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Alison by Edited 12601 NY Poughkeepsie, College, Vassar 92831; CA Fullerton, University, d Ingenier a io .LvnMteaia n opttoa oeigSine etr rzn tt nvriy ep,A 85287-3901; AZ Tempe, University, State Arizona Center, Sciences Modeling Computational and Mathematical Levin A. Simon eateto ahmtc,Clfri tt nvriy ulro,C 92831; CA Fullerton, University, State California Mathematics, of Department eprtr ydoe(AS pdmc h 09H1N1 2009 the acute epidemic, severe 2003 (SARS) pandemic, syndrome HIV the respiratory from learned essons aBiom ´ ıa dc,Uiesddd o ne,Bgt,Clmi 111711; Colombia Bogota, Andes, los de Universidad edica, ´ | risk | ope dpiesystems adaptive complex a,b,c,1,2 rvd potnte to opportunities provide aegypti, Aedes f colo ieSine,AioaSaeUiest,Tme Z82740;and 85287-4501; AZ Tempe, University, State Arizona Sciences, Life of School ediBichara Derdei , | mobility d,e,1 o n climato- and no) ˜ n ejmnR Morin R. Benjamin and , | behavior c fc fteRco,Yca ehUiest,Ucqi cao 100119; Ecuador Urcuqui, University, Tech Yachay Rector, the of Office e etrfrCmuainladApidMteais aionaState California Mathematics, Applied and Computational for Center hsatcei NSDrc umsin ...i us dtrivtdb h Editorial 2 the by invited Editor Guest a 1 is A.P.G. Submission. Direct PNAS Board. a is article This interest. of conflict no declare authors performed The B.R.M. and D.B., C.C.-C., and research. research designed C.C.-C. contributions: Author n ie eodnso otpeettosaeaalbeo h A est at website NAS program the complete on The available DC. are Washington, presentations in most 2016, of Sciences 14–15, www.nasonline.org/Coupled recordings of March video Academy held Systems,” and National Environmental the and Human at “Coupled Sciences, of varied, extraordinarily dispersal arti- with are deal review populations that a distributed “Models disturbances spatially In that, localized and observe 21). al. (20, that et biodiversity Kareiva role cle, maintaining the in had establish have to used, was example, 19) microscopic (18, metapopulations for from of scales, theory arise The the (17). different features processes establish macroscopic at to the how operating used on highlighting processes focusing models paper between contributions and seminal relationships joint a techniques in of of together series development all a it integrated is put information that He how scales. knowing across to carried a tied was make phenomena that dependent scales spread the at derail diseases, to reemergent ability difference. and infor- our perceived emergent limits prop- or risks, of and real disease measure manner, on to timely mation officials a combined health in when communicate, public public risk, erly of ability guiding about risk the hosts’ in information world, with critical diffuse today’s In is or spread. mobile, knowledge disease highly to partic- responses the movement, health individuals’ of of those humans patterns to ularly between the population and humans, vectors, between human and interactions understanding the Therefore, of anywhere. of patterns nearly the proportion trade or increasing move, live, an of ity Information.” ecoepidemi- of by “Era reshaped the being in socioepisphere ology In a systems. in adaptive live complex we proto- define short, are that feedbacks dynamics the disease reemergence; of to and typical responses emergence human disease coevolving of the timing human importance the the of assessing recognizing role of while the dynamics quantify disease and to identify responses individu- help that may decisions undertake risk-aversion outbreak als the disease past on of 16) role (15, the a experiences or 14), required along 13, information (9, that of connections impact social scenarios the central- (12), recent of information of impact three sources the ized addressing but Studies cases are response. 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SUSTAINABILITY POPULATION COLLOQUIUM SCIENCE BIOLOGY PAPER partly because they employ three distinct characterizations of populations (see also ref. 46). Recent studies have also addressed space: as ‘islands’ (or ‘metapopulations’), as ‘stepping-stones’, the issue of homogeneity of contacts in epidemiology through or as a continuum” (22). We choose to deal with mobility network-based analyses that identify host contact patterns and using a metapopulation approach (7, 10, 18), that is, popu- clusters (refs. 47–49 and references therein). An extensive rela- lations exist on discrete “patches” defined by some charac- tively recent review paper is ref. 50. This approach by focusing teristic(s) (i.e., location, disease risk, water availability, etc.). on how each individual is connected within the population has Patches are connected by their ability to transfer relevant infor- been able to address the effects of host behavioral response on mation between one another, which, in the context of disease disease (see refs. 51, 57, and 58 for a review). Other dynamics, is modeled by the ability of individuals to move approaches included the effect behavioral changes triggered between patches. Patches may be constructed (defined) by by “fear” and/or awareness of disease (52–54, 56). Although species (human and mosquito) with movement explicitly mod- this stress-induced behavior may be of benefit to public health eled via patch-specific residence times and under a framework efforts in some cases, it can also cause somewhat unpredictable that sees disease dynamics as the result of location-dependent outcomes (55). interactions (23, 24). However, the fact remains that our ability to determine (hence The movement/behavior of individuals within and between define) what an effective contact is in the context of communica- these patches may be driven by real or perceived personal ble diseases, that is, our ability to measure the average number economic risk and accompanying social dynamics. Embedding of contacts that a typical patch resident has per unit of time and behavioral-driven decisions within epidemiological models has where, has been hampered by high levels of uncertainty. There- shed new perspectives on the modeling of disease dynamics fore, when we ask, what is the average number of contacts that an (25), expanding the options available to manage infectious dis- individual has while riding a packed subway in Japan or Mexico eases (5, 26). Economic epidemiological modeling (EEM) has City, or what is the average number of contacts that an individual previously addressed the role of individuals’ behavior when fac- has at a religious event involving hundreds of thousands of peo- ing the risk of disease, albeit it has often failed to incorpo- ple, including pilgrimages, one quickly arrives at the conclusion rate within host– feedback mechanisms (27–33). The that different observers are extremely likely to arrive at a highly focus on EEMs that account for host–pathogen feedback mech- distinct understanding and quantification of the frequency, anisms has propelled their study of the ways that contact intensity, and levels of heterogeneity involved. In short, this per- decisions impact disease emergence or alter “expected” infec- spective puts emphasis on the use of a different currency (res- tious disease- dynamics. The class of decisions idency times) because measuring contacts at the places where involved may include the determination to engage in trade along the risk of is the highest, pilgrimages, massive religious particular routes (34–37), or to travel to particular places (5, 38– ceremonies, “Woodstock time events,” packed subways, and 40), or to make contact with or to avoid particular types of peo- other forms of mass gathering or transportation have not been ple (25, 41, 42). EEMs advance the view that the emergence done to the satisfaction of most researchers. The risk of acquir- of novel zoonotic diseases, such as SARS or the Nipah virus, ing an infectious disease within a flight can be measured at least depend on the choices that bring people into contact with other in principle as a function of the time that each individual of species (43, 44). EEMs are typically built under the assump- x-type spends flying, the number of passengers, and the likeli- tion that associated disease risks are among the factors that hood that an infectious individual is on board. For example, mea- individuals must consider when making decisions. Therefore, suring the risk of acquiring tuberculosis, an airborne disease that individual decision-making process, within epidemic outbreaks, may spread by air circulation in a flight, may be more a func- may require the incorporation of humans’ cost–benefit-driven tion of the duration of the flight and the seating arrangement adaptive responses to risk. than the average number contacts per passenger within the flight (see ref. 59 and references therein). Furthermore, replication A Lagrangian Approach of Modeling Mobility and Infectious studies that measure risk of infection in a given environment may Disease Dynamics indeed be possible under a residency times model. In short, the Differences in disease risk exist between countries as a function risks of acquiring an infection can be quantified as a function of localized poverty, sanitary/phytosanitary conditions, access to of the time spent (residency time) within each particular envi- healthcare, levels of education, cultural practices, and norms ronment. The Lagrangian modeling approach builds (epidemi- with travel and trade overcoming the natural boundaries pro- ological) models by tracking individuals’ patch-residence times vided by these factors in limiting the spread of pests and or by budgeting their contacts according to the time spent on . The negative impact of the use of cordons san- each environment (60). The value of these models increase when itaires to limit the spread of Ebola in West Africa high- we have the ability to assess risk as a patch-specific characteris- light the importance of developing and implementing novel tic. In short, the lack of preference on the use of contacts is not approaches aimed at ameliorating the impact of disease out- tied to their proven intellectual value or the use of a Lagrangian breaks in areas of the world that cannot respond in a timely modeling perspective but rather to the difficulties that must be manner to novel disease outbreaks. Therefore, the identification faced when the goal is to measure the average number of contacts of a theoretical explanatory framework that systematically dis- per type-x individual in the environments that facilitate transmis- entangles the role of epidemiological and socioeconomic per- sion the most. spectives on disease dynamics becomes not only evident but The Lagrangian approach is highlighted here via the formu- necessary. lation of a disease model involving the joint dynamics of an Classical mathematical epidemiology uses per capita contact n-patch geographically structured population with individuals rates (who mixes with whom or who interacts with whom) as moving back and forth from their place of residence to other the social dynamics currency responsible for the transmission patches. Each of these patches (or environments) is defined by its dynamics of communicable diseases. We envision disease trans- associated risk of residency-time infection. Patch risk measure- mission as the result of the “collisions” between individuals or ments account for environmental, health, and socioeconomic as a consequence of the movement/relocation of individuals, conditions. The Lagrangian approach (61–63) keeps track of the never identified by placed of residence, from patch to patch. identity of the host regardless of their geographical/spatial posi- This approach has had great practical and theoretical successes; tion. The use of Lagrangian modeling in living systems was, to the scholarly and extensive review in ref. 45 addresses this view the best of our knowledge, pioneered and popularized by Okubo within homogenous and (heterogeneous mixing) age-structure and Levin (62, 63) in the context of animal aggregation. Recently,

2 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.1604994113 Castillo-Chavez et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 [S follows: as modeled is term infection the fore, i e aiantrldah n eoeyrts epciey in respectively, rates, recovery and death, Patch natural capita per atloCae tal. et Castillo-Chavez is, that patches, between movement no is there which where for case case the high-mobility represent the curves represent blue The curves. black dashed and 1. Fig. Letting area. ular vector “transmission/risk” the by B defined risks, environmental matrix times residence where a of help of the with context itored the in human and model (64–67) to (59). behavior bioterrorism used and been movement also crowd have approaches Lagrangian o ikPth1i ohapoce with approaches both in 1 Patch risk low N ( equations: differential nonlinear this of tem Patch captures an in to model approach Lagrangian visitors (SIS) susceptible-infected-susceptible and account cal residents must patch the each effec- within for population that effective conclude the we is, That isolated, Patch are in size patches population when tive is, that sal, Patch ntepthcnetvt tutr.Teeoe fteresidence the if Therefore, structure. matrix time connectivity patch the on I System rewrite to allows ˙ ˜ , I S = i i ˙ ( = where h ieiodo neto nec ac ste nt the to in tied is patch each in infection of likelihood The mon- is patches across mobility and status host-residence Here h yaiso h ies nalo h ace depends patches the of all in disease the of dynamics The ˙ i i 1 = netdi Patch in infected = = = diag( β j P b p P i , . . . , 1 Letting . yaiso h ies nPth1frtreseilcss h ymti eiec ie (p times residence symmetric The cases. special three for 1 Patch in disease the of Dynamics i h fetv population effective The . ij β , t N − j n b ¯ N =1 stepooto ftm eiet fPatch of residents time of proportion the is ¯ 2 i , , d β , . . . , P n − (S i d d S sirdcbe ace r togycnetd then connected, strongly are patches irreducible, is i includes I i i and , (d = )Pdiag(B netdi Patch in infected + N I i n γ ) (I = eoetepplto fPatch of population the denote i t j I 1 i = ] γ n h rprino ieseti partic- in spent time of proportion the and , − i d 10 20 30 40 50 60 70 80 P 1 2 0 1 0 eoetecntn erimn,the recruitment, constant the denote )diag( , P , . . . , h iko neto nPatch in infection of risk the × × j n ntefloigsnl etra form vectorial single following the in I =1 2 n j n , . . . , rprino netdi Patch in infected of Proportion Patch from Susceptible i =1 ac etn i h olwn sys- following the via setting Patch ttime at , p d N ij (S ˜ n | I P |{z} ) ) P j i j I R −1 β t ) n netdi Patch in infected and , 0 1 k n netdidvdas There- individuals. infected j ) k n − p P = =1 P t =1 12 , {z 10 t t γ d j n I = sgvnby given is , p 1 =1 i N p β ¯ +γ I kj − kj p 1 γ i i 21 N p = 1 − I diag(d | p k ij (γ = = k = i ttime at } j ij {z  d N h r urnl nPatch in currently are who 66 h etclai ersnstepeaec ftedsaei ac .Fgr oreyo e.24. ref. of courtesy Figure 1. Patch in disease the of prevalence the represents axis vertical The 0.7636. σ S i P d b j I 21 } i 1 1 i ( = , , nec Patch each in = j 1 . + d b γ , σ P 2 2 20 i p j 21 , . . . , γ 2 t ij ) predisper- i j n γ , . . . , )I typi- A . = =1 ) pn in spend 1≤i .I hr sn oeetbtentepths(lecre) h ies isoti the in out dies disease the curves), (blue patches the between movement no is there If 1. . p d b ij ,j n n ≤n N n [1]  [2] ) j t t , , . j Time 30 ecaatrzduigtefloigpthseicbscrepro- basic patch-specific numbers: following duction the using matrix characterized the when be system the of dynamics where by given aede u rprit i l ace)weee h ai repro- basic the whenever number patches) all duction (in persists or out dies ease System oiiyi elgbeadpoed oicroaetelf history life the incorporate to vector proceeds that and assumed dis- negligible is is the it mobility Here, transport mosquitoes. native Travelers infect patterns. and the ease mobility strongly human and is to States) America, United linked the and Central Mexico hitting America, (now Caribbean in Latin curves throughout blue spread the (see dynamics sup- 2). populations same and two 1 the the Figs. expected, change, behavioral as no port, is there is, (p that patches between movement con- under generated those stant from dynamics alter disease can expected approach the mobility modeling density-dependent crude, that such are natural placing by captured areas of is entries in the effect time on restrictions less This spend prevalence. or is high avoid dynamics of individuals approach disease that phenomenological to assumes a via that response crudely, in rather also play captured, matrix may mobility behavior constant human a Lyapunov of use average the the details). more using for 24 ref. established (see (68) be theorem can persistence Patch in k out dies disease the p oa epne neapeo uhdpnec ol ecap- be functions: could following dependency the such by of tured example An response. ioral p ij ij = 6 h pe twihtevco-on iavrsdsaehas disease virus Zika vector-borne the which at speed The Model In Patch in persists disease The ( ∂ (I I I i i i 1 ,I , , P j R ρ provided I ) j 2 Fg.1ad2.I h pca ae hr hr sno is there where case, special the In 2). and 1 (Figs. 0 i ≥ eoe h pcrlrdu and radius spectral the denotes = ) R = (P) uprsasaptrsodpoet.Ta s h dis- the is, That property. threshold sharp a supports for 0, 0 40 ua eairi rdl noprtdthrough incorporated crudely is behavior human 2, = P σ ij I I ρ(diag 1 1 γ p j 2 Statedependantresidencetime Stateindependantresidencetime R 1+I =1 (i i ij 1+I β 0 + 12 , > i σ i j sls hno rae hnuiy(24). unity than greater or than less is = +I d ) ij 0 i i p {1, ∈ N j ¯ 1 = 21 × for , and 50 )Pdiag(B = X j =1 n σ h iuainblwsoshwa how shows below simulation The . urne h xetdbehav- expected the guarantee 2}, h inequalities The P. 12 R (i p i  12 = 12 , 0 i if i j β (P) β = σ ) j p i whenever 21 )diag( p p = {1, ∈  kj 21 ii = < p 0 = (I = p )aedsrbdb h solid the by described are 0.5) h oeta adaptive that role The P. ij ac-pcfi disease Patch-specific 1. 60 21 i σ , V 2}   N P 12 ˜ I o all for NSEryEdition Early PNAS j = ) = P = sntirdcbecan irreducible not is = ) −1 and R σ  k n −diag(d σ 21 P p =1 0 i 12 (P) t ij = k σ p V σ ij p d b ij 1 = ii .Terdcurves red The 0. i i = 1+I kj ( −1 ∂ +σ > I  I i = j d b ,I .., , ), whereas 1, k σ ii k i j +I + I ) 21   p i +I ≤ j n γ ij . | The ). , (0, 0), = j 0 R f7 of 3 and and and 0 0), is

SUSTAINABILITY POPULATION COLLOQUIUM SCIENCE BIOLOGY PAPER 80 I State independant residence time 2 I State dependant residence time 70 2

60

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30

20

10

0 0 10 20 30 40 50 60 Time

Fig. 2. Dynamics of the disease in Patch 2. In the high-mobility case p12 = p21 = σ12 = σ21 = 1 (and then p11 = p22 = σ11 = σ22 = 0), the disease dies out β (solid red curve) for constant, with R˜2 = 1 = 0.8571. For the constant residence times matrix, the system is strangely decoupled because individuals P 0 γ2+d2 of Patch 1 spend all their time in Patch 2, whereas individuals of Patch 2 spend all their time in Patch 1. Hence, Patch 2 individuals (d2 and µ2) are subject ˜2 β1 exclusively to the environmental conditions that define Patch 1 (β1), and so the basic reproduction of the “isolated” Patch 1 is R = and the disease 0 γ2+d2 ˜2 1+I1 1+I2 I2 dies out because R = 0.8571. The disease persists if state-dependent (dashed red curve) as p12(I1, I2) = , p21(I1, I2) = , p11(I1, I2) = 0 P 1+I1+I2 1+I1+I2 1+I1+I2 I1 and p22(I1, I2) = . Figure courtesy of ref. 24. 1+I1+I2

and epidemiology of mosquitoes (69–74), which can be effec- recovery and natural mortality rates are given, respectively, by tively captured by decoupling host–vector mobility (71, 75). Fig. 3 γ and µ. Finally, the matrix P represents the proportion of time and System 3 illustrate the approach. A Lagrangian model based host of Group i, i = 1, . . ., n, spend in Patch j , j = 1, . . ., m. on residence times has been proposed recently for vector-borne The basic reproduction number of Model 3, with m patches and 2 diseases like Dengue, , and Zika (23). The appropriate- n groups, is given by R0(m, n) = ρ(Mvh Mhv ), where ness of the Lagrangian approach for the study of the dynamics of t −1 t −1 vector-borne diseases lies also in its assessment of the life-history Mhv = βhv diag(a)diag(P Nh ) diag(Nv )P diag(µ + γ) specifics of the vector involved (75). Lagrangian approaches have been used to model vector-borne and diseases (refs. 76–80 and other references contained therein), t −1 −1 M = β diag(N ) diag( N ) diag(a)diag(µv + δ) . albeit these researchers have not considered the impact that the vh vh h P P h residency–time matrix P may have on patch effective popula- tion size. Specifically, in refs. 76 and 78, the effects of move- If the host–vector network configuration is irreducible, then ment on patch population size at time t are ignored, namely, System 3 is cooperative and strongly concave with an irreducible Jacobian, and so the theory of monotone systems, particularly the population size in each patch j is fixed at Nj . In ref. 77, it is assumed that human mobility across patches does not produce Smith’s results (81), guarantee the existence of a sharp thresh- any “net” change on the patch population size. On the other old. That is, the disease-free equilibrium is globally asymptoti- cally stable if R2(m, n) is less than unity and a unique globally hand, in Model 3 the relationship between each patch population 0 asymptotic stable interior endemic equilibrium exists otherwise. and mobility are dynamic and explicitly formulated. Moreover, The effects of various forms of heterogeneity on the basic repro- the limited (vector mobility is ignored) Lagrangian approach duction number has been explored in ref. 23, and we have found, used here to model the dynamics of vector borne diseases cap- for example, that the irreducibility of the residence time matrix tures some unique features because the “spatial” structure of is no longer sufficient to ensure a sharp threshold property, mosquitoes is not the same as that of humans. Mosquitoes are P albeit the irreducibility of the host–vector network configuration stratified into m patches (that may represent, for example, ovipo- is necessary for such property (23). sition or breeding sites or forests) with infection taking place still  within each patch j , characterized by its particular risk βvh aj I˙ = β (N − I ) (a) ( t N )−1I − (µ + γ)I  h vh diag h h Pdiag diag P h v diag h for j = 1,..., m. Here, βvh represents the infectiousness of . human to mosquitoes bite with aj denoting the per capita bit- ˙ t −1 t Iv = βhv diag(a)diag(Nv − Iv )diag( Nh ) Ih − diag(µv + δ)Iv ing rate in Patch j . Hosts, on the other hand, are structured by P P social groups or age classes (n). This residency habitat division [3] can be particularly useful in the study of the impact of target control strategies. The Lagrangian approach of disease modeling can use con- The model in ref. 23 describes the interactions of n host groups tacts (60) or residency times or both as its currency. Here, we t choose time–spatial-dependent risk, that is, we choose to han- in m patches via System 3, where Ih = [Ih,1, Ih,2, . . ., Ih,n ] , t t ¯ dle social heterogeneity by keeping track of individuals’ social Iv = [Iv,1, Iv,2, . . ., Iv,m ] , Nh = [Nh,1, Nh,2, . . ., Nh,n ] , Nv = or geographical membership. In this context, it is possible to ¯ ¯ ¯ t t t [Nv,1, Nv,2, . . ., Nv,m ] , δ = [δ1, δ2, . . ., δm ] , a = [a1, a2, . . ., am ] , include adaptive responses, for example, via the inclusion of t and µ = [µ1, µ2, . . ., µn ] . The infected hosts are denoted by the prevalence-dependent dispersal coefficients. In this setting, the vector Ih and the host population by Nh . The infected vectors are underlying hypothesis is that host behavioral responses to dis- denoted by Iv and the mosquito population by Nv . The parame- ease are automatic: either constant or following a predefined ters ai , δi , and µv denote the biting, death rate of control, and function. The average residence time P incorporates the aver- natural death rate of mosquitoes in Patch j , for j = 1, . . ., m. age behavior of all hosts in each patch. This assumption is rather The infectiousness of human to mosquitoes is βvh , whereas the crude because it implicitly assumes that hosts have accurate infectiousness of mosquitoes to humans is given by βhv . The host information on health status and patch prevalence and respond

4 of 7 | www.pnas.org/cgi/doi/10.1073/pnas.1604994113 Castillo-Chavez et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 i rhrbhvo n h eair ftoeaon i or him around those of behaviors the alter and to decision-maker behavior a her for or capi- available his human mechanisms infectious the the and/or to and resource tal; coupling natural the the goals; of function, dynamics decision-makers’ objective such the appropriate to with linked an payoffs congruent of the stipulation of of the specification description proper behaviors; a a or and on pests, rests costs mod- EEM behavioral invasive for an of methodology within The control behavior system. eling the forestry (85– or in of species species, questions management the wild of harvesting of optimal exploitation context addresses the the that on literature work a influenced ongoing 87), strongly and been past has by construction EEM (82–84). ature and EEMs useful potentially challenging. of quite building is the to central is dis- risk of ease analyses rational cost–benefit modeling individualized of to avoid- connections ways values’ dis- effective disease finding population-level and Thus, underlying dynamics. disease ease by of constrained by) cost decisions (driven perceived, ance: on or all are based real prob- problems formulations may personal, decision decision rational-value individual individual where by that an assumed and generated to further is whom, solutions It the with lem. as of in, envisioned amount participates be and the social one that of assume activity result EEMs the indi- interactions. stages: of environmental disease susceptible, transition recovered the and through orderly disease, infected, the communicable epidemio- a for compartmental facing account viduals classical that on models built logical are EEMs Simple discussed Epidemiology Economic is and 25) disease (2, population addressed tradeoffs, overall recently in and the been choices for has these account dynamics on that place systems role individuals and within the value that individuals of what cost of incorporation function the The a as accordingly. decisions, human infection that of risk to 23. ref. of courtesy Figure structure. vectors’ the from decoupled 3. Fig. atloCae tal. et Castillo-Chavez E prahshv rcrosi h pdmooia liter- epidemiological the in precursors have approaches EEM cnmcEpidemiology. Economic lwdarmo arninmdli hc h otsrcueis structure host the which in model Lagrangian a of diagram Flow e fcnat,teotmlcoc fcnat steslto to solution the is contacts problem: of num- programing choice the dynamic optimal a on the depends contacts, status of health ber transitioning infected of to probability susceptible the If from contacts. in increasing and ness status function health the the is, of by that example, individual make, for representative they that a contacts sta- of infec- the health utility the and of on individual risk depend the to increased of assumed tus the is function against utility contact The the tion. a balancing of cost), value less maximize modeling marginal benefit to the (rudimentarily, chosen are utility Hence, contacts expected or infection. may volume an but activity that with contacts assumes making associated assumes from costs modeling The benefits incur 88). derive (5, par- individuals one a substituting riskier that by a or for engages behavior one ticular whom with and place takes contact (lowering in coming infection of chances with one’s reducing to identical logically (lowering infects. successfully contact such that susceptible probability the and the takes individuals/material, activity infectious with such contact of in unit individual a that probability the is ease, where by given is time of given a expectation at instantaneous new inci- the of simple number A captures the rate. that incidence infec- so-called of function the cases dence is time, secondary of it of unit generation because per of tion point, rate starting the classic to connected a provides behavior in causing economic, them call literature. to published continue previous with we keeping nature, are in infection monetary against mitigation not for motivations all Although her. l ihntemdl nti xml,ipc ies outcomes individu- Eqs. disease incidence). where impact the example, system in this adaptive changes in (through complex model, the a class, within have health als own we decision-makers’ short, the In classes, in health individuals other of ior in individuals of system, ior the of state current the on state health from where ordc otcs yicniiigtemitnneo contacts of maintenance individuals the incentivizing immune By such contacts. nontargeted reduce induce by to nullified that be policies turn, in social-distancing may, with associated but externality immunity positive acquired the benefits include The immunity once (41). herd generate of state to non–disease-transmitting contacts due immune, that individuals’ management an known the disease in is to externality critical it positive fact, yet the in and is, interventions, behavior health their over- individu- public generally been recovered in has policy of looked individuals public role susceptible of the protecting in design Indeed, als the costs. those to affects under- critical disease that of Thus, is costs prevention 90). relative disease 89, to and responds 26, self-quarantine individual (25, the to provi- welfare-enhancing how the individuals standing be that to infectious shown likely for infec- been is incentives of also spread of has the it sion in However, role conditional important disease. distancing, an tious plays individual status, that health shown on have con- individual EEMs this text, Within perspective. individual an from mized h tlt ucini sue ob ocv,dcesn nill- in decreasing concave, be to assumed is function utility The in engages one activity of volume the reduce to decision A infection undertakes individual an not or whether Modeling c r V S steaeaeaon fatvt hyegg in, engage they activity of amount average the is stedson aeand rate discount the is t (t (h c ) a ensoni aycsst ephenomeno- be to cases many in shown been has ) max = ) stenme fidvdasssetbet h dis- the to susceptible individuals of number the is C h s ohat state health to ( U U t S P h  (t SI h = )cP t (t , U C yatrn hr h activity the where altering by ) ρ SI (h t hj h  (t {S , stepoaiiyo transition of probability the is C + )ρ, j (t hspoaiiydepends probability This . h r ). ), X I j NSEryEdition Early PNAS 4 (t C and ρ ), −h hj R V n h behav- the and , 5 (t t +1 r ohopti- both are h behav- the )}, (j h ) ) sgiven, is , P | SI f7 of 5 C ¯ ρ [4] [5] (t h is ) .

SUSTAINABILITY POPULATION COLLOQUIUM SCIENCE BIOLOGY PAPER by recovered individuals policy may lower the probability of sus- within our framework, to health status and population-level ceptible individuals contacting infected individuals and/or allow dynamics, the components of a complex adaptive system. Con- susceptible and infected individuals to individually increase con- necting the Lagrangian movement-modeling approach with what tacts without changing the probability of infection. we have described here as EEMs seems promising, albeit com- putationally and mathematically challenging. However, as dis- Lagrangian and EEMs cussed in ref. 91, the perception that the benefits of disease Theoretical epidemiology aims to disentangle the role of epi- control are limited by the capacity of the weakest link in the chain demiological and socioeconomic forces on disease dynamics. to respond effectively is not a basic result of EEM models, which However, the role of behavior and individual decisions in actually show that it may not be in within the individual in a poor response to a changing epidemic landscape has not been tackled community/country to do actually more risk mitigation. In fact, systematically. In this rather succinct and biased perspective, the need for richer communities or nations to find ways to incen- we expand on alternative ways for modeling disease transmis- tivize greater levels of disease-risk mitigation in poor countries sion that can use contacts as its currency or residency times may be, in fact, the best approach. or both. Despite the overwhelming use of contacts as the most Simon Levin, in his address as the 2004 recipient of the common currency of transmission and its undeniable theoreti- Heineken award, placed our narrow perspective in a broader cal value, it seems evident to these researchers that contacts, in powerful context: the context of influenza, Ebola, tuberculosis, or other commu- A great challenge before us is thus to understand the dynamics of nicable diseases (as opposed to sexually transmitted diseases), social norms, how they arise, how they spread, how they are sus- cannot be measured effectively in settings where the risk of tained and how they change. Models of these dynamics have many acquiring such infections is the highest. In fact, when contact- of the same features as models of epidemic spread, no great sur- based models are fitted to data, it has become clear that con- prise, since many aspects of culture have the characteristics of being tact rates play primarily the role of fitting parameters; in other social diseases. 1998 Heineken award winner Paul Ehrlich and I have words, if the goal is connecting models to data that include been directing our collective energies to this problem, convinced that transmission mechanisms, then the use of contacts has serious it is as important to understand the dynamics of the social systems shortcomings. Therefore, if we are to advance the role of the- in which we live as it is to understand the ecological systems them- ory, we need models that are informed by data, and the need selves. Understanding the links between individual behavior and soci- to reinvest efforts to bring forth alternative modes of model- etal consequences, and characterizing the networks of interaction and ing becomes pressing (Lagrangian approaches that extends the influence, create the potential to change the reward structures so functionality of classical models while requiring only “functional that the social costs of individual actions are brought down to the contacts” whenever infection takes place). Modeling approaches level of individual payoffs. It is a daunting task, both because of the that require parameters like residence times and average time amount we still must learn, and because of the ethical dilemmas that to infection for a given environment (risk), that is, parame- are implicit in any form of social engineering. But it is a task from ters that can be measured, should be further investigated and which we cannot shrink, lest we squander the last of our diminishing their analyses contrasted to those that involve contacts. We resources. believe that the use of Lagrangian models parametrized in this fashion are likely to increase the give and take necessary for ACKNOWLEDGMENTS. This work was supported by NIH National Institute theory and data to modify, expand, or even reinvent the way of General Medical Sciences (NIGMS) Grant 1R01GM100471-01. This study that we look and think about the dynamics and evolution of was partly supported by a United States–United Kingdom collaborative infectious diseases. grant between the joint National Science Foundation–NIH–US Department The SARS, influenza, and Ebola have shown the of Agriculture Ecology and Evolution of Infectious Diseases Program (Grant DEB-1414374) and the UK Biotechnology and Biological Sciences Research dramatic role that individual decisions play on the dynamics of Council (Grant BB-M008894-1). The contents of this article are solely the infectious diseases. We have revisited recent work that equates responsibility of the authors and do not necessarily represent the official behavior with cost-benefit decisions, which, in turn, are linked, views of NIGMS.

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