Unique Positive Solution Research Articles

Dynamical behaviors of a stochastic SIVS epidemic model with the Ornstein-Uhlenbeck process and vaccination of newborns.

In this paper, we study a stochastic SIVS infectious disease model with the Ornstein-Uhlenbeck process and newborns with vaccination. First, we demonstrate the theoretical existence of a unique global positive solution in accordance with this model. Second, adequate conditions are inferred for the infectious disease to die out and persist. Then, by classic Lynapunov function method, the stochastic model is allowed to obtain the sufficient condition so that the stochastic model has a stationary distribution represents illness persistence in the absence of endemic equilibrium. Calculating the associated Fokker-Planck equations yields the precise expression of the probability density function for the linearized system surrounding the quasi-endemic equilibrium. In the end, the theoretical findings are shown by numerical simulations.

Global mathematical analysis of a patchy epidemic model

The dissemination of a disease within a homogeneous population can typically be modeled and managed in a uniform fashion. Conversely, in non-homogeneous populations, it is essential to account for variations among subpopulations to achieve more precise predictive modeling and efficacious intervention strategies. In this study, we introduce and examine the comprehensive behavior of a deterministic two-patch epidemic model alongside its stochastic counterpart to assess disease dynamics between two heterogeneous populations inhabiting distinct regions. First, utilizing a specific Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic model is globally asymptotically stable. For the stochastic model, we establish that it is well-posed, meaning it possesses a unique positive solution with probability one. Subsequently, we ascertain the conditions necessary to ensure the total extinction of the disease across both regions. Furthermore, we explicitly determine a threshold condition under which the disease persists in both areas. Additionally, we discuss a scenario wherein the disease persists in one region while simultaneously becoming extinct in the other. The article concludes with a series of numerical simulations that corroborate the theoretical findings.

Extinction Dynamics and Equilibrium Patterns in Stochastic Epidemic Model for Norovirus: Role of Temporal Immunity and Generalized Incidence Rates

Norovirus is a leading global cause of viral gastroenteritis, significantly affecting mortality, morbidity, and healthcare costs. This paper develops and analyzes a stochastic SEIQR epidemic model for norovirus dynamics, incorporating temporal immunity and a generalized incidence rate. The model is proven to have a unique positive global solution, with extinction conditions explored. Using Khasminskii’s method, the model’s ergodicity and equilibrium distribution are investigated, demonstrating a unique ergodic stationary distribution when R^s>1. Extinction occurs when R0E<1. Computer simulations confirm that noise level significantly influences epidemic spread.

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Dynamics analysis of a delayed stochastic SIRS epidemic model with a nonlinear incidence rate

. This article proposes a stochastic delayed SIRS epidemic model with nonlinear incidence to describe disease transmission in a stochastic environment. First, it proves that the model has a unique global positive solution and derives sufficient conditions for the extinction and persistence of the epidemic in the population under appropriate conditions. Then, by constructing suitable Lyapunov functions, the asymptotic behavior of the model around the disease-free equilibrium and endemic equilibrium points of the deterministic model is analyzed, and it is concluded that under certain conditions, the solution of the stochastic system randomly oscillates around these two equilibrium points. Finally, several numerical examples are provided to validate the theoretical results, illustrating the significant impact of stochastic perturbations and time delays on disease control.

Competitive exclusion and coexistence in a general two-species stochastic chemostat model with Markov switching

In this paper, a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment. The model incorporates both white noise and telegraph noise, the latter being described by Markov process. The existence of a unique global positive solution for the stochastic chemostat model is established. Subsequently, by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis, the critical value differentiating between persistence in mean and extinction for the microorganism species is explored. Moreover, the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis. Finally, numerical simulations are introduced to support the obtained results.

Structure of positive solutions for a reaction-diffusion model with additional food and protection zone

The additional food and protection zone provided to predator and prey respectively are introduced into a reaction-diffusion predation model with nonlinear functional response in this paper. We mainly concentrate on the combined effects of additional food and protection zone on the existence and uniqueness of solution for the model. It is shown that, compared with the models with classical Beddington-DeAngelis functional response, the integration of additional food and protection zone into Beddington-DeAngelis functional response causes the model to produce new critical values for the existence of positive solutions. Moreover, we derive that the model permits a unique positive solution when the strength of mutual interference among predators or the quantity of additional food is large, where, the effect of quantity of additional food on the positive solution is a new finding. The results shed light on that the combination of additional food and protection zone is an effective and environmentally-friendly strategy in preserving biodiversity.

Analysis of a stochastic SEIuIrR epidemic model incorporating the Ornstein-Uhlenbeck process

This article aims to analyze a stochastic epidemic model SEIuIrR (Susceptible-exposed-undetected infected-detected infected (reported -recovered) assuming that the transmission rate at which people undetected become detected is perturbed by the Ornstein–Uhlenbeck process. Our first objective is to prove that the stochastic model has a unique positive global solution by constructing a nonnegative Lyapunov function. Afterward, we provide a sufficient criterion to prove the existence of an ergodic stationary distribution of the mode by constructing a suitable series of Lyapunov functions. Subsequently, we establish sufficient conditions for the extinction of the disease. Finally, a series of numerical simulations are carried out to illustrate the theoretical results.

Analysis of a stochastic hybrid Gompertz tumor growth model driven by Lévy noise

This article investigates a stochastic hybrid Gompertz tumor growth model driven by Lévy noise. The global existence of a unique positive solution, extinction, non-persistence in mean, and persistence in mean are well explored. Then, stochastic permanence is obtained by using stochastically ultimately upper bounded and lower bounded. An interesting fact is that the white noise, Markovian noise and Lévy noise can all affect the persistence and extinction of the Gompertz tumor growth model.

Stability analysis and Hopf bifurcation of a fractional order HIV model with saturated incidence rate and time delay

In this paper, a fractional order HIV model with saturated incidence rate and time delay is proposed and analyzed. Firstly, the existence and uniqueness of positive solutions are proved. Secondly, the basic reproduction number and the sufficient conditions for the stability of two equilibriums are obtained. Thirdly, by using time delay as the bifurcation parameter, it is found that Hopf bifurcation may occur when the time delay passes through a sequence of critical values. After that, some numerical simulations are performed to verify the theoretical results. Finally, some discussions and conclusions are listed.

Dynamical Behaviors of a Stochastic Susceptible-Infected-Treated-Recovered-Susceptible Cholera Model with Ornstein-Uhlenbeck Process

In this study, a cholera infection model with a bilinear infection rate is developed by considering the perturbation of the infection rate by the mean-reverting process. First of all, we give the existence of a globally unique positive solution for a stochastic system at an arbitrary initial value. On this basis, the sufficient condition for the model to have an ergodic stationary distribution is given by constructing proper Lyapunov functions and tight sets. This indicates in a biological sense the long-term persistence of cholera infection. Furthermore, after transforming the stochastic model to a relevant linearized system, an accurate expression for the probability density function of the stochastic model around a quasi-endemic equilibrium is derived. Subsequently, the sufficient condition to make the disease extinct is also derived. Eventually, the theoretical findings are shown by numerical simulations. Numerical simulations show the impact of regression speed and fluctuation intensity on stochastic systems.

Exponential stability of periodic solutions for a hematopoiesis model with two time delays

In applied mathematics, many practical problems involving heat flow, species interaction, microbiology, neural networks, and more are associated with delay differential equations. To elucidate regulation and control mechanisms in physiological systems, Mackey and Glass introduced new mathematical models as a representation of hematopoiesis (blood-cell formation) in their influential and highly referenced 1977 paper. The presence of multiple delays, rather than a single delay, can lead to a new range of dynamics: an equation that was stable with one delay may become unstable when the delays are not equal, resulting in sustained oscillations. The dynamics of the non-autonomous Mackey-Glass model with two variable delays have not been thoroughly explored yet, which is presented as an open problem by Berezansky and Braverman. This paper focuses on the existence, uniqueness, and exponential stability of positive periodic solutions of a generalized class of the Hematopoiesis Model with two variable delays. Initially, using properties of cones and a fixed point theorem for a concave and increasing operator, we establish conditions ensuring the existence of a unique positive periodic solution for the model with almost periodic coefficients and delays. Subsequently, we employ a suitable Lyapunov functional method and differential inequality techniques to derive sufficient conditions guaranteeing the exponential stability of the unique positive periodic solution of the system. This approach covers a broad range of models that were previously studied. Additionally, an illustrative example with a numerical simulation is provided to validate the efficiency and reliability of the theoretical results. Our approach can be applied to the case of positive almost periodic solutions and positive pseudo almost periodic solutions of the model under consideration.

Data-driven adaptive optimal control for discrete-time linear time-invariant systems

In this paper, the problem of data-driven quadratic optimal control is investigated for discrete-time linear systems without the exact knowledge of system dynamics. A value iteration-based algorithm is proposed to obtain the optimal feedback gain. The main idea of the proposed approach is to obtain two sequences for the approximation of the unique positive definite solution of the corresponding algebraic Riccati matrix equation and the optimal feedback gain by using the collected data of the system states and control inputs. The proposed algorithm could be activated by an arbitrary bounded control input and a simple initial value of the Lyapunov matrix. An important feature of this algorithm is that an original stabilising feedback gain is not required. Finally, the effectiveness of the proposed approach is verified by two examples.

Deterministic vs. stochastic dynamics of a predator–prey system with disease in prey: Impact of fear and supplementary foods

In this study, we delve into the dynamics of a generalist predator–prey system with prevalence of a disease in the prey population and the presence of predator-induced fear. In this intricate ecological web, the fear response triggered by predators influences the birth rate of susceptible prey population and also escalates intra-specific competition among them. In an effort to encapsulate a more realistic scenario, we extend our deterministic model to its stochastic counterpart by introducing environmental white noise that impacts the mortality rates of both prey and predator species. Comprehensive mathematical analyses are performed on both the deterministic and stochastic systems to elucidate their qualitative behaviors. Specifically, we derive conditions under which the stochastic system exhibits a unique global positive solution and explore the potential extinction of species within the ecosystem. To unravel the intricate dynamics numerically, we perform sensitivity of parameters, and construct one- and two-parameter bifurcation diagrams. Our investigations reveal that when the costs of fear surpass specific thresholds and disease incidence is high, both the susceptible and infected prey populations face extinction. Notably, the introduction of supplementary foods can lead to system’s destabilization, potentially pushing it towards a state where only predators can persist. Moreover, we observe that lower intensities of white noise have minimal impact on the system’s dynamics, while higher noise intensities introduce substantial fluctuations and may precipitate the extinction of species within the ecosystem. Furthermore, we elucidate the abundances of prey and predator species within the ecosystem through histogram plots.

Dynamics analysis of an influenza epidemic model with virus mutation incorporating log-normal Ornstein–Uhlenbeck process

A stochastic influenza epidemic model where influenza virus can mutate into a mutant influenza virus is established to study the influence of environmental disturbance. And the transmission rate of the model is assumed to satisfy log-normal Ornstein–Uhlenbeck process. We verify that there exists a unique global positive solution to the stochastic model. By constructing proper Lyapunov functions, sufficient conditions under which the stationary distribution exists are obtained. In addition, we discuss the extinction of the disease. Furthermore, we get the accurate expression of probability density function near the endemic equilibrium of the stochastic model. Finally, several numerical simulations are carried out to verify theoretical results and examine the influence of environmental noise.

High multiplicity of positive solutions in a superlinear problem of Moore–Nehari type

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number κ≥1 of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter λ and on κ.Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in λ and the profiles of positive solutions.Finally, we give additional numerical results suggesting that high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.

Based on epidemiological parameter data, probe into a stochastically perturbed dominant variant of the COVID-19 pandemic model

During the COVID-19 pandemic, the SARS-CoV-2 gene mutation has been rapidly emerging and spreading all over the world. Experts worldwide regularly monitor genetic mutations and variants through genome-sequence-based surveillance, laboratory testing, outbreak investigation, and epidemiological probing. Clinical pathologists and medical laboratory scientists prefer developing or endorsing COVID-19 vaccines with a broader immune response involving various antibodies and cells to protect against mutations or new variants. Randomness plays an enormous role in pathology and epidemiology. Hence, based on epidemiological parameter data, we construct and probe a stochastically perturbed dominant variant of the coronavirus epidemic model with three nonlinear saturated incidence rates. We reveal the existence of a unique global positive solution to the constructed stochastic COVID-19 model. The Lyapunov function method is used to determine the presence of a stationary distribution of positive solutions. We derive sufficient conditions for the coronavirus to be eradicated. Eventually, numerical simulations validate the effectiveness of our theoretical outcomes.

Data‐driven adaptive optimal control for discrete‐time periodic systems

AbstractIn this paper, a problem of data‐driven optimal control is studied for discrete‐time periodic systems with unknown system matrices and input matrices. For this problem, a value iteration‐based adaptive dynamic programming algorithm is proposed to obtain the suboptimal controller. The core of the algorithm proposed in this paper is to obtain an approximation of the unique positive definite solution of the algebraic Riccati equation and the optimal feedback gain matrix by using the collected real‐time data of the system states and control inputs. Without an initial stabilizing feedback gain, the proposed algorithm could be activated by an arbitrary bounded control input. Finally, the effectiveness of the proposed approach is demonstrated by two examples.

Finite Difference Method for Infection Model of HPV with Cervical Cancer under Caputo Operator

In this paper, a fractional model in the Caputo sense is used to characterize the dynamics of HPV with cervical cancer. Generalized mean value theorem has been used to examine whether the infection model has a unique positive solution. The model has two equilibrium points: the disease-free point and the endemic point. The examination of the system’s local and global stability is provided in terms of the basic reproductive number Rp°. The global stability analysis has been carried out using an appropriate Lyapunov function and the LaSalle invariant principle. The results demonstrate that in the infection model, if Rp°<1, then the solution converges to the disease-free equilibrium, which is both locally and globally asymptotically stable. Whilst Rp°>1, the endemic equilibrium is considered to exist. Simulations are implemented via a finite difference method with Grünwald-Letnikov discretization approach for Caputo derivative operator to define how changes in parameters impact the dynamic behavior of the system using Matlab.

Investigating the Dynamics of Bayoud Disease in Date Palm Trees and Optimal Control Analysis

The fungus Fusarium oxysporum (f.sp. albedinis) causes Bayoud disease. It is one of the epiphytotic diseases that affects a wide range of palm species and has no known cure at present. However, preventive measures can be taken to reduce the effects of the disease. Bayoud disease has caused enormous economic losses due to decreased crop yield and quality. Therefore, it is essential to develop a mathematical model for the dynamics of the disease to propose some affordable methods for disease management. In this study, we propose a novel mathematical model that describes the transmission dynamics of the disease in date palm trees. The model incorporates various factors such as the contact rate of the fungi with date palm trees, the utilization of fungicides, and the introduction of a quarantine compartment to prevent disease dissemination. We first prove a few key properties of the proposed model to ensure that the model is well-posed and suitable for numerical investigations. We establish that the model has a unique positive solution that is bounded and stable over time. We use sensitivity analysis to identify the parameters that have the greatest effect on the reproduction number R0 and illustrate this effect graphically. We then formulate an optimal control problem to identify the most suitable and cost-effective disease control approaches. As a first approach, we solely focus on the application of fungicide to susceptible trees and determine the best spray rates for a greater decrease in exposed and infected trees. Secondly, we emphasize quarantining exposed and infected trees at optimal quarantine rates. Finally, we explore the combined effect of fungicide spraying and isolating infected trees on disease control. The findings of the last approach turn out to be the most rewarding and cost-effective for minimizing infections in date palm trees.