Department of Mathematics, School of Science, University of Management and Technology, Lahore, Pakistan.
Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.
Sci Rep. 2023 Sep 22;13(1):15840. doi: 10.1038/s41598-023-42953-x.
The terms susceptibility, exposure, infectiousness, and recovered all have some inherent ambiguity because different population members have different susceptibility levels, exposure levels, infectiousness levels, and recovery patterns. This uncertainty becomes more pronounced when examining population subgroups characterized by distinct behaviors, cultural norms, and varying degrees of resilience across different age brackets, thereby introducing the possibility of fluctuations. There is a need for more accurate models that take into account the various levels of susceptibility, exposure, infectiousness, and recovery of the individuals. A fuzzy SEIR model of the dynamics of the measles disease is discussed in this article. The rates of disease transmission and recovery are treated as fuzzy sets. Three distinct numerical approaches, the forward Euler, fourth-order Runge-Kutta, and nonstandard finite difference (NSFD) are employed for the resolution of this fuzzy SEIR model. Next, the outcomes of the three methods are examined. The results of the simulation demonstrate that the NSFD method adeptly portrays convergent solutions across various time step sizes. Conversely, the conventional Euler and RK-4 methods only exhibit positivity and convergence solutions when handling smaller step sizes. Even when considering larger step sizes, the NSFD method maintains its consistency, showcasing its efficacy. This demonstrates the NSFD technique's superior reliability when compared to the other two methods, while maintaining all essential aspects of a continuous dynamical system. Additionally, the results from numerical and simulation studies offer solid proof that the suggested NSFD technique is a reliable and effective tool for controlling these kinds of dynamical systems.The convergence and consistency analysis of the NSFD method are also studied.
术语“易感性”、“暴露”、“传染性”和“康复”都存在一定的固有模糊性，因为不同人群的易感性水平、暴露水平、传染性水平和康复模式都有所不同。当研究具有不同行为、文化规范和不同年龄段弹性程度的人群亚组时，这种不确定性变得更加明显，从而增加了波动的可能性。需要更准确的模型来考虑个体的不同易感性、暴露、传染性和康复水平。本文讨论了麻疹疾病动力学的模糊 SEIR 模型。疾病传播和恢复的速度被视为模糊集。采用三种不同的数值方法，即前向欧拉法、四阶龙格-库塔法和非标准有限差分法（NSFD）来求解这个模糊 SEIR 模型。然后，检查了这三种方法的结果。模拟结果表明，NSFD 方法能够在不同时间步长下准确地描绘出收敛解。相反，传统的欧拉法和 RK-4 方法只有在处理较小的步长时才表现出正定性和收敛性解。即使考虑较大的步长，NSFD 方法也能保持一致性，展示其有效性。这表明 NSFD 方法比其他两种方法更可靠，同时保持了连续动力系统的所有基本方面。此外，数值和模拟研究的结果为所提出的 NSFD 技术是控制这类动力系统的可靠有效的工具提供了确凿的证据。还研究了 NSFD 方法的收敛性和一致性分析。
