Partial Differential Equations in Applied Mathematics最新文献_第3页

Generalised Kolmogorov-Petrovskii-Piskunov equation of fractional order: Power series and shifted Legendre collocation methods 分数阶广义Kolmogorov-Petrovskii-Piskunov方程：幂级数和移位Legendre配置方法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-20 DOI: 10.1016/j.padiff.2025.101318

Richard Olu Awonusika, Abayomi Samuel Oke

The classical Kolmogorov-Petrovskii-Piskunov (KPP) equation describes physical phenomena such as combustion, chemical reaction, evolution of dominant genes, and propagation of nerve pulses. In this paper, we present solutions of a time-fractional order generalised KPP equation using a power series method and Legendre collocation method. The fractional order derivative is described in the Caputo sense. The proposed power series method assumes that the solution of the governing problem can be represented by a fractional power series in time variable, with space-variable expansion coefficients. The nonlinear term is assumed to be analytic in the unknown, and thus, admits a power series representation. The generalised Cauchy product is applied to transform the power series in the unknown to one in the time-dependent variable. An explicit recursion formula for the variable expansion coefficient is then constructed. On the other hand, the shifted Legendre collocation method assumes that the solution of the proposed problem can be expressed as a shifted Legendre polynomial series, with constant expansion coefficients to be determined. Collocating at the shifted Legendre-Gauss nodes, we obtain a set of nonlinear algebraic equations. These algebraic equations are then solved for the unknown Legendre expansion coefficients using Newton’s iteration method. Convergence analyses of both methods are presented. Examples of the proposed problem involving quadratic, cubic, and exponential nonlinearities are considered to demonstrate the efficiency, accuracy, and reliability of the proposed techniques. The results obtained from both methods are in excellent agreement with the exact solutions.

经典的Kolmogorov-Petrovskii-Piskunov （KPP）方程描述了燃烧、化学反应、优势基因的进化和神经脉冲的传播等物理现象。本文利用幂级数法和勒让德配置法给出了一类时间分数阶广义KPP方程的解。分数阶导数是在卡普托意义上描述的。所提出的幂级数方法假定控制问题的解可以用一个时变分数阶幂级数来表示，其展开系数是空变的。假定非线性项在未知情况下是解析的，因此可以用幂级数表示。应用广义柯西积将未知的幂级数变换为时变的幂级数。然后构造了变膨胀系数的显式递推公式。另一方面，移位勒让德配置法假定所提问题的解可以表示为移位的勒让德多项式级数，其展开系数为常数。在移位的legende - gauss节点处，我们得到了一组非线性代数方程。然后用牛顿迭代法求解未知的勒让德展开系数。给出了两种方法的收敛性分析。所提出的问题涉及二次、三次和指数非线性的例子被认为是为了证明所提出的技术的效率、准确性和可靠性。两种方法得到的结果与精确解非常吻合。

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引用次数: 0

Invariant formulation of nonclassical symmetries and explicit solutions of Rosenau-Hyman equation along with bifurcation analysis Rosenau-Hyman方程非经典对称的不变量表述和显式解及其分岔分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-19 DOI: 10.1016/j.padiff.2025.101314

M.A. El-Shorbagy , Sonia Akram , Mati Ur Rahman , Hossam A. Nabwey

This study focuses on the Rosenau–Hyman equation, which is a fundamental model in nonlinear wave dynamics, and investigates it through the lens of nonclassical symmetry analysis. The approach employs symbolic computation to derive determining equations and uncover new invariant formulations, from which several explicit exact solutions are constructed. To further understand the system’s behavior, dynamical tools such as bifurcation analysis, sensitivity tests, Lyapunov exponents, and phase portraits are applied, highlighting the presence of stability transitions, multistability, and chaotic regimes. In addition, travelling wave solutions are obtained using the enhanced modified extended tanh function method (eMETFM), providing complementary wave structures. The findings deepen our understanding of nonlinear dispersive wave propagation and soliton interactions, with particular relevance to shallow water dynamics. More broadly, the developed solutions and their graphical interpretations contribute valuable insights for theoretical studies and applied research in fluid dynamics and wave modeling.

本文以非线性波动动力学的基本模型Rosenau-Hyman方程为研究对象，从非经典对称分析的角度对其进行了研究。该方法采用符号计算来推导确定方程并揭示新的不变公式，并从中构造出几个显式精确解。为了进一步了解系统的行为，应用了诸如分岔分析、灵敏度测试、李雅普诺夫指数和相位画像等动力学工具，突出了稳定性转变、多稳定性和混沌状态的存在。此外，使用增强修正扩展tanh函数法（eMETFM）获得行波解，提供互补波结构。这些发现加深了我们对非线性色散波传播和孤子相互作用的理解，特别是与浅水动力学有关。更广泛地说，开发的解决方案及其图形解释为流体动力学和波浪建模的理论研究和应用研究提供了宝贵的见解。

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引用次数: 0

Improved explicit finite difference method for extended shallow water partial differential equation 扩展浅水偏微分方程的改进显式有限差分法

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-17 DOI: 10.1016/j.padiff.2025.101316

Saeed Ahmed Rajput , Shakeel Ahmed Kamboh , Khuda Bux Amur , Afaque Ahmed Bhutto

The shallow water and Boussinesq equations, being highly nonlinear coupled partial differential equations, form a significant basis for the simulation of flow phenomena within the surface and subsurface domains. The application of these equations extends into several fields in the area of engineering and science, including the simulation of Kelvin wake waves, scenarios involving dam break, propagation of flood waves and flow over a bump. In this paper, the shallow water equation for surface flow regions and the Boussinesq partial differential equation for subsurface flow regions are unified into a single set of extended shallow water nonlinear partial differential equations which is applicable for both flow domains. These equations are solved numerically using an improved two-step Lax-Wendroff method with sixth-order accuracy. The method is validated by benchmarking against existing experimental data, which shows a percentage error of less than 1%, confirming its high accuracy and reliability. Moreover, the method is applied to two test cases. The first case is flow over a bump, where the relative error for surface and subsurface flow regions is as low as 10⁻⁸ at t = 100. In the second case, Kelvin wake wave is investigated with the improved sixth-order Lax-Wendroff scheme predicting an arm angle of 20.6955°, while a theoretical particle angle of 19.47° was observed. The results show that this method is good enough and useful for the simulation of surface and subsurface flows phenomena.

浅水方程和Boussinesq方程是高度非线性耦合的偏微分方程，是模拟地表和地下区域内流动现象的重要基础。这些方程的应用扩展到工程和科学领域的几个领域，包括开尔文尾流的模拟，涉及大坝溃坝的场景，洪水波的传播和流过凸起的水流。本文将表面流区的浅水方程和地下流区的Boussinesq偏微分方程统一为一组适用于两个流域的扩展浅水非线性偏微分方程。采用改进的六阶精度的两步Lax-Wendroff方法对这些方程进行了数值求解。通过对已有实验数据的基准测试，验证了该方法的准确性和可靠性，误差小于1%。此外，还将该方法应用于两个测试用例。第一种情况是通过凸起的流动，在t = 100时，表面和地下流动区域的相对误差低至10−8。在第二种情况下，采用改进的六阶Lax-Wendroff格式研究开尔文尾流，预测臂角为20.6955°，而理论粒子角为19.47°。结果表明，该方法可以很好地模拟地表和地下流动现象。

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引用次数: 0

Dynamics and modeling of Malaria disease with vector mortality rate and host transmission by using Piecewise Fractional Operator 基于分段分数算子的疟疾病媒死亡率和宿主传播动力学与建模

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-16 DOI: 10.1016/j.padiff.2025.101309

Muhammad Farman , Saba Jamil , Evren Hincal , Ali Akgul , Muhammad Umer Saleem , Dumitru Baleanu

Malaria remains one of the most persistent vector-borne diseases, requiring advanced modeling tools to capture its complex transmission dynamics. This study introduces a novel piecewise Caputo fractional operator with a singular kernel to model malaria transmission between humans and mosquito vectors. The proposed operator allows the system to switch between different fractional dynamics across subintervals, thereby capturing sudden behavioral or environmental changes an ability not present in traditional fractional- or integer-order models. The existence and uniqueness of the systems solutions are rigorously established using the Arzelà–Ascoli and Schauder fixed-point theorems, ensuring mathematical validity. The basic reproduction number is derived via the next-generation matrix approach and analyzed through sensitivity indices to identify key epidemiological parameters influencing transmission. Furthermore, the generalized Ulam–Hyers stability confirms robustness under small perturbations. Numerical simulations based on the Newton polynomial scheme reveal crossover behavior between subintervals and demonstrate that lower fractional orders intensify memory effects, leading to delayed but more stable epidemic responses. Overall, the piecewise Caputo framework enhances the modeling of malaria dynamics by integrating memory-dependent and regime-switching properties, offering a more biologically realistic approach to designing and timing intervention strategies.

疟疾仍然是最顽固的病媒传播疾病之一，需要先进的建模工具来捕捉其复杂的传播动态。本文引入了一种新颖的带奇异核的分段Caputo分数算子来模拟人与蚊子媒介之间的疟疾传播。所提出的算子允许系统在子区间的不同分数动态之间切换，从而捕获突然的行为或环境变化，这是传统分数阶或整数阶模型所不具备的能力。利用Arzelà-Ascoli和Schauder不动点定理严格地建立了系统解的存在唯一性，保证了系统解的数学有效性。通过新一代矩阵法导出基本繁殖数，并通过敏感性指数进行分析，确定影响传播的关键流行病学参数。此外，广义Ulam-Hyers稳定性证实了小扰动下的鲁棒性。基于牛顿多项式格式的数值模拟揭示了子区间之间的交叉行为，并表明较低分数阶强化了记忆效应，导致延迟但更稳定的流行病响应。总的来说，分段Caputo框架通过整合记忆依赖和状态切换特性增强了疟疾动力学的建模，为设计和定时干预策略提供了一种更现实的生物学方法。

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引用次数: 0

On nonlinear ordinary differential system for infectious disease: A neuro-swarming intelligence scheme 传染病的非线性常微分系统：一种神经群智能方案

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-16 DOI: 10.1016/j.padiff.2025.101317

Farhad Muhammad Riaz , Junaid Ali Khan , Khalil Ur Rehman , Wasfi Shatanawi

It is believed that nonlinear ordinary differential systems are essential for epidemic modeling because they can narrate the complex and unpredictable aspects of the spread of infectious diseases. Therefore, one can use nonlinear differential systems to propose control measures for the spread of diseases, and it remains a challenging task for researchers to obtain the best solution for such nonlinear systems. The present article offers the best solution remedy for nonlinear differential systems. To be more specific, to address the nonlinear dynamics of the spread of COVID-19, we propose an intelligent computational framework based on single-layer feed-forward artificial neural networks (FF-ANNs) and the optimization techniques of global and local search approaches. The SEIR-NDC model is solved by using a global-local search strategy called PSONM, which combines Particle Swarm Optimization (PSO) and the Nelder-Mead Simplex (NM). The differential nonlinear mathematical model based on SEIR-NDC and initial conditions is used in the hybrid PSONM combination to optimize an error-based fitness function. Ten neurons are used to demonstrate the numerical performance of the SEIR-NDC nonlinear model using ANN methods in conjunction with PSO-SQP. The correctness of the developed scheme is testified through the comparative analysis of the reference solution and the obtained outcomes. The absolute error performances are reported within appropriate ranges for every class of the SEIR-NDC model. It is found that the AE lies in the range 10^–14 and 10^–17. The statistical analysis is presented to verify the developed scheme's stability, convergence, and robustness. The statistical measures, i.e., mean square error falls between 10^–8 and 10^–13, while the mean absolute deviation falls between 10^–8 and 10^–11. We believe that the outcomes of the present analysis will be a helping hand in encountering nonlinear differential systems that are subject to practical applications.

人们认为，非线性常微分系统对于流行病建模是必不可少的，因为它们可以描述传染病传播的复杂和不可预测的方面。因此，人们可以利用非线性微分系统来提出控制疾病传播的措施，但如何获得这种非线性系统的最佳解仍然是研究人员面临的一个挑战。本文给出了非线性微分系统的最佳解补救方法。具体而言，为了解决COVID-19传播的非线性动力学问题，我们提出了一种基于单层前馈人工神经网络（ff - ann）和全局和局部搜索方法优化技术的智能计算框架。SEIR-NDC模型采用粒子群算法（PSO）和Nelder-Mead单纯形算法（NM）相结合的全局局部搜索策略PSONM进行求解。将基于SEIR-NDC和初始条件的微分非线性数学模型应用于混合PSONM组合中，对基于误差的适应度函数进行优化。采用10个神经元对SEIR-NDC非线性模型进行了数值模拟，并结合PSO-SQP方法对模型进行了仿真。通过对参考解和所得结果的对比分析，验证了所提方案的正确性。在适当的范围内报告了每一类SEIR-NDC模型的绝对误差性能。发现声发射分布在10-14和10-17范围内。通过统计分析验证了所提方案的稳定性、收敛性和鲁棒性。统计度量，即均方误差在10-8 ~ 10-13之间，平均绝对偏差在10-8 ~ 10-11之间。我们相信，本分析的结果将有助于遇到实际应用的非线性微分系统。

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引用次数: 0

Buoyancy-induced nanofluid circulation in a novel configuration of a porous square cavity 一种新型多孔方腔结构中浮力诱导的纳米流体循环

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-14 DOI: 10.1016/j.padiff.2025.101315

Muhammad Faisal , Talha Anwar , Farah Javed

Efficient thermal management is vital in modern mechanical and energy systems, where conventional engine oils often exhibit limited heat transfer capabilities. This study investigates the enhancement of thermal convection in engine oil by dispersing molybdenum tetrasulfide nanoparticles (MoS₄) to form a high-performance nanofluid. The natural convection behavior of this nanofluid is analyzed within a square porous cavity featuring uniformly heated horizontal walls and isothermally cooled vertical walls. The governing equations are developed using scaling variables and the Boussinesq approximation and solved numerically through the finite element method. The effects of nanoparticle volume fraction (0–0.07), Rayleigh number (10³–10⁶), and Darcy number (10⁻⁵–10⁻²) are systematically examined. Results show that increasing the MoS₄ nanoparticle concentration substantially enhances convective heat transfer, with the average Nusselt number rising by up to 28 % and the peak stream function reaching 17.0 at a volume fraction of 0.07 under low Darcy and Rayleigh conditions. These findings demonstrate that even minimal nanoparticle addition can significantly improve the heat transport capability of engine oils in porous enclosures. The study introduces a novel combination of molybdenum tetrasulfide-based nanofluids and porous media analysis, extending beyond prior work by quantifying the coupled effects of nanoparticle concentration and porous resistance on buoyancy-driven flow performance.

在现代机械和能源系统中，高效的热管理是至关重要的，传统的发动机油通常表现出有限的传热能力。本研究通过分散四硫化钼纳米颗粒（MoS₄）形成高性能纳米流体来增强机油中的热对流。在具有均匀加热的水平壁面和等温冷却的垂直壁面的方形多孔腔内，分析了这种纳米流体的自然对流行为。利用尺度变量和Boussinesq近似建立了控制方程，并通过有限元方法进行了数值求解。系统地考察了纳米颗粒体积分数（0-0.07）、瑞利数（103-10⁶）和达西数（10 -10⁻2）的影响。结果表明：在低达西和瑞利条件下，当体积分数为0.07时，mos4纳米颗粒浓度的增加显著增强了对流换热，平均Nusselt数提高了28%，峰值流函数达到17.0；这些发现表明，即使是最小的纳米颗粒添加也可以显著提高多孔外壳中发动机油的传热能力。该研究引入了基于四硫化钼的纳米流体和多孔介质分析的新组合，通过量化纳米颗粒浓度和多孔阻力对浮力驱动流动性能的耦合影响，扩展了之前的工作。

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引用次数: 0

Numerical exploration for bioconvective nanofluid flow towards a rotating surface with chemical reaction and radiative effects 具有化学反应和辐射效应的旋转表面生物对流纳米流体流动的数值探索

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-14 DOI: 10.1016/j.padiff.2025.101310

Tooba Sadaf , Ali B.M. Ali , Sami Ullah Khan , M. Ijaz Khan , Nidhal Ben Khedher

This investigation explored the bioconvection applications in rotatory disk nanofluid flow with implementation of magnetic field. The heat transfer analysis involved the significance of radiated effects while chemical reactive species are utilized to the concentration equation. The investigation accounts the convective thermal constraints to analyze the heat transfer impact. The problem is simplified by using the appropriate variables and set of dimensionless equations has been obtained. For solution methodology, shooting technique is adopted. A detailed physical analysis is performed in view of modeled flow parameters. It has been observed that azimuthal velocity component increases due to ratio of stretching to rotation parameter. Change in ratio of stretching to rotation parameter enhances declines the temperature profile.

本研究探讨了磁场作用下生物对流在旋转圆盘纳米流体流动中的应用。传热分析中考虑了辐射效应的重要性，而浓度方程则采用了化学反应物质。研究考虑了对流热约束来分析换热影响。采用适当的变量对问题进行了简化，得到了一组无因次方程。求解方法采用射击法。针对模型流动参数进行了详细的物理分析。观察到，由于拉伸与旋转参数之比，方位角速度分量增大。拉伸与旋转参数之比的变化增强了温度分布。

引用次数: 0

Hemodynamic analysis of Jeffrey blood flow with two-layered model through a multiple stenoses in a diverging narrow channel with a porous layer under slip conditions 滑移条件下多孔层发散窄通道内多个狭窄通道的双层Jeffrey血流动力学分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-06 DOI: 10.1016/j.padiff.2025.101307

K. Rajyalakshmi, G. Ravi Kiran, N. Lavanya

This study provides an analytical examination of the hemodynamic characteristics of two-layered blood flow in a diverging narrow channel featuring multiple symmetrical stenoses, porous wall effects, and slip boundary conditions. The central region, characterized by a concentration of RBCs, is modeled as a Jeffrey fluid, whereas the peripheral region is considered Newtonian. Under the assumption of mild stenosis and incompressible, completely developed laminar movement, the governing equations are precisely formulated and solved through direct integration. Closed-form expressions for velocity, mean hematocrit, core hematocrit and effective viscosity have been obtained. Parametric analysis indicates that velocity escalates with the Jeffrey parameter and slip, whereas effective viscosity diminishes with elevated Jeffrey parameter and Darcy number values, but augments with slip and stenosis height. The core and mean hematocrit diminish with most parameter variations, yet increase with the Jeffrey parameter. These findings improve comprehension of pathological conditions such as arterial occlusions and illustrate microcirculatory effects, including the Fåhraeus–Lindqvist phenomenon. The integrated modeling framework enhances physiological relevance and facilitates biomedical applications in the diagnosis and treatment of vascular diseases.

本研究分析了具有多重对称狭窄、多孔壁效应和滑移边界条件的发散狭窄通道中两层血流的血流动力学特征。以红细胞浓度为特征的中心区域被建模为杰弗里流体，而外围区域被认为是牛顿流体。在轻度狭窄、不可压缩、层流运动完全发展的假设下，精确地建立了控制方程，并采用直接积分法求解。得到了流速、平均红细胞压积、核心红细胞压积和有效粘度的封闭表达式。参数分析表明，速度随杰弗里参数和滑移量的增大而增大，有效粘度随杰弗里参数和达西数值的增大而减小，但随滑移和狭窄高度的增大而增大。核心和平均红细胞压积随大多数参数的变化而减小，随Jeffrey参数的变化而增大。这些发现提高了对动脉闭塞等病理条件的理解，并说明了微循环效应，包括fastraeus - lindqvist现象。集成的建模框架增强了生理相关性，促进了血管疾病诊断和治疗的生物医学应用。

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引用次数: 0

Optimal design problem with thermal radiation 热辐射优化设计问题

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-03 DOI: 10.1016/j.padiff.2025.101304

Kosuke Kita , Kei Matsushima , Tomoyuki Oka

This paper is concerned with configurations of two-material thermal conductors that minimize the Dirichlet energy for steady-state diffusion equations with nonlinear boundary conditions described mainly by maximal monotone operators. To find such configurations, a homogenization theorem will be proved and applied to an existence theorem for minimizers of a relaxation problem whose minimum value is equivalent to an original design problem. As a typical example of nonlinear boundary conditions, thermal radiation boundary conditions will be the focus, and then the sensitivity of the Dirichlet energy will be derived, which is used to estimate the minimum value. Since optimal configurations of the relaxation problem involve the so-called grayscale domains that do not make sense in general, a perimeter constraint problem via the positive part of the level set function will be introduced as an approximation problem to avoid such domains, and moreover, the existence theorem for minimizers of the perimeter constraint problem will be proved. In particular, it will also be proved that the limit of minimizers for the approximation problem becomes that of the relaxation problem in a specific case, and then candidates for minimizers of the approximation problem will be constructed by employing a nonlinear diffusion-based level set method. In this paper, it will be shown that optimized configurations deeply depend on force terms as a characteristic of nonlinear problems and will also be applied to real physical problems.

本文研究了主要由极大单调算子描述的非线性扩散方程中使Dirichlet能量最小的双材料热导体的构型。为了找到这样的构型，我们将证明齐次化定理，并将其应用于最小值相当于原始设计问题的松弛问题的最小值的存在性定理。作为非线性边界条件的典型例子，以热辐射边界条件为重点，推导狄利克雷能量的灵敏度，并以此估计最小值。由于松弛问题的最优构型涉及所谓的灰度域，通常没有意义，因此将通过水平集函数的正部分引入周长约束问题作为近似问题来避免这些域，并且证明了周长约束问题的最小化存在性定理。特别地，还将证明在特定情况下，逼近问题的极小值极限会变成松弛问题的极小值极限，然后利用基于非线性扩散的水平集方法构造逼近问题的极小值候点。本文将证明优化构型作为非线性问题的一个特征深深地依赖于力项，并将应用于实际的物理问题。

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引用次数: 0

Analyzing dynamics of a heterogeneous reaction convection diffusion COVID-19 model with vaccination effects 考虑疫苗接种效应的非均质反应对流扩散COVID-19模型动力学分析

Q1 Mathematics

Partial Differential Equations in Applied Mathematics

Pub Date : 2025-10-01 DOI: 10.1016/j.padiff.2025.101308

Tasmia Hoque , Samir Kumar Bhowmik

Mathematical models are fundamental tools for understanding the dynamics of infectious disease transmission and for guiding effective control strategies. In this study, we extend existing COVID-19 models by incorporating a risk-dependent (variable) vaccination policy, heterogeneity in individual susceptibility, and spatial diffusion effects. The model is formulated through a system that combines ordinary and partial differential operators, allowing us to capture both population-level dynamics and spatial variability. Specifically, we introduce vaccination rates that vary with individual risk, reflecting real-world prioritization strategies where highly vulnerable groups are targeted first. This extension provides a more realistic representation of epidemic control measures and allows the study of how different vaccination efforts alter disease trajectories. Numerical simulations demonstrate that risk-based vaccination strategies significantly influence epidemic patterns, including the emergence of rebounds, shoulders, and oscillations in infection prevalence. Our findings highlight the critical role of variable vaccination, heterogeneous risk structures, and spatial diffusion in shaping epidemic outcomes. They also provide insights into how adaptive and risk-sensitive vaccination strategies can mitigate transmission more effectively under realistic conditions of variability.

数学模型是理解传染病传播动力学和指导有效控制战略的基本工具。在本研究中，我们通过纳入风险依赖（可变）疫苗接种政策、个体易感性异质性和空间扩散效应，扩展了现有的COVID-19模型。该模型是通过一个结合了普通和偏微分算子的系统制定的，使我们能够捕获种群水平的动态和空间变异性。具体而言，我们引入了因个体风险而异的疫苗接种率，反映了现实世界的优先战略，即首先针对高度脆弱群体。这个扩展提供了流行病控制措施的更现实的表示，并允许不同的疫苗接种工作如何改变疾病轨迹的研究。数值模拟表明，基于风险的疫苗接种策略显著影响流行病模式，包括感染流行率的反弹、肩部和振荡的出现。我们的研究结果强调了可变疫苗接种、异质风险结构和空间扩散在形成流行病结果中的关键作用。它们还提供了关于适应性和风险敏感型疫苗接种策略如何在变异性的现实条件下更有效地减轻传播的见解。

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引用次数: 0