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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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AffiliationsDepartment of Physiology, Genetics, and Microbiology, University of Alicante, Alicante, SpainFrancisco Martinez-Hernandez, Inmaculada Garcia-Heredia & Manuel Martinez-GarciaDepartment of Biology, University of North Carolina at Greensboro, Greensboro, NC, USAAwa Diop & Louis-Marie BobayAuthorsFrancisco Martinez-HernandezAwa DiopInmaculada Garcia-HerediaLouis-Marie BobayManuel Martinez-GarciaCorresponding authorCorrespondence to
Manuel Martinez-Garcia. More

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

The result of this study is presented in three categories, namely; the descriptive statistics, the water quality test result and the ANN model and the model evaluation performance, respectively.The descriptive statistics result is presented in Tables 1, 2, 3, 4. This describes the basic features of the data in this study. They provide simple summaries about the sample and the measures such as the mean, median, maximum, minimum and standard deviation, respectively.Table 1 Descriptive statistics of the analyzed water quality at point 1.Full size tableTable 2 Descriptive statistics of the analyzed water quality at point 2.Full size tableTable 3 Descriptive statistics of the analyzed water quality at point 3.Full size tableTable 4 Descriptive statistics of the analyzed water quality at point 4.Full size tableThe descriptive statistics in Tables 1,2, 3, 4 shows that the mean values of the data set ranges from 6.29 to 6.34, 1956.21 to 2458.19, 3.35 to 7.39 and 39.13 to 51.06 for Ph, TDS (mg/l), EC (dS/m) and Na (mg/l), respectively. The median values of the data set ranges from 6.31 to 6.39, 2010.00 to 2439.50, 3.14 to 4.24 and 39.13 to 51.06 for pH, TDS (mg/l), EC (dS/m) and Na (mg/l), respectively. The Maximum values data set ranges from 6.48 to 6.64, 2286.00 to 2742.00, 2.21 to 5.82, and 64.50 to 88.45 for Ph, TDS (mg/l), EC (dS/m) and Na (mg/l), respectively. The minimum values dataset ranges from 6.00 to 6.09, 1367.00 to 2199.00, 2.01 to 3.18, and 21.21 to 40.24 for Ph, TDS (mg/l), EC (dS/m) and Na (mg/l), respectively. The standard deviation values ranges from 0.08 to 0.16, 114.47 to 213.04, 0.23 to 31.49 and 14.06 to 8.16 for Ph, TDS (mg/l), EC (dS/m) and Na (mg/l), respectively. The low values of standard deviation recorded in this study shows that data set were very close to the mean of the dataset.The water quality analysis test result indicates the level of concentrations of the TDS (mg/l), EC (dS/m) and Na (mg/l) in the Ele river in Nnewi, Anambra State Nigeria. The FAO standard for irrigation water quality for TDS, EC and Na are 0–2000, 0–3 and 0–40, respectively. The water quality results show that the pH values which ranges from 6.01 to 6.87 were within the FAO standard in all the points for both rainy and dry seasons, whereas the TDS (mg/l), EC (dS/m) and Na (mg/l) parametric values range from 2001 to 2506, 3.01 to 5.76, and 40.42 to 73.45 respectively, were above the FAO standard from point 1 to point 3 and falls within the FAO standard at point 4 with values ranging from 1003 to 1994, 2.01 to 2.78 and 31.24 to 39.44, respectively. However, during the dry season, the TDS, EC, and Na values range from 2002 to 2742, 3.04 to 5.82 and 40.14 to 88.45 respectively, were all above the FAO standard. Anthropogenic pollution emitted into water bodies has recently been identified as a significant source of pollutants that need immediate action in order to avoid serious environmental effects11.The results equally revealed that the concentrations decrease along the sampling points going downstream. It is noteworthy that irrigation water with a pH outside the normal range may cause a nutritional imbalance or may contain a toxic ion which is harmful to crops19. The high concentrations of TDS as observed in this study are likely to increase the salinity of the river water, change the taste of the water, and as well decrease the dissolved oxygen level of the surface water making it difficult for the survival of plants and aquatic organisms7.Moreover, these anions and cations which increase the electric conductivity in water affect irrigation adversely since salts settle at crop root zones making it difficult for infiltration, absorption of moisture and nutrients necessary for crop production.The ANN model and forecast for the water quality parameters are shown from Figs. 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19. Considering the water quality permissible range, River quality modeling and forecast shows different variations seasonally such that the pollution level during dry season was higher than the rainy season.Figure 4(A and B): pH model and forecast graph at point 1.Full size imageFigure 5(A and B): TDS model and forecast graph at point 1.Full size imageFigure 6(A and B): EC model and forecast graph at point 1.Full size imageFigure 7(A and B): Na model and Forecast graph at point 1.Full size imageFigure 8(A and B): Ph model and Forecast graph at point 2.Full size imageFigure 9(A and B): TDS model and Forecast graph at point 2.Full size imageFigure 10(A and B): EC model and Forecast graph at point 2.Full size imageFigure 11(A and B): Na model and Forecast graph at point 2.Full size imageFigure 12(A and B): Ph model and Forecast graph at point 3.Full size imageFigure 13(A and B): TDS model and Forecast graph at point 3.Full size imageFigure 14(A and B): EC model and Forecast graph at point 3.Full size imageFigure 15(A and B): Na model and Forecast graph at point 3.Full size imageFigure 16(A and B): pH model and Forecast graph at point 4.Full size imageFigure 17(A and B): TDS model and Forecast graph at point.Full size imageFigure 18(A and B): EC model and Forecast graph at point 4.Full size imageFigure 19(A and B): Na model and Forecast graph at point 4.Full size imageGenerally, the artificial neural network model the actual data set very well. At various sampling points, the developed ANN models descriptively show insignificant values in deviation for the actual data set. There were continues variations in the developed models and forecasts over time. The feed-forward Multilayer Neural Network (FFMNN) Model Performance Evaluation Results are shown in Table 5. The model performance evaluation was carried out based on the developed ANN model training, Testing and forecast, respectively. The model performance evaluation was carried out using the coefficient of multiple determination R2 and Root Mean Squared Error (RMSE).Table 5 Statistical measurement of the trained, test and forecast model.Full size tableThe R2 values were generally observed to have varied in the second decimal place for the training, testing and forecast model, respectively.The training performance evaluation shows that R2 values ranges from 0.981 to 0.990, 0.981 to 0.988, 0.981 to 0.989 and 0981 to 0.989, for pH, TDS, EC, and Na, respectively. The training results shows that the pH model have the best performance followed by EC, and Na.Also, the testing performance shows that the R2 value ranges from 0.952 to 0.967, 0.953 to 0.970, 0.951 to 0.967 and 0.953 to 0.968, for pH, TDS, EC and Na, respectively. However, the testing performance evaluation shows that TDS had the best performance. The forecast performance evaluation shows that the R2 values ranges from 0.945 to 0.968, 0.946 to 0.968, 0.944 to 0.967 and 0.949 to 0.965 for pH, TDS, EC and Na respectively. It was however discovered that the TDS made best forecast followed by the pH. The water quality forecast performance was further evaluated using the Root Mean Squared Error (RMSE) which ranges from 0.022 to 0.088, 0.012 to 0.087, 0.015 to 0.085and 0.014 to 0.084 for pH, TDS, EC and Na, respectively. The ANN model performed very well as their coefficient of multiple determinations R2 were very close 1, which is in agreement with the study of Awu et al. (2017) and Abrahart et al., (2005). On comparing the performance of the training model to the testing model and forecast, it shows that the training set performed better than the testing set followed by the forecast as its coefficient of multiple determinations, R2, was much closer to 1. More

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