Advances in Difference Equations最新文献

Industrial development has made air pollution increasingly severe, and many respiratory diseases are closely related to air quality in terms of infection and transmission. In this work, we used the classic stochastic susceptible–infectious–recovered (SIR) model to reflect the spread of respiratory disease, coupled with the diffusion process of air pollutants to the infectious disease model, and we investigated the impact of various environmental noises on the process of disease transmission and air pollutant diffusion. The value of this study lies in two aspects. Mathematically, we define threshold (mathcal{R}_{1}^{s}) for extinction and threshold (mathcal{R}_{2}^{s}) for persistence of the disease in the stochastic model ((mathcal{R}_{2}^{s}<mathcal{R}_{1}^{s})) when the parameters are constant, and we show that (i) when (mathcal{R}_{1}^{s}) is less than 1, the disease will go to stochastic extinction; (ii) when (mathcal{R}_{2}^{s}) is larger than 1, the disease will persist almost surely and the model has a unique ergodic stationary distribution; (iii) when (mathcal{R}_{1}^{s}) is larger than 1 and (mathcal{R}_{2}^{s}) is less than 1, the extinction of the disease has randomness, which is demonstrated through numerical experiments. In addition, we derive the exact expression of the probability density function of the stationary distribution by solving the corresponding Fokker–Planck equation under the condition of disease persistence and analyze the effects of random noises on stationary distribution characteristics and the disease extinction. Epidemiologically, the change of the concentration of air pollutants affects the conditions for disease extinction and persistence. The increase in the inflow of pollutants and the increase in the clearance rate have negative and positive impacts on the spread of diseases, respectively. We found that an increase in random noise intensity will increase the variance, reduce the kurtosis of distribution, which is not conducive to predicting and controlling the development status of the disease; however, large random noise intensity can also increase the probability of disease extinction and accelerates disease extinction. We further investigate the dynamic of the stochastic model, assuming that the inflow rate switches between two levels by numerical experiments. The results show that the random noise has a significant impact on disease extinction. The data fitting of the switching model shows that the model can effectively depict the relationship and changes in trends between air pollution and diseases.

This article proposes a class of nonsmooth Filippov pest–predator ecosystems with intermittent control strategies based on the pest’s antipredator behavior. aiming to investigate the influence of control strategies and switching thresholds on pest control. First, a comprehensive theoretical analysis of various equilibria within the Filippov system is undertaken, emphasizing the presence and stability of sliding mode dynamics and pseudoequilibrium. Secondly, through numerical simulations, the article discusses boundary-focus, boundary-node, and boundary-saddle bifurcation. Finally, the nonexistence of limit cycles in the Filippov system is theoretically studied. The research indicates that the solution trajectories of the model ultimately stabilize either at the real equilibria or at pseudoequilibrium on the model’s switching surface. Moreover, when the model has multiple coexisting real equilibrium and pseudoequilibrium, the pest-control strategy is correlated with the initial density of both the pest and the predator population.

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers’ comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus.

In this article, we are working on an SEIR-SI type model for dengue disease in order to better observe the dynamics of infection in human beings. We calculate the basic reproduction number (mathcal{R}_{0}) and determine the equilibrium points. We then show the existence of global stability in each of the different states depending on the value of (mathcal{R}_{0}). Moreover, to support the theoretical work, we present numerical simulations obtained using Python. We also study the sensitivity of the parameters included in the expression of (mathcal{R}_{0}) with the aim of identifying the most influential parameters in the dynamics of dengue disease spread. Finally, we introduce two functions u and v, respectively indicating the treatment of the infected people and any prevention system minimizing contact between humans and the disease causing vectors. We present the curves of the controlled system after calculating the optimal pair of controls capable of reducing the dynamics of the disease spread, still using Python.

Vaccination is an important tool in disease control to suppress disease, and vaccine-influenced diseases no longer conform to the general pattern of transmission. In this paper, by assuming that the infection rate is affected by the Ornstein–Uhlenbeck process, we obtained a stochastic SIRV model. First, we prove the existence and uniqueness of the global positive solution. Sufficient conditions for the extinction and persistence of the disease are then obtained. Next, by creating an appropriate Lyapunov function, the existence of the stationary distribution for the model is proved. Further, the explicit expression for the probability density function of the model around the quasi-equilibrium point is obtained. Finally, the analytical outcomes are examined by numerical simulations.

We consider optimal control problems for a system governed by a stochastic differential equation driven by a d-dimensional Brownian motion where both the drift and the diffusion coefficient are controlled. It is well known that without additional convexity conditions the strict control problem does not admit an optimal control. To overcome this difficulty, we consider the relaxed model, in which admissible controls are measure-valued processes and the relaxed state process is governed by a stochastic differential equation driven by a continuous orthogonal martingale measure. This relaxed model admits an optimal control that can be approximated by a sequence of strict controls by the so-called chattering lemma. We establish optimality necessary conditions, in terms of two adjoint processes, extending Peng’s maximum principle to relaxed control problems. We show that relaxing the drift and diffusion martingale parts directly as in deterministic control does not lead to a true relaxed model as the obtained controlled dynamics is not continuous in the control variable.

The coronavirus disease 2019 (COVID-19) presents a severe and urgent threat to global health. In response to the COVID-19 pandemic, many countries have implemented nonpharmaceutical interventions (NPIs), including national workplace and school closures, personal protection, social distancing, contact tracing, testing, home quarantine, and isolation. To evaluate the effectiveness of these NPIs in mitigating the spread of early COVID-19 and predict the epidemic trend in the United Kingdom, we developed a compartmental model to mimic the transmission with time-varying transmission rate, contact rate, disease-induced mortality rate, proportion of quarantined close contacts, and hospitalization rate. The model was fitted to the number of confirmed new cases and daily number of deaths in five stages with a Markov Chain Monte Carlo method. We quantified the effectiveness of NPIs and found that if the transmission rate, contact rate, and hospitalization rate were approximately equal to those in the second stage of the most strict NPIs, and the proportion of quarantined close contacts increased by 3%, then the epidemic would die out as early as January 12, 2021, with around 1,533,000 final cumulative number of confirmed cases, and around 55,610 final cumulative number of deaths.

Breast cancer is the most common type of cancer in women. Chemotherapy is primarily used for patients with stage 2 to 4 breast cancer. Most chemotherapy drugs are effective at destroying rapidly growing and proliferating cancer cells. However, drugs also damage normal, rapidly growing cells, which can lead to serious side effects. Breast cancer treatment with chemotherapy can affect heart health. Side effects of chemotherapy on the heart are called cardiotoxicity. Therefore, we have constructed a mathematical model from the breast cancer patient population. In this article, we utilize the Caputo–Fabrizio fractional order derivative for mathematical modeling of the breast cancer stages in chemotherapy patients. The use of Caputo–Fabrizio fractional derivative provides a more valuable insight into the complexity of the breast cancer model. The stability of the fractional order model is also proven by the (mathscr{P})-stable approach of the fixed point theorem. Also, the numerical simulations are performed via Laplace Adomian decomposition method to establish the dependence of the breast cancer dynamics on the order of the fractional derivatives. Based on the geometric results in the figures, we can conclude that the magnitude of the fractional order has a considerable impact on the days, which the maximum or minimum of the system solutions are reached, with a shift in the time at which this happens as the fractional order decreases from 1. However, it is obvious that the solutions of Caputo–Fabrizio fractional model approach the relevant results of the classical integer order system, when the fractional order approaches to 1.

In this paper, we study the solvability and generalized Ulam–Hyers (UH) stability of a nonlinear Atangana–Baleanu–Caputo (ABC) fractional coupled system with a Laplacian operator and impulses. First, this system becomes a nonimpulsive system by applying an appropriate transformation. Secondly, the existence and uniqueness of the solution are obtained by an F-contractive operator and a fixed-point theorem on metric space. Simultaneously, the generalized UH-stability is established based on nonlinear analysis methods. Thirdly, a novel numerical simulation algorithm is provided. Finally, an example is used to illustrate the correctness and availability of the main results. Our study is a beneficial exploration of the dynamic properties of viscoelastic turbulence problems.

Abstract

In this paper, the controllability concept of a nonlinear fractional stochastic system involving state-dependent delay and impulsive effects is addressed by employing Caputo derivatives and Mittag-Leffler (ML) functions. Based on stochastic analysis theory, novel sufficient conditions are derived for the considered nonlinear system by utilizing Krasnoselkii’s fixed point theorem. Correspondingly, the applicability of the derived theoretical results is indicated by an example.