On the space-time evolution of a cholera epidemic

E. Bertuzzo,1,4,5 S. Azaele,2,4 A. Maritan,2,6 M. Gatto,3 I. Rodriguez-Iturbe1,4

and A. Rinaldo1,5

Received 30 May 2007; revised 22 September 2007; accepted 5 October 2007; published 19 January 2008.

[1] We study how river networks, acting as environmental corridors for pathogens, affectthe spreading of cholera epidemics. Specifically, we compare epidemiological data fromthe real world with the space-time evolution of infected individuals predicted by atheoretical scheme based on reactive transport of infective agents through a biasednetwork portraying actual river pathways. The data pertain to a cholera outbreak in SouthAfrica which started in 2000 and affected in particular the KwaZulu-Natal province. Theepidemic lasted for 2 years and involved about 140,000 confirmed cholera cases.Hydrological and demographic data have also been carefully considered. The theoreticaltools relate to recent advances in hydrochory, migration fronts, and infection spreadingand are novel in that nodal reactions describe the dynamics of cholera. Transport throughnetwork links provides the coupling of the nodal dynamics of infected people, who areassumed to reside at the nodes. This proves a realistic scheme. We argue that thetheoretical scheme is remarkably capable of predicting actual outbreaks and, indeed, thatnetwork structures play a controlling role in the actual, rather anisotropic propagation ofinfections, in analogy to spreading of species or to migration processes that also use riversas ecological corridors.

Citation: Bertuzzo, E., S. Azaele, A. Maritan, M. Gatto, I. Rodriguez-Iturbe, and A. Rinaldo (2008), On the space-time evolution of a

cholera epidemic, Water Resour. Res., 44, W01424, doi:10.1029/2007WR006211.

1. Introduction

[2] Cholera is an intestinal disease caused by the bacteriumVibrio cholerae, which colonizes the human intestine. Thedynamics of cholera epidemics have been studied since the1800s, when John Snow established the link between choleracases and exposure to contaminated water of a well inLondon. V. cholerae is also a natural member of the aquaticmicrobial community [Colwell, 1996; Lipp et al., 2002].Thus the spatial and temporal patterns of cholera epidemicsare strongly related to the ecology of the bacterium in theenvironment which is itself driven by meteorological andclimatic variability. Time series analyses of cholera cases inendemic regions, such as Bangladesh, show a time variabilitywith subannual, annual, and interannual components. Thelow-frequency variability has been established to be relatedto long-term climatic oscillations [Pascual et al., 2000;Koelle et al., 2005]. In nonendemic regions the annualcomponent is much more important. Also, the spatial distri-bution of the disease plays a fundamental role, which is,

however, usually neglected in the existing cholera models.The aim of this paper is to understand the spatio-temporalevolution of cholera by explicitly accounting for the envi-ronmental matrix within which the disease can spread.[3] The model will be tested against a very well docu-

mented case. In 2000, after several years without choleraoutbreaks a new epidemic spread in South Africa, affectingin particular the KwaZulu-Natal province. The epidemiclasted for 2 years with only a few cases recorded during thethird year, and ultimately involved about 140,000 confirmedcholera cases. The epidemic was caused by the 01 el Torstrain [Mugero and Hoque, 2001], which is more easilytransmitted through contamination of aquatic environmentsthan the classical biotype of Vibrio cholerae.[4] Models of cholera dynamics are relatively recent.

Capasso and Paveri-Fontana [1979] proposed a mathemat-ical model to describe the 1973 cholera epidemic in Bari(Italy) with two equations describing the dynamics of theinfected population and the free-living pathogens. Codeco[2001] extended Capasso and Paveri-Fontana’s model, add-ing an equation for the dynamics of the susceptible popu-lation, and studied the role of the aquatic reservoir in theendemic-epidemic dynamics of cholera. In the model pro-posed by Pascual et al. [2002], another equation is added todescribe the temporal evolution of the volume of waterhosting the free-living bacteria. Recent laboratory findingssuggest that passage of the bacterium through the gastroin-testinal tract results in a short-lived hyperinfectious state thatcan enhance the human-to-human versus environmental-to-human transmission of cholera. Hartley et al. [2006] incor-porate the hyperinfectious state into Codeco’s model toachieve a better explanation of explosive cholera outbreaks.

1Dipartimento di Ingegneria Idraulica, Marittima Ambientale e Neotei-nia, Universita di Padova, Padua, Italy.

2Dipartimento di Fisica ‘‘G. Galilei,’’ Universita di Padova, Padua, Italy.3Dipartimento di Elettronica e Informazione, Politecnico di Milano,

Milan, Italy.4Now at Department of Civil and Environmental Engineering, Princeton

University, Princeton, New Jersey, USA.5International Center for Hydrology ‘‘Dino Tonini,’’ Universita di

Padova, Padua, Italy.6Now at Consorzio Interuniversitario per le Scienze Fisiche della

Materia, Istituto Nazionale Fisica della Materia, Padua, Italy.

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[5] All the above models do not consider space explicitly.They assume a unique community of people who interactand share the same resources. We believe, however, that thespatial distribution of the communities and how theyinteract is crucial to understanding the spatial spreading ofthe epidemic in a disease-free region, particularly if traveltimes of pathogens are comparable with the characteristictime of the virulent infection. The spatial distribution ofdifferent communities, along with the distribution of theirpopulation size and how they are interconnected, couldindeed affect the dynamics of the process, especially inthe case of a nonendemic region.[6] V. cholerae can survive in the aquatic environment in

associations with chitinaceous zooplankton like copepodsand shellfish and also with the aquatic vegetation [Colwell,1996]. Therefore V. cholerae (and the disease) can spreadfrom the coastal region, where it is autochthonous, to theinland area through waterways and river networks. In thesame manner the infection can spread from inland regionswith epidemic outbursts into the surrounding areas.[7] We propose a model that explicitly accounts for the

role of the river network in transporting and redistributingV. cholerae between several human communities, and weproceed to apply the model to the real conditions of the2000 cholera epidemic in the KwaZulu-Natal province. Themodel explicitly recognizes a role for network structuresacting as support for the infection, in analogy to recentstudies on migrating fronts constrained by landscape heter-ogeneities or spreading of species along riparian ecologicalcorridors [Campos et al., 2006; Bertuzzo et al., 2007;Muneepeerakul et al., 2007].[8] The paper is organized as follows. Section 2 describes

the theoretical approach and the model used in detail. Thecomplete data set and the case study are presented in aspecific chapter (section 3). The main results and the

discussion are collected in section 4. A set of conclusionscloses the paper.

2. Theoretical Approach

[9] Spreading of epidemics in networks is addressed byviewing the environmental matrix as an oriented graph (i.e.,a directed graph having no symmetric pair of directededges). Nodes represent human communities (cities, towns,and villages) in which the disease can be diffused and grow.The edges represent links between the communities, typi-cally hydrological links. Edge direction is chosen accord-ingly to the flow direction. The model is assembled bycoupling two models: (1) a local epidemic model at nodesof the graph and (2) a transport model for the spreading ofthe disease vector through the edges of the support. Detailsof the two models follow.[10] As for the local dynamics, we use a continuous

model of the susceptible, infected, and recovered class witha reservoir of free-living infective propagules. It is obtainedby a slight modification of the cholera epidemic modelintroduced by Codeco [2001]. The model has three statevariables: the number of susceptibles (S), the number ofinfected (I), and the concentration of V. cholerae in theaquatic environment (B), whose respective temporal dy-namics are described by the following system of first-orderdifferential equations:

dS

dt¼ n H � Sð Þ � a

B

K þ BS

dI

dt¼ a

B

K þ BS � r þ mþ nð ÞI

dB

dt¼ nBBþ

p

WI

ð1Þ

[11] Themeaning of the parameters is explained in Table 1.The first equation describes the dynamics of susceptibles in

Table 1. Description of the Symbols Used in the Text

Symbol Description Value Note

Si number of susceptibles at node i equation (1)Ii number of infected at node i equation (1)Bi concentration of V. cholerae in aquatic environmenta equation (1)Ci number of cumulated cases of node i equation (7)Hi total human population size of node i at the disease-free equilibrium input datan population natality and mortality rateb 5 � 10�5 estimateda rate of exposure to contaminated waterb 1 estimatedK concentration of V. cholerae in water that yields 50% chance of being infected with choleraa

r rate at which people recover from cholerab 0.2 estimatedm mortality rate due to cholerab 4 � 10�4 estimatednB net growth rate (usually negative) of V.cholerae in the aquatic environmentb �0.228 calibratedp rate of production by one person infected of V. cholerae that reach the water bodyc

Wi volume of waterd at node ISCi

critical threshold of node i equation (3)b transport bias (Pout � Pin) 0.08 calibratedl V. cholera mobilityb 3.5 calibratedVi vulnerability of node i: Hi/SCi

equation (6)c per capita water volumee

HT threshold on the node population 29,000 calibratedp/(Kc) combined parameters ratiob 4.76 � 10�6 calibrated

aMeasured in cells per m�3.bMeasured per day.cMeasured in cells per day per person.dMeasured in m3.eMeasured in m3 per person.

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a community of size H. Susceptible individuals are born anddie on average at rate n. Newborn individuals are consideredsusceptible. Susceptible people become infected at a rate aB/(K + B), where a is the rate of contact with contaminatedwater and B/(K + B) is a logistic dose response curve thatlinks the probability of becoming infected to the concentra-tion of vibrios B in water. Infected people (whose dynamicsare described by the second equation) die at a rate which isthe sum of natural mortality n and disease-caused mortalitym and recover with rate r. The third equation describes thedynamics of the free-living infective propagules in thereservoir. Infected people contribute to the concentrationof vibrios at a rate p/W, where p is the rate at which bacteriaare produced by one infected person and W is the volume ofthe contaminated water body. The growth rate nB of the free-living bacteria in the water body is usually negative becausebacteria mortality in natural environments exceeds reproduc-tion. If nB is positive, the model would predict an exponentialgrowth of the vibrios concentration, and all the susceptiblepopulation would be affected by the disease. The hiddenequation for the recovered is (dR/dt) = rI � nR. Peoplerecovered from cholera are considered immune. The modeldoes not take into account any loss of immunity (i.e., a fluxfrom recovered to susceptibles) because immunity usuallylasts for a period longer than the 2 years of the epidemic weconsider here [Koelle et al., 2005]. Nevertheless, the immu-nity loss could play an important role in the dynamics ofcholera in regions where it is endemic [Koelle et al., 2005].[12] As long as we are not interested in the numerical value

of the concentration B of V. cholerae, we can introduce thedimensionless concentration B* = B/K, thus obtaining fromequation (1) the system of equations

dS

dt¼ n H � Sð Þ � a

B*

1þ B*S

dI

dt¼ a

B*

1þ B*S � r þ mþ nð ÞI

dB*

dt¼ nBB* þ

p

KWI :

ð2Þ

Notice that equation (2) has the advantage of merging theparameters K, p, and W (which can hardly be directlyestimated) into a unique ratio. This ratio will be the controlparameter of the process jointly with the V. cholerae growthrate nB. On the contrary, the mortality rates, both natural (n)and due to cholera (m); the recovery rate (r); and the exposurerate (a) can reasonably be estimated from demographic andepidemiological studies, as we show below.[13] A linear stability analysis shows that, given an initial

condition of the type S(0) = H; I(0) > 0; B*(0) = 0, themodel predicts an epidemic outbreak only if the populationsize is greater than a certain critical threshold SC given by[Codeco, 2001]

H > SC ¼� r þ nþ mð ÞKnBW

ap; ð3Þ

otherwise the infected population decreases to zero. It isimportant to remark on the dilution effect: The larger thevolume of the water body is, the higher the critical thresholdwill be.

[14] We model the spreading of V. cholerae through thenetwork with a biased random walk process on orientedgraph [Bertuzzo et al., 2007]. For a detailed discussion ofthe process, see also Johnson et al. [1995]. An infectiouspropagule can move with some probability from a node toone of the adjacent nodes, which are all the nodes that areconnected to it through an inward or outward edge. Weassign to each edge of the graph an orientation according tothe flow direction. Consider first a particular case of thenetwork in which every node has only one inward and oneoutward edge (i.e., a one-dimensional lattice). We define asPout (Pin) the probability that a propagule leaving a nodemoves to another node along an outward (inward) edge. Wehave then Pout + Pin = 1.[15] We now turn to the analysis of a random walk

process on a generic-oriented graph in which every nodecan have an arbitrary number of inward and outward edges.We assume that a propagule can move following an outwardor inward edge with a probability proportional to Pout andPin, respectively. In this case, the probability Pij for apropagule to be transported from node i to node j can beexpressed as follows:

Pij ¼

Pout

dout ið ÞPout þ din ið ÞPin

if i! j

Pin

dout ið ÞPout þ din ið ÞPin

if i j

0 if i 6$ j;

8>>>>><>>>>>:

ð4Þ

where dout(i) and din(i) are the outdegree and indegreeof node i, respectively (i.e., the number of outward andinward edges of node i, respectively). Since Pout + Pin = 1,

one hasPN

j¼1 Pij = 1, where N is the total number of nodes.

We term b = Pout � Pin = 2Pout � 1 the bias of the transport.[16] When we apply the local epidemic model at each

node of the network, we have 3N state variables Si, Ii, andB*i, where the subscript i identifies the nodes. We assumethat vibrios are removed at every node with a certain ratel (d�1) and transported through the network following thetransition probabilities from equation (4). Then the equa-tions that describe the coupled process are

dSi

dt¼ n Hi � Sið Þ � a

B*i

1þ B*i

Si

dIi

dt¼ a

B*i

1þ B*i

Si � r þ mþ nð ÞIi

dB*idt

¼ nBB*i þ

p

KWi

Ii � lB*i þXNj¼1

l PjiB*j

Wj

Wi

;

ð5Þ

for i = 1, 2, . . ., N. Note that all the parameters are node-independent except for the population size Hi and the watervolume Wi. The latter represents the whole set of watersupplies available for that community, not only the oneprovided by the river. The network acts as a link throughwhich different sets of water supplies of different commu-nities can be connected and contaminated. In order tofurther minimize the number of parameters, we assume thatthe water volume is a nondecreasing function of thepopulation size: Wi = f(Hi). Different choices of the functionf can lead to different scenarios of the epidemic. In fact,

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consider the ratio Vi between the population size and thecritical threshold equation (3).

Vi ¼Hi

SCi

¼ Hi a p

� r þ nþ mð ÞK nBf Hið Þ/ Hi

f Hið Þ: ð6Þ

[17] This is an index of the node vulnerability to anepidemic. Let us first analyze the case in which the wateravailability is constant for all the nodes (Wi = constant).This corresponds to assuming that the communities utilizewater resources that are quite uniformly distributed in space.In this case, the vulnerability of the nodes increases linearlywith the population size, namely, Vi / Hi. In this scenariothen, if an epidemic occurs, the most affected communitieswould be the most populated. Another different scenarioderives from assuming that larger communities manage toincrease their own water supply so that the per capita

available water is kept constant. This is equivalent toassuming that the water volume of a node is proportionalto the population size: Wi / Hi, and then Vi = constant.Under such a scenario an epidemic would affect, even ifwith different dynamics, all the communities regardless oftheir size. This assumption seems reasonable for large anddeveloped communities, but it is unsatisfactory for thenodes with small population size because it would implya small water body associated with the node regardless ofthe natural presence of water. A more general and realistichypothesis could derive from the combination of the twoassumptions described above. In particular, we assume thatthe nodes with population size Hi smaller than a certainthreshold HT have a constant water volume associated withthem, whereas the nodes with Hi > HT have a constant percapita water availability. Summarizing, we have Wi =max(cHT, cHi) = c � max(HT, Hi), where c is the per capitavolume of water resources.[18] Substituting the last relationship into the second term

of the right-hand side of the third equation of system (5), weget pIi/(K c � max(HT, Hi)). Thus the parameters to beseparately estimated are the ratio p/(K c) and the populationthreshold HT. These are parameters that depend on socialand environmental factors, like hygiene, health conditions,eating habits, and lifestyle, and on how these variables varywith population density. Nevertheless, the particular choiceof parameters that link vulnerability to the actual distribu-tion of population has to be evaluated for each case.

3. Case Study

[19] We apply the model to a well-documented case ofcholera epidemics that occurred in South Africa. The datawere provided by the KwaZulu-Natal Health Departmentand consist of a record of each single cholera case specifiedby the date and health subdistrict where it occurred. Thespatial representation of the districts is shown in Figure 1a.The record starts from August 2000 and runs continuouslyuntil present time. Our analysis focuses on the two largestepidemic outbreaks which occurred during the 2000–2001and 2001–2002 summers and involved 135,000 choleracases. The data set also provides the population size of eachdistrict. The total population of the province is about8.5 million inhabitants. The temporal evolution of theweekly cholera cases is reported in Figure 1b. The dataexhibit a clear seasonality with the outbreaks occurringduring the warmest months of the austral summer. This isprobably due to the increased growth rate of vibrios at warmtemperatures in association with plankton blooms. Evidencefor this phenomenon comes from the higher rates ofisolation of the bacterium in the environment during warmperiods [Lipp et al., 2002].[20] In order to apply the model, we need to build the

network along which infection is transported. First of all, wehave derived the mathematical model of the river networksfrom the hydrological geographic information systems dataprovided by the South Africa Department of Water Affairsand Forestry shown in Figure 2. All the channels ofperennial rivers are considered edges, and all the endpointsof these channels are considered as nodes. Second, we hadto transfer the information from districts to network nodes.This is done by assigning the population and the choleracases of a subdistrict to the nearest network node, with the

Figure 1. (a) Spatial representation of the health districtsof the KwaZulu-Natal province, South Africa. Colorsrepresent the percentage incidence of cholera cases overthe population size of each district. (b) Also shown istemporal evolution of the new weekly cholera cases for thewhole province.

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distances being computed from the centroid of the subdis-trict to the node. The results of this interpolation for thepopulation and the total cumulated cholera cases are shownin Figure 3, where the color coding is obtained by spatiallinear interpolation of the node values. Comparing thespatial distribution of population sizes and total cases inthe nodes, one can note that the high population densityareas recorded few cases of cholera as did the low-densityones. The most affected nodes are those with intermediatepopulation size. This clearly appears by plotting the choleraaverage incidence (i.e., the total number of cases divided bythe population size) as a function of the population size (seeFigure 4). The highest incidence was recorded for populationsizes between 2000 and 30,000. This is probably due to thefact that the highest population density regions correspond, inthis particular case, with the most developed ones. Thesecities can then rely on wastewater treatment and treated watersupply that help to reduce cholera transmission.[21] The framework described in this paper addresses the

spread of an epidemic in a single river basin, and for thetime being we avoid any modeling of the flux of bacteriaacross different catchments. For this reason, in order to testthe validity of the model, we have applied it to the basin ofriver Thukela, the largest of the region (see Figure 2). The29,000 km2 area drained by this river is populated by1.5 million people, and cholera cases recorded thereamounted to 29,000 (21% of the total cases of the wholeprovince) during the two epidemic outbreaks considered.[22] We estimated the birth and mortality rate of the

population as the inverse of the average lifetime for thisregion (about 60 years), so n ’ 5 � 10�5 d�1. Because theaverage duration of the cholera disease in an infected personis approximately 5 d [Codeco, 2001; Hartley et al., 2006],we set the recovery rate at r = 0.2 d�1. The deaths due tocholera for the epidemic analyzed were 0.2% of the cholera

cases. Thus we can estimate the cholera mortality rate m byassuming that after the duration of the disease, 99.8% of theinfected population survive, that is, exp(�m/r) = 0.998, andthen m = 4 � 10�4 d�1. From this simple analysis weconclude that, given the order of magnitude of the parametersinvolved, we can simplify the model setting r + m + n ’ r.Following Codeco [2001], we assume that people ingestcontaminated water or food once a day (a = 1 d�1).[23] To model the seasonality of the bacterium ecology as

discussed in section data, we let the net growth rate ofV. cholerae in the aquatic environment vary periodically in

Figure 2. Hydrographic map of KwaZulu-Natal provincewith the Thukela river basin evidenced. The dot reports thelocation of the first epidemic outbreak in the basin studied.

Figure 3. Spatial linear interpolation of network nodesvalue of (a) cholera cases and (b) population size.

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time according to nB(t) = nB(1 + sin(2pt/365)) (with t indays and t = 0 corresponding to 1 October) following thewater temperature cycle [Jury, 1998]. We implicitly assumethat the free-living bacteria are in demographic equilibrium(nB(t) = 0) during the warmest season. After the aboveconsiderations, the model parameters to be calibrated are themean net V. cholerae growth rate in the aquatic environment(nB), the rate at which V. cholerae are removed and trans-ported (l), the ratio (p/(K c)), the threshold (HT), and the biasof the transport (b).

4. Results and Discussion

[24] In order to calibrate the model, we subdivided the setof parameters into two groups: The first contains theparameters related to the transport and spatial distributionof cases (l, b, and HT), while the second set groups theparameters closely related to the local model and thetemporal dynamics (nB and p/(Kc)). For every randomlychosen combination of the three parameters of the firstgroup, we calibrated the two parameters of the second groupby minimizing the mean square error between the data ofthe temporal evolution of cumulated cases in the wholebasin and the simulations. Note that the temporal evolutionof the cumulated cases for each node can be obtained via theequation

dCi

dt¼ a

B*i

1þ B*i

Si; ð7Þ

which takes into account only the flux of susceptibles thatactually become infected. By adding the cumulated cases attime t throughout all the nodes, we get the cumulativeevolution for the entire catchment. Every simulation startswith the initial conditions of one infected person in a singlenode where, according to data, the first case of cholera wasrecorded and runs continuously for 2 years (the location ofthis node is reported on the map in Figure 2). Then, for eachcombination of the five parameters (the triplet l, b, HT andthe corresponding calibrated pair nB, p/(K c)) we computedthe mean square error between the total cumulated cases ofcholera at each node after 2 years obtained from the

simulations and from the data. Note that the error from thefirst calibration measures the likelihood of the temporalpatterns, while the second calibration compares simulatedand recorded spatial patterns of the epidemic. Finally, wechose the combination of parameters that minimizes theweighted sum of the two errors. The weight for each error iscomputed as the inverse of the minimum value of the erroritself found among all the realizations. The values of thecalibrated parameters are listed in Table 1.[25] Figure 5 shows the comparison between data (dots)

and model simulation (solid line) of the temporal dynamicsof the weekly (Figure 5a) and cumulated (Figure 5b) choleracases in the whole Thukela river basin. Figure 6 comparesthe spatial distribution of data with that obtained viasimulation. It shows the distribution of the cumulatedcholera cases after the first and second epidemic outbreak.As in Figure 3, the colors are graded via spatial linearinterpolation of nodal values.[26] The model does well in reproducing the distribution

of the cholera cases during the two outbreaks as well as theirspatial spreading. It is interesting to note that cases from thesecond outbreak are mainly located in new regions withrespect to the first one. This is related to the spread ofcholera from the regions involved in the first epidemicoutbreak into disease-free ones. This supports our hypoth-esis that the dynamics of cholera epidemics in nonendemicregions depend on the spatially anisotropic spreading alongan environmental matrix defined by river corridors as wellas on inner local dynamics. Similar results in a differentcontext were obtained by Campos et al. [2006], Bertuzzo etal. [2007], and Muneepeerakul et al. [2007].

Figure 4. Average cholera incidence (i.e., total choleracases over population size). Nodes have been grouped in alogarithmic bin on the basis of their population size.

Figure 5. Comparison between the data (dots) andsimulated (solid line) temporal evolution of (a) weeklycholera cases and (b) cumulated cases for the Thukela riverbasin of the KwaZulu-Natal province.

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W01424 BERTUZZO ET AL.: ON THE SPACE-TIME EVOLUTION OF A CHOLERA EPIDEMIC W01424

[27] Finally, we have checked whether our model is ableto reproduce the relation discussed in section 3 betweencholera incidence and population size of each node. Thecomparison for the Thukela river basin is reported inFigure 7. It demonstrates that the model results regardingthe incidence distribution agree quite well with the data.

5. Conclusions

[28] The following main conclusions can be drawn fromour results:[29] 1. Our model of cholera dynamics explicitly

accounts for the spatial distribution of the communitiesand their interconnections, proving capable of reproducingthe main spatial and temporal patterns of the spreading for awell-documented epidemic into a disease-free region in theKwaZulu-Natal province of South Africa.[30] 2. A significant role emerges for the ecological

corridors defined by waterways and river networks. Suchhydrologic control derives from the transportation andredistribution of the free-living infective propagules. Inparticular, because vibrios can spread along stream bothupstream and downstream with a slightly biased propaga-tion downstream (which is to be expected, of course), theinfection patterns are anisotropic.[31] 3. Despite its satisfactory performance, the model is

not completely reliable in reproducing secondary peaks ofinfections in the tail of both annual outbreaks. We speculatethat they might be the combined results of seasonality in theepidemiological parameters and the presence of short-livedhyperinfectious bacteria. This hypothesis remains to betested and will be the object of future work.

[32] In conclusion, we suggest that this approach repre-sents a first step toward understanding how hydrology andpopulation distribution along the water network control thespreading of water-borne diseases.

Figure 6. Comparison between data and simulated spatial distribution of the cumulated cholera casesafter the first and second epidemic outbreak for the Thukela river basin. Colors are obtained via spatiallinear interpolation of the node values.

Figure 7. Comparison between data and simulated choleraincidence (i.e., the ratio of total cholera cases to populationsize) for the Thukela river basin. Nodes have been groupedin a logarithmic bin on the basis of their population size.Bars represent the mean incidence for each bin, while theerror bars represent the standard error of the mean. TheKolmogorov-Smirnov goodness of fit test value is 0.0781,and that leads us to accept the hypothesis that the model fitsthe data with a significance level of 0.05.

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[33] Acknowledgments. Special thanks go to the KwaZulu-NatalDepartment of Health for providing the data set that made this workpossible. Research funding was provided by AQUATERRA EU projectGOCE-505428, PRIN 2006 ‘‘Fenomeni di trasporto sul ciclo idrologico,’’Politecnico di Milano, and the University of Padua. I.R.-I. gratefullyacknowledges the support of the James S. McDonnell Foundation underthe grant ‘‘Studying Complex System’’ (220020138).

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����������������������������S. Azaele and I. Rodriguez-Iturbe, Department of Civil and Environ-

mental Engineering, Princeton University, Princeton, NJ 08540, USA.E. Bertuzzo and A. Rinaldo, Dipartimento di Ingegneria Idraulica,

Marittima Ambientale e Geotecnica, Universita di Padova, via Loredan 20,I-35131 Padova, Italy. (bertuzzo@idra.unipd.it)

M. Gatto, Dipartimento di Elettronica e Informazione, Politecnico diMilano, via Ponzio 34, I-20133 Milano, Italy.

A. Maritan, Consorzio Interuniversitario per le Scienze Fisiche dellaMateria, Istituto Nazionale Fisica della Materia, via Marzolo 8, I-35131Padova, Italy.

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