Kumar Arjun, Dubey Uma S, Dubey Balram
Department of Mathematics, BITS Pilani, Pilani Campus, Pilani 333031, Rajasthan, India.
Department of Biological Sciences, BITS Pilani, Pilani Campus, Pilani 333031, Rajasthan, India.
Chaos. 2025 Mar 1;35(3). doi: 10.1063/5.0251992.
This study introduces an epidemic model with a Beddington-DeAngelis-type incidence rate and Holling type II treatment rate. The Beddington-DeAngelis incidence rate is used to evaluate the effectiveness of inhibitory measures implemented by susceptible and infected individuals. Moreover, the choice of Holling type II treatment rate in our model aims to assess the impact of limited treatment facilities in the context of disease outbreaks. First, the well-posed nature of the model is analyzed, and then, we further investigated the local and global stability analysis along with bifurcation of co-dimensions 1 (transcritical, Hopf, saddle-node) and 2 (Bogdanov-Takens, generalized Hopf) for the system. Moreover, we incorporate a time-delayed model to investigate the effect of incubation delay on disease transmission. We provide a rigorous demonstration of the existence of chaos and establish the conditions that lead to chaotic dynamics and chaos control. Additionally, sensitivity analysis is performed using partial rank correlation coefficient and extended Fourier amplitude sensitivity test methods. Furthermore, we delve into optimal control strategies using Pontryagin's maximum principle and assess the influence of delays in state and control parameters on model dynamics. Again, a stochastic epidemic model is formulated and analyzed using a continuous-time Markov chain model for infectious propagation. Analytical estimation of the likelihood of disease extinction and the occurrence of an epidemic is conducted using the branching process approximation. The spatial system presents a comprehensive stability analysis and yielding criteria for Turing instability. Moreover, we have generated the noise-induced pattern to assess the effect of white noise in the populations. Additionally, a case study has been conducted to estimate the model parameters, utilizing COVID-19 data from Poland and HIV/AIDS data from India. Finally, all theoretical results are validated through numerical simulations. This article extensively explores various modeling techniques, like deterministic, stochastic, statistical, pattern formation(noise-induced), model fitting, and other modeling perspectives, highlighting the significance of the inhibitory effects exerted by susceptible and infected populations.
本研究介绍了一种具有Beddington-DeAngelis型发病率和Holling II型治疗率的流行病模型。Beddington-DeAngelis发病率用于评估易感个体和感染个体实施的抑制措施的有效性。此外，我们模型中选择Holling II型治疗率旨在评估疾病爆发情况下有限治疗设施的影响。首先，分析了模型的适定性，然后，我们进一步研究了该系统的局部和全局稳定性分析以及余维1（跨临界、霍普夫、鞍结）和余维2（博格达诺夫-塔肯斯、广义霍普夫）的分岔情况。此外，我们纳入了一个时滞模型来研究潜伏期延迟对疾病传播的影响。我们严格证明了混沌的存在，并建立了导致混沌动力学和混沌控制的条件。此外，使用偏秩相关系数和扩展傅里叶振幅灵敏度测试方法进行了灵敏度分析。此外，我们利用庞特里亚金极大值原理深入研究了最优控制策略，并评估了状态和控制参数延迟对模型动力学的影响。再次，使用连续时间马尔可夫链模型对传染病传播建立并分析了一个随机流行病模型。使用分支过程近似对疾病灭绝的可能性和疫情的发生进行了解析估计。空间系统给出了全面的稳定性分析和图灵不稳定性的产生准则。此外，我们生成了噪声诱导模式来评估白噪声在种群中的影响。此外，利用波兰的COVID-19数据和印度的艾滋病毒/艾滋病数据进行了案例研究以估计模型参数。最后，所有理论结果都通过数值模拟进行了验证。本文广泛探索了各种建模技术，如确定性、随机性、统计性、模式形成（噪声诱导）、模型拟合以及其他建模视角，突出了易感人群和感染人群所施加的抑制作用的重要性。
