The conducted research showed that adopting four variants in the models of the activity of agents in the selected test powiats enabled the obtainment of statistically significant results. It also enabled the precise determination of the impact of the level of restrictions on the numbers of cases, hospitalizations, and deaths. By using a geographic information system and multiagent modeling in the modeling process and a detailed database of topographic objects, it was possible to simultaneously investigate when and where an infection occurs and determine the impact of spatial location and land cover on the development of a pandemic.

To ensure the credibility of the research, the base case model was calibrated independently for all three test powiats based on sanitary and epidemiological data of the number of cases on individual days. The proposed model has many parameters set individually for each powiat, for instance, housing and population density, the residents’ level of mobility, and the level of public transport use. Since each of these parameters may affect the accuracy of calculations in various ways, it was necessary to recalibrate the model for each powiat using source epidemiological data.

As an objective function during calibration, we assumed the minimization of the difference between the number of infected according to the real data and the model’s result after 1 month of simulation. During the calibration, the assumption was that the only variable of the objective function would be the probability coefficient of infection during agent interaction. The adopted method was the bisection algorithm.

For the Gołdap powiat, the accuracy (MPE error) was 1.64%; for Ropczyce-Sędziszów, the accuracy was 0.68%; and for Pruszków, the accuracy was 0.64%. Due to the model’s high computational complexity, the calibration process was nontrivial; thus, the researchers did not choose complete automation of the process. The probability coefficient of infection during agent contact was calibrated. Because of the nature of the model, this coefficient required only minor adjustments to calibrate to actual data. However, depending on the powiat, the calibration process took several days to 2 weeks. Tables 3, 4 and Figs. 5, 6, 7, 8 and 9 present the numerical results obtained by the multiagent model, while Figs. 6, 8 and 10 show the spatial distribution of the number of cases, their locations, and differences between individual models.

Table 3 Numerical results obtained in the multiagent model for the tested powiats.
Table 4 Numerical results obtained in the multiagent model for the selected powiats: place of infection.
Figure 5
figure 5

Number of symptomatic infected agents in the Ropczycko-Sędziszowski powiat.

Figure 6
figure 6

Spatial distribution of the number of cases, their locations, and the differences between individual models in the Ropczyce-Sędziszów powiat. (a) Ropczyce-Sędziszów powiat: land cover. (b) The number of cases in 4 models in the gminas in relation to the number of inhabitants of these gminas. (c) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): the base case model. (d) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): the no restrictions model. (e) The location of cases (black dots) and the density of cases by place of infection: the base case model. (f) The location of cases (black dots) and the density of cases by place of infection: the no restrictions model. (g) The Pearson correlation coefficient between the place of infection and the number of inhabitants in the area (white < 0.5, yellow < 0.75, red ≥ 0.75): the base case model. (h) The Pearson correlation coefficient between the place of infection and the number of inhabitants in the area (white < 0.5, yellow < 0.75, red ≥ 0.75): the no restrictions model (developed by the authors in QGIS ver. 3.22.5).

Figure 7
figure 7

Number of symptomatic infected agents in the Pruszków powiat.

Figure 8
figure 8

Spatial distributions of the number of cases, their locations, and differences between the individual models in the Pruszków powiat. (a) Pruszków powiat: land cover. (b) The number of cases in the 4 models in the gminas in relation to the number of inhabitants of these gminas. (c) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): Stringency Index + 20 model. (d) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): No restrictions model. (e) The location of cases (black dots) and the density of cases by place of infection: Stringency Index + 20 model. (f) The location of cases (black dots) and the density of cases by place of infection: No restrictions model. (g) The Pearson correlation coefficient between the place of infection (developed by the authors in QGIS ver. 3.22.5).

Figure 9
figure 9

Number of symptomatic infected agents in the Gołdap powiat.

Figure 10
figure 10

Spatial distributions of the numbers of cases, their locations, and the differences between the individual models in the Gołdap powiat. (a) Gołdap powiat: land cover. (b) The number of cases in the 4 models in the gminas in relation to the number of inhabitants of these gminas. (c) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): Stringency Index − 20 model. (d) The number of cases in a 1 × 1 km grid (pie chart) and the percentage of cases (intensity of red color): No restrictions model. (e) The location of cases (black dots) and the density of cases by the place of infection: Stringency Index − 20 model. (f) The location of cases (black dots) and the density of cases by place of infection: No restrictions model. (g) The Pearson correlation coefficient between place of infection and the number of inhabitants of the area (white < 0.5, yellow < 0.75, red ≥ 0.75): Stringency Index − 20 model. (h) The Pearson correlation coefficient between the place of infection and the number of inhabitants of the area (white < 0.5, yellow < 0.75, red ≥ 0.75): No restrictions model (developed by the authors in QGIS ver. 3.22.5).

It is worth noting that the process of multiagent modeling, which included tens of thousands of agents interacting with each other in a virtual topographic space with a level of detail (LoD) corresponding to analog maps at a 1:10,000 scale, was lengthy and computationally demanding. The calculations were performed in the CENAGIS computing cluster with 16 Intel (R) Xeon (R) Silver 4216 CPU @ 2.10 GHz processors with 128 GB RAM. The calculations for a single case for the base variant took 5 h 25 m for the Gołdap powiat, 10 h 34 m for the Ropczyce-Sędziszów powiat, and 33 h 45 m for the Pruszków powiat. It should be stressed that the spatial interpretation of the obtained results required complex SQL querying of the database with spatial operators. To highlight various aspects of the obtained results for individual models, the authors provide a summary for entire powiats, gminas constituting powiats, and 1 km2 units, as used in the official statistics. Additionally, to analyze the spatial relationship between individual parameters, e.g., the number of people living in a given region and the number of cases in this area, the authors used proprietary tools to determine the Pearson correlation coefficient in a moving (circular) window of a given size. The authors obtained discrete results (e.g., point information on the percentage of incidence in a 1 km2 area) and interpolated them to show a continuous statistical surface illustrating the spatial distribution of individual phenomena (Figs. 6, 8 and 10).

When analyzing the data in Table 3, it should be emphasized that due to the specificity of the multiagent model used, the analysis of “exposed” cases is of crucial importance. “Exposed” means contact with the potential to cause infection; some infected agents will develop symptoms characteristic of infection only after the incubation period (median incubation period of 5.1 days27). This is important because not only the moment when the symptoms arise but also the moment of infection are considered, and the duration between such events may be several to several dozen days. Naturally, the numbers of symptomatic infected, asymptomatic infected, hospitalized, and deceased agents are also essential for analysis. The results (Table above) indicate that changing the level of restrictions significantly affects the number of cases. Obviously, due to the different number of inhabitants in individual powiats, it is crucial to compare the values between particular models. In almost all cases, the incidence rate in the PLUS model is several percentage points lower than that in the base case. The opposite was observed for the MINUS model with a lower level of restrictions. The no restrictions model showed very significant differences, sometimes exceeding the reference value by threefold. The results show the importance of restrictions such as social distancing, remote learning, movement limitations, and mask use.

It is interesting to compare the spatial distribution of the “exposed” locations. While the analysis indicates the role of the level of restrictions, it also reveals the impact of topography, building density, recreational areas, quality of public transportation, and resident mobility. The three analyzed powiats represent individual regions of Poland and enable the determination of the impact of topographic factors on the course of the pandemic. As the table below (belowTable 4) shows, the inhabitants of the Gołdap powiat have a level of mobility close to zero, and there is almost no public transportation. As a result, the number of cases related to traveling does not exceed 0.3%. In the Pruszków powiat, where a approximately 12% of inhabitants commute to the capital every day (mainly by suburban railway), the level of infection connected to public transportation reaches 20.1% in the PLUS model. In the MINUS model for the Ropczyce-Sędziszów powiat, which has good connections to the city of Rzeszów, the percentage of cases related to public transportation exceeds 24%.

Workplaces account for the highest percentages in the number of cases:

  • 73.4% in the Gołdap powiat (the PLUS model),

  • 65.2% for the base case in the Pruszków powiat,

  • 66.2% in the Ropczyce-Sędziszów powiat (the PLUS model).

It should be emphasized that depending on the model, the number of people who become infected at their workplace differ significantly. Mobility restrictions and requisite remote work may increase the number of infections related to recreational areas (city parks and forests). In the Pruszków powiat, characterized by a relatively small area of parks, applying more significant restrictions (the PLUS model) causes a percentage decrease in the number of cases in particular places from 6% (buildings) to 75% (clinics), except for recreational facilities, which are associated with an increase in the number of cases (compared to the base case) by 45% (!). What is equally significant is the percentage increase in the number of cases related to commercial facilities in this powiat in the MINUS model (156% of the base case model) and the no restrictions model (970% of the reference value), demonstrating the vital role of shops and malls in spreading COVID-19 when there are no social distancing restrictions in place.

The conducted analysis also shows that infection occurs in residential buildings in nearly 10% of cases. Introducing restrictions leads to almost complete elimination of cases related to public health care facilities (0.1% in the PLUS model in the Pruszków powiat), with very low absolute values (5 people in the Ropczyce-Sędziszów powiat in the PLUS model and 13 people in the Pruszków powiat in the same analytical variant). Closing schools and transitioning to remote learning are also of great importance; lack of such restrictions resulted in 43 infected students in the Gołdap powiat, 47 in the Ropczyce-Sędziszów powiat, and 635 in the Pruszków powiat, with 0, 7, and 91 corresponding values in the base model and 0, 3, and 25 in the PLUS model, respectively.

The maps show the results of the no restrictions model and one of the models including the spatial distancing policy (Figs. 6, 8 and 10), illustrating the differences resulting from topographic or demographic differentiation and from adopting a specific restrictive policy for each of the three analyzed powiats. For the Ropczyce-Sędziszów powiat, the maps show the base case. For the Pruszków powiat, the maps show the Stringency Index + 20 (the PLUS variant). For the Gołdap powiat, the maps show the Stringency Index − 20 (the MINUS variant). Such an approach enables the analysis of the spatial differentiation of the development of the COVID-19 pandemic and the verification of research hypotheses indicating the crucial role of restrictive policies.

Ropczyce-Sędziszów powiat

The no restrictions model for this powiat had the highest increase in the total number of infections (326% compared to the base model) out of all the analyzed powiats. The greatest increase in the number of infections occurred in recreational areas (as high as 1594%). The bar chart in Fig. 6b shows the absolute values of the “exposed” agents in the individual models in relation to the number of inhabitants of individual gminas comprising the powiat. The values related to the no restrictions model are dominant; the bar sizes indicate two towns, Ropczyce and Sędziszów Małopolski, with the highest number of cases.

The maps in Fig. 6c,d show a different approach: the number of cases in individual 1 km2 units (the size of the pie chart) and the percentage of people who fell ill in a given areal unit in relation to the number of inhabitants of a given square. One should note that this analysis shows the number of infected people living in a given square unit, regardless of the place of infection. Comparison of the base case and the no restrictions models shows considerable differentiation in the disease prevalence and incidence. In the absence of restrictions, over 40% of the inhabitants who fell ill occupied approximately 1/3 of the powiat’s area. In the reference model (base case), the value exceeds 15% for only a dozen areal units. Black dots on the maps in Fig. 6e,f show where infections occurred, while the statistical surface layer indicates the number of cases in a given region, represented by varying intensities of red. This map also shows the primary role of areas with dense industrial or residential development in the progression of the pandemic. The maps (Fig. 6g,h) show a linear Pearson correlation between where infections occurred and the number of inhabitants in that area. The correlation coefficient value is calculated in a moving window with a radius of 2.5 km, which serves as a spatial filter. In a given areal unit, the values of the number of cases and the number of inhabitants in the individual squares of the official statistical grid are analyzed. The obtained point values (discrete) are then interpolated to a continuous statistical surface. White indicates no correlation, yellow indicates a weak linear correlation (Pearson’s correlation coefficient of 0.5), and red indicates a strong correlation (correlation coefficient > 0.75). The strongest spatial correlations occur in densely populated areas where many people fall ill. When analyzing the obtained results, the level of spatial generalization of the results should be considered; each dot on the map represents a value assigned to its circular surroundings with a radius of 2500 m (nearly 20 km2).

The map (Fig. 6) reveals interesting conclusions: in both models, infection cases in Sędziszów Małopolski are more concentrated, while infection cases in Ropczyce are more dispersed. One way to explain this is that Sędziszów Małopolski is smaller and less populated than Ropczyce, but the population density is double (Sędziszów Małopolski: 838 people/km2, Ropczyce: 336 people/km2), with a railway station in its center.

Pruszków powiat

The Pruszków powiat is inhabited by the largest number of people, with high labor mobility. A significant number of the inhabitants commute to work in neighboring areas (mainly Warsaw) via public transportation (Warsaw Commuter Railway, train); therefore, public transportation and work constitute the most significant infection sources. Additionally, in this powiat, the highest percentage increase in infections in all the analyzed models occurred further from the house (more than 3 km); accordingly, the proportion of people who become infected at home is lower than those in the other analyzed powiats. In the no restrictions model, the highest increase in infections was recorded in recreational areas (1219% of the base case model value), while in the MINUS model, the highest increase in infections was recorded in trade-related areas. On the other hand, in the PLUS model, despite a significant decrease in the number of infections compared to the base variant (by an average of 21.4%), there was an increase in the number of infections in recreational areas (45%). The reason is the increase in professional restrictions (remote work and learning) and the population’s willingness to visit open natural areas.

As seen in the maps in Fig. 8, in the case of Pruszków powiat, the authors present the results of the no restrictions model and the most restrictive PLUS model. The pie charts show that the highest number of cases occur in the main urban centers, Pruszków and Piastów, and more than half of the Pruszków powiat contains areas in which over 50% of residents will fall ill in the no restrictions model. Moreover, in the no restrictions model, regions with high infection densities are strongly correlated with places with high population densities (Brwinów, Raszyn, Nadarzyn, Michałowice). Notably, the largest numbers of railway stations are in the gminas of Pruszków, Piastów, and Brwinów. Interestingly, there were many infections in the former two gminas, while the latter gmina (Brwinów) had the smallest number of infections. Brwinów has a low degree of industrialization, and the existing industrial centers (mainly warehouses) are located outside the city.

Comparison of the maps in Fig. 8 indicates that the model in which the level of safety was increased by 20% in relation to the restrictions implemented in Poland (the PLUS variant) showed a significant reduction in the number of cases and complete elimination in areas with an incidence rate higher than 50%. The no restrictions model has a stronger correlation between the number of cases in a given area and the population density, which is evident in the eastern (Michałowice and Raszyn), western (Brwinów), and southern (Nadarzyn) parts of the powiat. In the PLUS model, the correlation is almost zero, while it exceeds 0.5 in the no restrictions model.

Gołdap powiat

The Gołdap powiat has the smallest population and the lowest level of resident mobility; most residents work on their own farms. Consequently, this powiat had the lowest total number of infections in all the analyzed scenarios. In relation to the base model, there is an increase in the number of infections in the no restrictions model by 204% (the smallest increase among all the analyzed powiats). Contrary to the other gminas, there is no significant increase in the number of infections in recreational areas (0.5%). The MINUS model shows the smallest increase in infections (8.3% in total) compared to the base model. On the other hand, only in the Gołdap powiat is the PLUS model characterized by a slight increase (0.4% on average) in infections (in the other powiats, the total number of infections in this model decrease).

The conducted analyses show that among the three gminas that make up the Gołdap powiat, a significant increase in the number of cases occurs mainly in the town of Gołdap (Fig. 10b). It should be emphasized that the comparison of the modeling results in Fig. 10 relates to the analysis of two models with low levels of restrictions: the Stringency Index -20 (MINUS) and the no restrictions models. In both models, the primary infection outbreaks occur in the powiat‘s capital, where infections occur at home, at work, in shops and in schools. Due to the agricultural nature of this powiat, characterized by scattered housing developments and low resident mobility, the overall number of cases is relatively low, even in the model without restrictions. However, the percentage of infections in some units of the 1 km2 statistical grid exceeds 40%, indicating significant roles of topography, scattered single-family housing, the level of economic development of the region, and the mobility of residents over the course of the pandemic.