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NEWS AND VIEWS
15 September 2021

A spotlight on seafood for global human nutrition

What role might seafood have in boosting human health in diets of the future? A modelling study that assesses how a rise in seafood intake by 2030 might affect human populations worldwide offers a way to begin to answer this.

Lotte Lauritzen

 ORCID: http://orcid.org/0000-0001-7184-5949

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Lotte Lauritzen

Lotte Lauritzen is in the Department of Nutrition, Exercise and Sports, University of Copenhagen, 1958 Frederiksberg C, Denmark.

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An adequate and sustainable supply and intake of nutritious food is essential to tackle major global health issues such as dietary deficiencies. Seafood, which in this context includes fish, shellfish and marine mammals, is rich in micronutrients (such as vitamin A, iron, vitamin B12 and calcium) needed to combat the most common such deficiencies. Seafood is also the dominant source of marine omega-3 fatty acids, which have many health-promoting effects. Writing in Nature, Golden et al.1 present ambitious research that puts seafood centre stage.

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doi: https://doi.org/10.1038/d41586-021-02436-3

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We first identify the location of organic crop fields in Kern County and then estimate whether status as organic versus conventional fields determines pesticide use (Fig. 5).Fig. 5: Methodology overview.Figure outlines the main method steps from identifying organic fields to creating the analysis data to performing the statistical analyses. All images shown are simplified, visual representations of the datasets. CDFA refers to the California Department of Food and Agriculture, while APN is the Assessor’s Parcel Number and TRS is the Township-Range-Section. Identifying organic fields combines the created CDFA organic APN, CDFA organic TRS, and organic pesticides data layers together to create the final organic versus conventional fields layer used in the analysis data section. All analysis data layers are then inputted into the various statistical analyses.Full size imageIdentifying organic fieldsWe identified organic fields using a combination of California Department of Food and Agriculture (CDFA) records and Kern County Agricultural Commissioner’s Office spatial data (“fields shapefiles”) and pesticide use records. No single source was complete, and as such, we evaluated several different approaches to identifying organic fields.California Department of Food and Agriculture (CDFA) recordsData on the location of organic fields, per the California State Organic Program, for 2013–2019 was obtained by request from the California Department of Food and Agriculture (CDFA). The CDFA, through the State Organic Program, requires annual registration of certified organic producers who have an expected gross sale of over $5000. We were specifically interested in the pesticide aspects of organic production, which is governed in our study region by the USDA’s National List of Allowed and Prohibited Substances. The National List of Allowed and Prohibited Substances delineates which synthetic substances can be used and which natural substances cannot be used for pest control in US organic production. Besides substances specifically (dis)allowed on the National List, allowed substances include non-synthetic biological, botanical, and mineral inputs. Field location data were in the form of either Assessor’s Parcel Number (APN) or PLS System Township-Range-Section (TRS) values, though data were reported without systematic formatting. We harmonized the CDFA APN values to merge with the Kern County Assessor’s parcel shapefile (2017), which we then spatially joined with the Kern fields shapefiles. We followed a similar process with PLSS TRS values, which were then merged with the Kern County PLS Section shapefile, and joined to Kern field shapefiles. We refer to our final organic designation as “CDFA Organic”. Details of the data cleaning process are described in the Ancillary Data Processing Methods section below.Using pesticide use reports to refine organic field identificationAfter spot-checking pesticide use on CDFA Organic fields, it became clear we had not entirely eliminated conventional fields. This was likely due to variation in polygon geometries between PLSS Sections, Kern County Assessor parcels, and Kern agricultural fields data. To further refine our classification, we used field-level pesticide use, again from the Kern County Agricultural Commissioner’s Office. As thousands of pesticide products (active ingredients + adjuvants) are in use in Kern County, we took an iterative approach to eliminate fields using conventional pesticides. We first limited the universe of pesticides to those applied to fields that were CDFA Organic. We then identified the 50 most commonly used pesticide products by a number of applications, and manually classified each as organic or conventional. Having identified these products as described below, we matched them back in, eliminating fields that used conventional products and identifying as “PUR Organic” those that used only organic products. We repeated this process, hand identifying the most commonly used products and eliminating fields using conventional products until we had isolated fields using only organic products.To classify a product as organic or conventional, we first searched for each product’s U.S. EPA-registered product label, using the exact product name and EPA product registration number. If there was any indication on the label that the product was certified as organic by the Organic Materials Review Institute (OMRI), or said “for use in organic production” or “organic”, then the pesticide was identified as organic (n = 132). If there was no organic indication on the product label, we searched the OMRI certification database for products with identical names and manufacturers, and identified products as organic if such certifications existed (n = 39). If all ingredients were defined (i.e., no inert or undefined ingredients) and were known organic active ingredients, then the pesticide was identified as organic (n = 1) (Supplementary Data 1). We failed to find EPA-registered labels for three products and confirmed on the California Department of Pesticide Regulation website that they are either inactive or out of production (EPA registration numbers: 52467-50008-AA-5905, 36208-50020-AA, 2935-48-AA-120). Each of the three was rarely used (n  0) to be the same as the mechanisms determining the amount sprayed when some pesticides are used (pesticides when pesticides  > 0). Double-hurdle models64 are an alternative to the Tobit model that allows for the separation of these two decisions.The mechanisms underlying the two decisions (to spray, how much to spray if spraying) can differ such that different covariates can describe each process, and the same covariates are allowed to influence the two processes in different ways (i.e., sign and magnitude can differ). The first, binary decision is usually modeled with a probit model.$${{{{{rm{P}}}}}}left(y=0|{{{{{bf{x}}}}}}right)=1-Phi left({{{{{bf{x}}}}}}gammaright)$$
(1)
Then, the second decision is modeled as a linear model with pesticide use following a lognormal distribution, conditional on positive pesticide use64$$log (y)|{{{{{bf{x}}}}}},y , > , 0 sim {{{{{rm{Normal}}}}}}({{{{{bf{x}}}}}}{{{{{mathbf{upbeta }}}}}},{sigma }^{2})$$
(2)
where Φ is the standard normal cdf, x is a vector of explanatory variables including organic status, y is pesticide use, and ({{{{{mathbf{upbeta }}}}}}) is a vector of coefficients. We use a lognormal hurdle model rather than a truncated normal hurdle model since pesticide use is highly non-normal, and Q-Q plots suggested substantial model improvement using a lognormal rather than normal distribution. In contrast to the panel data models described in the Ancillary Statistical Methods below, our estimation equation used natural log-transformed variables for pesticides (and field and farm size) rather than inverse hyperbolic sine (IHS) transformation since only positive observations are included in the second hurdle model. Following insights derived from our panel data models (Supplementary Notes), we build on the basic hurdle model concept using the farm-by-crop family interaction as a random intercept in both the first and second hurdle. We chose the farm-by-crop family interaction rather than a crossed random effect due to computational feasibility with thousands of permits and hundreds of crops, due to similarity of results to the within estimator model (i.e., fixed effects in causal inference terminology; Supplementary Notes, Supplementary Table 2), and due to AIC/BIC (Supplementary Table 3). Further, we find evidence of heteroskedasticity from both visual inspection and Levine’s test, which adds additional complications to computing crossed random effects. Thus, we proceed with the farm-by-crop family interaction in a random intercept model with cluster robust standard errors clustered at the same grouping. In doing so, observations, where the taxonomic family of the crop was unclear, were dropped. Of the 7367 fields that were dropped due to missing crop families, 6684 were uncultivated agriculture.Our data are effectively repeated cross-sections rather than a true panel since fields are defined by the farm-site-year combination and thus generally change year-to-year or when crops rotate. We model it as such. This implies we do not require observations to have no spray in all time periods, as would be the case in a double hurdle panel model. Linking field IDs over time through spatial processing is complicated by crop rotations of different size areas. Since farmers may farm multiple fields under different management systems, as we illustrate here, and have different contractual obligations at a sub-farm level, requiring farms to never use pesticides on all fields is not reflective of on-the-ground decisions.We repeated the lognormal hurdle models individually for carrots, grapes, oranges, potatoes, and onions, which were the only widely-grown crops with more than 100 organic fields. This allowed for a different slope and intercept by crop type.We conduct several robustness checks. First, we do not have data on crop yields. However, to assess the potential implications of a yield gap on our results, we modify our per hectare rates following Ponisio et al.15 as a robustness check. We group commodities into cereals, roots and tubers, oilseeds, legumes/pulses, fruits, and vegetables and assign them the Ponisio et al.15 yield gap estimates for that group. Crops that did not fall into any of the above groups (e.g., cannabis) were provided the all-crop average from Ponisio et al.15. Second, we analyze how conventional and organic differ with respect to soil quality using a within estimator approach to account for crop-specific differences in soil quality. Third, binary toxicity metrics, while valuable given the number of chemicals and endpoints of interest here, nevertheless fail to distinguish gradations of toxicity for chemicals above (or below) the regulatory threshold. As mentioned above, the data needed to calculate many aggregate indices (e.g., Pesticide Load57 and Environmental Impact Quotient58) are not readily available for all of the chemicals in our study. For completeness, we attempted to calculate the Pesticide Toxicity Index for one well-studied endpoint, fish. We supplemented data provided in Nowell et al.41 with data from Standartox42. However, only about 70% of the chemicals used in our study matched, and pesticide products used on organic fields were more likely to lack toxicity information for one or more chemicals. We briefly discuss the highly preliminary investigation, given the non-random missing toxicity data.All spatial analyses were performed in R Statistical Software v 3.5.3 and all statistical analyses were performed Stata 16 MP. For all tests, statistical significance was based on two-tailed tests with (alpha =0.05.)Ancillary data processing methodsCleaning parcel dataTo spatially locate organic fields, we needed to match the Assessor’s parcel numbers (APNs) provided in the CDFA tabular data to APNs in the Kern County Parcel shapefile (from 2017). Over 90% of the APN entries in the CDFA data were in the format [xxx-xxx-xx], though multiple APNs were often provided in the same cell separated by line breaks, semi-colons, commas, and/or spaces. We made initial edits separating values into individual cells in Microsoft Excel since formatting was highly inconsistent. Observations whose APNs were not in the [xxx-xxx-xx] were modified so that their format matched. In the R environment, dashes were inserted after the third, sixth, and eighth characters (1234567895 became 123-456-78-95) for APNs that did not already contain them. Occasionally, APN numbers were provided with dashes, but with segments of incorrect length (e.g., 12-34-567). In these instances, APN segments were either trimmed from the right or padded with a zero on the left so they matched the [xxx-xxx-xx] format. This approach yielded the greatest number of matches and was checked for accuracy as described below. Additional segments (from APNs with more than two dashes and eight numeric characters) were dropped. A handful of APNs with fewer than eight numeric characters and no dashes were dropped entirely.The edited CDFA APNs were then joined with the Kern County Assessor’s parcel shapefile, creating the “CDFA organic shapefile”. In total, 1637 of 1829 individual CDFA records joined successfully. To evaluate the accuracy of joins between CDFA tabular data, Kern County parcel, and Kern County agricultural spatial data, we spot-checked ownership information using “Company” (CDFA) and “PERMITTEE” (Kern County agricultural data) values.To then identify the crop fields within the organic parcels, we performed a spatial join between the CDFA organic shapefile and the Kern County fields shapefiles. Prior to performing the join, the CDFA parcels’ dimensions were reduced with a 50-m buffer to eliminate spatial joins between CDFA parcels and crop fields that were only touching the parcel margins. Of five different buffer widths evaluated, 50 m reduced the number of false positives and negatives, as determined by comparing the “Company” and “PERMITTEE” values. We refer to the fields that match as “APN Organic”.Cleaning PLSS Township-Range-Section valuesEach year several producers reported Township, Section, and Range (TRS) values, consistent with the PLS System (PLSS), rather than APN values. We used these TRS values to identify PLSS Sections that contained organic fields.We separated any cell containing multiple TRS values and removed any prefixes such as “S”, “Section”, “Sec.”, “T”, and “R” that would prevent joining to Kern County PLSS spatial data in Excel. In the R environment, we padded the left side of the “S” value with a 0 if it was a single digit, then concatenated the three columns into a “TRS” column. We joined TRS from the CDFA tabular data to PLSS spatial data, which identified 563 Sections as containing organic fields, from 2013 to 2019, out of a total of 664 unique TRS codes in the CDFA dataset. We then performed a spatial join between PLSS Sections that contain organic fields and Kern County fields shapefiles, to identify all agriculture fields that overlap with those Sections. Additional processing using the Pesticide Use Reports is described above.Ancillary statistical methodsWe began with a pooled ordinary least squares (OLS) model that, as the name suggests, pools observations over farms, years, and crop types. However, there may be attributes of crops or farms that may be systematically different between organic and conventional, and this systematic difference could bias our pooled OLS results. To address this, we first considered propensity score approaches but were unable to find a sufficient balance of our covariate distribution between organic and conventional fields. As an alternative, we limited our sample to fields with overlapping farmers and crop types. In other words, we focused on the subset of fields that are grown by farmers producing both organic and conventional fields and to crops that are produced both conventionally and organically. However, this shrunk our dataset by two-thirds.To leverage more of our data, we next considered panel data models as a means to address unobserved variables. We consider both within-estimator models (also known as “fixed effects” in causal inference terminology, but different from the biostatistical use of the term) and random effects models (with random intercepts), seeking to capture characteristics of the crop, grower, and year. The advantage of a within-estimator approach is that the omitted variables are removed (through differencing) and thus, they can be correlated with covariates without biasing the estimation. In other words, pesticide use and all covariates are differenced from their crop-specific mean (or crop family, farmer, etc. specific mean, depending on the model). In doing so, the propensity for certain crops (crop family, farmer) to be grown organic or to be fast or slow adopters of new technologies is removed. The disadvantage is that characteristics shared by all fields of a crop (e.g., value) are lost in the differencing, and more importantly, that the differencing is not easily translated to nonlinear models that we employ later in the analysis. Random effects are more easily translated to nonlinear models. The disadvantage of random effects is the strong assumption that the unobserved variables are uncorrelated with the covariates18,65, which is required for random effects coefficient estimates to be unbiased. Here, we see the difference in coefficient estimates between the within-estimator and random effects models are quite small (Supplementary Table 2).Random effects particularly crossed random effects with thousands of permits and hundreds of crops, introduce computational challenges due to large, sparse matrices. Further, we find evidence of heteroskedasticity from both visual inspection and Levine’s test, which adds additional complications to computing crossed random effects. We proceed using the farm-by-crop family interaction in a random intercept model with cluster robust standard errors clustered at the same grouping based on AIC/BIC (Supplementary Table 3), computational feasibility, and similarity to the within-estimator results (Supplementary Table 2). Observations, where the taxonomic family of the crop was unclear, were dropped in any models including family in either the random effects or the cluster robust standard errors. Of the 7367 fields that were dropped due to missing crop families, 6684 were uncultivated agriculture.In the panel data models, we used IHS transformations to accommodate highly non-normal pesticide (and field and farm size) data. IHS is very similar to natural log transformation66 but is defined at zero, which is important given a sizable fraction of our observations have zero pesticide use. As with log–log transformations, IHS–IHS transformation can be interpreted as elasticities. We pre-multiply pesticide use by 100 to improve estimation66, though this does not affect interpretation. As described above, we leverage insights on model specification from the panel data models, but rely on the double hurdle models to parse apart the decision to spray from the decision of how much to spray.Reporting summaryFurther information on research design is available in the Nature Research Reporting Summary linked to this article. More

The absence of the first principles for biological systems in general, and in particular for vegetation populations where phenomena are interconnected makes their mathematical modelling complex. The theory of vegetation pattern formation rests on the self-organisation hypothesis and symmetry-breaking instability that provoke the fragmentation of the uniform cover. The symmetry-breaking instability takes place even if the environment is isotropic31,33,35. This instability may be an advection-induced transition that requires the pre-existence of the environment anisotropy due to the topography of the landscape34,39,40. Generally speaking, this transition requires at least two feedback mechanisms having a short-range activation and a long-range inhibition. In this respect, we consider three different vegetation models that are experimentally relevant systems: (i) the generic interaction redistribution model describing vegetation pattern formation which incorporates explicitly the facilitation, competition and seed dispersion nonlocal interactions (ii) the local nonvariational partial differential model described by a nonvariational Swift–Hohenberg type of model equation, and (iii) the reaction–diffusion system that incorporate explicetely water transport.The interaction-redistribution approachThe integrodifferential modelThis approach consists of considering a well-known logistic equation with nonlocal plant-to-plant interactions. Three types of interactions are considered: the facilitative (M_{f}(mathbf {r},t)), the competitive (M_{c}(mathbf {r},t)), and the seed dispersion (M_{d}(mathbf {r},t)) nonlocal interactions. To simplify further the mathematical modelling, we consider that the seed dispersion obeys a diffusive process (M_{d}(mathbf {r},t)approx nabla ^{2}b(mathbf {r},t)), with D the diffusion coefficient, b the biomass density, and (nabla ^{2}=partial ^2/partial x^2+partial ^2/partial y^2) is the Laplace operator acting in the (x,y) plane. The interaction-redistribution reads$$begin{aligned} M_{i}=expleft{ frac{xi _{i}}{N_{i}}int b(mathbf {r}+mathbf {r}’,t)phi _i(r,t)dmathbf {r}’right} , { text{ with } } phi _i(r,t)= exp(-r/L_{i}) end{aligned}$$
(1)
where (i=f,c). (xi _i) represents the strength of the interaction, (N_i) is a normalisation constant. We assume that their Kernels (phi _i(r,t)) are exponential functions with (L_i) the range of their interactions. The facilitative interaction (M_{f}(mathbf {r},t)) favouring vegetation development. They involve the accumulation of nutrients in the neighbourhood of plants, the reciprocal sheltering of neighbouring plants against climatic harshness which improves the water budget in the soil. The range of the facilitative interaction (L_f) operates on the crown size. The competitive interaction operates over a length (L_c) and involves the below-ground structures, i.e., the rhizosphere. In nutrient-poor or/and in water-limited territories, lateral spreading may extend beyond the radius of the crown. This extension of roots relative to their crown size is necessary for the survival and the development of the plant in order to extract enough nutrients and/or water from the soil. When incorporating these nonlocal interactions in the paradigmatic logistic equation, the spatiotemporal evolution of the normalised biomass density (b(mathbf {r}, t)) in isotropic environmental conditions reads14$$begin{aligned} partial _{t} b(mathbf {r},t)=b(mathbf {r},t)[1-b(mathbf {r},t)]M_{f}(mathbf {r},t)- mu b(mathbf {r},t)M_{c}(mathbf {r},t)+Dnabla ^{2}b(mathbf {r},t). end{aligned}$$
(2)
The normalisation is performed with respect to the total amount of biomass supported by the system. The first two terms in the logistic equation with nonlocal interaction Eq. (2) describe the biomass gains and losses, respectively. The third term models seed dispersion. The aridity parameter (mu) accounts for the biomass loss and gain ratio, which depends on water availability and nutrients soil distribution, topography, etc. The homogeneous cover solutions of Eq. (2) are: (b_{o}=0) which corresponds to the state totally devoid of vegetation, and the homogeneous cover solutions satisfy the equation$$begin{aligned} mu =(1-b)exp (Delta b), end{aligned}$$
(3)
with (Delta =xi _{f}-xi _{c}) measures the community cooperativity if (Delta >0) or anti-cooperativity when (Delta 0). The solution (u_{-}) is always unstable even in the presence of small spatial fluctuations. The linear stability analysis of vegetated cover ((u_{+})) with respect to small spatial fluctuations, yields the dispersion relation$$begin{aligned} sigma (k)=u_{+}(kappa -2u_{+})-(nu -gamma u_{+})k^{2}-alpha u_{+}k^{4}. end{aligned}$$
(8)
Imposing (partial sigma /partial k|_{k_{c}}=0) and (sigma (k_{c})=0), the critical mode can be determined$$begin{aligned} k_{c}=sqrt{frac{gamma -nu /u_{c}}{2alpha }}, end{aligned}$$
(9)
where (u_{c}) satisfies (4alpha u_{c}^2(2u_{c}-kappa )=(2gamma u_{c}-nu )^2). The corresponding aridity parameter (eta _{c}) can be calculated from Eq. (7).The reaction–diffusion approachThe second approach explicitly adds the water transport by below ground diffusion. The coupling between the water dynamics and the plant biomass involves positive feedbacks that tend to enhance water availability. Negative feedbacks allow for an increase in water consumption caused by vegetation growth, which inhibits further biomass growth.The modelling considers the coupled evolution of biomass density (b(mathbf {r},t)) and groundwater density (w(mathbf {r},t)). In its dimensionless form, this model reads33$$begin{aligned} frac{partial b}{partial t}= & {} frac{gamma w}{1+omega w}b-b^{2}-theta b+nabla ^{2}b, end{aligned}$$
(10)
$$begin{aligned} frac{partial w}{partial t}= & {} p-(1-rho b)w-w^{2}b+delta nabla ^{2}(w-beta b). end{aligned}$$
(11)
The first term in the first equation describes plant growth at a constant rate ((gamma /omega)) that grows linearly with w for dry soil. The quadratic nonlinearity (-b^{2}) accounts for saturation imposed by poor nutrients soil. The term proportional to (theta) accounts for mortality, grazing or herbivores. The mechanisms of dispersion are modelled by a simple diffusion process. The groundwater evolves due to a precipitation input p. The term ((1-rho b)w) in the second equation accounts for the evaporation and drainage, that decreases with the presence of vegetation. The term (w^{2}b) models the water uptake by the plants due to the transpiration process. The groundwater movement follows the Darcy’s law in unsaturated conditions; that is, the water flux is proportional to the gradient of the water matric potential41. The matric potential is equal to w, under the assumption that the hydraulic diffusivity is constant41. To model the suction of water by the roots, a correction to the matric potential is included; (-beta b), where (beta) is the strength of the suction. More

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The Thau lagoon dataThe measurement campaign concerned four wastewater treatment plants (WWTP) in the Thau lagoon area in France, serving the cities of Sète, Pradel-Marseillan, Frontignan and Mèze. The measurements were obtained by using digital PCR20 (dPCR) to estimate the concentration of SARS-CoV-2 virus in samples taken weekly from 2020-05-12 to 2021-01-12. We provide further details about the measurement method in the “Methods” section.Figure 1Concentrations of SARS-CoV-2 (genome units per litre in logarithmic scale) from four WWTPs in Thau lagoon, measured weekly with dPCR technology from May 12th 2020 to January 12th, 2021. Note that there are some missing points.Full size imageFigure 1 shows the outcomes in a logarithmic scale over a nine months period. We summarise now their main features.

1.

An exponential phase starts simultaneously in Mèze and Frontignan WWTPs in early June.

2.

The genome units concentration curves in these two places reach, again simultaneously, a plateau. It has stayed essentially stable or slightly decreasing since then.

3.

The evolution at Sète and Pradel-Marseillan remarkably followed the previous two zones in a parallel way, with a two weeks lag. The measurements at Sète and Pradel-Marseillan continued to grow linearly (recall that this is in log scale, thus exponentially in linear scale), while the Mèze and Frontignan figures have stabilised ; then, after two weeks, they too stabilised at a plateau with roughly the same value as for the other two towns.

4.

The measurements seem to show a tendency to increase over the very last period.

The epidemiological model with heterogeneity and natural variability of population behaviourThe appearance of such plateaus and shoulders need not be explained either by psychological reactions or by public health policy effects. Indeed, the regulations were roughly constant during the measurement campaign and awareness or fatigue effects do not seem to have altered the dynamics over this long period of time. Witness to this is the fact that two groups of towns saw the same evolution, but two weeks apart one from the other. To understand this phenomena we propose a new model.Given the complexity and multiplicity of behavioural factors favouring the spread of the epidemic, we assume that the transmission rate involves a normalised variable (a in (0,1)) that defines an aggregated indicator of risky behaviour within the susceptible population. Thus, we represent the heterogeneity of individual behaviours with this variable. We take a as an implicit parameter that we do not seek to calculate. The classical SIR model is macroscopic and the type of model we discuss here can be viewed as intermediate between macroscopic and microscopic.The initial distribution of susceptible individuals (S_0(a)) in the framework of a SIR-type compartmental description of the epidemic can be reasonably taken as a decreasing function of a. We take the infection transmission rate (a mapsto beta (a)) to be an increasing function of a. In the Supplementary Information (SI) Appendix, the reader will find a more general version of this model involving a probability kernel of transition from one state to another. The model here can be derived as a limiting case of that more general version.Likewise, the behaviour of individuals usually changes from one day to another21. Many factors are at work in this variability: social imitation, public health campaigns, opportunities, outings, the normal variations of activity (e.g. work from home certain days and use of public transportation and work in office on others) etc. Therefore, the second key feature of our model is variability of such behaviours: variations of the population density for a given a do not only come from individuals becoming infected and leaving that compartment but also results from individuals moving from one state a to another21. In the simplest version of the model, variability is introduced as a diffusion term in the dynamics of susceptible individuals.The modelWe denote by S(t, a) the density of individuals at time t associated with risk parameter a, by I(t) the total number of infected, and by R(t) the number of removed individuals. We are then led to the following system:$$begin{aligned} frac{{partial S(t,a)}}{{partial t}} & = d{mkern 1mu} frac{{partial ^{2} S(t,a)}}{{partial a^{2} }} – beta (a)S(t,a)frac{{I(t)}}{N} \ frac{{{text{d}}I(t)}}{{{text{d}}t}} & = frac{{I(t)}}{N}{mkern 1mu} intlimits_{0}^{1} beta (a)S(t,a);da – gamma I(t), \ frac{{{text{d}}R(t)}}{{{text{d}}t}} = & gamma I(t), \ end{aligned}$$
(1)
where (gamma) denotes the inverse of typical duration (in days) of the disease and d a positive diffusion coefficient. System (1) is supplemented with initial conditions$$begin{aligned} S(0,a) = S_0(a), quad I(0) = I_0, quad hbox {and} quad R(0) = 0, end{aligned}$$
(2)
and with zero flux condition in a at (a=0, 1). In the “Methods” section below, we discuss the relation of this model with other current works.A more general modelIn a more general version of our model, we can consider the population of infected as also structured by the parameter a. The equations are as before but now we keep track of the class a in the infected population. The mechanism here is that a susceptible individual from class a can be infected by infectious from any class I(t, b) but then gives rise to an individual I(t, a) of the same parent class. We also assume that there is a diffusion of the infected behaviours. We denote by ({mathfrak {B}}(a,b)) the transmission rate of S(t, a) by I(t, b). For simplicity and because it is natural, we will assume that it is of the form$$begin{aligned} {mathfrak {B}}(a,b)= beta (a) beta (b) end{aligned}$$where (beta) is as before. For full generality, we can also envision multi-dimensional parameters (ain {mathbb {R}}^d), with (a_iin (0,1)). We are then led to the system:$$begin{aligned} frac{{partial S(t,a)}}{{partial t}} & = d;Delta _{a} S(t,a) – S(t,a)frac{{beta (a)}}{N}intlimits_{0}^{1} beta (b)I(t,b);db \ frac{{partial I(t,a)}}{{partial t}} & = d^{prime}Delta _{a} I(t,a) + S(t,a)frac{{beta (a)}}{N}intlimits_{0}^{1} beta (b)I(t,b)db – gamma I(t,a), \ frac{{{text{d}}R(t)}}{{{text{d}}t}} & = gamma intlimits_{0}^{1} I (t,b){mkern 1mu} db, \ end{aligned}$$
(3)
In the SI we write further, more general, forms of this model, with ({mathfrak {B}}(a,b)) and more general diffusion of behaviours, that can include jumps or non-local variations. The type of models we discuss here may also shed light on the initial phase of the epidemic. We plan to investigate these questions in future work.Patterns generated by the modelIn the next section, we will discuss how the model fits the data observed in the Thau lagoon measurements. But before that, we start by showing that the above model (1) can generate the different patterns we mentioned. For this we rely on numerical simulations without fitting real data. And indeed we obtain plateaus, shoulders, and oscillations. The latter can be interpreted as epidemic rebounds.The key parameter here is the diffusion coefficient d, which controls the amplitude of behavioural variability (see Fig. 2). Large values of d rapidly yield homogenised behaviours, leading to classical SIR-like dynamics of infectious individuals. For very small values of d, the system also has a simple dynamics, in the sense that I(t) has a unique maximum, and therefore has no rebounds. We derive this in the limit (d=0) for which we show in the SI that there are neither plateaus nor rebounds.For some intermediate range of the parameter d, plateaus may appear after an exponential growth, like in the initial phase of the SIR model. A small amplitude oscillation, called “shoulder”, precedes a temporary stabilisation on a plateau, followed by a large time convergence to zero of infectious population. We also show that for small enough d, time oscillations of the infectious population curve, i.e. epidemic rebounds, may be generated by Model (1). Such oscillations also appear after a plateau, in a similar way to what one can see in observations.Simulations in Fig. 2 illustrate the various patterns obtained on the dynamics of infected population as a function of the diffusion parameter. For small enough d, in the top left graph of Fig. 2, one can see oscillations of the fraction of infectious individuals. These oscillations cannot be achieved in the classical SIR model. In fact, the two lower graphs of that figure, for somewhat larger values of d, exhibit the SIR model outcomes. Indeed, for sufficiently large d, the system becomes rapidly homogeneous (i.e. constant with respect to a). Yet, such oscillations are standard in the dynamics of actual epidemics, like the current Covid-19 pandemic. The intermediate value of d, represented in the upper right corner of Fig. 2 shows the typical onset of a plateau at a rather high value of I. Note that this plateau is preceded by a first small dip and then a characteristic “shoulder-like” oscillation.Figure 2Model behaviour depending on diffusion parameter values: infected rate dynamics in logarithmic scale. From left to right and then top to bottom, graphs are associated with (d=10^{-5}), (d=5times 10^{-5}), (d=10^{-3}) and (d=5times 10^{-3}) (in (day^{-1}) unit).Full size imageSecondary epidemic peaks are of lower amplitude than the first one, as shown in the top graphs of Fig. 2. This empirical observation leads us to conjecture that, at least in many cases, it is a general property of this model (with (beta) independent of time). This property would then reflect a kind of dissipative nature of Model (1). It is natural to surmise that a change of behaviours in time may generate oscillations with higher secondary peaks. Such changes result for instance from lifting social distancing measures or from fatigue effects in the population.We illustrate this with numerical simulations in Fig. 3. We assume a collective time modulation of the (beta (a)) transmission profile. That is, we replace (beta (a)) by (beta (a)varphi (t)) for some time dependent function (varphi), the other parameters are the same as in the simulations shown in Fig. 2. We look at the effect of a “lockdown exit” type effect. Then, (varphi (t)) takes two constant values, 1 from (t=0) to (t={1000}) and 1.2 after (t={1100}). In between, that is, for (tin ({1000}, {1100})), (varphi (t)) changes linearly from the value 1 to 1.2.Figure 3Multiple epidemic rebounds: susceptible individuals are divided into 50 discrete groups in the case where relaxation of social distancing measures starts on Day (t=1000) and ends up on Day (t=1100). The fraction of infected individuals in the population is represented in the left graph in logarithmic scale and in linear scale in the right graph.Full size imageOne can clearly see a secondary peak with higher amplitude than the first one. Note that after this peak, a third one occurs, with a lower amplitude than the second one. This third peak happens in the regime when (beta) is again constant in time.The effect of variantsAnother important factor that yields secondary peaks with higher amplitudes is the appearance of variants. Consider the situation with two variants. We denote by (I_1(t)) and (I_2(t)) the corresponding infected individuals. The first variant, which we call the historical strain, is associated with (beta _1) and (I_1(0)) and starts at (t=0). The variant strain corresponds to (beta _2) and (I_2) and starts at Day (t=1000). In this situation, the system (1) is extended by the following system:$$begin{aligned} frac{{partial S(t,a)}}{{partial t}} & = d{mkern 1mu} frac{{partial ^{2} S(t,a)}}{{partial a^{2} }} – left( {beta _{1} (a)I_{1} (t) + beta _{2} (a)I_{2} (t)} right)frac{{S(t,a)}}{N}, \ frac{{{text{d}}I_{2} (t)}}{{{text{d}}t}} & = frac{{I_{2} (t)}}{N}{mkern 1mu} intlimits_{0}^{1} {beta _{2} } (a)S(t,a){mkern 1mu} da – gamma _{2} I_{2} (t), \ frac{{{text{d}}I_{1} (t)}}{{{text{d}}t}} & = frac{{I_{1} (t)}}{N}{mkern 1mu} intlimits_{0}^{1} {beta _{1} } (a)S(t,a){mkern 1mu} da – gamma _{1} I_{1} (t) \ frac{{{text{d}}R(t)}}{{{text{d}}t}} & = gamma _{1} I_{2} (t) + gamma _{1} I_{2} (t), \ end{aligned}$$
(4)
The total infected population is (I(t)=I_1(t)+I_2(t)). Figure 4 shows a simulation of this system. Before the onset of the second variant, i.e. for (t< 1000), we observe a peak, followed by a small shoulder and a downward tilted plateau. The second variant corresponds to a higher transmission coefficient: namely, we take here (beta _2(a) equiv frac{3}{2} beta _1(a)). When it appears at time (t=1000), initially there is no effect, because the initial number of infectious with variant 2 is very small. Then, there is an exponential growth caused by this second variant gaining strength. The secondary peak is then higher than the first one. A very small shoulder precedes another stabilisation on a downward plateau.Figure 4 also shows the dynamics of fractions of infected with each one of the variants. Note that the infectious with variant 1 very rapidly all but disappear at the onset of the second exponential growth phase. One might have expected that the historical strain would be gradually replaced by the new strain, merely tilting further downward the plateau. But that does not happen. Thus, it is remarkable that the historical strain gets nearly wiped out at the very beginning of the second exponential growth.Figure 4Multiple epidemic rebounds due to a variant virus: susceptible individuals are divided into 50 discrete groups in the case where a new variant appears at Day (t=1000). The transmission rate (beta _2) is taken as (beta _2(a) = 1.5 , beta _1(a)), (d=0.0002), (gamma _1=0.1) and (gamma _2= 0.05). The fraction of infected individuals in the population is represented in the left graph in logarithmic scale. The total infected population is represented in linear scale in the right graph (black curve), variant 1 in red and variant 2 in green.Full size imageApplication to the Thau lagoon measurementsModel (1) describes the dynamics of the fraction of infectious in the population, that is (t mapsto I(t)/N). Therefore, we need to derive this fraction from the wastewater measurements. To this end, we use an “effective proportionality coefficient” between the two quantities. This coefficient itself is derived from the measurements (compare Section “SARS-CoV-2 concentration measurement from wastewater with digital PCR” in the “Methods” part below). Calibration of model (1) also requires fitting the values of (gamma), the profiles (a mapsto beta (a)) and the initial distribution of susceptible individuals in terms of a.We carried this procedure and the resulting fitted curve is displayed in Fig. 5. Note that the outcome correctly captures the shoulder and plateau patterns.Figure 5Calibrated model on Sète area: blue dots are measures of SARS-CoV-2 genome units and black curve represents the total infected individuals as an output of the model discretized into (n_g=20) groups in a. Initial distribution of susceptible individuals and (beta) function are taken as described in supplementary information. Parameters d and (gamma) are taken as follows: (d=2.5 times 10^{-4}) (day^{-1}), and (gamma =0.1) (day^{-1}).Full size imageThe underlying dynamics of the rate of susceptible individuals is given in Fig. 6 below for (n_g=20) groups. The lower curve illustrates the competition phenomenon between diffusion and sink term due to new infections, depending on the level of risk a of each state.Figure 6Calibrated model on Sète WWTP: density of susceptible individuals of each group for (n_g=20). The densities of susceptible of each group is represented in colour curves as functions of time. The curves are ordered from top to bottom according to increasing a. The resulting average total susceptible population is represented in black. Susceptible individuals of highest a trait, which are represented in the bottom light blue curve exhibit a non monotonic behaviour.Full size image More