Cholera incidence in some regions of the Indian subcontinent may exhibit two annual peaks although the main environmental drivers that have been linked to the disease (e.g., sea surface temperature, zooplankton abundance, river discharge) peak once per year during the summer. An empirical hydroclimatological explanation relating cholera transmission to river flows and to the disease spatial spreading has been recently proposed. We specifically support and substantiate mechanistically such hypothesis by means of a spatially explicit model of cholera transmission. Our framework directly accounts for the role of a model river network in transporting and redistributing cholera bacteria among human communities as well as for spatial and temporal annual fluctuations of river flows. The model is forced by seasonal environmental drivers, namely river flow, temperature and chlorophyll concentration in the coastal environment, a proxy for Vibrio choleraeconcentration. Our results show that these drivers may suffice to generate dual-peak cholera prevalence patterns for proper combinations of timescales involved in pathogen transport, hydrologic variability and disease unfolding. The model explains the possible occurrence of spatial patterns of cholera incidence characterized by a spring peak confined to coastal areas and a fall peak involving inland regions. Our modeling framework suggests insights on how environmental drivers concert the generation of complex spatiotemporal infections and proposes an explanation for the different cholera patterns (dual or single annual peaks) exhibited by regions that share similar hydroclimatological forcings.
 The seasonality and inter-annual variability of endemic and epidemic cholera and the interplay between environmental drivers and disease dynamics are receiving increasing interest and research efforts [de Magny et al., 2008; Pascual et al., 2008; Emch et al., 2010], yet they remain unsatisfactorily understood. Inter-annual variability has been studied in relation to rainfall [Ruiz-Moreno et al., 2007; Hashizume et al., 2008], river discharge [Akanda et al., 2009], sea surface temperature [Pascual et al., 2000; Koelle et al., 2005] and sea surface height [Lobitz et al., 2000]. In areas of endemicity, although the number of cases can greatly change from year to year, the disease exhibits a clear annual cycle. Seasonal patterns vary depending on geographic location [Emch et al., 2008] and environmental forcings [Lipp et al., 2002]. Endemic recurrence, however, does not relate one-to-one to climatic forcings. For instance rainfall has been found to enhance cholera insurgence in dry regions while it can buffer the propagation of the disease in wet regions due to a dilution effect [Ruiz-Moreno et al., 2007]. However a complete understanding of the causative link between the seasonality of environmental drivers and cholera dynamics is still elusive. A striking example is the annual occurrence of cholera incidence in the Bengal region. At the regional scale, the disease exhibits two peaks per year [Bouma and Pascual, 2001] although all known major environmental drivers peak once per year. A first outbreak occurs in spring and is followed, after a period of lower incidence during the monsoon, by a second, usually larger, peak in autumn (Figure 1a). This particular temporal pattern is further complicated when looking at its spatial distribution (Figures 1b–1e). Spring outbreaks usually occur in coastal areas while fall peaks dominate further inland regions [Sack et al., 2003; Akanda et al., 2011]. Local incidence patterns showing a dual peak can be found in regions of transition between coast and inland (e.g., Matlab district, Figure 1d).
Akanda et al.  have recently proposed a hydroclimatological hypothesis for this pattern: low river flows during spring favor the intrusion of brackish water (the natural environment of Vibrio cholerae, the causative agent of the disease) which, in turn, triggers the first outbreak in the coastal region. During summers, rising river discharges prompt a temporary dilution effect and the repulsion of contaminated water which lowers disease incidence. However, monsoon flooding, together with the induced crowding of the population and the failure of sanitation systems, can possibly facilitate the spatial transmission of the disease and promote the fall outbreak. Similar mechanisms have also been suggested by Pascual et al.  and Ruiz-Moreno et al. .
 Here we specifically test this hypothesis by means of a spatially explicit model of cholera transmission that has proven able to reproduce complex spatiotemporal epidemic patterns, such as the cholera outbreak in KwaZulu-Natal [Bertuzzo et al., 2008] and Haiti [Bertuzzo et al., 2011]. The model is here extended in order to account for the spatio-temporal fluctuations of river discharge, level and velocity. The importance of developing models that account for the seasonal variation of the environmental water reservoir of pathogens was first pointed out byPascual et al. . Also Ruiz-Moreno et al.  called for the integration of the effects of droughts and floods on cholera transmission models. As shown by Righetto et al. the effects of fluctuations of the hydrological regime can profoundly influence the temporal patterns of disease occurrence. We deem the present study a step forward towards a more comprehensive spatiotemporal analysis of the process. Starting from the particular dual-peak pattern typical of the Bengal region, we then explore the range of epidemiological and hydrological conditions that can lead, in the general case, to single- or dual-peak patterns in regions that share similar hydroclimatological forcings [e.g.,de Magny et al., 2008; Pascual et al., 2002].
2.1. Theoretical Framework
 Spreading of cholera epidemics along river networks is addressed by viewing the environmental matrix as an oriented graph. Nodes represent human communities (cities, towns, villages), while edges are hydrologic links between communities [Bertuzzo et al., 2007, 2010]. Edge direction indicates water flow direction. The model is assembled by coupling: i) a local epidemic model at node level, ii) a hydrologic model for the water balance of each node and iii) a transport model for the spreading of the disease agent through the river network. Cholera dynamics in a generic node iof the network is described via a compartmental SIR-like model with five state variables, namely the abundance of susceptible to the disease (Si), infected individuals (Ii), recovered from the disease (Ri) and the abundance of V. cholerae (Vi) in the local water resources (Wi). The dynamics of the system can be described by the following system of nonlinear differential equations:
The evolution of the susceptible compartment (equation (1)) is described by the balance between population demography, infections due to contact with V. cholerae and immunity losses. The host population is assumed to be at a demographic equilibrium with a constant recruitment μHi, where Hi is the size of the local community, and a constant mortality rate μ. Susceptible people become infected at a rate β(t)(Vi/Wi)/(K + Vi/Wi), where β is the transmission parameter accounting for contact with contaminated water, and (Vi/Wi)/(K + Vi/Wi) is the logistic dose-response curve (sensuCodeço ). The dynamics of the infected compartment (equation (2)) is described as the balance between newly infected individuals and losses due to recovery and natural/cholera-induced mortality, withγ and α being the rates of recovery and mortality due to cholera, respectively. Recovered from the disease (equation (3)) lose their immunity and become susceptible again at a rate ρ. The environmental water reservoir of a generic node i (equation (4)) depends on the balance between local rainfall Ji, losses due to infiltration and evapotranspiration Li, and on the difference between outflowing (Qi) and inflowing discharge, where niup is the number of nodes immediately upstream of i.Equation (5)describes the dynamics of the free-living bacteria in the local water reservoir. Infected people contribute to vibrios abundance at aper capita rate p.Free-living bacteria are also assumed to die at a constant rateμV. The last two terms in equation (5) implement the transport model of V. cholerae through the river network as explained in the following.
 A complex set of processes is known to drive the dispersion of vibrios and the consequent spreading of the disease among communities [Colwell, 1996; Pascual et al., 2002; Lipp et al., 2002]. A primary mechanism of propagation is related to the dispersion through surface waters. V. cholerae can in fact survive outside the human host in the aquatic environment and may also live in symbiosis with phytoplankton and zooplankton [Islam et al., 1994; Colwell, 1996]. As a result, the bacteria, and therefore the disease, can spread through hydrologic pathways among human communities. This mechanism is modeled assuming that vibrios are transported downstream by advection at a rate (third term of the RHS of equation (5)). The advection rate is assumed to be proportional to the local river velocity ui(t) and therefore, in general, it can vary in space and time due to hydrological fluctuations. This baseline mechanism can be superimposed by other possible transport pathways related, e.g., to the short-range distribution of water for irrigation or consumption or to the movement of bacteria attached to phyto- and zoo-plankton species (like algae and copepods) that can in turn be transported by larger organisms [Lipp et al., 2002]. Moreover, human mobility can also enhance the spreading of the disease because susceptibles and/or infecteds can act as vectors of pathogens [Bertuzzo et al., 2011; Chao et al., 2011]. All these processes can promote transport also against the flow direction and therefore they are conceptually modeled as isotropic dispersion, i.e., vibrios diffuse from any node i to each nearest neighbor at a rate lD (last term of the RHS of equation (5)). In the equations ni is the number on nearest neighbors of node i. Note that in a river network each node has only a downstream node, therefore ni = niup + 1.
 Seasonal variations of the hydrologic regime produce changes in the volume of the local water reservoir Wi(t), in the river velocity and, hence, in the vibrio advection rate They are thus expected to significantly affect disease dynamics. The effect of hydrologic fluctuations on the dispersion rate lD is instead hardly predictable because of the various potential underlying mechanisms. In absence of field/experimental evidence on its behavior, we decided to keep it constant in the present analysis.
 In order to broaden the applicability of the analysis, we adopt Optimal Channel Networks (OCNs) [Rinaldo et al., 1992] as a general model of hydrological networks. They are constructions shown to yield forms indistinguishable from real-life river networks (Figure 2, right). Moreover, we impose a uniform distribution of population, i.e., Hi = H ∀ i.These two assumptions allow us to single out the hydroclimatologic controls on the prevalence patterns in a non-specific geographical context and to exclude other sources of variability. Also, in the case of uniform population, gravity-like models that have successfully been employed to model human mobility [Mari et al., 2011] can be effectively approximated by the dispersion mechanism introduced before.
 The nature of the exercise suggests some simplifications of the hydrological model. In particular, for the monsoon-like climate analyzed herein the hydrological dynamics of a local node(4) is fast with respect to temporal variations of precipitation. Therefore, one can reasonably assume that at any time the water reservoir of each node is in instantaneous equilibrium with the external forcings (i.e., dWi/dt = 0). If in addition we consider that precipitation (Ji) and loss (Li) terms are quite uniformly distributed in space, for any pair of nodes i and j we obtain the relation Qi(t)/Qj(t) = Ai/Aj, where Ai is the drainage area of node i, i.e., the total number of nodes upstream of i. At any time, also the river velocity can safely be assumed to be uniform (ui(t) = u(t)), a reasonable approximation for many landscapes [see Leopold et al., 1964]. As a result, also the advection rate is uniform and it can be expressed as , where the overline indicates time averaging. The water volume of each node Wiis proportional to the river cross-sectional areaQi(t)/u(t), thus we finally obtain the relation Wi(t) ∝ Qi(t) ∝ Ai: at a fixed time, the water reservoir increases downstream in proportion to the drainage area. Focusing on a single node, instead, river velocity, cross section area and, in turn, water volume vary in time following the seasonal variation of the hydrologic regime. Adopting a stage-discharge approach [Leopold et al., 1964], one gets u(t) ∝ Qi(t)a and consequently Wi(t) ∝ Qi(t)1−a. In the following we will employ the exponent a = 0.4 which is derived under the hypothesis of a uniform flow in a large floodplain [Leopold et al., 1964]. Overall, the hydrologic conditions explained above are deemed representative of a significant range of cases of interest. Moreover, they allow to derive the spatiotemporal evolution of water volume Wi(t) directly from the discharge observed in a single node, without explicitly solving equation (4) and thus without further assumptions on the processes of precipitation, evapotranspiration and infiltration. Specifically, we employ the discharge time series observed in the Bengal area (Figure 1). The model resulting from the above assumptions is detailed in the auxiliary material.
 During and after the monsoon period, a large fraction of the region under study, up to 70% during the most severe floods, is inundated and people crowd the unaffected areas. This is typically accompanied by a large failure of the sanitation and sewage systems, which results in a reduced access to treated water and in an increased environmental contamination. From a modeling standpoint, this translates into higher values of the exposure rates β, the rate at which susceptibles ingest contaminated water and food, and p, the per capita rate at which stools reach and contaminate the water resource [Codeço, 2001]. We assume that at each node these rates are proportional to the fluctuations of the local water volume around the mean value (see auxiliary material).
 To run the model, we also employ climatological forcings typical of the Indian subcontinent. In particular, we model the effects of the temperature annual cycle (Figure 1) on the survival of cholera bacteria in the environment. Warmer temperatures are known to favor the attachment, growth, and multiplication of V. cholerae in the surface water [Colwell, 1996; Lipp et al., 2002]. This is accounted for by assuming that the mortality of the bacteria in freshwater environments depends on temperature T as follows:
where is the average vibrio mortality and Tmax and are the maximum and the average temperature, respectively. The parameter 0 ≤ ϵ ≤ 1 quantifies the effect of temperature on V. cholerae mortality. With ϵ = 0, the mortality rate is assumed to be constant throughout the year, whereas with ϵ = 1 the net mortality rate of cholera pathogens is null during the warmest period.
 Finally, to model the natural presence of V. cholerae in the coastal aquatic environment, we impose at the outlet of the river network a concentration of pathogens proportional to the measured chlorophyll concentration (a proxy for phyto–zooplankton and, in turn, V. Cholerae concentration [Lobitz et al., 2000; Jutla et al., 2010] (Figure 1a)). We analyze the long-term behavior of the system by simulating the process for 30 years in which each year has the same hydroclimatological pattern. To ensure that the system has reached an endemic state without memory of its initial conditions, we discard the first 10 years and analyze the patterns of cholera prevalence for the remaining 20 years.
3. Results and Discussion
Figure 2shows an example of spatial and temporal prevalence patterns generated by our scheme. Despite the single-peak forcings (Figure 1), the model originates a dual-peak global cholera pattern (Figure 2a) characterized by spring and fall outbreaks. Subdividing the system into coastal, transitional and inland regions on the basis of their distance to the outlet (Figure 2b) it can be noticed that the spring outbreak affects the coastal region while the fall outbreak is concentrated within inland areas. Regions of transition between the two exhibit a mixed behavior characterized by a dual peak. The local prevalence patterns shown in Figure 2b are indeed qualitatively very similar to those reported in three different areas of Bangladesh (Figures 1b and 1e) with a sharp fall peak in the inland areas (Figure 1d) and basal cholera prevalence throughout the year along the coast (Figure 1c).
 The modeling tools developed can also quantify the conditions under which a single or a dual global cholera incidence peak is expected in regions with hydroclimatological forcings similar to those presented in Figure 1. Several simulations have been performed to that end by varying the average advection rate , the dispersion rate lD and the seasonality of the vibrio mortality rate ϵ. Four behaviors of the global prevalence pattern can be identified depending on ϵ and on the ratio (Figure 3), where Δtis an arbitrary time-span. The quantity can be thought of as the ratio between the travel distance due to advection and the characteristic displacement due to dispersion. In fact, if we term d the characteristic distance among nodes, the former can be expressed as and the latter as . In settings dominated by advection (high ), the summer flood washes out the system and thus the epidemic cycle exhibits only a spring outbreak. A dual-peak pattern emerges for intermediate values of and for seasonality ϵ greater than about 0.1. For low advection and high seasonality the annual prevalence pattern is controlled by the temperature cycle and therefore it is basically characterized by a single peak in the warmest period with a slight perturbation due to the monsoon flood. Finally, for low advection and low seasonality the annual cycle is dominated by a single fall peak. To quantify the relevance of the modeled processes on the formation of the prevalence patterns, we have repeated the analysis of Figure 3 relaxing, one at a time, the hypotheses of varying discharge, of varying vibrios concentration at the outlet and also setting to zero the phase difference between discharge and temperature cycles. Results (Figure S3 of the auxiliary material) show that the assumption of varying vibrio concentration in the ocean, although it can profoundly impact local prevalence patterns, especially along the coast, it has a marginal effect on the parametric region where the system exhibits global (i.e., coastal and inland) dual-peak patterns. Notice that the model accounts for a reduced advection during low-flow periods which favors the intrusion ofV. cholerae, however it does not simulate the effects of low flows on coastal salinity, water temperature and phyto-zooplankton dynamics. The inclusion of these processes could possibly enhance the effect of varying vibrio concentration at the outlet.Figure S3also shows that the time-lag between the maxima of temperature and discharge holds some explanatory power for the dual-peak pattern. Notably, no dual peaks can be obtained without accounting for hydrologic fluctuations.
 The insightful hydroclimatological hypothesis proposed by Akanda et al.  tested here was based on the observation of a negative correlation between spring low flow and spring outbreaks (correlation coefficient r = − 0.65) and a positive correlation between summer floods and fall outbreaks (r = 0.55). Moreover, Akanda et al.  measured a significant correlation between spring and fall outbreaks (r = 0.73). The analyses presented so far were performed forcing the system every year with the same hydroclimatological cycle (Figure 1a). We now instead let the annual discharge cycle vary according to a realistic sequence of droughts and floods (Brahmaputra river, Bahadurabad station, record length: 36 years) in order to test whether the model is also able to reproduce the correlation structure observed in real data. Indeed, model results exhibit a positive correlation between flood and fall outbreak (r = 0.52) and between spring and fall outbreak (r = 0.66). The correlation values refer to the same model parametrization adopted in Figure 2. We have repeated the analysis for the range of parameters that lead to a dual-peak pattern (Figure 3) finding similar correlation coefficients. This result suggests that the model not only is able to capture the hydrologic control on the intra-annual dual-peak pattern, but also partially explains inter-annual variability of prevalence. On the contrary, the model underestimates the negative correlation between low flow during spring droughts and spring outbreaks (r = − 0.20 vs r = − 0.65). This is somehow expected because the model, as discussed above, does not fully account for the complex set of coastal processes that drive salinity, water temperature and phyto-zooplankton dynamics. However, these environmental forcings are known to control the ecology of the bacteria in the ocean [Miller et al., 1982; Colwell, 1996; Jutla et al., 2010, 2011], which in turn is tightly related to the spring outbreak.
 Overall the results presented support the crucial role of hydrologic and climatologic fluctuations in controlling the seasonal –from spring to fall– and spatial –from coast to inland– progression of the disease in the Bengal region. The analysis isolates the effects of river discharge and temperature from other environmental drivers which are deemed to drive V. cholerae population dynamics in the coastal aquatic environment (e.g., salinity, water temperature and plankton abundance). These factors are crucial to understand and possibly forecast spring cholera insurgence in coastal areas and have so far received the largest share of interest and research effort. Mechanisms underlying the subsequent spatiotemporal progression of the disease have received less attention. Our results suggest that in order to develop quantitative tools for cholera prediction and early warning systems this gap must be filled by integrating models of the ecology of V. cholerae in the coastal environment with a model of its spatial spreading in the environment and in the human hosts.
 EB, LM, LR and AR gratefully acknowledge the support provided by ERC advanced grant program through the project RINEC-227612 and by the SFN/FNS project 200021_124930/1. RC and MG acknowledge the support provided by Politecnico di Milano. MG worked at the project while being visiting professor at EPFL. IRI gratefully acknowledges the support of the James S. McDonnell Foundation through a grant for Studying Complex Systems (220020138).
 The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.