It was found that the Ti68Nb7Ta3Zr4Mo18 alloy changes from a single crystal to a polycrystal during the tensile process, and the transition of the Ti68Nb7Ta3Zr4Mo18 (at.%) high-entropy alloy from the BCC phase to the FCC phase occurs. At high temperatures and low strain rates, grain boundary slip is the main deformation mechanism, and at low temperatures and high strain rates, dislocation slip replaces grain boundary slip as the dominant deformation mechanism, which improves the strength of the alloy. Moreover, when the grain size is too small, the strength of the alloy decreases, which does not satisfy the fine grain strengthening theory and shows an inverse Hall–Petch relationship. This study offers a new compositional window for the additive manufactured lightweight high-strength material categories for various applications including the aerospace industry.Intense research efforts are focused on the development of advanced high-entropy alloys intended for premium aerospace components and other applications, where high strength and good formability are crucial. The mechanical properties of these alloys are closely related to the phase transformation, dislocation evolution, and grain size, and these factors are affected by the deformation temperature. The response of the retained austenite to strain-induced martensitic transformation at various temperatures was studied in an advanced Ti High-entropy alloys (HEAs) were first proposed and named in 2004 . It is g0.1  Shockley, yellow represents 1/3[1 0 0] Hirth, and red represents the other. It is clear that the main type of dislocation in the alloy is Shockley dislocation, which is a continuous long dislocation at low temperatures and a discrete short dislocation at high temperatures. Dislocations occur mainly within the grain, and most remain parallel to each other. The grain boundary has a certain hindrance effect on the dislocation slip, so at the grain boundary, the dislocations accumulate and wrap to form dislocation nodes, as shown in the black coils in In order to reveal the influence of temperature and strain rate on the deformation mechanism of the high-entropy alloy, the structural changes and dislocation evolution of the high-entropy alloy under different tensile conditions were studied. After stretching to ultimate strength, the structure and dislocation of the high-entropy alloy at 77 K and 1000 K temperatures are shown in 10 s−1, the monocrystalline is transformed into polycrystalline in addition to molecular structure transformation. In order to study the effect of grain size on the mechanical properties, the Voronoi structure method was used to modify the model [3 in size and contained different quantities of randomly oriented particles, which were controlled by this method. When the strain rate of 300 K is 1 × 10he model ,26, as sIn order to further investigate the cause of the inverse Hall–Petch phenomenon, a snapshot of the grain and dislocation with the tensile strain of the high-entropy alloy at 0.15 was taken, as shown in 68Nb7Ta3Zr4Mo18 (at.%) HEA model underwent BCC–FCC phase transition in uniaxial Y-axis stretching along the NVT system, which is somewhat different from the single crystal Fe model studied by Ma et al. [X-axis. During the stretching process, the transition from BCC to FCC is divided into two stages. First, under tensile load, the BCC close-packed surface atoms change from “ABABAB” stacking mode to “ABCABCABC” stacking mode, as shown in The Tia et al. , which uAs can be seen from the results in Summary I confirm 3.2 of this paper, obvious grain boundary slip occurs in the high-entropy alloy at high temperatures. Due to the high temperature, the thermal motion of atoms is intensified, and a large number of disordered atoms are formed from the original position of atoms, and these disordered atoms are mostly concentrated near the crystallization, as shown in It is generally believed that dislocation occurs in the deformation process of materials, and dislocations intertwine with each other to form dislocation knots, resulting in increased dislocation density and relatively difficult dislocation movement. Because the free travel of dislocation movement decreases, the interaction between dislocation hinders the slip movement of dislocation, and the stronger the material resistance to deformation, the higher the strength of the material. However, when the grain size is less than a certain degree, it is difficult for the dislocation to move within the grain, and the phenomenon of intertwining will not appear. In addition, grain boundaries are more likely to break down during deformation due to the small grain size. The above two reasons will lead to the reduction of material strength and reverse the Hall–Petch phenomenon at the realization point.68Nb7Ta3Zr4Mo18 (at.%) HEA were studied by the molecular dynamics method, and the effects of temperature, strain rate, and grain size on the mechanical properties were considered. The following conclusions were obtained.High temperatures can soften the alloy and reduce the elastic modulus and ultimate strength of the alloy. The strain rate has little effect on the elastic modulus of the high-entropy alloy but a great effect on the ultimate strength of the high-entropy alloy, that is, the higher the strain rate, the greater the ultimate strength.68Nb7Ta3Zr4Mo18 (at.%) HEA. The main deformation mechanism at high temperatures is grain boundary slip, while the main deformation mechanism at low temperatures and high strain rates is dislocation slip.The transition from an FCC structure to a BCC structure occurs during the tensile process of the Ti68Nb7Ta3Zr4Mo18 (at.%) HEA is too small due to the internal dislocation of grain growth and slippage, and with the instability of grain boundaries, the strength of the alloy will reverse the Hall–Petch law phenomenon.The grain size of the TiThe micromechanical properties of a Ti


Around 42,000 years ago, anatomically modern humans appeared in Western Europe to the detriment of indigenous Neanderthal groups. It is during this period that new techno-cultural complexes appear, such as the Châtelperronian that extends from northern Spain to the Paris Basin. The Grotte du Renne (Arcy-sur-Cure) is a key site for discussing the biological identity of its makers. This deposit has yielded several Neanderthal human remains in its Châtelperronian levels. However, the last inventory of the paleoanthropological collection attributed to this techno-complex allowed the identification of an ilium belonging to a neonate (AR-63) whose morphology required a thorough analysis to assess its taxonomic attribution. Using geometric morphometrics, we quantified its morphology and compared it to that of 2 Neanderthals and 32 recent individuals deceased during the perinatal period to explore their morphological variation. Our results indicate a morphological distinction between the ilia of Neanderthals and anatomically modern neonates. Although AR-63 is slightly outside recent variability, it clearly differs from the Neanderthals. We propose that this is due to its belonging to an early modern human lineage whose morphology differs slightly from present-day humans. We also explore different hypotheses about the presence of this anatomically modern neonate ilium among Neanderthal remains. Recently, Gravina et al.23 conducted a taphonomic, spatial, and typo-technological reassessment of Châtelperronian lithic material from the EJOP sup level of La Roche-à-Pierrot  . In the context of this debate, the Grotte du Renne (GDR) of Arcy-sur-Cure 27, located in the Yonne Basin, 35 km south of Auxerre (France)  and the beginning of the Upper Paleolithic (UP) coincides with the decline of the Neanderthals and the expansion of the first anatomically modern human (AMH) groups in Western Europe. It was during this period, around 42,000 years ago , that new techno-cultural complexes appeared in this area, discovered at the interface of underlying MP and overlying UP occupations. This episode of human plurality therefore raises questions about the biological identity of the bearers of these techno-complexes. Among these, the Châtelperronian, present in the north of Spain, the south-west, the center of France and in the Paris Basin between around 43,760 and 39,220 BP30, Figure 1 p. 427 and Fig. 32 and laminar lithic pieces37 associated with human remains identified as Neanderthals29.Located at the opening of a karstic gallery oriented to the south on the left bank of the Cure river Fig. c and d, 41, the discovery of these Neanderthal fossils in layers containing a lot of material with features considered as "typical" of the UP  is a result of inter-stratigraphic disturbances that mixed Mousterian human remains with remains from overlying levels reported to the UP.According to some authors42. Although the results of this study show the existence of post-depositional processes that displaced remains in certain sectors, notably through the identification of refitting between some pieces from distinct subsets44 or the detection of runoff, it was emphasized that the movements occurred within the subsets of the same layer. Except in small localized areas, these indications of post-depositional movements do not call into question the major stratigraphic subdivisions initially defined and therefore the association of Châtelperronian material with layers X, IX and VIII.Based on field notes and surveys from the excavation notebooks of A. Leroi-Gourhan's team, a taphonomic analysis of different archaeological assemblages of the GDR was conducted for a first assessment of their integrity48. A recent synthesis of these results, carried out by Banks and d’Errico49 proposed a new chronology for the GDR archaeological sequence using OxCal 4.250 and the IntCal13 calibration curve51. By combining the results of the two most recent studies, obtained by Accelerator Mass Spectometry (AMS) from remains from A. Leroi-Gourhan's excavations, and by employing a new probabilistic model based on all of the previous dating, Banks and d’Errico propose a chronological interval in agreement with the results of Hublin et al.14 regarding the Mousterian of layers XII and XI, i.e., between about 46 and 44.5 kya cal BP, and the early phase of the Châtelperronian, i.e., layers X and IX, which would have begun around 44.5 kya cal BP. However, the authors point out some differences between their model and that of Hublin et al. regarding the transition between the Châtelperronian and early Aurignacian: 39.5 kya cal BP for Banks and d’Errico and 41 kya cal BP for Hublin et al. Note that Higham et al.52 assign an age to the Châtelperronian occupations similar to that obtained by Hublin et al. The interpretations of these results stay questionable taking into account the analysis questioning the chronocultural integrity of the dated material53.To assess the age of the archaeological levels, seven radiocarbon dating projects have been conducted22) assigned to the Neanderthal lineage on the basis of its mitochondrial DNA, has been dated and provided an age between 40,680 and 42,335 cal BP 54.Concerning the age of the human remains unearthed in the Châtelperronian levels, only one piece  showed clear Neanderthal affinities13. More recently, two cranial bone fragments (AR-14 and AR-30), first identified by ZooMs (Zooarchaeology by Mass Spectrometry), have been shown to carry Neanderthal-like mitochondrial DNA22.To date, studies of the human fossils of the GDR identified 64 remains attributed to layers X, IX and VIII. Among these 64 remains, 38 of them exhibit morphological and genetic traits assigning them to the Neanderthal lineage29 leading to the identification of new human pieces. While all were previously reported to fall within the Neanderthal range of variation, a right ilium  and to that of a recent neonate from Classic Kerma  it was clear that the morphology of AR-63, for some traits, is more similar to T41 than to LM2  in form space , being very close to each other. According to PC2, LM2 and MZ1 plot outside the RH variation in the negative values. MZ1 is the most distant while LM2 is much closer to the RH cluster. AR-63 is found on the periphery of the RH variation in the negative values.To further explore the position of fossils relative to RH we analyzed their distribution according to PC3 reflecting 10.2% of the total variance Fig. b. Along According to the plot of PC2 and PC3, we note that the three fossils are all excluded from the variability of the RH with AR-63 which is the closest, then LM2 and MZ1. On PC3, AR-63 shows the highest positive value and, as such, is clearly opposite of the two Neanderthals placed at the minimum end of the negative values with LM2 corresponding to the lowest value.The computation of the extreme morphologies of the two first PCs allows us to visualize the shape variation along each axis Fig. . With thPC2 discriminates individuals based on the outline of the acetabular area. In lateral view, the anterior half of this surface shows a more inferior orientation in the two Neanderthals accompanied by a more superior posterior half compared to the other specimens. By superimposing the Neanderthal mean shape calculated from the Procrustes coordinates of LM2 and MZ1, on that of the mean of the RH, it is clear, when positioning the ilium in lateral view, that the two Neanderthals have an acetabular area that is more posterolateral in orientation than that of the RH, for which it is placed anterolaterally Fig. a. In addAR-63, placed at the periphery of the RH cluster, is clearly distinct from Neanderthals by a more anterolateral acetabular area Fig. a but witvice versa. This morphological distinction is stronger posteriorly with a significant distance observed between the two posterior-superior iliac spines, with that of AR-63 and RH oriented more laterally than that of the Neanderthals, which is characterized by a particularly curved posterior region  such as the left ilium of La Ferrassie 8  or that of Dederiyeh 161 we can observe that their shape is very close to the two perinatal Neanderthals. We observe that this morphological distinction on the ilium is mainly noticeable in the orientation of the acetabular area, which is more posterolateral in the two Neanderthals. In our opinion, this morphological feature is in line with what has been described on the ilium of the Neanderthal adult Kebara 270. This morphological difference between Neanderthals and AMH has been proposed to be related to distinct posture and locomotor biomechanics in these two human groups70. In addition, Wolpoff71 describes, in relation to the pelvis of a much older fossil, that of Australopithecus africanus STS 1473 an orientation of the acetabulum more lateral to that of AMH. The "more laterally facing" acetabular area orientation has also been reported in chimpanzees (Pan troglodytes) relative to RH74. The identification of such a trait in ancient hominins, such as the Neanderthals, and great apes raises the question of the derived or plesiomorphic condition of the orientation of the acetabulum in RH. In any case, we consider that the more lateral orientation of the acetabulum in Neanderthal neonates testifies to its presence before any biomechanical or postural influences.The proximity observed between LM2 and MZ1 indicates a Neanderthal morphological homogeneity of the ilium in these two fossils representing groups separated chronologically by about 20,000 years and geographically by more than 3000 km , Peștera cu Oase (Romania), Bacho Kiro (Bulgaria), and Zlatý kůň (Czech Republic) unearthed in Western Eurasia and predating 37 kya cal BP  presents in superior view a curvature of the iliac crest clearly closer to that of AR-63 than to that of LM2 and MZ1, a feature that we have described as discriminating between RH and Neanderthals. For the Gravettian, only the two unpublished newborns from Krems-Wachtberg  would be available, as to date, no human fossil ilium deriving from the different phases of the Aurignacian or Protoaurignacian has been discovered. Concerning older fossils, the left ilium of the perinatal Qafzeh 13 , only partially well preserved, has an acetabular area with an anterolateral orientation on the basis of Figure 11 , therefore closer to the morphology of our RH sample. Altogether, these previous descriptions suggest that the lack of knowledge about the variability of Neanderthal and ancient AMH perinatal ilia does not limit us in our taxonomic interpretation of the ilium of AR-63.Regarding the chronological antiquity of AR-63, it would be interesting to include other fossil perinatal ilia from the UP. Based on previously published figures, the Epigravettian ilia of the Ostuni 1b fetus . This hypothesis would be consistent with the suggestion by some that layer admixture occurred at the GDR42. The bone could have come from overlying layer of the Aurignacian, as a result of human or large carnivore activities42.If we assume that Neanderthals (acculturated or not) are the only makers of the Châtelperronian49. However, taking into account the post-excavation history  and the very thin compact bone of the specimen, to date, no direct dating was undertaken.Only a direct dating of the piece could resolve this issue105, then the AR-63 ilium could be in situ but all the Neanderthal remains of the layer should be intrusive and derived from the underlying Mousterian (hypothesis H2). We consider H2 unlikely. Although a large number (n=38) of Neanderthal human remains were unearthed at the GDR in association with Châtelperronian material, much less were discovered in the underlying Mousterian layer (n=6). This situation contrasts with that observed in the nearby Grotte du Bison where many Neanderthal remains were found in the late Mousterian levels (n = 50) and none in the Châtelperronian. Furthermore, the surface distribution of Neanderthal remains in the Châtelperronian layers at the GDR seems hardly compatible with H229. Finally, and even more critically, H2 is at odds with the direct dating of one genetically identified Neanderthal from the Châtelperronian layer of the GDR matching the radiocarbon dates obtained in this layer14 and postdating the Mousterian layers at the site.If one assumes that AMH are the makers of the Châtelperronian42 have documented such significant disturbances.A third hypothesis (H3) could be that all human remains, both AR-63 and the Neanderthal fossils, are intrusive. Vertical movements, from top to bottom, in the case of AR-63, and from bottom to top, in the case of the Neanderthal pieces, would explain the presence of these remains within the Châtelperronian sequence. If such disturbances had occurred, they would have necessarily mobilized all the archaeological material, including the most characteristic Châtelperronian lithic pieces. However, none of the previous published studies that question the integrity of the archaeological levels of the GDRIf we assume that the integrity of the Châtelperronian levels is as good as that of the underlying Mousterian and those of the overlying UP, and that the perinatal ilium is indeed contemporary with the Neanderthal human remains of layer Xb2, then AR-63 would attest the presence of AMH in this area of western Europe during the Châtelperronian period. The makers of the Châtelperronian could then be human groups where Neanderthals and AMH coexisted (hypothesis H4) or the GDR could have been occupied alternately by distinct human groups, makers of the same techno-complex (hypothesis H5).104.If validated, H4 would bring strong support to the notion that the development of an UP like assemblage such as the Châtelperronian associated to Neanderthal makers at the time of the transition resulted from cultural diffusion or acculturation processes with possible population admixture between the two groupsHomo naledi . Clearly different from Neanderthal morphology, AR-63 also shows morphological peculiarities, such as its very "laterally offset" posterior-superior iliac spine, that exclude it from RH variability. This reflects, in our view, an ancient biologically modern phenotypic expression not previously documented within RH variability.The 3D morphometric analysis of the perinatal ilium from the Châtelperronian layer Xb2 of the GDR compared to two perinatal Neanderthal ilia and the RH collections allows us to attribute AR- 63 to the AMH as well as to discuss the presence of features on the ilium that distinguishing AMH and Neanderthals from the perinatal period. In addition, the curvature of the iliac crest distinguishes the two perinatal Neanderthals from the other ilia analyzed in this study, a feature that has not been described in immature individuals in detail previously. It would be interesting to integrate the older individual La Ferrassie 8 in order to reconstruct the growth trajectories of the Neanderthal ilium and to apply surface semi-landmarks on the whole bone, such as what was recently done for the ilium of the immature individual 52. Only a direct dating of AR-63 would give reliable information on its antiquity. Until now, its taxonomic identification documents in an unprecedented way the population dynamics of AMH groups that settled the middle latitudes of Eurasia since at least 45 kya cal BP107. Furthermore, additional analyses must be conducted to discuss the archaeological integrity of the Châtelperronian sequence of the GDR such as what has been done at Saint-Césaire23. Indeed, it incites to undertake taphonomic and spatial studies of the GDR remains since it is now the only site delivering human remains in Châtelperronian layers for which these kind of studies have not been carried out. Finally, if we accept the presence, on the same territory and associated with the same techno-complex, Neanderthals and AMH, it becomes crucial to test the hypothesis of a potential AMH genetic contribution within the genome of the Neanderthal individuals from Arcy-sur-Cure (hypotheses H4 and H5). Although the existence of genetic introgressions from ancient population of African origin into Middle Pleistocene Neanderthal genomes have already been identified110 , a similar phenomenon remains to document for the time period of the Neanderthal replacement in Europe.We are aware that these results will need to be complemented by an expanded study of newborn ilia from other RH populations and fossils from the late Upper Pleistocene and Early Holocene.AR-63 has been unearthed in the level Xb2, i.e., at the base of the Châtelperronian sequence of the GDR, within which 11 Neanderthal fossils were also found. We consider the most parsimonious hypothesis, in the absence of any direct absolute dating, that this ilium originates from the chronological period corresponding to the transition between the MP and UP in this region of western Europe, between 45 and 41 kya cal BPWe selected the ilia  of 4 highly premature individuals, 32 full-term and newborn individuals and 9 young children Table .Table 1N62.The fossil sample consists of 2 Neanderthal individuals: LM2 and MZ1 and the individual AR-63. The very limited fossil comparative sample is due to the scarcity of perinatal remains from between the end of MIS 5 to MIS 4 and 3, and especially of well-preserved onesAR-63 is a very well preserved right ilium of a perinatal individual, with a maximum height of 29 mm and a maximum length of 32 mm.Regarding its contour morphology, in lateral or medial view Fig. a, its an112, a Levallois techno-complex according to Jaubert et al. (2011)88, Layer J has been dated by thermoluminescence and provided an age of approximately 40.3 kya +/− 2600 BP113. Electron spin resonance (ESR) dating of the underlying layer H yielded more recent mean ages114. In 2014, Higham et al. dated several bone remains from the Le Moustier sequence. Regarding the 5 pieces from layer J that provided a result, the dating fell between 45,100 +/− 2300 and 37.6 kya +/− 900 cal BP (Ref.115). The authors concluded that across the 20 horizons analyzed, the late Mousterian chronological interval was between 41,030 and 39,260 cal BP (at 95.4% probability). These latter results should be viewed with caution given the new understanding of the stratigraphy of this shelter116.LM2 is an almost complete skeleton of a Neanderthal newborn from the Lower shelter of Le Moustier, located in the Dordogne, about 45 km southeast of Périgueux (France) Fig. a. Only tcirca 2 years old) with a well-preserved iliac crest, such as the left ilium of Dederiyeh 1 , we can state that this depression in the iliac crest of LM2 is related to a taphonomic alteration and not an anatomical feature. As viewed from superior, the curvature of its iliac crest clearly differs from that of AR-63 by being more straight  Fig. a. The skWith a maximum height of 32 mm and a maximum length of 38 mm, Mezmaiskaya 1 is more damaged than AR-63 and the left ilium of LM2, particularly at the anterior-superior margin, which is devoid of its iliac spine. The entire upper margin of the iliac crest is severely eroded, revealing cancellous bone down to the posterior-superior spine. Despite this alteration, it is possible, in superior view, to see the curvature of its iliac crest which is straight as in LM2 120. We selected the ilia of 25 individuals (17 left and 8 right). The historical collection comes from a French medieval parish cemetery from Provins 121. The ilia of 20 individuals (17 left and 3 right) were included in this study.The RH sample consists of a proto-historical and historical collections Table  from recBoth collections are curated at the laboratory of PACEA laboratory .2 within an ancient alluvial terrace and buried under a layer of 20 to 40 cm of sediment120. Among these graves, 64 immature individuals were found, 54 of which had an estimated age-at-death between 24 and 46 weeks of amenorrhea based on the estimated stature from the maximum length of long bones67.Located on the Nile between the second and third cataract in northern Sudan, this site has yielded a complex of 66 burials related to the end of the Classical Kerma period (1750-1500 B.C.), spread over an area of 150 m122. This site yielded a total of 111 primary individual burials including those of 90 children123. 35 individuals have an age at death that has been estimated to be between 26 and 46 weeks of amenorrhea67.Located in the Parisian Basin, 77 km east of Paris, the church and parish cemetery of Saint-Ayoul de Provins were the subject of programmed excavations that uncovered in-place burials with a chronology ranging from the 13th to 18th centuries AD124, i.e. the method of Fazekas and Kósa (1978)55 revised by Sellier (cited in Ref.125). Since these formulae have been developed for various long bones, they can be applied to individuals from the fetal stage up to four years of age126 and provide confidence intervals. We then distinguish sub-age groups among the age group of individuals who died during the perinatal period lato sensu: highly premature individuals with mean age-at-death ranging between 24 and 34 completed weeks of amenorrhea, slightly premature and full-term individuals with mean age-at-death ranging between 35 and 40 completed weeks of amenorrhea and newborns with mean age-at-death ranging between 41 and 48 completed weeks of amenorrhea.As part of the constitution of our comparison sample, we based our age-at-death assessment methodology for very young individuals on the one chosen by one of us (C.P.)In an attempt to place the remains of AR-63 and the two Neanderthal neonate ilia within the morphological variability of RH ilia, we undertook a 3D geometric morphometric analysis that required the microtomographic acquisition of these specimens.Arcy's ilium was scanned at the Department of Human Evolution of the Max Planck Institute Leipzig  using a tabletop micro-CT scanner (SkyScan 1173) with a resolution of 31 μm.127.The acquisition of LM2 was performed at the Musée National de Préhistoire in Les Eyzies-de-Tayac (France) using an industrial μCT scanner (BIR ACTIS 225/300) from the Max Planck Institute for Evolutionary Anthropology  with an isotropic voxel resolution of 70 μm118.The 3D scanning of MZ1 was performed at Pokrovskaya Hospital in St. Petersburg (Russia) using helical CT Finally, the RH ilia were scanned at the PLACAMAT platform  using X GE™ V/TOME/X S microtomographic equipment with a resolution of 30 μm.® 7.0.1. software , the ilium surfaces were first segmented using the density based thresholding using the histogram tool, then they isolated using the tool extract subvolume. From these surfaces, the contour of the iliac bone, when oriented in lateral or medial view, and that of the acetabular area were quantified by manually applying a template of 72 landmarks (5 fixed landmarks and 67 semilandmarks) on each ilium according to a protocol available in Supplementary Information, Supplementary Methods. Using the Avizo® landmark module, anatomical landmarks were placed on the iliac spines: anterior-superior, posterior-superior and posterior-inferior, and two others were positioned anterior-posteriorly on either side of the acetabulum. Using the Avizo® B-spline function, the semilandmarks were placed on the outline of the iliac bone through the outline of the acetabular area  interpolation function . After mirroring the landmark configurations of the 15 right ilia, due to the higher number of left ilia in our comparative sample, the bending energy was minimized by sliding the semilandmarks which can then be considered homologous133. The landmark coordinates were then converted to shape variables via a generalized Procrustes analysis . Results of the PCA were visualized by computing extreme shapes on each axis (±5 SD from the mean).The landmark coordinates were then imported and processed in RStudio139 and discuss the taxonomic attribution of the right ilium of the Châtelperronian perinatal AR-63. Finally, to identify which individuals are most similar in shape to AR-63 we used the Neighbor joining method between pairwise Procrustes distances to identify nearest neighbors based on shape similarity140.A PCA was conducted to visualize the distribution of specimens in form space and shape space to analyze their size and shape variationSupplementary Information.


Duplex stainless steel is a unique material for cast products, the use of which is possible in various fields. With the same chemical composition, melting, casting and heat treatment technology, pitting and crevice corrosion were observed at the interphase boundaries of non-metallic inclusions and the steel matrix. To increase the cleanliness of steel, it is necessary to carefully select the technology for deoxidizing with titanium or aluminum, as the most common deoxidizers, and the technology for modifying with rare earth metals. In this work, a comprehensive analysis of the thermodynamic data in the literature on the behavior of oxides and sulfides in this highly alloyed system under consideration was performed. Based on this analysis, a thermodynamic model was developed to describe their behavior in liquid and solidified duplex stainless steels. The critical concentrations at which the existence of certain phases is possible during the deoxidation of DSS with titanium, aluminum and modification by rare earth metals, including the simultaneous contribution of lanthanum and cerium, was determined. Experimental ingots were produced, the cleanliness of experimental steels was assessed, and the key metric parameters of non-metallic inclusions were described. In steels deoxidized using titanium, clusters of inclusions with a diameter of 84 microns with a volume fraction of 0.066% were formed, the volume fraction of which was decreased to 0.01% with the subsequent addition of aluminum. The clusters completely disappeared when REMs were added. The reason for this behavior of inclusions was interpreted using thermodynamic modeling and explained by the difference in temperature at which specific types of NMIs begin to form. A comparison of experimental and calculated results showed that the proposed model adequately describes the process of formation of non-metallic inclusions in the steel under consideration and can be used for the development of industrial technology. The key property of duplex stainless steels (DSS) is resistance to pitting and crevice corrosion . The useThe influence of NMI types on pitting formation has been studied in detail ,13,14,15There are many factors that influence the type of NMIs: the amount of introduced rare earth metals, the technology of preliminary deoxidation, and the oxygen and sulfur content in the melt before modification. Depending on these factors, the formation of higher REM oxides ,22, variIn an earlier work by the authors, three steels deoxidized with titanium, aluminum, and rare earth metals were studied and the effect of deoxidizers on resistance to pitting and crevice corrosion was shown . HoweverThe experimental steel was melted in an open induction furnace. It was necessary to evaluate the effect of deoxidation and modification technology, with other parameters being equal in the experiment. Thus, one melt was carried out, producing ingots weighing 1 kg with a cross-sectional width of 40 mm after each addition of deoxidizers. This made it possible to obtain three ingots of the same chemical composition, with different concentrations of deoxidizing agents. Three deoxidation schemes were used: (1) titanium; (2) titanium and aluminum; and (3) complex deoxidation with titanium, aluminum and rare earth metals. Thus, deoxidizers and modifiers were introduced in order to increase the steels’ affinity for oxygen. The same experimental technique made it possible to trace the evolution of NMIs’s transformation during their modification. Since during the production of cast casings of centrifugal pumps, solidification proceeds slowly, molds for experimental ingots were made from a quartz sand with liquid silica. The melt was poured off from the furnace at a temperature of 1440 °C. The chemical composition of the experimental ingots was determined using an optical emission spectrometer Experimental ingots with a cross-sectional width of 40 mm were cooled in sand molds at a rate of 1–5 °C/s , so a σ-The chemical composition of non-metallic inclusions was determined using a TESCAN Mira-3M  scanning electron microscope equipped with an energy-dispersive spectrometer (SEM-EDS).[R] and [X] represent the concentration, respectively, of the deoxidizer and oxygen, sulfur or nitrogen in the melt, mass. %; n and m are stoichiometric coefficients.Thermodynamic modeling was used to predict the type and composition of NMIs. To predict the equilibrium types of NMIs during deoxidation of DSS with titanium, aluminum and rare earth metals, stability diagrams of equilibrium inclusions were created. This method for calculating isothermal sections of multicomponent phase diagrams was proposed in ,31,32, d[j] is the concentration of the element, weight %.The activity coefficients of the components of the metal melt were calculated using the Wagner formalism. In this work, we took into account first-order interaction parameters and their temperature dependences (3) :(3)lgfRTaking the logarithm of expression (2), we obtain the transcendental Equation (4):Similar transformations can be carried out for all reactions of the formation of the corresponding oxide or sulfide, taking into account the interaction parameters for all elements included in the steel ,36,37 A. As a reAlong with the formation of pure stoichiometric inclusions, thermodynamic modeling considered the formation of complex inclusions in solid solutions. These are the particles most often found in steel . Due to Based on the law of the conservation of mass in the system, a balance model was developed for the NMIs’s formation during deoxidation, the cooling of liquid steel and the solidification, similar to the approach in ,40. The S is the area, µm2, and P is the perimeter, µm. Thus, the shape factor is defined so that 1 corresponds to an ideal round inclusion, and the lower its value, the further the shape of the inclusion is from being round.A preliminary analysis of the NMIs was carried out using an optical microscope. Optical metallography makes it possible to accurately determine the volume fraction of inclusions, their sizes and other metric parameters in ingots. The assessment results are shown in In Steel 2 with titanium and aluminum, the volume fraction of single NMIs is smaller (0.014% with an average size of 5 µm). However, in steel 2, clusters were also found at a volume fraction of 0.01%, with an average size of 58 µm, and the size of the largest cluster reaches 128 µm. After treatment with titanium, aluminum and rare earth metals in Steel 3, the volume fraction of single inclusions practically did not change and their volume fraction was 0.016%, while the average size of the inclusions was constant and amounted to 5 µm. There were no clusters in steel 3. It should be noted that the distribution density of inclusions in all cases changed, as did the volume fraction of inclusions. However, in steel with rare earth metals, the distribution density is higher, since due to the lower interfacial energy of rare earth metal inclusions and liquid steel, the nucleation processes of inclusions proceed more intensely .The shape factor of single inclusions in Steel 1 with titanium and in Steel 2 with titanium and aluminum is almost the same, at 0.65–0.66. However, the shape factor for clusters is 0.08–0.12. This is due to their branched morphology and large size. In Steel 3 with titanium, aluminum and rare earth metals, the shape factor of single inclusions increases to 0.75, and the clusters are completely removed a,b. A smThe formation of single inclusions and clusters is associated with differences in chemical composition. Chemical composition of non-metallic inclusions and their electronic images presented in 2O3·Al2O3 ; in region V, it is solid Ti3O5 particles; in area VI it is solid Ti2O3 particles; and in area VII, it is TiN. At a concentration of [Ti] ≥ 0.016%, non-metallic inclusions of the Ti3O5 type are formed. When only 0.00004% aluminum is added to the melt, 3Al2O3∙2SiO2 inclusions are formed, and when the aluminum concentration reaches 0.0001%, Al2O3 inclusions reach an equilibrium in almost all titanium concentration ranges. The second case is the reduction of titanium from the oxide, and its rate is limited by diffusion through the solid oxide layer. In the second case of preliminary deoxidation with titanium and aluminum and the subsequent addition of almost any amount of lanthanum or cerium, the formation of aluminum and titanium oxides is suppressed. According to 2O3∙11Al2O3; region II consists of solid particles of Ce2O3∙Al2O3; region III consists of cerium sulfide, Ce2S3; and region IV consists of lanthanum sulfides, LaS. In a melt containing aluminum, at low concentrations of cerium, complex inclusions of the Ce2O3∙11Al2O3 type occur. At a concentration of [Ce] ≥ 0.0013%, NMIs of the Ce2O3∙Al2O3 type are formed. The formation of lanthanum sulfide (LaS) is also possible at a concentration of [La] ≥ 0.0011%.The phase regions for the first case of deoxidation using titanium and aluminum are considered. In Thus, when adding any amount of aluminum into this steel, the subsequent addition of titanium will not have a significant effect on the processes of oxide formation. Conversely, if steel is first deoxidized with titanium, its oxides will form, and with further addition of aluminum, the transformation of inclusions will begin. This process can occur according to two mechanisms. The first is the dissolution of a nonequilibrium inclusion and subsequent formation in an equilibrium form, the rate of which is determined by the melt mixing conditions that determine the kinetics of the process . The add2O3  were formed , lanthanum oxides (La2O3) and lanthanum sulfides (LaS) are formed. As the temperature decreases, the amount of lanthanum sulfide increases slightly. Clusters are not found in this steel because titanium and aluminum are completely reduced by rare earth metals from oxides and nitrides. Titanium nitrides were found along the boundaries of the inclusions, apparently forming below the solidus temperature. Note that, as in the case of Steel 2, even after REM treatment, untransformed inclusions can be found in the ingot, partially preserved until the complete solidification of Steel 3. In In Steel 3, deoxidized with titanium, then aluminum, and then modified with cerium and lanthanum, almost all inclusions are primary c. At a tIn all three steels, at the end of solidification, when the liquation of sulfur reaches a significant value, the formation of manganese sulfide occurs. But since the temperature of its formation is already close to the solidus temperature, its formation occurs incompletely and in small quantities, and most of the MnS inclusions are precipitated in the solid phase.Thus, thermodynamic modeling almost completely predicts the types of non-metallic inclusions observed in experimental steels and makes it possible to explain their nature, including the formation of clusters and different degrees of transformation of preliminary deoxidation products. However, non-metallic inclusions characteristic of earlier stages of processing were found in all experimental steels. Their nature is associated with the incomplete occurrence of transformation processes of inclusions upon sequential introduction of deoxidizers. Based on the results obtained in this work, it can be concluded that the optimal treatment of liquid DSS should include preliminary deoxidation with titanium or aluminum, aimed at reducing the initial oxygen content. However, in practice, there is some delay in the transformation of inclusions, caused by the relatively low rate of diffusion of strong elements through the layer of solid inclusions. In this case, when most of the oxygen is in the oxides of titanium and aluminum, the introduction of rare earth metals is aimed primarily at modifying existing inclusions, and not at deoxidation.


Selective laser melting (SLM) has significant advantages in the near net shape manufacturing of metal parts with complex geometries. However, SLM parts usually have problems such as poor surface quality and low dimensional accuracy, which require post-processing. This paper focuses on the research around the influence of ultra-precision micro-grooving the SLM Ti6Al4V alloy on the cutting force and serrated chips. The influence of the processing parameters on the cutting force and surface processing quality was analyzed in detail, and the cutting simulation model of the SLM Ti6Al4V alloy was established. The formation process of the serrated chip was successfully simulated, and the experiments verified the reliability of the established model. The research results show that the dynamic cutting force and surface processing quality are mainly related to the depth of cut, and the two trends are consistent. It is also shown that the serrated chip begins on the free surface of the workpiece and propagates deeply in the shear zone, forming a shear band, and its serrated nodules move upward and forward to form periodic serrated chips. Additive manufacturing (AM) technology has significant advantages in the rapid prototyping and near net line manufacturing of complex-shaped parts and has great potential in aerospace, biomedical and other fields ,2,3. TheIn recent years, relevant scholars have conducted processing research on the post-processing of additive manufacturing ,9,10,11.In addition, SLM Ti6Al4V has a low thermal conductivity and a relatively slow heat dissipation. It is relatively easy to form adiabatic shear bands and serrated chips in traditional cutting, which accelerates the tool wear and affects the surface quality of the machined parts. To reduce the experimental cost and time constraints, it is necessary to conduct a finite element numerical analysis to overcome the experimental limitations and better explain the necessary methods of the experimental mechanism . P. DidiAlthough much research has been done on traditional titanium alloy processing, there needs to be more research on the machinability of the SLM Ti6Al4V alloys, especially the finite element modeling analysis of the SLM Ti6Al4V alloys in serrated chips and the cutting force. Therefore, this paper investigates the serrated chip formation process of additively manufactured parts and the influence of the machining quality during micro-grooving.The experimentally processed sample SLM Ti6Al4V was prepared using a 67.5° scanning strategy. The laser power was 200 W, the spot size was 75 μm, the thickness of each layer was 30 μm, the laser scanning speed was 1500 mm/s and the laser scanning spacing was 65 μm. The detailed preparation strategy is shown in The material’s microstructure directly determines the material’s mechanical properties and affects the cutting force. Since the SLM Ti6Al4V alloy has a line-by-line layered stacking structure, the sample microstructures in each direction are different. After the sample was processed, a JEOL JSM-5500 scanning electron microscope (SEM) was used to observe the chip morphology. A Veeco3D confocal microscope was used to observe the profile of the processed surface to evaluate the surface processing quality. In addition, for the reliability of the FEM model established in this paper, the accuracy of the FEM was verified by comparing the geometric characteristics and the cutting force of the chips in the experimental and simulation processes as the evaluation criteria. In this section, the chips generated under different cutting parameters were collected and mounted using epoxy resin, and after polishing and etching, the chip morphology was obtained using scanning electron microscopy (SEM).In this section, Abaqus/Explicit was used to model the cutting process model of SLM Ti6Al4V, and the cutting model was divided into meshes using Lagrangian meshing technology. As shown in p, mT and rT represent the melting temperature and room temperature, respectively. A, B, C, n and m represent the initial yield stress, hardening modulus, strain rate sensitivity coefficient, strain rate dependency coefficient and thermal softening coefficient, respectively. The detailed parameters of the JC constitutive are shown in Johnson and Cook first proposed the Johnson–Cook model, which has been widely adopted in metal cutting simulations for considering the effects of strain hardening, strain rate hardening and thermal softening. At the same time, the model reflects the mechanical stress of metals under high strain, a high strain rate and high temperature, and is usually used for ductile materials. The flow stress is expressed by Equation (1) .(1)σ=A+In order to simulate the failure model of the material, the paper adopted the JC damage fracture model, and the specific formula is as follows (2)ε¯JC=Surface topography is an important characteristic that can be used to evaluate the surface quality of SLM parts. However, since the additive manufacturing process is a top-down, layer-by-layer processing method, the environment and temperature fields on the sides of the sample are different during the manufacturing process. The scanning traces and unmelted particles adhering to the sample also affect surface roughness. The cutting force is an important factor in evaluating the machinability of materials, and it is closely related to the materials’ mechanical properties and processing parameters. Compared to the traditional Ti6Al4V alloy, the hardness, yield strength and ultimate tensile strength of SLM Ti6Al4V were significantly higher than that of the forged Ti6Al4V alloy, which also meant that the cutting force was different from the traditional titanium alloy, which had specific research significance .During ultra-precision machining, the cutting force is closely related to the mechanical properties and microstructure of the material. In order to further understand the difference between the titanium alloy and the traditional cast titanium alloy, it was necessary to compare the differences in the cutting force between them. The roughness is an important index to evaluate the processing quality of the micro-groove. In this paper, the surface roughness (Sa) was used to evaluate the quality of the processed surface. In addition, to evaluate the profile roughness of the micro-grooves and avoid the errors caused by the grooves and inclinations, the sample profile should be flattened (cylinder and tilt) before the roughness evaluation. The chips in ultra-precision machining are a critical evaluation of machining performance. The state of the chips can provide important information during the cutting process. In addition, the cutting force and surface roughness are also affected by chips . Figure To further quantify the formation law of the serrated chips and verify the accuracy of the finite element model, the chip samples were prepared and polished to obtain a cross-sectional view of the chips. Due to the serrated chip morphology and cutting force being closely related to the field variables , the chip geometrical characteristics and cutting forces were generally employed as an evaluation criterion to verify the accuracy of the FEM . The chiThe selective laser melting (SLM) process has significant advantages in the near net shape manufacturing of metal parts with complex geometries. Unfortunately, SLM parts usually suffer from poor surface quality and low dimensional accuracy, requiring post-processing. Therefore, this paper focused on the influence of ultra-precision micro-grooving of the SLM Ti6Al4V alloy on the cutting force and serrated chip, established a finite element cutting simulation model of the SLM Ti6Al4V alloy and verified the reliability of the established model through experiments. The specific research is as follows.(1)The cutting simulation prediction model of the SLM Ti6Al4V alloy was established and the cutting force and serrated chip parameters predicted under the same cutting parameters agreed with the experimental results. The research results showed that the serrated chip began on the free surface of the workpiece and propagated deeply in the shear zone to form a shear band. The nodules moved upward and forward, forming the serrated chips.(2)The dynamic cutting force was mainly related to the depth of cut and gradually increased. The dynamic cutting force gradually increased and the range was between 0–10 N. However, the cutting speed parameters had less influence on the dynamic cutting forces.(3)The surface roughness (Sa) of ultra-precision micro-grooving of the SLM Ti6Al4V alloy fluctuated around 0.1 μm and its Sa gradually increased with the increase in the depth of cut. However, the correlation with the cutting speed was small and its changing trend was consistent with the cutting force.


To describe the bibliometric characteristics of the world scientific production in inguinal hernia surgery using the Web of Science Core Collection (WoS) database.A search for original articles published in the WoS up to December 2021 was carried out. Bibliometric analysis was performed using the Bibliometrix data package in R and VOSviewer, version 1.6.17.A total of 3,795 articles were identified in the database, with an annual average of 81 published articles and sustained growth with the highest number of publications in 2021. The article “Wide Variation and Excessive Dosage of Opioid Prescriptions for Common General Surgical Procedures” published in 2017 was the most cited (517 citations). The United States was the country of the corresponding author with the highest number of published articles (725 articles). The author with the highest number of published articles was Rosenberg J., affiliated with the University of Copenhagen, with 52 articles and 765 citations. The journal “Hernia” had the highest number of articles published on inguinal hernias in the WoS, representing 18% of the total number of articles. Finally, the keywords most used were “herniorrhaphy” and “hernioplasty” in 2008, and more recently: “single incision” “society guidelines” and “single port”.The scientific production of original articles on inguinal hernia surgery has increased. There is currently great interest in research on minimally invasive surgical methods and standardization of procedures based on clinical practice guidelines. This abBibliometric studies make it possible to measure and analyze the scientific production of different countries or regions and provide a general overview of certain research topics. This type of research is necessary to systematically verify and summarize this information due to the increase in the number of publications . LikewisCurrently, bibliometric studies cover abdominal wall hernias; however, no bibliometric analysis of the publications on inguinal hernia, which is the most frequent condition among abdominal hernias, has been developed . LikewisThe present study aimed to evaluate the bibliometric characteristics of the global scientific production of original articles on inguinal hernia surgery in the WoS Core Collection database. This analysis may be useful for professionals seeking guidance related to specialization opportunities in the field of inguinal hernia surgery and may be used for decision-making at the level of governments, universities, and research centers/groups to reorient funding and research priorities in this area.2.A bibliometric analysis of original articles on inguinal hernia indexed in the bibliographic database of the WoS Core Collection from inception up to May 12, 2022, was performed . All the2.1.The following search terms were used: #1, TS = ; #2, TS = (“groin hernia”); #3, TS = ; #4, TS = (groin AND hernia*); #5, #1 OR #2 OR #3 OR #4, where TS means “title and subject”; and were searched in the categories of “Surgery” and filtered to those of the type “Article”; we used the Science Citation Index Expanded (created in 1964) of WoS Core Collection . The sea2.2.The metadata from the full records obtained from the search were collected and downloaded as “ciw” files. Subsequently, using Rayyan , two aut2.3.The bibliometric indices of the articles published for the study were obtained using the VOSviewer software and the data package Bibliometrix in R , 15. AlsWe aimed to report the following outcomes: total articles, total citations, average number of articles published annually, trends of articles and citations over the years, the ten most cited articles on inguinal hernia , the twenty corresponding authors countries with the highest number of published articles , the ten authors with the highest production of articles on inguinal hernia , the twenty journals with the highest number of articles published , co-authorship network according to authors, co-authorship network according to institutional affiliations, and co-occurrence network of keywords. The Normalized Total Citations (NTC) is calculated by the Bibliometrix in R by dividing the total number of citations of an article by the average number of citations of all documents published in the same year; this measure takes into account differences in citation practices across disciplines and provides a more meaningful comparison of citation impact . Also, wIn addition, the GDP per capita of the countries that are among the twenty corresponding authors countries with the highest number of published articles and the population size of each country were collected from the World Bank website , 18; the2.4.The conduct of the study did not require the approval of an ethics committee because it was an analysis of the records of studies published in a database. The Universidad Científica del Sur approved the execution of this study than will serve as a partial requirement for two of the authors to obtain their bachelor's Degree (EICC and JYEQ).3.We conducted a search in the WoS and found a total of 4,025 articles on inguinal hernia. After evaluating the titles and abstracts in Rayyan, we obtained 3,795 articles in the database, with an average of 81 articles published annually. These articles received a total of 71,349 citations. The 10 most cited articles on inguinal hernia in the WoS had a total citation range between 270 and 517. The article with the highest number of citations (517) was titled “Wide Variation and Excessive Dosage of Opioid Prescriptions for Common General Surgical Procedures” , publishThe United States had the highest number of published articles with a corresponding author affiliated with the country, with 725 articles, followed by the United Kingdom and China, with 284 and 280 articles. In terms of publications with authors from the same country, the United States led with 673, followed by China with 265 and the United Kingdom with 261. In relation to multiple country publications, Brazil led with 92 articles, followed by United States with 52, and Italy with 26. Rosenberg J., affiliated with the University of Copenhagen, had the highest production of articles on inguinal hernia, with a total of 52 published articles and 765 citations, followed by Kockerling F. , and Nordin P.  . The topHernia had the highest number of articles published on inguinal hernia in the WoS, with 687 articles, representing 18% of the total articles included in the analysis. Surgical Endoscopy and other Interventional Techniques was in second place with 351 articles  followed by the Journal of Laparoendoscopic & Advanced Surgical Techniques with 173 articles . Annals of Surgery had the highest impact factor of 12.96, as compiled from Journal Citation Reports in Clarivate, followed by the British Journal of Surgery with 6.94 and the Journal of The American College of Surgeons with 6.113. According to the WoS classification, most journals were Q1, 4 belonged to Q2, 5 to Q3, and 3 to Q4, , followed by Surgical Endoscopy and Other Interventional Techniques , both based in the United States (The United States had the most publications in our analysis, likely due to its high investment in research and development, which reaches 3.45% of the GDP per capita . Factorsd States , 31. Thed States .Regarding the co-occurrence network of terms in the WoS Core Collection database, “herniorrhaphy” and “hernioplasty” were initially the most commonly used terms, both referring to open surgical approaches involving incisions and suturing layers of tissue or using a mesh to correct hernial defects . BetweenAmong the limitations of the study, the total number of citations of an article does not necessarily reflect its impact on disease management. The duration during which citations are accumulated is also an important factor to consider since recent articles may not yet have enough citations to surpass older publications. Furthermore, the search was only conducted in WoS, which may exclude studies published in other databases, such as Scopus or MEDLINE. However, WoS offers a variety of metrics and indicators beyond the impact factor, allowing for a more comprehensive and unbiased analysis of the journals and the quality of the information. WoS also provided vast information which has allowed the standardization and updating of these topics.In conclusion, the evolution of research on inguinal hernia surgery in recent years is significant, with a shift from focusing on early surgical techniques to the development of new technologies and minimally invasive methods, as well as the creation of clinical practice guidelines. Knowledge of current trends in research in inguinal hernia surgery can be useful for the adoption and proposal of new lines of research, at the level of universities and research centers/groups. Likewise, these results can guide budgetary priorities for government agencies and industry to finance the development of new techniques, materials, and devices that seek to optimize the surgical management of inguinal hernias.


The real-world vaccine protection rates (VPRs) against the severe acute respiratory syndrome coronavirus 2 (SARS‑CoV‑2) infection are critical in formulating future vaccination strategies against the virus. Based on a varying coefficient stochastic epidemic model, we obtain 7 countries’ real-world VPRs using daily epidemiological and vaccination data, and find that the VPRs improved with more vaccine doses. The average VPR of the full vaccination was 82% (SE: 4%) and 61% (SE: 3%) in the pre-Delta and Delta-dominated periods, respectively. The Omicron variant reduced the average VPR of the full vaccination to 39% (SE: 2%). However, the booster dose restored the VPR to 63% (SE: 1%) which was significantly above the 50% threshold in the Omicron-dominated period. Scenario analyses show that the existing vaccination strategies have significantly delayed and reduced the timing and the magnitude of the infection peaks, respectively, and doubling the existing booster coverage would lead to 29% fewer confirmed cases and 17% fewer deaths in the 7 countries compared to the outcomes at the existing booster taking rates. These call for higher full vaccine and booster coverage for all countries. Severe acute respiratory syndrome coronavirus 2 (SARS‑CoV‑2) has been circulating globally with a sequence of emerging variants since the start of the pandemic. Particularly, the Delta and Omicron variants have contributed to surges in the infected cases across the globe due to their high transmissibility ,2. To prThe booster dose had been shown to increase protection against infection. For homologous or heterologous booster doses of Pfizer, AstraZeneca, and Moderna, the effectiveness was 82.3%–97.0% against Delta and 55.6%–73.9% against Omicron, with higher effectiveness using Moderna as the heterologous booster dose , which iDifferent from vaccine efficacy and effectiveness, the real-world VPR is defined as one minus the ratio of the infection rate of the vaccinated over the infection rate of the unvaccinated population of a country. The VPR measures the combined effectiveness of vaccines administrated in a country at a particular age distribution and nonpharmaceutical intervention (NPI) measures against COVID-19. The impacts of these factors are not necessarily evaluated in the homogeneous clinical trials, cohort studies, or case–control studies. Indeed, the conventional vaccine efficacies are pegged to a specific vaccine or a mix of vaccines in the clinical trials after excluding a certain part of the population, which may not conform to the population characteristics of the country. Therefore, the available vaccine efficacy or effectiveness does not necessarily reflect the vaccine immunity level of the whole population against different variants of the SARS-CoV-2 virus. Hence, it is crucial to obtain the real-world VPRs of a country to timely evaluate the effect of receiving the full and the booster vaccination on protecting the population in the whole country, which can provide quantitative evidence for the effectiveness of booster vaccination, dispel public doubts about the necessity of booster vaccination, and contribute to the public health policy decision-making.Various models were proposed to study the spread of COVID-19, the behavior of SARS-CoV-2 cells’ infection, and some related biological mechanisms, such as the fractional order models ,15, the We considered 7 countries that are representative for different types of vaccines with a sufficient number of confirmed cases  after vaccination. Specifically, the results of the US may be used to show the effect of mRNA vaccines (Pfizer and Moderna); the 3 European countries, UK, Italy, and Germany, mainly used a mixture of nonreplicating viral vector vaccines (AstraZeneca) and mRNA vaccines; the 2 south American countries, Brazil and Peru, utilized the inactivated vaccines (Sinopharm and Sinovac), AstraZeneca, and Pfizer; Turkey used the inactivated vaccines at the beginning and then started Pfizer. In this paper, the full dose means one dose for Janssen and 2 doses for the other brands to complete the primary vaccination. Those who have not completed the full vaccination are called partially vaccinated. The booster shot means one dose after full vaccination. The coverage rates of the partial, full, and booster vaccines in population are reported in Fig. For each country, we estimate the VPRs in the 6 consecutive nonoverlapping post-vaccine periods: the pre-Delta, Intervening I, Delta-dominated, pre-Omicron, Intervening II, and Omicron-dominated periods. Details of these periods are provided in Materials and Methods.The real-world VPRs for the partial, full, and booster vaccination in the 7 countries in the 6 post-vaccine periods are reported in Fig. The booster shot was started in the pre-Omicron period when the Delta was dominant. Our study shows that it readily restored the VPRs to 78.8% to 97% , which means that the booster vaccination’s VPRs were 20.5% to 31.8%  higher than those of the full vaccination against the Delta variant. In the Omicron-dominated period, the booster shot’s VPRs ranged from 55.6% to 67.0% , largely staying above the 50% threshold. These suggest that the booster shot provided enhanced and effective protection against both the Delta and Omicron variants.The VPR for a specific vaccine brand is similarly defined as the relative reduction in the infection rate of the population with that type of vaccine compared to the population without any vaccine protection in a country. It is noted that the information on the number of vaccine intakes for different brands of vaccines is unavailable in the national level statistics, which prevents us from directly measuring the VPR for a specific vaccine brand in a country. However, our approach is readily applicable, should the data information be available. In that case, we would conduct a regression analysis on the relationship between the estimated VPRs of each country and the intake proportions of different vaccine brands of this country in a certain time period.To further evaluate the protection of the COVID-19 vaccination, we investigate the impacts of the full and the booster vaccination on the size of the epidemics and deaths. Five vaccination scenarios were designed: (a) no vaccination at all; (b) receiving the partial but no full vaccination; (c) receiving the partial and full vaccination but no booster shots; receiving the booster shots only at half (d) and twice (e) of the actual daily booster coverage rates. The impacts of these scenarios were projected using the stochastic epidemic model with the estimated parameters for each country. See the specific designs of the scenario analysis (SA) in Materials and Methods.The projected cumulative confirmed cases and deaths during the pre-booster vaccine periods from the start of vaccination to the start of boosters for the 7 countries  under the no-vaccination (a) and only the partial vaccination (b) scenarios are shown in Fig. The lower increase in the confirmed cases and deaths under the no- and partial vaccination scenarios in Peru was due to its low and slow pace of vaccination, with only 6% and 3% of the population having received the partial and full vaccination within the first 100 days of vaccination. In contrast, 13.4% to 18.9% and 5.5% to 13.9% of the populations had been partially and fully vaccinated in Germany, Italy, Brazil, and Turkey, and the US and UK had the highest vaccination rates of 27.7% to 48.1% and 15.5% to 15.9% for the partial and full vaccination over the same period  and 94  thousands in total confirmed cases and deaths for the 7 countries in the post-booster period. It is noted that the relatively large reductions in the confirmed cases and deaths in Germany, Italy, Turkey, and UK under the double-booster scenario were due to their actual higher (more than 40%) rates of taking the booster shots by 2022 March 15  under the no-booster scenario and decrease by 26% to 63% mean 41%, SE 5%) relative to the observed peaks under the twice booster scenario, which verifies that the booster doses can further relieve the pressure on the healthcare system in the Omicron era due to its higher VPRs against the Omicron variants as reported in Fig. %, SE 5% It is noted that, in Italy, the projected peak under the partial vaccination scenario was higher than that under the no-vaccination scenario in the Intervening II period. This was due to the fact that a considerable proportion of population would have been infected in the Delta-dominated period under the no-vaccine scenario, and Italy had the highest rate of partial vaccination before the Intervening II period among the 7 countries Fig. A. HoweveThis study targets the population protection rates of vaccines. Although the vaccine efficacies are different between different age groups, our results reveal the overall protection rates of a country, which are informative on the total infection size and the demand on the health resources of a country. It is shown that the real-world VPRs of the partial, full, and booster vaccination decreased with time. The full vaccination was effective before Omicron with the VPRs remaining above 50%, which became insufficient when the Omicron was dominant. The booster shot was effective in slowing down the epidemics in both the Delta- and Omicron-dominated periods with the average VPRs well above the 50% threshold. Our results on the real-world VPRs were consistent with the vaccine effectiveness in the existing cohort or case–control studies ,12,21. Ttθ and the average time duration rμ from recovery to loss of natural immunity (average duration for reinfection) on the estimated VPRs. The sensitivity analyses show that the average absolute differences between the VPRs by using different values of the daily asymptomatic rate and the average duration for reinfection were 1.26% (SE: 0.23%) and 0.94% (SE: 0.17%), respectively, indicating the robustness of the estimated VPRs with respect to the 2 key parameters.Two sets of sensitivity analyses have been conducted to explore the impact of the uncertainty associated with model parameters for the daily asymptomatic rate 1 − Despite the effectiveness of the booster shot in the Omicron era, the booster vaccine coverage had a rather slow pace of growth with less than 9% increase from 2022 March 15 to 2022 July 17 in Italy, Turkey, the UK, and the US, as shown in Fig. We obtained the publicly available nationwide epidemiological and vaccination data from 2020 February 23 to 2022 March 15 for the 7 countries considered in this study. The daily cumulative numbers of the confirmed cases and deaths were obtained from “the 2019 Novel Coronavirus Visual Dashboard" at Johns Hopkins University. The daily recovered cases were imputed using 14 days as the average time of recovery from diagnosis by The study period in this research is from the start of the vaccine in a country to 2022 March 15, while part of the pre-vaccine period was considered for model parameter estimation. For each country, we divide the post-vaccine era into 6 consecutive nonoverlapping periods: the pre-Delta period from the start of vaccination till the Delta variant was first detected in the country, the following intervening period (Intervening I) until the Delta variant became predominant (more than 50% of the daily detected cases), the Delta-dominated period when the majority of the cases were caused by the Delta variant till the start of booster shots, the pre-Omicron period from the start of booster shots till the Omicron variant was first detected, the intervening period (Intervening II) till Omicron became predominant, and the Omicron-dominated period when the majority of the cases were caused by the Omicron variant. It is noted that the dominant variant was still Delta in the pre-Omicron period in the 7 countries. Since the start of the booster shot (2021 June 21) was close to the date (2021 June 29) when the Delta variant began to dominate in Turkey, we merged its Delta-dominated period and pre-Omicron period, which resulted in Turkey having only 5 periods. The 6 key dates for determining the post-vaccine periods, including the start dates of vaccination and boosters, the dates for the first detection of the Delta and Omicron variants, and the dates when Delta and Omicron began to dominate, are provided in Table We estimate the real-world VPR by building a stochastic epidemiological model with 11 compartments to quantify the epidemic process and developing a novel estimation procedure for its parameters.V0), partial (V1), full (V2), and booster (V3) vaccine immunity and the ones who have been vaccinated but lost vaccine immunity or have recovered from previous infection but lost natural immunity (eV), in the top row of Fig. S) consists of unvaccinated people (V0), the vaccinated people with expired vaccine immunity, and the recovered with expired natural immunity (eV). Let ϕt1,, ϕt2,, and ϕt3, be the time-varying vaccination rates from V0(t) to V1(t), V1(t) to V2(t), and V2(t) to V3(t), respectively, at day t. We consider the temporary vaccine-induced immunity and natural immunity with the average lengths of immunity after the partial, full, and booster doses of vaccines and the recovery from previous infection being 1/μ1, 1/μ2, 1/μ3, and 1/rμ days, respectively. We set μ1 = 1/60, μ2 = 1/240, μ3 = 1/300 and μ1 = 1/56, μ2 = 1/90, μ3 = 1/140 before and after the emergence of the Omicron variant according to the existing studies  serves as a substitution for V1(t), V2(t), V3(t), and S(t) byV1, V2, and V3 being zero at the start of the vaccination, whereNote that tβ can be estimated by nonparametric regression of The infection rate tβ is the same estimated as For pre-vaccine eras, B) of times to obtain B independent bootstrap estimates for the parameter. The sample standard deviation and the 2.5% and 97.5% percentiles of the bootstrap estimates can be used to estimate the standard error of the estimates obtained in the above 4 steps of the estimation and to construct confidence intervals for the parameters.Bootstrap procedure is used to obtain confidence intervals for the estimated parameters concerned in the aforementioned 4 steps of the estimation. Given the estimates We use the relative errors between the mean projected numbers of cumulative confirmed cases by 1,000 simulations with the estimated parameters and the observed numbers of cumulative confirmed cases to evaluate the fitting performance of our proposed model. The relative errors in the main analysis of the 7 countries shown in Fig. tθ and the diagnosis rate α of the pre-symptomatic cases in our model. While α is empirically estimated by the proposed method, tθ is determined from existing studies. Murray [tθ was no more than 8.6% and the average of the absolute differences was 1.26% (SE: 0.23%). We have also conducted a sensitivity analysis on the estimated VPRs with respect to rμ, the time duration from recovery to loss of natural immunity. It shows that the differences between the estimated VPRs in the 7 countries with rμ being 16 months as in the main analysis and those with rμ being 6 months were at most 5% apart with the average absolute differences being 0.94% (SE: 0.17%). Table The number of reported cases is influenced by the severity of symptoms, public willingness to do testing, and the testing capacity. These factors determine the pre-symptomatic proportion . Murray  suggesteSA is conducted to evaluate the impacts of different vaccination strategies on the confirmed cases and death of the pandemic. Five vaccination scenarios were designed: (a) no vaccination at all; (b) receiving the partial but no full vaccination; (c) receiving the partial and full vaccination but no booster shots; receiving the booster shots only at half (d) and twice (e) of the actual daily booster vaccination rate.Specifically, in the no- and partial vaccination scenarios (a and b) designed for the effects of the full vaccination, simulations were performed from the start of vaccination to 2022 March 15. During this period, the sub-population having received the full and booster doses of vaccines were set to zero while maintaining the actual numbers for having received partial vaccination in the partial vaccination scenario, and the sub-population receiving at least one dose was set to zero in the no-vaccination scenario. Similarly, for the last 3 scenarios (c to e) designed for evaluating the effects of the booster shot, simulations were performed from the start of the booster vaccination with the daily observed numbers of people having received the partial and full vaccination, while the numbers of people having received the booster doses were set to be zero, half and twice the observed vaccine up-takes, respectively. In the twice booster up-take scenario, the would-be numbers of people having received booster doses were truncated to the numbers of full vaccination at the time if the doubling up exceeds the latter number of the country. Under each of the scenarios, the daily adjusted numbers of vaccinated people and the empirically estimated VPRs of partial, full, and booster vaccination