Department of Mathematics, Institute of Numerical Sciences, Gomal University, Dera Ismail Khan, 29050, KPK, Pakistan.
Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, 83111-55181, Iran.
Sci Rep. 2023 Sep 15;13(1):15284. doi: 10.1038/s41598-023-42244-5.
Typhoid fever is a contagious disease that is generally caused by bacteria known as Salmonella typhi. This disease spreads through manure contamination of food or water and infects unprotected people. In this work, our focus is to numerically examine the dynamical behavior of a typhoid fever nonlinear mathematical model. To achieve our objective, we utilize a conditionally stable Runge-Kutta scheme of order 4 (RK-4) and an unconditionally stable non-standard finite difference (NSFD) scheme to better understand the dynamical behavior of the continuous model. The primary advantage of using the NSFD scheme to solve differential equations is its capacity to discretize the continuous model while upholding crucial dynamical properties like the solutions convergence to equilibria and its positivity for all finite step sizes. Additionally, the NSFD scheme does not only address the deficiencies of the RK-4 scheme, but also provides results that are consistent with the continuous system's solutions. Our numerical results demonstrate that RK-4 scheme is dynamically reliable only for lower step size and, consequently cannot exactly retain the important features of the original continuous model. The NSFD scheme, on the other hand, is a strong and efficient method that presents an accurate portrayal of the original model. The purpose of developing the NSFD scheme for differential equations is to make sure that it is dynamically consistent, which means to discretize the continuous model while keeping significant dynamical properties including the convergence of equilibria and positivity of solutions for all step sizes. The numerical simulation also indicates that all the dynamical characteristics of the continuous model are conserved by discrete NSFD scheme. The theoretical and numerical results in the current work can be engaged as a useful tool for tracking the occurrence of typhoid fever disease.
伤寒是一种传染病，通常由沙门氏菌 Typhi 引起。这种疾病通过粪便污染食物或水传播，感染未受保护的人。在这项工作中，我们的重点是数值研究伤寒非线性数学模型的动态行为。为了实现我们的目标，我们利用条件稳定的四阶龙格库塔（RK-4）和无条件稳定的非标准有限差分（NSFD）方案来更好地理解连续模型的动态行为。使用 NSFD 方案求解微分方程的主要优点是，它能够在离散连续模型的同时保持解收敛到平衡点的关键动态特性，并且对于所有有限步长都是正的。此外，NSFD 方案不仅解决了 RK-4 方案的缺陷，而且还提供了与连续系统解一致的结果。我们的数值结果表明，RK-4 方案仅在较小的步长下是动态可靠的，因此不能准确保留原始连续模型的重要特征。另一方面，NSFD 方案是一种强大而有效的方法，它准确地描绘了原始模型。开发微分方程的 NSFD 方案的目的是确保它具有动态一致性，即通过离散化连续模型来保持重要的动态特性，包括所有步长下平衡点的收敛性和解的正性。数值模拟还表明，离散 NSFD 方案保留了连续模型的所有动态特征。当前工作中的理论和数值结果可以作为跟踪伤寒疾病发生的有用工具。
