key: cord-0530930-67glpvbw
authors: Biernacki, Szymon; Malarz, Krzysztof
title: Does social distancing matter for infectious disease propagation? A SEIR model and Gompertz law based cellular automaton
date: 2022-03-28
journal: nan
DOI: nan
sha: 49e4e7d6ff9305413194d222233d5aa8554ab5ad
doc_id: 530930
cord_uid: 67glpvbw

In this paper, we present stochastic synchronous cellular automaton defined on a square lattice. The automaton rules are based on SEIR model with probabilistic parameters gathered from real-world data on human mortality and SARS-CoV-2 disease characteristics. With computer simulations, we show the influence of the neighbourhood radius on the number of infected and deceased agents in the artificial population. The increase of the neighbourhood radius favors the spread of the epidemic. However, for a large range of interactions of exposed (who are both inward for and/or untested against) agents, even isolation of infected agents cannot prevent successful disease propagation. This supports aggressive testing against disease as one of the useful strategies preventing large peaks of infection in SARS-CoV-2-like disease spreading.

Currently, the death rate of SARS-CoV-2 [1] for 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] are based on cellular automata technique [18] [19] [20] [21] .

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. 22-25 for mini-reviews) including, for instance, those based on the percolation theory [26] .

The history of the application of compartmental models to the mathematical modelling of infectious diseases dates to the first half of the 20-th century and works of Ross [27] , Ross and Hudson [28, 29] , Kermack and McKendrick [30, 31] and Kendall [32] , see for instance Ref. 33 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 * 0000-0001-9980-0363; malarz@agh.edu.pl from susceptible (S) via infected (I) to recovered (R) one [30, 31, 34] . The infected agents may transmit the disease to their susceptible neighbours (S → I) with 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

dn I dt = k p 1 n S n I − p 2 n I , (1b)

where k is a mean numbers of agents' contacts in the neighbourhood and n S , n I , n R stand for the fraction of susceptible, infected, recovered agents, respectively. Typically, the initial condition for Equation (1) is

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 arbitrary or they may correspond to the reciprocal of agents' residence times in selected states. In the latter case, the residence time may be estimated by clinical observations [35, 36] . The probability p 1 describes speed of disease propagation (infecting rate) while value of p 2 is responsible for the frequency of getting better (recovering rate). In such approximation, the dynamics of the infectious class depends on the reproduction ratio:

The case of R 0 = 1 separates phase when the disease dies out and the phase when the disease propagates among population members. Equations (1) describe a mean-field evolution, which simulates a situation when all agents interact directly with each other. In low dimensional spatial networks, the mean-field dynamics (1) is modified by diffusive mechanisms [37] . In a realistic situation the diffusive mode of epidemic spreading is mixed with the mean-field dynamics, corresponding to non-local transmissions resulting from the mobility of agents [38] .

We utilise the SEIR model [39, 40] , where additional compartment (labeled as 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 E and p I corresponding to transition rates (probabilities) S → E after contact with exposed agent in state E and S → E after contacting with infected agent in state I, respectively. We would like to emphasise 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 real world observations. 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. The mortality tables [41] show some correlations between the probability of death and age [42] . This observation was first published by Gompertz in 1825 [43] . According to Gompertz's law, the mortality f increase exponentially with age a of the individual as

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 instance, an epidemic of fatal disease increases the mortality rate.

To take care about these factors in the modeling of disease propagation, we consider removing agents from the population. This happens with agents' age-dependent probabilities f G and f C for health 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 neighbourhoods 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. 44 and 45, where also age-structured populations were studied with SEIR-based and multicompartments models, respectively. We utilize the cellular automata technique [18] [19] [20] [21] for modelling of 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 F of synchronous site states update.

The rule F defines state s i of site i at time (t + 1) basing on this site state s i at the time t and the state of sites in i-th site's neighbourhood N

We adopt SEIR model for the simulation of disease spreading by probabilistic synchronous cellular automata on a square lattice with various neighbourhoods N . The considered neighbourhoods are presented in Figure 1 . Every agent may be in one of four available states: susceptible (S), exposed (E), infected (I) or recovered (R). Initially (at t = 0), every agent is in S state, their age a is set randomly from normal distribution with mean value a = 50 · 365 days and dispersion σ 2 = 25 · 365 days. The 'Patient Zero' in E state is placed randomly at a single site of a square grid with L 2 = 100 2 nodes.

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

• The susceptible agent may be infected (S → E) by each agent in state E or I present in his/her neighbourhood N with radius r E or r I ≤ r E , respectively. We set r I ≤ r 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 E and/or r I as presented in Figure 1 . The infection of the susceptible agents occurs with probability p E (after contacting with agent in state E) or p I (after contacting with agent in state I), respectively.

• The incubation (i.e., the appearance of disease symptoms) takes τ E days-every agent in E state is converted to infected state (E → I) with probability 1/τ E . The exposed agent may die with agedependent probability f C (a). In such a case, he/she is replaced (E → S) with newly born agent (a = 0).

• The disease lasts for τ I days. The ill agent (in state I) may either die (and be replaced by a newly born child I → S) with age specific probability f C (a) or he/she can recover (and gain resistance to disease I → R) with probability 1/τ I .

• A healthy agent (S or 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 S and in age of a = 0).

The agents' state modifications are applied synchronously to all sites. A single time step of system evolution is presented in Algorithm 1.

Basing on American data on the annual death probability [41] we predict the daily death probability as f G (a) = 184 · 10 −10 exp(0.00023(a + 40259)),

where a is the agent's age expressed in days. The data follows Gompertz's exponential law of mortality [43, 46] . We assume a daily death probability for infected people of age a (expressed in days) as

These probabilities are calculated as a chance of death during SARS-CoV-2 infection (based on Polish statistics [47] ) divided by (τ E + τ I ). We assume that infection lasts τ I ≈ 14 days and an incubation process takes τ E ≈ 5 days [36] . Exponential fits [Equations (5) and (6)] to the real-world data are presented in Figure 2 .

In Figure 3 the snapshots from single run simulation are presented. They give quantitative picture of the influence of interaction range (neighbourhood radius) on disease propagation. The snapshots in Figures 3a to 3e show situation for fixed parameters p E = 0.03 and p I = 0.02 at t = 150 time step which corresponds to five months after introducing (at random site) 'Patient Zero'. The last subfigure (Figure 3f ) presents situation after very long time of simulations (t > 20000) where recovered agents die due to their age (according to Equation (5)) and they are subsequently replaced by newly born children. For the interaction limited to the first coordination zone ( Figure 1a ) the disease propagation stays limited Figure 3e ). The direct system evolution based on Ref. 49 may be simulated and observed with JavaScript application available at [48] .

In Figure 4 the fraction n I of infected agents (in state I) is presented. The figure shows ten different simulations' results for values of neighbourhood radius r E = r I = 1.5, p E = 0.03, p I = 0.02. In addition, the results of averaging over R = 10 simulations are presented. In two among 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 to 7 ) over ten independent simulations. Diagrams in Figures 5 and 6 show the evolution of epidemic. Namely, they show the number of agents in each state in every day of the epidemic as well as the cumulative fraction n D of deaths (D). The fraction n S of susceptible agents and the fraction n R of recovered agents are shown on the left vertical axis, while the fractions n E of exposed and n I of infected agents and the cumulative fraction n D of deaths caused by infection are displayed on the right vertical axis.

A. rE = rI = 0

The case of r E = r I = 0 (corresponding to the total lockdown) leads to immediate disease die out as only In Figure 5 the dynamics of states fractions n for various values of neighbourhood radius r E = r I are presented. We assume infection rates p E = p I = 0.005. The assumed transition rates p E and p I are very low. As a result, the disease has a very limited possibility of spreading in society. 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 lasts a maximum of forty days, the number of agents who are ill at the same time is less than one (in average, in ten simulations), there were only two deaths out of ten simulations, the population was not affected by the disease. The case presented in Figure 5b , illustrates the situation when 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 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 5a . We would like to emphasise that the term "duration of the epidemic" determines the time of the longest duration of the epidemic among the ten carried out simulations. Figure 5c shows the situation when there are twelve agents in the neighbourhood of each cell. The transition rates (p E and p I ) turn out to be so low, thatdespite extending the neighbourhood-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 neighbourhood radius to 2.5 (see Figure 5d ) increased the number of exposed and infected agents more than twice in comparison with Figure 5c . On the peak day of the epidemic, five agents were sick, and seven died during the epidemic (in 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 neigh- borhood 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 epidemic, around 75% of the population got infected.

In the left column of Figure 6 the dynamics of states fractions n for various values of neighbourhood radius r E = r I are presented. We assume p E = 0.03, p I = 0.02. As we still keep r I = r E setting p E > p I simulates the fact that exposed agents (who are not aware about 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 are presented. As for earlier cases with r = 1, the epidemic stops quite quickly. The largest number of agents who are ill (n 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 R is almost not visible).

The Figure 6d shows that contact with eight agents in the neighborhood (for assumed values of infection rates p E and p 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 6g ) 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 as many as 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 over 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 due to the disease that day, and if you add about 800 agents who were in exposed state 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 6m) 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 E state on the same day, the effect is that the disease affected nearly 40% of agents on the same day.

In the middle column of Figure 6 the dynamics of state fractions n S , n E , n I , n R for various values of neighbourhood radius r E are presented. We assume p E = 0.03 and p I = 0.02. For this set of plots, r I has been predefined as 1 and only r 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 I state can infect only four of their closest neighbors.

The case illustrated in Figure 6b was analyzed in the Section III B in Figure 6a .

The plot in Figure 6e shows the situation when agents in the state E can infect up to eight agents in their neighbourhood. We see irregularities in the shapes of the curves showing the fraction of agents in the state I and 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 due to the disease was recorded in an average of 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 state S from the top of the graph is almost imperceptible.

In Figure 6h the plots for further extension (to the third coordination zone, with z = 12 neighbours) of the radius of the neighbourhood of exposed agents (in the state E) are presented. The curves are much smoother than those presented in Figure 6e , however, we can still see some irregularities in the curves describing agents in the E and I states. The duration of the longest simulation was about 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 S and R almost perfectly line up on the right side of the chart, meaning half of the population contracted the disease while the other half remained healthy throughout the epidemic.

The case where agents in the state E can infect agents within a radius of 2.5 is shown in Figure 6k . The epidemic was definitely more dynamic than in the previously studied case (Figure 6h ), 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 E increased to three (see ure 6n) we do not observe too much changes compared to the previous case (Figure 6k ). 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 I state). The results are therefore almost identical to that for r E = 2.5, except that disease propagation is faster. The right column of Figure 6 shows cases with predetermined r E = 3 and various r I . We keep p E = 0.03 and p I = 0.02

All graphs are very similar to each other, as a large range of infections of agents in the state E makes influence of r 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 E = 3 is large enough (at least for assumed values of transmission rates p E and p I ) to prevent any observable influence of r I on epidemic trajectories. Let us start the discussion with comparing the most left (Figures 6a, 6d, 6g, 6j and 6m ) and the middle (Figures 6b, 6e, 6h, 6k and 6n ) columns of Figure 6 . The comparison reveals that quarantining or limiting the connectivity of agents in the 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 E = 1.5, this completely brought the epidemic to a halt, which would otherwise affect more than half the population. With r E = 2, it makes it possible to reduce the share of infected agents in the population from 75% to 50%. The effects on r E = 2.5 and r E = 3 are similar -instead of the total population less than 90% of population became infected. When r 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 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 E = p 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 E and/or p 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 I = r E = 3) for low values of p E = p I = 0.005 nearly 75% of population reached the R state (Figure 5e ) while increasing infection rates to p E = 0.03 and p I = 0.02 caused the entire population to fall ill (Figure 6m ). The summed values of fractions n E of exposed and n I infected agents at the peaks of disease are 6.6% (for p E = p I = 0.005, Figure 5e ) and 39% (for p E = 0.03 and p I = 0.02, Figure 6m ) of the population.

The results presented in Figures 5 and 6 are summarized in Figure 7 , where the maximum fraction of agents in state 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 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 I.

In this paper, we present stochastic synchronous cellular automaton defined on a square lattice. The automaton rules base on SEIR model with probabilistic parameters gathered from real-world data on human mortality and SARS-CoV-2 disease characteristics. The automaton rules are presented in Algorithm 1. With computer simulations we show the influence of the neighbourhood radius on the number of infected and deceased agents in the artificial population.

The increase of the neighbourhood radius (and the number of interacting agents) favors the spread of the epidemic. However, for a large range of interaction of exposed agents (who are not aware of being infected) even isolation of infected (who are aware of their infec-tion) agents cannot prevent successful disease propagation. This supports aggressive testing against disease as one of useful strategies preventing huge peaks of infection in SARS-CoV-2-like disease spreading. The latter can have devastating consequences for the health care system, in particular for the availability of hospital beds for both SARS-CoV-2 and other diseases.

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