A non-linear reaction–diffusion system with cross-diffusion describing the COVID-19 outbreak is studied using the Lie symmetry method. A complete Lie symmetry classification is derived and it is shown that the system with correctly specified parameters admits highly non-trivial Lie symmetry operators, which do not occur for all known reaction–diffusion systems. The symmetries obtained are also applied for finding exact solutions of the system in the most interesting case from applicability point of view. It is shown that the exact solutions derived possess typical properties for describing the pandemic spread under 1D approximation in space and lead to the distributions, which qualitatively correspond to the measured data of the COVID-19 spread in Ukraine.

使用李对称方法研究了描述 COVID-19 爆发的具有交叉扩散的非线性反应扩散系统。导出了完整的李对称分类，并表明具有正确指定参数的系统允许高度非平凡的李对称算子，这不会出现在所有已知的反应扩散系统中。从适用性的角度来看，获得的对称性也适用于在最有趣的情况下找到系统的精确解。结果表明，所导出的精确解具有描述大流行在空间中一维近似下传播的典型特性，并导致分布与乌克兰 COVID-19 传播的测量数据定性对应。