Anticipating Persistent Infection
NNonlinear dynamics and Chaos
Anticipating Persistent Infection
Promit Moitra , Kanishk Jain and Sudeshna Sinha ∗ Indian Institute of Science Education and Research Mohali,Sector 81, PO Manauli 140306, Punjab, India Department of Physics, Emory University, Atlanta GA 30322
Abstract
We explore the emergence of persistent infection in a closed region where the disease progression of the individuals is givenby the SIRS model, with an individual becoming infected on contact with another infected individual within a given range. Wefocus on the role of synchronization in the persistence of contagion. Our key result is that higher degree of synchronization,both globally in the population and locally in the neighborhoods, hinders persistence of infection. Importantly, we find thatearly short-time asynchrony appears to be a consistent precursor to future persistence of infection, and can potentially providevaluable early warnings for sustained contagion in a population patch. Thus transient synchronization can help anticipatethe long-term persistence of infection. Further we demonstrate that when the range of influence of an infected individual iswider, one obtains lower persistent infection. This counter-intuitive observation can also be understood through the relationof synchronization to infection burn-out.
The spread of infectious diseases in a population isa field of wide-spread inquiry and continues to attractintense research activity [1]. One of the outstandingproblems in this area has been obtaining reliable earlywarning signals for persistence of infection in a region.This is a problem of obvious significance, as it can poten-tially influence strategies of long-term control of disease.Mathematically this is a challenging problem, as one hasto consider large interactive complex systems that arestrongly nonlinear and typically not well mixed. In thiswork we attempt to uncover what dynamical features atearly times are strongly correlated to long-term charac-teristics, such as the continued presence of infection in apopulation patch. Such features, if found to exist, canpotentially provide important early warning signals forpersistent infections.Mathematically, epidemiological models have success-fully captured the dynamics of infectious disease [2, 3].One well known model for non-fatal communicable dis-ease progression is the SIRS cycle. This model appro-priately describes the progression of diseases such as ty-phoid fever, tetanus, cholera, small pox, tuberculosis andinfluenza [4, 5]. The SIRS cycle is described by the fol-lowing stages: • Susceptible (denoted by symbol S ) - An individualin this state remains susceptible until they contractthe infection from another infected person in theirneighbourhood. At the end of the refractory stage(namely the stage of temporary immunity) of thedisease cycle, the individuals return to this state. • Infected (denoted by I ) - In this stage of the diseasecycle, the individual is in an infected state, whichsignifies they can infect others around them. ∗ Electronic address: e-mail: [email protected] • Refractory (denoted by R ) - At the end of the infec-tious stage, the individuals acquire temporary im-munity to the disease. In this stage they neither getinfected by infectious neighbors, nor do they infectanyone in their surroundings. Typically, this stagelasts longer than the infected stage, and at the endof this stage the individual loses the temporarilyacquired immunity and becomes susceptible to theinfection once again.So the progression of an individual from the susceptiblestage, to the infected stage, onto the refractory stage andback to susceptible ( S → I → R → S ) is the SIRS diseasecycle . Cellular automata models [6] of this cyclic diseaseprogression have provided very good test-beds for study-ing infection spreading [7–9]. In this class of models weconsider individuals located on a plane, namely each in-dividual is indexed on a 2 dimensional lattice by a pair ofsite indices ( i, j ). The state of each individual is charac-terized by a integer-valued counter τ i,j ( t ) that describesits phase in the cycle of the disease, at discrete time step t [7]. Here τ can take values 0 , . . . τ I , τ I + 1 . . . , τ I + τ R .At any instant of time t (where t is integer-valued), ifphase τ i,j (t) = 0, then the individual at site ( i, j ) is sus-ceptible ( S ); if 1 ≤ τ i,j ( t ) ≤ τ I , then it is infected ( I ); ifphase τ i,j ( t ) > τ I , it is in the refractory stage ( R ).The dynamics is given as follows: for infected or refrac-tory individuals whose phase τ i,j ( t ) (cid:54) = 0, τ i,j increasesby 1 at the subsequent time step. Additionally, at theend of the refractory period i.e. when namely τ = τ ,where τ = τ I + τ R , the individual becomes susceptibleagain (characterized by τ = 0). So this implies that if τ i,j ( t ) = τ , then τ i,j ( t + 1) = 0. The complete set ofevolution rules can then be summarized mathematicallyas: τ i,j ( t + 1) = τ i,j ( t ) + 1if 1 ≤ τ i , j (t) < τ (1)= 0 if τ i , j (t) = τ (2)The total length of the disease cycle, denoted by τ D ,1 a r X i v : . [ n li n . C G ] M a y s equal to τ I + τ R + 1, including the state τ = 0 theindividual returns to at the end of the refractory period.In this work we consider the typical condition where therefractory stage is longer than the infective stage, i.e. τ R > τ I . Spatiotemporal evolution of infection:
We now inves-tigate the spread of disease in a spatially distributedgroup of individuals, where at the individual level thedisease progresses in accordance with the SIRS cycle de-scribed by the Cellular Automaton model above. In par-ticular, we consider a population of individuals on a 2-dimensional square lattice of linear dimension L , whereevery node represents an individual [10]. Unlike manyearlier studies, we are interested in a closed patch ofindividuals. So instead of the commonly used periodicboundary conditions, the boundaries of our system arefixed, with no individuals outside the boundaries. Wewill focus on the emergence of persistent infection in suchan isolated patch.We consider a following condition for spread of in-fection: if one or more of its nearest neighbours ofa susceptible individual is infected, then the suscep-tible individual will become infected. That is, if τ i,j ( t ) = 0, (namely, the individual is susceptible), then τ i,j ( t + 1) = 1, if any 1 ≤ τ x,y ( t ) ≤ τ I where x, y belong to a neighbourhood consisting of 4 individu-als, namely the von Neumann neighborhood, given by: x, y ∈ { ( i − , j ) , ( i, j + 1) , ( i + 1 , j ) , ( i, j + 1) } . Further,we will also consider a neighbourhood comprising of 8individuals, namely the Moore neighbourhood, given by: x, y ∈ { ( i − , j ) , ( i, j + 1) , ( i + 1 , j ) , ( i, j + 1) , ( i − , j − , ( i + 1 , j + 1) , ( i − , j + 1) , ( i + 1 , j − } We denotethe number of neighbours by K , with the von Neumannneighbourhood having K = 4, while the Moore neigh-bourhood has K = 8. Larger K implies that an infectedindividual can affect individuals in a larger zone aroundit, namely the infected individual has a larger range ofinfluence. So the dynamics of this extended system com-bines deterministic, as well as probabilistic elements. Thedisease progression of an infected individual is determin-istic, with the infected period of length τ I , followed bythe refractory period of length τ R . However, the processof contracting the infection is probabilistic, arising fromthe interplay of the localized nature of the interactionsand the random initial states of the individuals.Now the infection in this closed patch can either dieout, or it can persist. So it is of considerable significanceto find the conditions that lead to sustained infection, aswell as to uncover the salient features that characterizethe persistent state. The relevant quantity here is theasymptotic fraction of infected individuals in the popu-lation. To obtain an appropriate measure of this we firstfind the fraction of infecteds at time t , denoted by I ( t ).In order to gauge asymptotic trends, we consider thisfraction of infected individuals, after long transient time,averaged over several disease cycles, denoted by (cid:104) I (cid:105) . Thisquantity serves as an order parameter for persistent in-fection, with non-zero (cid:104) I (cid:105) indicating persistent infection, while (cid:104) I (cid:105) = 0 indicates that infection has died out in thepatch. Further we consider the ensemble averaged (cid:104) I (cid:105) ,denoted by (cid:104)(cid:104) I (cid:105)(cid:105) . This quantity reflects the the size ofthe basin of attraction of the persistent state, and indi-cates the probability of persistent infection arising froma generic random initial condition of the population. So (cid:104)(cid:104) I (cid:105)(cid:105) is non-zero when persistent infection arises fromtypical initial states and zero otherwise.By studying the dependence of (cid:104)(cid:104) I (cid:105)(cid:105) on the initial frac-tion of infecteds I , susceptibles S and refractory indi-viduals R in the population, it was found in Ref. [11]that for sustained contagion in a population, the initialpopulation needed to be a well mixed heterogeneous col-lection of individuals, with sufficiently large number ofboth susceptible and refractory individuals. Further, itwas found that in a population composed of an admix-ture of susceptible and refractory individuals, persistentinfection emerged in a window of reasonably low I , with I → correlation betweensynchronization and persistent infection . We ask twocomplementary questions: First, does lack of synchro-nization characterize the state of the population whereinfection is sustained. Secondly, and more significantly,does the lack of synchronization in the early stages ofdisease spreading lead to persistent infection at latertimes. We will explore this question by introducing localand global measures of synchronization . Lastly, we willdemonstrate that when the range of infection transmis-sion of an infected individual is wider, one obtains lowerpersistent infection. We will account for this counter-intuitive observation through the relation between syn-chronization and infection burn-out. Synchronization characterizes populations withsustained infection : We first explore the degree ofglobal synchronization in the system, by calculating thequantity: σ ( t ) = | N Σ N exp iφ m,n ( t ) | (3)where φ m,n = 2 πτ m,n /τ D is a geometrical phase corre-sponding to the disease stage τ m,n of the individual atsite ( m, n ). Here the indices m and n run from 1 to L ,namely over all N = L × L individuals in the populationpatch. We use Eqn. 3 to obtain the asymptotic time av-eraged synchronization order parameter, denoted by (cid:104) σ (cid:105) ,by averaging σ ( t ) over time, of the order of several diseasecycles, after transience. This reflects the synchronizationin the emergent system, namely the asymptotic degreeof synchronization in the population arising from a spe-cific initial state. So when the phases of disease of theindividuals are uncorrelated, that is the disease cycles of2he individuals in the population are not synchronized, (cid:104) σ (cid:105) is close to 0. On the other hand when the individ-ual disease cycles are quite synchronized, (cid:104) σ (cid:105) tends to 1.We will then go on to calculate the ensemble averagedasymptotic synchronization order parameter denoted by (cid:104)(cid:104) σ (cid:105)(cid:105) , obtained by further averaging the time-averagedasymptotic synchronization order parameter (cid:104) σ (cid:105) over alarge number of initial states characterized by a specific( I , S , R ). This order parameter indicates the proba-bility of synchronization arising from a generic randominitial condition of the population. We will use this mea-sure, alongside the ensemble averaged persistence orderparameter (cid:104)(cid:104) I (cid:105)(cid:105) , to help us gauge the broad correlationbetween synchronization in the emergent population (orlack thereof) with persistent infection. I ›› σ fifi ›› I fifi FIG. 1:
Dependence of the ensemble averaged asymptotic synchro-nization order parameter (cid:104)(cid:104) σ (cid:105)(cid:105) (black solid line) on the initial fractionof infecteds in the population I (with equal initial fractions of sus-ceptible and refractory individuals: S = R ). Here system size is100 ×
100 and K = 4. The figure also shows the variation of (cid:104)(cid:104) I (cid:105)(cid:105) (reddashed line) with respect to I , where (cid:104)(cid:104) I (cid:105)(cid:105) is an ensemble averagedorder parameter reflecting the degree of persistence of infection in thepopulation. Fig. 1 shows (cid:104)(cid:104) I (cid:105)(cid:105) and (cid:104)(cid:104) σ (cid:105)(cid:105) , for different initialfraction of infecteds I in the population. As mentionedbefore, one observes persistent infection (i.e. (cid:104)(cid:104) I (cid:105)(cid:105) (cid:54) = 0),in a window of I [11]. Further, it is now clearly evidentthat in this same window of persistent infection, theglobal asymptotic synchronization order parameter isthe lowest. So higher persistence of infection is consis-tently correlated with lower degree of synchronization ,distinctly implying that a population where infection ispersistent is generally characterized by low synchroniza-tion among the individuals. Specifically, for instancefor the case of persistent infection with (cid:104)(cid:104) I (cid:105)(cid:105) ∼ , wefind 0 < (cid:104)(cid:104) σ (cid:105)(cid:105) < . On the other hand, for cases wherethe infection eventually dies out, i.e. (cid:104)(cid:104) I (cid:105)(cid:105) ∼
0, we have (cid:104)(cid:104) σ (cid:105)(cid:105) ∼
1. So it is evident that there is clear inversedependence of infection persistence as reflected by (cid:104)(cid:104) I (cid:105)(cid:105) and degree of synchronization of the disease cycles ofthe individuals in the emergent population as reflectedby (cid:104)(cid:104) σ (cid:105)(cid:105) . So one can infer that a population wherepersistent infection emerges, is quite unsynchronized.
Transient synchronization results in weaker per-sistence of infection:
We have shown above that lackof synchronization is a key feature of populations with sustained infection, and the asymptotic synchronizationorder parameter (cid:104) σ (cid:105) successfully characterizes popula-tions with different degrees of persistence of infection.This motivates us to explore the second question: is syn-chronization in the initial (transient) stage, which we willcall transient synchronization here, an indicator of futurepersistence of infection in the population?First, we show in Figs. 2a-b illustrative examples inorder to visually examine the state of the system at vari-ous instances of time within the first disease cycle, arisingfrom two distinct initial conditions. The first example isa population with initial fraction of infecteds I = 0 . I = 0 . first few time steps . We would nowlike to investigate if this qualitative observation holdsconsistently, quantitatively, over a large range of initialstates. Quantifying finite-time transient synchroniza-tion:
In order to quantify the early time synchroniza-tion in the system, we introduce a finite time average ofthe synchronization order parameter σ ( t ), from the ini-tial time ( t = 0) up to a specific time t = T denoted by (cid:104) σ T (cid:105) . Such a measure reflects the degree of synchroniza-tion over short time-scales, at early times. We furtherconsider the ensemble average of this quantity, where theaverage is over a large set of initial states with a specificinitial partitioning ( I , S , R ). This quantity reflects thedegree of synchronization typically arising up to time T inthe population, from a generic initial state, for a specific( I , S , R ), and is denoted by (cid:104)(cid:104) σ T (cid:105)(cid:105) . When T ∼ τ (ofthe order of a single disease cycle), this quantity reflectsthe transient synchronization or early-time synchroniza-tion, namely synchronization of the population withinthe first cycle of disease. In this work we will aim to ex-plore if this quantity can offer a consistent early warningsignal for persistence of infection in the patch of popula-tion. Specifically we will now investigate (cid:104)(cid:104) σ (cid:105)(cid:105) , namely thecase where T = τ D + 1, where τ D is the length of thedisease cycle. So this quantity reflects the synchroniza-tion of the individual disease cycles in the population atearly times, and can serve as an useful order parameterfor transient synchronization . When (cid:104)(cid:104) σ (cid:105)(cid:105) →
1, com-plete synchronization of the individual disease cycles inthe population is obtained soon after one disease cycle.Fig. 3a shows the dependence of the degree of transientsynchronization (cid:104)(cid:104) σ (cid:105)(cid:105) , namely the degree of synchro-nization right after completion of the first cycle of disease,on the fraction of infecteds I in the initial population.It is evident that the onset of the persistence window isclearly indicated by minimum (cid:104)(cid:104) σ (cid:105)(cid:105) , i.e. the transientsynchronization is the lowest when persist infection be-gins to emerge in the population. So, the early synchro-nization properties of the system allows one to gauge the
20 40 60 80 100020406080100
S: 0.46 I: 0.10 R: 0.44 t=0
S: 0.07 I: 0.24 R: 0.70 t=5
S: 0.43 I: 0.07 R: 0.50 t=10
S: 0.62 I: 0.21 R: 0.17 t=15
SIR Cycle
S: 0.25 I: 0.50 R: 0.25 t=0
S: 0.00 I: 0.01 R: 0.98 t=5
S: 0.26 I: 0.00 R: 0.74 t=10
S: 1.00 I: 0.00 R: 0.00 t=15
SIR Cycle (a) (b)
FIG. 2:
Snapshots of the infection spreading pattern at very early times t = 0 , , ,
15, in an initial population comprising of a random admixtureof individuals, with S = R and (a) I = 0 . I = 0 .
5. The colour bar shows the relative lengths of the susceptible (S), infected (I) andrefractory (R) stages in the disease cycle, where τ I = 4, τ R = 9 and the total disease cycle τ D is 14. The red box shows the fraction of S, I and Rindividuals in the population at that instant of time. Notice that the population appears to lack of synchrony in the individual states, and has anon-uniform distribution. The infection persists in (a) and dies out in (b). I ›› σ fifi ›› I fifi ›› σ fifi ›› I fifi neighbours × × (a) (b) FIG. 3: (a) Dependence of the transient synchronization order param-eter (cid:104)(cid:104) σ (cid:105)(cid:105) on the initial fraction of infecteds I ∈ [0 ,
1] (with S = R and system size 100 × I is obtained by averaging over 100 random initial conditions.(b) Correlation between the asymptotic persistence parameter (cid:104)(cid:104) I (cid:105)(cid:105) and the ensemble averaged transient synchronization order parameter (cid:104)(cid:104) σ (cid:105)(cid:105) . The quantities are obtained by averaging over I ∈ [0 , S = R . Here K = 4. future persistence of contagion . A valuable consequenceof this observation is that early-time synchronization canserve as an early warning signal for sustained infection ata much later time.Now we examine the explicit correlation between (cid:104)(cid:104) σ (cid:105)(cid:105) and the asymptotic fraction of infecteds in thepopulation (cid:104)(cid:104) I (cid:105)(cid:105) . This is shown in Fig. 3(b), from whereone can clearly see a well-defined transition to long-termpersistent infection as the transient states get more syn-chronized. So the asymptotic fraction of infecteds de-creases sharply at short-time synchronization order pa-rameter values close to 2 /
3. Namely, there exists acritical transient synchronization order parameter σ (cid:63)T ,beyond which persistent infection does not occur (i.e. (cid:104)(cid:104) I (cid:105)(cid:105) ∼ σ (cid:63)T reflects early-time properties, while offering a clear correlation with anasymptotic phenomena. It quantitatively confirms our intuition that when the system is more synchronous atearly times, there is greater propensity of the infectiondying out.So we conclude that greater degree of synchronizationat early times hinders the sustenance of infection. Thus early short-time asynchrony appears to be a consistentprecursor to future persistence of infection, and canperhaps provide valuable early warning signals foranticipating sustained contagion in a population patch . Transient Local synchronization:
Now we explorethe correlation of transient local synchronization , namelysynchronization in a local neighbourhood of an individ-ual. This is important, as infection spread is a local con-tact process and so the composition of its local nigh-bourhood is most crucial for an individual. In order tocapture finite-time local synchrony, we introduce the fol-lowing synchronization parameter: σ ( i,j ) K ( t ) = | K + 1 Σ m,n exp iφ m,n ( t ) | (4)where φ m,n is a geometrical phase corresponding to thedisease stage τ m,n of the individual at site ( m, n ). Herethe indices m and n run over the site index and all K sites contained within the neighbourhood of ( i, j ). Theaverage of σ ( i,j ) K ( t ) over all sites ( i, j ) in the system isdenoted by σ K ( t ).The focus of our investigation is the finite time averageof σ K ( t ) from initial time ( t = 0) to time T , where T isof the order of one disease cycle length. We denote thismeasure of finite-time local synchronization as (cid:104) σ K,T (cid:105) .The ensemble averaged (cid:104) σ K,T (cid:105) is denoted by (cid:104)(cid:104) σ K,T (cid:105)(cid:105) ,and this quantity reflects the typical transient local syn-chronization present in the system.4 .6 0.7 0.8 0.9 1.0 ›› σ L, fifi ›› I fifi neighbours × × RMSD ( ›› σ L, fifi ) ›› I fifi FIG. 4:
Dependence of the asymptotic persistence parameter (cid:104)(cid:104) I (cid:105)(cid:105) onthe ensemble averaged transient local synchronization order parameter (cid:104)(cid:104) σ K, (cid:105)(cid:105) . The quantities are obtained by averaging over I ∈ [0 , S = R and K = 4. Inset shows the dependence of (cid:104)(cid:104) I (cid:105)(cid:105) on theroot mean square deviation (RMSD) of σ K, . We show the explicit correlation between the transientlocal synchronization order parameter (cid:104)(cid:104) σ K, (cid:105)(cid:105) and theasymptotic fraction of infecteds in the population (cid:104)(cid:104) I (cid:105)(cid:105) in Fig. 4. It is clearly evident that there exists a sharptransition to infection burn-out as transient local syn-chronization goes beyond a critical value σ (cid:63)K,T ∼ / localneighbourhoods are synchronized beyond a critical degreeduring early stage of disease spreading, persistent infec-tion does not occur . So, though critical σ (cid:63)K,T dependson early-time spatially local information, it offers a clearindication of asymptotic phenomena.Further notice that the spread in transient local syn-chronization across initial states, as reflected by the rootmean square deviation (RMSD) of (cid:104) σ K,T (cid:105) in the insetof Fig. 4, also exhibits a sharp transition from the caseof non-persistent infection (i.e. (cid:104)(cid:104) I (cid:105)(cid:105) = 0) to persistentinfection (where (cid:104)(cid:104) I (cid:105)(cid:105) ∼ / Dependence of persistence of infection on therange of infection transmission:
Lastly, we explorethe influence of the range of infection transmission on thepersistence of infection. Specifically we investigate thecase of K = 8, namely the case where infected individualscan affect eight neighbours. So now the range of influenceof the infected individual is double that presented earlier,where K was 4. Fig. 5 shows the dependence of the per-sistence order parameter (cid:104)(cid:104) I (cid:105)(cid:105) on the fraction of infectedindividuals I in the initial population, with S = R .It is clearly evident from the figure that persistent in-fection is lower when the infected individual influences alarger number of neighbouring individuals. That is, sur-prisingly, a larger range of infection transmission hinderslong-term persistence of the disease. ›› σ fifi ›› I fifi neighbours ›› σ L, fifi ›› I fifi neighbours RMSD ( ›› σ L, fifi ) ›› I fifi FIG. 6:
Dependence of the asymptotic persistence parameter (cid:104)(cid:104) I (cid:105)(cid:105) on (left) ensemble averaged transient synchronization order parameter (cid:104)(cid:104) σ (cid:105)(cid:105) and (right) the ensemble averaged transient local synchroniza-tion order parameter (cid:104)(cid:104) σ K, (cid:105)(cid:105) . The quantities are obtained by aver-aging over I ∈ [0 , S = R . Here system size is 100 ×
100 and K = 8. I ›› σ fifi neighbours ›› I fifi›› σ fifi ›› I fifi neighbours ›› I fifi›› σ fifi FIG. 5:
Dependence of (cid:104)(cid:104) I (cid:105)(cid:105) on the initial fraction of infected indi-viduals I in the population, for the case of K = 8 (solid red line) andthe case of K = 4 (red dashed line) for reference. Here system size is100 × (cid:104)(cid:104) σ (cid:105)(cid:105) on I , for the caseof K = 8 (solid black line) and the case of K = 4 (black dashed line)for reference. However, this counter-intuitive result is completely inaccordance with our earlier observation, namely highersynchronization implies lower persistence of infection.This is clearly bourne out by the asymptotic synchroniza-tion order parameter, which is also displayed in Fig. 5alongside the persistence order parameter (cid:104)(cid:104) I (cid:105)(cid:105) . Fromthe figure it can be seen that for K = 8 the synchro-nization is enhanced, and so (cid:104)(cid:104) σ (cid:105)(cid:105) is low only in a verysmall range of I . It is this precise range that supportspersistent infection. Since the range of low synchroniza-tion is significantly smaller for K = 8 vis-a-vis K = 4, wecorrespondingly have a significantly smaller range of per-sistent infection when the range of infection transmissionis larger.Further, we again examine the explicit correlationbetween the transient synchronization, as reflected by (cid:104)(cid:104) σ (cid:105)(cid:105) , as well as the local transient synchronization,as reflected by (cid:104)(cid:104) σ K, (cid:105)(cid:105) , and the asymptotic fraction ofinfecteds in the population (cid:104)(cid:104) I (cid:105)(cid:105) . These are shown inFigs. 6a-b, from where one can again clearly see a well-defined transition to long-term persistent infection as thetransient states get more synchronized both locally andglobally. So again, quantitatively it can be seen thatearly-time local and global properties offer a clear indi-cation of asymptotic persistence properties. This lendsfurther credence to our central observation, and demon-5trates the robustness and generality of the phenomenonwith increasing range of infection transmission.Also interestingly, as in the case of K = 4, theasymptotic fraction of infecteds again decreases sharplyat transient synchronization order parameter valuesclose to 2 / /
4. However we observe thatthe precise value of the critical transient synchronizationorder parameters, σ (cid:63)T and σ (cid:63)K,T , beyond which persistentinfection does not occur (i.e. (cid:104)(cid:104) I (cid:105)(cid:105) ∼ K , as evidentin Fig. 5. Discussion : In summary, we have explored the emer-gence of persistent infection in a closed region where thedisease progression of the individuals is given by the SIRSmodel, with an individual becoming infected on contactwith another infected individual within a given range.We focussed on the role of synchronization in the per-sistence of contagion. Our key result is that higher de-gree of synchronization, both globally in the populationand locally in the neighborhoods, hinders persistence ofinfection. Importantly, we found that early local asyn-chrony appears to be a consistent precursor to future per-sistence of infection , and can potentially provide valuable early warnings for sustained contagion in a populationpatch. Thus transient local synchronization can help an-ticipate the long-term persistence of infection. Furtherwe demonstrated that when the range of influence of aninfected individual is wider, one obtains lower persistentinfection. This counter-intuitive observation can also beunderstood through the relation of synchronization to in-fection burn-out.Lastly, our results also have broad relevance in thecontext of large interactive excitable systems. For in-stance, the system we study here is reminiscent of modelsof reaction-diffusion systems [14], heterogeneous cardiactissue [15] and coupled neurons [16]. The self-sustainedexcitations in these systems are analogous to the stateof persistent infection we have focused on in this work.Specifically, persistent chaotic activity in a patch of tis-sue is characteristic of atrial fibrillation, and so our ob-servations may have potential relevance to such phenom-ena arising in cardiac tissue. In the context of brainfunctions, neuronal circuits are able to sustain persis-tent activity after transient inputs, and studies have sug-gested that the asynchronous phase of synaptic trans-mission plays a vital role in the this persistent activitywhich is of considerable importance to motor planningand memory. Further, in the context of metapopulations[17], there exists research which argues that enhancedcoherence would decrease the probability of species per-sistence [18]. So our demonstration of the potential of early short-time local and global synchronization as anearly warning signal for anticipating persistent activity,has relevance to such phenomena as well. [1] C. McEvedy,
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