Need help?

800-5315-2751 Hours: 8am-5pm PST M-Th;  8am-4pm PST Fri
Medicine Lakex

MODELLING THE SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE URGERA, GERARDO CHOWELLB, PEP MULETC, AND LUIS M. VILLADAD Abstract. A spatial-temporal transmission model of 2009 A/H1N1 pandemic influenzaacross Chile, a country that spans a large latitudinal gradient, is developed to characterizethe spatial variation in peak timing of the 2009 A/H1N1 influenza as a function of spatialconnectivity assumptions across Chilean regions and the location of introduction of the virusinto the country. The resulting model is a SEIR (susceptible-exposed-infected-removed)compartmental model with local diffusion and optional non-local terms to describe themigration of individuals of the S, E and R classes and the effect of a "hub region". Thismodel is used along with epidemiological data to explore the spatial-temporal progressionof pandemic influenza in Chile by assuming a range of transmission scenarios. Numericalresults indicate that this relatively simple model is sufficient to characterize the south-northgradient observed during the 2009 influenza pandemic in Chile, and that the "hub region"corresponding to the capital region plays the critical role in keeping the population wellmixed in a relatively short period of time.
1.1. Spatial-temporal variation of influenza. Increasing our understanding of the spa-tial dissemination patterns of influenza is essential for public health surveillance and theimplementation of reactive social distancing measures. Factors that have been associatedwith the spatial-temporal variation in seasonal influenza activity at the city or regional levelinclude local environmental characteristics (e.g., temperature, specific humidity [1, 2] thatenable local transmission, school cycles [3,4] whereby influenza transmission rates tend to de-cline during school breaks, as well as regional and global population mobility patterns [5–7]).
For instance, a study based on 30 years of influenza-related mortality found a significant cor-relation between influenza activity across US states and the rates of movement of people toand from their workplaces (workflows) compared with geographical distance [5]. Another Date: January 27, 2014.
Key words and phrases. SEIR model, influenza pandemic, spatial-temporal model, peak timing, trans- mission dynamics, hub region.
aCI2MA and Departamento de Ingenier´ıa Matem´atica, Facultad de Ciencias F´ısicas y Matem´aticas, Uni- versidad de Concepci´ on, Casilla 160-C, Concepci´ on, Chile. E-Mail: [email protected].
bMathematical and Computational Modeling Sciences Center, School of Human Evolution and Social Change, Arizona State University, Tempe, AZ 85287, USA, and Fogarty International Center NationalInstitutes of Health, Bethesda, MD 20892, USA. E-Mail: [email protected].
cDepartament de Matem atica Aplicada, Universitat de Val encia, Av. Dr. Moliner 50, E-46100 Burjassot, Spain. E-Mail: [email protected].
dCI2MA and Departamento de Ingenier´ıa Matem´atica, Facultad de Ciencias F´ısicas y Matem´aticas, Uni- versidad de Concepci´ on, Casilla 160-C, Concepci´ on, Chile. E-Mail: [email protected].
URGER, CHOWELL, MULET, AND VILLADA Figure 1. (a) Regions of Chile, (b) pandemic onset (denoted by symbol I)and pandemic peak (denoted by symbol N) timing across the 15 Chilean regionssorted from north (top) to south (bottom) Chile [14].
study using influenza hospitalization records among older adult populations across US statesfound a significant gradient in the peak timing of influenza at the state level whereby westernstates tended to peak earlier than northeastern states [8]. Similarly, another study basedon weekly laboratory-confirmed influenza A from Canadian and US influenza surveillancesystems showed a slight gradient in peak timing from the southwest regions in the US tonortheast regions of Canada and the US. This study also found that regional influenza epi-demics were more synchronized across the US (3–5 weeks) compared with Canada (5–13weeks) [9].
1.2. The 2009 A/H1N1 pandemic influenza in Chile. In the context of the recent 2009A/H1N1 influenza pandemic, population contact rates linked to school cycles or interventionstrategies [10–12], demographic factors [13], local transmissibility [10, 11, 14], and globalmobility patterns, which drives the timing of a virus's seeding across countries [15], have beenassociated with the complex spatial and temporal evolution of the 2009 A/H1N1 influenza SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE Arica y Parinacota Table 1. Official data for population and area for each region and approx-imate longitudinal length. Note that we number regions consecutively fromnorth to south, and that these numbers do not correspond to the official Romannumerals assigned to the administrative regions of Chile.
pandemic. The 2009 A/H1N1 influenza pandemic spread as a single wave of transmissionin Chile during the winter of 2009 soon after the first cases were confirmed in Mexico andCalifornia, USA [16]. The first two cases of novel 2009A/H1N1 influenza in Chile wereconfirmed in metropolitan Santiago on May 17, 2009 [16]. However, a retrospective studybased on emergency room visit and laboratory viral surveillance conducted in southern cityof Puerto Montt, capital of Los Lagos region, suggested that this city could have experienceda faster transmission rate and earlier pandemic onset by the end of April 2009 comparedto the metropolitan area of Santiago [16]. Indeed, a recent study set in Chile showed thatthis country experienced a strong latitudinal gradient in pandemic peak timing in 2009,with southern regions experiencing earlier pandemic activity than northern regions [14], seeFigure 1. Specifically, the southernmost regions (Biob´ıo, Araucan´ıa, Los R´ıos, Los Lagos,Ays´ en, and Magallanes) peaked on average 16–39 days earlier relative to the northernmost region (Arica y Parinacota). This geographical variation in pandemic peak timing in Chilewas found to be associated with differences in latitude and climatic conditions, with latitude,maximum temperature and specific humidity accounting for 69–80% of this variability inpeak timing [14]. This south-north gradient in pandemic peak timing reported for Chile isconsistent with a decreasing trend in transmissibility in the same direction, which was foundto be statistically associated with maximum temperature and specific humidity. This isconsistent with experimental studies suggesting that influenza transmission is more efficientunder dry and cold conditions [2, 17–21].
1.3. This contribution. Here we develop a spatial-temporal transmission model of the2009 A/H1N1 pandemic influenza across Chile, a country that spans a large latitudinal URGER, CHOWELL, MULET, AND VILLADA gradient, to characterize the spatial variation in peak timing of influenza as a function ofspatial connectivity assumptions across Chilean regions and the location of introduction ofthe virus into the country. We use epidemic modeling together with epidemiological data toexplore the spatial-temporal progression of pandemic influenza in Chile by assuming a rangeof transmission scenarios to investigate the robustness of the south-north gradient observedduring 2009 influenza pandemic in Chile.
The remainder of the paper is organized as follows. In Section 2, we describe the spatial- temporal SEIR (susceptible-exposed-infected-removed) model. In particular, in Section 2.1we define its spatial version including local diffusion and, owing to the special geographyof Chile, reduce the model to one space dimension. In Section 2.2 we perform a stabilityanalysis for the case of constant diffusivity, starting from the well-known non-spatial versionof the SEIR model which is given by a system of four coupled ordinary differential equations,with the result that whether the basic reproductive ratio R0 is smaller or larger than onedecides whether the disease-free state is locally asymptotically stable or unstable. Then, inSection 2.3 we define a generalization of the model to include a so-called "hub region", whichis defined by a non-local convolution term. In Section 3 we outline the numerical scheme usedfor simulations. Then, in Section 4, we compare simulations obtained with the continuousmodel with diffusion and with/without migration via a hub region, which corresponds to theChilean capital (metropolitan) region. We compute the number of infected individuals andestimate the "peak time", when the maximum concentration of infected is observed, in eachregion . These results are compared with the peak times observed in [14]. These results arediscussed in Section 5.
2. Spatial-temporal SEIR model Classical epidemiological models describe disease transmission on single populations of individuals by aggregating all of the members into one of four different classes. However,these single population models rely on the strong assumption that the entire population ismixing homogeneously and they offer no information about the spread of the disease acrossregions, which is reasonable in small areas but does not well reflect reality across largegeographic regions.
2.1. Spatial domain and governing equations. We consider one country identified by abi-dimensional domain Ω ⊂ R , where x = (x, y) and x and y are latitudinal and longitudinal coordinates, respectively, measured in kilometers. We assume that Ω is simply connectedand has a continuous, piecewise smooth boundary ∂Ω with a normal n = n(x).
We consider the classical epidemical SEIR model proposed by Kermack and McKendrick [22] with standard incidence [23] (see also [24–29]). This model keeps track of four classes ofindividuals at time t and the location x ∈ Ω, the density of susceptibles S(x, t), the densityof exposed E(x, t) in which individuals are in the latent period, the density of infectivesI(x, t) in which individuals are infectious and the density of recovered R(x, t) that keepstrack of individuals removed from the infected compartment. We start from the followingassumptions.
SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE (1) The population is constant (without births or deaths, and the disease is assumed non-letal), i.e.
where the population density is N = S + E + I + R.
(2) The transmission is described by local standard incidence. Thus, for each x ∈ Ω, once an infected individual is introduced into the susceptible area and contacts asufficient number of susceptibles at time t, the fraction of new infected individualsper unit time is β(x)S(x, t)/N (x, t), where β(x) is the local transmission coefficient.
The number of new infected individuals per unit time resulting in each point x ∈ Ωat time t is β(x)S(x, t)I(x, t)/N (x, t).
(3) The number of individuals recovered from the infected class per unit time at x ∈ Ω and at time t is γI(x, t), where γ is the rate constant for recovery, corresponding toa mean infectious period of 1/γ.
(4) Individuals of each class disperse by means of Fickian diffusion throughout Ω. Pre- cisely, and let −dP (x)∇P be the population flux, where dP (x) ≥ 0 is the space-dependent diffusion coefficient for P ∈ {S, E, R}. Thus, individuals are assumed tomove an in undirected manner across the entire region and to contact only thoseindividuals in their immediate area. Infected individuals are not allowed to move.
(5) At any time the population in Ω is isolated.
Based on these assumptions, we obtain the following model: + ∇ · dS(x)∇S, − κE + ∇ · dE(x)∇E, ∂I = κE − γI, ∂R = γI + ∇ · dR(x)∇R.
A sufficient condition for an isolated population is a zero-flux boundary condition for P ∈{S, E, I, R}, which leads to the homogeneous Neumann boundary conditions for x ∈ ∂Ω and t > 0, where n denotes the unit exterior normal vector to the boundary ∂Ω of Ω at position x and,as usual, ∂S/∂n = ∇S · n, etc. Thus, the total number of individuals at time t, is actually constant, i.e., Ntot(t) = Ntot(t0) = N0, since URGER, CHOWELL, MULET, AND VILLADA For the particular geographic reality of countries that mainly extend in the direction of onlyone of the coordinate axes (an assumption obviously motivated by the geography of Chile,but which may include a few other places as well), a significant model reduction but withan acceptable loss of realism can be achieved if we reduce the model (2.1) to one spacedimension. Thus, we obtain the following system: ∂I = κE(x, t) − γI(x, t), which is supplemented by initial conditions.
2.2. Stability analysis. To discuss the stability properties of (2.1), let us briefly recallwell-known results for the standard (non-spatial) SEIR model, which is recovered by settingdP ≡ 0, P ∈ {S, E, R}, in (2.1). We select the version of the model defined at position x (asby the choice of β(x)), and consider for a moment x as a parameter, i.e. we study the ODEmodel dI(t; x) = κE(t; x) − γI(t; x), dR(t; x) = γI(t; x).
For the model (2.3) we define the basic reproductive ratio associated with position x by R0(x) := β(x)/γ.
If R0(x) > 1 for x ∈ Ω, then the disease-free steady state at position x is unstable so that anepidemic may potentially occur. If that happens then I(t; x) first increases to a maximumattained when S(t; x) = γN (t; x)/β(x), and then decreases to zero. On the other hand, ifR0(x) < 1 for x ∈ Ω, then the disease-free steady state at x is stable so that the disease diesout, i.e., I(t, x) decreases to zero at x ∈ Ω.
We now perform a linearized stability analysis for the one-dimensional system (2.2) with dP (x) := d for ∈ {S, E, R} and β(x) = β. With this goal, we rewrite (2.2) in the form SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE where u = (S, E, I, R)T, g(u) corresponds to the reactive term in the SEIR model, and The linearized equation for a small perturbation v about an equilibrium point u(0) of the dynamical system u0 = g(u) is obtained by substituting u = u(0) + v into (2.4) andneglecting second order terms in v. This yields the following linearized version of (2.4): vt = g0(u(0))v + dDv.
We consider u(0) = (s0, 0, 0, 0)T with s0 > 0.
We now seek solutions of (2.6) of the form v(x, t) = z(t; ξ) exp(iξx) for a fixed frequency ξ. The vector function z satisfies the system of ordinary differential equations z0 = M u(0); ξz, where we define the matrix M u(0); ξ := g0 u(0) − ξ2dI∗, where I∗ = diag(1, 1, 0, 1) and The linearized asymptotical stability of u(0) is equivalent to limt→∞ z(t, ξ) = 0 for all ξ, and this is equivalent to Re λ < 0 for any eigenvalue λ of M (u(0); ξ) and any ξ ∈ R. Theeigenvalues of M (u(0); ξ) are λ1,2 = −ξ2d and − (γ + κ + dξ2) ± p(γ + κ + dξ2)2 − 4(κ(γ − β) + γdξ2), so that Re λ3,4 < 0 if and only if κ(γ − β) + γdξ2 > 0. Therefore we have proved the followinglemma.
Lemma 1. The equilibrium point u(0) is locally asymptotically stable for the model (2.2) ifR0 = β/γ < 1 and unstable if R0 > 1.
2.3. Continuous model with a hub region. For certain subpopulation u ∈ {S, E, R}, aconstant non-local migration can be represented by the following equation: K(z, x)u(z, t) dz − K(x, z)u(x, t) dz, where [xa, xb] is certain hub region and a, xb] or y ∈ [xa, xb], URGER, CHOWELL, MULET, AND VILLADA is the function that indicates the rate of individuals at x that move to y. The first integral in(2.7) corresponds to migration from the hub region to a point x at rate ν, while the secondintegral corresponds to migration of population from x to the hub region at rate ν. Observethat K(z, x)u(z, t) dz − K(x, z)u(x, t) dz u ∈ {S, E, I, R}, therefore the total population remains constant.
3. Numerical scheme We consider the domain [0, L] of length L = 4200, which is approximately the length of continental Chile in kilometers. For grid points xj := (j − 1 )∆x for j = 1, . . , M , where ∆x := L/M , and tn := n∆t for n ∈ N0, we calculate approximate values un ≈ u(x u ∈ {S, E, I, R}. We denote un = (Sn, En, In, Rn)T and un = (un, . . , un )T.
The continuous model (2.2), supplied with suitable initial conditions, is solved numerically by an explicit-implicit Euler method, for which the reaction is discretized explicitly, while animplicit discretization is used for the diffusive part in the continuous model. (This methodis the simplest variant of the so-called IMEX schemes for convection-diffusion problems,see [30].) The resulting scheme is un+1 = un + ∆t G(un, x) + dD∆xun+1, where G(un, x) is a vector of 4M components given by β(x)SI/N − κE g (S, E, I, R)T, x =  and D∆x is the (4M ) × (4M ) matrix that discretizes (2.5): D∆x = L∆x ⊗ I∗, I∗ = diag(1, 1, 0, 1), where L∆x denotes the usual discretization of the 1D Laplacian with Neumann boundaryconditions on an M -grid over the spatial domain [0, L] with cell size ∆x = L/M .
The model with non-local migration (2.7) is discretized in a similar way. In this case the non-local migration term is treated explicitly: un+1 = un + ∆t G(un, x) + dD∆xun+1 + Hhub(un), SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE where Hhub(un) is a vector of 4M components that approximates the integral terms in (2.7)by the midpoint rule: (Hhub(un))i = ∆x In the simulations we use M = 1000 and ∆t = 0.1 d, since we have checked that more resolution does not significantly change the simulation.
4. Numerical simulations 4.1. Epidemiological and population data. We relied on a large individual-level datasetcomprising all hospitalizations for severe acute respiratory infection (hereafter referred to asSARI) reported by all public and private hospitals to the Chilean Ministry of Health during01-May to 31-December 2009 to characterize peak timing across Chilean regions.
A total of 1809 SARI hospitalizations (29.4%) were laboratory confirmed with A/H1N1 pandemic influenza. We obtained regional estimates of population size for 2009 from theInstituto Nacional de Estad´ısticas [31].
We consider the initial population according to official data to December 2011 (see [14]) and assume a constant distribution of population along its geographical localization for eachregion based on official data that are tabulated in Table 1.
4.2. Initial values and constants. It is difficult to determine suitable values for E(x, 0)and I(x, 0) that would allow us to obtain similar results as those reported in [14]. We there-fore tested four different initial conditions corresponding to different scenarios. "Scenario i"corresponds to one infected individual uniformly distributed along region i and zero infectedindividuals in all other regions. The number of individuals of class P ∈ {S, E, I, R} in regioni ∈ {1, . . , 15} at time t is computed as where li is the latitude interval corresponding to region i. The number of individuals newlyinfected in the time interval [t − 1, t] (t measured in days) in region i is calculated as The constants are chosen as 1/κ = 2 d and 1/γ = 4 d, κ = 0.5 d−1 and γ = 0.25 d−1.
Moreover we choose d = 10 km2/d and ν = 10−5 d−1.
URGER, CHOWELL, MULET, AND VILLADA Figure 2. Simulation of disease spread for four different initial scenarios withd(x) = 10, ν = 1 × 10−5 and R0(x) > 1: (a) Scenario 3, (b) Scenario 7, (c)Scenario 10, (d) Scenario 13.
4.3. Case R0(x) > 1 for x ∈ [0, L]. In this case the local transmission coefficient β(x) is setsuch that R0(x) = β(x)/γ satisfies R0(0) = 1.2 (northernmost region) and increases linearlyas a function of space until R0(L) = 1.6 (southernmost region) i.e.
0 ≤ x ≤ L.
According to Lemma 1 in Section 2.2 any point in the domain is a endemic point.
In Figures 2 and 3 the disease spread for different initial scenarios is shown. In Figure 4 the peak times of simulations for different scenarios are shown and compared with the peaktiming reported in [14]. Scenarios 10 and 13 give a good fit to the experimental data, thuspointing that the epidemic might have started in southern Chile. Since the local reproductivenumber increases from southern to northern regions in our model, the attack rates shown inFigure 3 also increase in the same direction as R0.
SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE Figure 3. Simulation of disease spread for four different initial scenarios;same results as in Figure 2 but normalized by the number of inhabitants ineach region: (a) Scenario 3, (b) Scenario 7, (c) Scenario 10, (d) Scenario 13.
Finally, we display in Figure 5 the computed peak timing for the model (2.2) (without a hub region) and the diffusion coefficient d = 1000 km2d−1, which is much larger than the oneused before and has been chosen so that the time frame of the peak timing of the epidemicresembles that of the experimental data.
4.4. Case R0(x) < 1 for x ∈ [0, L/2) and R0(x) ≥ 1 for x ∈ [L/2, L]. In this case, thelocal transmission coefficient β(x) is set so that R0(0) = 0.8 (northernmost region) and thatincreases linearly as a function of space until R0(L) = 1.2 (southernmost region), i.e., 0 ≤ x ≤ L.
The parameters for diffusion and local-migration are d = 1000 km2d−1 and ν = 2 × 10−4 d−1.
URGER, CHOWELL, MULET, AND VILLADA Peak times (days) 50 Peak time (days) 50 scenario 13observed data 10 11 12 13 14 15 10 11 12 13 14 15 North <== regions ==> South North <== regions ==> South Figure 4. Simulated region-wise peak timing for different scenarios with d =10 km2d−1, R0(x) > 1 for x ∈ [0, L], and including a hub region (region 7)with ν = 10−5d−1.
scenario 3scenario 7 scenario 10scenario 13 10 11 12 13 14 15 North <== regions ==> South Figure 5. Simulated region-wise peak timing for different scenarios with d =1000 km2d−1, R0(x) > 1 for x ∈ [0, L], without a hub region.
In Figure 6 the newly infected individuals at each regions are shown. It is remarkable that an epidemic in the northern regions is triggered from an epidemic from the southern regions,even when the corresponding R0 is less than one.
SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE Normalized new infected per day Figure 6. Simulation new infected Inew(t) for each region i for scenario 3 (dashed curves) and scenario 13 (solid curves) for d = 1000 km2d−1, a hubregion with ν = 2 × 10−4 d−1, R0(x) < 1 for x ∈ [0, L/2) and R0(x) ≥ 1 forx ∈ [L/2, L].
5. Conclusions and discussion We have used epidemic modeling to gain a better understanding of the spatial-temporal pattern associated with the 2009 A/H1N1 influenza pandemic in Chile. Our results show thata relatively simple spatial SEIR transmission model with a single hub representing the highlyconnected metropolitan region is able to reproduce the qualitative characteristics of thespreading pandemic wave in Chile (see Figures 2 and 3). These results further support thatthe SEIR model is a suitable basis for the description of the 2009 influenza A/H1N1; see [32]for a compartmental version of this model applied to describe the spread of this disease inJapan. Moreover, our findings suggest that the south-north gradient in pandemic peakingtiming observed in Chile in 2009 is robust to variations in the initial conditions (e.g., location URGER, CHOWELL, MULET, AND VILLADA of initial infectious cases) as long as the local basic reproduction number follows an increasingtrend from south to north based on earlier estimations in [14]. Importantly, our resultssuggest that future influenza pandemics could follow similar spatial temporal dynamics tothat of the 2009 A/H1N1 influenza pandemic. Our findings could have implications forpandemic preparedness and control of future influenza pandemics.
Our results indicate that the hub region that corresponds to the metropolitan region of Chile plays the critical role in keeping the entire population well mixed in a relatively shortperiod of time. Hence, the infection is quickly spread across the entire territory as soonas the initial cases are seeded in any region of the country (Figure 4). Overall, the peaktiming tended to occur first in the region where the initial cases were first introduced, but itthen rapidly spread throughout the country and was locally modulated by the correspondingtransmissibility level as measured by R0 in each region, and followed a decreasing trend ofR0 from the southernmost to the northernmost regions of Chile. Our results also showedthat even when the region-specific R0 is set to values less than one, it is possible to generatesmall outbreaks via frequent importation of exposed individuals, through the hub, from otherregions that support epidemic outbreaks, see Figure 6. In contrast to the transmission modewith hub, our transmission model with spatial local diffusion alone was not able to generatepandemic profiles that were qualitatively consistent with the 2009 A/H1N1 pandemic datafrom Chile obtained in [14]. Specifically, the results obtained using the diffusion model werequite sensitive to the initial region where the first cases were introduced as shown in Fig 5.
Perhaps not surprisingly the spatial model with diffusion generated the best fits to peaktiming data when the initial cases were introduced in the southernmost regions of Chile.
However, even in these scenarios the spreading pandemic wave took a considerable amountof time to reach the northernmost regions (over 140 days compared with about 78 days fromactual pandemic data).
It is interesting that the south-north spreading wave of 2009 pandemic activity in Chile is reminiscent of the spread of the 2009 pandemic in Brazil, with the southernmost regions ofthis country being hit earlier and experiencing greater severity than northern regions [33]. Wehypothesize that our transmission model with a hub represented by the highly connectedareas of the south of Brazil (e.g., S˜ ao Paulo, Rio do Janeiro) and a similar south-north gradient in transmissibility could be able to generate a qualitatively similar pattern to thatobserved in 2009 in that country. By contrast, seasonal influenza has been observed tooriginate from low-population regions in the equatorial north of Brazil and travel to highlypopulous regions in the subtropical south over a 3-month period [34], together with a weaktransmissibility gradient [35].
Although we have focused on characterizing the pandemic peak timing of the spreading wave, the timing of pandemic onset could not be well characterized using out dataset ofsevere acute respiratory infections (SARI), which tend to capture the highest levels of theseverity pyramid. As previously reported [14], the metropolitan region experienced earlyintroductions of the A/H1N1 influenza virus in May 2009, but local outbreaks did not im-mediately followed, which suggest that local climatic conditions at the time did not enablewidespread transmission in the region. Instead, epidemiological investigations revealed thatthe well-connected southern city of Puerto Montt experienced full-scale transmission of novel SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE A/H1N1 influenza as of late April 2009 before the confirmation of the first case in the coun-try [16]. This suggests that local climatic conditions played a significant role in facilitatingthe onset of the 2009 A/H1N1 influenza pandemic by modulating the timing of a shift of thebasic reproduction number from R0 < 1 to R0 > 1.
It is worth noting that we did not attempt to quantify the exact magnitude and progression of the spread of the 2009 A/H1N1 influenza pandemic because our SARI data only allowedthe approximate identification of the timing of evolution of the pandemic (e.g. peak timing)rather than an accurate assessment of the onset, peak timing, and duration. Furthermore,the quantification of the magnitude of the pandemic in terms of attack rates would requiremore complex models than those employed here. For instance, we did not model the effectof winter vacation periods although it has been reported that the school break took placeafter the pandemic had reached peak levels in most parts of the country [14]. In addition,we did not account for the high rates of antiviral use in Chile, a country where treatmentwith oseltamivir was recommended for all symptomatic individuals older than 5 years thatcomplied with the influenza clinical case definition [16].
Our findings could have important implications for pandemic preparedness as our results suggest that future influenza pandemics could follow similar spatial temporal dynamics tothat of the 2009 A/H1N1 influenza pandemic. Intensified surveillance strategies in southernregions could lead to earlier detection of novel influenza viruses.
RB acknowledges support by Fondecyt project 1130154; Conicyt project Anillo ACT1118 (ANANUM); Red Doctoral REDOC.CTA, MINEDUC project UCO1202 at Universidad deConcepci´ on; BASAL project CMM, Universidad de Chile and Centro de Investigaci´ Ingenier´ıa Matem´ atica (CI2MA), Universidad de Concepci´ on; and Centro CRHIAM Proyecto Conicyt Fondap 15130015. PM is supported by Spanish MINECO project MTM2011-22741.
[1] J. Tamerius, M.I. Nelson, S.Z. Zhou, C. Viboud, M.A. Miller, W.J. Alonso, Global influenza seasonality: reconciling patterns across temperate and tropical regions, Environ. Health Perspect. 119 (2011) 439–445.
[2] J. Shaman, V. Pitzer, C. Viboud, B. Grenfell, M. Lipsitch, Absolute humidity and the seasonal onset of influenza in the continental United States, PLoS Biol. 8 (2010), paper e1000316.
[3] C. Jackson, E. Vynnycky, J. Hawker, B. Olowokure, P. Mangtani, School closures and influenza: sys- tematic review of epidemiological studies, BMJ open 3 (2) (2013), paper e002149.
[4] S. Cauchemez, N. Ferguson, C. Wachtel, A. Tegnell, G. Saour, B. Duncan, A. Nicoll, Closure of schools during an influenza pandemic, Lancet Infect. Dis. 9 (2009) 473–481.
[5] C. Viboud, O.N. Bjornstad, D.L. Smith, L. Simonsen, M.A. Miller, B.T. Grenfell, Synchrony, waves, and spatial hierarchies in the spread of influenza, Science 312 (2006) 447–451.
[6] D. Balcan, H. Hu, B. Goncalves, P. Bajardi, C. Poletto, J.J. Ramasco, D. Paolotti, N. Perra, M. Tizzoni, W. Van den Broeck, V. Colizza, A. Vespignani, Seasonal transmission potential and activity peaks ofthe new influenza A(H1N1): a Monte Carlo likelihood analysis based on human mobility, BMC Med. 7(2009), paper 45 (12pp).
[7] V. Colizza, A. Barrat, M. Barthelemy, A.J. Valleron, A. Vespignani, Modeling the worldwide spread of pandemic influenza: baseline case and containment interventions, PLoS Med. 4 (2007), paper e13(16pp).
URGER, CHOWELL, MULET, AND VILLADA [8] J.B. Wenger, E.N. Naumova, Seasonal synchronization of influenza in the United States older adult population, PLoS One 5 (2010), paper e10187 (11pp).
[9] D.L. Schanzer, J.M. Langley, T. Dummer, S. Aziz, The geographic synchrony of seasonal influenza: a waves across Canada and the United States, PLoS One 6 (2011), paper e21471 (8pp).
[10] G. Chowell, S. Echevarr´ıa-Zuno, C. Viboud, L. Simonsen, J. Tamerius, M.A. Miller, V.H. Borja-Aburto, Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico, PloS Med. 8 (2011),paper e1000436 (13pp).
[11] G. Chowell, C. Viboud, C.V. Munayco, J. Gomez, L. Simonsen, M.A. Miller, J. Tamerius, V. Fiestas, E.S. Halsey, V.A. Laguna-Torres, Spatial and temporal characteristics of the 2009 A/H1N1 influenzapandemic in Peru, PLoS One 6 (2011), paper e21287 (10pp).
[12] H. Yu, S. Cauchemez, C.A. Donnelly, L. Zhou, L. Feng, N. Xiang, J. Zheng, M. Ye, Y. Huai, Q. Liao, Z.
Peng, Y. Feng, H. Jiang, W. Yang, Y. Wang, N.M. Ferguson, Z. Feng, Transmission dynamics, borderentry screening, and school holidays during the 2009 influenza A (H1N1) pandemic, China, Emerg.
Infect. Dis. 18 (2012) 758–766.
[13] L. Opatowski, C. Fraser, J. Griffin, E. de Silva, M.D. Van Kerkhove, E.J. Lyons, S. Cauchemez, N.M.
Ferguson, Transmission characteristics of the 2009 H1N1 influenza pandemic: comparison of 8 Southernhemisphere countries, PLoS Pathog. 7 (2011), paper e1002225 (10pp).
[14] G. Chowell, S. Towers, C. Viboud, R. Fuentes, V. Sotomayor, L. Simonsen, M. Miller, M. Lima, C.
Villarroel, M. Chiu, The influence of climatic conditions on the transmission dynamics of the 2009A/H1N1 influenza pandemic in Chile, BMC Infect. Dis. 12 (2012), paper 298 (12pp).
[15] K. Khan, J. Arino, W. Hu, P. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Liauw, A. Chan, M. Gardam, Spread of a novel influenza A (H1N1) virus via global airline transportation, NewEngl. J. Med. 361 (2009) 212–214.
[16] E. Pedroni, M. Garcia, V. Espinola, A. Guerrero, C. Gonzalez, A. Olea, M. Calvo, B. Martorell, M.
Winkler, M. Carrasco, Outbreak of 2009 pandemic influenza A(H1N1, Los Lagos, Chile, April–June2009, Eurosurveillance 15 (2010), paper 19456 (9pp).
[17] A. Lowen, S. Mubareka, J. Steel, P. Palese, Influenza virus transmission is dependent on relative hu- midity and temperature, PLoS Pathog. 3 (2007) 1470–1476.
[18] A. Lowen, J. Steel, S. Mubareka, P. Palese, High temperature (30 degrees C) blocks aerosol but not contact transmission of influenza virus, J. Virol. 82 (2008) 5650–5652.
[19] S. Mubareka, A. Lowen, J. Steel, A. Coates, A. Garcia-Sastre, P. Palese, Transmission of influenza virus via aerosols and fomites in the guinea pig model, J. Infect. Dis. 199 (2009) 858–865.
[20] J. Shaman, M. Kohn, Absolute humidity modulates influenza survival, transmission, and seasonality, Proc. Natl. Acad. Sci. USA 106 (2009) 3243–3248.
[21] J. Steel, P. Palese, A. Lowen, Transmission of a 2009 pandemic influenza virus shows a sensitivity to temperature and humidity similar to that of an H3N2 seasonal strain, J. Virol. 85 (2011) 1400–1402.
[22] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy.
Soc. A 115 (1927) 700–721.
[23] Y. Takeuchi, Y. Iwasa, K. Sato, Mathematics for Life Science and Medicine. Springer-Verlag, Berlin, [24] R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford Science Publications, 1991.
[25] E. Vynnycky, R.E. White, An Introduction to Infectious Disease Modelling, Oxford University Press, [26] F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Second Ed., Springer, New York, 2012.
[27] P. van den Driessche, Deterministic compartmental models: extensions of basic models. In F. Brauer, P.
van den Driessche, J. Wu (Eds.), Mathematical Epidemiology, Springer-Verlag, Berlin, 2008, 147–157.
[28] O. Diekmann, H. Heesterbeek, T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton Series in Theoretical and Computational Biology, Princeton University Press,2012.
SPATIAL-TEMPORAL EVOLUTION OF THE 2009 A/H1N1 INFLUENZA PANDEMIC IN CHILE [29] L. Sattenspiel, The Geographic Spread of Infectious Diseases: Models and Applications, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2009.
urger, P. Mulet, L.M. Villada, Regularized nonlinear solvers for IMEX methods applied to diffusively corrected multi-species kinematic flow models, SIAM J. Sci. Comput. 35 (2013) B751–B777.
[31] Instituto Nacional de Estad´ısticas (INE). Estad´ısticas Demogr´ aficas y Vitales, 2009. Available from: estadistico/demografia y vitales/demo y vita.php [32] M.M. Saito, S. Imoto, R. Yamaguchi, H. Sato, H. Nakada, M. Kami, S. Miyano, T. Higuchi, Extension and verification of the SEIR model on the 2009 influenza A (H1N1) pandemic in Japan, Math. Biosci.
246 (2013) 47–54.
[33] C. Schuck-Paim, C. Viboud, L. Simonsen, M.A. Miller, F.E. Moura, R.M. Fernandes, M.L. Carvalho, W.J. Alonso, Were equatorial regions less affected by the 2009 influenza pandemic? The Brazilianexperience. PLoS One 7 (2012), paper e41918 (10pp).
[34] W.J. Alonso, C. Viboud, L. Simonsen, E.W. Hirano, L.Z. Daufenbach, M.A. Miller, Seasonality of influenza in Brazil: a traveling wave from the Amazon to the subtropics, Amer. J. Epidemiol. 165(2007), 1434–1442.
[35] G. Chowell, C. Viboud, L. Simonsen, M. Miller, W.J. Alonso, The reproduction number of seasonal influenza epidemics in Brazil, 1996–2006, Proc. Biol. Sci. 277 (2010) 1857–1866.



Microbiology (2007), 153, 1394–1404 ScbA from Streptomyces coelicolor A3(2) hashomology to fatty acid synthases and is able tosynthesize c-butyrolactones Nai-Hua Hsiao,13 Johannes So¨ding,2 Dirk Linke,2 Corinna Lange,3Christian Hertweck,3 Wolfgang Wohlleben1 and Eriko Takano13 Mikrobiologie/Biotechnologie, Eberhard-Karls-Universita¨t Tu¨bingen, Auf der Morgenstelle 28,