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Combining agent mobility patterns and SIR modelTo take into account agent mobility19 in a scenario compatible with a SIR model, we developed the model pictorially illustrated in Fig. 1. As explained in details in the Methods Section, the agents can move on a lattice through jumps processes, modelled using a Lévy walk of jump parameter (beta)36,37,38. When (beta) becomes large, i.e., for (beta rightarrow 2), agents tend to perform a Brownian random walk with very short jumps. As (beta rightarrow 1), agents can travel long distances in just one step. There are no constraints on the number of agents that can occupy a single cell. In each cells, agents can be infected by neighbours according to the SIR rules. Thus, the parameters that control the model are the jump parameter (beta) plus the standard SIR parameters, infection rate (alpha) and removal rate (gamma). The agent-based lattice model considered here reduces to a standard SIR model when the well-mixed population condition is satisfied, i. e. when large jumps dominate the dynamics (Fig. 2).Figure 1Agent-based SIR model on a lattice. (a) Agents of different colors, representing the SIR states, move on a lattice. White cells represent empty sites. Green cells are occupied by susceptible (S) agents, blue cells contain only removed (R) agents. Red cells contain only infected (I) agents. Shaded cells contain agents in a mixture of states. Agents can move among cells performing jumps (black arrows) whose length follows Lévy statistics. The letters i and j, with (i=1,..,N_b) and (j=1,…,N_b) define the location of the cell (i, j). (b,c) Agents in the same cell undergo a SIR dynamics: (b) S become I at a rate (alpha); (c) I become R at rate (gamma). (d) The jump dynamics allows an agent to move from the cell (i, j) to ((i+k,j+l)). The probability to perform a large/small jump is controlled by the parameter (beta in [1.0,1.99]). Large (beta) values correspond to small jumps, i. e., a random walk that gives rise to Brownian motion. Small (beta) values correspond to large jumps.Full size imageFor reproducing the kinetics of real data we made the following assumptions:

In the absence of containing strategies, the infection is characterized by a high infection rate (we take (alpha =0.9)) and a low removal rate ((gamma =0.025) or 0.05). Using as a unit of time the update of all agent positions (see Methods for details), the removal rate introduce a time scale (tau _I = gamma ^{-1}=40) or (20). This characteristic time scale represents the average time an agent remains infected and can thus spread the infection. This condition ensures that we are in an epidemic regime, i. e., the mean-field value is (R_t gg 1). We stress that, since the SIR dynamics with only three sub-populations is a simplification of the real chain of epidemic transmission, the parameters we choose for the epidemic spreading are not strictly related to those of Covid-19. Because we are interested in the effect of mobility restriction on epidemic spreading, we fix the epidemic parameters in a way that, without mobility restrictions, we are sure to stay in the worst-case scenario with an exponentially fast spreading of the infection.

The parameter (beta in [1,1.99]) tunes the intensity of mobility restrictions. The higher its value, the stricter the limitations. (beta) is one of the fitting parameters.

Other interventions that mitigate the epidemic spreading tend to increase the removal rate (gamma). We thus assume that (gamma) is another fitting parameter. This is because typical measures, for instance, quarantine, remove infected agents from the system. In this way, we reabsorb the presence of many hidden sub-populations into an effective value of (gamma).

We define the parameter (delta), i. e., the fraction of infected agents at the epidemic peak with respect to the entire population, that provides a quantitative measure of the reduction of the epidemic peak. In other words, the parameter (delta) represents the efficiency of a given containing strategy compared to the uncontrolled situation where all the agents turn out to contract the infection (which is the case of our model for (gamma ll alpha), (alpha =0.9), and (beta =1)).

To detail how mobility restrictions induce deviations from the SIR model, we calculate, via numerical simulations, the epidemic curves as a function of time for different values of (beta) as illustrated in Fig. 2a. Here, the SIR parameters are (alpha =0.9) and (gamma =0.025), i. e., the corresponding SIR model is in the fully blown epidemic regime. For small (beta) the epidemic growth is well captured by the exponential function, indicating that we are in the epidemic regime. As (beta) increases the curve turns out to be flattened and the peak reduces to (80%). Moreover, the growth of the epidemic for the largest (beta) examined is well described by the power law (I(t) sim t^{2}). The value of the exponent is comparable with those measured in different countries during the COVID(-19) epidemic wave23. The model considered here suggests that the crossover from exponential growth to power-law might be related to changes of the mobility patterns that, in our picture, shift from being dominated by large jumps to small ones. This finding is consistent with the observation that a sub-exponential growth in the number of infected people is a consequence of containing strategies23. Moreover, in the microscopic description adopted here, the crossover in the kinetics of I(t) is driven by just one parameter.Figure 2Agent dynamics impacts the epidemic spreading process. (a) The graph shows the dependency of the epidemic curves on (beta =1.20,1.50,1.75,1.80,1.85,1.87,1.90,1.92,1.95,1.97,1.99) (increasing values of (beta) from yellow to violet). As (beta) decreases, the epidemic grows exponentially fast (dotted black curve) and approaches the evolution of SIR model in well-mixed population (dashed red curve). The dash-dot blue curve is a power law (sim t^2). The parameters of the SIR reactions are (alpha =0.9) and (gamma =0.025). (b–g) Typical configurations taken at the same fraction of infected agents (I/N sim 0.25) for increasing values of (beta =1.0,1.2,1.4,1.6,1.8,1.9) (red are infected sites, green the susceptible ones, we keep white the sites populated by removed agents). (h) The probability distribution function of the local density of infected sites. (i) Radius of the cluster of infected agents ((beta =1.99)) as a function of time. The red dashed line is a linear fit.Full size imageThe crossover from exponential to power-law growth reflects the drastic change in the structure of clusters of infected agents, as illustrated in Fig. 2b–g, where typical configurations with the same fraction of infected agents are shown ((I/N=0.25, alpha =0.9, gamma =0.025)). As one can see, in the high mobility region ((beta = 1)), infected agents are spread almost everywhere in the system. As (beta) increases, infected sites tend to form a single cluster. This phenomenology is consistent with the literature of mobile agents undergoing SIR dynamics39,40. This structural change is quantitatively documented by the density distribution of infected sites shown in panel (h) of the same figure (see section Methods for details). As one can appreciate, the distribution becomes double-peaked as (beta) increases. The first peak around zero indicates the presence of an extended region of susceptible agents. The peak at high values is due to the growing cluster of infected agents. As highlighted in panel (i), the cluster grows linearly in time and thus the number of infected grows with (t^2).Another interesting aspect to understand with this model is the trade off between mobility restrictions and and other kind of interventions that have the effect of increasing the removal rate. In particular in Asian countries41, NPIs applied during the COVID-19 waves have relied mostly on contact tracing and/or preventive quarantine, with little mobility reduction, leading to effective and durable control of epidemic spreading, as reviewed by Ref.21. To understand if there is an optimal balance between containing strategies (characterized by (beta)) and efficiency in removing infected agents (denoted by (gamma)), we calculate the fraction of infected population at the epidemic peak (the maximum of I(t)) as a function of the jump parameter (beta) and of the removal rate (gamma). As above, the initial occupation number of each site is, on average, one. The infection rate is (alpha =0.9). The resulting phase diagram is shown in Fig. 3. The color indicates the fraction of infected population: in the violet region, this fraction goes to zero (epidemic is suppressed) while in the yellow region such a value goes to one, indicating an epidemic regime. The phase diagram fully recapitulates the effectiveness of the two strategies used to mitigate the infection spread, a strong lockdown with limited contact tracing, or an efficient contact tracing a moderate reduction of the mobility.Figure 3Effect of different containment strategies. The phase diagram is obtained considering as control parameters (beta), that represents mobility restrictions, and (gamma), the efficiency in removing infected agents. The color scale represents the fraction of the initial susceptible population that becomes infected, ranging between 0 (epidemic suppression, violet region) and 1 (fully-blown epidemic, yellow region). Containment is achieved as (beta) increases (corresponding to increasing mobility restrictions) even with low removal rate, or increasing (gamma) (effective removal of infected agents), even with limited mobility restrictions.Full size imageHowever, even under the strictest lockdown, several activities could not be stopped (hospitals, food supply chain, …), meaning that a single mobility parameter cannot fully describe this varied situation. To understand what could be the impact of heterogeneous motility patterns on the evolution of the epidemic, we introduce in the model some regions characterized by a high mobility (jump parameter, (beta _2)), while the majority of the the cells have restricted mobility, with a jump parameter (beta _1=1.99) (see Methods for more details). By varying (beta _2) and the density of more mobile cells (parameter (rho)) we are able to draw the phase diagram shown in Fig. 4.Figure 4Sites of different mobility affect epidemic spreading. (a) Each cell labelled by (i, j) is characterized by its own mobility parameter (beta _{ij}). We consider the special case of a binary mixture ((beta _{ij} = beta _{1,2})) of high and low mobility regions. Changing the density (rho) of (beta _2) sites and the value of (beta _2), we obtain the the phase diagram presented in panel (b), obtained for (beta _1=1.99), (alpha =0.9), and (gamma =0.05), conditions that grant contained epidemic spreading thanks to the low-mobility group. A small amount of sites with small values of (beta _2) can trigger the epidemic spreading.Full size imageAs in the previous case, in the violet area the epidemic spreading is stopped, while in the yellow area the epidemic peak reaches the entire population. Epidemic spreading takes place above a critical curve: for a given value of mobility (beta _2 More

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SettingThis study was conducted in Cantabria, a region of the North coast of Spain.Design and patientsA cross-sectional study with prospective data collection was performed at the Allergy Services of the Marqués de Valdecilla University Hospital in Santander and the Sierrallana Hospital in Torrelavega (Cantabria, Spain).98 patients diagnosed of rhinoconjunctivitis, asthma or both, caused by sensitization to grass pollen, were included in a sequential way from October 2015 to March 2016.Written informed consent was obtained from all patients before entering the study. The study met the principles of the 1975 Helsinki declaration and was reviewed and approved by the local Research Committee of Cantabria (CEIC reference number 2015.207).A serum sample was obtained from each patient and stored at – 20 °C until used.Pollen extract preparationAll methods were performed in accordance with the relevant guidelines and regulations.Cortaderia selloana (CS) pollen was obtained commercially (Iber-Polen, Jaén, Spain) and then extracted at a 1:10 (w/v) ratio in PBS pH 6.5 with magnetic stirring for 90 min. at 5 °C. The soluble fraction was separated by centrifugation. After dialysis against PBS, the extract was filtered through 0, 22 µm filters. Protein content was determined by Bradford method (BioRad, Hercules, CA, USA). Two different batches were obtained (07 and 09) with consistent results.Part of the extract was adjusted to 0.25 mg protein/ml and formulated in PBS with 50% glycerol, phenol 0.51% (SPT buffer). The remaining extract was stored in aliquots at − 20 °C.Phleum pratense (Phl) pollen extract was made as described for CS. The origin of the pollen in this case was ALK Source Materials, Post Falls, Idaho, USA.The protein profiles of the CS or the Phl extracts were determined by polyacrylamide electrophoresis in the presence of sodium dodecyl sulphate (SDS-PAGE) under reducing conditions (Invitrogen-Novex tricine gels 10–20% acrylamide, Fisher Scientific, SL, Madrid Spain).Skin prick testPatients were skin prick tested (SPT) with a commercial extract (ALK-Abelló, S.A. Madrid, Spain) of Phl and the CS extract. Histamine dihydrochloride solution (10 mg/ml) and SPT buffer were used as positive and negative control (no reaction), respectively.The SPT wheal areas were measured by planimetry. A cut-off area of 7 mm2 (about 3 mm average diameter) or higher was considered a positive test result (histamine).The CS extract was tested in 10 control subjects, that were not sensitised to grass pollen, with negative result (no reaction).IgE assaysSerum samples were tested for IgE antibodies against Phleum pratense (Phl) pollen extract and the allergens Phl p 1, Phl p 5, Phl p 7 (polcalcin) and Phl p 12 (profilin) (ImmunoCap FEIA, Thermo Fisher Scientific, Barcelona, Spain).In addition, specific IgE against Phl and CS pollen extracts was determined by RAST (Radio Allergo Sorbent Test). Paper discs were activated with CNBr and sensitised with the pollen extracts as described by Ceska et al.21. Phl and CS discs were incubated overnight with 50 µL of the patient’s serum and after washing (0.1% Tween-20 in PBS), with approximately 100,000 cpm of the iodine 125–labeled anti-IgE mAb HE-2 for 3 h as described22. Finally, the discs were washed, and their radioactivity was determined in a gamma counter. sIgE values in kilounits per litre were determined by interpolating in a standard curve built up with Lolium perenne—sensitised discs and 4 dilutions of a serum pool from patients with grass allergy, which was previously calibrated in arbitrary kU/l.A cut-off value of 0.35 kU/l was considered positive for both ImmunoCap and RAST. There was a very significant correlation between the sIgE against Phl determined by both methods (r Spearman = 0.8874, p  More