One of the most important crops in Mexico, in economic terms, is avocado (Persea americana). In recent years, the number of hectares planted with this crop has increased considerably (231,028 ha) (^{SIAP, 2018}); the avocado production in the country is focused mainly in the state of Michoacán, which is known as the main producing and exporting region in the country, followed by the states of Jalisco, Nayarit and the State of Mexico, the latter having a production focused mainly on the municipalities of Coatepec Harinas, Temascaltepec, Tenancingo, Villa de Allende, Donato Guerra, and recently, Ocuilan (^{SENASICA, 2019}).

Avocado is one of the products that Mexico exports to the rest of the world, and the international commercial demand for this fruit is growing every day, which requires high quality standards in production. Phytosanitary management and control aspects have become one of the main concerns for the avocado production sector, since pests and diseases found in orchards have limited the quality of the fruits and their commercialization (^{Orozco et al., 2017}). One of the phytosanitary problems that limit avocado production is anthracnosis, one of the most frequent diseases, particularly in post-harvest (^{Maeda, 2014}). The symptoms of this disease are caused by species of phytopathogenic fungi of the deuteromycetes or fungi imperfecti, and mostly by those belonging to the genus Colletotrichum (anamorphic), such as C. hymenocallidis and C. siamense, which have recently been known to cause anthracnosis in avocado crops (^{Trinidad, 2017}). However, the species considered to be highly pathogenic for this crop is Colletotrichum gloeosporioides (^{Rojo-Báez et al., 2017}).

Colletotrichum gloesosporiodes has been widely documented as a phytopathogen that remains in a state of latency in avocado orchards, waiting for adequate conditions for its development and dispersal, such as a relative humidity above 90% and a temperature below 29 °C (^{Basulto et al., 2011}). Some of the characteristic symptoms that appear on leaves, flowers and fruits are usually wilted tips, clove or smallpox, measles, and others, producing, in turn, irregular, salmon-colored spots that later turn grayish, maroon or black due to the presence of apressories, acervuli and conidia (^{Maeda, 2014}).

^{Kermack and McKendric (1927}) are known in science for their significant contributions to epidemiology, since their work has led to an understanding of the dynamics of the diseases, developing a system of mathematical equations with a reach that is currently applied in different fields of research, including agriculture, where these mathematical bases help understand, compare and predict the spatial and temporary behavior of plant diseases (^{Torres et al., 2010}).

Authors such as ^{Breilh (2007}) refer to epidemiological studies related to the importance of agricultural activity in diverse ecosystems, as well as their negative impacts. Likewise, ^{Monsalve (2013}) refers to the fact that in agriculture it is highly important to know and control the spatial variability that appears in most of the phenomena observed, and therefore the search for new alternatives of spatial modelling becomes necessary. Due to this, methodologies derived from spatial statistics, mainly Geostatistics, have been widely used in the analysis of agricultural pest and disease distribution, as well as their interaction with the environment and other abiotic variables, considering the geographic space as “a set of locations covered by diseased, healthy, exposed or removed plants” where it is crucial to understand the spatial dynamics they follow (^{Riley et al., 2016}). Due to the importance of the avocado crop and anthracnosis for the State of Mexico, the aim of this work was to analyze their spatial distribution in four producing municipalities, using tools derived from spatial statistics, i.e., Geostatistics.

The study was carried out in Coatepec Harinas, Tenancingo, Donato Guerra and Temascaltepec, considering 200 avocado trees of the Hass cultivar in each municipality, all of which were selected at random and georeferenced using a Trimble brand GPS. One criterion followed for the selection of these trees included the age range, which fluctuated between 5 and 10 years. In addition, we verified that the agronomic management was carried out in a similar fashion in the four municipalities, observing that the owners of the trees only carried out culture controls occasionally.

Likewise, out of each georeferenced tree, 12 branches were considered, four in each of the strata of the trees (upper, middle and lower), and in turn, 12 fruits with symptoms of anthracnosis were considered (four for every stratum of the trees), in which the incidence was obtained by taking 0 to 12 diseased fruits per tree as a reference (^{Ávila et al., 2004}). It is worth mentioning that in sampled plots, avocado is planted only for local commerce, and its agricultural management is minimum, with a traditional control of pests and diseases, as mentioned above. Samples were taken fortnightly between July and December of the year 2018, for a total of 48 samplings (modified from ^{Rivera et al., 2018}).

To identify the causal agent, samples were taken from infected fruits, then taken to the Phytopathology Lab of the Centro de Investigación y Estudios Avanzados en Fitomejoramiento de la Facultad de Ciencias Agrícolas of the UAEM for the identification and observation of Colletotrichum gloeosporioides, which were placed in wet chambers and placed in an incubator at 24 °C. After 7 days, the samples were checked and diseased tissue was planted in a potato-dextrose-agar (PDA) medium, following the methodology by ^{Morales and Ángel (2007}), performing a morphological characterization of monoconidial strains, which were replanted in PDA and placed in an incubator at 24 °C.

The characteristics considered for the morphological identification of the strains were color, the consistency and the type of myceliar growth, direction and length of hyphae on the edge of the culture, color and shape of culture, and the presence of concentric rings. It is worth mentioning that this procedure was carried out only to corroborate that this was the fungal species reported in literature as the causal agent of anthracnosis for avocado in Mexico (^{Rojo-Báez et al., 2017}).

In the samplings performed, we were able to identify symptoms of anthracnosis in trees from all four municipalities, therefore it is convenient to infer that this disease has a wide distribution. The symptoms found in the trees ranged from small dark spots on the pericarp of the fruits to larger, maroon to dark brown lesions, in which small acervuli-conidia-forming structures could be noticed at plain sight. Additionally, larger necrotic lesions with irregular and sunken edges were noticed, coinciding largely with reports by ^{Morales and Ángel (2007}), and by ^{Aquino et al. (2008}), who mention the presence of black, sunken, irregularly-shaped lesions with salmon-colored spores (conidia) on the fruits.

On the other hand, it is worth pointing out that anthracnosis symptoms appeared in the first stages of fruit development, confirming the proposal by ^{Juárez et al. (2010}), who states that the infections take place in the early growth stages, although the lesions typical of the disease appear when the fruits reach the stage of maturity.

After integrating the data bases obtained in the samplings of avocado fruits with anthracnosis, the geostatistical analysis was carried out, starting with the estimation of the experimental semivariograms from the incidence obtained in the samplings, using the software Variowin 2.2 (^{Software for the analysis of spatial data in 2D. Primavara Verlag, New York; U.S.A}.) (^{Maldonado et al., 2017}). The experimental semivariograms were adjusted to theoretical models, which tend to be spherical, gaussian, exponential, with a pure nugget effect, logarithmical, monomic and with a hole effect, where the plateau, range and effect parameters and nugget effect are modified until validation statistics are obtained which are adequate and help mathematically approve the adjusted models, such as the Standard Error of the Mean (SEM), Mean Squared Error (MSE) and the Dimensionless Mean Squared Error (DMSE) (^{Ramírez, 2012}; ^{Acosta et al., 2018}).

The experimental value of the semivariogram was calculated using the following expression (^{Isaaks and Srivastava, 1989}; ^{Journel and Huijbregts, 1978}):

γ*(h) = 12N(h)SN(h)i - 1[z(xi+h) z(xi) ]²

Where γ*(h) is the experimental value of the semivariogram for the distance interval h, N(h) is the number of pairs of sample points separated by the distance interval h, z (xi) is the value of the variable of interest for the sample point xi, and z(xi+h) is the value of the variable of interest in the sample point xi+h.

This work included the production of 48 semivariograms, which show that anthracnosis in the avocado crops presented a spatial distribution of an aggregate sort, adjusting to gaussian and exponential models in the four municipalities, although spherical models also appeared in several sampling dates (Table 1), therefore interpreting that anthracnosis presents accelerated growth in time; it is possible that it remains constant and with a tendency to increase throughout the surface of the area of study. ^{Quiñones et al. (2016}) mention that the gaussian models may help explain the variability and capacity of the diseases to spread, and hence this study explains that the spatial continuity of anthracnosis was explained with the radial extension it covered, i.e., that the appearance of the disease in a tree led to the infection of adjacent trees.

This aggregation of the disease became particularly evident in the months of August, September, October, November and December, when changes in rainfalls, temperature and humidity were variable. However, in the remaining samplings, the disease remained constant in the sampled areas, mainly in Coatepec Harinas. This helps infer the existence of favorable weather conditions (ideal temperature and humidity, physiographical differences that allow for the accumulation of rain or irrigation water, etc.). Likewise, ^{Fisher et al. (2012}), mention that the high incidences of diseases such as anthracnosis are widely related to the resistance of phytopathogens such as C. gloeosporioides.

Regarding the adjusted parameters that helped validate the models, it is appropriate to mention the importance of the range, since it lies in explaining the distance at which there is association between the data sampled. The nugget effect represents the origin of the semivariogram, while the plateau is the highest point in which the data intersect; the level of spatial dependence was obtained by dividing the nugget effect between the plateau, explaining the result as a percentage (^{Ramírez, 2012}). In this way, the values of the reach of the disease for the exponential, spherical and gaussian samplings displayed values between 19.2 m and 28 m (Table 1).

The nugget effect for all the models adjusted was equal to zero (Table 1), which, according to ^{Twizeyimana et al. (2008}), can be interpreted as a high level of aggregation of the incidence of the disease. This study created maps through kriging, to estimate the percentages of the surface infected and the values related to non-sampled points. In this way, it was possible to view the spatial distribution of the disease for each municipality and sampling date. These maps were obtained using the software Surfer 9 (^{Surface Mapping System, Golden Software Inc. 809, 14th Street. Golden, Colorado 80401-1866. U.S.A.}).

The 48 maps created clearly show points or centers of aggregation, which remained constant from beginning to end of the sampling period. This indicates that the disease is present in these areas, and is latent and constant, waiting for favorable conditions that help it proliferate. These infection patches in the map are assumed to be the main sources of infection from which the disease originates, and from where it is distributed to the entire sampled area (Figures 1 A, B, C, D). In regard to this, ^{Cárdenas et al. (2017}) mention that the semivariograms and the maps created with kriging help identify sources of infection of diseases. At the same time, they suggest that Geostatistics is a tool that helps explain the spatial arrangement followed by the diseases in plantations, thus contributing to accurate and timely decision-making processes, contributing to the creation of integrated management strategies.

The highest percentage of infected surface in the municipality of Coatepec Harinas was 100% in July, August, November and December, whereas in Donato Guerra, it was 100% only for the first sampling in November, although for the second sampling in the same month, the percentage of infection decreased by 21% in that municipality.

In the municipality of Temascaltepec, as in the municipalities mentioned above, percentages of 100% for the infection of the disease were also displayed. In July, October, November and December, the rest of the samplings, these percentages of infection also remained above 90%, with the exception of the last sampling, whose infection value was 86%. Likewise, in the municipality of Tenancingo, the percentages of infection by C. gloeosporioides remained between 87 and 100%, the latter percentage found in the second sampling of September and the first one in October.

Due to the above, in all the samplings performed in the four municipalities, the level of spatial dependence was high in all cases, suggesting that a correct sampling scale was used and the error was minimal (Table 1). This is also confirmed by the high spatial dependence displayed in all samplings, which proved the existing correlation between the data. Therefore, these results coincide with reports by ^{Quiñones et al. (2016}), who suggest that the high spatial dependence is an indicator of the relationship between the georeferenced data and the nature of the variable under study, considering the size, shape and configuration of the spatial units.

In conclusion, anthracnosis in avocado displayed an aggregate spatial behavior; we found clearly defined aggregation centers, which remained constant during the six months of sampling in all the areas of study. The spatial distribution was adjusted to gaussian, exponential and spherical models, which helped explain the spatial dependence of the anthracnosis found in the four municipalities. Geostatistics has proven to be one of the methodologies used in the agricultural sector which has been efficient for the analysis of spatial distributions of crop diseases, helping, in turn, to make adequate, pertinent and timely decisions on the integrated management. The results suggest carrying out preventive applications in the initial infection points, since it would have an effect on the change of the spatial patterns of the disease. Likewise, we propose applying curative fungicides at the appearance of the symptoms and on focus points in order to avoid the disease spreading onto the rest of the crop. Finally, this type of investigations contribute widely to minimizing both investment costs and the environmental impact produced by the use of agrochemicals in the avocado producing areas of the State of Mexico.

Figure 1. Anthracnose infection incidence maps in avocado fruits in the State of Mexico by municipality A) Coatepec Harinas, B) Donato Guerra, C) Temascaltepec and D) Tenancingo.