## 1 Introduction  
Soy is a grain used in the preparation of food in many families. It is cultivated in all Brazilian regions, and thus, it has become one of the country's most exported products in recent decades. Asian soybean rust is one of the diseases that most worry producers. If left unchecked, it can cause severe economic damage to legume crops.  
Asian soybean rust is a plant disease that has been present in Brazilian agribusiness for 20 years [1]. It causes early grain ripening, reducing crop productivity significantly. It is caused by the fungus _Plukopsora pachyrhizi_ (PP), which has soybeans as its principal host. The wind carries its spores, and volunteer soybeans on the sides of highways guarantee the disease's continuity from one harvest to the next [2, 3]. This research was dedicated to simulating the spatiotemporal dynamics of this fungus using a new approach.  
Transport models have evolved for airborne diseases with a spore dispersal component based on atmospheric emissions models that calculate the dispersal of suspended aerosols emitted by a point source. These spores are treated as polluting particles, except for the issue of spore viability. Thus, long-distance dispersion, like any physical-dynamic problem in which both advection and diffusion are dominant, can be modeled using an Eulerian diffusion-advection approach [4]. Eulerian diffusion-advection models assume that atmospheric diffusion is analogous to molecular diffusion. Furthermore, they comply with Fick's Law, which determines that the diffusion rate is proportional to the concentration gradient of the diffusing material [5].  
A diffusion-advection equation was constructed to estimate the fungus concentration, and researchers used a differential equation to model the behavior of pollutants. From this equation, [PERSON] (2008) [6] modeled the behavior of foot-and-mouth disease in cattle; [PERSON] (2009) and [PERSON] and [PERSON] (2011) [7; 8] simulated the evolution of pollutants in Lago Manso, in Mato Grosso, in two and three dimensions; and [PERSON] and [PERSON] (2007) [9] modeled the dispersion of pollutants in an air-water system. The equation involves a diffusive term, which refers to the dispersion capacity of the pollutant; a transport term, represented by the vector field that transfers the pollutant from one location in the system to another; the decay, which can be interpreted as a mortality rate; and the term source, which guarantees the production of this pollutant.  
The domain chosen for the construction of numerical approximations was the state of Mato Grosso, as it is the largest soybean producer in the country. The model was fed with data from the National Institute of Meteorology--INMET, and the results were compared with records of disease occurrences made available by the Antitrust Consortium.  
The text initially presents the problem's classical and variational modeling and the development of spatial, temporal, and domain discretizations. It also explains how the terms diffusive, source, transport, and decay were constructed, constituting the equation. In sequence, it explains how the parameters used were defined and presents the results of the numerical simulations. Finally, the impressions left regarding the solutions obtained are recorded.