**1. Introduction**

Currently, the death rate of SARS-CoV-2 [1] in the whole world reached around 1.2% of the infected population and more than 5 × <sup>10</sup><sup>8</sup> cases have been confirmed [2,3], (tab 'Closed Cases') . Thus, one should not be surprised of the publication rash in this subject giving (i) theoretical bases of SARS-CoV-2 spreading, (ii) practical tips on preventing plagues or even (iii) clinical case studies making it easier to recognize and to treat cases of the disease. The Web of Science database reveals over 80,000 and over 110,000 papers related to this topic registered in 2020 and in January–November 2021, respectively, in contrast to only 19 papers in 2019. Among them, only several [4–20] are based on the cellular automata technique [21–24].

The likely reason for this moderate interest in using this technique to simulate the spread of the COVID-19 pandemic is the large degree of simplification of 'rules of the game' in cellular automata. To fill this gap, in this work, we propose a cellular automaton based on a compartmental model, the parameters of which were adjusted to the realistic probabilities of the transitions between the states of the automaton. Let us note that modeling the spread of the pandemic is also possible with other models (see Refs. [25–28] for mini-reviews) including, for instance, those based on the percolation theory [29].

The history of the application of compartmental models to the mathematical modeling of infectious diseases dates to the first half of the 20th century and works of Ross [30], Ross and Hudson [31,32], Kermack and McKendrick [33,34] and Kendall [35]; see, for instance, Ref. [36] for an excellent review. In the compartmental model, the population is divided into several (usually labeled) compartments so that the agent only remains in one of them and the sequences of transitions between compartments (label changes) are defined. For instance, in the classical SIR model, agents change their states subsequently from *susceptible* (S) via *infected* (I) to *recovered* (R) [33,34,37]. Infected agents can transmit the disease

**Citation:** Biernacki, S.; Malarz, K. Does Social Distancing Matter for Infectious Disease Propagation? An SEIR Model and Gompertz Law Based Cellular Automaton. *Entropy* **2022**, *24*, 832. https://doi.org/ 10.3390/e24060832

Academic Editor: Adam Gadomski

Received: 25 March 2022 Accepted: 11 June 2022 Published: 15 June 2022

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

to their susceptible neighbors (S→I) with a given probability *p*1. The infected agent may recover (I→R) with probability *p*2. After recovering, the agents are immune and they can no longer be infected with the disease. These rules may be described by a set of differential equations:

$$\begin{split} \frac{dn\_{\mathcal{S}}}{dt} &= -\langle k \rangle p\_1 n\_{\mathcal{S}} n\_{\mathcal{T}'} \\ \frac{dn\_{\mathcal{T}}}{dt} &= \langle k \rangle p\_1 n\_{\mathcal{S}} n\_{\mathcal{T}} - p\_2 n\_{\mathcal{T}'} \\ \frac{dn\_{\mathcal{R}}}{dt} &= p\_2 n\_{\mathcal{T}'} \end{split} \tag{1}$$

where *k* is the mean number of agents' contacts in the neighborhood and *n*S, *n*<sup>I</sup> , *n*<sup>R</sup> represent the fraction of susceptible, infected, and recovered agents, respectively. Typically, the initial condition for Equation (1) is:

$$\begin{array}{l} n\_{\mathcal{S}}(t=0) = 1 - n\_{0\prime} \\ n\_{\mathcal{T}}(t=0) = n\_{0\prime} \\ n\_{\mathcal{R}}(t=0) = 0, \end{array} \tag{2}$$

where *n*<sup>0</sup> is the initial fraction of infected agents (fraction of 'Patients Zero').

The transition rates between states (i.e., probabilities *p*<sup>1</sup> and *p*2) may be chosen arbitrarily or they may correspond to the reciprocal of agents' residence times in selected states. In the latter case, residence time may be estimated by clinical observations [38,39]. The probability *p*<sup>1</sup> describes the speed of disease propagation (infecting rate) while the value of *p*<sup>2</sup> is responsible for the frequency of getting better (recovering rate). In this approximation, the dynamics of the infectious class depends on the reproduction ratio:

$$R\_0 = \frac{\langle k \rangle p\_1}{p\_2}.\tag{3}$$

The case of *R*<sup>0</sup> = 1 separates the phase when the disease dies out and the phase when the disease spreads among the members of the population.

Equation (1) describe a mean-field evolution, which simulates a situation in which all agents interact directly with each other. In low-dimensional spatial networks, the mean-field dynamics (1) is modified by diffusive mechanisms [40]. In a realistic situation, the diffusive mode of pandemic spreading is mixed with the mean-field dynamics, corresponding to nonlocal transmissions resulting from the mobility of agents [41].

We use the SEIR model [42,43], upon extending the SIR model, where an additional compartment (labeled E) is available and it corresponds to agents in the *exposed* state. The exposed agents are infected but unaware of it—they neither have symptoms of the disease nor have been diagnosed by appropriate tests. This additional state requires splitting the transition rate *p*<sup>1</sup> into *p*<sup>E</sup> and *p*<sup>I</sup> corresponding to transition rates (probabilities) S→E after contact with the exposed agent in state E and S→E after contact with the infected agent in state I, respectively. We would like to emphasize that both exposed (in state E) and infected (in state I) agents *may transmit* disease.

According to Equation (1), after recovering, the convalescent in state R lives forever, which seems contradictory to the observations of the real world. Although the Bible Book of Genesis (5:5–27; 9:29) mentions seven men who lived over 900 years, in modern society—thanks to public health systems (and sometimes in spite of them)—contemporary living lengths beyond one hundred years are rather rare. Mortality tables [44] show some correlations between probability of death and age [45]. This observation was first published by Gompertz in 1825 [46]. According to Gompertz's law, mortality *f* increases exponentially with the age *a* of the individual as:

$$f(a) \propto \exp(b(a+c)),\tag{4}$$

where *b* and *c* are constants. Moreover, as we mentioned in the first sentence of the Introduction, people can also die earlier than Gompertz's law implies. For example, an epidemic of fatal diseases increases the mortality rate. To take care of these factors in modeling disease propagation, we consider removing agents from the population. This happens with agents' age-dependent probabilities *fG* and *fC* for healthy and ill people, respectively. The removed individual is immediately replaced with a newly born baby.

In this paper, we propose a cellular automaton based simultaneously on SEIR model of disease propagation and Gompertz's law of mortality. In Section 2, the cellular automaton, its rules and the available site neighborhoods are presented. Section 3 is devoted to the presentation of the results of simulations based on the proposed cellular automaton. Discussion of the obtained results (Section 4) and conclusions (Section 5) end the paper. We note [47,48], where age-structured populations were also studied with SEIR-based and multicompartments models, respectively.
