*3.2. r*<sup>E</sup> = *r*<sup>I</sup> > 0

In Figure 5 the dynamics of states fractions *n* for various values of the neighborhood radius *r*<sup>E</sup> = *r*<sup>I</sup> are presented. We assume infection rates *p*<sup>E</sup> = *p*<sup>I</sup> = 0.005. The assumed transition rates *p*<sup>E</sup> and *p*<sup>I</sup> are very low. As a result, the disease has a very limited chance of spreading in society.

**Figure 5.** *Cont*.

**Figure 5.** Dynamics of states fractions for various values of the neighborhood radius *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> . *<sup>p</sup>*<sup>E</sup> = *<sup>p</sup>*<sup>I</sup> = 0.005, *<sup>R</sup>* = 10.

Figure 5a illustrates the situation where an infected person (independently either in the E or I states) can only infect the four closest neighbors. The epidemic lasted a maximum of forty days, the number of agents who were ill at the same time was less than one (on average, in ten simulations), there were only two deaths out of ten simulations, and the population was not affected by the disease.

The case presented in Figure 5b illustrates the situation in which each person can infect up to eight neighbors. This does not cause significant changes during the epidemic compared with Figure 5a; the average number of simultaneously ill agents remained below one, this time only one person in ten simulations died due to SARS-CoV-2 infection, and the duration of the epidemic was approximately 75 days—twice longer than presented in Figure 5a. We would like to emphasize that the term 'duration of the epidemic' determines the time of the longest duration of the epidemic among the ten simulations carried out.

Figure 5c shows the situation where there were twelve agents in the neighborhood of each cell. The transition rates (*p*<sup>E</sup> and *p*<sup>I</sup> ) turned out to be so low, that—despite extending the neighborhood—the epidemic vanished quickly. This time the fractions of exposed and infected agents were slightly higher, the maximum number of sick agents in one day was more than one, the longest simulation lasted 80 days, and four agents died within ten iterations.

Increasing the radius of the neighborhood to 2.5 (see Figure 5d) increased the number of exposed and infected agents more than twice compared to Figure 5c. On the day of the peak of the epidemic, five agents were sick and seven died during the epidemic (on average). The epidemic lasted about 600 days, but only about 1–2% of the population became infected throughout the epidemic.

Only an increase in the number of agents in the neighborhood to 24 (as presented in Figure 5e) caused a smooth and rapid development of the epidemic. In this case, it lasted about 550 days, the largest number of sick agents in one day was approximately 500, and the same number of agents also died throughout the epidemic. During the epidemic, around 75% of the population became infected.

In the left column of Figure 6 the dynamics of states fractions *n* for various values of the radius of the neighborhood *r*<sup>E</sup> = *r*<sup>I</sup> are presented. We assume *p*<sup>E</sup> = 0.03, *p*<sup>I</sup> = 0.02. As we still keep *r*<sup>I</sup> = *r*<sup>E</sup> , setting *p*<sup>E</sup> > *p*<sup>I</sup> simulates the fact that exposed agents (who are not aware of their infection) are more dangerous to those around them than those who know that they are sick, and therefore avoid contact if they show severe symptoms of the disease.

In Figure 6a the results for the smallest possible neighborhood radius *r* > 0 is presented. As in the previous cases with *r* = 1, the epidemic stopped quite quickly. The largest number of agents who were ill (*n*<sup>I</sup> *<sup>L</sup>*2) at the same time was two, the longest simulation time was 120 days, the average number of deaths was well below one, and totally only a few agents were infected (the curve representing agents in the state of R is almost not visible).

Figure 6d shows that contact with eight agents in the neighborhood (for assumed values of infection rates *p*<sup>E</sup> and *p*<sup>I</sup> ) is enough for the pandemic to affect society to a very large extent and last for a long time. The longest simulation took about 1000 days. During the epidemic, almost half of the population was infected, approximately 350 agents died, and at epidemic peak there were just over 180 sick agents in a single day.

Increasing the range of interaction to the next coordination zone (see Figure 6g) caused a significant reduction in the duration of the epidemic, as well as causing greater havoc among agents. In approximately 350 days, roughly 75% of the population was infected, approximately 500 died, and up to 800 were sick on the day when this number was highest.

Increasing the radius to 2.5, as shown in Figure 6j, further shortened the epidemic time, this time to just two hundred days. Almost all agents were infected, slightly more than 600 agents died, and the maximum number of patients on one day was 2000, which is as much as 20% of the population. Every fifth person was infected by the disease on that day, and if one adds about 800 agents who were in the exposed state E, we get a situation where more than one quarter of the population is under the influence of this disease at the same time.

Further increasing the range of interactions (to the fifth coordination zone and 24 neighbors in it, shown in Figure 6m) gave very similar results to the previous one, except that the virus spread even faster, in less than 150 days. The number of deaths was similar (a little below 600), almost all agents had contact with the disease at some stage of the pandemic, and the maximum number of sick agents in one day was 2800. If one adds over 1100 agents in the E state on the same day, the effect is that the disease affected nearly 40% of agents on the same day.

**Figure 6.** Dynamics of states fractions for various values of the neighborhood radius (**left**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> , (**middle**) *<sup>r</sup>*<sup>E</sup> ≥ *<sup>r</sup>*<sup>I</sup> = 1, (**right**) 3 = *<sup>r</sup>*<sup>E</sup> ≥ *<sup>r</sup>*<sup>I</sup> . *<sup>p</sup>*<sup>E</sup> = 0.03, *<sup>p</sup>*<sup>I</sup> = 0.02, *<sup>R</sup>* = 10.
