††thanks:  0000-0001-9980-0363

# Does social distancing matter for infectious disease propagation? A SEIR
model and Gompertz law based cellular automaton

Szymon Biernacki  Krzysztof Malarz<EMAIL_ADDRESS>AGH University of
Science and Technology, Faculty of Physics and Applied Computer Science,
al. Mickiewicza 30, 30-059 Kraków, Poland

(August 27, 2024)

###### Abstract

In this paper, we present stochastic synchronous cellular automaton defined on
a square lattice. The automaton rules are based on the SEIR (susceptible $\to$
exposed $\to$ infected $\to$ recovered) model with probabilistic parameters
gathered from real-world data on human mortality and the characteristics of
the SARS-CoV-2 disease. With computer simulations, we show the influence of
the radius of the neighborhood on the number of infected and deceased agents
in the artificial population. The increase in the radius of the neighborhood
favors the spread of the epidemic. However, for a large range of interactions
of exposed agents (who neither have symptoms of the disease nor have been
diagnosed by appropriate tests), even isolation of infected agents cannot
prevent successful disease propagation. This supports aggressive testing
against disease as one of the useful strategies to prevent large peaks of
infection in the spread of SARS-CoV-2-like disease.

##  I Introduction

Currently, the death rate of SARS-CoV-2 [1] in the whole world reached around
2% of population [2, tab ’Closed Cases‘]. Thus, one should not be surprised to
the publication rash in this subject giving both: _i_) theoretical bases of
SARS-CoV-2 spreading, _ii_) practical tips on preventing plague or even _iii_)
clinical case studies allowing 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
[3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] are based on
cellular automata technique [20, 21, 22, 23].

The likely reason for this moderate interest in using this technique to
simulate the spread of the COVID epidemic 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 epidemic is also possible with other models (see Refs. 24,
25, 26, 27 for mini-reviews) including, for instance, those based on the
percolation theory [28].

The history of the application of compartmental models to the mathematical
modeling of infectious diseases dates to the first half of the 20-th century
and works of Ross [29], Ross and Hudson [30, 31], Kermack and McKendrick [32,
33] and Kendall [34], see, for instance, Ref. 35 for 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 ($\mathcal{S}$) via infected ($\mathcal{I}$) to recovered
($\mathcal{R}$) one [32, 33, 36]. Infected agents can transmit the disease to
their susceptible neighbors ($\mathcal{S}\to\mathcal{I}$) with a given
probability $p_{1}$. The infected agent may recover
($\mathcal{I}\to\mathcal{R}$) 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

$\dfrac{dn_{\mathcal{S}}}{dt}=-\langle k\rangle
p_{1}n_{\mathcal{S}}n_{\mathcal{I}},$ (1a)
$\dfrac{dn_{\mathcal{I}}}{dt}=\langle k\rangle
p_{1}n_{\mathcal{S}}n_{\mathcal{I}}-p_{2}n_{\mathcal{I}},$ (1b)
$\dfrac{dn_{\mathcal{R}}}{dt}=p_{2}n_{\mathcal{I}},$ (1c)

where $\langle k\rangle$ is the mean number of agents‘ contacts in the
neighborhood and $n_{\mathcal{S}}$, $n_{\mathcal{I}}$, $n_{\mathcal{R}}$
represent the fraction of susceptible, infected, recovered agents,
respectively. Typically, the initial condition for Equation 1 is

$n_{\mathcal{S}}(t=0)=1-n_{0},$ (2a) $n_{\mathcal{I}}(t=0)=n_{0},$ (2b)
$n_{\mathcal{R}}(t=0)=0,$ (2c)

where $n_{0}$ is the initial fraction of infected agents (fraction of
’Patients Zero‘).

The transition rates between states (i.e., probabilities $p_{1}$ 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 [37, 38]. The probability $p_{1}$ describes
the speed of disease propagation (infecting rate) while the value of $p_{2}$
is responsible for the frequency of getting better (recovering rate). In this
approximation, the dynamics of the infectious class depends on the
reproduction ratio:

$R_{0}=\dfrac{\langle k\rangle p_{1}}{p_{2}}.$ (3)

The case of $R_{0}=1$ separates the phase when the disease dies out and the
phase when the disease spreads among the members of the population.

Equations (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
[39]. In a realistic situation, the diffusive mode of epidemic spreading is
mixed with the mean-field dynamics, corresponding to nonlocal transmissions
resulting from the mobility of agents [40].

We use the SEIR model [41, 42], upon extending the SIR model, where an
additional compartment (labeled $\mathcal{E}$) is available and it corresponds
to agents in 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 $p_{1}$ into $p_{\mathcal{E}}$ and $p_{\mathcal{I}}$ corresponding to
transition rates (probabilities) $\mathcal{S}\to\mathcal{E}$ after contact
with the exposed agent in state $\mathcal{E}$ and $\mathcal{S}\to\mathcal{E}$
after contact with the infected agent in state $\mathcal{I}$, respectively. We
would like to emphasize that both exposed (in state $\mathcal{E}$) and
infected (in state $\mathcal{I}$) agents may transmit disease.

According to Equation 1, after recovering, the convalescent in state
$\mathcal{R}$ 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 [43] show some
correlations between probability of death and age [44]. This observation was
first published by Gompertz in 1825 [45]. According to Gompertz‘s law,
mortality $f$ increases exponentially with age $a$ of the individual as

$f(a)\propto\exp(b(a+c)),$ (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 $f_{G}$ and $f_{C}$ 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 II,
the cellular automaton, its rules and the available site neighborhoods are
presented. Section III is devoted to the presentation of the results of
simulations based on the proposed cellular automaton. Discussion of the
obtained results (Section IV) and conclusions (Section V) end the paper. We
note Refs. 46, 47, where age-structured populations were also studied with
SEIR-based and multicompartments models, respectively.

##  II Model

(a)

(b)

(c)

(d)

(e)

Figure 1:  Sites in various neighborhoods $\mathcal{N}$ on a square lattice.
(a) von Neumann‘s neighborhood ($r=1$, $z=4$), (b) Moore‘s neighborhood
($r=\sqrt{2}$, $z=8$), (c) neighborhood with sites up to the third
coordination zone ($r=2$, $z=12$), (d) neighborhood with sites up to the
fourth coordination zone ($r=\sqrt{5}$, $z=20$), (e) neighborhood with sites
up to the fifth coordination zone ($r=2\sqrt{2}$, $z=24$).

We use the cellular automata technique [20, 21, 22, 23] to model disease
propagation. The cellular automata technique is based on several assumptions,
including:

  * •

discrete (geometrical) space (i.e. regular lattice) and time;

  * •

discrete and finite set of available states of the single lattice‘s site;

  * •

local rule $\mathcal{F}$ of synchronous site states update.

The rule $\mathcal{F}$ defines the state $s_{i}$ of the site $i$ at time
$(t+1)$ basing on this site state $s_{i}$ at time $t$ and the state of the
sites in the $i$-th site‘s neighborhood $\mathcal{N}$

$s_{i}(t+1)=\mathcal{F}\Big{(}s_{i}(t);\mathcal{N}(s_{i}(t))\Big{)}.$

We adopt SEIR model for the simulation of disease spreading by probabilistic
synchronous cellular automata on a square lattice with various neighborhoods
$\mathcal{N}$. Every agent may be in one of four available states: susceptible
($\mathcal{S}$), exposed ($\mathcal{E}$), infected ($\mathcal{I}$) or
recovered ($\mathcal{R}$). The agents in $\mathcal{E}$ and $\mathcal{I}$ are
characterized with different values of the probability of infection
($p_{\mathcal{E}}$ and $p_{\mathcal{I}}$) and different range of interactions
(radii of neighborhoods $r_{\mathcal{E}}$ and $r_{\mathcal{I}}$). The
considered neighborhoods (and their radii) are presented in Figure 1.

Initially (at $t=0$), every agent is in $\mathcal{S}$ state, their age $a$ is
set randomly from normal distribution with mean value $\langle
a\rangle=50\cdot 365$ days and dispersion $\sigma^{2}=25\cdot 365$ days. The
’Patient Zero‘ in $\mathcal{E}$ state is placed randomly at a single site of a
square lattice with $L^{2}=100^{2}$ nodes.

Every time step (which corresponds literally to a single day in real-world)
the lattice is scanned in typewriter order to check the possible agent state
evolution:

  * •

The susceptible agent may be infected ($\mathcal{S}\to\mathcal{E}$) by each
agent in state $\mathcal{E}$ or $\mathcal{I}$ present in his/her neighbourhood
$\mathcal{N}$ with radius $r_{\mathcal{E}}$ or $r_{\mathcal{I}}\leq
r_{\mathcal{E}}$, respectively. We set $r_{\mathcal{I}}\leq r_{\mathcal{E}}$
as we assume that infected (and aware of the disease) agents are more
responsible than exposed (unaware of the disease) ones. The latter inequality
comes from our assumption that the exposed agent is not careful enough in
undertaking contacts with his/her neighbours while the infected agent is
serious-minded and realizing the hazard of possible disease propagation and
thus he/she avoids these contacts at least on the level assigned to exposed
agents. The number and position of available neighbours who may infect the
considered susceptible agent depend on the value of radius $r_{\mathcal{E}}$
and/or $r_{\mathcal{I}}$ as presented in Figure 1. The infection of the
susceptible agents occurs with probability $p_{\mathcal{E}}$ (after contacting
with agent in state $\mathcal{E}$) or $p_{\mathcal{I}}$ (after contacting with
agent in state $\mathcal{I}$), respectively.

  * •

The incubation (i.e., the appearance of disease symptoms) takes
$\tau_{\mathcal{E}}$ days—every agent in $\mathcal{E}$ state is converted to
infected state ($\mathcal{E}\to\mathcal{I}$) with probability
$1/\tau_{\mathcal{E}}$. The exposed agent may die with age-dependent
probability $f_{C}(a)$. In such a case, he/she is replaced
($\mathcal{E}\to\mathcal{S}$) with newly born agent ($a=0$).

  * •

The disease lasts for $\tau_{\mathcal{I}}$ days. The ill agent (in state
$\mathcal{I}$) may either die (and be replaced by a newly born child
$\mathcal{I}\to\mathcal{S}$) with age specific probability $f_{C}(a)$ or
he/she can recover (and gain resistance to disease
$\mathcal{I}\to\mathcal{R}$) with probability $1/\tau_{\mathcal{I}}$.

  * •

A healthy agent ($\mathcal{S}$ or $\mathcal{R}$) may die with a chance given
by age dependent probability $f_{G}(a)$. In such a case, it is replaced with a
newly born susceptible baby (in state $\mathcal{S}$ and in age of $a=0$).

[htbp] Single time step in automaton

1: $t\leftarrow t+1$

2: tmp_pop $\leftarrow$ pop

3: tmp_pop $\leftarrow$ pop for all $i\in$ pop do

4: age[$i$] $\leftarrow$ age[$i$]$+1$

5: age[$i$] $\leftarrow$ age[$i$]$+1$ if tmp_pop[$i$] = $\mathcal{S}$ then

6: if tmp_pop[$i$] = $\mathcal{S}$ then for all $j\in$ pop, $j\neq i$ do

7: for all $j\in$ pop, $j\neq i$ do if tmp_pop[$j$] =
$\mathcal{I}\land\|i,j\|\leq r_{\mathcal{I}}\land$ random()$\leq
p_{\mathcal{I}}$ then

8: pop[$i$]$\leftarrow\mathcal{E}$ $\triangleright$
$\mathcal{S}\to\mathcal{E}$ after contacting with $\mathcal{I}$

9: $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}-1$

10: $n_{\mathcal{E}}\leftarrow n_{\mathcal{E}}+1$

11: break break

12: break if tmp_pop[$j$] = $\mathcal{E}\land\|i,j\|\leq r_{\mathcal{E}}\land$
random()$\leq p_{\mathcal{E}}$ then

13: pop[$i$]$\leftarrow\mathcal{E}$ $\triangleright$
$\mathcal{S}\to\mathcal{E}$ after contacting with $\mathcal{E}$

14: $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}-1$

15: $n_{\mathcal{E}}\leftarrow n_{\mathcal{E}}+1$

16: break break break break break break

17: break if tmp_pop[$i$] = $\mathcal{E}\,\land$ random()
$<1/\tau_{\mathcal{E}}$ then

18: pop[$i$]$\leftarrow\mathcal{I}$ $\triangleright$
$\mathcal{E}\to\mathcal{I}$

19: $n_{\mathcal{E}}\leftarrow n_{\mathcal{E}}-1$

20: $n_{\mathcal{I}}\leftarrow n_{\mathcal{I}}+1$ $n_{\mathcal{I}}\leftarrow
n_{\mathcal{I}}+1$

21: $n_{\mathcal{I}}\leftarrow n_{\mathcal{I}}+1$ if tmp_pop[$i$] =
$\mathcal{I}\,\land$ random() $<1/\tau_{\mathcal{I}}$ then

22: pop[$i$]$\leftarrow\mathcal{R}$ $\triangleright$ Recovered

23: $n_{\mathcal{I}}\leftarrow n_{\mathcal{I}}-1$

24: $n_{\mathcal{R}}\leftarrow n_{\mathcal{R}}+1$ $n_{\mathcal{R}}\leftarrow
n_{\mathcal{R}}+1$

25: $n_{\mathcal{R}}\leftarrow n_{\mathcal{R}}+1$ if tmp_pop[$i$] =
$\mathcal{S}\,\lor$ tmp_pop[$i$] = $\mathcal{R}$ then

26: if tmp_pop[$i$] = $\mathcal{S}\,\lor$ tmp_pop[$i$] = $\mathcal{R}$ then if
random() $<f_{G}($age[$i$]) then

27: pop[$i$]$\leftarrow\mathcal{S}$ $\triangleright$ Removed…

28: age[$i$]$\leftarrow 0$ $\triangleright$ …then Susceptible

29: age[$i$]$\leftarrow 0$ $\triangleright$ …then Susceptible if
tmp_pop[$i$]=$\mathcal{S}$ then $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}-1$
if tmp_pop[$i$]=$\mathcal{S}$ then $n_{\mathcal{S}}\leftarrow
n_{\mathcal{S}}-1$

30: if tmp_pop[$i$]=$\mathcal{S}$ then $n_{\mathcal{S}}\leftarrow
n_{\mathcal{S}}-1$ if tmp_pop[$i$]=$\mathcal{R}$ then
$n_{\mathcal{R}}\leftarrow n_{\mathcal{R}}-1$ if tmp_pop[$i$]=$\mathcal{R}$
then $n_{\mathcal{R}}\leftarrow n_{\mathcal{R}}-1$

31: $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}+1$ $n_{\mathcal{S}}\leftarrow
n_{\mathcal{S}}+1$

32: $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}+1$ $n_{\mathcal{S}}\leftarrow
n_{\mathcal{S}}+1$ $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}+1$ else
$\triangleright$ tmp_pop[$i$] = $\mathcal{E}\,\lor$ tmp_pop[$i$] =
$\mathcal{I}$

33: else $\triangleright$ tmp_pop[$i$] = $\mathcal{E}\,\lor$ tmp_pop[$i$] =
$\mathcal{I}$ if random() $<f_{C}$(age[$i$]) then

34: pop[$i$]$\leftarrow\mathcal{S}$ $\triangleright$ Removed…

35: age[$i$]$\leftarrow 0$ $\triangleright$ …then Susceptible

36: age[$i$]$\leftarrow 0$ $\triangleright$ …then Susceptible if
tmp_pop[$i$]=$\mathcal{E}$ then $n_{\mathcal{E}}\leftarrow n_{\mathcal{E}}-1$
if tmp_pop[$i$]=$\mathcal{E}$ then $n_{\mathcal{E}}\leftarrow
n_{\mathcal{E}}-1$

37: if tmp_pop[$i$]=$\mathcal{E}$ then $n_{\mathcal{E}}\leftarrow
n_{\mathcal{E}}-1$ if tmp_pop[$i$]=$\mathcal{I}$ then
$n_{\mathcal{I}}\leftarrow n_{\mathcal{I}}-1$ if tmp_pop[$i$]=$\mathcal{I}$
then $n_{\mathcal{I}}\leftarrow n_{\mathcal{I}}-1$

38: $n_{\mathcal{S}}\leftarrow n_{\mathcal{S}}+1$

39: $n_{\mathcal{D}}\leftarrow n_{\mathcal{D}}+1$ $\triangleright$ cumulative
number of deaths caused by infections $n_{\mathcal{D}}\leftarrow
n_{\mathcal{D}}+1$ $\triangleright$ cumulative number of deaths caused by
infections $n_{\mathcal{D}}\leftarrow n_{\mathcal{D}}+1$ $\triangleright$
cumulative number of deaths caused by infections $n_{\mathcal{D}}\leftarrow
n_{\mathcal{D}}+1$ $\triangleright$ cumulative number of deaths caused by
infections

40:
return$(t,n_{\mathcal{S}}/L^{2},n_{\mathcal{E}}/L^{2},n_{\mathcal{I}}/L^{2},n_{\mathcal{R}}/L^{2},n_{\mathcal{D}}/L^{2})$
return$(t,n_{\mathcal{S}}/L^{2},n_{\mathcal{E}}/L^{2},n_{\mathcal{I}}/L^{2},n_{\mathcal{R}}/L^{2},n_{\mathcal{D}}/L^{2})$

The agents‘ state modifications are applied synchronously to all sites. A
single time step (from $t$ to $t+1$) of the system evolution described above
is presented in Section II. The $L^{2}$-long vector variables tmp_pop[] and
pop[] represent the current population (at time $t$) and the population in the
next time step ($t+1$), respectively. The presence of two such variables in a
model implementation is caused by the synchronicity of cellular automaton. The
$i$-th element of these vectors keeps information on the state (either
$\mathcal{S},\mathcal{E},\mathcal{I}$ or $\mathcal{R}$) of the $i$-th agent in
the population. The age $a$ (measured in days) for the $i$-th agent is kept in
$L^{2}$-long vector age[] and its $i$-th element is either incremented (line 4
of Section II) or reset (lines 28 and 35 of Section II) in the case of
removal. The random() function returns a real pseudo-random number uniformly
distributed in $[0,1)$. The function $\|i,j\|$ measures the Euclidean distance
between agents $i$ and $j$. The age-dependent daily death probability
functions $f_{C}(\cdot)$ and $f_{G}(\cdot)$—based on real-world data—are
defined in the next paragraph in Equations 5 and 6. The numbers
$n_{\mathcal{S}}$, $n_{\mathcal{E}}$, $n_{\mathcal{I}}$ and $n_{\mathcal{R}}$
of agents in various states must be initialized at $t=0$ basing on initial
conditions. The cumulative number $n_{\mathcal{D}}$ of deceased agents—earlier
either in the exposed ($\mathcal{E}$) or in the infected ($\mathcal{I}$)
state—is incremented in line 39 of Section II. The numbers $n_{\mathcal{S}}$,
$n_{\mathcal{E}}$, $n_{\mathcal{I}}$, $n_{\mathcal{R}}$ and $n_{\mathcal{D}}$
are normalized to the total number $L^{2}$ of agents in the system before
their return.

Basing on American data on annual death probability [43] and assuming 365 days
a year, we predict the daily death probability as

$f_{G}(a)=184\cdot 10^{-10}\exp(0.00023(a+40259)),$ (5)

where $a$ is the agent‘s age expressed in days. The data follow Gompertz‘s
exponential law of mortality [45, 48].

Using the same trick, we estimate the probability of daily death for infected
people of age $a$ (expressed in days) as

$f_{C}(a)=\begin{cases}5\cdot 10^{-5}&\iff a\leq 30\leavevmode\nobreak\ y,\\\
2\cdot 10^{-6}\exp(0.0003a)&\iff a>30\leavevmode\nobreak\ y.\end{cases}$ (6)

These probabilities are calculated as the chance of death during SARS-CoV-2
infection (based on Polish statistics [49]) divided by
$(\tau_{\mathcal{E}}+\tau_{\mathcal{I}})$. We assume that infection lasts
$\tau_{\mathcal{I}}\approx 14$ days and that an incubation process takes
$\tau_{\mathcal{E}}\approx 5$ days [38]. Exponential fits [Equations 5 and 6]
to real-world data are presented in Figure 2.

Figure 2: Daily death probability $f(a)$ for patients infected by the
coronavirus (SARS-CoV-2, $\times$, $f_{C}(a)$, [49]) and natural death
probability ($+$, $f_{G}(a)$, [43]).

##  III Results

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: Snapshots from direct simulation [50] for $p_{\mathcal{E}}=0.03$,
$p_{\mathcal{I}}=0.02$, $R=1$. The assumed ranges of interactions are (a)
$r_{\mathcal{E}}=r_{\mathcal{I}}=1$, (b)
$r_{\mathcal{E}}=r_{\mathcal{I}}=1.5$, (c)
$r_{\mathcal{E}}=r_{\mathcal{I}}=2$, (d, f)
$r_{\mathcal{E}}=r_{\mathcal{I}}=2.5$, (e)
$r_{\mathcal{E}}=r_{\mathcal{I}}=3$. The simulation takes $t=150$ time steps
except of Figure 3(f), where situation after $t>20000$ time steps is
presented.

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 Figures 3(a), 3(b),
3(c), 3(d) and 3(e) show the situation for fixed parameters
$p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$ at the $t=150$ time step
which corresponds to five months after introducing (at random site) ’Patient
Zero‘. The last subfigure (Figure 3(f)) presents a situation after a very long
time of simulations ($t>20000$) 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
1(a)) the disease propagation stays limited to the nearest-neighbors of the
’Patient Zero‘. On the other hand, for the neighborhood with sites up to the
fifth coordination zone (Figure 1(e)) for the same infection rates
($p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$) the disease affects all
agents in the population (see Figure 3(e)). The direct evolution of the system
based on Ref. 51 can be simulated and observed with the JavaScript application
available at [50].

Figure 4: Ten different simulations for values of neighbourhood radius
$r_{\mathcal{E}}=r_{\mathcal{I}}$=1.5. $p_{\mathcal{E}}=0.03$,
$p_{\mathcal{I}}=0.02$.

In Figure 4 the fraction $n_{\mathcal{I}}$ of infected agents (in state
$\mathcal{I}$) is presented. The figure shows the results of ten different
simulations for values of neighborhood radius
$r_{\mathcal{E}}=r_{\mathcal{I}}=1.5$, $p_{\mathcal{E}}=0.03$,
$p_{\mathcal{I}}=0.02$. 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. Basing on this test [for
(not shown) roughly twice large statistics, $R=25$, which do not reveal
significant deviations], we decided to average our results (presented in
Figures 5, 6 and 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 $n_{\mathcal{D}}$ of deaths ($\mathcal{D}$).
The fraction $n_{\mathcal{S}}$ of susceptible agents and the fraction
$n_{\mathcal{R}}$ of recovered agents are shown on the left vertical axis,
while the fractions $n_{\mathcal{E}}$ of exposed agents and $n_{\mathcal{I}}$
of infected agents and the cumulative fraction $n_{\mathcal{D}}$ of deaths
caused by infection are shown on the right vertical axis.

###  III.1 $r_{\mathcal{E}}=r_{\mathcal{I}}=0$

The case of $r_{E}=r_{I}=0$ (corresponding to total lockdown) leads to
immediate disease dieout as only $n_{0}L^{2}$ ’Patients Zero‘ at $t=0$ are
infected and recover after about $\sim 1/q_{C}(a)$ time steps (days) depending
on the agent‘s age $a$.

###  III.2 $r_{\mathcal{E}}=r_{\mathcal{I}}>0$

(a) $r_{I}=r_{E}=1$

(b) $r_{I}=r_{E}=1.5$

(c) $r_{I}=r_{E}=2$

(d) $r_{I}=r_{E}=2.5$

(e) $r_{I}=r_{E}=3$

Figure 5: Dynamics of states fractions for various values of the neighborhood
radius $r_{\mathcal{E}}=r_{\mathcal{I}}$.
$p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$, $R=10$.

In Figure 5 the dynamics of states fractions $n$ for various values of the
neighborhood radius $r_{\mathcal{E}}=r_{\mathcal{I}}$ are presented. We assume
infection rates $p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$. The assumed
transition rates $p_{\mathcal{E}}$ and $p_{\mathcal{I}}$ are very low. As a
result, the disease has a very limited chance of spreading in society.

Figure 5(a) illustrates the situation where an infected person (independently
either in the $\mathcal{E}$ or $\mathcal{I}$ states) can only infect the four
closest neighbors. The epidemic lasts a maximum of forty days, the number of
agents who are ill at the same time is 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 5(b), 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 5(a), the average number of
simultaneously ill agents remains 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 5(a). 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 5(c) shows the situation where there are twelve agents in the
neighborhood of each cell. The transition rates ($p_{\mathcal{E}}$ and
$p_{\mathcal{I}}$) turn out to be so low, that—despite extending the
neighborhood—the epidemic vanishes quickly. This time the fractions of exposed
and infected agents are slightly higher, the maximum number of sick agents in
one day is 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 5(d)) increased
the number of exposed and infected agents more than twice compared to Figure
5(c). 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 5(e)) 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.

(a) $r_{\mathcal{I}}=r_{\mathcal{E}}=1$

(b) $r_{\mathcal{E}}=1$

(c) $r_{\mathcal{I}}=1$

(d) $r_{\mathcal{I}}=r_{\mathcal{E}}=1.5$

(e) $r_{\mathcal{E}}=1.5$

(f) $r_{\mathcal{I}}=1.5$

(g) $r_{\mathcal{I}}=r_{\mathcal{E}}=2$

(h) $r_{\mathcal{E}}=2$

(i) $r_{\mathcal{I}}=2$

(j) $r_{\mathcal{I}}=r_{\mathcal{E}}=2.5$

(k) $r_{\mathcal{E}}=2.5$

(l) $r_{\mathcal{I}}=2.5$

(m) $r_{\mathcal{I}}=r_{\mathcal{E}}=3$

(n) $r_{\mathcal{E}}=3$

(o) $r_{\mathcal{I}}=3$

Figure 6: Dynamics of states fractions for various values of neighbourhood
radius (left) $r_{\mathcal{E}}=r_{\mathcal{I}}$, (middle) $r_{\mathcal{E}}\geq
r_{\mathcal{I}}=1$, (right) $3=r_{\mathcal{E}}\geq r_{\mathcal{I}}$.
$p_{\mathcal{E}}=0.03$, $p_{\mathcal{I}}=0.02$, $R=10$.

In the left column of Figure 6 the dynamics of states fractions $n$ for
various values of the radius of the neighborhood
$r_{\mathcal{E}}=r_{\mathcal{I}}$ are presented. We assume
$p_{\mathcal{E}}=0.03$, $p_{\mathcal{I}}=0.02$. As we still keep
$r_{\mathcal{I}}=r_{\mathcal{E}}$ setting $p_{\mathcal{E}}>p_{\mathcal{I}}$
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 6(a) the results for the smallest possible neighborhood radius $r>0$
are presented. As in the previous cases with $r=1$, the epidemic stops quite
quickly. The largest number of agents who are ill ($n_{\mathcal{I}}L^{2}$) at
the same time is two, the longest simulation time is 120 days, the average
number of deaths is well below one, and totally only a few agents are infected
(the curve representing agents in the state of $\mathcal{R}$ is almost not
visible).

The Figure 6(d) shows that contact with eight agents in the neighborhood (for
assumed values of infection rates $p_{\mathcal{E}}$ and $p_{\mathcal{I}}$) 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,
while 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
6(g)) caused a significant reduction in the duration of the pandemic, as well
as the greater havoc it caused 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 6(j), 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 you add about 800
agents who were in exposed state $\mathcal{E}$, we get a situation when 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 6(m)) gives very similar results to
the previous one, except that the virus spreads even faster, in less than 150
days. The number of deaths is 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 you add over 1100 agents in the
$\mathcal{E}$ state on the same day, the effect is that the disease affected
nearly 40% of agents on the same day.

###  III.3 $r_{\mathcal{E}}\geq r_{\mathcal{I}}$

In the middle column of Figure 6 the dynamics of state fractions
$n_{\mathcal{S}}$, $n_{\mathcal{E}}$, $n_{\mathcal{I}}$, $n_{\mathcal{R}}$ for
various values of the neighborhood radius $r_{\mathcal{E}}$ are presented. We
assume $p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$. For this set of
plots, $r_{\mathcal{I}}$ has been predefined as 1 and only $r_{\mathcal{E}}$
changes. Translating this into a description of the real world, sick agents
who have disease symptoms are aware of their illness (infected) and limit
their contact with the environment to a minimum, while oblivious agents
(exposed) do not realize that they can transmit the disease. In each of the
tested parameter sets, agents in $\mathcal{I}$ state can infect only four of
their closest neighbors.

The case illustrated in Figure 6(b) was analyzed in the Section III.2 in
Figure 6(a).

The plot in Figure 6(e) shows the situation where agents in the state
$\mathcal{E}$ can infect up to eight agents in their neighborhood. We see
irregularities in the shapes of the curves showing the fraction of agents in
the state $\mathcal{I}$ and $\mathcal{E}$, the duration of the epidemic is
relatively long, over 400 days. However, the values shown in the graphs show
that the pandemic was not dangerous for the entire population. Death from the
disease was recorded on average in less than eight agents, the maximum number
of sick agents in one day is just over six, while the total number of infected
agents in the entire epidemic was so low that the deviation of the curve
representing the number of agents in the state $\mathcal{S}$ from the top of
the graph is almost imperceptible.

In Figure 6(h) the plots for further extension (to the third coordination
zone, with $z=12$ neighbors) of the radius of the neighborhood of exposed
agents (in the state $\mathcal{E}$) are presented. The curves are much
smoother than those presented in Figure 6(e), however, we can still see some
irregularities in the curves describing agents in the $\mathcal{E}$ and
$\mathcal{I}$ states. The duration of the longest simulation was approximately
700 days, the number of deaths was less than 330, and at the peak of the
pandemic, approximately 220 agents were simultaneously ill. The curves
presenting the fraction of agents in the states of $\mathcal{S}$ and
$\mathcal{R}$ almost perfectly line up on the right side of the chart, meaning
that half of the population contracted the disease while the other half
remained healthy throughout the epidemic.

The case where agents in the state $\mathcal{E}$ can infect agents within a
radius of 2.5 is shown in Figure 6(k). The epidemic was definitely more
dynamic than in the previously studied case (Figure 6(h)), lasting only a
little over two hundred days and the maximum number of sick agents in one day
$n_{I}L^{2}$ reached almost 1500 agents. The sum of those who died from the
coronavirus is less than 600, and during the pandemic, approximately 85% of
the population became recovered (and earlier exposed and/or infected).

Finally, for the radius $r_{\mathcal{E}}$ increased to three (see Figure 6(n))
we do not observe too many changes compared to the previous case (Figure
6(k)). The epidemic is even shorter, it lasts a little more than 150 days, the
number of deaths is very similar (it can be estimated at 600), slightly more
than 85% of agents in the population were infected, while at the hardest
moment of the pandemic, there were at the same time about 2,370 sick agents
(in $\mathcal{I}$ state). The results are therefore almost identical to those
of $r_{\mathcal{E}}=2.5$, except that disease propagation is faster.

The right column of Figure 6 shows cases with predetermined
$r_{\mathcal{E}}=3$ and various $r_{\mathcal{I}}$. We keep
$p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$

All graphs are very similar to each other, as a large range of infections of
agents in the state $\mathcal{E}$ makes influence of $r_{\mathcal{I}}$ on
epidemic evolution only marginal. It can be easily observed by comparing the
scales presented in all subfigures. The assumed range of infection
$r_{\mathcal{E}}=3$ is large enough (at least for assumed values of
transmission rates $p_{\mathcal{E}}$ and $p_{\mathcal{I}}$) to prevent any
observable influence of $r_{\mathcal{I}}$ on epidemic trajectories.

##  IV Discussion

Let us start the discussion with comparing the most left (Figures 6(a), 6(d),
6(g), 6(j) and 6(m)) and the middle (Figures 6(b), 6(e), 6(h), 6(k) and 6(n))
columns of Figure 6. The comparison reveals that quarantining or limiting the
connectivity of agents in the $\mathcal{I}$ state (both, infected and
informed) may bring good or even very good results in preventing disease
propagation, depending on the arrangement of the other parameters. For
$r_{\mathcal{E}}=1.5$, this completely brought the epidemic to a halt, which
would otherwise affect more than half the population. With
$r_{\mathcal{E}}=2$, it makes it possible to reduce the share of infected
agents in the population from 75% to 50%. The effects on $r_{\mathcal{E}}=2.5$
and $r_{\mathcal{E}}=3$ are similar — instead of the total population less
than 90% of population became infected. When $r_{\mathcal{E}}=1.5$ pandemic
duration was significantly shortened, it is because the virus was unable to
survive, and the disease was extinct. In other cases, the duration of the
epidemic increased, the restrictions introduced for agents in the
$\mathcal{I}$ state did not completely extinguish the disease, but allowed to
slow it down and mitigate its effects.

For low values of the probability of infection
($p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$), that is, in a situation where the
transmission of the virus is not too high, even a slight limitation of the
contact among agents allows for a complete inhibition of the disease and
protection of the society against its negative effects (see Figure 5). We note
that manipulating in $p_{\mathcal{E}}$ and/or $p_{\mathcal{I}}$ parameters may
reflect changes in disease transition rates with their low values
corresponding to wild variant of SARS-CoV-2 virus while higher values to the
fiercer (including delta and specially omicron) variants of SARS-CoV-2 virus.
For instance, for a fixed radius of interaction
($r_{\mathcal{I}}=r_{\mathcal{E}}=3$) for low values of
$p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$ nearly 75% of population reached the
$\mathcal{R}$ state (Figure 5(e)) while increasing infection rates to
$p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$ caused the entire population
to fall ill (Figure 6(m)). The summed values of fractions $n_{\mathcal{E}}$ of
exposed and $n_{\mathcal{I}}$ infected agents at the peaks of disease are 6.6%
(for $p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$, Figure 5(e)) and 39% (for
$p_{\mathcal{E}}=0.03$ and $p_{\mathcal{I}}=0.02$, Figure 6(m)) of the
population.

The results presented in Figures 5 and 6 are summarized in Figure 7, where the
maximum fraction of agents in state $\mathcal{I}$ vs. the neighbours number
$z$ for various sets of parameters is presented. We observe a gradual increase
in the maximum number of infected agents (up to 30%) as the number of agents
in the neighborhood increases. An exception to this rule is observed only when
$r_{\mathcal{E}}=3$, when the maximum level of infections is constant and it
does not change with the increase of the range of the interaction of agents in
state $\mathcal{I}$.

Figure 7: Maximal fraction $n_{\mathcal{I}}$ of agents in state $\mathcal{I}$
as dependent on the number of agents‘ neighbours $z$ in the neighbourhood. (a)
$p_{\mathcal{E}}=p_{\mathcal{I}}=0.005$, $z_{E}=z_{\mathcal{I}}=z$, (b)
$p_{\mathcal{E}}=p_{\mathcal{I}}=0.01$, $z_{E}=z_{\mathcal{I}}=z$, (c)
$p_{\mathcal{E}}=0.03$, $p_{\mathcal{I}}=0.02$, $z_{E}=z_{\mathcal{I}}=z$, (d)
$p_{\mathcal{E}}=0.03$, $p_{\mathcal{I}}=0.02$, $z_{E}=z$,
$z_{\mathcal{I}}=4$, (e) $p_{\mathcal{E}}=0.03$, $p_{\mathcal{I}}=0.02$,
$z_{E}=24$, $z_{\mathcal{I}}=z$.

##  V Conclusions

In this article we present a stochastic synchronous cellular automaton defined
on a square lattice. The automaton rules are based on the SEIR model with
probabilistic parameters collected from real human mortality data and SARS-
CoV-2 disease characteristics. Automaton rules are presented in Section II.
With computer simulations, we show the influence of the radius of the
neighborhood on the number of infected and deceased agents in the artificial
population.

The study presented in this paper is based on static automaton. Thus our
approach is equivalent to disease propagation described in terms of
conduction-like processes (i.e. the position of each cell is fixed and can
infect a neighbor, at distance $r_{\mathcal{E}}$ or $r_{\mathcal{I}}$). The
latter is a natural way to model with the cellular automata technique.
However, we note that also convection-like processes (i.e. the population can
flow within the system) may play a crucial role at both large [52] and local
scale [18, 53].

Further enrichment of the model can lead to the introduction of additional
components, including the compartment $\mathcal{V}$ that describes the
vaccinated agents. This model improvement allows us to study scenarios with
limited and unlimited vaccine supply [19] or the existence and stability of
steady states [54, 55]. Vaccinations seem to be particularly effective when
the vaccination campaign starts early and with a large number of vaccinated
individuals [19]. Moreover, the realistic models should take into account
people‘s attitudes to vaccination programs [56, 57].

In conclusion, increasing the radius of the neighborhood (and the number of
agents interacting locally) favors the spread of the epidemic. However, for a
wide range of interactions of exposed agents, even isolation of infected
agents cannot prevent successful disease propagation. This supports aggressive
testing against disease as one of the useful strategies to prevent large peaks
of infection in the spread of SARS-CoV-2-like disease. The latter can have
devastating consequences for the health care system, in particular for the
availability of hospital beds for SARS-CoV-2 and other diseases.

###### Acknowledgements.

Authors are grateful to Zdzisław Burda for critical reading of the manuscript
and fruitful discussion.

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