Comparing Metapopulation Dynamics of Infectious Diseases

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(7). that migration patterns describ- population datasets describe movement detailed human highly of large, because ing of possible now availability is recent ranges the geographical wide across ring modeling processes. movement dynamical disease of various choice on have the focuses can which framework that of study consequences model a the conducted a then on have with we is coupled Here, which and dynamics. disease data mobility mod- mobility human underlying with simulating an parameterized for choose framework must dynamics diseases spatial eling infectious the sep- human of geographically studies of between Real-world disease driver populations. important infectious arated an of be spread to the proved long-distance of hosts cases, human these infected of of each travel In (6). coronavirus novel often 2019 are as models such across such syndrome pandemics, spread Many recent that 2). by epidemics motivated (1, study regions to geographical mod- of applied large mathematical effectiveness and such the adapted many and els been epidemics have of There impact interventions. and timing, size, M movement population human epidemiology mathematical results. intel- Bioko produces model of on the predictions one importation ligible that using and finding Guinea, model transmission Equatorial movement malaria Island, of appropriate case an applied choosing impact of impor- the profound tance demonstrate a further has We outcomes. another find- epidemiological choosing of on where results, instead regimes model’s parameter model are movement disease there one choice cases, each all the on in that that has effect ing model the candi- movement describe two We of of models. either movement incorporate date to model, Ross–Macdonald the and model, susceptible–infected– susceptible–infected–susceptible model the the models model, dynamics, recovered movement disease parameterize well-studied infectious of three of to adapt choice models We data outcomes. one’s disease new whether modeled impacts the and is movement, leverage question of to global-scale unanswered best understanding remaining of dis- how a infectious purposes Nevertheless, of the models epidemics. realistic for the construct dynamics to to ease possible distances. now contributes geographical is large movement It across diseases population infectious of 2020) human spread 21, April inves- review how to for opportunities (received tigate exciting 2021 8, present March datasets approved available and Newly CA, Davis, California, of University Hastings, Alan by Edited and 20910; MD nttt o elhMtisadEauto,Uiest fWsigo,Sate A98121; WA Seattle, Washington, of University Evaluation, and Metrics Health for Institute aaeeiigraitcmdl fdsaetasiso occur- transmission disease of models realistic Parameterizing 01Vl 1 o 8e2007488118 18 No. 118 Vol. 2021 ies rnmsin aigi osbet siaethe estimate to possible it making transmission, understanding for disease tools important are models athematical 3,Eoavrsdsae() iafvr() n the and (5), fever Zika (4), disease virus Ebola (3), c iiino isaitc pdmooy nvriyo aiona ekly A94720 CA Berkeley, California, of University Epidemiology, & Biostatistics of Division a,1 a R alsA Guerra A. 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J.M.H., and D.T.C. contributions: Author the mod- dynamics, of disease purposes infectious the has understand For study to 23). a movement (9, of eling conclusions attention the less affect much would received model of choice the movement-related model. suffi- the transmission as disease set mechanistic to evaluated a used been of be parameters has then it may evalu- 20–22). once accurate, 18, been ciently model, (13, settings movement have different fitted of which variety A models, a in radiation data against and ated two gravity models, between movement as candidate volume such many flow already or are travel There accurately locations. of data, frequency movement to statisti- the well a predicting adequately finding fits is that model consideration cal important An data? available models parameterize and build to data movement. the host use representing to ques- synthesize unanswered best to an how or remains data tion It the sources. in many gaps from build fill coming to to data necessary true movement is host of it of picture model studies a partial many for a and provides patterns, only movement collected host data human of how each method quantify Nevertheless, outcomes. to epidemiological models affects use movement to opportunity participants’ new ing study track to used been (19). activities recently movement also have gers enL Wu L. 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ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY structure of the movement model—the rules which govern how second is a Lagrangian movement model which we call the Sim- hosts’ movement patterns are represented—affect the model’s ple Trip model. We directly compare the Flux and Simple Trip quantitative behavior and subsequent predicted epidemiological models by setting parameters such that the total flux of travelers outcomes. The same travel datasets used to calibrate different between each subpopulation remains constant, thus emulating a movement models can have different disease dynamics as a con- case where one might have a single dataset from which one could sequence of the movement model choice (9). One’s choice of calibrate either movement model. We examine how disease movement model structure defines the set of parameters which model outcomes can change with different movement models require fitting and remains distinct from the question of how best for each transmission model. For all three transmission models, to fit those parameters. we find that the modeled quantities of interest relating to dis- There is an important structural difference between two of the ease dynamics can differ dramatically depending on one’s choice simplest and most commonly used classes of movement models. of movement model. We conclude by applying this analysis to a Eulerian movement models specify the rates at which hosts from model of malaria transmission and importation on Bioko Island one location travel to any other location. Eulerian models do not in Equatorial Guinea (EG) and use intuition based on our prior track individual behavior, such that after a host moves it has no analysis to frame our understanding for why the Flux model sur- memory of its previous location. Lagrangian movement models, prisingly fails to produce meaningful predictions of transmission by way of contrast, follow individuals. The class of Lagrangian intensity in that context. movement models includes any model that specifies how frequently hosts travel away from home before returning (23, 24). Methods The authors of ref. 24 formulate models of vector-borne disease Host Movement Models. We use metapopulation models to represent a with Eulerian and Lagrangian host movement but they do not network of geographically isolated populations of hosts. Each population directly compare the models’ quantitative behavior, inviting fur- occupies a location where local conditions may affect transmission intensity. ther questions of when one host movement model might be more Each population i contains Ni hosts. We assume that the population at each location remains stable over time—in applied contexts each population may appropriate than another. Depending on the level of detail con- be calibrated using census or other population data. Disease transmission is tained in the available data, one might be able to use the data assumed to be completely local, such that hosts from different subpopula- to select a class of candidate mechanistic movement models, but tions come into contact with one another only if they travel to occupy the the context for a problem also provides some useful information same location. We describe and compare two simple models of host move- about which model might be more appropriate (23). Modelers ment, each of which represents a different set of rules governing how hosts whose aim is to understand how movement influences epidemio- move from one location to another. logical outcomes must choose one such mechanistic description The Flux model is a Eulerian movement model which describes hosts as to represent movement. diffusing from one metapopulation to another (24):

More generally, there has never been a thorough exploration K K dNi X X of how the modeler’s choice of movement model may influence = − fi,jNi + fj,iNj, [1] dt modeled epidemiological outcomes. Most early discussions of j=1 j=1 metapopulation modeling (e.g., ref. 25) primarily focused on for- mulating the models, without discussing how one might make where Ni counts the number of hosts currently located at site i. The total number of hosts remains constant over time (N = PK N ). The constant f use of travel data for movement model calibration or how to i=1 i i,j represents the rate at which hosts located at i travel to j, where fi,i = 0 for select a movement model depending on epidemiological context. all i. The fully specified Flux model requires K(K − 1) parameters. More recently, the authors of ref. 26 offer the heuristic advice The Simple Trip model is a Lagrangian movement model which assigns that a Eulerian movement model is suited to describing animal home locations to each host and describes how hosts travel temporarily to migration while a Lagrangian movement model is more suited to other locations before returning home (23, 24). Unlike the Flux model, the describing human commuting behavior, but they do not provide Simple Trip model differentiates between the residents and the visitors cur- supporting quantitative analysis. At least one study has used the rently located at the same site. Visitors are able to interact with residents at same travel datasets to calibrate different movement models only a given site, but they return home at fixed rates:

to ﬁnd that the disease dynamics can change as a consequence of K K dNi,i X X the movement model choice (9). In some cases, it may be possible = − φi,jNi,i + τi,jNi,j dt to derive and compare simpler models of disease transmission by j=1 j=1 . [2]

carefully rescaling movement parameters to capture the essen- dNi,j = −τi,jNi,j + φi,jNi,i tial features of spatial dynamics in a generalized metapopulation dt (27). Generally speaking, however, there has not yet been a thorough quantitative investigation into whether the modeler is free The notation in Eq. 2 has changed to account for hosts retaining memory of their home sites: A host from i who is currently located at j will to substitute one model for another for a given epidemiological be counted as belonging to the Ni,j population, and the number of hosts setting without affecting the model’s results and conclusions. whose home is i remains constant over time, even if members visit other When it comes to incorporating movement data into an epi- PK sites (Ni = j=1 Ni,j). The constant φi,j represents the rate at which hosts demiological model, the problem becomes whether choosing one whose home is i travel to j, while the constant τi,j is the rate at which hosts mechanistic movement model over another can impact mod- visiting j from i return home to i. Both φi,i = 0 and τi,i = 0 for all i. The fully eled epidemiological outcomes, and whether there are certain speciﬁed Simple Trip model requires 2K(K − 1) parameters. epidemiological settings where one type of movement model When the movement equations reach a steady state and the derivatives may be more appropriate or accurate. In the present analysis on the left-hand side of Eq. 2 equal zero, the population Ni is distributed we explore this problem using compartmental metapopulation across the K metapopulation sites as follows (23): models which integrate together host movement and infectious 1 N* = N disease dynamics. We use three infectious disease transmis- i,i φ i 1 + PK i,k k=1 τi,k sion models—the susceptible–infected–recovered (SIR) model, . [3] φ 1 the susceptible–infected–susceptible (SIS) model, and the Ross– * = i,j Ni,j φ Ni Macdonald model—which represent a suite of tools for modeling τi,j PK i,k 1 + k=1 τ the transmission dynamics of a wide variety of pathogens. We i,k

adapt each of the disease transmission models to incorporate two We can use Eq. 3 to deﬁne a “time at risk matrix” ψ where ψi,j ≡ Ni*,j/Ni, the mechanistic representations of movement: The ﬁrst is a Eule- average number of individuals from i found at site j in the steady state, or rian movement model which we call the Flux model, and the the fraction of time on average that hosts who live in site i spend in site j.

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Trip Simple Flux the the parameterize to model required is Flux information requires the detailed only model, model Trip Flux Simple ified the in symmetric become fluxes parameters the match to popula- parameters steady-state Eq. of the property that Another model model. circumstances Flux these the under in tions shown be can It odrcl opr h uniaiebhvo fbt oes ewill we models, both of behavior quantitative the compare directly To obnn h ai I oe ihteSml rpmdl2 eobtain we 2, model Trip Simple the with model SIR basic the Combining i nteSml rpmodel: Trip Simple the in dR dR dS dS dI dI u eoddsaetasiso oe steSSmdl which model, SIS the is model transmission disease second Our dt dt dt dt h etse st noprt ohmvmn oesinto models movement both incorporate to is step next The dt dt i i i i i i ,j ,i ,j ,j ,i ,i ======f i β ,j β β −β −β γ γ dR dS = = dt dt dI dt and j i I I i i i S S ,j ,i → i i i φ 1 j j i P P i i ,j ,i + − = = = S S i h ubrof number the + ,j + qain (23): equations 2 P P P P i i N j N ,j ,i k K k K φ X k γ γ β −β =1 =1 nteFu oe stesm stenme of number the as same the is model Flux the in qain eciigtasiso ewe hosts between transmission describing equations i P * =1 P P i K β * i ,i k K k K i k K k K I ,j i =1 =1 /N =1 =1 φ S i nEq. in qa h tbeppltosi h ipeTrip Simple the in populations stable the equal R N − k K i N N k K k K i ovr coslctos o cosindividuals, across not locations, across vary to φ i i =1 I =1 =1 ,j i S ,i i N k k i i N N I I dR i X dt dS ,k dt j dt ,j ,i dI k k − I K − + f =1 i i K ,j ,i k k i 4 R I I φ ,j τ (K ,j ,i k k − τ i γ τ i − − ,j ,i i = = = ,i sta hnw osri h lxmodel Flux the constrain we when that is = ,k ,k f i j I ,j 6 ,i i − + X i + − ,j j I γ γ γ β −β N =1 f N − i K R ,j j ersn h rnmsinrt and rate transmission the represent I I I ,i aaees enn htmore that meaning parameters, 1)/2 j X k φ * X SI i i k i i N ,i ,j ,i ne hscniinteflyspec- fully the condition this Under . =1 X K =1 + + /N K j f i =1 ,j SI + − N K i j − ,j S X eiettaeesrtrighome returning travelers resident τ S j 1 φ i i =1 φ X k ,i K i i f γ ,k =1 + i K i i + ,k − ,j ,j I R . I I S f P i i i X j τ φ ,i i j ,k ,i + ,i =1 K i φ R i − ,j ,k k K + S j j X =1 ,i j I =1 i f τ i K ,j X k ,i j i ,i =1 ,j K + S φ τ I f i j γ j j ,j j ,k ,k ,i X τ k ycnrs,i held is contrast, by , I =1 i K j ,k N . S j τ . i ,k i ,k I i ,k . [5] [4] [7] [6] i eetteevrnetsspoto h etr hc rnmtdisease. transmit which vectors the of support environment’s the reflect infection tracks (M model parameters Ross–Macdonald (X infected hosts The human from among transmission hosts. dynamics vec- of human a mechanism includes susceptible a it as to except serves model, which The SIS population 29). the tor to (28, similar transmission is malaria model describe Ross–Macdonald to developed was which model, Models. 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X i j ,j =1 1 K X I j o entos which definitions) for =1 i + K ,i r f − j ,w banaset a obtain we 1), N X ,i ac f g i i i I τ ,j j . N i X X R ,j β i i i PNAS I i 0 salwdto allowed is ) i . i   ,j ,w obtain we 2), safunction a as aisacross varies . . | f9 of 3 [10] [11] [14] [13] [12] [9] [8] 13

ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY We also extend the basic Ross–Macdonald model by allowing human hosts We set γ = 1 in all locations, hold β2 constant, and measure the to move from one metapopulation to another as described by the Simple residual population S∞ in location 1 as β1 varies. We identify two Trip model. Following the analysis of refs. 17, 24, and 30, we simplify the transmission parameter regimes: β2 > γ, which is high enough to analysis of the equilibrium behavior of the metapopulation Ross–Macdonald cause an outbreak in location 2, and β2 < γ, which in isolation model using the time-at-risk matrix ψ. We rewrite the force of infec- does not lead to an outbreak in location 2. We also deﬁne two tion experienced by human hosts as a weighted average which combines travel parameter regimes, one where travel occurs frequently and local transmission and the average fraction time spent at risk in different locations: lasts short periods of time (τi,j > γ) and one where travel occurs K infrequently and lasts long periods of time (τi,j < γ). We specify dXi X Zj = (Ni − Xi) ψi,jba − rXi the parameter values for φi,j and τi,j in each of these regimes dt PK ψ N j=1 k=1 k,j k for the Simple Trip model and use Eq. 4 to also parameterize the dZi −gn . [15] Flux model, such that we emulate using the same data inputs to = acκi(Mie − Zi) − gZi dt parameterize both movement models. PK ψk,iXk Fig. 1 compares how the two SIR models with metapopula- κ = k=1 i PK k=1 ψk,iNk tion movement behave in different parameter regimes. In all

The expression κi represents the fraction of infectious humans located at cases the outcomes are dramatically different, showing that SIR i, accounting for both local residents as well as visitors from other loca- model predictions depend strongly on how host movement is tions. From the endemic equilibrium of Eq. 15, and following the analysis of represented in the metapopulation model. The two movement {i} ref. 17, we obtain an expression for R0 which reflects how the movement models do agree when transmission parameters are equal in both model alters transmission at each metapopulation site: locations (β1 = β2), but there are large quantitative differences whenever there is some heterogeneity in the local transmission K ! {i} X − rXk rκi R = ψ 1 / . [16] conditions across the different locations. 0,ST i,k N − X 1 + ac κ k=1 k k g i The major difference between the Flux and Simple Trip models is that the Simple Trip model constrains the amount of time that a traveler spends away. For example, the larger residual pop- Results ulation for the Simple Trip model, seen in the right-hand column SIR Models. We begin by directly comparing the predicted out- of Fig. 1, follows from a limit on the amount of time that resi- comes of the two versions of the SIR model with Flux and dents of population 1 spend in the high-transmission location 2. Simple Trip movement. We simulate epidemics in two coupled Another difference between the Flux and the Simple Trip mod- metapopulations of Ni = 500 human hosts by numerically inte- els is that hosts in the Flux model have no home residence and grating Eqs. 6 and 7 and measuring the size of the outbreak. continue to move freely between populations after the outbreak The population of susceptibles remaining after the outbreak has ends. As a result, as the epidemic dies out the fractions of sus- ended S∞ quantifies the size of the outbreak and allows one to ceptibles become constant across all locations. In contrast with infer R0 (26). Furthermore, S∞ quantifies the population at risk the Simple Trip model, the Flux model effectively erases any evi- in subsequent future epidemics, which is relevant for modeling dence that the transmission intensity differed between the two seasonally recurring pathogens such as influenza (31). locations.

Fig. 1. Comparing SIR model results. We numerically integrate Eqs. 6 and 7 to ﬁnd the size of the residual population of susceptibles following the outbreak. ∞ We plot the residual population fraction in location 1 (S1 /N1) as a function of transmission intensity in location 1 (β1/γ) while holding transmission intensity in location 2 (β2/γ) constant. We explore four different parameter regimes deﬁned by the transmission intensity of location 2 and the duration of travel. In all four parameter regimes there is a dramatic separation between the predicted outbreak sizes that depends on whether one uses the Flux (red) or the Simple Trip (blue) model. As long as residents of location 1 travel from a low- to a high-transmission environment (β2 > β1) the Simple Trip model’s outbreak will be smaller and the residual population will be larger; the opposite is true when residents of location 1 travel from a high- to a low- transmission environment (β2 < β1). (A) Short travel duration regime (φi = 1, τi,j = 10); low transmission in location 2 (β2 = 0.5). (B) Short travel duration regime (φi = 1, τi,j = 10); high transmission in location 2 (β2 = 2.5). (C) Long travel duration regime (φi = 0.01, τi,j = 0.1); low transmission in location 2 (β2 = 0.5). (D) Long travel duration regime (φi = 0.01, τi,j = 0.1); high transmission in location 2 (β2 = 2.5).

4 of 9 | PNAS Citron et al. https://doi.org/10.1073/pnas.2007488118 Comparing metapopulation dynamics of infectious diseases under different models of human movement Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 oprn eaouaindnmc fifciu iessudrdfeetmdl fhuman of models different under diseases movement infectious of dynamics metapopulation Comparing al. et Citron ( duration (β short travel 2 is When location duration prevalence. (φ in travel and regime transmission When intensity duration travel. transmission travel of between Short duration relationships (B) different the dramatically and produce 2 (Lower models location long (blue) of is Trip intensity Simple transmission and the (red) by Flux defined regimes parameter different four 1 location 2. Fig. location in locally and transmission 2, endemic sustain to enough high is intensity (β transmission 1 the location of function in a as model) Trip Simple the (I 1 prevalence, location now in in equilbria is endemic the model for SIS Eqs. solving numerically the by find for we interest which of outcome The N (Eq. model (Eq. metapopulation model metapopulation Flux Models. SIS o rnmsinlcto .Smlry h ipeTi model Trip Simple the the in Similarly, risk 2. at time location 2 A their transmission of Fig. less in low spend prevalence travelers endemic its because model and elevated Trip away, Simple spent The time away. of Eq. amount Flux (from The the rates home. movement specify from on not away constraint spends does extra traveler model a an time imposes of model amount the Trip Simple the because changes used. is disease, model endemic movement sustained which on has depending 1 location which intensity. above transmission 2A local Fig. rela- and In predicted prevalence models’ noticeable between movement a is tionships two there the locations between two separation the between differs travel 2 short intensity the (Fig. In regimes. regime parameter four duration the of each in models model. Eq. Flux use the and of model for Trip periods Simple the long for over values and (τ infrequently (τ time occurs time of travel periods where short travel one over where and one regimes, frequently parameter occurs travel two define again We i edfietotasiso aaee regimes: parameter transmission two define We h iceac ntesottae uainrgm arises regime duration travel short the in discrepancy The Flux and Trip Simple the both from results the shows 2 Fig. 500 = 9 β and oprn I oe eut.W ueial ov Eqs. solve numerically We results. model SIS Comparing i 2 ,j I γ < 1 ∗ ope oehr gi,w set we Again, together. coupled γ < /N ehold We 10. ti vntecs htteaprn rtclpoint, critical apparent the that case the even is it enx opr h epciebhvo fteSIS the of behavior respective the compare next We hr rvldrto eie(φ regime duration travel Short (A) converge. results models’ movement two the ), hc antlclyssanedmctransmission. endemic sustain locally cannot which , 1 .I ahrgm eseiytae aeparameter rate travel specify we regime each In ). safnto ftasiso nest nlcto (β 1 location in intensity transmission of function a as 1 1 ∗ ). /N 1 o h lxmdland model Flux the for β 2 A = 2 edt vrsiaetetm spent time the overestimate to tend 4) osatadslefrteprevalence the for solve and constant ogtae uainrgm,9φ regime, duration travel Long (D) 0.5). and for 10) i = ,a oga h transmission the as long as B), 0.5, n h I ipeTrip Simple SIS the and 9) τ i K ,j = 2 = 0;hg rnmsini oain2(β 2 location in transmission high 40); γ eaouain of metapopulations 1 = (I 4 1,1 ostparameters set to ∗ tallocations. all at + β 2 I i 1,2 ,j ∗ γ > γ > 9 )/N which , and and ) 1 1 10 for /γ i = is hl odn rnmsinitniyi oain2(β 2 location in intensity transmission holding while ) o h rvlnea nei qiiru.W ltteedmcpeaec in prevalence endemic the plot We equilibrium. endemic at prevalence the for 0.005, ec n rnmsinitniyfrtetoRoss–Macdonald two the for intensity transmission and lence and prevalence between R relationships predicted dura- models’ travel two the short 3 the (Fig. in regime particularly tion results, and Flux model the Trip between separation Simple dramatic a is there different, are to from defined Eq. using are parameters parameters model Trip model based Simple Flux the regimes before, parameter As (φ 1/4,000). of trips long-duration pair low-frequency, versus another (φ trips and short-duration high-frequency, on 2 regimes location (X parameter in high two define on we based Again, respectively. models, two the 14 compare We N using levels. models prevalence endemic movement observed sustain to at required is intensity malaria model transmission local with Ross–Macdonald the of pathogen fies the malaria-causing period in a infectious interest We represent mean movement. to Trip a model Simple the and a Flux use with of models less Macdonald makes Models. home Ross–Macdonald from away the spent because converge, time difference. regime, models Trip parameter on Simple this preva- constraint and In the model 2: to Flux location equal the of become residents time to among enough travelers lence is among there period prevalence infectious for disease high the long than the the longer in In 2, is smaller. risk (Fig. be regime at to duration spend constrained travel 1 is location 2 location from transmission travelers that time of 2B Fig. in prevalence endemic 0 i i.3ilsrtshwpeaec nlcto aisi relation in varies 1 location in prevalence how illustrates 3 Fig. τ CD AB R 500 = i oapoc n nte.Terltosisbtenpreva- between relationships The another. one approach do ,j and 0 = 2 nlcto .A oga h rvlne nec location each in prevalences the as long As 1. location in = ) ihtasiso nlcto ( β 2 location in transmission high 0.4); ) (C 2.5). 16 ua ot yseiyn rvlneaduigEqs. using and prevalence specifying by hosts human ocalculate to ogtae uainrgm,9φ regime, duration travel Long ) i = 0.5, A 0 = τ and i ,j . esslw(X low versus 25) = eeaietebhvo fteRoss– the of behavior the examine We 0;lwtasiso nlcto (β 2 location in transmission low 40); .I rvldrto sln,however, long, is duration travel If B). K R 0 2 = ssprse eas h amount the because suppressed is o h lxadSml Trip Simple and Flux the for Lower r https://doi.org/10.1073/pnas.2007488118 −1 once uppltosof subpopulations connected 200 = ,hwvr fteti time trip the if however, ), 2 = 2 i /γ 2.5). 1 = .Teotoeof outcome The d. 0 = i = i osat eexplore We constant. ) 4. R 1 = /360, 0.005, . 0 prevalence 025) hc quanti- which , /72,000, PNAS τ τ i ,j i ,j Upper = 1 = ) low 0.4); 2 | = τ f9 of 5 /10) ,the ), i 0.5). ,j =

ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY AB

Fig. 3. Comparing Ross–Macdonald model results. We numerically solve Eqs. 14 and 16 to find the relationship between prevalence in location 1 (X1/N1) and transmission intensity (R0). We plot the relationship between location 1 prevalence (X1/N1) as a function of transmission intensity in location 1 while holding prevalence constant (X2/N2) in location 2. Note that we plot prevalence on the y axis and transmission intensity on the x axis, to emphasize how the Ross–Macdonald model behaves similarly to the SIS model (Fig. 2). We explore four different parameter regimes defined by the prevalence of location 2 and the duration of travel in relation to the average duration of infection (r−1 = 200 d). In the short travel duration regime, there is a large quantitative difference in the relationship between prevalence and R0 for the two different movement models. In the regime with short travel duration {i} and high location 2 prevalence (Upper Right), for a given prevalence of 0.2 using the Flux model would suggest low transmission (R0,Flux < 1), whereas using {i} the Simple Trip model would suggest high transmission (R0,ST > 1). In the long travel duration regime, the quantitative disagreement between the two movement models is less significant. (A) Short travel duration regime (φi = 1/180, τi,j = 1/10); low prevalence in location 2 (X2/N2 = 0.025). (B) Short travel duration regime (φi = 1/180, τi,j = 1/10); high prevalence in location 2 (X2/N2 = 0.25). (C) Long travel duration regime (φi = 1/72,000, τi,j = 1/4000); low prevalence in location 2 (X2/N2 = 0.025). (D) Long travel duration regime (φi = 1/72,000, τi,j = 1/4,000); high prevalence in location 2 (X2/N2 = 0.25).

models are qualitatively similar to the one shown previously for veys (MIS) (34–37) has used geostatistical techniques (38) to the SIS model in Fig. 2. map the estimated prevalence of malaria (Fig. 4A) and the esti- There are some ranges of prevalence that are not allowed as mated probability of a resident leaving (Fig. 4B) across different long as R0 > 0, such as the Flux model predictions in Fig. 3B. This locations on Bioko Island (14). The analysis showed an elevated occurs when incidence and, by consequence, prevalence among risk of infection among people who reported traveling recently travelers to location 2 is so high that it raises the base preva- (14), a pattern corroborated by other studies (33, 39). In partic- lence in location 1. For example, in the Flux model shown in ular, this suggests that travelers who had spent time in mainland Fig. 3B, the relationship between transmission, movement, and EG, which still has an estimated mean prevalence of around 0.43 prevalence specified by the model makes it impossible to rec- (40, 41), are very likely to bring back infections with them that oncile high prevalence in location 2 with very low prevalence would contribute to the sustained endemic prevalence measured in location 1. The transmission model creates an effective lower on Bioko Island. bound on prevalence in location 1, and we interpret prevalence We can use the Ross–Macdonald model to infer local trans- values below that lower bound as having no valid solution which mission intensity, described by the local R0, based on the known is consistent with the transmission model. prevalence and travel behavior. We treat each of the pixels shown in the maps in Fig. 4 as its own subpopulation with its own local Malaria Modeling Example. We further illustrate the quantitative transmission environment, which allows the model to contain the disagreements between the Flux and Simple Trip models by spatial heterogeneity known from the data. We use the popula- applying the Ross–Macdonald models with host movement to tion data to set the population Ni and the geospatial estimates to the problem of estimating local R0 (i.e., computed as if there set the prevalence Xi /Ni at each location. For the travel param- was no host movement) in a real-world setting. Bioko Island eters we use trip destinations reported in the MIS (37, 42) to fit is located in the Atlantic Ocean about 225 km to the north- a multinomial probability model of destination choice. Together west of mainland EG in Western Africa. From population data with the travel frequency data (Fig. 4B), we find a set of travel collected during a bednet mass-distribution campaign in 2015, frequency parameters {φi,k }, illustrated in Fig. 4C. We use MIS human population count was estimated to be 239,000 people, data on trip duration (37, 42) to fit an exponential model of trip about 85% of whom lived in EG’s capital city, Malabo, in the duration and find that travel within Bioko Island and to mainland −1 −1 north (32). Following the recent efforts of an intensive malaria equatorial had a mean duration of τi,BI = 21.2 and τi,EG = 10.3 d, control and elimination program (Bioko Island Malaria Elimina- respectively (further details can be found in SI Appendix, section tion Program, BIMEP), the average malaria parasite rate (PR, a 2). The rate parameters fi,j for the Flux model were calculated measure of prevalence) in children 2 to 14 y of age has fallen using Eq. 4. from an average of 0.43 to 0.11 between 2004 and 2016, with In this way, we construct a fully parameterized pair of Ross– further progress appearing to have stagnated since 2015 (33). A Macdonald models with movement between subpopulations, recent analysis of data collected through malaria indicator sur- from which we can infer the transmission intensity in each

6 of 9 | PNAS Citron et al. https://doi.org/10.1073/pnas.2007488118 Comparing metapopulation dynamics of infectious diseases under different models of human movement Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 i.5. Fig. 4. Fig. oprn eaouaindnmc fifciu iessudrdfeetmdl fhuman of models different under diseases movement infectious of dynamics metapopulation Comparing al. et Citron (Inset solution. island. the across (Eq. culated model Trip Simple the (Eq. model R Flux the using culated R mission (Eq. the model model, Trip of Simple volume the using the 5B and Fig. prevalence infections. local imported local reconcile with to while model intensity transmission infected the transmission become for people high too which is at traveling rate the locations, these pixels destination the of all to 5A Fig. travel Island. of Bioko rate on total location sum the or regions, destination seven weights Arrow the EG. of mainland each to to travel represent pixel ( off-island period. origin pointing survey from Arrows region. 8-wk travel. travel each an recent within of during reporting 1- home rate data a leaves MIS to resident within from proportional a estimated prevalence are that pixels, estimated probability example the four mean of shows the each pixel for each shows where pixel 14, ref. Each from 14. permission ref. from data surements, n h w eso rdcin n ute lutaighwthe areas. how some in illustrating results further meaningful and produce to predictions fails of model Flux sets two 5C the travelers Fig. ing infections. to of imported attributable distributions with the be compares mainland may the area from urban returning this in prevalence for solution valid no is there where areas (Eq. model represent Flux the using lated BC AB ABC 0,Flux 0,ST enn htacrigt h lxmdltaeesaesbett o ihtasiso ikwe ntemiln.(B) mainland. the on when risk transmission high too to subject are travelers model Flux the to according that meaning , ≤ rvlne(aaierate, (parasite Prevalence (A) data. from model Ross–Macdonald Parameterizing oee siae of estimates Modeled 1 R nmn lcsi aao ugsigta h sustained the that suggesting Malabo, in places many in 0,ST R 0,ST ≥ 1 siae r airt nepe.Lcltrans- Local interpret. to easier are estimates ln h otws os n rudRiaba. around and coast northwest the along aaeeie ihteiptdt hw nFg .(C 4. Fig. in shown data input the with parameterized 16) R h ulrneo h lxmdlrsls h akga ouno auscut h ra where areas the counts values of column gray dark The results. model Flux the of range full The ) 0 l oee siae eegnrtduigtesm rvlneadtae aa lutae nFg .(A) 4. Fig. in illustrated data, travel and prevalence same the using generated were estimates modeled All . asetmtsof estimates maps R aaeeie ihteiptdt hw nFg .Dr rydntsaesweeteei ovldslto for solution valid no is there where areas denotes gray Dark 4. Fig. in shown data input the with parameterized 14) 0,Flux .Tedr ieso h map the of pixels dark The 14). asetmtsof estimates maps .I otatwt h Flux the with contrast In 16). and R 0,ST ute contrast- further , R 0,ST R 0,Flux calculated R 0,Flux calcu- In . httemdlbek onadde o lo meaningful allow high not so does is how and infections see down imported we breaks of of regime values model rate parameter the include the this not that locations in does travel- some and model from therefore in Flux constraint, home and The a return infections. EG such who malaria mainland people with on of ing spend number amount the the the travelers values constrains reduces that model prevalence Trip time low Simple of for The model. how underestimate shown Trip to see ple results tends we the model where recall Flux (τ We 3A, prevalence. short Fig. local is in the duration than travel higher where exists regime Island r parameter Bioko a on subpopulation in Each model. Macdonald K n h rvlnei h f-sadtae etnto is destination travel off-island the in prevalence the and ) efaeorudrtnigo hs eut ntrso the of terms in results these of understanding our frame We 2 = upplto xml icse rvosyfrteRoss– the for previously discussed example subpopulation itgascmaigtedsrbtosof distributions the comparing Histograms ) X R i /N 0,Flux × i 1-km a,etmtdfo h I olce aaiei mea- parasitemia collected MIS the from estimated map, ) . 2 rvl“rvlne a,rpoue with reproduced map, “prevalence” Travel (B) area. https://doi.org/10.1073/pnas.2007488118 C R oee ae ftravel of rates Modeled ) 0 oprdt h Sim- the to compared R 0,ST R R 0,Flux acltdusing calculated , 0,Flux PNAS and a ovalid no has R 0,Flux R | 0,ST {φ f9 of 7 i cal- , ,j cal- i ,k > }

ECOLOGY BIOPHYSICS AND COMPUTATIONAL BIOLOGY Discussion individuals who travel from one location to another. Often, We have shown that models of infectious disease dynamics for the sake of preserving subscriber privacy, CDRs are aggre- are sensitive to the choices that modelers make when includ- gated across individual users, time, and geographical area before ing movement-mediated interactions between subpopulations. becoming public. That is to say, publicly available CDR datasets To do so, we have adapted each of three transmission models to count the number of mobile phones traveling from one service include two types of movement between subpopulations. For all tower catchment area to another on a daily or weekly basis; these three transmission models we have identified parameter regimes datasets do not record an individual’s rate of leaving home or where choosing to incorporate one movement model instead of time spent while traveling away from home. The loss of indi- the other will result in dramatically different results. Specifically, vidual identity from datasets has already been shown to impact when there is a difference in transmission intensity between two the predicted outcome of disease models in other contexts (9). interacting locations, and when mean travel duration is short While the limits and biases inherent to CDR datasets have been compared to the mean duration of infection. well-documented (43), the present study suggests an additional In all instances we set movement parameters to match the drawback when it comes to using CDR datasets for epidemi- total flux of individuals moving between each pair of subpop- ological modeling: Because it is not clear that a Flux model is ulations. In this way, we have emulated a situation where the sufficient to produce accurate modeling results in some contexts, modeler has a single movement dataset and must choose which it is likewise not clear that flux data are sufficient to quantify movement model to implement. The full impact of the choice of travel behavior when building a movement model. movement model became clear from the example of modeling This is not to say that CDR data or flux models are useless malaria transmission and importation on Bioko Island: We cali- in every context. Our aim is that the present study serves as a brated each of the two movement models using the same dataset starting point for understanding when the flux model and Simple and found that each movement model produced dramatically dif- Trip model are interchangeable. In some settings, the conclu- ferent results. In some areas, the Flux model and was unable to sions may not be affected by the choice of a model, but one needs produce meaningful values of R0,Flux. to carefully consider the epidemiological context before choos- The Simple Trip model, by specifying the rate at which trav- ing a movement model. More generally, the question of which elers return home, adds a constraint on the amount of time that model is appropriate to use in context or how to model the spatial travelers spend while away from home. As a consequence, the dynamics of pathogens merits greater attention (9, 23, 27). Simple Trip model constrains the amount of risk experienced The present work is restricted to discussing two basic deter- by travelers. The Flux model has no such constraint: As long ministic models of movement. In theory, one may imagine simi- as there are variations in the transmission intensity across dif- larly comparing more detailed and complicated movement mod- ferent locations there will be significant differences between the els, such as incorporating travel behavior with multiple stops, two movement models’ results. adding stochastic effects, or using agent-based models to incor- The way that each of the movement models contributes to the porate movement behavior heterogeneity among the travelers. It infectious disease model results can lead to real confusion over remains to be seen the extent to which such additional model the true relationship between transmission intensity and epi- features might further affect the epidemiological outcomes of demiological outcomes. In some applications, as with the Bioko interest. In practice, however, the data required for calibrating Island example, the purpose of the model is to infer transmis- such models may not be readily available, and using a Flux or sion conditions, which may be difficult to measure empirically, Simple Trip description of travel behavior may be more practical. based on more readily available prevalence or incidence mea- Even for such simple movement models there is a risk of misus- sures. Returning to Fig. 2B, we can imagine trying to determine ing the travel data that are available to calibrate a movement the transmission intensity β1 based on a measured prevalence model which cannot be reconciled with the data and produces of 0.25. From the Flux model we infer β1 < 1, which suggests outputs that are difficult or impossible to interpret. that local transmission risk is low. From the Simple Trip model, however, we infer β1 > 1, which suggests the opposite. A simi- Supporting Datasets lar inconsistency is illustrated for the Ross–Macdonald model in Refer to SI Appendix, section 4 to find code and data supporting Fig. 3B and appears again in the Bioko Island example in Fig. 5. the analysis presented in Malaria Modeling Example and Figs. 4 In the case of modeling malaria transmission and importation and 5. on Bioko Island we have shown how the Flux model is not suit- able for producing meaningful results. Despite this drawback, Data Availability. Data and code supporting the analysis presented in the Flux model remains an attractive option for creating a basic Malaria Modeling Example have been deposited in Figshare (https://doi.org/ quantitative description of host movement in other epidemio- 10.6084/m9.figshare.12084831.v2). All study data are included in the article and/or . logical contexts. The Flux model requires fewer parameters to SI Appendix specify completely. For this reason, parameterization is possible ACKNOWLEDGMENTS. We acknowledge support from the Bill & Melinda with relatively simple datasets which count the number of indi- Gates Foundation, Grant OPP1110495. Additionally, we thank the partici- viduals moving from one each location to each other location pants of Bioko Island who have taken part in the surveys from which the Bioko data were drawn, the BIMEP field surveyors and supervisors who col- over time. Unfortunately, datasets which record flux volumes do lected these data, as well as the National Malaria Control Program and the not necessarily lend themselves to parameterizing Simple Trip Ministry of Health and Social Welfare of Equatorial Guinea, Marathon Oil, models, as they do not distinguish between the rates of leaving Nobel Energy, and partners for their continued support in producing these and returning home. datasets. We thank the following collaborators for their helpful insights and discussion: Joel C. Miller, Nick W. Ruktanonchai, M. U. G. Kraemer, Amy For example, mobile phone call data records (CDRs) are a Wesolowski, John R. Giles, Guillermo A. Garc´ıa, Dianna Hergott, Su Yun source of high-volume datasets for measuring the number of Kang, Katherine E. Battle, and Daniel J. Weiss.

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