Locally self-organized quasi-critical percolation in a multiple disease model
LLocally self-organized quasi-critical percolation in a multiple disease model
Jeppe Juul and Kim Sneppen
Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark (Dated: October 16, 2018)Diseases emerge, persist and vanish in an ongoing battle for available hosts. Hosts, on the otherhand, defend themselves by developing immunity that limits the ability of pathogens to reinfectthem. We here explore a multi-disease system with emphasis on mutual exclusion. We demonstratethat such a system develops towards a steady state, where the spread of individual diseases self-organizes to a state close to that of critical percolation, without any global control mechanism orseparation of time scale. For a broad range of introduction rates of new diseases, the likelihood oftransmitting diseases remains approximately constant.
PACS numbers: 89.75.Fb, 64.60.ah, 64.60.al, 05.65.+b
I. INTRODUCTION
Many phenomena within materials science, physics,and biology are associated with percolation theory [1, 2].In particular, it has been shown that bond percolation isequivalent to the class of susceptible/infectious/recovered(SIR) epidemic models on a network [3–8].In such an SIR model, all nodes start out susceptibleto a new disease. If a node is infected, it will try to infectits neighbors for a fixed time τ , after which it recoversand becomes immune to the disease. How widely eachdisease is spread on the network depends on the proba-bility p , with which a node infects each of its neighborsbefore it recovers. The probability p will, therefore, bedirectly given by the disease time τ . If p is greater than acritical percolation threshold p c , there is a finite probabil-ity that the disease will span the entire infinite network,thus becoming an epidemic.In nature, percolation phenomena are often found nearthe critical probability p c [9]. A possible explanationof this is the concept of self-organized criticality (SOC),where complex systems drive themselves to critical stateswithout the need for fine-tuning of the parameters [10–12]. Many models for self-organized percolation havebeen studied [13–17]. In these, the self-organization ei-ther arises as a result of very different time scales orthrough dynamics involving a global control mechanism.For instance, a percolation system can self-organize to thecritical threshold by dynamically adjusting the probabil-ity p , such that the percolation cluster keeps growing ata specific rate [14]. However, this requires that all nodeson the network ’know’ how fast the cluster is growingglobally; a condition that is rarely fulfilled.In this paper we study an SIR model for the spread ofmultiple diseases that compete with each other. Whenmany diseases are present, they may well influence eachother by weakening of host immunity, or they may inhibiteach other through cross-immunization or by mutual ex-clusion [18–21]. Considering mutual exclusion only, wehere show that the system self-organizes to a state closeto the critical percolation threshold for a wide range of in-put parameters. That is, it exhibits self-organized quasi-criticality without any global control mechanism or sep- Figure 1. At each time step, a random node ’coughs’ on arandom neighbor and thereby transmits a random one of itsdiseases. If the neighbor is not already immune to the disease,it is infected and becomes infectious for a time τ . aration of time scale. II. MODEL
In our model, diseases are spread on a 2-d square latticewith periodic boundary conditions and N = L sites,each representing a host. At each time step the followingactions take place: • With the small probability αN , a new disease origi-nates in a random node on the network. • A random node i and one of its four neighbors j are selected. If i carries any disease(s), a randomdisease is selected. If j is not already infected orimmune to this disease, it is transmitted to j (seefig 1). • After τ sweeps over the lattice, j will be cured fromthe disease.The model uses a framework similar to the one recentlydeveloped in [22], except that the present model allowseach node to be infected by several diseases at the sametime. Also, the present model has two input parameters; α is the introduction rate of new diseases on the network a r X i v : . [ n li n . AO ] J u l while the disease time τ corresponds to the duration anynode is infectious with a disease.A key element is that each node can only transmit oneof its diseases at any given time step. Thus, a disease isless likely to spread from a node that carries many otherdiseases. The model can be run online as a java-appletat cmol.nbi.dk/models/disease/MultipleDiseases.html .If a node is constantly infected with k diseases, theprobability that it tries to infect a given neighbor with agiven disease before it is cured, can be found to be p = 1 − exp (cid:16) − τ k (cid:17) . (1)This probability corresponds to the percolation probabil-ity of a bond percolation system. When α ≈
0, diseasesare rare and infected nodes will have only one disease.When τ is large, this disease will have plenty of time toinfect its neighbors, and will therefore spread in a circularmanner with a broad rim of infected nodes and a solid in-terior of recovered nodes, very similar to the well-studiedEden growth [23–25]. When τ is small, this disease willrarely manage to infect a neighbor before dying out. For τ = 4 ln(2) ≈ .
77, we see from (1) that the probability ofinfecting a given neighbor is exactly p c = , which is thecritical threshold of bond percolation on a 2d-lattice [26].In this case, the spread of the disease becomes fractal-like and the regions of infected nodes are only one nodethick. This behavior, which is shown in figure 2(a)-2(c)for a system of size L = 256, is well-studied for singledisease models [3].When α > k and, therefore, p . For τ > .
77, the system will initially be supercritical, andall diseases will grow rapidly. Consequently, the averagenumber of diseases per node (cid:104) k (cid:105) will increase and the av-erage probability (cid:104) p (cid:105) over the lattice will decrease. If (cid:104) p (cid:105) becomes less than , most new diseases will only spreadto a handful of nodes before dying out. Thus, (cid:104) k (cid:105) willdecrease and (cid:104) p (cid:105) will increase. This negative feedbackmechanism will drive the system to a state, with (cid:104) p (cid:105) closeto the critical percolation threshold, where disease sizesof all orders of magnitudes occur. In this state, the num-ber of diseases per node is close to Poisson distributedacross the lattice, but with both spatial and temporalcorrelations in k . A high τ will result in many diseasesper node and a high α will make the system self-organizefaster, but for a wide range of both parameters diseaseswill spread in fractal-like shapes, as can be seen in figure2(d)-2(f).The average probability (cid:104) p (cid:105) of transmitting a diseasecan be measured directly by monitoring how many neigh-bors each node on average tries to infect with a disease,before it is cured. In figure 2(g)-2(i) the development (cid:104) p (cid:105) is shown for various input parameters. It is seen that (cid:104) p (cid:105) converges to a value close to (cid:104) p (cid:105) = . Figure 2. Spread of diseases on a lattice of size L = 256for different input parameters. Black nodes are healthy, andbright nodes carry many diseases. Accentuated clusters repre-sent a particular disease. In ( a − c ) only one disease is present( α ≈ τ ≈ .
77, diseases will spread in fractal shapes ofcritical percolation clusters. In the supercritical ( τ > . τ < .
77) cases, diseases will grow to spanthe entire network or quickly die out, respectively. In ( d − f ), α > p c = . Thus, disease clustershave fractal shapes for a wide range of input parameters. Thedevelopment of p is shown in ( g − i ). III. CRITICAL EXPONENTS
To compare the model to a percolation system, theclusters of recovered and immune nodes were investigatedfor different sets of input parameters. For each disease,its cluster diameter, mass and exterior perimeter weremeasured.For critical percolation the cluster mass scales with thediameter giving a fractal dimension of D = ≈ . when ατ is small and vice versa.The exterior perimeter is defined as the number of sitesin the cluster that have one or more neighbors strictlyoutside the cluster. For critical percolation, it scales withthe diameter with the critical exponent D e = [2]. Infigure 3(b) this is seen to be very close to the scaling ofdisease clusters.At the critical point, the cluster size distribution is ex-pected to fall off with the critical exponent − ≈ − . (a)Cluster mass scales with the diameter with exponent closeto the fractal dimension of critical percolation D ≈ . D e = .(c)Cluster size distribution falls off with exponent broadlydistributed around that of critical percolation − . Figure 3. Critical exponents for disease clusters when α = 2and τ = 5. Black lines show the exponents for critical per-colation, while the fitted values for the exponents are shownin the tables for different input parameters. The inserts showthe experimental data normalized with respect to the criticalpercolation exponents. Figure 4. Steady state average transmitting probability (cid:104) p (cid:105) shown for different input parameters. When τ > .
77, thenumber of diseases per node self-organizes to a value, suchthat (cid:104) p (cid:105) is close to the critical probability of percolation. Forlow α , the system is slightly supercritical and, conversely,when α is high, the system is slightly subcritical. Fine-tuningof a parameter is necessary in order for the system to be trulycritical. Note that the critical threshold is not necessarily at p c = due to correlations in the number of diseases per node. with exponents broadly distributed around this value,with steeper exponents when ατ is small. Here, the chanceof getting a disease spanning the entire network is large,but the chance of a large disease suddenly dying out islow. IV. DISCUSSION
In figure 4, the steady state probability (cid:104) p (cid:105) is shownas a function of the input parameters. The observationsagree well with the characteristics of quasi-criticality [27].The system self-organizes to a near-critical state, but afine-tuning of a parameter (e.g. α ) is necessary in or-der for the system to be truly critical. When α is toolow, diseases are transmitted with a probability some-what larger that p c , and the system is supercritical - thedisease clusters become ’heavy’ with a fractal dimensionabove D e = , an external perimeter dimension below D = and high probability of forming an epidemic.When α is too high, the system is subcritical with low (cid:104) p (cid:105) and D , high D e , and with low probability of formingan epidemic.It should be emphasized that, due to correlations inthe number of diseases per node, the critical thresholdof the multiple disease model is not necessarily equal tothat of critical percolation p c = . Diseases will tend to’get stuck’ and accumulate in regions where there are al-ready many diseases, while they will quickly ’sweep over’regions with few diseases.In the model, a node carrying k diseases will have aprobability of k to pass on each of them. This couplingmechanism between diseases is not based on empirical ev-idence, but merely reflects that diseases compete againsteach other. However, the self-organization is robust tochanges in the dynamics as long as diseases inhibit eachother. For instance, if a node carrying k diseases has aprobability of k to pass on each of them, the case α = 1and τ = 5 gives clusters with D = 1 . D e = 1 . V. CONLUSION
Using a recently developed framework for spread ofmany diseases [22] we have presented a simple multipledisease model that exhibits self-organized quasi-criticalpercolation [27]. The model is based only on local in-formation, having no global control mechanism or sepa-ration of time scale. Furthermore, the main feature ofthe self-organization is robust to changes in the extentto which multiple diseases inhibit each other. The basicmechanism employed in this model may be applicable toother systems, where “new” has an intrinsic advantageover “old”, and where transmission capacity is limited.Thus the model may be equally valid as a rumor spread-ing model, where rumors compete for attention and be-come locally outdated. In this framework the predictedself-organization may help to explain the broad distribu-tions found in human social activities. [1] M. Sahimi,
Applications of percolation theory (Taylor &Francis, 1994) ISBN 9780748400751.[2] D. Stauffer and A. Aharony,
Introduction to percolationtheory (Taylor & Francis, 1992) ISBN 9780748400270.[3] P. Grassberger, Mathematical Biosciences, , 157(1983), ISSN 0025-5564.[4] J. C. Miller, Phys. Rev. E, , 020901 (2009).[5] M. E. J. Newman, I. Jensen, and R. M. Ziff, Phys. Rev.E, , 021904 (2002).[6] M. E. J. Newman, Phys. Rev. E, , 016128 (2002).[7] L. M. Sander, C. P. Warren, and I. M. Sokolov, Phys-ica A: Statistical Mechanics and its Applications, , 1(2003), ISSN 0378-4371, stochastic Systems: From Ran-domness to Complexity.[8] M. A. Serrano and M. Bogu˜n´a, Phys. Rev. Lett., ,088701 (2006).[9] C. P. Warren, L. M. Sander, and I. M. Sokolov,ArXiv Condensed Matter e-prints (2001), arXiv:cond-mat/0106450.[10] P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett., , 381 (1987).[11] P. Bak and K. Sneppen, Phys. Rev. Lett., , 4083(1993).[12] M. Paczuski, S. Maslov, and P. Bak, EPL (EurophysicsLetters), , 97 (1994).[13] D. Wilkinson and J. F. Willemsen, Journal of Physics A:Mathematical and General, , 3365 (1983).[14] A. M. Alencar, J. S. Andrade, and L. S. Lucena, Phys.Rev. E, , R2379 (1997).[15] P. Bak, K. Chen, and C. Tang, Physics Letters A, ,297 (1990), ISSN 0375-9601. [16] C. L. Henley, Phys. Rev. Lett., , 2741 (1993).[17] S. Zapperi, K. B. Lauritsen, and H. E. Stanley, Phys.Rev. Lett., , 4071 (1995).[18] C. M. Pease, Theoretical Population Biology, , 422(1987), ISSN 0040-5809.[19] V. Andreasen, J. Lin, and S. A. Levin, Journal ofMathematical Biology, , 825 (1997), ISSN 0303-6812,10.1007/s002850050079.[20] J. Gog and J. Swinton, Journal of Mathematical Biology, , 169 (2002), ISSN 0303-6812, 10.1007/s002850100120.[21] S. Kryazhimskiy, U. Dieckmann, S. A. Levin, andJ. Dushoff, PLoS Comput Biol, , e159 (2007).[22] K. Sneppen, A. Trusina, M. H. Jensen, and S. Bornholdt,PLoS ONE, , e13326 (2010).[23] M. Eden, in Proc. 4th Berkeley Sympos. Math. Statist.and Prob., Vol. IV (Univ. California Press, Berkeley,Calif., 1961) pp. 223–239.[24] D. Mollison, Journal of the Royal Statistical Society.Series B (Methodological), , pp. 283 (1977), ISSN00359246.[25] H. M´artin, J. Vannimenus, and J. P. Nadal, Phys. Rev.A, , 3205 (1984).[26] M. F. Sykes and J. W. Essam, Journal of MathematicalPhysics, , 1117 (1964).[27] J. A. Bonachela and M. A. Mu˜noz, Journal of Statisti-cal Mechanics: Theory and Experiment,2009