Unique Positive Solution Research Articles

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.

Dynamic analysis of a fuzzy Bobwhite quail population model under g-division law

This paper is concerned with a kind of Bobwhite quail population model xn+1=A+Bxn+xnxn-1xn-2,n=0,1,⋯,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} x_{n+1}=A+Bx_n+\\frac{x_n}{x_{n-1}x_{n-2}},\\ \\ n=0,1,\\cdots , \\end{aligned}$$\\end{document}where the parameters and initial values are positive parabolic fuzzy numbers. According to g-division of fuzzy sets and based on the symmetrical parabolic fuzzy numbers, the conditional stability of this model is proved. Besides the existence, boundedness and persistence of its unique positive fuzzy solution. When some fuzzy stability conditions are satisfied, the model evolution exhibits oscillations with return to a fixed fuzzy equilibrium no matter what the initial value is. This phenomena provided a vivid counterexample to Allee effect in density-dependent populations of organisms. As a supplement, two numerical examples with data-table are interspersed to illustrate the effectiveness. Our findings have been verified precise with collected northern bobwhite data in Texas, and will help to form some efficient density estimates for wildlife populations of universal applications.

On an Anisotropic Logistic Equation

We consider a nonlinear Dirichlet problem driven by the (p(z),q)-Laplacian and with a logistic reaction of the equidiffusive type. Under a nonlinearity condition on a quotient map, we show existence and uniqueness of positive solutions and the result is global in parameter λ. If the monotonicity condition on the quotient map is not true, we can no longer guarantee uniqueness, but we can show the existence of a minimal solution uλ* and establish the monotonicity of the map λ⟼uλ* and its asymptotic behaviour as the parameter λ decreases to the critical value λ^1(q)>0 (the principal eigenvalue of (−Δq,W01,q(Ω))).

Positive solutions for the Riemann–Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervals

In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda >0$\\end{document} by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.

Existence of Solutions for the Initial Value Problem with Hadamard Fractional Derivatives in Locally Convex Spaces

In this paper, we investigate an initial value problem for a nonlinear fractional differential equation on an infinite interval. The differential operator is taken in the Hadamard sense and the nonlinear term involves two lower-order fractional derivatives of the unknown function. In order to establish the global existence criteria, we first verify that there exists a unique positive solution to an integral equation based on a class of new integral inequality. Next, we construct a locally convex space, which is metrizable and complete. On this space, applying Schäuder’s fixed point theorem, we obtain the existence of at least one solution to the initial value problem.

Stochastic dynamic effects of media coverage and incubation on a distributed delayed epidemic system with Lévy jumps

A stochastic SIAM (Susceptible individual–Infected individual–Aware individual–Media coverage) epidemic system with Lévy jumps and distributed time delay is established, which is utilized to investigate hybrid dynamic effects of media coverage and incubation on infectious disease transmission. Stochastically ultimate boundedness of the positive solution is discussed. Existence of a unique global positive solution is studied. By constructing appropriate stochastic Lyapunov functions, existence of a unique ergodic stationary distribution is investigated. Sufficient conditions are derived to discuss exponential ergodicity based on verifying a Foster–Lyapunov condition. Furthermore, sufficient conditions for persistence in mean and extinction of infectious disease are investigated. Finally, numerical simulations are carried out to show consistency with theoretical analysis.

On existence of positive solutions of p-Laplace equations

In this paper, we consider the existence of positive solutions of an equation with p-Laplacian where n ≥ 3, 1 < p < n, q > 0, Δ p u = div(|∇u| p−2∇u), and K(x) is a double bounded function. We will prove the following results. When 0 < q < p − 1, the equation has no solution satisfying inf Rnu = 0 for any double bounded function K(x). When q = p − 1, the equation has a positive radial singular solution for K(x) ≡ Const.. In addition, q = p − 1 is a necessary condition such that this equation (with K(x) ≡ 1) has finite energy solutions. In addition, the existence of positive solutions are also derived for the system where u, v > 0 in Rn \\ {0}, n ≥ 3, 1 < p 1, p 2 < n, q 1, q 2 > 0, and K 1(x), K 2(x) are double bounded functions.

Dynamic Analysis of a Ratio-Dependent Food Chain Model with Prey-Taxis

In this paper, we consider a food chain model with ratio-dependent functional response and prey-taxis. We first investigate the global existence and boundedness of the unique positive classical solutions of the system in a bounded domain with smooth boundary and Neumann boundary conditions. Then, we analyze the local stability of the system and the existence of Hopf bifurcation. In addition, we prove the global asymptotic stability of steady states under some conditions by constructing a Lyapunov functional, and investigate convergence rates. Finally, we present several numerical simulations to illustrate the results.

Dynamic Analysis of the M/G/1 Stochastic Clearing Queueing Model in a Three-Phase Environment

In this paper, we consider the M/G/1 stochastic clearing queueing model in a three-phase environment, which is described by integro-partial differential equations (IPDEs). Our first result is semigroup well-posedness for the dynamic system. Utilizing a C0—semigroup theory, we prove that the system has a unique positive time-dependent solution (TDS) that satisfies the probability condition. As our second result, we prove that the TDS of the system strongly converges to its steady-state solution (SSS) if the service rates of the servers are constants. For this asymptotic behavior, we analyze the spectrum of the system operator associated with the system. Additionally, the stability of the semigroup generated by the system operator is also discussed.

Stochastic SIR Epidemiological Model With Two Levels of Severity

In this paper, we develop an epidemic model to analyze the spread of an infectious disease with two levels of severity within a population of varying size. Additionally, we examine the stability of the disease-free equilibrium.&#x0D; Such a model is suitable for the dynamics of COVID-19 disease spread. If the two types of infected individuals have different recovery rates, then there is no endemic equilibrium; only the disease-free equilibrium will be the subject of our study. We then study both the deterministic model and a stochastic version. The stochastic model is obtained by perturbing the contact rate using white noise. For the deterministic model, we have shown that if the basic reproduction number R_0 &amp;lt; 1, then the equilibrium state is globally asymptotically stable by using Lyapunov function. This implies that the disease will disappear, and the entire population will become susceptible. For the stochastic version, we demonstrate that the system admits a unique positive global solution that exists within a positively invariant domain. Under suitable conditions on the intensity of the white noise perturbations, we prove that the number of infectious individuals converge almost surely exponentially to zero and the disease-free equilibrium of system is stochastically asymptotically stable in the large provided. Finally, we give some numerical simulations to illustrate our theoretical results.

Modified Riccati iterative algorithms for stochastic coupled algebraic Riccati equations of linear stochastic Markovian jump systems

AbstractThis paper investigates the stochastic coupled algebraic Riccati equations (SCAREs) associated with linear stochastic Markovian jump systems. First, two existing algorithms utilized to solve the SCAREs are reviewed and discussed. Second, based on the analysis of these two algorithms, a modified Riccati iterative algorithm is proposed with a tuning parameter added. It is verified that the proposed algorithm with zero initial values converges to the unique positive definite solution of the SCAREs with fast speed. Furthermore, two numerical examples are given to verify the efficiency and superiority of the proposed algorithm.

Positive solutions for Hadamard-type fractional differential equations with nonlocal conditions on an infinite interval

The purpose of this paper is to analyse the local existence and uniqueness of positive solutions for a Hadamard-type fractional differential equation with nonlocal boundary conditions on an infinite interval. The technique used to arrive our results depends on two fixed point theorems of a sum operator in partial ordering Banach spaces. The local existence and uniqueness of positive solution is given, and we can make iterative sequences to approximate the unique positive solution. For the illustration of the main results, we list two concrete examples in the last section.

A new fixed point approach for solutions of a $ p $-Laplacian fractional $ q $-difference boundary value problem with an integral boundary condition

&lt;p&gt;We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.&lt;/p&gt;

On positive definite solutions of the matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $

&lt;p&gt;In this paper, we used the outstanding properties of the Thompson metric to conclusively demonstrate the existence of a unique positive definite solution for the nonlinear matrix equation $ X-\sum_{i = 1}^{m}A_{i}^{\ast}X^{-p_{i}}A_{i} = Q $ without any additional assumptions. Furthermore, we designed an iterative algorithm to compute this unique positive definite solution, and derive its corresponding error estimate formula. Additionally, we presented three refined existence intervals for positive definite solutions of this equation. Finally, numerical examples were employed to validate the practicability of our iterative algorithm.&lt;/p&gt;

Besicovitch almost periodic solutions for a stochastic generalized Mackey-Glass hematopoietic model

&lt;p&gt;This article aimed to investigate the existence and stability of Besicovitch almost periodic ($ B_{ap} $) positive solutions for a stochastic generalized Mackey-Glass hematopoietic model. To begin with, we used stochastic analysis theory, inequality techniques, and fixed point theorems to prove the existence and uniqueness of $ \mathcal{L}^p $-bounded and $ \mathcal{L}^p $-uniformly continuous positive solutions for the model under consideration. Then, we used definitions to prove that this unique positive solution is also a $ B_{ap} $ solution in finite-dimensional distributions. In addition, we established the global exponential stability of the $ B_{ap} $ positive solution using reduction to absurdity. Finally, we provided a numerical example to verify the effectiveness of our conclusions.&lt;/p&gt;

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;

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

A Levy process approach coupled to the stochastic Leslie–Gower model

This paper focuses on a two-dimensional Leslie–Grower continuous-time stochastic predator–prey system with Lévy jumps. Firstly, we prove that there exists a unique positive solution of the system with a positive initial value. Then, we establish sufficient conditions for the mean stability and extinction of the considered system. Numerical algorithms of higher order are elaborated. The obtained results show that Lévy jumps significantly change the properties of population systems.

Positive solutions for the periodic-parabolic problem with large diffusion

&lt;p&gt;In this paper, we study the positive solutions of the periodic-parabolic logistic equation with indefinite weight function and nonhomogeneous diffusion coefficient. By employing sufficient conditions to guarantee negative principal eigenvalue, we obtain the existence, uniqueness, and stability of the positive periodic solutions. Moreover, we prove that the positive periodic solution tends to the unique positive solution of the corresponding non-autonomous logistic equation when the diffusion rate is large.&lt;/p&gt;