Ecology

Morphology and phenotypic plasticity
The morphological features of the investigated colepid strains differed from those described for C. hirtus, C. spetai and even for N. nolandi1 (Fig. 1). Characteristics that matched the descriptions were the ciliate cell length and width, the barrel-shaped cell (except for strain CIL-2017/7, which was pear-shaped and strain CCAP 1613/15 that had a cylindrical shape), a number of six armor tiers, the structure of the armor tiers (hirtus-type or nolandia-type, respectively), and one caudal cilium (Table S2). Variations (CV > 20%) were found (i) in the number of plate windows in the posterior/anterior main plates even within individual cells, and (ii) in the presence/absence of anterior and posterior spines (Tables 1 + S2, Fig. 1). This phenotypic plasticity of the ciliate could also be observed in freshly collected Coleps specimens and was therefore not an artifact resulting from cultivation conditions (Fig. 1A). Wickham and Gugenberger43 hypothesized that the formation of the spines was a response to grazing pressure on C. hirtus; however, this could not be confirmed by respective experiments. Nevertheless, spineless specimens of C. hirtus have obviously been found before44,45,46,47. Luckily, we were able to investigate two strains (CCAP 1613/1 and CCAP 1613/2) that had been kept in the CCAP culture collection since the 1950ies and the 1960ies and which did not bear any spines or symbionts and could be clearly assigned to C. hirtus (Fig. 1T–V). These observations suggest that without predation pressure, colepid ciliates probably do not need to synthesize spines avoiding ingestion by a predator.
The presence/absence of green algal endosymbionts, one of the diagnostic features for the discrimination among C. hirtus subspecies and C. spetai, was also not a stable feature (Table S2). Under culture conditions, some strains lost their endosymbionts completely, other strains consisted of symbiotic and aposymbiotic individuals, and some strains showed only symbiont-bearing individuals (e.g., CCAP 1613/5 and CIL-2017/6). This indicates that the symbiosis is facultative and might be probably influenced by cultivation or environmental conditions (presumably, though not tested, food availability). Consequently, the morphological separation of C. hirtus into the two subspecies may no longer be valid. We clearly demonstrated that the morphological features used for species descriptions can vary and have severe consequences for colepid species identification. Moreover, even the strains belonging to the groups 1 and 2 discovered by the phylogenetic analyses (Fig. 2) cannot discriminate morphotypes because they can neither be assigned to a certain cell morphology nor to the possession of algal endosymbionts. This questions the traditional morphology-based taxonomy. The separation of Coleps hirtus hirtus, C. hirtus viridis and C. spetai, which Foissner et al.1 differentiated by the presence of zoochlorellae in the latter two species and the number of windows in the armor plates, could not be supported by our analyses. C. hirtus viridis was originally described by Ehrenberg48,49 as C. viridis and later transferred as synonym of C. hirtus by Kahl50 based on almost identical morphological features. However, Foissner22 described C. spetai for the green Coleps because of the morphological discrepancies to the Ehrenberg’s C. viridis (presence of only 11 windows per plate row and smaller cell size in C. viridis; see Table 1 for comparison). Our study has clearly demonstrated that most of the morphological features are variable and the limits for species separation were too narrow. Therefore, we propose the re-establishment of C. viridis for group 1 and C. hirtus for group 2, both with emended descriptions as follows. Considering our findings, the morphological descriptions of C. spetai, C. hirtus viridis and C. hirtus hirtus cannot be applied for (sub-) species separation any more. Consequently, we deal with a cryptic species complex, i.e., two genetically different groups that are fused in a highly variable morphotype including features of all three (sub-) species. To solve this taxonomic problem, two possible scenarios can be proposed: (1) We merge the three morphotypes under C. hirtus, the type species of Coleps. As a consequence, two new species needed to be proposed for both groups 1 and 2, which could be done following the suggestion of Sonneborn51 for the P. aurelia-complex. However, Sonneborn based his new descriptions on results of mating experiments, which are not applicable for Coleps here because conjugations have not been reported and the conditions for the induction of sexual reproduction are unknown. (2) To avoid confusion by introducing new species names, we propose keeping the already existing names, i.e., C. viridis for group 1 and C. hirtus for group 2 including the synonyms (see below).
Clonal cultures of both genetically varying Coleps groups have been deposited in the CCAP culture collection. Future studies may therefore be able to investigate, for example, sibling among strains or predator-prey experiments revealing spine- or wing-formation.
Coleps viridis Ehrenberg 1831 (printed 1832), Abh. Königl. Akad. Wiss. Berlin 1832: 101.
Synonym: Coleps spetai Foissner 1984, Stapfia 12: 21-22, Fig. 7, SP: 1984/10 and 1984/11 (lectotype designated here deposited in LI, see Aescht 2008: Denisia 23: 179), Coleps hirtus sensu Kahl 1930, Tierwelt Deutschlands 18: 134.
Diagnosis: Differed from other colepid ciliates by their SSU and ITS rDNA sequences (MT253680).
Lectotype (designated here): Fig. II, Tab. XXXIII, 3 in Ehrenberg 1838, Infusionsthierchen als vollkommene Organismen, p. 314.
Improved Description (specifications in brackets apply to our reference strain CCAP 1613/7): Coleps with conspicuous armor composed of six tiers with plate windows of the hirtus-type. With or without green algal endosymbionts. Cell size 44–63 × 21–35 μm (52–54 × 35–36 μm). Total number of windows in length rows 12–16 (14–16), number of windows of anterior primary plates 3–6 (4–6), number of windows of anterior secondary plates 2–3 (2), number of windows of posterior primary plates 4–5 (4–5), number of windows of posterior secondary plates 2–3 (2–3). One caudal cilium (1). With 0-2 anterior (0–1) and 0–5 posterior (1–4) spines, respectively.
Reference material (designated here for HTS approaches): The reference strain CCAP 1613/7 permanently cryopreserved at CCAP in a metabolically inactive stage.
Locality of reference strain: Plankton of Lake Mondsee, Upper Austria, Austria (47° 50′ N, 13° 23′ E).
Coleps hirtus (O.F. Müller) Nitzsch ex Ersch & Gruber 1827, Allgemeine Encyclopädie der Wissenschaften und Künste 16: 69, NT (proposed by Foissner 1984, Stapfia 12: 22, fig. 8): 1984/12 and 1984/13 (LI, in Aescht 2008: Denisia 23: 159).
Protonym: Cercaria hirta O.F. Müller 1786, Animalcula Infusoria: 128, tab. XIX, fig. 17, 18 (lectotype designated here).
Diagnosis: Differed from other colepid ciliates by their SSU and ITS rDNA sequences (MT253687).
Improved Description: Coleps with spiny armor composed of six tiers with plates of the hirtus-type. Without green algal endosymbionts. Cell size 42–52 × 23–28 μm. Total number of windows in length rows 12-13, number of windows of anterior primary plates 3-5, number of windows of anterior secondary plates 2, number of windows of posterior primary plates 4-5, number of windows of posterior secondary plates 2. One caudal cilium. Without anterior and 1-4 posterior spines, respectively.
Reference strain (designated here for HTS approaches): The strain CCAP 1613/14 permanently cryopreserved at CCAP in a metabolically inactive stage.
Locality of reference strain: Plankton of Lake Piburg, Tyrol, Austria (47° 11′ N, 10° 53′ E).
Molecular phylogeny of the Colepidae (Prostomatea)
The colepids belonging to the Prostomatea form a monophyletic lineage in the phylogenetic analyses of SSU rDNA sequences (Fig. 2). Mixotrophic as well as heterotrophic Coleps strains that resembled C. hirtus and C. spetai clustered in group 1 whereas group 2 included only two specimens which were identified as C. hirtus. These findings confirm the results of Barth et al.29 with one exception. The authors found a clear separation into mixotrophic and heterotrophic species, which were therefore assigned to a C. spetai-(with endosymbionts) and a C. hirtus-group (without endosymbionts), respectively. Despite the difficulties of identifying these species by morphology, both groups clearly differed in their SSU and ITS rDNA sequences (Fig. 3). The ITS-2/CBC approach introduced for green algae (details in Darienko et al.52) clearly demonstrated that both groups represented two separate ciliate species from a molecular point of view, which was also confirmed by analyses of the V9 region of the SSU, a region commonly used for metabarcoding (Figs. 4 and 5).
Our study also confirmed the findings of Chen et al.7, Lu et al.9, and Moon et al.28, showing that the generic concept of colepid ciliates needs to be revised. None of the genera represented by more than one species is monophyletic. For example, the three species of Nolandia belonged to separate lineages. Nolandia nolandi was a sister to our studied strains, whereas both other species were closely related to taxa of Apocoleps, Pinacocoleps, and Tiarina (Fig. 2). The genus Levicoleps and Coleps amphacanthus formed a monophyletic clade representing another example that the generic conception is artificial and needs to be revised. However, to provide a new generic concept of colepid ciliates, it is necessary to study more of the described species by using an integrative approach including experimental approaches on, e.g., the formation of spines. For example, we clearly demonstrated that one key feature, which is the presence/absence of anterior/posterior spines, is highly variable and can therefore not be used to separate colepid genera as indicated by Foissner et al.12 (Fig. 1). There is a need for more experimental studies with colepids belonging to the Cyclidium viridis and C. hirtus morphotype. Therefore, we deposited all clones used in this study in the CCAP culture collection. One option would be to incorporate all species into one genus, i.e., Coleps in revised form.
Endosymbiosis in Coleps
Some strains of Coleps are known to bear green algal endosymbionts1. These green algae have Chlorella-like morphology (Fig. 6) and were identified as Micractinium conductrix (Fig. 7). So far, this alga was only known as endosymbiont of the ciliate Paramecium bursaria34. All green algal endosymbionts of Coleps harbored this Micractinium species. In contrast, Pröschold et al.34 found that one ciliate strain identified as C. hirtus viridis had Chlorella vulgaris as endosymbiont (the algae has been deposited in the Culture Collection of Algae and Protozoa under the number CCAP 211/111). Unfortunately, this ciliate strain is not available anymore53.
Ecology and distribution
For limnological studies, the preservation with Bouin’s solution and QPS is an appropriate method for quantifying and identifying ciliate species in environmental samples54. However, the quality in characterization of ciliates at the species level is sometimes limited as, in case of Coleps, the characteristic armored calcium carbonate plates are dissolved by the acidified fixation solution. Therefore, in our study, we could only distinguish between algal-bearing (mixotrophic) and non-algal-bearing (heterotrophic) Coleps. Despite that limitation, we could clearly see that the heterotrophic ones were only found in the deepest zones of both lakes (Fig. 8A). Not surprisingly, Coleps is often observed in nutrient- and ion-rich and also oxygen-depleted freshwater habitats or areas, e.g., sulfurous and crater lakes1,5,6,27,55,56 or even in the sludge of wastewater treatment plants57. Mixotrophic individuals of Coleps were mainly found in the upper layers of both lakes, whereas in Lake Mondsee we could also detect specimens down to 40 m depth (Fig. 8). In contrast to the mero- and monomictic Lake Zurich4,10,58,59, Lake Mondsee is holo- and dimictic60. During mixis events, algal-bearing Coleps specimens can be transferred passively from the upper layers into the deeper zones and vice versa. Although morphotype countings and HTS analyses reads matched quite well, we found discrepancies that have already been discussed before10,61 (Fig. S2).
Biogeographic aspects (haplotype network)
Our metabarcoding approach showed that C. viridis was found in both lakes as a common ciliate (Fig. S2). In contrast, C. hirtus could not be detected during the sampling period. To obtain more information about the distribution of both species, we used the BLASTn search algorithm62 (100 coverage, >97% identity) for the V4 and the V9 regions of the SSU and the ITS-2 sequences. No records using the V9 and the ITS-2 approaches could be discovered in GenBank, but 25 reference sequences using the V4 (Table S3). Together with the newly sequenced strains, we therefore constructed a V4 haplotype network (Fig. 10). Both groups are obviously widely distributed and subdivided into five (group 1) and four (group 2) haplotypes, respectively. All reference sequences were collected from freshwater habitats except for two marine records63 (EU446361 and EU446396; Mediterranean Sea) and showed no geographical preferences.
Figure 10

TCS haplotype network inferred from V4 sequences of Coleps viridis and C. hirtus. This network was inferred using the algorithm described by Clement et al.64,65. Sequence nodes corresponding to samples collected from different geographical regions and from different habitats.

Full size image

Co-occurrence networks
In the sub-networks of C. viridis in both lakes, we found several significant correlations that pointed to either potential prey items, e.g., diverse flagellated autotrophic or heterotrophic protists or co-occurring ciliates (Fig. 9). Also, the smaller ciliates such as Cinetochilum margaritaceum or Cyclidium glaucoma may as well be considered as food for the omnivorous C. viridis (for a compilation of the food spectrum; see Foissner et al.1). However, we identified the endosymbiont M. conductrix and its host C. viridis from both sub-networks of Lake Mondsee but not of Lake Zurich (Fig. 9). Despite this result, we want to point out that we may probably not find M. conductrix free-living in a water body because outside their ciliate host the algae were immediately attacked and killed by so-called Chlorella-viruses66. Therefore, the HTS-detection of M. conductrix was probably only together with a host ciliate. This might further explain why the green algae were detected in Lake Mondsee even in the aphotic 40 m zone where photosynthesis was impossible and individuals probably passively transferred into the deeper area by lake mixis.
Outlook
As demonstrated in our study, the combination of traditional morphological investigations, which includes the phenotypic plasticity of the cloned strains, and modern molecular analyses using both SSU and ITS sequencing as well as HTS approaches advise a taxonomic revision of the genus Coleps. This comprehensive and integrative approach is also applicable for other ciliate species and genera and will provide new insights into the ecology and evolution of this important group of protists.
Experimental procedures
Study sites, lake sampling and origin of the Coleps strains
Our main study sites were Lake Mondsee (Austria) and Lake Zurich (Switzerland), two pre-alpine oligo-mesotrophic lakes that were sampled at the deepest point of each lake (Table S4). Water samples were taken monthly from June 2016 through May 2017 over the whole water column and additionally biweekly at two main depths, i.e., 5 m in both lakes, 40 m in Lake Mondsee, and 120 m in Lake Zurich, respectively. A 5-L-Ruttner water sampler was used for Lake Zurich and a 10-L-Schindler-Patalas sampler (both from Uwitec, Austria) for Lake Mondsee. Twelve Coleps strains were isolated from Lake Mondsee and one from Lake Zurich. Another six clones could be obtained either from already successfully cultivated own strains, fresh isolates or from culture collections. Detailed information about sampling sites, dates and strain numbers is given in Table S2.
Seasonal and spatial distribution and abundance
For quantification, subsamples (200-300 mL) were preserved with Bouin’s solution (5% f.c.) containing 15 parts of picric acid, 5 parts of formaldehyde (37%) and 1 part of glacial acetic acid54. The samples were filtered through 0.8 μm cellulose nitrate filters (Sartorius, Germany) equipped with counting grids. The ciliates were stained following the protocol of the quantitative protargol staining (QPS) method after Skibbe54 with slight modifications after Pfister et al.67. The permanent slides were analyzed by light microscopy up to 1600x magnification with a Zeiss Axio Imager.M1 and an Olympus BX51 microscope. For identification of Coleps and Nolandia cells, the identification key of Foissner et al.1 was used. Microphotographs were taken with a ProgRes C14 plus camera using the ProgRes Capture Pro imaging system (version 2.9.0.1, Jenoptik, Jena, Germany).
Cloning, identification and cultivation of ciliates and endosymbionts
Single cells of Coleps were isolated and washed using the Pasteur pipette method68. The isolated strains were cultivated in 400 μl modified Woods Hole medium69 (MWC; modified) and Volvic mineral water in a mixture of 5:1 and with the addition of 10 μl of an algal culture (Cryptomonas sp., strain SAG 26.80) as food in microtiter plates. These clonal cultures were transferred into larger volumes after successful enrichment. All cultures were maintained at 15–21 °C under a light: dark cycle of 12:12 h (photon flux rate up 50 mol m−2 s−1).
For the isolation of their green algal endosymbionts, single ciliates were washed again and transferred into fresh MWC medium. After starvation and digestion of any food, after approx. 24 hrs, cells were washed again and the ciliates transferred onto agar plates containing Basal Medium with Beef Extract (ESFl; medium 1a in Schlösser70). Before placement of the ciliates onto agar plates, 50 μm of an antibiotic mix (mixture of 1% penicillin G, 0.25% streptomycin, and 0.25% chloramphenicol) were added to prevent bacterial growth. The agar plates were kept under the same conditions as described. After growth (6–8 weeks), the algal colonies were transferred onto agar slopes (1.5%) containing ESFl medium and kept under the described culture conditions.
For light microscopic investigations of the algae, Olympus BX51 and BX60 microscopes (equipped with Nomarski DIC optics) were used. Microphotographs were taken with a ProgRes C14 plus camera using the ProgRes Capture Pro imaging system (version 2.9.0.1, Jenoptik, Jena, Germany).
PCR, sequencing and phylogenetic methods
Single-cell PCR was used to obtain the sequences of the Coleps strains. Before PCR amplification, single cells of Coleps were washed as described above. After starvation followed by additional washing steps, cells were transferred into 5 μm sterile water in PCR tubes and the prepared PCR mastermix containing the primers EAF3 and ITS055R71 was added. After this primary PCR amplification and subsequent PCR purification, a nested PCR was conducted using the primer combinations EAF3/N1400R and N920F/ITS055R71.
The sequences of the Coleps strains were aligned according to their secondary structures of the SSU and ITS rDNA (see detailed folding protocol described in Darienko et al.52) and included into two data sets: (i) 34 SSU rDNA sequences (1,750 bp) of representatives of all members of the Prostomatea and (ii) 19 ITS rDNA sequences (538 bp) of the investigated strains. Genomic DNA of the green algae was extracted using the DNeasy Plant Mini Kit (Qiagen GmbH, Hilden, Germany). The SSU and ITS rDNA were amplified using the Taq PCR Mastermix Kit (Qiagen GmbH, Hilden, Germany) with the primers EAF3 and ITS055R. The SSU and ITS rDNA sequences of the isolated green algae (aligned according to the secondary structures) were included into a data set of 31 sequences (2,604 bp) of representatives of the Chlorellaceae (Trebouxiophyceae).
GenBank accession numbers of all newly deposited sequences can be found in Table S2 and in Fig. 7, respectively. For the phylogenetic analyses, the datasets with unambiguously aligned base positions were used. To test which evolutionary model fit best for both data sets, we calculated the log-likelihood values of 56 models using Modeltest 3.772 and the best models according to the Akaike criterion by Modeltest were chosen for the analyses. The settings of the best models are given in the figure legends. The following methods were used for the phylogenetic analyses: distance, maximum parsimony, maximum likelihood, and Bayesian inference. Programs used included PAUP version 4.0b16473, and MrBayes version 3.2.374.
The secondary structures were folded using the software mfold42, which uses the thermodynamic model (minimal energy) for RNA folding.
Haplotype networks
The haplotypes of the V4 region were identified among the groups of Coleps (see Fig. S1). The present haplotypes and the metadata (geographical origin and habitat) of each strain belonging to the different haplotypes are given in Table S3. To establish an overview on the distribution of the Coleps groups, the V4 haplotypes were used for a BLASTn search62 (100% coverage, >97% identity). To construct the haplotype networks, we used the TCS network tool64,65 implemented in PopART75.
High-throughput sequencing of the V9 18S rDNA region and subsequent bioinformatic analyses
On each sampling date, water samples for a high-throughput sequencing approach (HTS) were taken in depths of 5 m and 40 m at Lake Mondsee and 5 m and 120 m depths in Lake Zurich. DNA extraction, amplification of the V9 SSU rDNA, HTS and quality filtering of the obtained raw reads was conducted as described in Pitsch et al.10. After quality filtering, all remaining reads were subjected to a two-level clustering strategy76. In the first level, replicated reads were clustered in SWARM version 2.2.2 using d=177. In the second level, the representative sequences of all SWARM OTUs were subjected to pairwise sequence alignments in VSEARCH version 2.11.078 to construct sequence similarity networks at 97% sequence similarity. The network sequence clusters (NSCs) resulting from the second level of clustering were then taxonomically assigned by running BLASTn analyses against NCBI’s GenBank flat-file release version 230.0 and the Coleps SSU sequences obtained from single-cell sequencing. Network sequence clusters were assigned to Coleps, if the closest BLAST hit of the NSC representative sequence was a Coleps reference sequence. Furthermore, the NSC representative sequence had to share a fragment of at least 48 consecutive nucleotides and at least 90% sequence similarity to a reference sequence in order to be assigned to Coleps.
Co-occurrence networks
With the protist community data matrix resulting from HTS, we further conducted co-occurrence network analyses to assess biotic and abiotic interactions of Coleps. For each lake and depth, we ran network analyses with NetworkNullHPC (https://github.com/lentendu/NetworkNullHPC) following the null model strategy developed by Connor et al.79. This strategy was especially designed for dealing with HTS datasets and allows for inferring statistically significant correlations between NSCs while minimizing false positive correlation signals. We screened the resulting networks for Coleps nodes and extracted their subnetworks including all directly neighbouring co-occurrence partners as well as all edges between Coleps and its neighbours and the neighbours themselves. More

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Overview
In earlier work34, epidemiological models are broadly divided into two large categories, called forecasting and mechanistic. The former models fit a specific curve to the data and then attempt to predict the dynamics of the quantity under consideration. The most well known mechanistic models are the SIR-type models. As noted by Holmadahl and Buckee34, the mechanistic models involve substantially more complicated mathematical machinary than the forecasting models, but they have the advantage that they can make predictions even when the relevant circumstances change. In our case, since our goal is to make predictions after the situation changes due to the lifting of the lockdown measures, we need to consider a mechanistic model. However, it is widely known that the main limitation of mechanistic models is the difficulty of determining the parameters specifying such models. In this direction, a methodological advance was presented by the authors35, filling an important gap in the relevant literature: it was shown in35 that from the knowledge of the most reliable data of the epidemic in a given country, namely the cumulative number of deaths, it is possible to determine suitable combinations of the constant parameters (of the original model) which specify the differential equation characterizing the death dynamics. Furthermore, a robust numerical algorithm was presented for obtaining these parameters. One of these constants, denoted by c, is particularly important for the analysis of the effect of easing the lockdown conditions, because it is proportional to the number of contacts between asymptomatic individuals that are infected by SARS-CoV-2 and susceptible ones. Specifically, as the equations presented below will indicate, this coefficient is measured in units of inverse population (where the population represents the number of individuals to which we assign no units) times inverse days. This constant reflects the probability of infection given a contact which is proportional to the viral load (i.e., the viral concentration in the respiratory-tract fluid) of expelled respiratory droplets36. Easing the lockdown will lead to an increase of the value of this constant. Thus, in order to quantify this effect we assumed that the post-lockdown situation could be described by the same model but with c multiplied by an integer number (zeta), such as (zeta =2), or 3, etc. Assuming a fixed viral load emission (i.e., no face mask or similar protective measures), this would be tantamount to doubling or tripling the number of contacts per day. To put things into perspective, it is relevant to mention here that in the relevant literature a ballpark estimate for daily contacts of an individual is about 13.437.
We first applied the above algorithm to the case of the COVID-19 epidemic in Greece. However, the novelty and the main interest of the present work consists of the extension and application of the above methodology to two subpopulations. This situation is significantly more complicated than that of2 and is described by 12 ODEs involving 18 parameters (details are discussed in the “Methods” section). Using this extended formulation, we analysed the effect of easing the lockdown measures under two distinct possible scenarios: in the first, we examined what would happen if the interactions between older persons, namely persons above 40 years of age, as well as between older and younger persons, namely those below 40, continue to be dictated by the same restrictions as those of the lockdown period. However, we assumed that the interaction among the young was progressively more free. In the second case, we analysed the effect of easing the lockdown measures in the entire population without distinguishing the older from the young. In principle, the effect on deaths in the above two scenarios could be analyzed by the extension of the rigorous results of35. However, due to the sparsity of the deaths data (especially for the younger population), this approach is practically not possible at present. Thus, we supplemented the data for deaths for the two subpopulations with data for the cumulative numbers of reported infected.
Using four sets of data, namely the number of deaths and the number of reported infected for the older and the younger population we found that the above two alternatives would result in very different outcomes: in the first case, the total number of deaths of the two sub-populations and the number of total infections would be relatively small. In the second case, these numbers would be prohibitively high. Specifically, in the case of Greece, if the lockdown was to be continued indefinitely, our analysis suggests that the total numbers of deaths and infections would finally be around 165 and 2550, respectively. These numbers would remain essentially the same even if the lockdown measures for the interaction between the young people were eased substantially, provided that the interactions of older-older and older-young would remain the same as during the lockdown period. For example, even if the parameter measuring the effect of the lockdown restrictions on the young-young interactions were increased fourfold, the number of deaths and infections would be (according to the model extrapolation) 184 and 3585, respectively. On the other hand, even if the parameters characterizing all three interactions were increased only threefold, the relevant numbers would be 48144 and 1283462. It is clear that the latter numbers are prohibitive, suggesting that a generic release of the lockdown may be catastrophic.
In our view, the explanations provided in the “Methods” section for the assumptions of our model, which show that these assumptions are typical in the standard epidemiological models, substantiate the qualitative conclusions (and notes of caution) regarding the impact of the above two different types of exit policies. This may provide a sense of how a partial restoration of regular life activities can be achieved without catastrophic consequences, while the race for pharmacological or vaccine-based interventions that will lead to an end of the current pandemic is still ongoing. Importantly, we also offer some caveats emphasizing the qualitative nature of our conclusions and possible factors that may substantially affect the actual outcome of the lifting of lockdown measures.
Model setup: single population versus two age groups
We divide the population in two subpopulations, the young (y) and the older (o). In order to explain the basic assumptions of our model we first consider a single population, and then discuss the needed modifications in our case which involves two subpopulations. Let E(t) denote the exposed (but not infectious) population. An individual in this population, after a median 4-day period (required for incubation — see e.g.38) will either become sick or will be asymptomatic; an interval of 3-10 days captures 98% of the cases. The sick (infected) and asymptomatic populations will be denoted, respectively, by I(t) and A(t). The rate at which an exposed person becomes asymptomatic is denoted by a; this means that each day aE(t) persons leave the exposed population and enter the asymptomatic population. Similarly, each day sE(t) leave the exposed population and enter the sick population. These processes, as well as the subsequent movements are depicted in the flowchart of Fig. 1.
Figure 1

Flowchart of the populations considered in the model and the rates of transformation between them. The corresponding dynamical equations are Eqs. (1)–(6).

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The asymptomatic individuals recover with a rate (r_1), i.e., each day (r_1A(t)) leave the asymptomatic population and enter the recovered population, which is denoted by R(t). The sick individuals either recover with a rate (r_2) or they become hospitalized, H(t), with a rate h. In turn, the hospitalized patients also have two possible destinations; either they recover with a rate (r_3), or they become deceased, D(t), with a rate d.
It is straightforward to write the above statements in the language of mathematics; this gives rise to the equations (1)–(5) below:

$$begin{aligned} frac{dA}{dt}= a E – r_1 A end{aligned}$$
(1)

$$begin{aligned} frac{dI}{dt}= s E – (h + r_2) I end{aligned}$$
(2)

$$begin{aligned} frac{dH}{dt}= h I – (r_3+d) H end{aligned}$$
(3)

$$begin{aligned} frac{dR}{dt}= r_1 A + r_2 I + r_3 H end{aligned}$$
(4)

$$begin{aligned} frac{dD}{dt}= d H end{aligned}$$
(5)

$$begin{aligned} frac{dE}{dt}= c left[ T – (E+I+A+H+R+D)right] left( A + b Iright) – (a+s) E end{aligned}$$
(6)

It is noted that our model is inspired by various expanded versions of the classic SIR model adapted to the particularities of COVID-19 (such as the key role of the asymptomatically infected). It is, in particular, inspired by, yet not identical with that of14. In order to complete the system of equations (1)–(6), it is necessary to describe the mechanism via which a person can become infected. For this purpose we adopt the standard assumptions made in the typical epidemiological models, such as the SIR (susceptible, infected, recovered) model: let T denote the total population and let c characterize the number of contacts per day made by an individual with the capacity to infect (c is thought of as being normalized by T). Such a person belongs to I, A or H. However, for simplicity we assume that the hospitalized population cannot infect; this assumption is based on two considerations: first, the strict protective measures taken at the hospital, and second, the fact that hospitalized patients are infectious only for part of their stay in the hospital. The latter fact is a consequence of the relevant time scales of virus shedding in comparison to the time to hospitalization and the duration of hospital stay. The asymptomatic individuals are (more) free to interact with others, whereas the (self-isolating) sick persons are not. Thus, we use c to characterize the contacts of the asymptomatic persons and b to indicate the different infectiousness (due to reduced contacts/self-isolation) of the sick in comparison to the asymptomatic individuals.
The number of people available to be infected (i.e., the susceptible population) is (T-(E+I+A+H+R+D)). Indeed, the susceptible individuals consist of the total population minus all the individuals that are going or have gone through the course of some phase of infection, namely they either bear the infection at present ((E+A+I+H)) or have died from COVID-19 (D) or are assumed to have developed immunity to COVID-19 due to recovery (R). Hence, if we call the total initial individuals T, this susceptible population is given by the expression written earlier. The rate by which each day individuals enter E is given by the product of the above expression with (c(A+bI)). At the same time, as discussed earlier, every day ((a+s)E) persons leave the exposed population. It is relevant to note here that within this simpler model, it is possible to calculate the basic reproduction number (R_0), which is a quantity of substantial value in epidemiological studies32,33. In this model, this can be found to be33:

$$begin{aligned} R_0=frac{c T}{a+s}left[ frac{a}{r_1} + frac{b s}{r_2+h} right] . end{aligned}$$
(7)

This will be useful below for the purposes of finding the change in c (under lockdown) needed in order for transmission to cross the threshold of (R_0=1) and thus to lead to growth of the epidemic. In the particular case of the data shown in Table 1, (R_0=0.4084), in accordance with the lockdown situation associated with a controlled epidemic.
It is straightforward to modify the above model so that it can describe the dynamics of the older and younger subpopulations. Each subpopulation satisfies the same set of equations as those described above, except for the last equation which is modified as follows: the people available to be infected in each subpopulation are described by the expression given above where T, E, I, A, H, R, D have the superscripts (^o) or (^y), denoting older and young, respectively; (A+bI) is replaced in both cases by (A^o+A^y+b(I^o+I^y)) where for simplicity we have assumed that the infectiousness of the older and the young is the same. We have already considered the implications of the generalisation of the above model by allowing different parameters to describe the interaction of the older and young populations; this will be discussed in the “Methods” section. In what follows, we will discuss the results of this simpler “isotropic” interaction model.
Quantitative model findings
The parameters of the model are given in the flowchart of Fig. 1. Naturally, for the two-age model considered below, there is one set of such parameters associated with the younger population and one associated with the older one. The optimization routine used for the identification of these parameters is explained in detail in the “Methods” section. The parameters resulting from this optimization for the single population model are shown in Table 1, whereas for each of the two populations are given in Table 2. Clearly, many of these parameters are larger for the older population in comparison to the young, leading to a larger number of both infections and deaths in the older than in the young population.
Table 1 Optimized model parameters for the single population model, and the variation interval of each parameter within the optimization process (for further details, see “Methods” section).
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Table 2 Optimized (isotropic) model parameters for the young and older populations, and the variation interval of each parameter within the optimization process (for further details, see “Methods” section).
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Support for the validity of our model is presented in Fig. 2, which depicts its comparison (using the above optimized parameters) with the available data. The situation corresponding to keeping the lockdown conditions indefinitely, is the one illustrated in Fig. 2. In this case, the number of deaths and cumulative infections rapidly reaches a plateau, indicating the elimination of the infection. Here, we have optimized the model on the basis of data used from Greece39 between April 3rd and May 4th. It is noted that daily updates occurred at 3pm for the country of Greece, hence it is not clear up to what time the data are collected that are included in the daily report. We have assumed that the data reflect the infections and deaths present on that particular day. This possibly shifts the starting point of our count by a few hours, but should not change the overall result trends.
We next explain the implications of the model when different scenarios of ‘exit’ from the lockdown state are implemented. The relevant results are illustrated in Figs. 3, 4 and the essential conclusions are summarized in Table 3 for the numbers of deaths and cumulative infections, respectively. First, we need to explain the meaning of the parameter (zeta) appearing in the above tables: this parameter reflects the magnitude of the easing of the lockdown restrictions. Indeed, since the main effect of the lessening of these restrictions is that the number of contacts increases, we model the effect of easing the lockdown restrictions by multiplying the parameter c with a factor that we refer to as (zeta). The complete lockdown situation corresponds to (zeta)=1; the larger the value of (zeta), the lesser the restrictions imposed on the population. By employing the above quantitative measure of easing the lockdown restrictions, we consider in detail two distinct scenarios. In the first, which corresponds to the top rows of the Figures 3 and 4, we only allow the number of contacts of “young individuals with young individuals” (corresponding to the parameter (c^{yy}) mentioned in the “Methods” section) to be multiplied by the factor (zeta). This means that the lockdown measures are eased only with respect to the interaction of young individuals with other young individuals, while the interactions of the young individuals with the older ones, as well as the interactions among older individuals remain in the lockdown state. In the second scenario, corresponding to the bottom rows of the Figures 3 and 4, the restrictions of the lockdown are simultaneously eased in both the young and the older population; in this case all contacts are increased by the factor (zeta). It is noted that while we change c by this factor, we maintain the product cb at its previous value (i.e., we concurrently transform (crightarrow zeta c) and (brightarrow b/zeta)) considering that the sick still operate under self-isolation conditions and thus do not accordingly increase their number of contacts.
Table 3 Deaths D(t) and cumulative infections C(t) in the case of increasing of the number of contacts by (zeta). The second and fourth columns refer to the case for which the lockdown measures are eased for the young population, whereas the third and fifth column refer to the one where this occurs for both the young and older populations.
Full size table

Figure 2

Evolution of the current situation of deaths D(t) (left) and cumulative infections C(t) (right) in Greece, under the case of an indefinite continuation of the lockdown conditions. In this and all the figures that follow, the blue curve corresponds to the young population, while the red curve to the older population. The data for Greece from the 3rd of April to the 4th of May 2020 are depicted by dots. For the latter, alternate colors have been used (i.e., blue dots for the older population and red for the younger for clearer visualization).

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Fig. 3 corresponds to the case where the parameter (zeta) associated with the number of contacts between susceptible and asymptomatic individuals doubles. In this case, as also shown in Table 3, the situation does not worsen in a dramatic way. In particular, the number of deaths increases by 1, whereas the cumulative infections only increase by the small number of 58. In the second scenario where the number of contacts is doubled for both the young and the older populations, we find slightly larger (but not totally catastrophic) effects: the number of deceased individuals increases by 58 and the total number of infections grows by 1550.
Figure 3

Again the deaths D(t) and the cumulative infections C(t) are given for the case where the c factor (characterizing the number of contacts) amongst young individuals is doubled, but those of the older individuals (and of the young-older interaction) are kept fixed. This is shown in the top panels. In the bottom panels, the c’s of both young and old individuals are doubled.

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The situation becomes far more dire when the number of contacts is multiplied by a factor of 3 for both the young and older populations, meaning that the lockdown restrictions are eased significantly for the entire population. As shown in Table 3 and in Fig. 4, if the c’s of the young population only are multiplied by a factor of 3, then the deaths are increased by 3 and the infections by 198 (black line in the Figure and 3rd row of the Tables). This pales by comparison to the dramatic scenario when the c’s associated with both the young and older sub-populations are multiplied by 3; in this case, the number of deaths jumps dramatically to 48144, while the number of infections is a staggering 1283462, growing by about 500 times.
Figure 4

Same as reported in Fig. 3 but now where the contacts are multiplied by factors 3, 4 and 5. Full (dashed) lines hold for the young (older) population.

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An example corroborating the above qualitative trend can also be found in Fig. 4 and in the 4th and 5th rows of Table 3. Here, for e.g. (zeta =5), even the effect of releasing solely the young population leads to very substantial increases, namely to 6044 deaths and 306219 infections although of course it is nowhere near the scenarios of releasing both young and older populations. In the second scenario, the numbers are absolutely daunting: using the parameters of Table 2 we find that the number of deaths jumps to 83274 and the number of cumulative infections to 2221296.
Figure 5

Hospitalizations when only the young population (left) or both the young and older (right) population are released. Full (dashed) lines hold for the young (older) population.

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Finally, we show the prediction of the easing measures in the hospitalizations (i.e. daily occupied beds in hospitals). This is a crucial point to assess in order that the health system does not collapse because of COVID-19 patients. Figure 5 shows these trends for the above mentioned values of (zeta). In the case of releasing solely the young population (see left panel of the Figure), it is observed that the number of hospitalizations decreases monotonically except for (zeta =5), where the hospitalization peak is 523 for the young population and 1426 for the older one (values that are affordable by Greek health system); however, if both the young and older population are released (see right panel of the Figure), there is a monotonically decreasing behaviour only for (zeta =1) and 2. For higher (zeta) we observe that the height of the peak obviously increases with (zeta), while this peak also occurs earlier when the number of contacts is increased; for instance, for (zeta =3), the hospitalization peak number of the young population is 3844 whereas this value is 37030 for the older one, numbers that are, unfortunately, unaffordable for the Greek health system. These figures grow even further to 16869 and 163648 if (zeta =5).
In light of the above results, the significance of preserving the lockdown restrictions of the sensitive groups of the older population is naturally emerging. It can be seen that in the case where the number of contacts is roughly doubled, the behavior of release of young or young and older individuals is not dramatic (although even in this case releasing only the young population is, of course, preferable). Nevertheless, a more substantial release of the young population is still not catastrophic. On the other hand, the higher rates of infection, hospitalization and proneness to death of senior individuals may bring about highly undesirable consequences, should both the young and older members of the population be allowed to significantly increase (by 3 times or more) their number of contacts. More

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