**3. Results**

In Figure 3 snapshots from a single-run simulation are presented. They give a quantitative picture of the influence of the interaction range (neighborhood radius) on the spread of the disease. The snapshots in Figure 3a–e show the situation for fixed parameters *p*<sup>E</sup> = 0.03 and *p*<sup>I</sup> = 0.02 at the *t* = 150 time step, which corresponds to five months after introducing (at random site) 'Patient Zero'. The last subfigure (Figure 3f) presents a situation after a very long time of simulations (*t* > 20,000) where the recovered agents die due to their age (according to Equation (5)) and are subsequently replaced by newly born children. For the interaction limited to the first coordination zone (Figure 1a) the disease propagation stays limited to the nearest neighbors of 'Patient Zero'. On the other hand, for the neighborhood with sites up to the fifth coordination zone (Figure 1e) for the same infection rates (*p*<sup>E</sup> = 0.03 and *p*<sup>I</sup> = 0.02) the disease affects all agents in the population (see Figure 3e). The direct evolution of the system based on [51] can be simulated and observed with the JavaScript application available at [52].

In Figure 4 the fraction *n*<sup>I</sup> of infected agents (in state I) is presented. The figure shows the results of ten different simulations for values of the neighborhood radius *r*<sup>E</sup> = *r*<sup>I</sup> = 1.5, *p*<sup>E</sup> = 0.03, *p*<sup>I</sup> = 0.02. In addition, the results of averaging over *R* = 10 simulations are presented. In two out of ten cases, the epidemic died out right after the start, while in the remaining eight cases it lasted from about eight hundred to over a thousand time steps (days). The figure also shows that the averaging of the results allows for a significant smoothing of the curves, which fluctuate strongly for individual simulations. Based on this test (for (not shown) roughly doubly large statistics, *R* = 25, which do not reveal significant deviations), we decided to average our results (presented in Figures 5–7) over ten independent simulations.

The diagrams in Figures 5 and 6 show the evolution of the epidemic. Namely, they show the number of agents in each state on each day of the epidemic, as well as the cumulative fraction *n*<sup>D</sup> of deaths (D). The fraction *n*<sup>S</sup> of susceptible agents and the fraction *n*<sup>R</sup> of recovered agents are shown on the left vertical axis, while the fractions *n*<sup>E</sup> of exposed agents and *n*<sup>I</sup> of infected agents and the cumulative fraction *n*<sup>D</sup> of deaths caused by infection are shown on the right vertical axis.

**Figure 3.** Snapshots from direct simulation [52] for *<sup>p</sup>*<sup>E</sup> = 0.03, *<sup>p</sup>*<sup>I</sup> = 0.02, *<sup>R</sup>* = 1. The assumed ranges of interactions are (**a**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 1, (**b**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 1.5, (**c**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 2, (**d**,**f**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 2.5, and (**e**) *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 3. The simulation took *<sup>t</sup>* = 150 time steps except for Figure 3f, where the situation after *t* > 20,000 time steps is presented.

**Figure 4.** Ten different simulations for values of the neighborhood radius *<sup>r</sup>*<sup>E</sup> = *<sup>r</sup>*<sup>I</sup> = 1.5. *<sup>p</sup>*<sup>E</sup> = 0.03, *<sup>p</sup>*<sup>I</sup> = 0.02.

$$3.1. \ r\_{\mathcal{E}} = r\_{\mathcal{T}} = 0$$

The case of *rE* = *rI* = 0 (corresponding to total lockdown) leads to immediate disease dieout as only *<sup>n</sup>*0*L*<sup>2</sup> 'Patients Zero' at *<sup>t</sup>* = 0 are infected and recover after about ∼1/*qC*(*a*) time steps (days) depending on the agent's age *a*.
