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A novel fractional-order grey Euler prediction model and its application in short-term traffic flow 新型分数阶灰色欧拉预测模型及其在短期交通流中的应用

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-14 DOI: 10.1016/j.chaos.2024.115722

Yuxin Song , Huiming Duan , Yunlong Cheng

In intelligent transportation systems, short-term traffic flow prediction, as a core component, plays a crucial role in improving the operational efficiency and safety of the transportation system. To achieve accurate traffic flow prediction, a novel fractional-order grey Euler prediction model has been established. The new model utilizes the fractional-order accumulation technique and the characteristics of cycle truncation accumulated generating operation to develop a new fractional-order cycle truncation accumulating generation operator. By using this sequential operator for modeling, the new fractional-order operator can fully utilize new information promptly, reflect the dynamic and periodic characteristics of the traffic flow system, and flexibly capture short-term fluctuations in traffic flow data. By adjusting the parameters, the dynamic changes in the traffic flow system can be described more accurately. Meanwhile, the properties of this new fractional order operator are analyzed, the modeling conditions of this new sequential operator are verified, and the particle swarm algorithm is used to optimize the model parameters with the objective function of minimizing the average absolute percentage total error to improve the overall performance of the new model. Finally, the novel model is implemented to simulate and forecast traffic flow data on UK highways. Its performance is validated through a comprehensive analysis of traffic flows spanning three distinct periods, ensuring its robustness under varying traffic conditions. A comparative study with seven established grey prediction models reveals that our model surpasses them in both simulation and prediction outcomes, exhibiting remarkable stability and precision in both fitting and forecasting. Consequently, the integration of this new model into traffic flow analysis offers a potent tool to accurately depict traffic parameter trends, bolstering data adaptability and enhancing modeling capabilities significantly.

在智能交通系统中，短期交通流预测作为一个核心组成部分，对提高交通系统的运行效率和安全性起着至关重要的作用。为了实现精确的交通流预测，我们建立了一种新型的分数阶灰色欧拉预测模型。新模型利用分数阶累加技术和循环截断累加生成运算的特点，建立了一个新的分数阶循环截断累加生成算子。利用该序列算子建模，新的分数阶算子可以及时充分利用新信息，反映交通流系统的动态性和周期性特征，灵活捕捉交通流数据的短期波动。通过调整参数，可以更准确地描述交通流系统的动态变化。同时，分析了这种新的分数阶算子的特性，验证了这种新的序列算子的建模条件，并利用粒子群算法优化模型参数，以最小化平均绝对百分比总误差为目标函数，提高新模型的整体性能。最后，新模型被用于模拟和预测英国高速公路的交通流量数据。通过对三个不同时期的交通流量进行综合分析，验证了该模型的性能，确保其在不同交通条件下的稳健性。通过与七个成熟的灰色预测模型进行比较研究发现，我们的模型在模拟和预测结果方面都超过了它们，在拟合和预测方面都表现出了显著的稳定性和精确性。因此，将这一新模型整合到交通流分析中，为准确描述交通参数趋势提供了有力工具，增强了数据适应性，大大提高了建模能力。

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引用次数: 0

Numerical approach and physical description for a two-capacitive neuron and its adaptive network dynamics 双电容神经元及其自适应网络动力学的数值方法和物理描述

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-14 DOI: 10.1016/j.chaos.2024.115738

Yixuan Chen , Qun Guo , Xiaofeng Zhang , Chunni Wang

A simple neuron containing one capacitive variable can mimic the dynamical property of electrical activities in a biological neuron. Functional enhancement and activation of adaptive regulation must clarify the energy characteristic and nonlinear property of cell membrane of the neuron. In this work, two capacitors are connected via a nonlinear resistor for exploring the electrical activities in a double-layer nonlinear membrane, and the additive branch circuits are incorporated with piezoelectric ceramic and Josephson junction, which can perceive external acoustic wave and changes of magnetic field. The nonlinear equations for the neural circuit are converted into equivalent dimensionless neuron model in the form of nonlinear oscillator. The energy function for the neuron model is defined and proofed by using the Helmholtz theorem. Any mode transition is dependent on the shift of energy levels and coherence resonance is induced by noisy excitation. An adaptive control law under energy flow is proposed to regulate the firing patterns. Finally, the neuron is clustered to build a neural network with nearest neighbor coupling on a square array. Statistical synchronization factor is defined and calculated to predict the synchronization stability and wave propagation in the neural network. By activating the adaptive growth of coupling intensity and capacitance ratio for the outer and inner cell membrane, target like waves are developed in the neural network.

包含一个电容变量的简单神经元可以模拟生物神经元电活动的动态特性。功能增强和激活自适应调节必须明确神经元细胞膜的能量特征和非线性特性。在这项工作中，两个电容器通过一个非线性电阻器连接，以探索双层非线性膜中的电活动，并在加法支路中加入了压电陶瓷和约瑟夫森结，它们可以感知外部声波和磁场的变化。神经回路的非线性方程被转换成非线性振荡器形式的等效无量纲神经元模型。利用亥姆霍兹定理定义并证明了神经元模型的能量函数。任何模式转换都取决于能级的移动，而相干共振则是由噪声激励引起的。提出了一种能量流下的自适应控制法则来调节发射模式。最后，对神经元进行聚类，在方阵上建立一个近邻耦合神经网络。通过定义和计算统计同步因子来预测神经网络的同步稳定性和波传播。通过激活细胞外膜和内膜耦合强度和电容比的自适应增长，神经网络中会产生类似目标的波。

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引用次数: 0

Resonant responses and double-parameter multi-pulse chaotic dynamics of FG-GP reinforced rotating pre-twisted composite laminated blade under combined transverse aerodynamic force and parametric excitations FG-GP 增强旋转预扭曲复合材料层压叶片在横向空气动力和参数激励共同作用下的共振响应和双参数多脉冲混沌动力学特性

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-14 DOI: 10.1016/j.chaos.2024.115725

Y.Z. Lian , W. Zhang , Y.F. Zhang

The new functionally grated graphene platelets (FG-GP) reinforced rotating pre-twisted composite laminated blade may exhibit strong nonlinear vibrations under combined the transverse aerodynamic force and axial excitation. These nonlinear vibrations are major contributors to the overall failure of the rotating pre-twisted composite laminated blade. In this paper, the resonant responses, threshold surface, global bifurcations and double-parameter multi-pulse chaotic dynamics of the rotating pre-twisted composite laminated blade are investigated on the basis of the nonlinear dynamic model of the new FG-GP reinforced rotating pre-twisted composite laminated blade for the first time. Considering the primary parametric and 1:1 internal resonances, the averaged equations for the new FG-GP reinforced rotating pre-twisted composite laminated blade are derived by using the multiple scale perturbation (MSP) method. The amplitude-frequency and force-amplitude response curves are obtained to analyze the resonant responses of the new FG-GP reinforced rotating pre-twisted composite laminated blade. The extended Melnikov method is used to explore the threshold surface, global bifurcations and double-parameter multi-pulse chaotic dynamics of the new FG-GP reinforced rotating pre-twisted composite laminated blade under combined the transverse aerodynamic force and axial excitation. Through the theoretical analysis and numerical simulation, it is discovered that the new FG-GP reinforced rotating pre-twisted composite laminated blade exhibits the hard spring characteristics. The energy transfers between two-order modes are proved by the occurrence of the bubble response shape. The complex double-parameter multi-pulse chaotic vibrations are investigated for the new FG-GP reinforced rotating pre-twisted composite laminated blade under the influence of different parameters. These results have the significant theoretical guidance for the optimized design of the new FG-GP reinforced rotating pre-twisted composite laminated blade.

在横向空气动力和轴向激励的共同作用下，新型功能格栅石墨烯平板（FG-GP）增强旋转预扭曲复合材料层压叶片可能会表现出强烈的非线性振动。这些非线性振动是导致旋转预扭曲复合材料层压叶片整体失效的主要原因。本文在新型 FG-GP 增强旋转预扭曲复合材料层叠叶片非线性动力学模型的基础上，首次研究了旋转预扭曲复合材料层叠叶片的共振响应、阈值面、全局分岔和双参数多脉冲混沌动力学。考虑到主要参数共振和 1:1 内部共振，利用多尺度扰动（MSP）方法推导出了新型 FG-GP 增强旋转预扭曲复合材料层压叶片的平均方程。得到了振幅-频率和力-振幅响应曲线，从而分析了新型 FG-GP 增强旋转预扭曲复合材料层压叶片的共振响应。采用扩展梅利尼科夫法探讨了新型 FG-GP 增强旋转预扭曲复合材料层压叶片在横向气动力和轴向激励联合作用下的阈值面、全局分岔和双参数多脉冲混沌动力学特性。通过理论分析和数值模拟，发现新型 FG-GP 增强旋转预扭曲复合材料层叠叶片具有硬弹簧特性。气泡响应形状的出现证明了二阶模态之间的能量传递。研究了新型 FG-GP 增强旋转预扭曲复合材料层压叶片在不同参数影响下的复杂双参数多脉冲混沌振动。这些结果对新型 FG-GP 增强旋转预扭复合材料层压叶片的优化设计具有重要的理论指导意义。

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引用次数: 0

Analytical approximation of dynamic responses of random parameter nonlinear systems based on stochastic perturbation-Galerkin method 基于随机扰动-加勒金法的随机参数非线性系统动态响应的分析近似

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-13 DOI: 10.1016/j.chaos.2024.115724

Bin Huang , Cihang Ma , Yejun Li , Zhifeng Wu , Heng Zhang

The analytic approximation of dynamic responses is very significant for performing the optimization, parameter identification and reliability analysis of structural systems. However, the dynamic responses analysis of nonlinear random parameter systems is a challenging task due to the combined effects of randomness and strong nonlinearity. To address this problem, a new solution method based on the stochastic perturbation-Galerkin method is developed to obtain analytical solutions of the dynamic responses of single-degree-of-freedom nonlinear systems with random parameters. By combining the high-order perturbation and the Newmark-β method, the dynamic responses of systems are initially approximated using the power series expansions. Then a new approximation is defined for the Galerkin projection by utilizing the different orders of the power series expansion terms as trial functions. The employed Galerkin projection ensures the statistical minimization of the random error of the approximate series expansion. The numerical example shows that, for the first time, high-precision analytical expressions for the dynamic responses of a single-degree-of-freedom Duffing system with random parameters are obtained, even if the nonlinear coefficient reaches a value of 50. And it is found that the relationship between the dynamic responses and random variable is strongly nonlinear and constantly evolves over time, becoming increasingly complex along with the nonlinear coefficient. Numerical results further indicate that the new method owns superior computational accuracy compared with the perturbation method and the generalized polynomial chaos method of the same order, can better maintain convergence during long-time integration and has better efficiency than the direct Monte Carlo simulation method.

动态响应的解析近似对于结构系统的优化、参数识别和可靠性分析非常重要。然而，由于随机性和强非线性的共同作用，非线性随机参数系统的动态响应分析是一项具有挑战性的任务。为解决这一问题，我们开发了一种基于随机扰动-Galerkin 方法的新求解方法，以获得随机参数单自由度非线性系统动态响应的解析解。通过结合高阶扰动和 Newmark-β 方法，利用幂级数展开对系统的动态响应进行初步近似。然后，利用幂级数展开项的不同阶数作为试函数，为 Galerkin 投影定义新的近似值。所采用的 Galerkin 投影确保了近似级数展开随机误差的统计最小化。数值实例表明，即使非线性系数达到 50，也能首次获得随机参数单自由度 Duffing 系统动态响应的高精度解析表达式。结果发现，动态响应与随机变量之间的关系是强烈的非线性关系，并且随着时间的推移不断演变，随着非线性系数的增加而变得越来越复杂。数值结果进一步表明，与同阶的扰动法和广义多项式混沌法相比，新方法具有更高的计算精度，在长时间积分过程中能更好地保持收敛性，与直接蒙特卡罗模拟法相比具有更好的效率。

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引用次数: 0

Determination of a time dependent source in semilinear pseudoparabolic equations with Caputo fractional derivative 带有卡普托分数导数的半线性假抛物方程中时间相关源的确定

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-13 DOI: 10.1016/j.chaos.2024.115716

Kh. Khompysh

This paper devoted to study unique solvability of inverse source problem for a semilinear time fractional pseudoparabolic equation perturbed by a damping term. Inverse problem consists of recovering a solely time dependent coefficient of right-hand side under a measurement in an integral form. The damped term acts in the equation as a nonlinear source or as an absorption, depending on its sign of coefficient, where it is positive or negative, respectively. In these both cases, we have established sufficient conditions on data, where the inverse problem has a global or local in time unique weak and strong solution.

本文致力于研究受阻尼项扰动的半线性时间分数伪抛物方程的逆源问题的唯一可解性。反源问题包括在积分形式的测量条件下恢复右侧完全与时间相关的系数。阻尼项在方程中起非线性源或吸收的作用，这取决于其系数的符号，分别为正或负。在这两种情况下，我们都建立了数据的充分条件，即逆问题在时间上具有全局或局部唯一的弱解和强解。

引用次数: 0

Analytical properties and related inequalities derived from multiplicative Hadamard k-fractional integrals 由乘法哈达玛 k 分积分导出的分析性质和相关不等式

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-12 DOI: 10.1016/j.chaos.2024.115715

Ziyi Zhou , Tingsong Du

The present article is intended to address the properties and associated inequalities of multiplicative Hadamard

k

-fractional integrals. The core concept lies in introducing the multiplicative Hadamard

k

-fractional integrals. In this framework, various analytical characteristics they possess, such as

^{*}

integrability, continuity, commutativity, semigroup property, boundedness, and others, are examined herein. Subsequently, the Hermite–Hadamard-analogous inequalities are formulated for the novelly constructed operators. Meanwhile, an identity is inferred within multiplicative Hadamard

k

-fractional integrals, based on which a series of Bullen-type inequalities are derived in this article, where the function

Λ^{*}

is GG-convex and the function

{(ln Λ^{*})}^{s}

is GA-convex for

s > 1

, with a particular focus on discussing the case when

0 < s \leq 1

. To facilitate a more profound understanding of the outcomes, we offer illustrative examples together with numerical simulations to confirm the consistency of the theoretical results. Finally, applications of the proposed results in multiplicative differential equations, quadrature formulas, and special means for real numbers are investigated as well.

本文旨在探讨乘法哈达玛 k 分积分的性质和相关不等式。其核心概念在于引入乘法哈达玛 k 分数积分。在此框架下，研究了它们所具有的各种分析特性，如∗可整性、连续性、交换性、半群性质、有界性等。随后，对新构造的算子提出了赫米特-哈达玛（Hermite-Hadamard）类似不等式。同时，在乘法哈达玛 k 分数积分中推导出一个同一性，在此基础上，本文推导出一系列布伦型不等式，其中函数Λ∗是 GG-凸的，函数（lnΛ∗）s 在 s>1 时是 GA-凸的，特别着重讨论了 0<s≤1 时的情况。为了便于更深刻地理解这些结果，我们提供了示例，并通过数值模拟来证实理论结果的一致性。最后，我们还研究了所提结果在乘法微分方程、正交公式和实数特殊手段中的应用。

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引用次数: 0

Unraveling dengue dynamics with data calibration from Palu and Jakarta: Optimizing active surveillance and fogging interventions 利用帕卢和雅加达的数据校准揭示登革热动态：优化主动监测和雾化干预措施

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-12 DOI: 10.1016/j.chaos.2024.115729

Dipo Aldila , Joseph Páez Chávez , Chidozie W. Chukwu , Athaya Yumna Fathiyah , Juni Wijayanti Puspita , Kartika A. Dimar Setio , Ahmad Fuady , Putri Zahra Kamalia

Dengue fever is a complex infectious disease driven by multiple factors, including viral dynamics, mosquito behavior, environmental conditions, and human behaviors. The intricate nature of its transmission and outbreaks necessitates an interdisciplinary approach, integrating expertise from fields such as mathematics and public health. In this research, we examine the role of active case finding and mosquito population reduction through fogging in dengue control using a mathematical model approach. Active case finding aims to identify undetected dengue cases, both asymptomatic and symptomatic, which can help prevent further transmission and reduce the likelihood of severe symptoms by enabling earlier treatment. The model was developed using a system of nine-dimensional nonlinear ordinary differential equations. We conducted a mathematical analysis of the equilibria and their stability based on the basic reproduction number (

R_{0}

). Our analysis shows that the disease-free equilibrium is locally asymptotically stable when

R_{0} < 1

. Furthermore, when

R_{0} = 1

, the model may exhibit backward bifurcation , depending on the death rate induced by dengue. The higher the dengue-induced death rate, the greater the likelihood of backward bifurcation at

R_{0} = 1

. We used dengue incidence data from two Indonesian provinces, Jakarta and Palu, to calibrate the model parameter values. Our global sensitivity analysis on the basic reproduction number indicates that active case findings are more crucial in Palu compared to Jakarta. Conversely, Jakarta is more sensitive to the infection parameter than Palu. Our numerical continuation simulation shows that implementing fogging to control the mosquito population should carefully consider the intensity, timing, and duration of the intervention to achieve a more optimal results.

登革热是一种复杂的传染病，由多种因素驱动，包括病毒动态、蚊虫行为、环境条件和人类行为。其传播和爆发的复杂性要求采用跨学科的方法，整合数学和公共卫生等领域的专业知识。在这项研究中，我们采用数学模型方法研究了登革热控制中通过雾化主动发现病例和减少蚊子数量的作用。主动发现病例的目的是发现未被发现的登革热病例，包括无症状和有症状的病例，这有助于防止进一步传播，并通过尽早治疗降低出现严重症状的可能性。该模型是利用九维非线性常微分方程系统建立的。我们根据基本繁殖数（R0）对平衡及其稳定性进行了数学分析。我们的分析表明，当 R0<1 时，无疾病平衡是局部渐近稳定的。此外，当 R0=1 时，模型可能会出现向后分叉，这取决于登革热引起的死亡率。登革热引起的死亡率越高，在 R0=1 时出现向后分叉的可能性就越大。我们使用印度尼西亚雅加达和帕卢两个省的登革热发病率数据来校准模型参数值。我们对基本繁殖数的全球敏感性分析表明，与雅加达相比，帕卢的活动病例发现更为关键。相反，雅加达对感染参数的敏感度要高于帕卢。我们的数值持续模拟结果表明，实施雾化控制蚊虫数量应仔细考虑干预的强度、时间和持续时间，以达到更理想的效果。

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引用次数: 0

The impact of social noise on the majority rule model across various network topologies 不同网络拓扑结构下社会噪音对多数规则模型的影响

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-12 DOI: 10.1016/j.chaos.2024.115718

Roni Muslim , Didi Ahmad Mulya , Zulkaida Akbar , Rinto Anugraha NQZ

We explore the impact of social noise, characterized by nonconformist behavior, on the phase transition within the framework of the majority rule model. The order–disorder transition can reflect the consensus-polarization state in a social context. This study covers various network topologies, including complete graphs, two-dimensional (2-D) square lattices, three-dimensional (3-D) square lattices, and heterogeneous or complex networks such as Watts–Strogatz (W–S), Barabási–Albert (B–A), and Erdős–Rényi (E–R) networks, as well as their combinations (multilayer network). Social behavior is represented by the parameter

p

, which indicates the probability of agents exhibiting nonconformist behavior. Our results show that the model exhibits a continuous phase transition across all networks. Through finite-size scaling analysis and evaluation of critical exponents, our results suggest that the model falls into the same universality class as the Ising model.

我们在多数规则模型的框架内，探讨了以不服从行为为特征的社会噪音对相变的影响。有序-无序转换可以反映社会背景下的共识-极化状态。本研究涵盖各种网络拓扑结构，包括完整图、二维（2-D）方格、三维（3-D）方格和异构或复杂网络，如瓦特-斯特罗加茨（W-S）、巴拉巴西-阿尔伯特（B-A）和厄尔多斯-雷尼（E-R）网络，以及它们的组合（多层网络）。社会行为由参数 p 表示，该参数表示代理人表现出不守规矩行为的概率。我们的研究结果表明，该模型在所有网络中都表现出连续的相变。通过有限规模缩放分析和临界指数评估，我们的结果表明，该模型与伊辛模型属于同一普遍性类别。

引用次数: 0

True random number generator using stochastic noise signal of memristor with variation tolerance 使用具有变化容差的忆阻器随机噪声信号的真随机数发生器

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-11 DOI: 10.1016/j.chaos.2024.115708

Dayeon Yu , Suhyeon Ahn , Sangwook Youn , Jinwoo Park , Hyungjin Kim

Memristors are suitable for internet of things (IoT) edge devices due to their high scalability and low power consumption. Also, their stochastic noise signals can be used to produce entropy sources for information encryption in edge devices, which is essential during data exchange. Random telegraph noise (RTN) in memristors is a current fluctuation that occurs when electrons are captured or emitted from a trap, and its stochastic characteristics can be used as an entropy source for a true random number generator (TRNG). However, post-processing is essential because RTN tends to be biased toward specific time constants, and there is significant variation within each time constant. In this work, we present a TRNG circuit that can tolerate the time variation of RTN and demonstrate experimental results with a breadboard. The dependence of RTN signals on a read voltage is statistically verified, confirming a significant variation in a time constant of RTN signal (σ/μ > 700 %). Despite this variation, high randomness of generated random number stream can be obtained by a falling edge detector and is verified by NIST SP 800–22 test (with >14 tests passed) and auto-correlation function test (rate of exceeding confidence bound <1.5 %).

晶闸管具有高扩展性和低功耗的特点，适合用于物联网（IoT）边缘设备。此外，它们的随机噪声信号可用于产生边缘设备信息加密的熵源，这在数据交换过程中至关重要。忆阻器中的随机电报噪声（RTN）是电子从阱中捕获或发射时产生的电流波动，其随机特性可用作真随机数发生器（TRNG）的熵源。然而，后处理是必不可少的，因为 RTN 往往偏向于特定的时间常数，而每个时间常数内部又存在显著的差异。在这项工作中，我们提出了一种可容忍 RTN 时间变化的 TRNG 电路，并用面包板演示了实验结果。RTN 信号对读取电压的依赖性经过统计验证，证实 RTN 信号的时间常数变化很大（σ/μ > 700 %）。尽管存在这种变化，但仍可通过下降沿检测器获得所生成随机数流的高随机性，并通过 NIST SP 800-22 测试（通过 14 次测试）和自相关函数测试（超过置信区间的比率为 1.5%）进行验证。

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引用次数: 0

Analysis of a stochastic SEIS epidemic model motivated by Black–Karasinski process: Probability density function 以 Black-Karasinski 过程为动机的 SEIS 随机流行病模型分析：概率密度函数

IF 5.3 1区数学 Q1 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

Chaos Solitons & Fractals

Pub Date : 2024-11-11 DOI: 10.1016/j.chaos.2024.115713

Baoquan Zhou, Ningzhong Shi

This paper examines a stochastic SEIS epidemic model motivated by Black–Karasinski process. First, it is shown that Black–Karasinski process is a both biologically and mathematically reasonable assumption compared with existing stochastic modeling methods. By analyzing the diffusion structure of the model and solving the relevant Kolmogorov–Fokker–Planck equation, a complete characterization for explicitly approximating the stationary density function near some quasi-positive equilibria is provided. Then for the deterministic model, the basic reproduction number and related asymptotic stability are studied. Finally, several numerical examples are given to substantiate our theoretical findings.

本文研究了以 Black-Karasinski 过程为动机的 SEIS 流行病随机模型。首先，与现有的随机建模方法相比，Black-Karasinski 过程在生物学和数学上都是一个合理的假设。通过分析模型的扩散结构和求解相关的 Kolmogorov-Fokker-Planck 方程，为显式逼近某些准正平衡点附近的静态密度函数提供了完整的描述。然后研究了确定性模型的基本繁殖数和相关渐近稳定性。最后，给出了几个数值例子来证实我们的理论发现。

引用次数: 0