Fluctuating epidemics on adaptive networks

Transcription

1 PHYSCAL REVEW E 77, Fluctuating eiemics on aative networks Leah B. Shaw Deartment of Alie Science, College of William an Mary, Williamsburg, Virginia 23187, USA ra B. Schwartz US Naval Research Laboratory, Coe 6792, Nonlinear Systems Dynamics Section, Plasma Physics Division, Washington, DC 2375, USA Receive 3 January 28; ublishe 3 June 28 A moel for eiemics on an aative network is consiere. Noes follow a suscetible-infectiverecovere-suscetible attern. Connections are rewire to break links from noninfecte noes to infecte noes an are reforme to connect to other noninfecte noes, as the noes that are not infecte try to avoi the infection. Monte Carlo simulation an numerical solution of a mean fiel moel are emloye. The introuction of rewiring affects both the network structure an the eiemic ynamics. Degree istributions are altere, an the average istance from a noe to the nearest infective increases. The rewiring leas to regions of bistability where either an enemic or a isease-free steay state can exist. Fluctuations aroun the enemic state an the lifetime of the enemic state are consiere. The fluctuations are foun to exhibit ower law behavior. DO: 1.113/PhysRevE PACS number s : Hc, 87.1.Mn. NTRODUCTON The stuy of recurrent eiemics has a long history 1, an many moels, both eterministic an stochastic, have been consiere. Deterministic moels have been use since the time of Bernoulli an have exlaine some of the mechanisms in the srea of infectious iseases. However, eterministic moels are not sufficient to account for some of the imortant stochastic ynamics, such as extinction 2,3 an sustaine fluctuations 4. From the general theory of finite Markov chains 5, it was shown that in stochastic moels the robability of extinction is equal to one in the asymtotic time limit. Numerical 6 8 an analytic 9 comarisons of stochastic an eterministic moels have been erforme. The numerical results hol for very small amlitue noise as well as real finite noise. Deterministic suscetible-infectivesuscetible SS or suscetible-infective-recoveresuscetible SRS moels result in an equilibrium enemic resence of infectives for an aroriate choice of arameters. t is clear that stochastic effects may result in very ifferent ynamics from eterministic moels, articularly when fluctuations an/or extinction occur. More recently, the stuy of fluctuations an the srea of simle moels of eiemics have been simulate on large networks n almost all of these network moels, the eiemic roagates on a fixe network. The eiemic ynamics is tyically stuie as an SS or SR moel, in which the oulation is large an isolate. n aition to the ynamics on such fixe architectures, controls base on vaccination have been consiere as well 15,16. Several recent moels have consiere eiemics on a network that changes structure ynamically accoring to rules that o not een on the noes eiemic status 17,18. n contrast to the moels of a static network or moels with externally alie changes in structure, a new class of moels base on enemic SS oulations on an aative network has been recently introuce 19. Changes to the network structure are mae in resonse to the eiemic srea an in turn affect future sreaing of the eiemic. Here, the governing arameter is one that escribes the rewiring rate of the network, which is controlle by the fraction of suscetible S -infective links. The network alters ynamically when there are contacts between S an, an social ressures the esire to avoi illness rewire the contacts to be instea between S an S. nfections are reuce ue to isolation, an for aroriate choices of arameters, bistability between the isease-free equilibrium an enemic state has been observe. This is in contrast to static networks in a large oulation, where there is tyically only a single attracting enemic or isease-free state. A ifferent moel has been introuce in which suscetible-infective links are broken rather than rewire an later reconnect at ranom; this rule for network aatation also leas to bistability an other ynamics not observe in static network moels 2. n this aer, we introuce a recovere, immune class an consier this slight generalization of the SS moel on an aative network. We examine the structure of the network an the ynamics of the fluctuations of the eiemic. Our aroach is to combine Monte Carlo simulations an stochastic mean fiel moels for eiemic evolution on evolving networks. The layout of the aer is as follows. We introuce the moel in Sec. an resent its bifurcation structure in Sec.. Proerties of the network structure are iscusse in Sec. V. We iscuss ynamical roerties of the system, incluing fluctuations an lifetimes of the states, in Sec. V.. MODEL We stuy a suscetible-infective-recovere-suscetible SRS moel on an aative network. Eiemic ynamics on the noes is as follows. The rate for a suscetible noe to become infecte is N,nbr, where N,nbr is the number of infecte neighbors the noe has. The recovery rate for an infecte noe is r. A recovere noe becomes suscetible /28/77 6 / The American Physical Society

2 LEAH B. SHAW AND RA B. SCHWARTZ again with rate q, which we efine as the resuscetibility rate. While the eiemic sreas, the network is also being rewire aatively. f a link connects a noninfecte noe to an infecte noe, that link is rewire with rate w to connect the noninfecte noe to another ranomly selecte noninfecte noe. Self-links an multile links between noes are isallowe. n examining steay state solutions, it is sufficient to fix one of the rates, as time may be rescale accoringly. For this reason, we fix r=.2 throughout this aer. We erforme Monte Carlo simulations of this moel on a system with N=1 4 noes an K=1 5 links. Larger system sizes with the same noe-to-link ratio were also consiere. The major results of this aer o not een strongly on system size. n each Monte Carlo ste MCS, we ranomly select N noes an M links, where M is the number of links that may otentially rewire suscetible-infecte an recovere-infecte links, an the links are selecte from the ool of links that may rewire. nitial conitions are constructe in one of two ways. We either generate a ranom Erös-Rényi grah of suscetibles an convert a fraction f of them to infectives, or we use the final state of a revious run as an initial conition. Transients are iscare an simulations run long enough that the initial conitions o not affect the results. Following 19, we also eveloe a corresoning mean fiel moel for the system. The mean fiel moel tracks the ynamics of both noes an links. P A enotes the robability of a noe to be in state A, where A is either S suscetible, infecte, orr recovere. P AB enotes the robability that a ranomly selecte link connects a noe in state A to a noe in state B. We obtain the following mean fiel equations for the evolution of the noes: Ṗ S = qp R K N P S, Ṗ = K N P S rp, Ṗ R = rp qp R. 3 For examle, in the first equation, recoveres are converte to suscetibles with rate q, an infection sreas with rate along each suscetible-infecte link. Rewiring oes not aear irectly in the noe equations, since rewiring oerates on links, but it affects the system imlicitly through the number of suscetible-infecte links KP S. We next write a system of mean fiel equations for the links. To close the system, we follow 19 an make the assumtion for three oint terms that P ABC P AB P BC / P B. This assumtion leas to the following system of equations for links: Ṗ S =2 K N P S Ṗ SS = qp SR + w P S 2 K P S + P R N P SS P S P S 1 2 P SS P S P S, 4 + qp R rp S wp S P S + K N 2 P S P S, 5 Ṗ = P S + K N P R 2 P S, P S 2rP Ṗ SR = rp S + w P S +2qP RR qp SR P S + P R K P S P SR N + w P S P R, P S P S + P R Ṗ R =2rP + K N PHYSCAL REVEW E 77, P S P SR P S qp R rp R wp R, 8 Ṗ RR = rp R 2qP RR + w P R. 9 P S + P R The orinary ifferential equations of the mean fiel moel can be integrate easily with any well-known numerical integration technique. We choose initial conitions so that we are near an enemic state. We also note that the moel oes suort solutions with negative values, but these are unhysical, an so we ignore them. n the case of stochastic simulations, we have consiere the effects of both internal fluctuations, moele as multilicative noise, as well as external fluctuations, moele as aitive noise. We use a fourth orer Runge-Kutta solver for each of these cases to generate stochastic stimulations of the mean fiel. The noise strength was ket small for both aitive an multilicative cases, an our stochastic stuies were run on stable enemic branches which were far from the isease-free state comare to the noise levels consiere. Thus fluctuations in the variables i not rive them to unhysical values for the arameters use here. We also tracke the steay states as a function of arameters using a continuation ackage 21. P R. BFURCATON STRUCTURE We first consier the steay state bifurcation structure of the moel. n Fig. 1, we show examles of the average infecte fraction versus the transmission rate. Two steay states can occur, a isease-free state an an enemic state. n the absence of rewiring Fig. 1 a, the isease-free state loses stability for very small transmission rates, an only the enemic state is observe at larger values. When rewiring is introuce Fig. 1 b, the isease-free state is stabilize for larger values. A region of bistability, in which both enemic an isease-free states are observe, now occurs. Bistability was observe reviously for the SS moel 19. Monte Carlo simulation oints in Fig. 1 were comute as follows. To locate the uer branches enemic state, we swet from larger to smaller values, using the final state of each run as the initial state for the next run MCS of transients were iscare, an the steay state was average over 1 4 MCS. To locate the lower branch isease-free state, we began with a ranomly connecte network in which a fraction f of the noes were infecte, while the rest were suscetible. The system was simulate for MCS. f values between.25 an.9 were trie, an at least five runs were one for each value. f the eiemic ie

3 FLUCTUATNG EPDEMCS ON ADAPTVE NETWORKS PHYSCAL REVEW E 77, LP BP HB.2 w x1 3 V FG. 1. Average infecte fraction vs transmission rate. a Static network, w=. b Rewire network, w=.4. Black ots: Monte Carlo simulations; soli gray lines: mean fiel solution stable branches ; an ashe gray lines: mean fiel solution unstable branches. q=.16, r= x1 3 FG. 2. a Two arameter lot for w an of the bifurcation oints for steay states when q=.64. The heavy ashe line is the line of sale-noe oints limit oints. The soli line enotes the Hof bifurcation oints, an the light ashe line enotes the transcritical bifurcation oints BP. b A bifurcation iagram of the infective fraction as a function of, where w=.4. The squares enote the sale-noe oint an transcritical oint. Dashe lines are unstable branches. out in any of the runs, the isease-free state was consiere stable. Due to the stochastic nature of the Monte Carlo simulation, an because the isease-free state is absorbing, stability esignations are uncertain. t is ifficult to istinguish a weakly unstable state from a weakly stable state with a short lifetime. Lifetimes of the enemic state are consiere in more etail in Sec. V. The results in Fig. 1 show fairly goo agreement between the mean fiel aroximation an the Monte Carlo simulation of the full system, an we tyically see this level of agreement in the steay state values, although as we iscuss later, the stability an tye of bifurcation sometimes iffers. Using the mean fiel moel, we next exlore the bifurcation structure of the system for a wier range of arameters. An interesting roerty of the steay state instabilities aears when one consiers each of the steay state bifurcation oints of the mean fiel equations. There o exist branches of erioic orbits, but since they occur within a very small range of arameters, we ignore them in this aer. They will be treate elsewhere. f the resuscetibility rate q is hel fixe an a bifurcation iagram constructe, we fin the existence of at least two istinct regimes for ifferent q values, illustrate in Figs. 2 an 3. The instabilities aear as a transcritical bifurcation from the isease-free steay state, a sale-noe bifurcation of enemic steay states, an a Hof bifurcation, from which a branch of subcritical unstable erioic orbits emanates not shown. n the case where q=.64, eicte in Fig. 2 b for w =.4, we show a tyical bifurcation lot of stable an unstable steay states as the infection rate is increase. The ashe lines reresent the unstable steay states, while the soli lines eict the attractors. For low values of the isease-free steay state is stable. Tracking along the isease-free branch, at a critical value of, an unstable branch subcritical of enemic steay states aears. The enemic branch becomes stable at the sale-noe oint. There exists a clear region of bistability with coexisting enemic an isease-free states for a range of. fwenow vary the arameter w an track each bifurcation curve, we obtain the result in Fig. 2 a. We escribe the bifurcation regions in etail for w larger than.1. n region, we have only a stable isease-free equilibrium. As we cross into region for large w, the isease-free equilibrium is stable, an there exists an unstable enemic state. Region exhibits bistability between the isease-free equilibrium an enemic state, an region V has just a stable enemic equilibrium. We note that at w=.4, we have the simle sale-noe transition eicte in Fig. 2 b, since there is no Hof bifurcation to erioic cycles at that articular w value

4 LEAH B. SHAW AND RA B. SCHWARTZ PHYSCAL REVEW E 77, LP HB BP.1.8 w.1 V robability x robability x1 3 FG. 3. a Two arameter lot of the steay state bifurcation oints when q=.16. The heavy ashe line is the line of limit oints. The soli line enotes the Hof bifurcation oints, an the light ashe line enotes the transcritical bifurcations. b A bifurcation iagram of the infective fraction as a function of, where w=.4 same bifurcation iagram in Fig. 1. The revious iscussion resente a case for resuscetibility rate q where the limit sale-noe an Hof bifurcation branches are close to each other. The istinction between the sale-noe an Hof branches can be seen more easily if q is lowere to.16, as shown in Fig. 3 a. n Fig. 3 a, for sufficiently large w, as is increase in region the system first unergoes a limit oint bifurcation, an then a Hof bifurcation as it asses through region. The Hof curve is actually a close isola in two arameters. The limit oint here is a sale-sale oint, where a steay state having a two-imensional unstable manifol connects to a steay state with a one-imensional unstable manifol. n both cases, we have bistable behavior for w sufficiently large, but the region of bistability is much smaller since the Hof an transcritical branches are closer together for this value of q region. For w=.4, the mean fiel enemic steay state loses stability in a sale-noe bifurcation for q=.64 an in a Hof bifurcation for q =.16. n our iscussion of fluctuations in the enemic state in Sec. V, we will refer to q =.64 because the sale-noe bifurcation structure best corresons to the scaling of fluctuations that we observe in Monte Carlo simulations of the full system egree FG. 4. Degree istributions from Monte Carlo simulation for =.2, q=.16, an r=.2. a Static network, w=. All noe tyes have Poissonian egree istribution. b Rewire network, w=.4. Soli gray line: infectives; ashe line: recoveres; an black line: suscetibles. V. NETWORK GEOMETRY A. Degree istributions Rewiring leas to significant alterations in the network structure. We first consier the egree istribution. Figure 4 shows egree istributions for each tye of noe in the absence Fig. 4 a an resence Fig. 4 b of rewiring. n Fig. 4 b, we average over MCS. n Fig. 4 a, the network is static, so we average over ten searate runs to obtain better statistics. Using mean fiel ieas, we can unerstan the recovere an suscetible egree istributions as noes flow from infecte to recovere to suscetible. We outline the calculation briefly here. However, this aroach oes not accurately reict the egree istribution for infecte noes because correlations lay a more imortant role for these noes, as we will exlain below. For this reason, we cannot write a selfconsistent set of equations for the egree istributions that coul be solve without inutting simulation results. Let X,n be the number of noes in state X either S,, or R with egree n. Recovere noes originate when infectives recover an are lost when they become suscetible again. The egree of a recovere noe can only increase, as other suscetible an recovere noes wire to connect to it. This leas to the following equations for R,n : t R, = r, q R, k R,,

5 FLUCTUATNG EPDEMCS ON ADAPTVE NETWORKS t R,n = r,n q R,n k R,n + k R,n 1 for n, 1 where k is the average rate for noes to rewire to a given noninfecte noe. k is given by the ratio of the total rewiring rate to the number of otential target noes: k = wk P S + P R. 11 N P S + P R Given the egree istribution of the infectives,n an the robabilities aearing in k, Eqs. 1 can be solve for the egree istribution of the recoveres. Suscetible noes originate when recoveres become suscetible again, an they may be lost when they become infecte by a neighbor. As with the recoveres, the egree of a suscetible increases ue to rewiring. Thus the time evolution of the egree istribution for the suscetibles can be written as t S, = q R, k S,, t S,n = q R,n k n S,n k S,n + k S,n 1 for n, 12 where k is the infection rate er link into a suscetible noe. We assume that k is ineenent of egree, which we know from simulations is aroximately correct cf. Fig. 5, an write k =, 13 P S + P SR +2P SS where the fraction is the ratio of the number of links that can transmit infection to the total number of links into a suscetible. As with the recoveres, the steay state egree istribution for suscetibles can be comute from Eq. 12. The reicte egree istributions for suscetibles an recoveres are overlai on the actual istributions in Fig. 4 b, using Monte Carlo simulation averages for the infective egree istribution an the noe an link robabilities. Note that the noe an link robabilities coul instea be obtaine from the mean fiel system. Deviations between the reiction an simulation are smaller than the with of the curves in Fig. 4 b, so they are inistinguishable. The egree of infecte noes, however, cannot be reicte by this aroximate roceure. We might exect that where P S t, = k,1 r,, t,n = k n +1,n+1 k n,n + k n S,n r,n for n, 14 robabillity nbr. infecte fraction egree PHYSCAL REVEW E 77, P S + P R k = w 15 P S + P R +2P is the er link rewiring rate for links connecting to an infective i.e., the ratio between links that can otentially rewire an total links into infectives. Figure 5 a comares the actual egree istribution for infectives with the istribution reicte by Eq. 14. The number of low egree infectives is significantly overreicte. This occurs because the mean fiel aroximation in Eq. 15 is not accurate for infectives. Figure 5 b shows the fraction of infecte neighbors that a noe has, eening on its egree an isease status. Results are average over MCS. Low egree infecte noes ten to have a much higher fraction of neighbors that are also infecte, ue to transmission of the isease. Once infecte, these neighbors will not rewire away until recovere, so the rewiring rate er link is smaller than one woul exect from the mean fiel k in Eq. 15. Preicting the infective egree istribution accurately woul require a theory that accounts for these correlations in infection status of neighboring noes, which is beyon the scoe of the resent work. B. Distance from an infective We next consier the istribution of istances from a given noe to the nearest infective. These istances are of interest because they relate to the number of hos the isease must make in orer to reach an uninfecte iniviual. The isease cannot roagate through recovere noes until they become suscetible again, so the istance from the nearest egree FG. 5. a Actual black line an reicte ashe gray line egree istributions for infectes. b Average fraction of neighbors that are infecte vs egree. Soli gray line: infectes; ashe line: recoveres; an black line: suscetibles. =.2, q=.16, r =.2, an w=

6 LEAH B. SHAW AND RA B. SCHWARTZ number number istance istance FG. 6. Distribution of istances from the nearest infecte noe. inicates noes that are comletely isconnecte from an infective. a Soli black line: with rewiring w=.4, =.2, an q =.16 ; ashe gray line: no rewiring w=, =.2, an q =.9. b Black line: rewire case sameasin a ; an gray line: istribution for ranom grahs, as escribe in text. PHYSCAL REVEW E 77, infective oes not necessarily correson to a ath for isease roagation. However, we note that rewiring acts only on links to infectives, an thus the chains of suscetible an recovere links that this metric ientifies will ersist until the infection leas to their interrution. Figure 6 a shows the istribution of istances from the nearest infective in the resence an absence of rewiring. To islay the effect on the network geometry alone, rather than on the steay state number of infections as well, we have use a smaller q value for the w= case so that the total number of infectives is aroximately the same in both curves. Results were average over MCS, samle every 1 MCS after removing transients. Rewiring significantly ecreases the number of noes that are irectly connecte to an infective. However, esite the rewiring, only a small fraction of noes are fully isconnecte from the infection. Figure 6 b shows a semilog lot of the w=.4 case to islay the tail of the curve at larger istances. An aroximation base on ranom networks is also shown. Given how far from Poisson the egree istributions are when the network is rewire, it is somewhat surrising that the form of the ecay in the istribution of istances can be reicte from ranom networks. Beginning with S,, an R noes in numbers matching that observe in the average of Monte Carlo simulations, we generate 1 ranom networks an ae ranomly selecte links until the number of S-S, S-, S-R, etc. links also matche that from the average of Monte Carlo simulations. As Fig. 6 b shows, the istribution of istances for these ranom networks ecays in the same way as it oes in an aative network. Thus the form of the istribution of istances eens mainly on local ynamics noe an link ynamics rather than on the etails of longer range correlations. The main ifference is that the aative network has some noes fully isconnecte from infecte comonents, while the ranom networks o not. Most of these noes are recoveres with egree, which aear when infectives of egree recover. Each tye of noe in the ranom networks has a Poisson egree istribution, so they o not generally have noes of egree. V. FLUCTUATONS AND OTHER DYNAMCS n the revious sections, we have consiere steay states an long time averages of network roerties. We next consier fluctuations an ynamical roerties of the enemic state. A. Fluctuations near bifurcation oint Near the bifurcation oint where the enemic state loses stability, the number of infectives has larger fluctuations ue to noise overcoming weak attracting forces 22. Fluctuations in the SRS moel are significantly larger than those in the reviously stuie SS moel. We quantify the fluctuations by comuting the stanar eviation ivie by the mean for long time series in both the Monte Carlo an mean fiel simulations. n Fig. 7, we lot the fluctuations as the infection rate is swet towar the bifurcation oint. Monte Carlo results were comute from MCS time series samle every ten MCS, excet for the two smallest values, for which shorter time series were use ue to the shorter lifetimes of these states. All time series were longer than 1 5 MCS. For comarison, the mean fiel equations can also be consiere near equilibrium in stochastic form. n general, near equilibrium fluctuations can be moele as aitive noise 23, an we o so here. We o note that multilicative noise effects generate results similar to those reorte for aitive noise. We assume the mean fiel is of the following form: X = F X + t, 16 where F X is the mean fiel system in Eqs. 1 9, an t t = t t. is the noise strength, or amlitue. We have consiere both aitive noise an multilicative noise cases in the simulations of the stochastic attractors near the enemic state an have comute the stanar eviation ivie by the mean as escribe above for ten ranom initial conitions near equilibrium an ten realizations. We recall from Figs. 2 an 3 that eening on the value of the resuscetibility rate q, the bistability regions are vastly ifferent. Secifically, for q =.16, we saw that for sufficiently large wiring rates, the sale sale an Hof bifurcation branches were well-searate, whereas for q =.64, the branches were very close for small values of rewiring rate w. n Monte Carlo simulation, we have not observe the Hof bifurcations or stable erioic oscillations seen in the mean fiel, even for system sizes as large as noes. Although the value of q=.64 we use in the

7 FLUCTUATNG EPDEMCS ON ADAPTVE NETWORKS PHYSCAL REVEW E 77, (st)/(mean) (st)/(mean) (c) ln[(st)/(mean)] ln[(st)/(mean)] ln( ) c () FG. 7. Fluctuations in infectives stanar eviation ivie by mean vs infection rate near the bifurcation oint: a Monte Carlo an b mean fiel. Curves are to guie the eye. Log-log lots ata oints with best fit lines show ower law scaling for both Monte Carlo b an mean fiel. Monte Carlo arameters: q =.16, w=.4, an r=.2. Mean fiel arameters: q=.64, w=.4, r=.2, an = ln( ) c mean fiel fluctuation stuy is ifferent from that use in the Monte Carlo, the local bifurcation structure is similar when w=.4, in that it is a true sale-noe bifurcation oint. This has been checke by examining the local linear vector fiel at the sale-noe oint in question. Therefore, although the mean fiel has ifferent q value, the bifurcation structure is equivalent to that observe in Monte Carlo simulation, so we use q=.64 in the fluctuation stuy. The comutations reveal that the fluctuations exhibit ower law scaling, as shown in the log-log lots of Figs. 7 b an 7. On the horizontal axis, we lot ln c, where c is the critical oint at which the enemic state loses stability. For the mean fiel, the bifurcation oint is known exactly by examining the eigenvalues of the linearize vector fiel at the steay state. We aroximate the Monte Carlo bifurcation oint c as the value that rouces the most linear lot. Although both cases have ower law scaling, the exonents are ifferent:.59 for Monte Carlo an.27 for mean fiel. The scaling exonent for the full system eens on the number of noes. Whether it will aroach the mean fiel value in the limit of infinite system size is a subject for further stuy. We note that fluctuations near a Hof bifurcation oint, as occurs in the mean fiel for q=.16, woul rouce a very ifferent form of scaling from the salenoe case. One of the reasons for the ifference is that the instability is then two-imensional an unerame 24. This is known to cause very ifferent scaling laws in generic roblems, which can be much slower 25. To motivate the ower law scaling of the fluctuations, we consier scaling near a generic sale-noe bifurcation. The simlest generic case of a sale-noe bifurcation for equilibrium oints comes from solving for zeroes of the vector fiel at a arameter value where one eigenvalue of the Jacobian asses through zero. Stanar normal form analysis allows one to consier the generic roblem of a sale-noe bifurcation. n one imension, the stochastic ifferential equation of a sale-noe bifurcation may be moele as x t = a x 2 t t + W t. 17 The arameter is a, an we suose noise is aitive. Since noise in general may cause a shift in arameter values where the sale-noe oint isaears, we assume that the noise near the bifucation is sufficiently small, where W/ t is a white noise term, an W is a Brownian increment. We further assume that we are always near the attracting branch of the sale noe, so we are in a near equilibrium setting. Such an assumtion allows us to examine the stationary robability ensity function PDF of the stochastic ynamics by emloying the Fokker-Planck equation near steay state. For the stochastic ifferential equation, Eq. 17, the PDF is well known 22 an is given by a,x, = Ne 2 ax x3 /3 / Here N is a normalization constant. We comute the first an secon orer moments irectly using Eq. 18 an then take the ratio of the stanar eviation to the mean. Since a= is the value of the sale-noe oint, we examine the fluctuations in the neighborhoo of that value. The results, shown in Fig. 8, islay ower law scaling. B. Delaye outbreaks We next consier hase relationshis between the fluctuating variables. We tracke the number of infectives in the system at each time oint as well as the number of noes that neighbore an infective i.e., the number of S an R links. n the rewire system, fluctuations in the number of infectives lagge behin fluctuations in the number of infective neighbors that are not themselves infecte, as shown in Fig. 9 a

8 LEAH B. SHAW AND RA B. SCHWARTZ PHYSCAL REVEW E 77, PDF moel linear fit ln[(st)/(mean)] ln T ln ( ) 3/2 FG. 8. Fluctuation size of a generic sale-noe bifurcation as a function of bifurcation arameter a near the bifurcation oint using Eq. 18. The noise level use is.5. Both mean fiel an Monte Carlo simulations of the full system islaye this effect, an we stuie their eenence on the rewiring rate. Monte Carlo simulations were samle every 1 MCS for MCS after iscaring transients. The mean fiel was samle every 1 time unit for units. Aitive noise was inclue in the mean fiel equations with noise strength =.1. Cross correlations between the infectives an the infective neighbors were comute for varying shifts between the time series, an the lag maximizing the cross correlation was ientifie. As shown in lag (MCS) infectives an nbrs MCS w FG. 9. Delaye outbreaks ue to rewiring. a Monte Carlo time series. Black line: infectives an gray line: neighbors of infectives. Curves are scale in arbitrary units for comarison of eak times. =.65, w=.9, q=.16, an r=.2. b Time in MCS by which infectives lag behin infective neighbors vs rewiring rate. Soli black line: Monte Carlo an ashe gray line: mean fiel. =.65, q=.16, r=.2, mean fiel noise strength =.1. FG. 1. Deenence of enemic state average lifetimes on infection rate. Points: Monte Carlo simulations; line: best fit line. q=.16, r=.2, an w=.4. See text for etails. Fig. 9 b, rewiring leas to increasing lag times an elaye outbreaks. We also comute time lags for the mean fiel moel with multilicative noise an foun the same tren of increasingly elaye outbreaks with larger rewiring. C. Lifetime of enemic steay state The final ynamic effect we consier is the lifetime of the enemic steay state. Because the system is stochastic an the isease-free state is absorbing, all arameter values will lea to eventual ie out of the isease in the infinite time limit. These lifetimes become shorter an ie out is more easily observe in the bistable regime near the bifurcation oint where the enemic state has weak stability. We measure the eenence of the lifetimes on the infection rate. For each value, we reare a steay state initial conition an comute multile ulicate runs to obtain a istribution of lifetimes. We comute 1 ulicate runs for all but the two highest values an over 25 runs for the two highest. We then calculate the average lifetime T for each. For a generic sale-noe bifurcation in one imension, it is execte that ln T is a linear function of 3/2, where is the bifurcation oint 26,27. We obtaine surious results for our system with 1 4 noes, ossibly because the small system size le to rai ie out an we were unable to obtain goo statistics near the bifurcation oint. Away from the bifurcation oint, the system has a weakly ame oscillatory comonent an behaves like a focus. By switching to a system with noes an links, we were able to run longer simulations closer to the bifurcation oint an oerate in a regime where only one imension mattere an the oscillations coul be ignore. Preliminary scaling results are shown in Fig. 1. We use the bifurcation oint that gave the best fit line, which le to an R value of.99. The scaling results aear consistent with exectations, but further stuy an better statistics are neee. V. CONCLUSONS AND DSCUSSON We have exlore the stable states, network roerties, an ynamics of an SRS moel on a network with aative

9 FLUCTUATNG EPDEMCS ON ADAPTVE NETWORKS rewiring. As with the SS moel stuie reviously by Gross et al. 19, the rewiring leas to bistability of the enemic an isease-free states. A mean fiel version of the moel reicte the steay states with goo numerical accuracy an was also valuable in stuying the fluctuations of the system, with the caveat that one must be near the aroriate tye of bifurcation in the mean fiel to obtain corresoning results. With the aition of the recovere class an resuscetibility rate, we can control the with of the bistability region by maniulating the location of the bifurcation oints. The fluctuations in the infectives near the bifurcation oint showe ower law scaling. This agree with mean fiel results an our exectations for scaling near a sale-noe bifurcation. We stuie the effects of the rewiring on the network geometry. Degree istributions were altere, an mean fiel arguments were able to reict the istributions for suscetibles an recoveres. However, a new analytical aroach that inclues correlations is neee to fully unerstan the egree istributions. The other network roerty we consiere was the istribution of istances from noninfecte noes to the nearest infective, a quantity that may be imortant in isease sreaing an its control. This istribution eene rimarily on noe an link ynamics rather than on higher orer correlations, so it coul be reicte from ranom grahs. t is ossible to comute the istribution of istances from an infective analytically for ranom grahs, but this calculation is awkwar for a three secies system S,, R an oes not have a simle functional form, so we have omitte the iscussion of analytical results here. Delaye outbreaks were observe in the rewire system. Peaks in the infective fraction lagge behin eaks in the number of noes that neighbor an infective. For the arameter values stuie here, the lag time is on the orer of 1% of the mean infectious erio, which might be consiere a very short lag time. However, the arameters in this stuy were not selecte to correson to any secific isease. For most real iseases, we woul exect a much slower rate for immunity to wear off an recovere iniviuals to become suscetible again, comare to the mean infectious erio. This regime woul be more ifficult to stuy in Monte Carlo simulations, since the average number of infecte noes woul be much smaller than seen here. Further work is neee to etermine whether the observation of elaye outbreaks ue to rewiring woul ersist or erhas become more significant in a hysically realistic system. Finally, we consiere lifetimes of the enemic state near the sale-noe bifurcation where it loses stability. n orer to achieve extinction from a steay state, the isease must first overcome the attractive forces, which are weak near the PHYSCAL REVEW E 77, bifurcation oint. Due to the generic local toology of the sale-noe structure, the escae rate is well-characterize analytically 25. We foun that the Monte Carlo simulation agrees qualitatively with the escae times an yiels a wellknown ower law. However, this is for arameters in which enemic an extinction states are not too far aart. Such extinction regimes can be analyze using a Fokker-Planck aroach 28. Although reliminary results are in agreement with the execte scaling, further stuy of the lifetime scaling is neee, incluing in regimes that are hysically realistic, an where the usual extinction rates cannot be moele with a Fokker-Planck aroach. n aition to the irections for further research mentione above, a major challenge is to evelo network geometries an rewiring rules that are more consistent with human social networks. Real social networks are execte to have community structure 29. The networks stuie here i not. n our rewire networks, the number of connections from noninfecte noes to infecte noes was reuce in comarison to a ranom network, while the number of connections between noninfecte noes was increase. However, this rocess oes not inuce a community structure on the network in the Newman-Girvan Q-moularity sense 3. Rewiring was a nonlocal effect; new neighbors were chosen at ranom among all noninfecte noes in the network rather than introucing a local structure. t has been shown for static networks that community structure affects isease ynamics 31,32, an we exect an imact in aative networks as well. n the current work, an ieal setting was roose where noninfecte noes were assume to behave rationally an have erfect knowlege of the isease status of their current neighbors an otential new neighbors. t woul be of interest to consier a situation in which not all contagious iniviuals aear ill or know that they are contagious, as might be the case for a sexually transmitte isease ossessing asymtomatic iniviuals. This effect might be moele by simultaneous sreaing through the network of both the isease an information about the isease. f the current moel is extene with information an community structure, social ynamics coul be extraolate to imrove contact tracing an eiemic control in organize oulations with local structure. ACKNOWLEDGMENTS This work was suorte by the Office of Naval Research, Center for Army Analysis, an Arme Forces Meical ntelligence Center. The authors wish to thank Thilo Gross, Bern Blasius, Royce Zia, an Mark Dykman for helful iscussions

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