GGraphical Abstract
Percolation and the pandemic
Robert M. ZiffA system of non-planar extended-range bond percolation with z = 68nearest neighbors within a radius of √
20 around each site, on a square lat-tice at the bond threshold of p c = 0 . a r X i v : . [ c ond - m a t . d i s - nn ] J a n ighlights Percolation and the pandemic
Robert M. Ziff • Research highlight 1: For simple tree networks, the threshold corre-sponds to a critical infectivity of R ,c = ( (cid:104) z (cid:105) − p c = 1 • Research highlight 2: Adding cliques to the tree increases R ,c whenconsidering the outgoing bonds, thus suppressing the formation of anepidemic. • Research highlight 3: For extended-range site or bond percolation R ,c >
1, which implies clustering with finite-range connections alsosuppresses epidemics. ercolation and the pandemic
Robert M. Ziff
Center for the Study of Complex Systems and Department of Chemical Engineering,University of Michigan, Ann Arbor, Michigan USA 48109-2800
Abstract
This paper is dedicated to the memory of Dietrich Stauffer, who was a pioneerin percolation theory and applications of it to problems of society, such asepidemiology. An epidemic is a percolation process gone out of control, thatis, going beyond the critical transition threshold p c . Here we discuss how thethreshold is related to the basic infectivity of neighbors R , for trees (Bethelattice), trees with triangular cliques, and in non-planar lattice percolationwith extended-range connectivity. It is shown how having a smaller range ofcontacts increases the critical value of R above the value R ,c = 1 appro-priate for a tree, an infinite-range system or a large completely connectedgraph. Keywords:
1. Introduction
This paper is dedicated to the memory of Dietrich Stauffer, whose workand life had an enormous impact on me. His encouragement was unflagging,and his criticisms honest. We had many discussions about aspects of perco-lation including universality, thresholds, and algorithms over the years, forwhich I am immensely grateful.Dietrich was the master of a quickly but clearly written paper, sponta-neous, concise and to the point, something I have always tried to emulate.When explosive percolation first appeared in with the work of Achlioptaset al. [1], Dietrich suggested to me that I study it using the algorithm ofMark Newman and myself [2] on the square lattice rather than the complete
Email address: [email protected] (Robert M. Ziff)
Preprint submitted to Physica A January 5, 2021 raph as studied by the authors of [1]. I told him I was already thinking ofit, and he told me it was a here and now problem and that I had to do itimmediately—I had one week to get it done! In fact I finished it in four daysand it ended up as a PRL [3]. For one week I acted like Dietrich.One of the projects he repeatedly challenged me to was to find an exactexpression for the site percolation threshold on the square lattice, somethingI had found numerically to relatively high precision ( p c = 0 . p c = 0 . random numbers to be generated.) Whenpercolation was in its infancy, the best results were found using series, andthen Monte Carlo took over. Now an analytical approach (admittedly with alot of help from the computer) is again supreme over more brute-force MonteCarlo methods.Several years ago there was a meeting of the German Physical Society inBad Honnef, not far from Cologne where Dietrich lived. I visited him and hespent two days showing me around the area focusing on the political sites—things related to the war and the post-war period in a divided Germany,which interested him enormously. It was a wonderful visit. I didn’t know itwould be the last time I would see him.Dietrich was a pioneer in social physics, and in applying ideas relatedto percolation to various social problems. Of course a problem related topercolation is epidemiology, a subject of high relevance today.The basic problem of a spread of a disease is essentially a percolation2roblem, in which individuals spread an infection to contacts which hasthe prospect of going out of control and infecting a large fraction of thepopulation—exactly like a percolation transition.The conventional approach to studying epidemiology is through the SIR(Susceptible, Infected, Recovered) model of Kermack and McKendrick [13]which dates back almost to the time of the last pandemic. This model isessentially a mean-field model in which people are put into “compartments”representing the various categories of Susceptible, Infected, Recovered, andothers. When the basic reproduction number R , which represents the av-erage number of people ultimately infected by one infected individual, isgreater that 1, the infection can grow without bound. R ,c = 1 is essentiallythe percolation threshold for this mean-field model.When put on a lattice, the SIR model takes into account the geometryof contacts, and is in the same universality class as ordinary percolation[14, 15]). Normally one believes that a lattice model cannot be a good modelfor human disease spread because people travel both locally (work, shop-ping, entertainment, eating out) and globally (trains, air travel). During thepresent pandemic, however, this drawback may not be serious as people aregenerally less mobile (although not entirely, of course). Here, lattice-basedimmobile models may be more relevant. In this paper we discuss both Bethelattice models, which are effectively infinite-dimensional, and regular latticesin two and three dimensions. (a) (b) (c)(a) (b) (c) Figure 1: (a) The Bethe lattice, (b) Husimi tree with bonds, (c) Husimi tree (cactus) oftriangles.
2. Bethe lattice
First we review the case where the contacts do not interfere with eachother and the infection behavior is essentially as a tree. This corresponds3o the case of a Bethe lattice, or equivalently to low probability on a largefully connected graph (the Erd˝os-R´enyi model) where the chance of loopformation is low. The Bethe lattice is discussed in detail in Stauffer andAharony’s book [6].On a regular Bethe lattice, each node (site or vertex) is connected to z other nodes in a tree structure. An illustration of the Bethe lattice for z = 3 is shown in Fig. 1a. In the bond percolation problem, we assume allbonds (links) are open or occupied with probability p and closed otherwise.The bonds represent the transmission to the individual who is at the end ofthe bond. The percolation threshold can be found by the following simpleargument: Define Q as the probability that an outgoing bond from a nodedoes not connect to infinity (i.e., not be part of the percolating cluster orgiant component). Q satisfies the equation Q = 1 − p + pQ z − (1)where the term 1 − p represents the probability that the bond does notexist, while the second term represents the probability that the bond exists(probability p ) and the z − Q z − ). Eq. (1) can be written1 − Q = p (1 − Q z − )= p (1 − Q )(1 + Q + . . . + Q z − ) (2)which yields two solutions: Q = 1, and the solution for Q ≤ Q + Q + . . . + Q z − = 1 p (3)The transition is where the two curves intersect, which is at Q = 1 and p = p c given by p c = 1 z − z − p c ( z −
1) = 1 (5)4hich we can associate with the critical value R ,c of R = p ( z − R >
1, the epidemic will grow exponentially fast. Thus, for the simple Bethelattice, the critical R is equal to 1, consistent with the mean-field result that R > z , the probability ofinfecting a given neighbor p must be decreased to keep the system below thecritical point. But the net number of infected neighbors at the critical pointremains 1 no matter what z .The above development can be generalized to systems with a distributionof coordination number P z , z = 0 , , , . . . for each node, where (cid:80) z P z = 1.The distribution of neighbors from a node connected to a randomly chosenbond is q z = zP z / (cid:104) z (cid:105) P , where (cid:104) z (cid:105) P = (cid:80) z zP z , because a node with z bondsattached is z times more likely to be chosen than a node with one bond, andwe divide by (cid:104) z (cid:105) P to normalize the distribution. The analog to Eq. (1) isnow Q = 1 − p + p (cid:88) z ≥ q z Q z − = 1 − p + p ( q + q Q + q Q + q Q . . . ) (6)By the normalization of q z we can write q = 1 − q − q . . . (7)where q is the probability that a node reached through a bond has one neigh-bor, namely the neighbor where the bond came from, so this is a terminalbond. Putting Eq. (7) in Eq. (6) yields1 − Q = p [ q (1 − Q ) + q (1 − Q ) + q (1 − Q ) . . . ] (8)which has the roots Q = 1 and the solution to1 = p [ q + q (1 + Q ) + q (1 + Q + Q ) + . . . ] (9)At Q = 1, this equation yields the threshold [16] p c = 1 (cid:104) z (cid:105) q − (cid:104) z (cid:105) q = (cid:88) z ≥ zq z = 1 (cid:104) z (cid:105) P (cid:88) z ≥ z P z = (cid:104) z (cid:105) P (cid:104) z (cid:105) P (11)5he quantity (cid:104) z (cid:105) q represents the average coordination of nodes connectedto a randomly chosen bond. Again, this threshold corresponds to the pointwhere the outgoing bonds have an average occupancy of one: R ,c = p c ( (cid:104) z (cid:105) q −
1) = 1 . (12)Note that Eq. (10) also can be found by writing Q = 1 + (cid:15) and expandingEq. (6) about (cid:15) = 0, since that equation has a double root at Q = 1 whenyou are at the threshold p c . The relation between q z and P z is discussed morein Ref. [17].As a specific example, consider the case where P = 0 . P = 0 . (cid:104) z (cid:105) P = (0 . . . (cid:104) z (cid:105) P =18 . (cid:104) z (cid:105) q = 4 .
892 and p c = 0 . z = 3, then p c would be equal to 1 /
2. Thus, havingjust 10% of the nodes with z = 10 reduces the threshold almost in half andgreatly increases the risk of an epidemic occurring for a fixed p .Note that in the example above q = 27 /
37 and q = 10 /
37. Thus, if youpick a site randomly, its average number of neighbors is (cid:104) z (cid:105) P = 3 P +10 P =3 .
7, but if you look at a neighboring site of that randomly selected site (orthe site connected to a randomly selected bond), the average coordination ofthat site is (cid:104) z (cid:105) q = 3 q + 10 q = 4 . z around a given site follows a Poisson distribution with mean λ : P z = λ z z ! e − λ (13)which satisfies (cid:80) ∞ z =0 P z = 1. Then (cid:104) z (cid:105) P = λ , (cid:104) z (cid:105) P = λ + λ , and (cid:104) z (cid:105) q = λ +1,and it follows from Eq. (10) that p c = 1 /λ . Note that the average coordina-tion number of a randomly chosen node (cid:104) z (cid:105) P and the average coordinationnumber of a node connected to a randomly chosen bond (cid:104) z (cid:105) q differ by exactly1 here; so here it turns out that p c = 1 / (cid:104) z (cid:105) P , which is not generally the case.Another way to interpret the above result is that if one creates a networkwith a Poisson distribution with λ >
1, and each bond is occupied with6robability p , then the distribution of occupied bonds would also be a Poissondistribution, but with mean pλ and threshold p c = 1 /λ . If one makes anetwork (or Bethe lattice) with a Poisson distribution at λ = 1, then thesystem is right at the percolation point when all the bonds are occupied( p c = 1). If one has a Poisson distribution network with λ <
1, the systemwill never percolate.The complete graph, or Erd˝os-R´enyi model, is one in which all N nodesof a system are connected together by N ( N − / N − p ( N − p ( N − N this distribution becomes the Poissondistribution with mean λ = pN . and the threshold will be at p c = 1 /N . TheER graph is at the critical point if the mean number of outgoing bonds isequal to 1, just as in the Bethe lattice. For all examples of a Bethe lattice,the threshold corresponds to R ,c = p c ( (cid:104) z (cid:105) q −
1) = 1.
3. Percolation on the Bethe lattice with internal triangles.
Here we consider is a variation of the Husimi lattice or a cactus withtriangles connected to bonds in a tree structure, as shown in Fig. 1b. Wewill consider only a three-branched system here; the mathematics can begeneralized to a larger clique in the center and more branches. Such systemshave been used in the study of the spread of diseases (i.e., [18]. First weconsider a general system in which the triangle in the center is a 3-bond (or2-simplex) characterized by P = probability that none of the three verticesare connected together, P = the probability that one given pair of verticesare connected together (all three possible pairs having equal probability),and P = the probability that all three vertices are connected together, suchthat P + 3 P + P = 1 (14)Then, for Q we have Q = 1 − p + p ( P + P + 2 P Q + P Q ) (15)where P + P represents the events in which the outer vertices will notconnect to the incoming vertex of the triangle, 2 P represents the two ways7here will be one connection, and P represents the event in which bothoutgoing vertices connect to the incoming one.Writing P = 1 − P − P and substituting into Eq. (15), we find1 − Q = p [2 P (1 − Q ) + P (1 − Q )] (16)which once again has a root at Q = 1, and a second one at 1 /p = 2 P + (1 + Q ) P , and the threshold is determined by the crossing point where Q = 1,yielding the criticality condition1 = 2 p ( P + P ) (17)where P and P can also depend upon internal bond probabilities.When p = 1, we have simply that the threshold is determined by P + P =1 /
2, or, using Eq. (14), the condition P = P / / P = P that applies to a regularlattice hypergraph of 3-bonds that fall on a self-dual arrangement [19]. Thesystem here is a tree version of the generalized kagome lattice, which has beenstudied in Refs. [20] and [21], where it was found that P ( P ) ≈ . . P − . . . . which is an expansion about an exactly solublepoint, where the behavior to first-order is also linear.Now we consider that the 3-bond is indeed a simple triangle of threebonds, each occupied with probability r . Then P = (1 − r ) , P = r (1 − r ) and P = r + 3 r (1 − r ) and Eq. (17) yields the critical curve on the p - r plane as 1 = 2 p ( r + r − r ) (19)If p = r , that is all bonds on the network have the same probability, thenEq. (19) predicts p c = 0 . , ) whichon a regular lattice has a threshold of p c = [1 − π/ / = 0.807901[22], as expected a higher value than for the tree. For the two outgoing bondsleaving the triangle, the net R ,c is 2 p c = 1 . p = 1, we have the pure triangular husimi lattice with no bondslinking them together as shown in Fig. 1c, and the threshold is determinedby Eq. (19), with solution r c = 0 . R = 3 r c = 1 . R , and the closedloops attenuate the probability of an epidemic.Another model to consider is that either all three of the individuals inthe triangular clique become infected (probability P = r ), or none of thembecome infected ( P = 1 − r ), with P = 0. In other words, we are replacingthe triangle by a site and making this the site-bond percolation problem onthe Bethe lattice. Here we have from Eq. (17) simply pr = 1 / r = 1 we have the usual z = 3 Bethe lattice bond threshold p c = 1 / R c = 2 p c = 1. Now, however, by lowering r below 1, we canincrease R based on the outgoing bonds, R ,c = 2 p c = 1 /r . The minimumvalue of r is 1/2, at which point p c becomes 1 and the system becomes puresite percolation on the Bethe lattice, with R = 2. Having a probability lessthan 1 of infecting the other people in your group (i.e., decreasing r ) helpsin slowing the epidemic.While unrelated to the epidemic problem, we note that Eq. (17) is validfor correlated bonds or three-site interactions. (Of course, the site-bondproblem above is also an example of correlated bonds within the triangle.)For example, consider a system where P = 0, where every triangle has atleast one bond in it. The nomralization condition 3 P + P = 1 and thecriticality condition P + P = 1 / p = 1) implies P = P =1 /
4, or it is equally likely that a triangle is in each of the one-bond states orin the fully connected state. The fully connected triangles triangles ( P ) actas branches, and the other triangles act as single bonds, or dead-ends if thebond is one opposite side of a triangle from a given path. This is reminiscentto a model on the regular triangular lattice where each “up” triangle hasexactly one bond, which is also at the critical point [19].
4. Percolation on lattices with extended-range non-planar connec-tions
In ordinary bond percolation, the edges or bonds are independently occu-pied with probability p , and connected nodes (sites) make clusters or graphcomponents. When carried out on a regular lattice, this leads to the com-monly studied models of percolation. 9or a spread of an epidemic, a lattice percolation model has the virtuethat it can take spatial effects such as blocking into account. For most sit-uations, a simple two-dimensional lattice would not be appropriate, becauseof longer-range transmission through travel etc. However, in the presentCovid-19 pandemic, travel (both on large and small scales) has been drasti-cally curtailed, so lattice percolation may possibly be a useful model.For a spreading epidemic, we have the same situation as in the Bethelattice where the disease enters through one bond. The ( z −
1) remainingbonds are available for spreading. Assuming there is no other infection in thisregion of the lattice, the average number of bonds that carry the infection tonew sites will be p ( z − R with p ( z − R will be R ,c = p c ( z −
1) (20)For example, for bond percolation on the square lattice, where z = 4 and p c = 1 /
2, this implies that R ,c = 3 / . We see that this is significantly largerthan the critical value ( R ,c = 1) of the Bethe lattice, because many of thegrowth paths of the epidemic on a lattice are blocked by encounters withpreviously infected sites during the growth process.However, it is more relevant to have a longer range of contact includedin the model beyond the nearest neighbor. This can be incorporated intopercolation by adding long but finite-range bonds to the neighborhood ofthe individuals, who sit at the sites of the lattice. One can think of thelonger-range (or extended-range) bonds as representing a limited amount ofmobility for individuals.The study of extended-range non-planar percolation goes back to theequivalent-neighbor model of Domb and Dalton [25], where site percolationwas studied with several extended neighborhoods on a variety of lattices.Other site percolation models were studied by Gouker and Family [26], Galamand Malarz [27], d’Iribarne et al. [28], and many others [29, 30, 31, 32, 33, 34].Here we are more interested in bond percolation, where an occupied bondrepresents the spread of an infection to a susceptible individual within theallowed neighborhood. Bond percolation on lattices with extended neighbor-hoods has been studied more recently [35, 36, 34, 37, 38]. Another relatedmodel is long-range percolation which has an infinite range and a power-lawbond probability distribution [39, 40, 41].In table 1 we show results for p c (from different sources and calculationshere) and R ,c = ( z − p c for models of bond percolation with a neighborhood10 able 1: Bond percolation thresholds for square lattice with circular neighborhoods upto radius r , showing the coordination number z (number of nearest neighbors to a site)and the maximum nearest neighbor (NN) considered. Values come from the referencescited in superscripts, except z = 24, 44, 56, 68 and 80, which we determined here using asingle-cluster growth procedure and a method based upon a universal fluctuation ratio . r z NN p c ( z − p c r , namely with long range bonds that extendto all sites ( x, y ) about a given site ( x , y ) such that ( x − x ) +( y − y ) ≤ r .Note that for r > s = 16384 or 65536 occupied sites, and usedthe fact that near p c , the probability a site is connected to a cluster of sizegreater than or equal to s is expected to behave as P ≥ s ∼ As τ − [1 + B ( p − p c ) s σ + Cs − Ω ] (21)11here τ − / σ = 36 /
91, and Ω = 72 /
91 in 2D. This is valid for p very close to p c and s large. At p c , a plot of s τ − P ≥ s vs. s − Ω yields a straightline, as shown in Fig. 2 for z = 68. Note that we are finding consistency withthe usual 2D value of τ here; for very large z , one might expect Bethe-latticebehavior τ = 5 / s , but we did not see that here.We also verified this result by using the union-find simulation method [44],monitoring the largest cluster s max and its square, and finding their averagevalues and the ratio Q = (cid:104) s (cid:105) / (cid:104) s max (cid:105) . This quantity has the universalvalue of 1.04149 in 2D, for systems with a square periodic boundary [42].In a shorter amount of time than in the single-cluster simulations and in asingle run, taking a lattice of size 1024 × samples, we wereable to verify the value p c = 0 . Q = 1 . L = 2048 (270,000 samples). This method was also used for finding p c for z = 24, 44, 56, and 80 also. These simulation results support that non-planar percolation is in the normal universality class of ordinary percolation,including the ordinary critical exponents and universal fluctuation ratio Q .In Fig. 3 we plot R ,c = ( z − p c as a function of ln z , For large z , theasymptotic behavior R ,c = 1 is obtained, but for smaller z there is a markedincrease, with a peak of R ,c = 1 . z = 8 but then drops off fairlyslowly with z . This shows that by restricting possible contacts to a finiterange, the critical infection rate will be substantially larger than 1, so thatit is less likely for an epidemic to spread for a given R .Note that the large- z behavior of p c for bond percolation is expected tobehave as [45] ( z − p c ( z ) ∼ cz − / (22)(The authors of Ref. [45] consider zp c rather than ( z − p c , but to this orderin z the behavior is the same.) In Fig. 4 we plot ( z − p c as a function of z − / ,and it clearly can be seen that the large- z behavior satisfies this expectedform, with c = 2 . r are not exactly the same shape, because of thelattice effect, so one might expect some deviations from a smooth behavior.Nevertheless, the adherence to linear behavior on this plot is quite good.In Ref. [45], in Corollary 1.3, the authors reference results of a computersimulation that implies c = 1 . √ π = 2 .
13, which is a bit below the valuehere and the difference is likely a result of a lower level of statistics in thatwork. 12 .520.5220.5240.5260.5280.530.5320.5340.536 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 s t - P ≥ s s - W Figure 2: Determination of the bond percolation threshold for r = √
20 or z = 68 using thesingle-cluster growth algorithm. A plot of s τ − P ≥ s vs. s − Ω for different p . The two centralcurves suggest a threshold of p = 0 . s max = 16384except the two central curves with s max = 65536. Up to 10 runs were made for eachvalue of p , including the two systems with the larger cutoff. The line goes through theaverage of the points for p c = 0 . . In Fig. 5 we make a similar plot for 3D extended-range bond percolation,using data from Ref. [34] for smaller z and from more recent work [46] forlarger z up to z = 250. We find the data are consistent with a correction oforder z − / as predicted in Ref. [45], providing a better fit than assuming acorrection of z − / as in 2D (Eq. (22)) and was used in our previous paper[34]. Further work for even larger z is needed to demonstrate the correctionbehavior clearly.
5. Epidemics and extended-range site percolation
In extended-range site percolation with circular neighborhoods, all occu-pied sites that are within a circle of radius r of a given occupied site will beconnected to that site through long bonds of length greater than one latticespacing. In terms of epidemiology, site percolation corresponds to saying thatan infected individual will infect all susceptible neighbors within a radius r ,and the rest of the sites within that radius will never become susceptible orinfected. 13 ( z - ) p c ln z Figure 3: Plot of R ,c = ( z − p c vs. ln z using the data of table 1 (2D percolation) y = 2.7606x + 1 ( z - ) p c z -1/2 Figure 4: Plot of R ,c = ( z − p c vs. z − / using the data of table 1 (2D percolation).The linear fit for the data for z ≥
36 yields a slope of 2.76. = 2.6648x + 1 ( z - ) p c z -2/3 Figure 5: Plot of R ,c = ( z − p c vs. z − / for the 3D extended-range bond percolationdata of Refs. [34] and [46]. The peak in R = ( z − p c is lower here than the peak in 2Dsystems. The line with slope 2.665 fits the six points of data from z = 146 to 250, andsupports the prediction that the corrections go as z − / Extended-range site percolation differs from extended-range bond perco-lation in that the former is effectively planar while the latter can be non-planar — that is, paths of clusters can cross each other in extended-rangebond percolation, but cannot in extended-range site percolation. In site per-colation, one assumes that infected individual will infect all the people withinthe range of interaction and there are no additional susceptible individualswithin that range, which is not the case for bond percolation.For large r , the extended-range site percolation becomes equivalent tothe continuum percolation of overlapping disks of radius r/
2, because twodisks of that radius will just touch when their centers are a distance r apart,and the centers of the disks is where we consider the infected individuals toreside. In continuum percolation of disks, the critical coverage fraction ofdisks of radius r/ η c = π ( r/ NV (23)where N is the number of disks in a region of area V and η c = 1 . p = N/V is effectively the occupation of sites when thecontinuum percolation is superimposed on a lattice. The number of possibleneighbors on the lattice is z = πr , and putting this all together yields [34] zp c = 4 η c = 4 . z , but with a differentcoefficient than in bond percolation, where the coefficient is 1 as in Eq. (22).From the epidemic perspective, this means that you can infect up to 4.5people in your circle of influence and not have the epidemic spread. This isbecause those and subsequent people will have a smaller area of susceptiblepeople to infect, because of the overlap of the disks of influence. While inbond percolation, each individual has to infect just a little over one otherindividual to trigger an epidemic, but eventually more people within thecircle of radius r might eventually become infected by other bonds. It isan interesting question for future study to ask how many individuals withinthat radius r of the first site eventually become infected through the bondpercolation process, by means for direct paths and indirect paths.Thus, the site percolation result says that if people keep fewer than R ,c =4 .
6. Discussion
For a mean field or Bethe-tree behavior, one would expect that the criticalvalue of the infection number R is 1. Putting groups of people in a cliqueor otherwise making the tree have loops increases the critical R above 1.Furthermore, putting the individuals on a plane such as a square lattice witha limited range of infection also has the effect of increasing the critical R ,c above the value of 1, considering both site and bond percolation models.Clearly, from a public health perspective, clustering people and restrictingthe range that people can interact can inhibit the spread of a disease, and wecan see that percolation theory provides a quantitative explanation of this.It should be emphasized that in spite of the observation that for manysystems R ,c >
1, the mean-field SIR model with R ,c = 1 remains very usefulas a simple set of differential equations for understanding how various factorsaffect the spread of a disease and for easily finding the time dependence.Here we assume the individuals are stationary but interact over possiblylong but finite range bonds. The universality class of these models is still thatof ordinary percolation. It is possible that the class may change if movementof the individual is also added to the models, as was studied in [49].Here we considered just a square lattice. It may also be interesting tolook at a triangular lattice or even a random one such as the Voronoi latticeor the Delaunay triangulation, although one would expect the behavior to be16ualitatively similar. Higher dimensional lattices might be useful to modelmore complex interactions. A direct study of lattice versions of the SIRmodel with extended-range interactions should also be interesting. Anotherquestion to look at are the dynamics on various lattice structures such assmall-world and other fractal networks. This might explain the power-lawgrowth behavior seen in some regimes of Covid epidemics [41, 50, 51].
7. Acknowledgments
The author thanks Ivan Kryven for comments, and Chris Scullard forcomments and for preparing Fig. 1.
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