Puccinia graminis f. sp. tritici is a heteroecious airborne fungal pathogen responsible for stem rust of wheat. Wheat, the main host of P. graminis, constitutes a staple food for a great proportion of human population. It covers $2.15 \times 10^{8}$ hectares worldwide (1.4% of earth surface), representing the most abundant agricultural type of land cover [25].

In recent decades, yield losses caused by stem rust have been limited by planting resistant cultivars and fungicides application. Nonetheless, the emergence of new strains overcoming plant resistance may provoke severe yield losses, accounting for up to 50%–90% of the wheat production at the regional scale [19, 26]. An extraordinary outbreak in Ethiopia in 2013–2104 caused a complete yield loss in some cultivars [27], while concerns were raised in Europe after the detection of new virulent strains capable of infecting previously resistant cultivars [28].

We extracted the worldwide wheat distribution from the MapSPAM database [29] and computed the percentage of wheat cover on a regular grid with a resolution of 0.5^∘. We accounted only for cells containing at least 2% of wheat land cover. The resulting database contains 7814 cells (see section SI 1.1). We calculated backward air-mass Lagrangian three-dimensional trajectories of 120 hours [16] with the HYSPLIT (HYbrid Single-Particle Lagrangian Integrated Trajectory) model [30] from 1st January 2013 to 31st December 2018, using atmospheric GDAS data at a spatial resolution of 0.5^∘. We ran simulations from the centroids of each cell at 00:00, 06:00, 12:00 and 18:00 UTC +0, at an above-ground altitude set to the minimum between the base of the cloud and the mixed layer depth. Along each simulated trajectory, we recorded atmospheric variables at an hourly frequency. Overall, out of more than $6.8 \times 10^{7}$ potential trajectories, we computed only those $1.6 \times 10^{6}$ satisfying the criteria of host availability and environmental suitability specified in sections 2.3.1 and 2.3.4 (see also section SI 1.2).

We built a time-variant connectivity network for stem rust explicitly considering host (i.e. wheat) phenological phases, environmental conditions affecting spore aerobiology [31] and infection. In this network, nodes represent cells composing the worldwide wheat producing regions, while edges are computed from air-mass trajectories in order to model the dispersal of airborne spores. The nodes (or cells) of the networks remain fixed, while edges are re-computed for each of the 8764 (4 times a day for 6 years) simulations.

More specifically, edges result from the application of a set of biophysical filters to identify those air-mass movements that correspond to transport events (figure 1). We assume that, for a given time t, an edge exists between an arrival cell j and any cell i if the following conditions are satisfied:

• (a)  
  In cell i, which at time t − x ($x\in \{1,\ldots ,120\}$ h) is crossed by the air-mass trajectory arriving at j at time t:

  • The host is 'available', i.e. it is present and in a favourable phenological state for infection and sporulation, see section 2.3.1.
  • Environmental conditions are compatible for spore release, see section 2.3.2.
  • The altitude of the trajectory is lower than the planetary boundary layer.
• (b)  
  Environmental conditions along the trajectory between i and j allow spore survival, see section 2.3.3.
• (c)  
  The host in cell j is 'susceptible', which means:

  • The host is 'available'.
  • Environmental conditions are favourable for spore deposition and host infection, see section 2.3.4.

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2.3.1. Host availability

2.3.2. Spore release

2.3.3. Aerial spore transport and survival

2.3.4. Spore deposition and host infection

2.3.5. Time-variant connectivity networks

Since the direct observation of long-distance spores transport is beyond current technological capacities, we tested the validity of our model checking if we could use it to identify the well known North-American 'Puccinia pathway'.

We considered the yearly northward movement of the stem rust infection front in North America by tracking the average onset date (±1 standard deviation) during the 1922–1992 period along the 97th meridian west [3]. Then, we computed the weekly cumulative in-strength values of the nodes located along the same meridian in the US from the weekly connectivity networks C_w . The in-strength is a network metric considering the sum of the weights of the edges pointing a node i. In this case, the in-strength of a node i corresponds to the sum of the average weekly frequencies of connection from the other nodes toward i. We used the cumulative version of this metric, by assuming that local emergence of a disease is observable after sufficiently abundant inoculum has been deposited in that node. We graphically compared observed onset dates and average weekly cumulative in-strength in figure 3 (see section SI 1.6). Here, we built the frequency distribution of the cumulative in-strength values intersected by 1922–1992 onset date observations and computed the values corresponding to the interquartile range. Lastly, cumulative in-strength values falling in the interquartile range are shaded on the heat map.

2.5.1. Exploring 'Puccinia pathways' worldwide

2.5.2. Definition of efficient surveillance strategies

2.5.3. Measuring early-detection performance of the sentinel set

The duration of host susceptibility periods, i.e. the concurrence of the host availability and environmental conditions favorable to infection, varies considerably across the world (figures 2(a) and SI6). The majority of cells is susceptible for a period between a week and a month. The nodes where susceptibility occurs for more than a month per year are located in northeastern America, southern Brazil and Paraguay, central Europe and central China. The only nodes with susceptibility lasting for more than three months per year are located in Ethiopia (see section SI 2.1).

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Regions located in the Northern Hemisphere are generally well connected, in such a way that Europe, Asia and North Africa create a unique connected component (figure 2(b)). In spite of the obstacle represented by the Atlantic Ocean, extremely long distance connections may occur from North America to the Mediterranean basin. Conversely, clusters in the Southern Hemisphere are isolated between them, although internally connected. A first representation of the role played by each node within the epidemic network is given by the sum of the probabilities associated to trajectories going into (in-strength: figure 2(c)) and out from (out-strength: figure 2(d)) that node. In biological terms, in-strength (out-strength) is a proxy of the extent to which a region acts as a sink (source) of spore. In-strength appears to be higher in specific regions located in Europe and in the northeastern US, southern Brazil, the Himalaya and central China. Some regions exhibit great variability within relatively short distances, namely Ethiopia, Middle East and Central Asia. Other regions show a smoother and regular gradient, like the US. In this sense, a continuum of intermediate in-strength values extends from Europe to Western Siberia. Out-strength separates more sharply those regions characterized by high and low values and, in particular, Europe is characterized by large regions associated with high out-strength values.

Our global epidemic network permitted to recover the well known North-American Puccinia pathway. In figure 3 we compare the average onset date of outbreak along the 97th meridian west observed between 1922 and 1992 in North America [3], with the cumulative in-strength of network nodes along the same meridian. Most of the observed onset dates occur within a certain interval of values of cumulative in-strength (the interquartile range $[0.6,0.9]$), suggesting that the cumulative in-strength network metric can proxy the spatio-temporal progression of the 'Puccinia pathway' in North America. In other words, in this continent one would expect to observe the first signs of an epidemics when the cumulative in-strength has values between 0.6 and 0.9.

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Globally, we found that the airborne transport estimated via the center of mass of the monthly in-strength always moves poleward from tropical and temperate regions (figures 4 and SI10), except for the Ethiopian pathway, that follows a southward movement even if it is located in the Northern Hemisphere, likely due to its cropping calendar.

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The sentinel set obtained on the 2013–2016 network using the set cover algorithm allows for a complete coverage of the worldwide wheat producing regions by monitoring 1007 nodes, i.e. less than 13% of the total (figure SI12). Moreover, a reasonable coverage of 50% of the domain can be obtained monitoring 64 nodes (figure 5(a)), i.e. less than 1% of the total. The first selected nodes are those assuring the greatest coverage, and they do not distribute uniformly across continents (figure SI12—see section SI 2.4).

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The sentinel set selected to optimally cover the 2013–2016 epidemic network, via the set cover algorithm, provides satisfactory results also when applied to surveil the epidemic network obtained for the 2017–2018 period (figure 5(b)). In fact, it provides a coverage of $50\%$ of the domain by monitoring 114 nodes ($1.5\%$ of the total). For comparison, the same coverage of 50% would require 234 ($3\%$) random sentinels (on average: interquartile range [227; 242]), 475 ($6.1\%$) sentinels if ordered for increasing values of betweenness, or 611 ($7.8\%$) of PageRank.

In terms of DDR, the set cover algorithm outperformed all the other methods for sentinel sets containing between 20 and 650 nodes (figure 5(c)). In-strength provides better results for very small sentinel sets, while random sentinel sets are more suitable when larger sentinel sets have to be designed. The DDR associated to 275 sentinels is $19.2\%$, which means that 275 sentinels are able to detect an epidemic process started from any of $19.2\%$ of the world producing regions within a month. A similar DDR of 19.1% is obtained with 350 nodes chosen according to their betweenness, or more than 500 nodes according to their PageRank. Between 350 and 500 random sentinels are needed to achieve a DDR around 18%–22%.

We estimated the DDR associated to larger values of detection delay (3 month, 6 month, 1 year), showing that the set cover strategy improves its performances against the random sentinels sets (figure SI14).

Our results indicate that the duration of the susceptibility period greatly varies among regions (figures 2(a) and SI6). Eastern US, southern Brazil, central Europe and China, which are important areas of wheat production, experience the longest susceptibility periods. On the other hand, other important wheat-producing areas from the Middle East up to the Indian subcontinent have shorter periods of susceptibility. The reason could be the temporal mismatch between the occurrence of environmental conditions for infection and of host availability. This is particularly true for India and in Brazil, where, despite a long-lasting occurrence of environmental conditions for infection, the susceptibility period is constrained by relatively short host availability periods (figure SI6). Ethiopia, which is characterized by exceptional climatic diversity for tropical latitudes [51], exhibits a very heterogeneous behavior, with nodes that are never susceptible in close proximity to others that are highly susceptible (91–130 days per year). Since both host availability and favorable environmental conditions for spore deposition and host infection are climate dependent processes, it is likely that global crop epidemic dynamics will be affected by predicted climate changes [52]. Prank et al [19] predicted a worldwide increase in sporulation period under a RCP8.5 climate change scenario in 2100—potentially compensated by a general decrease of probability of germination. On the other hand, laboratory experiments conducted on other fungal species suggested that future climate conditions would rather inhibit sporulation, but increase mycelium growth rate [53].

To construct an epidemic network we had to explicitly account for biophysical conditions influencing spore aerobiology. This filtering enabled to discard uninformative air-mass connections and to reduce the complexity of the network, making it simultaneously more significant and manageable.

In principle, by tracking the movement of an air mass for an indefinite long time, one would obtain a completely connected network, i.e. a network with edge density equal to 100%. On the other hand, our epidemic network has an edge density equal to 3.7%, whereas the network of only the most recurring connections (i.e. those connections found at least 3 times out of 4 in the yearly networks 2013–2016) has a density of 0.4%. For comparison, previous studies using a similar technique but filtering only over a 48 hours time [54] obtained a density of 28% over the Mediterranean basin for the period 2011–2017. In essence, our filtering allows to make epidemic networks more manageable and informative, presenting itself as alternative to other network-based methods for the extraction of the truly relevant connections [55].

Our results on the frequency of connections (figure 2) indicate that regions in the Northern Hemisphere are more densely connected than in the Southern one. This can be explained by the fact that 90% of wheat fields are located in the Northern Hemisphere while the clusters below the Tropic of Cancer, i.e. South America, South Africa, Ethiopia and Australia are thousands of kilometers away from each other. Furthermore, our prediction of possible long-distance connections from North America to Europe, and from Europe towards Central Asia and Russia, are in accordance with previous theoretical and empirical findings [17, 19, 20, 56]. On the other hand, due to the time lag in the respective susceptibility seasons, no connection between the Northern and Southern Hemispheres is predicted by our epidemic network. For such a connection to exists, it is necessary the presence of a 'green bridge' [15] in a tropical or sub-tropical region located between the two hemispheres. This could be the case of Ethiopia, where, according to the FAO country brief [25], there exist two wheat growing seasons: Belg, from February to July and Meher, from May to December. Furthermore, according to our estimates on the host susceptibility duration (figure 2(a)), Ethiopia features some of the nodes with the longest susceptibility period (more than 90 days/year), whereas previous studies already identified this country (and more broadly East Africa and Yemen) as an important stepping stone [56] for the long distance transport of rust spores along the Rift Valley [15].

The in- and out-strength maps of the epidemic network (figure 2(c)) provide an overview of the epidemic role played by each region, sink rather than source of spores or both. Our results indicate that out-strength tends to be stronger towards the equator compared to in-strength, in line with previous findings [3] about the 'green' and 'golden' wave: regions closer to tropics are the first where infection is possible and therefore they are more likely to act as sources of spores rather than sinks. Among the recent seasonal epidemic events moving poleward, the 2013 regional outbreak of stem rust in Germany seems to have moved to Denmark and Sweden later in the season [7].

If one wanted to use our approach do derive effective measures for global crop protection it is worthy to note that in our exercise we privileged generality over local accuracy. Our assumptions to determine host availability and susceptibility are necessarily oversimplifications of a complex reality, as required by models. For instance, we neglected the presence of alternative hosts, such as spring wheat, whose cropping calendars would increase the host susceptibility period and the frequency of long distance connections. Also, we neglected the existence of the secondary hosts of P. graminis, the barberry (Berberis vulgaris), which is necessary for sexual reproduction, facilitating the emergence of new strains. It has been successfully eradicated in Western Europe and North America in the last century [7]. Yet, it is present in other regions which turns to be sources of new strains [27, 57] and it has been recently reintroduced in Europe [58].

Our model permitted to design an optimal surveillance strategy capable of covering 50% of wheat cultivated lands while monitoring only 64 sentinel nodes, i.e. less than 1% of the wheat cultivated cells. Although this proportion may seem optimistic for real applications, it is worth recalling the work of Sutrave et al [24] on P. pachyrhizi indicating that a reliable epidemiological status of soybean rust in the US could be achieved by reducing the sentinel set size from 500 to 12 network optimized nodes.

Of course, the success of the deployment of a sentinel system will also depend on the diagnostic ability of individual sentinels and on the communication between them. In this sense, efforts have recently been devoted to establish international protocols to improve the probability of detection and the timely communication of new outbreaks to a global network of scientists, public authorities and stakeholders (see the Global Cereal Rust Monitoring System [6, 59]). Our findings stress the importance of increasing efforts towards a trans-boundary perspective to efficiently contain the emergence of new virulent air-borne crop pathogens [23].