Epidemic Predictability in Meta-Population Models with Heterogeneous Couplings: the Impact of Disease Parameter Values

International Journal of Bifurcation and Chaos, Vol. 17, No. 7 (2007) 2491–2500 c World Scientiﬁc Publishing Company

EPIDEMIC PREDICTABILITY IN META-POPULATION MODELS WITH HETEROGENEOUS COUPLINGS: THE IMPACT OF DISEASE PARAMETER VALUES

V. COLIZZA∗,†, A. BARRAT∗,‡,M.BARTHELEMY´ § and A. VESPIGNANI† ∗Complex Networks Lagrange Laboratory (CNLL), Institute for Scientific Interchange (ISI ), 10133 Turin, Italy †School of Informatics and Biocomplexity Institute, Indiana University, Bloomington, IN 47408, USA ‡Laboratoire de Physique Théorique, (UMR du CNRS 8627), UniversitédeParis-Sud, Orsay, France §CEA-DIF Centre d’Etudes de Bruyères-Le-Châtel, BP12, F-91680, France

Received June 16, 2006; Revised September 12, 2006

We study the predictability of epidemic forecasts in a data-driven meta-population model considering the complete air transportation system and the associated urban areas. We define the predictability as the robustness of the system evolution with respect to the stochastic fluctuations. As a quantitative measure of predictability we consider the information similarity of the time series characterizing different epidemic outbreaks with the same initial conditions. We study the predictability as a function of the parameters describing the basic susceptible-latent- infected and recovered disease dynamics. We find that the overall predictability is determined by the level of sampling of the underlying travel pattern by infected and latent individuals.

Keywords: Epidemiology; complex networks; stochastic noise; predictability.

1. Introduction Lloyd & May, 1996; Grenfell & Bolker, 1998; Keel- Large scale epidemic forecast is crucially depen- ing & Rohani, 1995; Ferguson et al., 2003]. The dent upon the accurate and realistic modeling heterogeneities and details of the population struc- where the movement of individuals at various lev- ture have become increasingly important features els is taken into account. In the case of spa- and the introduction of large scale agent based mod- tially extended systems, modeling approaches have els has enabled the simulation of the propagation of evolved into meta-population schemes which explic- an infectious disease at the level of the single indi- itly include spatial structures and consist of mul- vidual [Chowell et al., 2003; Eubank et al., 2004; tiple subpopulations coupled by the movement of Ferguson et al., 2005; Longini et al., 2005]. If one people among those [Anderson & May, 1984; May focuses on the world-wide level, the large scale & Anderson, 1984; Bolker & Grenfell, 1993, 1995; and geographical impact of infectious diseases on

2491 2492 V. Colizza et al. populations in the modern era is mainly due to The paper is organized as follows: we recall the commercial air travel. For this reason, in the early modeling framework in Sec. 2; Sec. 3 presents the 80’s, models aimed at forecasting the geographi- results regarding the impact of disease parameter cal spread of epidemics by using information on values on quantitative forecasting of epidemic out- the airplane passenger fluxes among cities have breaks. Finally, we consider some conclusions and been developed [Rvachev & Longini, 1985], capi- directions for future work in Sec. 4. talizing on previous studies on the Russian airline network [Baroyan et al., 1969]. Similar modeling 2. Meta-Population Model approaches have been used to study specific out- We consider the general modeling framework pre- breaks such as pandemic influenza [Longini, 1988; sented in [Colizza et al., 2006a, 2006b], which Grais et al., 2004], HIV [Flahault & Valleron, 1991], describes the world-wide propagation of a dis- and SARS [Hufnagel et al., 2004]. More recently, ease through a meta-population approach, in the the availability of complete datasets [Guimerà& same spirit as in the models of [Longini, 1988; Amaral, 2004; Barrat et al., 2004; Guimerà et al., Grais et al., 2003; Flahault & Valleron, 1991; Huf- 2005] has allowed to build modeling frameworks for nagel et al., 2004]. The spatial structure is explicitly the computational study of epidemics taking into included in the model by considering multiple sub- account the full air transportation systems and the populations coupled by movements of individuals. census data of the corresponding urban areas [Col- More specifically, the sub-populations correspond izza et al., 2006a, 2006b]. In these approaches the to urban areas and individuals in each urban area only parameters in the models are the disease are allowed to travel according to the routes of the transition rates describing the etiology of the dis- airline transportation system. ease. All the couplings due to transportation terms, urban area size and the network connectivity pat- 2.1. Intra-city level: SEIR infection tern are fixed elements of the problem obtained dynamics from real data. While the ability to forecast the spread of The epidemic evolution inside each urban area is an infectious disease is inevitably entangled with described by a fully mixed population with a basic the accuracy and the level of realism introduced standard compartmentalization in which each indi- in the modeling approach, the impact of the vidual can only exist in one of the following discrete parameters and model features on the reliability states such as susceptible (S), latent (E), infected (I) of the predicted scenarios is difficult to assess. or permanently recovered (R). This set of states do Here we investigate how different communicable not correspond strictly to any particular disease but diseases — characterized by different epidemi- encompasses the most relevant features and param- ological parameters regarding infectiousness and eters of a variety of different virus transmission, transition rates — might affect our ability to by including specific stages of the disease dynam- predict their geographical and temporal propaga- ics such as e.g. the latency period. More refined and tion. In order to tackle this issue we consider extended compartmentalizations can be considered. the meta-population model defined in [Colizza In each city j the population is Nj and the num- et al., 2006a, 2006b] for the forecast of the world- bers of individuals in the various classes at time t wide spread of emerging diseases. We perform are denoted as Sj(t), Ej(t), Ij(t)andRj(t) respec- a systematic analysis of the effect of the dis- tively. By definition Nj = Sj(t)+Ej(t)+Ij(t)+ ease parameters on the statistical similarity of Rj(t). Individuals change compartment according the epidemic behavior evolution obtained with to the following rules: during the time interval the same initial conditions and different noise dt, the probability that a susceptible individual realizations. This amounts to the assessment of acquires the infection from any given infected indi- the robustness of the obtained predictions with vidual, and becomes latent, is proportional to βdt, respect to the inherent randomness of the epi- where β is the transmission parameter that cap- demic evolution. The analysis shows that the over- tures the etiology of the infection process. Latents all predictability of the epidemics is determined by become infected with probability dt, and infected competing effects related to the number of infected individuals recover with a probability µdt,where individuals and the heterogeneity of the traveling −1 and µ−1 are the average latency time and aver- pattern. age infection duration. Epidemic Predictability in Meta-Population Models with Heterogeneous Couplings 2493

The populations Nj of the large metropolitan 2.3. Global stochastic epidemic model areas served by the airports are obtained by differ- In each city we work directly with the master equa- ent census sources freely available on the Internet. tions for the dynamical rules given in Sec. 2.1. For large cities, Nj represents the population of the Under the assumption of large populations, we whole metropolitan area. The final data set contains obtain the Langevin equations for the numbers of the population of the 3100 largest airports. susceptible, latent, infected and recovered individu- 2.2. Transport operator als and we associate to each reaction process a noise term with amplitude proportional to the square root In addition to the infection dynamics taking place of the reaction term (for details, see [Colizza et al., inside each urban area, the epidemic evolution at 2006b; Gardiner, 2004; Marro & Dickman, 1998]). the global level is governed by the travel of individ- The noise term accounts for the stochastic nature of uals from one city to another by means of the airline the infection dynamics of directly transmitted dis- transportation network, therefore infecting different eases, with intrinsic fluctuations in the infectious regions of the world. The network is obtained by contact process, the generation of new cases and considering the International Air Transport Associ- 1 the removal of infectious individuals. These stochas- ation (IATA) database, which contains the world tic nonlinear differential equations are then coupled list of airport pairs connected by direct flights and through the transport operator, yielding the follow- the number of available seats on any given connec- ing set of stochastic discretized differential equa- tion. The resulting world-wide air-transportation tions for the worldwide spread of the infection: network is therefore a weighted graph compris- V ing = 3100 vertices denoting airports in 220 Sj(t +∆t) − Sj(t) different countries and E = 17 182 weighted edges whose weights represent the passenger flows IjSj IjSj between airports. This dataset accounts for 99% of = − β ∆t + β ∆tηj,1(t) Nj Nj the world-wide traffic and has been complemented by the population of the large metropolitan area +Ωj({S})(2) served by the airport as obtained by different Ej(t +∆t) − Ej(t) sources. The obtained network is highly heterogeneous both in the connectivity pattern and in IjSj IjSj the traffic capacities [Guimerà & Amaral, 2004; =+β ∆t − Ej∆t − β ∆tηj,1(t) Nj Nj Barrat et al., 2004]. In practice, the travel dynamics is described by + Ej∆tηj,2(t)+Ωj({E})(3) the transport operator Ωj({X})representingthe net balance of individuals in a given class X (S, Ij(t +∆t) − Ij(t) E, I or R) that entered and left each city j.This =+Ej∆t − µIj∆t − Ej∆tηj,2(t) operator is a function of the traffic flows wj per unit time, the city populations Nj,andmightalso + µIj∆tηj,3(t)+Ωj({I})(4) include transit passengers on connecting flights (see [Colizza et al., 2006b] for details). The number of Rj(t +∆t) − Rj(t) ξ Xj X passengers j( ) of each compartment travel- =+µIj∆t − µIj∆tηj,3(t)+Ωj({R}). (5) ing from a city j to a city is an integer random variable, extracted from a multinomial distribution Here ηj,1, ηj,2 and ηj,3 are statistically independent which considers pj = wj∆t/Nj as the probability normal random variables with zero mean and unit of traveling from j to in the time interval ∆t.Mean variance. The model is thus a compartmental sys- and variance of ξj(Xj)aregivenbyξj(Xj) = tem of 3100 × 4 stochastic differential equations pjXj and Var(ξj(Xj)) = pj(1 − pj)Xj, respec- whose integration provides the disease evolution in tively, and the transport operator for each city j the corresponding urban areas (see [Colizza et al., canbewrittenas 2006b] for more implementation details). A quan- Ωj({X})= (ξj(X) − ξj(Xj)). (1) tity commonly used to describe the epidemic behavior in time is the prevalence in a city j,whichis

1 http://www.iata.org. 2494 V. Colizza et al.

I II given by the fraction Ij/Nj of infectious individu- identical and sim(π , π ) = 0 when there is no als with respect to the city population. The global overlap at all. The normalization of π(t)how- prevalence at time t is defined as the world-wide ever means that two realizations differing only density of infectious individuals, j Ij/ j Nj, of a global multiplicative factor will generate the thus providing a measure of the global level of same vector π. In order to correctly take into infection. account the possible difference in the total epidemic prevalence between the two realizations, it 3. Predictability is thus necessary to consider as well the quantity aI , aII a a, − a a A central issue in modeling the spread of epidemic sim( )where =( 1 )and is the A /N diseases is to assess the reliability of the predicted worldwide density of active individuals j j patterns. The interplay of different aspects involved (N is the total population). This vector thus takes in the process — infection dynamics, travel, into account the similarity of the global preva- stochasticity, transportation network, etc. — might lence in two stochastic realizations. In summary, have a crucial role in the level of accuracy and relia- the overlap function between two different stochas- bility provided by the obtained epidemic forecasts. tic realizations of an epidemic with the same start- In [Colizza et al., 2006a, 2006b], we have investi- ing initial conditions and disease parameters is gated the role of the air-transportation network in defined as I II I II the emergence of spatiotemporal propagation pat- Θ(t)=simH (a , a ) × simH (π , π ). (7) terns and proposed to use a similarity measure to quantify the predictability of epidemic outbreaks. The overlap is maximal (Θ(t)=1)whenthevery Indeed, even when global average quantities have same cities have the very same number of active small fluctuations, different sub-populations might individuals in both realizations, and Θ(t)=0ifthe be affected in very different ways, resulting in very two realizations do not have any common infected different epidemic scenarios. We therefore need to cities at time t. assess the reliability of the epidemic forecast at the Here we focus on the influence of disease param- −1 local level however considering the complete set of eters such as the latency time , the infectious- sub-populations and geographical areas. More pre- ness β, and the initial percentage of susceptible cisely, the predictability of the epidemic pattern is individuals on the predictability. Different values −1 given by the statistical similarity of the detailed his- of β and point to different viruses or different tory of epidemics outbreaks starting with the same strains of the same virus, with variations in the virus initial conditions and subject to different noise real- infectiousness and average latency period; changes izations. Let us define the active individuals Aj in in the percentage of initially susceptible popula- city j as the sum of latent and infectious individ- tion are related to acquired immunity or vaccina- µ−1 uals, Aj = Ej + Ij, thus representing the infected tion campaigns. The average infectious period individuals still active, i.e. not yet recovered. We is kept fixed and equal to 3 days, as estimated for then consider the full vector π(t)whosecompo- seasonal influenza. In the following, we will study nents are πj(t)=Aj(t)/ A; i.e. the probability how the variation of the disease parameters affects that an active individual is in city j.2 The similarity the evolution of the overlap over time. As initial between two outbreak realizations is quantitatively conditions we consider five infected individuals in measured by the statistical similarity of two realiza- Frankfurt (which is the airport with the largest tions (I and II) of the global epidemic character- number of connections). Results are obtained as ized by the vectors πI and πII respectively. Among averages over 500 different realizations of the possible measures of similarity, we consider here the stochastic noise. We consider a set of parameter val- Hellinger affinity ues and initial conditions such that the epidemic outbreak always occurs and the spread of the infec- πI , πII πI πII. simH ( )= j j (6) tion out of the initially infected city is observed. j Cases in which the initial outbreak dies out in the This measure belongs to the interval [0, 1], with seeding city due to fluctuations are not considered sim(πI , πII) = 1 when the two distributions are in this analysis.

2 P Note that one could as well consider instead πj(t)=Ij(t)/ l Il, i.e. only infected individuals. Results are not aﬀected by the diﬀerent choice. Epidemic Predictability in Meta-Population Models with Heterogeneous Couplings 2495

3.1. Transmission rate overlap profile at short times is similar for all values of β, the decrease of Θ is more pronounced for We first focus on the influence of the transmission smaller β, as shown in Fig. 1(b). Such trend can be rate,atfixedlatencytimeandwithinafullysus- explained by the fact that, since the transmission ceptible population. This corresponds to a change rate is reduced the number of active individuals gen- in the virus infectiousness, i.e. its ability to be erated per unit time is lower. This has aprioritwo transmitted from an infectious individual to a sus- consequences which both tend to lower the overlap: ceptible one, and affects the value of the reproduc- first of all, even in the absence of any travel term, tive ratio R0, the key epidemiological parameter for the relative stochastic fluctuations associated to the the description of a disease. In an SEIR model, random contamination process are larger; moreover, the reproductive ratio is given by R0 = β/µ and the fluctuations due to travel are enhanced as well, represents the average number of secondary infec- since the smaller number of infected individuals can tions generated by a typical infectious individual spread the disease in very different “diffusion” pat- in a fully susceptible population. Increasing values R terns for different realizations. In other terms, the of 0 can be achieved by increasing the transmis- evolution of a large number of individuals on the sion rate or increasing the average infectious period network is statistically more predictable than that µ−1 , leading to the same qualitative results. In the of a small number. The effect can be quite drastic, following we will focus on changes in the virus since the minimal value reached by Θ in the initial β µ infectiousness while keeping constant. Larger epidemic stages varies from 70% to approximately β values of correspond to faster propagation, higher 30% for β varying from 0.50 to 1.055. prevalence and larger number of cases, as shown in In order to understand the separate roles of Fig. 1(a). these two effects, we have isolated the two aspects. The evolution of the overlap Θ is shown in In a first experiment we have computed Θ for an ini- Fig. 1(b). Starting from a value equal to 1, due tially infected isolated city, i.e. a dynamical setup in to setting identical initial conditions, Θ decreases which the travel term is absent. This corresponds in the early stage of the outbreak, when the num- to a single-population epidemic model with inter- ber of cases is still small. As discussed in [Colizza nal fluctuations. The overlap values obtained are et al., 2006a, 2006b], this decrease is essentially extremely high, with 1 − Θ of the order 10−4.In due to the fact that the initially latent or infected a second experiment, we have monitored the evolu- may travel and go out of the initially infected city tion of Θ when only the diffusion process is consid- through various routes. In particular, the overlap ered, with no infection dynamics. This corresponds decreases more if the seed of the infection is a air- to set the disease parameter β, and µ equal to port hub, from which many different possible paths 0. Θ(t) decreases exponentially at short times and for the virus propagation can be taken. The overlap reaches a plateau which depends on the number then increases again and reaches a maximum at the of individuals diffusing on the network. For larger epidemic peak: in the absence of containment mea- number of individuals, the travel pattern fluctuates sures, this peak corresponds indeed to a relatively less and the overlap is larger. uniform and predictable pattern of infection. The At finite β, two competing effects are therefore gradual decrease of the global prevalence can on the determining the overlap profile: the diffusion of indi- other hand be quite different from one realization to viduals decreases Θ, while the exponential increase the other, with the epidemic dying out in different in the number of active individuals tends to increase cities at different times, so that Θ decreases again it. This competition leads to the observed minimum towards 0. of Θ(t), which occurs at larger times for decreasing Although the global evolution of Θ always β (since the second effect is slower), and thus at follows the same qualitative pattern, a strong quan- lower values of Θ. After the minimum, the epidemic titative effect is observed when the disease parame- growth takes over so that the overlap increases; the ters are changed. A first observation is simply that values of Θ reached at the epidemic peak are essen- the time scale of the spatial propagation depends tially independent of β: although the prevalence is obviously on the time scale of the local infection smaller for smaller β, it is in any case large enough dynamics inside each urban area: the evolution is to have a large reproducibility. globally slower for smaller β. In the initial stages Let us finally note that, for smaller β, the fluc- however, the picture is more complicated: while the tuations associated to the average overlap profile 2496 V. Colizza et al.

(a)

(b) −1 −1 Fig. 1. Results obtained for different values of the virus infectiousness β, assuming =1.9daysandµ =3days. Sub-populations are considered to be fully susceptible. (a) Full lines and (b) symbols correspond to average values and shaded areas, representing 95% confidence intervals. (a) Prevalence profiles. Inset: percentage of the cumulative number of cases with respect to the total population. (b) Overlap profiles. Insets: overlap profiles with corresponding fluctuations for β =1.055 (top) and β =0.5 (bottom). increase, as shown in the insets of Fig. 1(b), due 3.2. Initial immunity to the smaller numbers of active individuals, which We next consider the influence of changes in lead to larger fluctuations from one couple of real- the initial fraction of susceptible individuals, izations to the other. taking this fraction uniform across cities for Epidemic Predictability in Meta-Population Models with Heterogeneous Couplings 2497

(a)

(b) Fig. 2. Results obtained for different values of the initial fraction of susceptible population S(t =0)=αN, assuming −1 −1 β =1.055, =1.9daysandµ = 3 days. (a) Full lines and (b) symbols correspond to average values and shaded areas representing 95% confidence intervals. (a) Prevalence profiles. Inset: percentage of the cumulative number of cases with respect to the total population. (b) Overlap profiles. Insets: overlap profiles with corresponding fluctuations for S(t =0)=N (top) and S(t =0)=0.6N (bottom).

simplicity: α = Sj(t =0)/Nj . Smaller values of α prevalence proﬁle assumes smaller values and the corresponds to situations in which a larger fraction time scale is longer for larger α [see Fig. 2(a)], as of individuals in each sub-populations is immune expected since the virus ﬁnds a reduced pool of to the virus at the start of the epidemic. The available susceptible individuals. 2498 V. Colizza et al.

(a)

(b) −1 −1 Fig. 3. Results obtained for different values of the average latency period , assuming β =1.055, µ =3daysanda fully susceptible population. (a) Full lines and (b) symbols correspond to average values and shaded areas representing 95% confidence intervals. (a) Prevalence profiles. Inset: percentage of the cumulative number of cases with respect to the total −1 −1 population. (b) Overlap profiles. Insets: overlap profiles with corresponding fluctuations for =1.5 days (top) and =6 days (bottom).

As shown in Fig. 2(b), a decrease in α has essen- of the predictability, due essentially to the larger tially the same effect as a decrease in the value of fluctuations in the diffusion pattern. The relevant β, and can be traced back to the same reason: a epidemiological parameter is indeed R1−α = αβ/µ smaller number of cases leads to stronger decrease which reduces to the reproductive ratio R0 in a fully Epidemic Predictability in Meta-Population Models with Heterogeneous Couplings 2499 susceptible population (α = 1). Changes in α will The predictability of the disease propagation pat- affect R1−α in the same way as changes in the virus tern is quantified by the overlap between realiza- infectiousness. tions having the same initial conditions but different In summary, smaller values of the infectious- noise realizations. While the fluctuations due to the ness β, or larger values of the initial immunity internal noise of the disease evolution inside each probability, correspond to less dangerous epidemic city are less relevant in affecting the reproducibil- outbreaks but at the same time more difficult to ity of an epidemic outbreak, the availability of a forecast. Note that we keep here β and α constant huge number of different traveling paths for latent over the whole propagation, while the application and infectious individuals lead to a strong decrease of containment measures can be modeled through of the predictability at short times. The overlap an effective reduction of β over time. Numerically reaches a minimum during the initial stage of the reproducing data of a successfully contained pan- outbreak and then increases until the epidemic peak demic in order to validate an epidemic model may is reached. Such a profile is due to the presence therefore be more challenging because the number of two competing effects: on the one hand, given of cases is drastically reduced thanks to efficient a certain number of individuals diffusing along the measures. transportation network from a given initial city, the overlap between two realizations decreases because 3.3. Latency time of the possible diversity in diffusion patterns; on the other hand, an increase in the number of indi- β S t Let us now consider, at fixed and ( = 0), the viduals diffusing (which occurs as the epidemics −1 influence of the latency time .Weshowdata evolves) leads to smaller fluctuations around the −1 . for =15 days and six days corresponding typ- average pattern. In other words, when higher is ically to seasonal influenza or SARS-like diseases, the number of infected and latent individuals, the respectively. smaller will be the effect of the noise in the dif- As Fig. 3(a) shows, increasing latency times fusion process. This readily implies that when the lead to slower propagation and a longer duration infectiousness is lower, or the latency time larger, of the epidemic. The number of cumulative cases is providing a slower injection of infected individuals unchanged [see inset of Fig. 3(a)] since the repro- in the system, the diffusion noise has a larger impact ductive ratio is constant, not depending on . and the predictability decreases. These results are Since the prevalence is lower at any given time a first assessment of the predictability that can be for larger latency times, the trend observed is the achieved with large scale computational approaches β same as when decreases: larger fluctuations in to epidemic forecasts and indicates that a careful −1 the travel pattern are observed as increases. theoretical analysis of the specific modeling frame- However, a supplementary effect takes place in this work is needed in order to define the appropriate −1 case: for larger values of latent individuals spend confidence limits for the obtained predictions. on average more time in traveling from one region to another before changing compartment and thus becoming infectious. The overall time since infec- Acknowledgments tion to recovery is longer, so that the disease will We thank the International Air Transport Associ- very likely propagate to different regions in differ- ation (IATA) for providing us with the world-wide ent realizations. In summary, the faster the onset of airline database. A. Barrat and A. Vespignani are symptoms and infectiousness, the more predictable partially funded by the European Commission — the propagation pattern. contract 001907 (DELIS). A. Vespignani is partially funded by the NSF IIS-0513650 award. 4. Conclusions In this paper, we have considered the problem References of the predictability of epidemic scenarios, within Anderson, R. M. & May, R. M. [1984] “Spatial, tempo- the global stochastic modeling framework devel- ral and genetic heterogeneity in host populations and oped in [Colizza et al., 2006a, 2006b]. This approach the design of immunization programs,” IMA J. 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