Analytical and cellular automaton approach to a generalized SEIR model for infection spread in an open crowded space
Andrea Nava, Alessandro Papa, Marco Rossi, Domenico Giuliano
aa r X i v : . [ phy s i c s . s o c - ph ] J un Analytical and Cellular Automaton approach to a generalized SEIR model forinfection spread in an open crowded space
Andrea Nava, Alessandro Papa, Marco Rossi and Domenico Giuliano
Dipartimento di Fisica, Universit`a della Calabria, Arcavacata di Rende I-87036, Cosenza, ItalyINFN - Gruppo collegato di Cosenza, Arcavacata di Rende I-87036, Cosenza, Italy (Dated: June 8, 2020)We formulate a generalized SEIR model on a graph, describing the population dynamics of an opencrowded place with an arbitrary topology. As a sample calculation, we discuss three simple cases,both analytically, and numerically, by means of a cellular automata simulation of the individualdynamics in the system. As a result, we provide the infection ratio in the system as a functionof controllable parameters, which allows for quantifying how acting on the human behavior mayeffectively lower the disease spread throughout the system.
PACS numbers: 64.60.aq, 87.23.Ge , 87.18.Bb
I. INTRODUCTION
Compartmental models provide a conceptually sim-ple and widely used mean to mathematically modelingthe dynamics of infection transmission in isolated pop-ulations . In such models, the population is dividedinto compartments, each one corresponding to a specifichealth status of the individuals that belong to it. Forinstance, the basic SIR model consists of three compart-ments, the
Susceptible compartment ( S ), to which notinfected, healthy individuals belong, the Infectious com-partment ( I ), that contains infectious individuals, andthe Recovered compartment ( R ), that contains individu-als who either recovered from the infection, or died. Inthe SIR model, the spread of the infection is described interms of a set of differential equations that describe thepopulation transfer from one compartment to anotherone. The key parameters of the model are the rates oftransfer between the compartments which, in general, arequantities to be experimentally fitted.In fact, while the SIR model is a good tool for longtime simulations, it does not take into account the latent phase, in which people are infected but not yet infec-tious. The simplest way to fix this flaw of the model isby adding the Exposed compartment ( E ). The E com-partment contains individuals who have been infected,but are not yet infectious. This improvement eventuallyleads to the SEIR model, more appropriate for short-timesimulations. For instance, the SEIR model has recentlybeen widely employed to describe the Covid-19 infection,though with some limitations .Despite their effectiveness in describing a numberof real-life infection dynamics, the SIR and the SEIRmodels, together with their generalization to a highernumber of different compartments , are in generalaffected by a limitation. Indeed, they describe, ingeneral, the average global behavior of the population,with no attention to local dynamics that can makethe level of infection strongly depend on the regionin real space that one is considering. Instead, thislast aspect has become of fundamental importance in, e.g. , countries recently affected by Covid-19 pan-demic, such as Italy, which show a strongly unevenspread of the infection across the country territory [see,for instance, https://en.wikipedia.org/wiki/COVID-19 pandemic in Italy for a detailed description of theCovid-19 infection distribution across Italy from thestart of March, 2020]. In addition, the SIR- and theSEIR-type models basically describe closed systems,without the possibility for people to enter or exit fromthe system. So, they are also expected not to be reliablewhen describing relatively small populations that canexchange individuals with the surrounding “environ-ment”, such as shopping centers or closed commercialareas. In systems as such, the infection dynamics isagain expected to be strongly space-dependent, due to, e.g. , the presence of very popular shops in a shoppingmall, where people spend on the average much moretime than in different areas.In general, it is well known that the spread of a diseasehas a strong dependence upon the topology of the systemand on how individuals and/or subgroups are connectedto each other. Specifically, the topology strongly affectsthe rates that eventually enter the differential equationsdescribing the population dynamics and, therefore, it de-termines the specific stationary solution, describing thesystem over long time scales . For this reason, in thelast years, a remarkable amount of work has been devotedto discuss the dynamics of epidemic processes on graphsand hypergraphs . In these systems, time- and space-inhomogeneity leads to nontrivial consequences on thepopulation dynamics . These are of the utmost im-portance over short-time, small-space scales, as in thecase of the disease dynamics on a sidewalk, in a shoppingcenter or running tracks, where the topology of the sys-tem strongly affects the effective contact rate, that is thenumber of contacts, per unit of time, between individu-als .Based on the above observations, in this paper wedescribe the population dynamics of an open crowdedplace in terms of a “local” SEIR model on a graph, withposition-dependent parameters. In particular, after writ-ing the set of differential equations describing our sys-tem, we first provide an approximate analytical solutionwithin a mean field approach to the full mathematicalproblem and, therefore, we resort to a cellular automata(CA) simulation of the people traffic in the system .In general, an analytical treatment of the problem interms of a collection of local compartmental models ina nonuniform nonequilibrium configuration is quite diffi-cult to deal with, as the number of differential equationsscales quickly with the size of the system. At variance,the CA is able to describe the pretty complicated behav-ior in terms of simple rules, that apply simultaneous to allthe nodes of the graph. Eventually, we employ the CAnumerical results to pertinently complement and checkthe reliability of the (approximate) analytical ones.To quantitatively describe the infection spread over thegraph, in all the cases we compute the (average) infec-tion ratio in the steady state of the system, C r , that is,the ratio between the variation in the average popula-tion in the compartment E over a time T long enoughfor the system to reach the steady state and the initialaverage population in the compartment S . C r measuresthe hazard for a healthy individual to get infected whengoing across the system in the presence of infected in-dividuals. Therefore, to quantify the level of infectionrisk for an individual in the system, we compute C r as afunction of various system parameters, some of which areparticularly important, as they can be readily acted onby, for instance, controlling the entrance rate of peoplein a shopping center, letting people move in one directiononly at each side of the aisles in the center, increasing ordecreasing the number of points at which it is possiblefor individuals to change their direction of motion, andso on.Considering all together in our model the effects ofboth effective parameters that depend on the (“intrin-sic”) disease dynamics and of parameters that can betuned by acting on the social behavior of individuals, weare able to quantify how an accurate control of human-depending effects, such as social distancing, use of per-sonal protective equipments and so on, can be effective inmitigating the effects of high contagion rates of diseases,such as the one due to Covid-19 .To present the main features of our model and to dis-cuss its implementation, both analytical and numerical,in the paper we focus on three simple, prototypical mod-els of open systems. Yet, as we eventually discuss in thepaper, within the CA approach, generalizations of ourmodels to more realistic situations are straightforward,and we plan to pursue them in forthcoming publications.The paper is organized as follows: • In Sec. II, we define our generalized SEIR modelon graphs. In particular, we present the (local)set of differential equations describing the (local)population dynamics on the graph and provide anexplicit analytical mean-field solution of the equa-tions. Eventually, after presenting the correspond-ing results, we highlight the main limitations of the analytical approach, which motivate our switch tothe numerical, CA approach; • In Sec. III, we present and discuss the CA rules de-scribing the spread of an infection within an open,finite connected graph. Therefore, we implementthem to obtain numerical results in the same sys-tems analytically studied in Sec. II. Eventually, weemploy the numerical results to check the reliabilityof the analytical approach we use in Sec. II; • In Sec. IV, we provide our main conclusions anddiscuss about possible further perspectives of ourwork; • In Appendix A we review the basic formulations ofthe compartmental SEIR model.
II. THE LATTICE LOCAL SEIR MODEL
In this section, we define and analytically study ourlattice model generalization of the SEIR model, suit-able to describe the spread of an infection throughouta number of open, finite connected graphs. Within ourmodel, we describe the dynamics of small populations,each one residing at the sites of a pertinently designed(quasi)one-dimensional lattice, and connected to eachother by means of a finite rate for individuals to “hop”from one site to the others. As we discuss in the following,our model is able to encompass several typical features ofinfection spreading in real world, particularly evidencing,on quantitative grounds, how the spreading depends onin principle tunable parameters of the system.Throughout this paper, we focus onto linear graphs,that is, finite one-dimensional lattices (chains), with openboundary conditions. Each chain corresponds to a sim-plified model of a straight way for pedestrians. Morecomplex (and, in many cases, more realistic) models canbe readily constructed by, e.g. , putting together finite,one-dimensional lattices to be used (with the appropriateboundary conditions) as “elementary building blocks”.Each lattice site j hosts a “local” population of individu-als that is characterized by the density of healthy people n S,λ ; j , that is, the number of healthy people per plaque-tte, by the density of exposed (infected, not contagious)people n E,λ ; j , that is, the number of exposed individu-als per plaquette, and by the density of infectious peo-ple n I,λ ; j , that is, the number of infectious individualsper plaquette. In this respect, each cell realizes a popu-lation whose dynamics is described by the SEIR modelreviewed in appendix A, though with a major simplifica-tion which we discuss next. In fact, the basic assumptionof the SEIR model that only an infectious individual canturn into a recovered one (either healed, or deceased),as well as with the observation that the duration of thetime that each individual spends within the graph is of afew hours, which is pretty short compared with the timescales for having a nonzero density of recovered people,enables us to set to zero the density of recovered peo-ple throughout the whole system. Furthermore, as themedian incubation period of a virus such as 2019-nCoVARD has been estimated to be about 3.0 days , we setto zero the probability for exposed people to become in-fectious. Accordingly, we assume that the total numberof infectious individuals in the system can only change byexchanges with the outside (infectious individuals eitherentering, or exiting, the system).Compared to the “standard” SEIR model, our frame-work allows for the net number of individuals in the pop-ulation at each lattice site to change, as a consequenceof individuals hopping between neighboring sites on thechain. To formalize this aspect of our system, when defin-ing the various local densities, we add a label λ , encod-ing various possible ways for individuals to move betweendifferent sites. In general, in our sample models, at anytime t , each individual at site j of a chain can either movetowards the right (corresponding to λ > λ < | λ | = 1 , ..., Λ, with rules andconstraints that depend on the specific lattice topology.Regardless of the specific lattice topology, the math-ematical description of the population dynamics on thelattice can be formalized by a set of differential equa-tions, whose parameters are determined as follows. Firstof all, to ease the mathematical formulation, it is usefulto label each cell by both the lattice site index, j , as wellas with an additional index λ encoding the informationabout the direction of motion. So, each “physically dis-tinct” lattice site j corresponds to several cells, whichwe label with the pair of indices j, λ . Within each cell, n S,λ ; j , n E,λ ; j , and n I,λ ; j can either be zero, or differentfrom zero.At any cell j, λ not residing at the endpoints of thechain (that is, with j = 1 , L ), on top of the “standard”SEIR dynamics for an isolated population, the numberof individuals in each compartment can either change be-cause individuals hop from-, and into-, neighboring cells moving in the direction defined by λ , or by changing thesign of λ (that is, the direction of motion) at a given j .To formally describe individual motion between differ-ent cells, we set ω h ; λ to be the rate (probability per unit time) of an individual to hop from cell- j, λ to cell- j ± , λ (where sign plus is for λ > λ <
0) and ω ℓ ; ( λ, ¯ λ ) to be the rate for an individual to hop from cell- j, λ to cell- j, ¯ λ , with ¯ λ = λ . If λ ¯ λ >
0, the individualis changing its walking lane but not its direction of mo-tion, while, if λ ¯ λ < j . Moreover, as there is apparently no rea-son for different types of individuals to move at differentrates, we assume that ω h ; λ and ω ℓ ; ( λ, ¯ λ ) are independentof whether the moving individual is S , E , or I .As we aim at eventually describing steady states of thesystem, without big local fluctuations in the various den-sities, consistently with the detailed balance principle,we assume ω ℓ ; ( λ, ¯ λ ) = ω ℓ ; ( ¯ λ,λ ). Finally, to account for the“local” SEIR dynamics, we introduce the parameter ω c ,which corresponds to the infection rate that determinesthe change in time of n S,λ ; j , n E,λ ; j , and n I,λ ; j at given j, λ . At the endpoints of the chains, that is, for j = 1,or j = L , the rate for individuals (of any type) to enterthe cell at fixed λ > λ <
0) is simply the entrancerate into the system, ω in (which is one of the tunableparameters of our system). Similarly, for j = L ( j = 1),the rate for individuals (of any type) to exit the cell atfixed λ > λ <
0) is given by the exit rate from thesystem, ω out . The rules listed above allow us to fullydetermine the set of equations describing our model. Inaddition, they also completely define the CA rules, oncethe rates are traded for the corresponding probabilitiesat each elementary time step, by multiplying all of themby the elementary time step of the CA, ∆ t (see Sec. IIIfor an extensive discussion of this point). Therefore, forthe sake of our presentation, we pictorially present all theabove rules in Fig. 5 of Sec. III which, as stated above,applies to both the mathematical model and to the CA.Putting the various ingredients listed above all to-gether, we obtain the following set of differential equa-tions for the local densities on the lattice, for individualsmoving in both directions and for 1 < j < L : dn S,λ ; j dt = ω h ; λ n S,λ ;[ j − sgn( λ )] − ω h ; λ n S,λ ; j + X ¯ λ = λ ω ℓ ; ( λ, ¯ λ )( n S, ¯ λ ; j − n S,λ ; j ) − ω c f λ ; j ( { n ν,λ ; j } ) , (1) dn I,λ ; j dt = ω h ; λ n I,λ ;[ j − sgn( λ )] − ω h ; λ n I,λ ; j + X ¯ λ = λ ω ℓ ; ( λ, ¯ λ )( n I, ¯ λ ; j − n I,λ ; j ) , (2) dn E,λ ; j dt = ω h ; λ n E,λ ;[ j − sgn( λ )] − ω h ; λ n E,λ ; j + X ¯ λ = λ ω ℓ ; ( λ, ¯ λ )( n E, ¯ λ ; j − n E,λ ; j ) + ω c f λ ; j ( { n ν,λ ; j } ) . (3)Consistently with the rules we discuss above, for j = 1 ( j = L ) and λ > λ < ω h ; λ n { S,I,E } ,λ ;[ j − sgn( λ )] at the right hand side of Eqs. (1)-(3) must be replaced with ω in; { S,I,E } ,λ . Similarly, for j = L ( j = 1) and λ > λ < − ω h ; λ n { S,I,E } ,λ ; j at the right hand side of Eqs. (1)-(3) must be replacedwith − ω out; λ n { S,I,E } ,λ ; L (1) .A general comment about the set of Eqs. (1)-(3) isthat, while they certainly apply for low values of the in-dividual densities at each cell, it is reasonable to assumethat the maximum total density of individual at each celldoes not go beyond a maximum value n max (which, bysymmetry, we assume to be cell-independent), that is, n S,λ ; j + n E,λ ; j + n I,λ ; j ≤ n max . Formally, such a con-straint can be easily implemented in the set of differentialequations above by, e.g. , substituting n { S,I,E } ,λ ; j at theright-hand site of the equations with n { S,I,E } ,λ ; j ( n max − P B n B,λ ; j ± ) /n max . In fact, while we definitely take intoaccount the constraint when solving the equations, aswell as when defining the cellular automata rules below,to ease the notation we prefer not to explicitly write itdown in Eqs. (1)-(3), by limiting that set of equationsto the low-density regime. The function f λ ; j in Eqs. (1),(3) is the joint probability to truly have infectious andhealthy people at the same time in the same cell. Ingeneral, this function is not simple to derive, especiallybecause it changes from scenario to scenario, that is, sinceit is strongly affected by the specific details of the lattice.However, at least in the sample cases we discuss below,we show that f λ ; j can be effectively estimated by meansof a simple, mean-field approximation. To be specific,we now present and discuss the form of the above rateequations in the sample cases we deal with in our work.In particular, in the following we consider three differentscenarios, that is • Scenario A: this describes a small sidewalk withindividuals that can move both ways within onechain only; • Scenario B: this describes, for instance, a wideshopping mall, or an aisle in a shopping center. Inthis case, individuals are more or less evenly dis-tributed between the two sides of the street, in frontof the shop windows, so that people on differentsides cannot infect each other; • Scenario C: this describes a couple of reverse one-way streets. Basically, in this case people with dif-ferent walking directions are forced to stay on dif-ferent sides of the street at a distance greater thanthe safety distance, so that people moving towardopposite directions cannot infect each other.Depending on the specific scenario we are focusingon, we resort to different mean-field decouplings for f λ ; j ( { n ν,λ ; j } ). In general, the mean-field approxima-tion for the joint probability function is grounded on theansatz f λ ; j ( { n ν,λ ; j } ) ≈ n S,λ ; j n I,λ ; j + X ¯ λ = λ µ ( λ, ¯ λ ) n I, ¯ λ ; j , (4)with the µ ( λ, ¯ λ )’s, that are either equal to zero, or to one,encoding all the specificities of each case. In particular,case-by-case we choose the µ ( λ, ¯ λ )’s as follows: • In scenario A we have just one chain on which indi-viduals can move in two opposite directions. Only one pathway is available in either direction, so, λ can only take the values ± µ ( λ, ¯ λ ) = 1. • In scenario B , in its simplest version, we considertwo pathways per each direction of motion of indi-viduals. Therefore, λ = ± , ± µ ( λ, ¯ λ ) = δ λ, − ¯ λ . • In scenario C again we have just one chain on whichindividuals can move in two opposite directions,such as in scenario A, but now spatial separationmakes it impossible for individuals moving towardopposite directions to infect each other. Accord-ingly, here again λ = ±
1, but now µ ( λ, ¯ λ ) = 0.In principle, given the appropriate initial conditionsand once the various rates have been pertinently es-timated, the system of differential equations reportedabove is enough to discuss in detail the evolution in timeof the individual flows across the system. In practice,knowing, for instance, the response in time to a suddenchange in the system parameters (the rates) can help topredict the increase/reduction in the infection diffusiononce local boundaries between regions in the same coun-try, or between different countries, are relaxed/enforced(that has recently become an ubiquitous procedure tokeep the infection under control all around the world).While we plan to discuss these features in a forthcomingpublication, here we are mostly focused onto infectionpropagation in an environment where people flow is ex-pected to shortly become stationary in time. For thisreason, here we just focus onto stationary solutions ofEqs. (1)-(3). In fact, while possible nonuniformities inthe stationary density distribution due to boundary ef-fects (which are in any case negligible in the large systemlimit), we numerically checked that at the system bound-aries, j = 1 , L , the asymptotic values of the populationdensities are the same as in the bulk of the system, thatis, for 1 < j < L .The main parameter characterizing a stationary solu-tion (after a relatively short-time transient) is the averagetime T that people spend from when they enter the sys-tem till they exit. To evaluate T , we make a number ofsimplifying assumptions. Specifically, we assume that the(stationary) flow in either direction does not depend onthe direction itself, that the densities at any cell are in-dependent of time and uniform, that is, that n { S,I,E } ,λ ; j is independent of both λ and j . As a result, droppingthe indices j and λ and making the other simplifying as-sumptions listed above, we see that Eqs. (1)-(3) reduceto dn S dt ≈ − ω c n S n I (1 + µ ) ,dn I dt ≈ ,dn E dt ≈ ω c n S n I (1 + µ ) , (5)with µ = 1 for scenario A and B and µ = 0 for scenarioC.In solving Eqs. (5), we assume n λ ; j = n S,λ ; j + n I,λ ; j + n E,λ ; j = ¯ n to be constant and independent of both j and λ . Accordingly, ¯ n only depends on the rate of peopleentering the system. The total number N of individualsin the system at time t is proportional to the numberof entering points, that is, to the number of differentvalues of λ , times the rate at which people enter the sys-tem, ω in = ω in; S,λ + ω in; I,λ + ω in; E,λ , minus the exit rate, ω out (which we assume to be equal to ω h , times the totalnumber of people in the last cells that, in the uniform,stationary regime, is just equal to N/L ). Therefore, oneobtains dNdt = ω in Λ − ω h NL , (6)with Λ being the number of different values of λ . Eq. (6)is solved by setting N ( t ) = ω in Λ Lω h (cid:16) − e − ωhL t (cid:17) . (7)Extrapolating from Eq. (7) the asymptotic value of N ( t )for t → ∞ , we can readily get the average density ofpeople in each cell for each value of λ , ¯ n , which is givenby ¯ n = N Λ L = ω in ω h . (8)Once ¯ n is fixed by the (asymptotic) system dynamics,it is still possible for individuals in the “local” populationto switch among compartments. This is, in fact, encodedin the “local” SEIR-like Eqs. (5). To discuss the SEIR-dynamics, we therefore solve Eqs. (5) by setting n S ( t =0) = n ,S , n I ( t = 0) = n ,I , and n E ( t = 0) = n ,E , with n ,S = δ S ¯ n ,n ,I = δ I ¯ n ,n ,E = (1 − δ S − δ I ) ¯ n , (9)and 0 ≤ δ I , δ S ≤ δ I + δ S ≤
1. Solving Eqs. (9), oneeventually obtains n S ( t ) = n ,S e − ω c n ,I (1+ µ ) t ,n I ( t ) = n ,I ,n E ( t ) = n ,E + n ,S (cid:16) − e − ω c n ,I (1+ µ ) t (cid:17) . (10)In our approach, the key observable to quantify the levelof infection due to individual motion through our systemis the infection ratio C r,λ : j ( t ), that is, the number of peo-ple that in cell- j, λ get infected in a time t normalized tothe number of healthy people that entered the system at t = 0. Clearly, within our stationary solution we expect C r,λ : j ( t ) to be independent of both λ and j . Accordingly,for the sake of simplicity we henceforth denote it simplyas C r ( t ). From Eqs. (10), we obtain that the infectionratio at time T , C r , is given by C r = n E ( T ) − n ,E n ,S = (cid:16) − e − ω c n ,I (1+ µ ) T (cid:17) . (11)Apparently, C r measures the hazard for a healthy indi-vidual to go across the system in the possible presenceof infected people. Among the parameters at the right-hand side of Eq. (11), n ,I depends on the environmentalconditions about the infection spillover, µ depends in aknown way on the system topology (see the previous dis-cussion) and, therefore, T is the only parameter that hasto be estimated. While, in general, T can be extractedfrom the results of the numerical simulation, for ω ℓ = 0it be analytically computed through a weighted average,as we discuss next.“Mimicking”, in a sense, the cellular automata ap-proach, to compute T , we assume that the evolutionin time of the system takes place via a discrete se-quence of elementary time steps, each one of duration∆ t . Accordingly, the fastest path taking an individualfrom the entrance to the exit of the system has duration T min = L ∆ t . In general, however, the topology of thesystem allows for backturns which make the actual timespent in the system larger than T min . As a result, oneobtains T = ∆ t ∞ X M =0 ( L + M ) p Lh (1 − p h ) M (cid:18) L − MM (cid:19) , (12)with the probability p h = ω h ∆ t . The right-hand sideof Eq. (12) corresponds to a weighted average, with theweight given by the product of the probability to exitfrom the system in ( L + M ) steps, that is p Lh (1 − p h ) M ,times the number of permutation of such probability, thatis (cid:18) L − MM (cid:19) , where the term − T = ∆ t Lp h = Lω h , (13)that is independent of ∆ t , as expected.As long as ω ℓ = 0, inserting Eq. (13) into Eq. (11) for C r , one gets C r = 1 − exp (cid:20) − δ I ω c ω in ω h (1 + µ ) L (cid:21) , (14)and then one can draw plots of the infection ratio as afunction of either ω h , at a given ω in , or of ω in , at a given ω h . In Fig. 1, we draw C r as a function of ω h for threesample values of ω in . Remarkably, we see that, at largeenough hopping rate between neighboring lattice cells, C r barely depends on ω in and keeps as low as 1% − C r as a function of ω in for three sample values of ω h . Here, we see that thedependence of C r on ω in is strongly affected by the valueof ω h . In particular, while keeping ω h as large as 0.4s − allows for maintaining C r of the order of 10%, even ω h C r ω =0.4 in ω =0.25 in ω =0.1 in
010 0.5
FIG. 1: C r as a function of ω h (expressed in s − ) for ω ℓ = 0and, respectively, ω in = 0 . − (red curve), ω in = 0 .
25 s − (green curve), ω in = 0 . − (blue curve). The other param-eters are L = 50, δ E = 0, δ I = 0 . ω c = 0 .
025 s − , µ = 0. C r ω =0.1 h ω =0.25 h ω =0.4 h ω in FIG. 2: C r as a function of ω in (expressed in s − ) for ω ℓ = 0and, respectively, ω h = 0 . − (red curve), ω h = 0 .
25 s − (green curve), ω in = 0 . − (blue curve). The other param-eters are L = 50, δ E = 0, δ I = 0 . ω c = 0 .
025 s − , µ = 0. at pretty large values of ω in , at variance, as soon as ω h becomes of the order of 0.25 s − , C r rises up to 40%when ω in ∼ . − . Finally, for ω h = 0 . − , we seethat C r is ∼
50% already for ω in ∼ .
15 s − . Thus, fromthe synoptic comparison of the plots in Fig. 1 and Fig. 2,one may on one hand infer how C r is poorly sensitiveto the value of ω in , provided ω h is large enough, whichbasically implies that, if the individual flow through theaisles is made fast enough, the entry rate of people intothe system from the outside is not a crucial parameterto keep C r low. On the other hand, one finds that, asa function of ω in , C r is strongly sensitive to the value of ω h . Indeed, Fig. 2 basically shows how easy is for C r tobecome already as large as 50%, if ω h is not kept largeenough to avoid crowding within the system.We discuss now the effects on C r of allowing individualsto change their direction of motion, once in the system.On intuitive grounds, one expects that letting ω ℓ = 0should lower C r , at fixed values of the other system pa- ω l C r ω in ω in ω in =0.4,=0.25, =0.1 ω h =0.25 ω h ω h =0.4=0.1, FIG. 3: C r as a function of ω ℓ (expressed in s − ) and, re-spectively, ω in = 0 . − , ω h = 0 . − (red curve), ω in =0 .
25 s − , ω h = 0 .
25 s − (green curve), ω in = 0 . − , ω h =0 . − (blue curve). The other parameters are L = 50, δ E = 0, δ I = 0 . ω c = 0 .
025 s − , µ = 0. rameters. This is reasonable because now individuals areallowed to exit the system from the same side they en-tered. This means that an individual who enters thesystem from say cell-1 , , j = L and back, thus consistently lowering the risk ofinfecting other people, or of being infected by other peo-ple meanwhile. In fact, allowing people to change theirwalking direction allows them to reach faster the shopthey are interested in by lowering the path accordingly.On the mathematical side, having ω ℓ = 0 does no moreallow for exactly computing T by means of a proceduresimilar to the one leading to Eq. (12). Therefore, to plot C r as a function of ω ℓ with the other system parametersbeing fixed, we numerically derived T at given ω ℓ andtherefore substituted the corresponding value in Eq. (11)for C r . In Fig. 3, we report the corresponding curves for C r as functions of ω ℓ , with the other system parametersset as discussed in the caption. Apparently, we see thatincreasing ω ℓ always acts to lower C r . Also, as we al-ready inferred from the plots in Fig. 1 and Fig. 2, we seethat C r is further lowered by simultaneously increasing ω h and lowering ω in , as we extensively discussed above.Finally, to investigate how, and to what extent, C r de-pends on a tunable parameter, in Fig. 4 we plot C r asa function of ω h for different values of the infection rate ω c . Indeed, ω c is a parameter one can effectively act onfrom the outside by, for instance, letting people enteringthe system to wear a mask and/or to keep social distanc-ing, et cetera . From the plots of Fig. 4 we see that while,as expected, at a given ω h , the lower is ω c , the loweris C r , at the lowest value of ω c , one sees that C r keepslower than 10% as soon as ω h ≥ . − . Apparently,this is a quantitative evidence of the effectiveness of us-ing as many measures to prevent infection as possible,such as masks and social distancing. A good combina- C r ω h ω =0.025 c ω =0.015 c ω =0.005 c
001 0.5
FIG. 4: C r as a function of ω h (expressed in s − ) and, re-spectively, ω in = 0 . − , ω ℓ = 0, ω c = 0 .
005 s − (red curve), ω c = 0 .
015 s − (green curve), ω c = 0 .
025 s − (blue curve).The other parameters are L = 50, δ E = 0, δ I = 0 . µ = 0. tion of prevention measures with a pertinent engineeringof the individual pathways inside a given system, as wellas with an appropriate regulation of the entrance rate inthe system, can apparently work as an effective mean tokeep the level of infection pretty low.To comment about our result, we note that our theoreti-cal model is definitely able to catch some interesting qual-itative behaviors. However, it overestimates the infectionratio, due the mean field approximation we employ toprovide an explicit form for the joint probability func-tion, f λ ; j . Indeed, for example, if we consider scenario Cwith a hopping probability equal to one, the contagionshould be exactly zero if less than an individual entersthe system at each turn. However, the theoretical modelis not able to reproduce this result because it totally ne-glects the actual people distribution in real space withinthe system. Furthermore, the mean field approximationis not able to properly distinguish between scenarios Band C. Indeed, as stated above, implementing the meanfield approximation fixes the parameter µ at µ = 1 in sce-nario B and at µ = 0 in scenario C. Yet, from the formulafor C r within mean field approximation, Eq. (11), we seethat the plots in the two cases just collapse onto eachother, provided one sets ω in in scenario C to be twice aslarge as ω in in scenario B. This is a consequence of thefact that, in both cases, people are divided into two sepa-rate groups: in scenario B they are divided into two lanes;in scenario C they are divided according to their direc-tion of motion. In both cases they interact with half ofthe people they would interact with in scenario A. How-ever, in scenario C, we have to consider the relative speedbetween people that, in the mean field approximation issimply thrown away (two people with opposite velocitymeet for sure while people moving in the same directiondo not). For this reason, as well as defining a playgroundto extend our approach to a systematic analysis of thetransient regimes and of a generic case of time-space de-pendent rates, in the following we resort to the cellular automata approach, by means of which we will be ableto implement the joint probability function in terms ofsimple rules. III. THE CELLULAR AUTOMATA RULES ANDTHEIR IMPLEMENTATION
In this section we discuss in detail the rules of the cellu-lar automata describing the spread of an infection withinan open, finite connected graph and how we implementthem. In particular, to keep consistent with the analyti-cal derivation of Sec. II, in the following we focus on thethree different scenarios we proposed there.Let us begin our discussion with scenario A. Refer-ring to Fig. 5, we model this system as a 2 × L squaregrid with von Neumann neighborhood. On indexing eachplaquette of the lattice with a “row” and a “column“ in-dex, we use the row index to store the information onthe direction of the motion of each individual, while thecolumn index keeps track of the spatial position. Ac-cordingly, we regard each cell as a portion of a “road” ofphysical length L . The first row of the matrix representspeople moving to the left (that corresponds to having λ = − , L ) tocell (1 , λ = +1 in thenotation of Sec. II), from cell (2 ,
1) to cell (2 , L ) (notethat, in scenario A, cells (1 , j ) and (2 , j ) overlap eachother in real space). To each cell ( λ, j ), we associatethree positive integers, n S,λ ; j , n E,λ ; j and n I,λ ; j , respec-tively corresponding to the number of healthy, exposedand infectious individuals moving in the direction λ , asdefined in Sec. II. While we allow more than a single in-dividual to occupy the same cell, to take into accountthat each cell corresponds to a finite region in space,we put a constraint on maximum limit of individualsin each cell. Letting d to be the side of each (square)cell, we are therefore limiting the maximum number ofpeople that can physically enter a d × d square regionof space. Accordingly, we require that, ∀ j = 1 , . . . , L ,we obtain P λ = ± ( n S,λ ; j + n E,λ ; j + n I,λ ; j ) ≤ n max , j , with n max , j = n max = 15 for all cells, as it appears to be rea-sonable for the cell size we consider (see below for thedetailed discussion about our choice of the system pa-rameters).The populations inside each cell are updated everytime step ∆ t . Each time step is composed by two phases:the movement turn and the contagion turn . At variancewith respect to our theoretical framework in Sec. II, inthis case we are considering integrated rates at each sin-gle turn, that is, probabilities. Therefore, referring tothe definition of the various rates in Sec. II, we denotewith p h = ω h ∆ t the probability that, in a single turn,an individual either moves backward from cell (1 , j ) tocell (1 , j − , i ) tocell (2 , j + 1) (focusing on the “inner” cells, that is,1 < j < L ). Going along the rate formalism of Sec. II, we scenario Ascenario Bscenario C FIG. 5: Scenario A: people moving over a narrow street ora sidewalk; they can walk in both directions and are closetogether at a distance less than the safety distance d . Thearrows pictorially encode the cellular automata rules that actevery time step ∆ t . In particular, the top row corresponds topeople moving to the left, the bottom row to people movingto the right. In the bulk of the system, individuals can makea step to the next cell with probability p h (red arrow), orchange their direction, with probability p ℓ (green arrow). Atthe boundaries, an individual can enter into the system witha probability p in (blue arrow). During the contagion phase,an healthy individual can be infected by infectious individu-als moving in either direction (here, as well as in the scenarioB and in scenario C, the yellow background represents therange of the infection). Scenario B is like scenario A, butwith people moving in both directions distributed along thetwo sides of the street (the “sublattices”). People can changetheir direction of motion or change street side or both (theprobabilities, p ℓ , associated to the three processes are, in prin-ciple, different) but, if they lie on different sublattices, theycannot infect each other. Scenario C is like scenario B, buteach sublattice hosts individuals moving in one direction only.The cellular automata probabilities are related to the rates ofthe theoretical model of Sec. II by p A = ω A ∆ t . define p ℓ = ω ℓ ∆ t to be the probability for an individualto change in a turn its direction of motion, that is, tomove from cell (1 , j ) to cell (2 , j ), or vice versa . Individ-uals, whether healthy, exposes or contagious, enter thesystem at each turn from the boundary cells (1 , L ) and(2 , p in = ω in ∆ t the probabilitythat one individual enters the system in a time step ∆ t and also letting δ S , δ E , δ I the average fraction of thetotal population corresponding to healthy, exposed andinfectious individuals respectively, we have that the prob-ability for an healthy, exposed and infectious individualto enter the system in a single time step is respectivelygiven by p in ,S = δ S p in , p in ,I = δ I p in and p in ,E = δ E p in ,clearly with δ S + δ E + δ I = 1 (note that, while p in isthe entrance probability for a single pedestrian, in gen-eral, more than a pedestrian can enter the system at eachturn). Finally, individuals can exit the system from thecells (1 ,
1) or (2 , L ). When a pedestrian exits the system,it is removed from the total count of people within thelattice and, at the same time, it triggers a counter thatkeeps trace of its health status. Likewise, another counterkeeps trace of the people that effectively entered the sys-tem. Each hopping, entering or change of direction isallowed only if the total population inside the target cellhas not saturated to the allowed value n max .The infection turn takes place right after the move-ment turn. In each cell ( λ, j ), the health status of eachpedestrian has a probability p c = ω c ∆ t to switch from S to E for each contagious individual that is present incells (1 , j ) and (2 , j ). Consistently with the assumptionsdiscussed in Sec. II and in Appendix A, we do not allowthe status of E - and I -individuals to change.Switching to scenario B, one readily sees that it is asimple “double copy” of scenario A. In this case, the ma-trix has four rows, rather than two, with two rows pereach “side of the street”. Therefore, one has two dif-ferent directions of motion per each side of the street,with the possibility, for a single individual, to switch, atthe same time walking side and/or direction. In order tocompare the results between scenarios A and B we keepfixed the incoming individual flow halving the incomingprobability p in .Scenario C is the same as scenario A, except for thefact that people moving in opposite directions are phys-ically separated in real space. Therefore, during thecontagion turn, an S -individual can only be infected byan E -individual moving in the same direction. Further-more, the constraint on the maximum allowed numberof individual in each cell must be satisfied separately foreach value of λ , that is, the constraint takes the form( n S,λ ; j + n E,λ ; j + n I,λ ; j ) ≤ n max ,λ .At the end of the simulation, we compute the infec-tious ratio C r as the number of E -individuals leavingthe system, minus the number of individuals already ex-posed before entering into the system, over the numberof healthy individuals entered into the system.The cellular automata parameters depend on themodel and on the physical system we are interested todescribe. In our simulation, we employ the following pa-rameters: • The cell dimension d : this should be pertinentlychosen to be of the order of the safety distance be-tween individuals, to keep consistent with our as-sumption that infection spread only happens insidea cell. Accordingly, we set d = 2m; • The number of cells L : this is not a crucial param-eters. It measures the length of the pathway we areconsidering in units of d . Throughout all out calcu-lations, we set L = 50 but, as stated above, it canbe readily changed to simulate longer, or shorter,paths; • The time step ∆ t : this measure the time requiredby a pedestrian to walk for d meters without slow-ing down. In the following, we set ∆ t = 2s; • The infection probability p c : in general, this de-pends on the effective virus transmission proba-bility, on the pedestrian health status (which, inturns, depends on age, gender, et cetera ), and bythe protective equipment wore by individuals, suchas masks and gloves; • Percentage of infectious individuals effectively en-tering the system, δ I : while, in principle, δ I is de-termined by the average percentage of infectiouspeople to the main population, at the entrance tothe system it can be strongly reduced (comparedto the outside) by, e.g. , checking the body tem-perature at the entrance and by forbidding peoplewith symptoms like fever or cough to enter the sys-tem. Indeed, in the case of infection by Covid-19, ithas been estimated that about 43 ,
8% of infectiouspeople have fever before hospitalization. Other dis-criminants, for checks at the entrance, can be ageand gender, indeed the median age was computedas 47 years, 58 .
1% were males, and only 0 .
9% ofpatients were aged below 15 years . • Entrance probability p in , “hopping” probabilitywithout changing direction, p h , and probability ofchanging direction, p ℓ : these depend on the num-ber of individuals in the system and on the timespent by them in units of ∆ t . Realistic estimates forthose probabilities (or for the corresponding rates)can be extracted from the crowd fluxes measured,for instance, by the transit crowdedness functionof Google Maps , in previous years. In addition,one should also take into account the possibilityof “artificially modifying” these probabilities by, e.g. , influencing people with disclaimers, turnstiles,watchmen or ”nudges” .As a main sample of CA results, in Fig. 6 we plot C r asa function of p h for all the three scenarios described inSec. II. To make the windows of values of the independent variable consistent with the one we use to draw Figs. 1,4,we let 0 ≤ p h ≤ p h = ω h ∆ t and since ∆ t = 2s, corresponds to 0 ≤ ω h ≤ . − of Figs. 1,4. As expected from the main features of thethree scenarios, at values of all the system parametersall equal in the three different cases, for what concernsthe infection rate, scenario A is the worst, since individ-uals moving toward opposite directions are not spatiallyseparated from each other, scenario C is the best, dueto the condition µ λ, ¯ λ = 0 (see the discussion in Sec. IIfor details), scenario B is halfway between the two ofthem. Remarkably, while we recover an acceptable qual-itative agreement with the results discussed in Sec. II, weobserve how, differently from the mean field approxima-tion, the CA approach is able to correctly discriminatebetween scenarios B and C and to reproduce the rightlimit C r → p h → t , theyexactly correspond to the ones we used to draw Fig. 1.In particular, Fig. 1 was drawn for µ = 0, which correctlydescribes scenario C only. For this reason, in Fig. 7 wedraw a synoptic plot of the curve of Fig. 1 correspondingto the parameters chosen to draw Fig. 6 and of the pointsof Fig. 6 corresponding to scenario C (note that we use p in as the sole independent variable for both plots, afterconverting ω in into p in using p in = ω in ∆ t ). Importantlyenough, the synoptic comparison shows that, at givensystem parameters, the mean field approximation sys-tematically overestimates C r at a given p h , which showsthat the simple analytical approach, in a sense, providesin a simple way a “safe” upper bound on the infectionrisk.To further compare the CA approach to the analyticalmean field approximation, in Fig. 8 we plot the numeri-cal data from the CA for C r as a function of p in at p ℓ = 0for various values of p h and with all the other parame-ters chosen as in the derivation of Sec. II (see the figurecaption for details). Qualitatively speaking, we see thatthe trend of the data in Fig. 8 is the same as we displayin Fig. 2, which was derived by applying the mean fieldapproximation to the generalized SEIR model. Specifi-cally, at a given value of p in , increasing the probabilityfor individuals to move from one cell to the neighboringone (that is, increasing the average speed of the pedes-trian motion in the pathway) determines a remarkablelowering of C r , as expected in view of the fact that, asdiscussed in the previous section, due the possibility forindividuals to exit from the same side of the street, themean time spend by each of them inside the system is re-duced, as well as the probability of infecting, or of beinginfected.Finally, to complement the results reported in Fig. 4with the corresponding analogs derived within CA frame-0 C r
10 0 1.0 h p scenario Ascenario Bscenario C FIG. 6: C r as a function of p h with, respectively, p in = 0 . p ℓ = 0, p c = 0 . δ E = 0, δ I = 0 .
05 computed within CA ap-proach for scenario A (red pointplot), scenario B (green point-plot), scenario C (blue pointplot). As expected, scenario Ais the worst, since individuals moving toward opposite direc-tions are not spatially separated from each other, scenario Cis the best, scenario B is halfway between the two of them . h p C r FIG. 7: C r as a function of p h computed in scenario C (greenpointplot) and within mean field approximation with µ = 0(full red line) with p in = 0 . p ℓ = 0, p c = 0 . δ E = 0, δ I = 0 .
05. Apparently, the mean field calculation alwaysoverestimates C r , compared to the “exact” CA result. work, in Fig. 9 we plot C r as a function of p h for scenarioC, for various values of p ℓ and all the other parametersset to quantitatively ground the comparison with Fig. 4(see the figure caption for details). As expected, the CAresults confirm that increasing p ℓ by keeping all the otherparameters fixed acts to lower C r , as it basically lowersthe average time spent by individuals in the system (seeSec. II for a detailed discussion about this point).Eventually, we recover an excellent consistency betweenthe (approximate) analytical results and the (basicallyexact) numerical ones obtained within the CA frame-work. This evidences that, on one hand, the mean fieldapproach we employ to solve the generalized SEIR equa-tion provides an acceptable level of qualitative descrip-tion of the system behavior, on the other hand, that the C r in p ppp hhh =0.1=0.5=0.8 FIG. 8: C r as a function of p in and, respectively, p ℓ = 0, p c = 0 . δ E = 0, δ I = 0 .
05 and p h = 0 . p h = 0 . p h = 0 . C r h p ppp lll =0.0=0.1=0.3 FIG. 9: C r as a function of p h and, respectively, p in = 0 . p c = 0 . δ E = 0, δ I = 0 .
05 and p ℓ = 0 (red pointplot), p ℓ = 0 . p ℓ = 0 . CA approach can be effectively implemented to improvethe quantitative reliability of the results, when necessary.
IV. CONCLUSIONS
In this paper we have formulated a generalized SEIRmodel on a graph that allowed us to describe the popula-tion dynamics of an open crowded place with an, in prin-ciple, arbitrary topology. To illustrate the effectiveness ofour model, we have discussed a few, simple paradigmaticcases, which we have treated both analytically, within amean field approach to the full set of SEIR model differ-ential equations, and numerically, by means of a cellularautomata simulation of the individual dynamics in thesystem. As a main result of our derivation, we were ableto provide the infection ratio C r as a function of “tun-able” system parameters, which eventually enables us toshow to what extent controlling human-depending effectsmay act to lower the disease spread in the system.1As an immediate further development of our work, wenote that, within our approach, one may readily extrap-olate the ratio between individuals that become exposedinside the system and infectious individual coming fromoutside as R = δ S δ I C r . (15)Dividing R , obtained within CA simulation, by the timeby which we run the simulation and then multiplyingthe result by the overall fraction of time usually spentby an individual into the system under analysis, in aperiod as long as the incubation time, one may obtain thenumber of people infected by each infectious individual,that is the basic reproduction number R , related toa specific environment. As a next development of ourwork, we plan to estimate this quantity, that is only afraction of the cumulative R (that is the sum of the R of all the places the individual spent time in), for differentscenarios, e.g. , a shopping center, a pedestrian track, agym, a school, and so on. Eventually, we plan to use itour results as a tool to evaluate the infection hazard ofa given place compared to others and to, e.g. , suggestwhich place would be safer to open first after a globallockdown. Finally, it is worth stressing that, as R is aneffective parameter that depends on the virus and on thesocial behavior, our approach allows for discriminatingthe contribution due the virus intrinsic properties from the human-depending effects like social distancing,personal protective equipment and so on, and to providea quantitative information about how to act to reducethe latter contribution.Finally, we remark that, while, in order to make thepresentation of our approach the straightest possible, weconfined ourselves to three simple sample scenarios, ourapproach can be easily generalized to more realistic and,unavoidably, more complex graphs, by means of a properimplementation of the neighborhood between cells withinCA simulation. Acknowledgements –
A.N. was financially sup-ported by POR Calabria FESR-FSE 2014/2020 - LineaB) Azione 10.5.12, grant no. A.5.1.
Appendix A: Review of the compartmental SEIRmodel
The compartmental SEIR model is a generalization ofthe widely used SIR model in epidemiology . Specif-ically, the SEIR model is based on parting a populationof N individuals (populating an isolated area, so that N is assumed to be constant) into four compartments, thatis • The
Susceptible compartment ( S ), that is made outof healthy individuals who can be affected by thecontagion; • The
Exposed compartment ( E ), that is made outof individuals who have been infected, but are notyet infectious; • The
Infectious compartment ( I ), that is made outof infectious individuals (the ones that can infectindividuals in the S compartment); • The
Recovered compartment ( R ), that is made outof individuals who either recovered from the infec-tion, or died.In its standard and simplest formulation, the SEIRmodel describes the evolution in time of the number ofindividuals in each sector by means of a set of differential,rate equations, given by dSdt = − ωN SIdEdt = ωN SI − αEdIdt = αE − γIdRdt = γI . (A1)As discussed in the main text, we describe each latticecell as a single SEIR model in which, however, the to-tal number of individuals can change in time, due tothe nonzero probability of hopping between a cell andthe nearest neighboring ones. In particular, this impliesthat, even when considering stationary solutions to thedynamical evolution equations for the local population,they are only on the average (in time) consistent with theconservation of the total number of individuals per eachcell. The various rates in Eqs. (A1) are clearly identifiedas follows: • ω is the “specific infection rate”, that is, the proba-bility per unit time that an individual belonging to S is infected by getting close to another individualbelonging to I ; • α is the probability per unit time that an individualswitches from E to I , that is, the rate that thevirus incubation ends and the individual becomesinfectious; • γ is the rate for an individual from I to switch to R . According to the specificities of the model, γ is identified with the healing-plus-death rate for anindividual from E .Even a rather simplified set of equations such as theones in Eqs. (A1) can be able to provide reliable infor-mations on the infection spillover, provided the variousrates at the right-hand side of Eqs. (A1) (the “parame-ters”) are pertinently estimated. For instance, in the caseof Covid-19 infection spreadout in Italy, we employ thenumerical values for the parameters rigorously estimated2in Ref. [35], that is, ω = 2 . − , α = 0 . − , γ = 0 . − , values that appear to fit well the in-fection dynamics in two among the mostly populated re-gions in Italy: Lombardy (Northern Italy) and Campania(Southern Italy).While, over a long enough time, the system ofEqs. (A1) always implies, given the parameters listedabove, a maximum in the contagion spreading curve andan asymptotic saturation of R ( t ) to R ∞ = R ( t → ∞ ) = N , throughout our work we set α = γ = 0 and, ac-cordingly, we neglect the equation for R ( t ). Indeed, inthe specific system we describe, we assume that the timespent by each individual inside the system is pretty shortcompared to the virus incubation and healing-plus-death rate. Accordingly, I ( t ) can become different from zeroonly because of people that come from outside, with anegligible change due to the switch in time from E ( t ) to I ( t ). 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