Analytical solutions are the most effective tools for understanding phenomena modeled with differential equations. On the other hand, analytical solvers may be limited in their ability to handle complex problems and we are forced to rely on approximate and numerical methods. Through the investigation of two types of fractional operators, this article aims to investigate both approaches to studying the interactive dynamics of an eco-epidemiological diffusion model. We consider two differential operators in our main model: local beta time and non-local Riesz space fractional derivative. Firstly, we propose novel classes of soliton wave solutions for the local time fractional version of the system using two efficient analytical methods. In the second part of the paper, we consider the case where the second-order space derivative operator in the problem is of the type of Riesz fractional operator that is non-local. A finite difference scheme is the main idea behind discretizing this fractional system. Throughout this article, we highlight the validity and efficiency of both analytical and numerical perspectives. The results can be examined by epidemiologists for a more comprehensive interpretation.

解析解是理解用微分方程建模的现象的最有效工具。另一方面，解析求解器处理复杂问题的能力可能有限，我们不得不依赖近似和数值方法。通过对两种类型的分数运算符的调查，本文旨在调查研究生态流行病学扩散模型的交互动力学的两种方法。我们在主模型中考虑了两个微分算子：局部 beta 时间和非局部 Riesz 空间分数阶导数。首先，我们使用两种有效的分析方法为系统的局部时间分数版本提出了新型孤子波解决方案。在论文的第二部分，我们考虑问题中的二阶空间导数算子是非局部的 Riesz 小数算子类型的情况。有限差分格式是离散化该分数系统的主要思想。在整篇文章中，我们强调了分析和数值角度的有效性和效率。流行病学家可以检查结果以获得更全面的解释。