inherent randomness in the transmission process and provide insights into the potential variations and uncertainties associated with the progression of the epidemic. Specifically, we show that the asymptotic behavior of our model is perfectly governed by two thresholds, Rσ,J and Rσ,J′. That is to say, if Rσ,J<1, the disease will be removed from the population, while it will persist if Rσ,J′>1. Our highlight lies in obtaining the necessary and sufficient conditions for extinction in the absence of jump noise, namely Rσ,0=Rσ,0′. This means that our sufficient conditions for extinction for the jump case are also almost necessary. Finally, we present a set of computational simulations to validate our theoretical findings, supporting the results developed throughout this article. Overall, this research contributes to our understanding of the COVID-19 pandemic and its impact on the global population.

Conditions for extinction and ergodicity of a stochastic Mycobacterium tuberculosis model with Markov switching

&lt;p&gt;This paper is concerned with a stochastic Mycobacterium tuberculosis model, which is perturbed by both white noise and colored noise. First, we prove that the stochastic model has a unique global positive solution. Second, we derive an important condition $ R_0^* $ depending on environmental noise for this stochastic model. We construct an appropriate Lyapunov function, and show that the model possesses a unique ergodic stationary distribution when $ R_0^* &amp;lt; 0 $, in other words, it indicates the long-term persistence of the disease. Finally, we investigate the related conditions of extinction.&lt;/p&gt;

Stationary distribution of stochastic COVID-19 epidemic model with control strategies

&lt;p&gt;In this research article, we investigated a coronavirus (COVID-19) epidemic model with random perturbations, which was mainly constituted of five major classes: the susceptible population, the exposed class, the infected population, the quarantine class, and the population that has recovered. We studied the problem under consideration in order to derive at least one, and only one, nonlocal solution within the positive feasible region. The Lyapunov function was used to develop the necessary result of existence for ergodic stationary distribution and the conditions for the disease's extinction. According to our findings, the influence of Brownian motion and noise effects on epidemic transmission were powerful. The infection may diminish or eradicate if the noise is excessive. To illustrate our proposed scheme, we numerically simulated all classes' findings.&lt;/p&gt;

Stochastic analysis of survival and sensitivity in a competition model influenced by toxins under a fluctuating environment

&lt;abstract&gt;&lt;p&gt;This paper proposed a stochastic toxin-dependent competition model to investigate the impact of environmental noise on species interaction dynamics. First, a survival analysis was conducted to establish the sufficient conditions for population extinction and persistence. Second, we proved the existence of a unique ergodic stationary distribution. Finally, the spatial arrangement of random states near the deterministic attractor was investigated using the stochastic sensitivity functions technique. This analytical approach facilitates constructing confidence ellipses and estimating critical noise intensity corresponding to the onset of transition. Both theoretical and numerical findings demonstrated that significant levels of noise experienced by one species lead to its extinction while promoting persistence in its competitor; conversely, negligible levels of noise did not alter the original competition outcomes in the deterministic model. However, when both species encounter moderate levels of noise, various modifications can occur in competition outcomes. These findings have significant implications for preserving ecosystem diversity.&lt;/p&gt;&lt;/abstract&gt;

Analysis of a stochastic fear effect predator-prey system with Crowley-Martin functional response and the Ornstein-Uhlenbeck process

&lt;p&gt;This paper studied a stochastic fear effect predator-prey model with Crowley-Martin functional response and the Ornstein-Uhlenbeck process. First, the biological implication of introducing the Ornstein-Uhlenbeck process was illustrated. Subsequently, the existence and uniqueness of the global solution were then established. Moreover, the ultimate boundedness of the model was analyzed. Then, by constructing the Lyapunov function and applying $ It\hat{o} $'s formula, the existence of the stationary distribution of the model was demonstrated. In addition, sufficient conditions for species extinction were provided. Finally, numerical simulations were performed to demonstrate the analytical results.&lt;/p&gt;