Quasi-Neutral theory of epidemic outbreaks
aa r X i v : . [ phy s i c s . b i o - ph ] S e p Quasi-neutral theory of epidemic outbreaks Quasi-Neutral theory of epidemic outbreaks
Oscar A. Pinto , , Miguel A. Mu˜noz Departamento de F´ısica, Instituto de F´ısica Aplicada, Universidad Nacional de San Luis - CONICET,5700 San Luis, Argentina. Departamento de Electromagnetismo y F´ısica de la Materia and Instituto de F´ısica Te´orica yComputacional Carlos I. Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.
Abstract
Some epidemics have been empirically observed to exhibit outbreaks of all possible sizes, i.e., to be scale-free or scale-invariant. Different explanations for this finding have been put forward; among them thereis a model for “accidental pathogens” which leads to power-law distributed outbreaks without apparentneed of parameter fine tuning. This model has been claimed to be related to self-organized criticality, andits critical properties have been conjectured to be related to directed percolation. Instead, we show thatthis is a (quasi) neutral model , analogous to those used in Population Genetics and Ecology, with the samecritical behavior as the voter-model, i.e. the theory of accidental pathogens is a (quasi)-neutral theory.This analogy allows us to explain all the system phenomenology, including generic scale invariance andthe associated scaling exponents, in a parsimonious and simple way.
Introduction
Many natural phenomena such as earthquakes, solar flares, avalanches of vortices in type II supercon-ductors, or rainfall, to name but a few, are characterized by outbursts of activity. These are typicallydistributed as power-laws of their size, without any apparent need for fine tuning – i.e. they are gener-ically scale invariant– [1–6]. This is in contrast to what occurs in standard criticality, where a controlparameter needs to be carefully tuned to observe scale invariance. The concept of self-organized critical-ity , which generated a lot of excitement and many applications in different fields, was proposed to accountfor generic scale invariance, i.e. to explain the “ability” of some systems to self-tune to the neighborhoodof a critical point [1–3].The spreading of some epidemics, such as meningitis in human populations, has been repeatedlyreported to exhibit scale-invariant traits, including a wide variability of both durations and sizes ofoutbreaks. Moreover, the ratio of the variance to the mean of the distribution of meningitis and measlesoutbreak sizes have been empirically found to be very large and to grow rapidly with population-size [7].This is the hallmark of anomalously large fluctuations such as those characteristic of the heavy tailsof power-law distributions [8]. Actually, power-laws have been proposed to fit the statistics of someepidemics such as measles, pertussis or mumps in some specific locations as the Faroe islands or Reykjavikfor which accurate long-term epidemiological data are available [9–12]. Remarkably, in some cases, morethan four orders of magnitude of scaling have been found [7]. Other features of scale invariance have beenreported for measles [7] and other infectious child diseases [13, 14], rabies and bovine tuberculosis [15], orcholera [16].At a theoretical level, as pointed out by Rhodes and Anderson [9,10], the lack of a characteristic scalein epidemic outbreaks is reminiscent of earthquakes and their associated (Guttenberg-Richter) power-law distribution. Actually, the presence of scale-invariance in measles, pertussis and others has beenjustified in [12] by exploiting the analogies between simple models for such epidemics and well-known(self-organized) earthquake models [1–3]. In what follows, we focus now on meningitis, for which a relatedself-organized mechanism has been recently proposed [17–22]. uasi-neutral theory of epidemic outbreaks
Neisseria meningiditis or meningococcus, is a human com-mensal: it is typically harmless and it is present in up to one fourth of the human population [23].Infection is transmitted through close contact with previously infected individuals. It is noteworthy thatkilling their hosts is a highly undesirable outcome for bacteria; therefore it makes sense that evolutionselected for hardly harmful bacteria strains. Nevertheless, the meningococcus can accidentally mutateinto a potentially dangerous strain, becoming highly damaging or even lethal for the host. This is anexample of a more general type of “accidental pathogens” that innocuously cohabit with the host butthat eventually –even if rarely– mutate causing symptomatic disease [23].Aimed at modeling accidental pathogens and to shed some light on the reasons for the emergence ofscale invariance in meningococcal epidemics, Stollenwerk, Jansen and coworkers proposed a simple andelegant mathematical model [17–22]. The
Stollenwerk-Jansen (SJ) model is a variant of the basicsusceptible-infected-recovered-susceptible (SIRS) model. In the SIRS, individuals can be in any of thefollowing states: “susceptible” (S), “infected” (I), or “recovered” (R) (also called “immune”) [24–26].Perfectly mixed populations are usually considered, i.e. every individual is neighbor of any other (whichis a mean-field assumption in the language of Statistical Physics or a “panmictic” one in the Ecologyjargon). The additional key ingredient introduced in the SJ model is a second, potentially dangerous,strain of infected individuals labeled Y . Dangerous strains appear at a very small (mutation) rate atevery contagion event. The dynamics of Y is almost identical to that of I except for the fact that ata certain rate ǫ they can cause meningococcal disease of newly infected hosts and eventually kill them, Y → X (see below) [17–22].By working out explicitly the analytical solution of this mean-field model as well as performingcomputer simulations, the authors above came to the counterintuitive conclusion that the smaller thevalue of ǫ the larger the total amount of individuals killed on average in a given outbreak [17, 18]. Thisapparent paradox is easily resolved by realizing that the total number of individuals infected with Y growsupon decreasing ǫ . Actually, it has been shown that, whilst for high pathogenicity ǫ the distribution ofthe number of observed Y -cases, s , is exponential, in the limit ǫ → P ( s ) ∼ s − ( τ =3 / F ( s/s c ) . (1)where F is some scaling function and s c a maximum characteristic scale controlled by 1 /ǫ [17, 18].Observe that the exponent τ = 3 / τ = 3 / τ = 3 / d = 2 and d = 1. The studyof more complex networks (as small-world networks or networks with communities), aimed at describingmore realistically the net of social contacts, is left for a future work.As already pointed out [17, 18], the SJ model can be straightforwardly made spatially explicit. Evenif analytical or numerical calculations have not been performed, it was conjectured in [17–20] that in d -dimensions the SJ model should be in the directed percolation class, i.e. the broad and robust universalityclass characterizing phase transitions between an active and an absorbing state [29–34]. In the presentcase, the absorbing state would be the Y -free state obtained for sufficiently small Y -infection rates, whilethe active or endemic phase would correspond to a non-vanishing density of Y ’s. In particular, if thecritical behavior was indeed that of directed percolation, then τ ≈ .
268 for two-dimensional populations(and τ ≈ .
108 for epidemics propagating in one dimension) (see [35] and [36]).A priori, the prediction of directed percolation scaling is somehow suspicious if the model is indeed uasi-neutral theory of epidemic outbreaks not in the directed percolation class [37–41]. Furthermore, models of self-organized criticality lacking of any conservation law (as is the case of the SJ model) have been shown not tobe strictly critical, i.e. they are just approximately close to critical points [42]; instead the exact solutionby Stollenwerk and Jansen proves that their model is exactly critical [17, 18]. These considerations castsome doubts on the conjecture of the SJ model being a model of self-organized criticality [1, 2, 37–41].Therefore, one is left with the following open questions: what is the SJ model true critical behavior?,what is the key ingredient why accidental pathogens – as described by the SJ model– originate scaleinvariant outbreaks?, does such an ingredient appear in other epidemics?The purpose of the present paper is to answer these questions by analyzing the SJ model in spatiallyextended systems.
Methods
The spatially explicit SJ model is defined as follows. Each site of a d -dimensional lattice is in one ofthe following states S, I, Y, R , or X . The dynamics proceeds at any spatial location x and any time t according to the following one-site processes: • Spontaneous recovery of the benign strain: I → R , at rate γ . • Spontaneous recovery of the dangerous strain: Y → R , at rate γ . • Loss of immunity: R → S at rate α . • Replacement or recovery of diseased: X → S , at rate ϕ ,and two-site processes: • Infection with the benign strain: S + I → I + I , at rate β − µ . • Infection with the dangerous strain: S + Y → Y + Y , at rate β − ǫ . • Mutation: S + I → Y + I , at rate µ . • Disease: S + Y → X + Y , at rate ǫ .Some comments on these reaction rules are in order. Accordingly to [21] it is assumed that beinginfected with one strain protects against co-infection with a second one. This assumption, which issupported by some empirical observations [21], prevents transitions as I + Y → Y + Y from appearing. X (diseased/dead) individuals are immediately replaced by new susceptible ones, so that the total populationsize is kept fixed; i.e. ϕ → ∞ . A “back mutation” ( S + Y → Y + I ) reaction could be introduced, butfor most purposes its rate is so small that it can be neglected. The dynamics is easily implemented incomputer simulations by using the Gillespie’s algorithm [43] or a variation of it in a rather standard way.In the well-mixed case writing the densities of the different species as S , I , Y , R , and X , one readilyobtains the following mean-field or rate equations:˙ S = αR + φX − βS ( I + Y )˙ I = ( β − µ ) SI − γI ˙ Y = ( β − ǫ ) SY − γY + µSI ˙ R = γ ( I + Y ) − αR ˙ X = ǫSY − ϕX (2) uasi-neutral theory of epidemic outbreaks S + I + Y + R + X = 1. Stollenwerk and Jansen [17, 18] worked out the exactsolution of this set of equations, concluding that it is critical in the limit ǫ →
0, and obtaining the explicitform of Eq.(1).Instead, in spatially explicit models this mean-field approach breaks down: (i) the densities need to bereplaced by spatio-temporal fields, as S ( x , t ), I ( x , t ), Y ( x , t ), etc; (ii) new (Laplacian) terms describingthe map of nearest-neighbor contacts appear; (iii) fluctuations become relevant and noise terms needto be added to account for demographic fluctuations. A full set of such stochastic Langevin equationscan be derived from the microscopic dynamics by using standard procedures [28, 44, 45] and numericallyinvestigated [46] (results not shown here). Results
We now report on extensive numerical simulations for the SJ model. We have chosen the followingparameter values: α = 0 . β = 0 . γ = 0 .
1, and have tried other different sets to confirm therobustness of the results. We consider the mutation rate µ to be sufficiently small, such that strainsgenerated by consecutive mutations do not overlap; i.e. outbreaks finish well before a new mutationappears. For this reason we take as initial condition for any outbreak a state with a single mutant of thepotentially dangerous strain Y and effectively fix µ = 0 during outbreaks. Two dimensions
We consider square lattices of linear size L = 32 , , , , L = 1024, take a random initialcondition with only S and I individuals, and run the dynamics with periodic boundary conditions, keeping µ = 0, until a steady state is reached. For instance, for L = 128, the steady state is characterized by < I > ≃ . < R > ≃ .
12 and < S > ≃ .
50 for the set of parameters above.Once the system sets into its steady state, we place a Y individual at the geometrical center of thelattice and study its spreading [31–34]. To avoid finite size effects, spreading experiments are stoppedonce the Y -strain touches the system boundary, and the described procedure is iterated. Dependingon system size we ran up to 10 independent realizations. As customarily done, we monitor: (i) theepidemics size distribution, analogous to Eq.(1), for both Y and X ; (ii) the average total number of Y as a function of time N ( t ); (iii) the surviving probability P s ( t ) that the Y strain is still present in thesystem at time t , and (iv) the average square radius from the origin of Y -infected individuals, R ( t ). Atcriticality, these quantities are expected to scale algebraically as N ( t ) ∼ t η , P s ( t ) ∼ t − δ , and R ( t ) ∼ t z ,while they should show exponential cut-offs in the sub-critical (or absorbing) phase.Fig.1 shows the epidemic outbreak size distribution for different values of ǫ and system size 128 ∗ Y -infected outbreaks, s Y , in the main plot, and size of X -infected outbreaks, s x , in the inset).The probability distribution of avalanches sizes s X for X -infected sites is observed to inherit the statisticsof Y -infected ones. Both distributions can be well fitted by Eq.(1) where F is a cut-off (exponential)function and s c ∼ ǫ − /σ determines the maximum size. The best fit gives τ ≈ . / /σ ≈ P ( s Y ) and P ( s X ). From these plots weconclude that the system becomes critical in the limit of vanishing ǫ , as occurs in mean-field. Observethat for small pathogeneicities, as ǫ = 0 .
02, the system, even if sub-critical, exhibits scaling along morethan three orders of magnitude.Both the mean and the variance of the above distributions are observed to diverge as power-laws inthe limit ǫ → + . Actually, as shown in Fig.2 the ratio of the variance σ s Y (resp. σ s x ) to the mean h s Y i (resp. h s X i ) diverges when ǫ → + as σ s Y / h s Y i ∼ σ s X / h s X i ∼ ǫ − . in the infinitely large system-sizelimit (otherwise, for finite L a size-induced cut-off appear as illustrated in Fig.2 by comparing simulationsfor two different sizes). uasi-neutral theory of epidemic outbreaks ǫ = 0 we measure the mean and variance as a function of system size; the ratio ofthe variance to the mean diverges very fast as L grows, σ s Y / h s Y i ∼ L . (results not shown). Observethat, trivially, in this case P ( s X ) = δ ( s X ), i.e. no death is produced.Fig.3 shows results obtained for spreading quantities for various sizes and ǫ = 0, i.e. at criticality.The best fits we obtain for the asymptotic behavior of these three magnitudes are N ( t ) ∼ , P s ( t ) ∼ log ( t ) t , R ( t ) ∼ t (3)An effective power-law, with exponent slightly smaller than unity can be fit to our numerical results for P s ( t ) (see Fig.3); however, such an effective value of the exponent can be seen to grow upon extendingthe maximum time, making the case for a logarithmic correction as described by Eq.(3) and illustratedin Fig.4. Observe also that the reported values, η = 0, δ = 1, z = 1 satisfy the hyper-scaling relation δ + η = dz/ d = 2 [35].Contrarily to a priori expectations and to a previous conjecture, these asymptotic laws have nothingto do with directed percolation values (which predicts pure power law behavior for the three spreadingquantities [31–34] with η ≈ . δ ≈ . z ≈ . τ ≈ . τ = 3 / /σ = 2. In particular, using thesetheoretical values, the variance to mean ratio should scale as L in good agreement with the numericalfinding above. One dimension
To further confirm the conclusion above, we have also performed studies of one-dimensional lattices, forwhich the expected behavior in the voter-model class is: N ( t ) ∼ t , P s ( t ) ∼ t − / , R ( t ) ∼ t, (4)while the avalanche size exponent is τ = 4 / , , ǫ = 0. The measured τ exponent, τ = 1 . L = 1024 , , η ≈ δ = 0 . z = 1 . η ≈ . δ ≈ . z ≈ . τ ≈ . Discussion
Neutral theories date back to the sixties when Kimura introduced them in the context of populationgenetics [52]. Kimura assumed, as a null or neutral model, that each allele of a given gene (in haploidpopulations) is equally likely to enter the next generation, i.e. allele-type does not affect the prospectsfor survival or reproduction. Inspired in this, Hubbell proposed an analogous neutral theory of (forest)bio-diversity, in which the prospects of death and reproduction do not depend on the tree species [53].Both theories lead to correct predictions and also to interpretation controversies (see [54] for a recentreview). uasi-neutral theory of epidemic outbreaks voter model [34,47–49]. This is defined by two symmetrical species(up/down, right/left, black/white, etc.) fully occupying a d -dimensional lattice (or, more in general, anarbitrary network). At some rate a randomly chosen individual is removed and replaced by any of itsnearest neighbors with homogeneous probability. This leads to a local tendency to create clusters of anyof the two symmetrical species. Clusters of each of the two species will occupy different positions untileventually one of the two species will take over the whole (finite) space, leading to fluctuation inducedmono-dominance in any finite system-size,Such a type of coarsening process has been studied in depth; the asymptotic scaling of the spreadingquantities, N ( t ), P s ( t ), and R ( t ) is analytically predicted to be given by Eq.(3). In particular, thelogarithmic corrections for the surviving probability correspond to the fact d = 2 is the upper criticaldimension of the voter universality class and are closely related to the marginality of the return time oftwo-dimensional random walkers [36, 47–49]. The key ingredients of this universality class turn out to bethe symmetry between the two existing absorbing (mono-dominated) states [51, 56, 57] and the absenceof surface tension [51].It is important to underline that, given the symmetry (neutrality) between the two species in the votermodel, a given population of any of the species can either grow or decline with the same probability; i.e.there is no deterministic bias. Using the field theoretical jargon, the “mass” or “gap” term vanishes; allthis is tantamount to the model being critical [58].Indeed, for the voter model in mean-field, calling φ the density in one of the two states (and 1 − φ the complementary density for the second species), then˙ φ = − φ (1 − φ ) + (1 − φ ) φ = 0 (5)where the first term represents the loss of an individual in the first state by contact with a neighboringin the second one, and the second represents the reverse process. Therefore, the average density of φ isconstant all along the system evolution. Actually, in the voter-model there is no control parameter to betuned; the model lies, by definition at a critical point: criticality is imposed by the neutral symmetry.Instead, introducing a bias or preference towards one of the species,˙ φ = − (1 − α ) φ (1 − φ ) + (1 + α )(1 − φ ) φ = α (1 − φ ) φ (6)where the constant α > φ to grow or to decrease depending on the sign of α ,i.e. deviating the system from criticality.Introducing spatial dependence and stochasticity, Eq.(5) transforms into the Langevin equation forthe voter class [51] ˙ φ ( x , t ) = ∇ φ ( x , t ) + D p φ ( x , t )(1 − φ ( x , t )) η ( x , t ) (7)where η ( x , t ) is a Gaussian white noise (observe that the equation is symmetrical under the change φ ↔ (1 − φ ) and that there are two absorbing states at φ = 0 and φ = 1, respectively).Despite of the coincidence in the asymptotic scaling, let us underline that the SJ model is not a votermodel. In the SJ model there are 5 different species and not just 2. In the case ǫ = 0, however, I and Y are perfectly symmetrical; but as processes as Y + I → Y + Y or Y + I → I + I do not exist, replacementof one species by the other occurs only if mediated by S particles. It is, therefore, only at sufficientlycoarse grained scales that the dynamics behaves as the voter-model; microscopically the two dynamicsdiffer significantly.Even if the SJ is not a voter model, the underlying reason for it to exhibit scale-invariance is that, inthe limit ǫ = 0 and µ = 0 the model is “neutral” (i.e. strains I and Y are perfectly symmetrical) and, uasi-neutral theory of epidemic outbreaks Z the total density of infected sites whichincludes both I and Y then, at a mean field level, keeping µ = 0 and ǫ = 0˙ S = αR − βSZ ˙ Z = βSZ − γZ ˙ R = γZ − αR, (8)and the steady state is Z st = ( αβ − αγ ) / ( αβ + βγ ), S st = γ/β , and R st = Z st γ/α . Let us suppose,for argument sake, that Z st >
0; i.e. that there is a non-vanishing stationary density of infected sites(otherwise epidemics would just extinguish in finite time) and that at time t = 0 is Y ( t = 0) = Y . Theevolution of Y is controlled by ˙ Y = βS st Y − γY = 0 , (9)implying Y ( t ) = Y , i.e. there is no bias for Y to grow or decay, or in other words, under these conditionsthe model is critical in what respects the Y variable. For example, in spreading experiments we fix Y = 1 /N (and I = Z st − /N ), implying that the total number of Y -sites is 1 on average, in agreementwith the numerical findings in Fig.3 and Fig.6. Switching on a non-vanishing ǫ is equivalent in thevoter model to introduce a bias, inducing deviations from criticality, i.e. exponential cut-offs for the sizedistribution and spreading quantities, as indeed observed in our numerical simulations for ǫ = 0 (see, forinstance, Fig.3 and Fig.6).It is noteworthy that in the perfectly symmetrical (neutral) case, ǫ = 0, µ = 0, the model is somehowdull: two benign strains compete in a critical way, but there is no observable consequence of this forthe population under study. The interesting behavior of the SJ model comes from slight deviations fromcriticality, this is, µ → ǫ → ǫ >> µ ; it is in this double limit that outbreaks leave behind anobservable scale-invariant distribution of sick/dead individuals.In summary, the Stollenwerk-Janssen model provides us with the most parsimonious explanationfor the appearance of scale-invariant epidemic outbreaks caused by accidental pathogens such as themeningococcus. Our main finding is that the system is critical in the limit of vanishing pathogenicity inall dimensions, and the reason for this is that the SJ is a neutral model , which turns out to be criticalfor the very same reasons as other neutral theories in Population Genetics and Ecology are critical.As a consequence of this, the critical behavior of the model is not described by directed percolationas previously conjectured, but instead by the voter-model universality class, representative of neutraltheories. Understanding the theory of accidental pathogens as a neutral theory gives us new insight intothe origin of generic scale invariance in epidemics in particular and in propagation phenomena in general.Similar ideas might apply to related problems as the spreading of computer viruses in the Internet orin the web of e-mail contacts: the largest overall damage is expected to occur for any type of spreadingagents if their probability to cause damage is as small as possible. Being close to neutrality warrantssuccess on the long term. Acknowledgments:
We acknowledge J.A. Bonachela, A. Maritan, and L. Seoane for useful discussions as well as for a criticalreading of early versions of the manuscript. uasi-neutral theory of epidemic outbreaks References
1. Bak P, Tang C, and Wiesenfeld K (1987) Self-Organized Criticality: An Explanation of 1/ f Noise.Phys.Rev. Lett. 59: 381-384.2. Bak P (1996) How Nature works: The science of self-organized criticality. Copernicus3. Jensen H J (1998) Self-Organized Criticality. Cambridge University Press.4. Grinstein G (1995) Generic Scale-Invariance and Self-Organized Criticality, in Scale-Invariance,Interfaces and Non-Equilibrium Dynamics Proc. 1994 NATO Adv. Study Inst., Eds. A. McKaneet al.5. Grinstein G (1991) Generic Scale Invariance in Classical Nonequilibrium Systems. J. Appl. Phys.69: 5441-5446.6. Dickman R, Mu˜noz MA, Vespignani A, Zapperi S (2000) Paths to Self-Organized Criticality. Braz.J. Phys. 30: 27-41.7. Keeling M and Grenfell B (1999) Stochastic dynamics and a power law for measles variability. Phil.Trans. R. Soc. Lond. B 354: 769-776.8. Newman MEJ (2005) Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46: 323-351.9. Rhodes CJ and Anderson RM (1996) Power laws governing epidemics in isolated populations.Nature 381: 600-602.10. Rhodes CJ and Anderson RM (1996) A scaling analysis of measles epidemics in a small population.Philos. Trans. R. Soc. Lond. B Biol. Sci. 351: 1679-1688.11. Rhodes CJ, Butler AR, Anderson RM (1998) Epidemiology of communicable diseases in smallpopulation. J. Mol. Med. 76: 111-116.12. Rhodes CJ, Jensen HJ, and Anderson RM (1997) it On the critical behaviour of simple epidemics.Proc. R. Soc. London Ser. B 264: 1639-1646.13. Trottier H and Philippe P (2005) Scaling properties of childhood infectious diseases epidemicsbefore and after mass vaccination in Canada. J. Theor. Biol. 235: 326-337.14. Philippe P, (2000) Epidemiology and self-organized critical systems: An analysis in waiting timesand diseases heterogeneity. Nonlinear Dyn. Psychol. Life Sci. 4: 275-295.15. Harnos A, Reiczigel J, Rubel F, Solymosi N (2006) Scaling properties of epidemiological time series.Appl. Ecol. and Environmental research, 4: 151-158.16. Azaele S, Maritan A, Bertuzzo E, Rodriguez-Iturbe I, and Rinaldo A, (2010) Stochastic dynamicsof cholera epidemics Phys. Rev. E 81: 051901- 051901(5).17. Stollenwerk N and Jansen VAA (2003) Meningitis, pathogenicity near criticality: the epidemiologyof meningococcal disease as a model for accidental pathogens. J. Theor. Biol. 222: 347-359.18. Stollenwerk N and Jansen VAA (2003) Evolution towards criticality in an epidemiological modelfor meningococcal disease Phys. Lett. A 317: 87-96.19. Stollenwerk N (2005) Self-organized criticality in human epidemiology in Modeling CooperativeBehavior in the Social Sciences. AIP Conf. Proc. Vol 779: 191-194 uasi-neutral theory of epidemic outbreaks uasi-neutral theory of epidemic outbreaks uasi-neutral theory of epidemic outbreaks -11 -9 -7 -5 -3 -1 -8 -6 -4 -2 P ( s X ) ~ s X - / P ( s Y ) s Y P ( s Y ) ~ s Y - / P ( s X ) s x Figure 1.
Main: Avalanche size distribution, P ( s ) for various values of ǫ in a two dimensional lattice of size size128 ∗ Y -infected individuals, s Y (main plot), and s X of X -infectedones (inset). s Y / < s Y > s X / < s X > s Y / < s Y > Figure 2.
Ratio of the variance ( σ s ) to the mean size ( s ) for Y (red) and for X (black), for and different values of andsystem sizes 128 ∗
128 (full symbols) and 256 ∗
256 (empty symbols). The ratio diverges in the double limit ǫ → L → ∞ .Inset: as the main plot, but in double logarithmic scale: as system-size increases the curves converge to straight lines, i.e.power-laws, but finite size effects are significant. uasi-neutral theory of epidemic outbreaks -5 -3 -1 L1024 512 256 128 64 32 16 P s ~ t -1 N ~ 1 < P s > , < N > , < R > t R ~ t Figure 3.
Log log plot of N ( t ), P s ( t ), and R ( t ), for spreading experiments in different two-dimensional systems (linearsizes L = 32 , , , , t . Dashed lines are a guide to the eye. L1024 512 256 128 64 t P s ln(t) Figure 4.
Plot of tP s ( t ) as a function of log( t ) for different system sizes (from L = 64 to L = 1024), illustrating thepresence of linear logarithmic corrections for the surviving probability in the two-dimensional SJ model. uasi-neutral theory of epidemic outbreaks -13 -11 -9 -7 -5 -3 -1 L8192 4096 2048 1024 512 P ( s Y ) ~ s Y - / P ( S Y ) s Y Figure 5.
Avalanche size distribution, P ( s Y ) for ǫ = 0 and various one-dimensional lattices, of size L = 512 , , , -5 -2 L8192 4096 2048 1024 P s ~ t -1/2 N ~ 1R ~ t t < P s > , < N > , < R > Figure 6.
Evolution of N ( t ), P s ( t ), and R ( t ) as a function of time t in log-log scale for spreading experimentsperformed on one-dimensional lattices of linear sizes 1024 , , α = 0 . γ = 0 . β = 1 ..