Mastering relational concepts and applying them to different contexts presupposes abstraction capacities and implies a high level of cognitive sophistication. One way to investigate extrapolative abilities is to assess cross-dimensional application of an abstract relational magnitude rule to new domains. Here we show that angelfish initially trained to choose either the shorter of two lines in a spatial task (line-length discrimination task) or the array with “fewer” items  spontaneously transferred the learnt rule to novel stimuli belonging to the previously unseen dimension demonstrating knowledge of the abstract concept of “smaller”. Our finding challenges the idea that the ability to master abstract magnitude concepts across domains is unique to humans and suggests that the circuits involved in rule learning and magnitude processing might be evolutionary conserved. Relational concepts are based on the relationship between stimuli and not on their physical resemblance, consistent with abstraction abilities4. Relational knowledge is indispensable to adapt and survive in complex and continuously changing environments and plays a key role in higher cognitive processes. The capacity to elaborate relational concepts and apply them to different contexts has long been considered a hallmark of human cognition controlled by the prefrontal cortex5. This belief has been challenged by studies showing that animals can learn relational concepts such as “same/different”, “above/below” or “larger than/smaller than” within a single domain7. There is also evidence that honeybees are able to transfer the concept of same-different from the olfactory domain to the visual domain8 and can also make a transfer from discrete (number) to continuous (size) magnitudes9.Concepts, the mental representations of classes of things, are the foundation of human knowledge. The ability to conceptualise promotes cognitive economy by reducing the need to learn each new stimulus encountered in everyday life and allows quick and adaptive responses10. It is widely acknowledged that humans share with other animals an approximate non-symbolic system for representing number (Approximate Number System) in which accuracy is affected by the ratio between the quantities to be compared. This ratio-dependence signature characterizes a variety of magnitude judgements  thus suggesting the existence of a common system for representation of both discrete  and continuous quantities  in both humans and non-human animals; the so-called “A Theory Of Magnitude” (ATOM)14. Although this theory is still controversial15, an ability to transfer a concept across magnitude domains would support the existence of a common system involved in quantity discrimination. However, spontaneous cross-dimensional transfer of a magnitude rule without specific training in animals is largely unexplored.The ability to discriminate between quantities is crucial to increase individual’s fitness 19. Here, we asked whether angelfish  can spontaneously transfer a relational magnitude rule acquired along one dimension  to another one  and vice versa thereby testing both their abstraction abilities and the existence of a common magnitude system. Half of the fish were trained to select the shorter of two lines with a 0.5 length ratio, the other half were trained to select the array containing “fewer” dots in 10 vs. 20 comparison . Numerical stimuli were controlled for non-numerical cues  to avoid the possibility of using the perceptual properties of the numerical stimuli to learn the discrimination  or the shorter line  during the first training session. Mean trials to learning criterion was 117.43 ± 38.86, with no difference between the fish trained on numbers and those trained on lines (independent t-test: t(12) = 0.079, p = 0.938, Fig. During the test phase, fish trained on number exhibited a significant preference for the shorter line (one-sample t-test: t(6) = 7.862, p < 0.001) and the fish trained on lines significantly chose the smaller number of dots (one-sample t-test: t(6) = 8.041, p < 0.001), with no difference between the two groups (independent t-test: t(12) = 0.283, p = 0.782, Fig. 20. In particular, abstract thinking, relational knowledge and numerical competence have long been the center-piece of theories of human intelligence as they represent higher cognitive functions primarily attributed to the prefrontal cortex22. The ability to make rule-based decisions on magnitude relations across dimensions is fundamental in technology and science. However, non-verbal precursors of this ability are poorly investigated, and its evolutionary origin is almost unknown.Over the past decades, the field of comparative cognition has dramatically grown and thrived as an area of inquiry in biological and social sciences to understand the evolution of the human mind23.Cross-dimensional transfer of relational concepts requires two levels of information processing. The first involves encoding the relationships between the stimuli beyond their physical attributes, and the second implies the ability to extrapolate and transpose the abstract rule to a novel context. Our results indicate that angelfish formed the concept of “smaller” as they spontaneously transferred the learnt rule  to the novel stimuli belonging to the previously unseen dimension thus suggesting a coherent representation of magnitude across contexts. Furthermore, the use of multiple pairs of training stimuli and the presentation of completely novel visual stimuli in the test phase excluded the possibility of simple stimulus generalisation and provide strong evidence of abstract concept learning5 that is already displayed at birth before the acquisition of language24. However, recent studies showed that other cognitive skills, traditionally ascribed to the mammalian neocortex, are supported by different neural circuitry in animals lacking a neocortex25. For instance, volitional orienting, which is linked to neocortical regions in humans, has been described in archerfish26. Here we propose that a functionally homologous circuit underlying relational magnitude conceptualisation may exist in non-mammalian species. This hypothesis is supported by our finding that magnitude representation across dimensions was symmetrical . The spontaneous ability to create mappings across different dimensions indicates that the fish brain is predisposed to treat number and space as related. These results are in line with the existence of a common magnitude system to process quantitative information, as proposed by ATOM, which would not be necessarily cortically dependent in animals as recently suggested in birds14. Thus, complex magnitude extrapolative capabilities may not be uniquely human, but rather may extend to other species with relatively simple neural systems.This study shows for the first time that fish are capable of spontaneously applying an abstract relational concept across dimensions despite lacking a defined neocortex. This is important as abstraction and conceptualisation of magnitude across dimensions have long been thought to be a uniquely human capability31. According to this hypothesis, it would not be necessary to invoke a unique magnitude system to explain our findings. Rather, each dimension (number and length) would have been processed separately by different systems and only the concept of ‘smaller’ was abstract enough to be applied to both dimensions thus providing similar behavioural patterns in the two tasks. However, the debate about the existence of a common system and whether it is evolutionarily pervasive is still an object of discussion15 with future studies being necessary to understand how quantities are represented more broadly.However, an alternative explanation may account for our results. In fact, although ATOM has been one of the most prominent theories in recent decades, an increasing number of studies providing behavioural and neural evidence of inconsistencies in spatial, temporal and number representations suggests that distinct modules are responsible for processing quantities38, one may argue that the fish used contour length as a discriminative cue, as previously suggested in infants39. In this scenario, fish choice of the smaller stimulus in the test phase would have been the result of a generalisation process based on contour length rather than the extrapolation of a concept from one dimension to another. However, several studies on both humans and animals have shown that contour length is not a relevant cue to discriminate quantities46. For instance, adult humans cannot make a discrimination between two arrays of multiple items differing in quantity based on contour length alone47. Similar results have been reported in fish48. Further, with respect to this issue, it has been argued that the presentation of multiple items can hinder the ability to represent continuous quantities, such as contour length, as the mechanisms for keeping track of distinct individuals would be overwhelmed50. However, we cannot formally exclude the possibility that the fish were at least partially using contour length as the discriminative cue in our study. Further experimentation controlling for this perceptual cue is needed before drawing any firm conclusion about the exact strategy adopted by fish.Lastly, despite controlling the stimuli for the non-numerical variables  usually controlled for in numerical cognition studies in animals35. Fish were stocked at the Laboratory of Comparative Psychology (University of Padua) for at least 15 days before the beginning of the experiment. The subjects were maintained in groups of 6–8 individuals in 150-l aquaria provided with an air filter and enriched with natural vegetation and gravel. Water temperature was kept at a constant temperature of 26 ± 1 °C and a 12:12 h light: dark (L:D) photoperiod with an 18-W fluorescent light. Before the experiment, fish were fed twice daily with commercial food flakes (GVG-Mix).Subjects were 20 juvenile angelfish obtained from a local commercial supplier. We used the same sample size previously used to study numerical abilities in angelfish with the same protocol here describedAll husbandry and experimental procedures complied with the national guidelines and regulations for the care and use of animals of the country  in which they were performed. The procedures employed were approved by the Scientific Committee on Animal Health and Animal Welfare  of the University of Padova (Protocol n.31/2015).51. Two 18-W fluorescent lamps illuminated the apparatus, one for each compartment. The area adjacent to the front wall of the experimental apparatus was divided into two identical choice areas (6 × 20 cm) by a green plastic divider (30 × 6 cm). The exterior walls of the tank were covered with green plastic material except the home compartment and the front wall where a green net acted as one-way screen to prevent the fish from being influenced by external visual stimuli.The experimental apparatus consisted of a glass tank (60 × 40 × 35 cm) filled with gravel and 30 cm of water maintained at a temperature of 26 ± 1 °C and was provided with two air filters. Green plastic partitions were used to divide the tank into a front “experimental compartment” and a back “home compartment” connected by a 10 × 8 cm corridor  was equal for the two numerosities. In another third the cumulative surface area was incongruent with the number of dots  and in the remaining third the cumulative surface area was congruent with the number of dots . Furthermore, half of the stimuli were controlled for the convex hull , whereas the other half was controlled for density  , whereas during the training and the test phase we used two sets of stimuli: line stimuli and numerical stimuli. Line stimuli consisted of pairs of horizontal black lines of different lengths (range 10–54 mm) on a white background (6 × 6 cm). Stimuli were extracted from a pool of 18 different pairs and the length ratio within each pair was always equal to 0.5 . Furthermore, some lengths could be either the shorter or the larger line within the pairs across trials  for a total of 12 trials per day, for 5 days a week. Inter-session interval lasted 4 h. Each trial started with the subject in the home compartment with a rectangular panel made with green net (36 × 8 cm) placed into the corridor and a green opaque barrier placed in front of it to prevent sight of the stimuli during setup of the trial. The stimulus (a white square) was hung outside the tank and the left/right position was counterbalanced within each session, with the stimulus not being presented on the same side more than 2 consecutive times in a row, to avoid the development of side bias. The opaque barrier was then removed to allow the subject to look at the stimulus through the net. In order to familiarize the fish with the procedure, the green net was removed as soon as the subject entered the corridor to allow it access to the experimental compartment. Entering the correct choice area led to a food reward (a mixture of water and food flakes) provided in close association with the stimulus with a Pasteur pipette. The stimulus was removed only after the subject had finished eating and was outside the choice area. The subject was then gently ushered into the home compartment using a transparent plastic panel to proceed to the next trial. Once the subject had approached the white square in 10 out of 12 trials (83%) in one day the pretraining was concluded. Training started the following day.We used a procedure similar to the one described in the pretraining but here the subjects were presented with two stimuli and they were allowed to look at them for 10 s before removing the net panel from the corridor. Fish were randomly assigned to one of two conditions: half of the fish (N = 10) were trained on the numerical discrimination task whereas the other half (N = 10) was trained on the line-length discrimination task. In the numerical task, all fish were trained to choose the smaller numerosity in a 10 vs. 20 comparison. The left/right position of the stimuli was counterbalanced over trials. If the fish entered the choice area in correspondence to the correct stimulus (10), it received the food reward. If the fish chose the wrong stimulus, it was immediately ushered toward the home compartment in order to start a new trial. There was no time limit for a response to occur. In the line-length task, all fish were trained to choose the shorter of the two lines. Correct and incorrect choices led to the same consequences as described for the numerical training. After reaching the learning criterion of 75% (9 out of 12) correct trials over two consecutive days, the fish started the test phase. As dependent variables, we determined the number of trials to reach the learning criterion and the proportion of choices for the correct stimulus. If the subject did not reach the criterion within 15 days, the training phase was interrupted and it did not start the test phase. Six fish were excluded from the experiment at this stage.53. In this phase, we computed the proportion of choices for the novel smaller stimulus presented .Fish underwent 6 probe trials intermingled with 8 standard trials for 4 consecutive days  for a total of 24 probe trials and 32 standard trials. In probe trials, the fish previously trained to select the smaller numerosity in the numerical discrimination task were presented with the line stimuli and the fish trained to select the shorter line in the line-length discrimination task were presented with 10 versus 20 dots. During the probe trials, the stimuli were removed as soon as the fish made a choice irrespective of the response and no food reward was provided (extinction procedure). In the standard trials, the fish were presented with the same stimuli used during the training phase and were normally rewarded with foodDuring the training phase, the number of trials required to reach the learning criterion was compared between the fish trained on the numerical task and those trained on the line-length task by using an independent t-test. To evaluate whether fish had a spontaneous preference for the smaller training stimulus, the proportion of correct choices in the first training session was analysed by means of one-sample Wilcoxon signed rank test to compare the performance to chance level (50%). To determine whether there was any difference in accuracy among the stimuli controls, a repeated measure ANOVA was used with area control as factor of analysis and a paired t-test was used to compare performance between convex hull and density controls. One-sample t-tests were used to assess whether the proportion of correct choices for each stimulus control was different from a random value of 50%. To evaluate whether fish transferred the learned rule to the new dimension in the test phase, the proportion of choices for the novel smaller stimulus was analysed by means of one-sample t-tests. An independent t-test was used to compare test performance of the two groups.All statistical tests were two tailed, α was set up at 0.05. Non-parametric and parametric statistics were used based on normality distribution of data verified by Shapiro–Wilk tests. Data were analysed using SPSS 25.0.


The topology concept in the condensed physics and acoustics is introduced into the elastic wave metamaterial plate, which can show the topological property of the flexural wave. The elastic wave metamaterial plate consists of the hexagonal array which is connected by the piezoelectric shunting circuits. The Dirac point is found by adjusting the size of the unit cell and numerical simulations are illustrated to show the topological immunity. Then the closing and breaking of the Dirac point can be generated by the negative capacitance circuits. These investigations denote that the topological immunity can be achieved for flexural wave in mechanical metamaterial plate. The experiments with the active control action are finally carried out to support the numerical design. These new kinds of structures have many extraordinary properties, e.g. the wave band gaps12, negative refraction14, acoustic/elastic wave cloaks17, etc. These interesting behaviors can be used to tune the wave propagation properties and have a wide variety of potential engineering applications. Acoustic couplers18, sensors19 and waveguides20 are several typical applications of phononic crystals and elastic wave metamaterials.Phononic crystals and elastic wave metamaterials are artificial structures which are arranged periodically and have received lots of attention21, zero refractive index25, Hall effect26 and topological edge states30, etc. It should be mentioned that in the photonic crystal, the spatial phase can be reconstructed by the degenerate Bloch modes at the Dirac point31. Moreover, the topological phase can also transform because of the band inversion32.In recent years, a lot of attention has been focused on the extraordinary transmission phenomena of acoustic/elastic waves near the Dirac point. The periodic structures with the Dirac point can present many interesting behaviors, e.g. carrier mobility37. To present the controllable flexural waves with the broadband characteristics by the active control, an experiment using the shunted piezoelectric patches was reported38. Furthermore, another experiment was performed to show the tunable waveguide of the phononic plate39. Although the active control action has been applied on the waveguide structures, its tunable effects on the topological properties of elastic wave metamaterials have not been considered. In this work, we propose a tunable topological state of the flexural wave with the active control. Based on the electrical control action, the Dirac point and its corresponding topological immunity can be achieved.On the other hand, some investigations have been reported on the phononic crystals and elastic wave metamaterials with the active control actiona is the lattice constant, R is the radius of the piezoelectric patch, h1–h3 are the thicknesses of the middle layer, groove and piezoelectric sheet, respectively. Each piezoelectric patch is connected by a negative capacitance circuit to behave as the active control load in Fig. Cp, compensation resistance R0, operational amplifier, fixed resistances R1 and sliding rheostat R2 are considered. The derivation of the negative capacitance circuit is presented in Supplementary Material. Both the dispersion curves and propagation properties are calculated by the finite element software COMSOL. Our attention is focused on the topological immunity of the elastic wave in which the energy bands of the flexural modes are presented.In this work, we focus on elastic wave metamaterials with double Dirac points locating at the center of the Brillouin zone. A hexagonal unit cell in the elastic wave metamaterial plate is showed in Fig. α = 0.9 in Eq.(A.4) is applied. The band structure with the Dirac point is shown in Fig. When the negative capacitance circuits are not connected, the band structure calculated by the finite element method is shown in Fig. 40 for elastic wave and an additional tuning approach. Due to both the P-symmetry and T-symmetry can be found, the Berry curvature and Chern number are zero. The periodic structure is excited by a shaker at the right boundary and the frequency response is considered.An interesting behavior is the extraordinary transmission at the Dirac cone. In this work, it is designed by the configuration and electrical tuning to show the immune property of the elastic wave to the object. Then the concept “topological immunity” is applied to present the propagation of the flexural wave around the circular defect. This phenomenon makes the active controllable structure can immunize to the object, which means that the metamaterial plate exhibits a robust property at a certain frequency. Then, we can design an elastic wave metamaterial plate as a new function to achieve the topological immunity of the flexural wave. This work demonstrates the similar physics to ref. Here we define the transmission coefficient as the ratio of the left to the right flexural displacements. From the transmission coefficient curve for the elastic wave metamaterial plate with a defect in Fig. In order to support these numerical results, the transmission response of the metamaterial plate without external circuits is calculated and shown in Fig. E = 2.5 GPa, density ρ = 1300 kg/m3 and Poisson’s ratio ν = 0.41. The material parameters of the P-4 piezoelectric patch are shown in TableThen, the active control experiments are performed on the elastic wave metamaterial plate in which the P-4 piezoelectric patches are boned at the corners. The elastic wave metamaterial plate is fabricated by the 3D printing technology with 8000 synthetic resin, in which the Young’s modulus The experimental setup is illustrated in Fig. Figures In this work, an elastic wave metamaterial plate with the active control system is design to illustrate the topological immune property. The band structure is calculated by the numerical simulation and the double Dirac point is achieved by the external negative capacitance circuits. Around the Dirac point, the transmission responses of the metamaterial plate with a defect are discussed. Experiments are performed with the 3D printing metamaterial plate being bonded by the external electrical circuits. The topological immunity is observed by the responses behind the defect and achieved by the control action.Supplementary Information.


The water surface expression of liftoff and its dependence on discharge are examined using numerical simulations with the Regional Ocean Modeling System (ROMS). Liftoff is the process by which buoyant river water separates from the bed and flows over denser saltwater. During low‐discharge conditions liftoff occurs in the river and is accompanied by a change in the surface slope. During high‐discharge conditions liftoff occurs outside the mouth and generates a ridge on the water surface. The location and height of the ridge can be described by analytical equations in terms of discharge, shelf slope, and river mouth aspect ratio. The offshore distance and height of the ridge are proportional to the river discharge and vary inversely with river mouth aspect ratio. For steep shelf slopes liftoff occurs close to the river mouth and generates a large ridge. The ridge is modified, but not eliminated, by the presence of tides. The water surface slope change at the ridge peak is large enough to be detected by the upcoming Surface Water and Ocean Topography (SWOT) altimeter and can be used to identify the liftoff location during high discharge. However, during low discharge the water surface slope change at the liftoff location is too small to be detected by SWOT. These results indicate that remote measurements of the presence or absence of the ridge may be useful to distinguish between low and high flows, and remote measurements of the ridge location or height could be used to estimate freshwater discharge. River discharge generates a ridge on the water surface at the liftoff location that may be remotely detectableLiftoff location and ridge height are predicted by equations in terms of discharge, shelf slope, and river mouth aspect ratioRemote measurements of the location or height of the ridge could be used to estimate discharge During low‐discharge conditions, the gravity current speed exceeds the depth‐averaged velocity, Frf<1 and the salt wedge propagates up into the channel a distance Lsw. Under these conditions, the liftoff location is defined as the location of the toe of the salt wedge ; however, understanding the dynamics during high discharge events is important because they are the primary drivers of morphological change 1/4/g′, that can be expressed in terms of the freshwater Froude number as Jones et al.  used scaB is the width of the estuary. If we assume that the width of the freshwater jet offshore is the same as the width of the estuary (b0=B) and that the depth at liftoff is hs+αLlo, then this equation predicts the following liftoff length outside the river mouth as This equation was tested with measurements at the Columbia River by Kilcher and Nash  who founb0 implies that the plume does not spread outside of the river mouth. While this assumption may hold for some cases, plume spreading may be important in determining the liftoff length as will be demonstrated in this work.The assumption that the width of the freshwater jet offshore is γ and n are dimensionless geometric constants that are assumed to vary with b0, hs, and α. Increasing the shelf slope was found to decrease γ which was found to be in the range of O(10−2−1). A reasonable range of n was 1<n<1.4 and it was found to be only weakly dependent upon b0, hs, and α. As γ and n are not known a priori, Equation Q, b0, hs, and α are known. The full model can be run to solve for the liftoff length based on specific values of Q, b0, hs, and α. A single equation that relates either liftoff length or ridge height to discharge in terms of b0, hs, and α would be useful for quantifying discharge if a remote measurement of the liftoff length or ridge height is possible. In order to test equations for the liftoff length and ridge height, we simulate river discharge into the coastal ocean using a three‐dimensional numerical model that captures more of the physical processes than the two‐layer hydraulic model.Recently, Poggioli and Horner‐Devine  used a tLlob0=γ is a parameter accounting for the spreading rate, shelf slope, and river mouth aspect ratio. Note that variations in the surface elevation, which are typically less than 2% of the flow depth , the integral simplifies to When plume spreading is negligible , the toe of the salt wedge is less than 10 km from the edge of the model grid; therefore, the slope upriver of the toe is calculated between the toe and the model grid edge. For the highest flow case (Frf=0.5), the toe of the salt wedge is less than 10 km from the river mouth; therefore, the slope downstream of the toe is calculated along the entire distance between the toe and the river mouth.The salt wedge length is calculated as the distance between the river mouth and the location upstream where the midriver salinity in the bottom S layer decreases below 2 psu, which indicates that the water in that layer is predominantly freshwater. Sea surface height, 4.2.2α, and RA values. The liftoff length is calculated as the distance between the river mouth and the location of the maximum gradient of the salinity in the bottom S layer at the same alongshelf location as the middle of the river. The surface slope signature is quantified as the slope change between upriver and seaward of the mouth, but additionally the slope change at the ridge peak is calculated. The ridge height is calculated as the height difference between the offshore peak water level and the water level at the river mouth.During high‐discharge conditions liftoff is outside the river mouth Figure  and the 4.2.3κ, which influences the liftoff length  and Frf ≥ 3, Equation RA is low and Frf is high. As a result, the bottom drag felt by the plume in the bottom‐attached region is higher than that predicted by the analytical model, and the ridge height is overpredicted. This is further explored by examining the momentum balance in section The dependence of the ridge height on discharge, s Figure  and we hs Figure . Both ths Figure . When thκ=0. It agrees best with Equation α=0.005  to illustrate the dynamics  and shelf slope 0.001, the plume starts spreading immediately after exiting the river mouth  and a steeper slope (0.005), the plume exiting the river mouth barely spreads before liftoff .Model results discussed so far do not take into account the role of barotropic tides, which change the dynamics of many coastal processes associated with riverine discharge  or large tidal amplitudes, the tidal signal may dominate the water surface elevation signal. A more detailed investigation of the influence of tides on the water surface signature of liftoff is left for future work.These results show that while the tide does affect the ridge height and position, it does not eliminate the ridge from the surface. For lower discharge conditions  when the shelf is steep (α ≥ 0.005).The predicted water surface elevation accuracy of SWOT measurements is 0.1 m/kmh Figure . In the y Figure . These dh Figure , which mT Figure . The ride − 5  and low‐ (Frf<1) discharge conditions. During high‐discharge conditions a ridge forms at the liftoff point on the water surface. The ridge height varies with discharge, but for most rivers it is too small to be detected by the elevation measurement of the upcoming satellite altimeter SWOT. The surface slope change at the ridge peak is detectable by SWOT's slope measurement, but it does not increase with discharge. A possible method to estimate high flow discharge from a SWOT measurement is to locate the ridge peak by its slope change and then use the equations derived here for liftoff length to calculate discharge. The equations derived here predict liftoff lengths and ridge heights that compare well with the results obtained from the numerical model. Plume spreading is found to be negligible between the river mouth and liftoff for shelf slopes steeper than 0.005, further simplifying the relationship between liftoff location and discharge. The liftoff ridge is still present with the addition of moderate amplitude tides, but the ridge location and height are modulated.We identified water surface elevation signals related to the liftoff process that could be useful for remote sensing of river discharge during high‐ (During low‐discharge conditions water surface slope and elevation changes can also be related to discharge; however, their magnitudes are smaller than will likely be detectable by SWOT. This study does not account for changes in water surface elevation due to waves, upwelling, and downwelling, which are expected to further influence the structure of the water surface elevation field near the river mouth. Investigation of these processes is left for future work. Finally, results from this study indicate that low‐ and high‐discharge conditions can be distinguished by the presence or absence of the ridge and SWOT measurements of the location or height of the ridge can be used to estimate discharge. Although these results show some conditions under which SWOT will not be able to measure discharge near the river mouth, we expect future altimeters to have even higher slope and vertical resolution capabilities than SWOT. Future geostationary satellites with high‐resolution altimeters could be positioned above key rivers in remote areas of the Arctic to measure discharge where it is difficult to maintain in situ gauges.


Laminated composites materials are mostly used in dynamically loaded structures. The design of these structures with finite element packages is focused on vibrations, elastic deformations and failure control. Damping is often neglected because of its assumed secondary importance and also because of dearth of information on relevant material properties. This trend is prone to change as it is now realised that damping plays an increasingly important role in vibration comfort, noise radiation and crash simulations. This paper shows in a first step how to identify the orthotropic elastic and damping properties of single layer fibre-reinforced composite material sheets using a new extended version of the Resonalyser procedure. The procedure is based on the elastic-viscoelastic correspondence principle and uses a mixed numerical experimental method. In a subsequent step, the complex laminate stiffness values are computed using the identified single layer material properties. To validate this approach, the modal damping ratios of arbitrary laminated plates of different materials at several resonance frequencies are predicted and experimentally verified. Composite materials are increasingly replacing more traditional materials like steel, wood and aluminium in a wide range of applications. From the eighties till the end of the previous century, composite materials were introduced successfully in dynamically loaded structures like satellites, military aircrafts, expensive cars, boats, sporting and many consumer goods. As a result of marked developments and improvements in manufacturing methods, robot assistance and quality control systems, composites are now employed in major structural parts of civil aircrafts and vehicles ,2. The sFinite element analysis of a structure often requires knowledge and input of both the damping behaviour and other relevant material properties into the FE program. Material damping can be defined as ‘the phenomenon of energy dissipation due to inelastic behaviour of a macroscopically uniform material’ Lazan . This deThe viscoelastic behaviour of the polymer matrix materials;The inelastic behaviour of the fibres;The inelastic behaviour of the interphase between fibre and matrix;The slip at the fibre/matrix interface in case of non-perfect adhesion;Thermo-elastic behaviour of fibres and matrix;The volume fractions of matrix and fibre.Material damping in fibre-reinforced polymers is a very complex phenomenon. It is due to a variety of contributions which can be distinguished into contributions from the following sources:Because of the complexity of these different mechanisms, theoretical models to predict quantitatively the material damping of composites are usually cumbersome. If good experimental equipment is available, it is more reliable to measure the damping properties on test specimens. For laminates the damping mechanism becomes even more complex because of the possible delaminations and other interlaminar effects between layers. All methods for predicting the damping of laminates require the damping properties of single layer materials. The quality of the prediction of the entire laminate damping hence relies strongly on the quality of the used single layer material properties.Many papers to predict laminate damping are based on the elastic-viscoelastic correspondence principle first introduced by Hashin ,9,10. AcIn an early paper, Ni and Adams  developePlates in composite materials are more natural test specimens than beams since they require less preparation and machining and show less influence of stress concentrations and delaminations at the free boundaries. Since analytical solutions for the vibration of orthotropic plates are not available, mixed numerical experimental methods must be used for material identification. In a mixed numerical experimental method, measured vibration quantities like resonance frequencies and damping ratios are compared with similar values computed with a numerical model. The material properties in the numerical model are tuned in such a way that the computed values match the experimental observations. The availability of powerful personal computers has promoted this new approach. The earliest article describing the use of rectangular test plates for the identification of elastic orthotropic moduli of composite material using a mixed numerical experimental method was published by De Wilde and Sol in 1985  and lateA major problem with most of the mixed numerical experimental methods is the necessity to know the correct sequence of mode shapes. To compare measured frequencies and damping ratios with numerical simulations , the sequence of computed mode shapes must be the same as in the experiment. This requires the identification of the mode shapes by modal analysis or at least by detecting the position of the nodal lines of the mode shapes. Basic knowledge of modal analysis is described among others in the book by He and Fu . Mode shThis paper describes as a first step the identification of the complex engineering constants of a single laminate layer. The capabilities of a new and improved version of the Resonalyser method to identify the orthotropic elastic and damping properties of single layer composite material sheets with minimal experimental equipment is demonstrated. Measured resonance frequencies and associated damping ratios of two test beams and a test plate are compared with similar values computed with the finite element method. The orthotropic elastic and damping material properties in the finite element models of the beams and the plate are simultaneously updated in such a way that the computed resonances and damping ratios match all the measured values. This paper introduces for the first time an improvement for the existing Resonalyser method by generating good starting values based on the virtual field method. The virtual field method is described in detail in the book by Pierron and Grediac in .In a next step, the paper shows how to compute the stiffness and damping properties of laminates with the elastic-viscoelastic correspondence principle using previously identified single layer properties. The complex laminate stiffness values are used to predict the modal damping ratio of rectangular plates by finite element simulation. Many experimental values of damping measurements on laminated test beams can be found in literature ,11,12,20The described approach to predict laminate damping values is validated experimentally on a test set of laminated plates. The test set includes laminated plates with arbitrary sizes and different composite materials .Materials like fibre-reinforced polymer composites exhibit viscoelastic behaviour. Viscoelastic behaviour is situated somewhere between pure elastic and pure viscous behaviour. Elastic solids have the capacity to store mechanical energy with no dissipation. Viscous fluids in a non-hydrostatic stress state have the capacity to dissipate energy but cannot store it. Viscoelastic materials can both store and dissipate energy. Most polymers do not show strictly linear viscoelastic behaviour but modelling by the linear viscoelastic theory is a ‘useful starting point’. If the strain history s is an arbitrary convolution variable with a dimension of time. Another form of the stress constitutive relation can be found through a change of variable (Repeated subscripts denote summation over their dimensions). Suppose that the material is subjected to a sinusoidal strain with time-independent amplitude Again, using the change of variable t < 0, the lower boundary for the integral can be set to If it is assumed that the sinusoidal strain function is acting on the material for an indefinitely long time so that all initial transient disturbances have died out and assuming that Expression (5) can be written as:In which:ij symmetric. Because of the symmetry of the strain tensor kl symmetric. These symmetries allow contracting the stress tensor 11->1  22->2  33->312->6  13->5  23->4The stress ij symmetric, the viscoelastic stiffness matrix ij symmetric. The stress–strain relation (9) and its inverse expression can be written explicitly for a two-dimensional state of plane stress, like it occurs in thin orthotropic composite sheets in a Cartesian axis system  with the 1- and 2- axis along the orthotropic material axes:, Hashin  has intrC * is the complex in-plane stiffness matrix, S* is the complex in-plane flexibility matrix, 1 is often fibre dominated while E2 is matrix dominated), Poisson’s ratios The real parts in (11) govern the elastic behaviour and the imaginary ‘tangents delta’ parts govern the damping behaviour of the complex engineering constants. Because of the symmetry relation, if the tangents delta of A powerful and yet simple method for the experimental identification of both the real and imaginary part of the orthotropic engineering constants is the impulse excitation technique (IET) as described in Heritage et al. . The impThe IET generates (assumed) uni-axial stress fields in the beamlike specimens of Resonance frequencies measured with IET on a test plate allow the identification of the real (thus elastic) part of the complex engineering constants. Resonance frequencies can be computed with the finite element model of the test plate by the solution of an eigenvalue problem as described in He and Fu . The meaMany researchers have used the principle of mixed numerical experimental methods in combination with vibration measurements on test plates in the past ,17,18,19The earlier mentioned two challenges can be resolved by extending the number of test specimens in the mixed numerical experimental method. An example of such a multi-specimen method is the ‘Resonalyser’ procedure . The Res1 and E2 can be found with the ASTM IET test on the two test beams. With these values of the Young’s moduli, a special aspect ratio of the test plate can be computed:A fair value of the Young’s moduli EIf Poisson’s ratio would be zero, this plate ratio would cause a double bending resonance in the 1- and 2-direction of the plate. A plate with such an aspect ratio is called a ‘Poisson plate’ because the frequencies of the second and third mode shapes are very sensitive for Poisson’s ratio as shown by Sol . The modHence, without the necessity to do an investigation towards the nature of the mode shapes, an IET on a Poisson plate automatically reveals the three first frequencies and their vibration behaviour. The iterative updating scheme in the mixed numerical/experimental method  can now d for a free vibration  in a Cartesian axis system .The a priori knowledge of the three mode shapes of a Poisson plate together with the knowledge of the basic flexural modal shapes of the two beams allows an additional improvement of the Resonalyser method by adopting the virtual field method. The application of the virtual field method can be used for generating good starting values for all the engineering constants. The principle of the virtual field method starts from the dynamic equilibrium equation of a statistical homogeneous orthotropic thin plate with constant thickness W is the out of plane transverse displacement, t is the time and M to plate curvatures X (14):w and angular frequency A general solution of (13) is a sinusoidal vibration with amplitude A finite element solution of (15) involves discretisation of the plate into a mesh of nodal points. The transverse displacement w is expressed as a summation of shape functions defined in the nodal points of the FE mesh. This leads to the formulation of set of homogeneous equations:a is the length and b is the width of the plate. The matrices A, B, C, E and H are finite element connectivity matrices depending on the choice of the shape functions. They are independent of material properties and sizes of the plate. Solutions of (16) can be found by the solution of an eigenvalue problem:if is the resonance frequency associated to the mode shapes.The solution of the eigenvalue problem yields eigenvalues Since the nature of the mode shapes of a Poisson plate is known , they ca(i) referring to the (i)-th eigenvalue and mode shape, following scalar functions (i), b(i), c(i)a and (i)e can be defined:With (23) can now be written as:This equation can be written for the first three mode shapes and eigenvalues of the Poisson plate:1 and E2 can be found with IET on the beams. By replacing the plate rigidity values ijD in (25), using the relationship (14) between the plate rigidities and the engineering constants, (25) can be solved for 12v and 12G. Hence the virtual field method provides starting values for all the engineering constants, without the necessity to actually measure the mode shapes. After obtaining the starting values, further identification with the mixed numerical experimental method shown in With the associated resonance frequencies of the torsion, saddle and breathing mode shapes measured by IET on the Poisson plate, good values for EAll the conditions for obtaining good results with the mixed numerical experimental method  are now The same three test specimens (two beams and 1 Poisson plate), as used for the identification of the elastic part of the engineering constants, are used for identifying the imaginary part. The IET is used again to measure the modal damping ratio associated with the fundamental bending mode shape of the test beams and the modal damping ratios of the three first mode shapes of the Poisson plate. The decaying signal after impact see  is curveCurve fitting (26) for obtaining the modal damping values is only possible if the resonance frequencies can be separated from each other. This is no problem for the fundamental bending resonance frequency of the test beams but requires a special procedure for the measurements of the modal damping ratios of the mode shapes of the Poisson plate. The procedure takes again profit from the special nature of the mode shapes of the Poisson plate . The nodThe suspension wires for the Poisson plate must be fixed on positions on nodal lines of the mode shapes for minimising the influence on the damping ratio measurement with the IET see .For measuring the decaying signals of the torsion and breathing mode shape after impact, the suspension wires are fixed in the intersection of both their nodal lines see a. The isDE dissipated during one cycle divided by the modal strain energy PE.The modal damping ratio of a mode shape PE stored in an elastic body is found by taking half of the volume integral of the contracted product of strain and stress columns:The strain energy The modal strain energy can be approximated for sufficiently small damping ratios using only the real part of the in-plane complex stiffness matrix PE can be expressed as a summation of four different stress–strain combinations:This is acceptable for polymer composite materials where the damping ratio is rarely higher than 1%. For an orthotropic thin plate, taking the symmetry into account, the strain energy DE during one oscillation cycle is:The dissipated energy DE can be expressed as a summation of four different stress–strain combinations:In a similar way as for the strain energy, the dissipated energy By entering (31) and (33) in (27), De Visscher  showed tThe strain energy portions of (31) can be computed with the finite element model of the Poisson plate. The IET on the test beams and the Poisson plate yields two sets of equations. In the first set (35), the tangents The second set (36) yields the tangents ijGW in (36) can be found using (34). The simultaneous solution of these two sets of Equations (35) and (36) requires the relation between the tangents The value of The relation for the in-plane shear modulus gives directly:With the introduction of a complex number The combined set of five Equations (35) and (36) can be solved, taking the relations (37)–(39) into account, resulting in the tangents The combination of the modified Resonalyser method with IET measurements hence provide in a simple way values for the complex engineering constants averaged over the test beams and test plate areas. This makes the results especially suitable for finite element models and laminate analysis programs .The stress–strain relation (10) in an orthotropic material axis system  can be transformed into a stress–strain relation (40) in a rotated axis system , see Jones (40)in The classical laminate theory for sufficient thin plates can now be applied to find the complex laminate stiffness values. The laminate is composed of N layers. The z-axis is taken perpendicular to the layer stacking .The symmetric extensional stiffness matrix A, the symmetric coupling stiffness matrix B and the symmetric bending stiffness matrix D are given by the following expressions:D. The transverse vibration mode shapes of a symmetric plate are hence dominated only by the plate rigidities. Making use of the definitions of the bending and torsion moments, it is possible to express the moments as a function of the global plate curvatures:Plates with a symmetric laminate configuration have no coupling between the in-plane rigidities A and the plate rigidities PE of an anisotropic vibrating thin plate can be expressed as: ijPE is partial portions related to the anisotropic plate rigidities ijD. The modal damping ratio of a mode shape of a laminated plate can be computed similar as (34):PE and the partial strain energy portions iPEj related to the plate rigidities can be calculated by the FE model of the plate.The total strain energy An UD 40 cm × 30 cm carbon/epoxy plate with all ten layers oriented in the 0° direction was prepared in an autoclave, according to the suppliers instructions . The layers were cut out of a 30 cm width bobbin CMP 200/300 CP0031 carbon/epoxy prepreg material. This UD test plate was used in a first step to identify the single layer orthotropic properties of the material using the extended Resonalyser procedure.Two test beams were cut respectively along the fibre direction and perpendicular to the fibre direction of the UD plate. The IET vibration results of the two carbon/epoxy beams are given in A Poisson plate with aspect ratio according to formula (12) was cut out of the UD carbon/epoxy plate. The sizes and mass of the Poisson plate are given in The resonance frequencies of the Poisson plate, measured with the IET are given in 1_Start = 1.125E + 11 [Pa]            E1_Final = 1.092E + 11 [Pa]E2_Start = 7.540E + 09 [Pa]            E2_Final = 7.303E + 09 [Pa]E12_Start = 4.907E - 01 [-]                v12_Final = 0.476 [-]v12_Start = 3.188E + 09 [Pa]          G12_Final = 3.660E + 09 [Pa]GThe virtual field method uses the resonance frequencies of the beams and Poisson plate to generate good starting values  for the elastic part of the engineering constants. Next, the mixed numerical experimental method as described in 2.2.1 computes iteratively the final elastic part of the engineering constants:With these elastic values of the engineering constants, the FE model of the Poisson plate can plot the nodal lines of the three first mode shapes see .The knowledge of these nodal lines allows finding the correct position of the intersection points to measure the damping ratios of the beams and Poisson plate using IET. The measured decaying sinusoidal signal can be curve fitted with the formula (26). A curve fitting example is shown in It can be seen that this apparently very low value for the damping ratio is able to damp out the signal completely in about 1 s. The obtained modal damping ratios by curve fitting of the measured signals with IET for the Poisson plate and the test beams are given in All experimental information is now available to compute the complex engineering constants with the procedure described in 2.2.2. The result is shown in With these single layer properties, the damping ratio of arbitrary laminated plates with arbitrary aspect ratios can be predicted using the classical laminate theory and the elastic-viscoelastic correspondence principle.S was prepared in an autoclave. The layers were cut out of the 30 cm width bobbin CMP 200/300 CP0031 carbon/epoxy prepreg material, the same as used for the single layer test. The measured average thickness of this laminate was 2.1 mm and the computed complex anisotropic plate rigidities of this laminated plate are shown in A 40 cm × 30 cm ten-layer symmetric laminate (0° −45° 45° −45° 0°)The computed resonance frequencies, mode shapes and damping ratios for a (0.2365 m × 0.1385 m × 0.0021 m) and a mass of 0.1011 kg laminated test plate using a thin plate finite element model are shown in These predicted values were compared with experimental measurements. The test plate was freely suspended, and the resonance frequencies were measured with IET and the DJB A/128/V1 micro accelerometer. Because the exact positions of all the nodal lines of the mode shapes of this laminated plate are not precisely known, the test plate could not be suspended with the nylon strings for each mode shape positioned on a nodal line. Also, the position of the hammer impact and the measurement position of the accelerometer required some trial and error. The measurement of the damping ratio was not possible with IET for the same reasons and additionally because the mode shapes could not be excited separately by a hammer impact. Therefore, to measure the damping ratios, acoustic excitation by a small Bluetooth loudspeaker (Philips model BT55B/00) was used. With a frequency generator, an acoustic sinusoidal signal with the targeted resonance frequency was sent as excitation signal and after cutting the excitation, the freely decaying signal was captured by the DJB A/128/V1 accelerometer. The experimental values are compared with the numerical predictions in It can be observed in 2 oriented in the 0° direction was prepared with hand layup, cured and compressed between two thick aluminium plates. The cured plate had a final thickness of 4.54 mm. The same measurement procedure for obtaining the single layer properties described in Experiment 1 is applied. The size and mass of the two test beams and the resulting Poisson plate are given in A 40 cm × 40 cm glass/polyester plate with eight UD glass textile layers of 600g/m1E and 2E are not as extreme as for the UD carbon epoxy. The tangents delta values are higher.With these single layer properties, the damping ratio of a test plate cut under an angle of 20° of the Poisson plate, is predicted and measured. The dimensions of the test plate are (0.177 m × 0.1337 m × 0.00454 m) and the mass is 0.19658 kg. The computed complex anisotropic plate rigidities of this glass/polyester plate are shown in The computed resonance frequencies, mode shapes and damping ratios for the 20° glass/polyester test plate using a thin plate finite element model are shown in Only three frequencies could be measured by the available IET and the DJB accelerometer. The associated damping ratios were identified using acoustic excitation with the Bluetooth loudspeaker. The glass/polyester plate was not laminated (excluding interlaminar effects) and was heavier than the previously tested carbon/epoxy plate (reducing the influence of wires). The measured values are however again higher than the predicted values. To further investigate this problem, a (0.129 m × 0.0217 m × 0.00454 m) beam in the x-direction was cut out of the plate.The predicted value for the beam in A 40 cm × 40 cm glass/polyester plate with three UD glass textile layers of 600 g/m2 oriented in the 0° direction was fabricated with an RTM manufacturing technique. The textile layers were separated by chopped strand mats of E glass of 225 g/m2. The same measurement procedure for obtaining the single layer properties as described in Experiments 1 and 2 is applied. The ‘single layer’ properties must now be regarded as the average over the thickness of the RTM layers. The obtained engineering constants are hence the apparent values in bending over the thickness of the test plate. The size and mass of the two test beams and the resulting Poisson plate are given in 1 and E2 are not as large as for the UD carbon epoxy. The tangents delta values are higher than for the UD carbon epoxy and the glass polyester plates in Experiments 1 and 2.With these single layer properties, the damping ratio of a test plate cut at an angle of 20° is predicted and measured. The dimensions of the test plate are (0.275 m × 0.235 m × 0.0042 m) and the mass is 0.3731 kg. The length to thickness ratio of the test plate is 65, which is closer to the thin plate assumptions of the test plate than in experiment 2. The computed complex anisotropic plate rigidities of this glass/polyester plate are shown in The computed resonance frequencies, mode shapes and damping ratios for the 20° glass/epoxy test plate using a thin plate finite element model are shown in Five resonance frequencies were measured using IET and the DJB accelerometer. The associated damping ratios were identified using acoustic excitation with the Bluetooth loudspeaker. The measured damping values are now closer to the predicted values than in the previous two examples. This can be explained because the mass of the test plate was now higher than in the previous cases (reducing the influence of wires) and the length to thickness ratio was higher (reducing the influence of transverse shear deformations). It is hence recommended that the ratio width/thickness of the Poisson plate is higher than 40.Measuring the decaying signal with a non-contacting laser Doppler velocity meter could partially solve the problem of additional structural damping contributions of accelerometer cables, but unfortunately a laser Doppler velocity meter is 20 times more expensive than a micro accelerometer and much more difficult to handle on freely suspended test plates (rigid body oscillations after impact).Measuring damping is much more difficult than measuring resonance frequencies. The physical reason for this fact is that in one vibration cycle the absorbed damping energy is very small (order of magnitude 1%) as compared with the strain energy. This paper showed in the first step, how to identify the orthotropic elastic and damping properties of the single layer fibre-reinforced composite material sheets using a new mixed numerical experimental method, which is an extension of the Resonalyser procedure. The extended procedure allows generating good starting values based on the virtual fields method. The procedure also takes the influence of the mass and wires of used accelerometers into account through corrections in the finite element model. However, the experimentalist needs to verify that the Poisson test plate mass is sufficiently large to allow for accurate mass compensation of the combined mass sensor-wires (ratio mass plate/mass sensor at least 200). Additionally, the value of the ratio width/thickness of the used Poisson test plates must be sufficiently high to limit the influence of transverse shear and rotation inertia in the used thin plate model (ratio width/thickness at least 40). With these precautions, the new extended Resonalyser procedure can identify accurately all the orthotropic elastic and damping properties of single layer fibre-reinforced composite material sheets. The single layer properties of three different composite materials are identified as examples.


Strawberry is one of the most perishable fruits, and precooling of strawberry increases the percentage of marketable fruits. To assess the sensitivity of strawberry cooling uniformity with respect to package vents and tray design, the previously proposed modified parallel airflow system (MPAS) was modified and a sensitivity analysis was conducted in this paper. Some improvements in homogeneous strawberry precooling process were made to give improved parallel airflow system (IPAS). To evaluate its performance, the cooling process of strawberries was simulated using the mathematical models of heat, momentum, and mass transfer, which were validated experimentally. Results showed that the IPAS was able to distribute cold air uniformly throughout the packages. A difference of 0.1°C was observed between the average fruit temperatures of the packages after 3 hr of cooling. Therefore, the cooling process of strawberry could be done at lower airflow rate, cooling time, and heterogeneity. In addition, the precooling process of strawberry was studied considering the moisture loss during the cooling process and comparing the data with the models without this term. The results indicated that the moisture loss of strawberries during the cooling process is not negligible and the cooling rate increased and the cooling time decreased (31%) by considering this term in the modeling. However, the moisture loss did not affect the heterogeneity of cooling process. Improved parallel airflow system was able to distribute cold air uniformly throughout the packages. The cooling process of strawberry could be done at lower airflow rate, cooling time, and heterogeneity compared with previous systems. To decrease the cooling heterogeneity, they developed a novel forced‐air cooling system that split the airflow in each layer of the pallet into two streams of parallel airflows  was improved by inserting an airflow restriction plate “ARP,” that is, a plate with holes on it Figure . They inMercier, Brecht, and Uysal  investigA review of lectures showed that, in the traditional cooling system of strawberry, there is considerable heterogeneity between the fruits temperature located in the individual packages and in different packages inside the tray, which leads to a decrease in the self‐life of fruits. The efforts of researchers could not solve this problem properly. However, Nalbandi et al.  showed tThe aims of this research were (a) to assess the sensitivity of strawberry cooling uniformity with respect to package vent and tray design, (b) to simulate the effectiveness of the proposed process using a previously validated model by Nalbandi et al. , and (c)22.1To eliminate the dead zone created between the packages in the previous study . The total package area increased to 33%. In addition, extra vents were created on side A of the tray wall  that occurs due to the vapor pressure gradient between the fruits and ambient atmosphere, which is calculated by Equation ps and pa and calculated with Equations Pw is saturation water vapor pressure (Pa) as a function of temperature (Equation g) is related to the air film mass transfer coefficient (kgconv) and fruit skin mass transfer coefficient (kgskin), which are calculated from Equation The effect of moisture loss was studied by the modified developed models with equations presented in Table Equation . Mass trm was obtained from Equation kgconvwas calculated by Equation The air film mass transfer coefficient is a function of airflow rate and was estimated by using Sherwood–Reynolds–Schmidt correlations  of the cold room Figure . Neverth3.2.3Average fruit temperature in the packages versus cooling time is shown in Figure The 7/8th cooling time of both packages was about 245 min. Therefore, the cooling time of fruits decreased by using suitable designs. Based on these observations, the cooling time and heterogeneity decreased by using the extra vents. The cooling times of fruits in all designed systems Figure  are pres3.3As expected, adding moisture loss term to the mathematical models had a considerable effect on an increase in cooling rate and decrease in cooling time, resulting from the effect of latent heat on cooling of fruit. Based on the results predicted by the simulator, the 7/8th cooling time of fruit (about 165 min) was 75 min lower than that of without moisture loss term (a decrease about 31%). Moisture loss only affected the cooling rate and time but did not affect the cooling uniformity. The cooling uniformity was obtained the same as previously developed model where the moisture loss was assumed negligible. The temperature variation of fruit is shown in Figure 3.4In order to validate the new model considering the effect of moisture loss, the fresh fruit was cooled immediately after harvesting. The central temperature of four fruit instrumented with K‐thermocouple wire was compared with that of the same ones obtained from the simulation process Figure . A good 4In the present work, the modified parallel airflow system proposed by Nalbandi et al.  was imprAuthors declare that they do not have conflict of interest.This study was approved by the University of Tabriz.Written informed consent was obtained from all study participants.


R0 is the average number of secondary infections arising from a single infectious individual during their infectious period in an entirely susceptible population.Plague is a re-emerging flea-borne infectious disease of global importance and in recent years, Zambia has periodically experienced increased incidence of outbreaks of this disease. However, there are currently no studies in the country that provide a quantitative assessment of the ability of the disease to spread during these outbreaks. This limits our understanding of the epidemiology of the disease especially for planning and implementing quantifiable and cost-effective control measures. To fill this gap, the basic reproduction number, R0 was estimated as a function of the average epidemic doubling time based on the initial exponential growth rate of the outbreak and the average infectious period for bubonic plague. R0 was estimated to range between 1.5599 [95% CI: 1.382–1.7378] and 1.9332 [95% CI: 1.6366–2.2297], with average of 1.7465 [95% CI: 1.5093–1.9838]. Further, an SIR deterministic mathematical model was derived for this infection and this estimated R0 to be between 1.4 to 1.5, which was within the range estimated above.Secondary epidemic data for the most recent 2015 Nyimba district bubonic plague outbreak in Zambia was analyzed. R0 for bubonic plague is an indication that each bubonic plague case can typically give rise to almost two new cases during these outbreaks. This R0 estimate can now be used to quantitatively analyze and plan measurable interventions against future plague outbreaks in Zambia.This estimated R st century, incidence has begun to sharply rise again mainly in developing nations including Zambia which recently recorded a five-fold increase in recorded cases by the turn of the century. For developing and resource strained nations like Zambia, such major public health challenges need to be addressed with the most cost-effective, efficient, and sustainable approaches appropriate to the region. This firstly requires developing a sufficient and comprehensive epidemiological understanding of the disease. This study therefore estimated the basic reproduction number (R0) for bubonic plague which is an estimate of the average number of secondary infections expected to arise from a single infectious individual during their infectious period assuming the entire population is susceptible to the infection. Based on data from the Nyimba district bubonic plague outbreak in Zambia, we estimated R0 for the infection to range between 1.5599 to 1.9332 which shows that each infectious individual with bubonic plague can potentially infect up to two other individuals during their infectious period. We also observed that this R0 estimate was comparable to that previously estimated for large medieval bubonic plague epidemics. Estimating this important epidemiological parameter for bubonic plague greatly enhances existing knowledge of the epidemiology and transmission dynamics of the infection in Zambia and creates prospects for better management of the disease through informed planning.Bubonic plague is a historically important and neglected bacterial infectious disease that has claimed more than 170 million lives from multiple devastating epidemics especially during mediaeval and earlier times. Despite a recorded decline in the disease for much of the 21 Yersinia pestis. It primarily infects rodents and their fleas but infection can spread to humans causing outbreaks with bubonic plague being the most common clinical form [0, for bubonic plague was estimated in this study based on data from the latest 2015 Nyimba district bubonic plague outbreak in Zambia, for which data was available. R0 is defined as the average number of secondary infections arising from a single infectious individual during their infectious period in an entirely susceptible population [0 is used to guide the magnitude of control measures that will be required to control the disease [Plague is a re-emerging infectious disease that has claimed more than 170 million lives in three waves of devastating pandemics in human history. The current third pandemic which begun in the late 1800s has spread the farthest globally and continues to affect vulnerable communities mainly in developing nations . Plague cal form –8. In Zacal form . Recent cal form –11. Howepulation –14. It ipulation ,14. More disease ,20, chol disease , Severe  disease , and fooNyimba district is located in the Eastern province of Zambia . In 2015Online secondary epidemic incidence data for the 2015 Nyimba district bubonic plague outbreak in Zambia, made publicly available in 2016 , was retth and May 5th 2015 [Y. pestis was detected in six (29%) of the cases through polymerase chain reaction (PCR). It was reported however that 12 of these cases actually tested positive for malaria during initial evaluation and this includes all six who tested PCR-positive for plague [th–7th April 2015 after which the incidence seemed to suddenly drop and the outbreak generally died out with the last case being recorded on 1st May 2015.The Nyimba bubonic plague outbreak occurred between March 265th 2015 . During 5th 2015 . Out of 5th 2015  and thes5th 2015 . Y. pestr plague . In factr plague . Three or plague . Apart fr plague . The epir plague  which sh0, for bubonic plague was estimated using a simplified mathematical method as described by others [0 directly from the epidemic incidence data. This method was chosen as the primary method of estimation for the study as it is quick and robust even in the absence of abundant epidemiological data. To give credence to the results of this primary method, a second alternative method was also used to estimate R0 for the same bubonic plague outbreak in Nyimba by means of creating a complete compartmental mathematical model for the infection. Both these methods used however relied on the following two underlying assumptions made about the Nyimba bubonic plague outbreak: (1) Since various disease interventions were applied during the course of this outbreak , it was assumed in this study that the disease transmission must have occurred naturally in the community only during the early stages of the outbreak which caused the initial peak in cases around the 7th of April 2015 [0 estimation used in this study relied only on analysis of the data from the initial growth phase of the Nyimba bubonic plague outbreak leading up to its peak; i.e before the provided interventions took full effect [Due to limited epidemiological data for the outbreak in Nyimba, the basic reproduction number, Ry others ,19,20,39ril 2015 ,24–28. Fl effect . (2) Furl effect –35,47. Il effect ,36–38. Nl effect , it was 0 in this natural system can be estimated using Td is the epidemic doubling-time, D is the average infectious period for bubonic plague, and ln2 is simply the natural log of 2. The epidemic doubling time (Td) was estimated using the exponential epidemic growth rate of the initial growth segment of the epidemic curve as summarized in the following steps: Step 1. A plot of the cumulative number of cases plotted against time for the period of initial rise of cases for the Nyimba bubonic plague outbreak was generated. An initial 17-day period from the 23rd March–8th April, 2015 was taken to approximately represent this period. Step 2. To assess the assumption of initial exponential growth of the outbreak, an exponential curve was fitted to the plot generated in step one. The R2 (coefficient of determination) value was used to assess how well the exponential model fit this observed data [C(t) is the total number of cases at a given time (t), C(0) is the initial number of cases at the start time, e is the mathematical constant and α is the epidemic growth rate [α), a log transformation of the plotted epidemic curve in step one was made by simply plotting the natural log of the cumulative number of cases against time. This produced a linear relationship between the log cumulative number of cases and time. This linear relation was arithmetically derived by simply taking the natural log of both sides of the general expression for the exponential function given above to produce the following linear expression:Briefly, this method assumes that in a homogeneously mixing population of susceptible subjects, the number of new cases increases exponentially during the initial stages of the epidemic. The basic reproduction number is then estimated from the initial exponential growth rate of the epidemic based on the analysis of the initial segment of the epidemic curve ,19,20,39wth rate . To obtaα (gradient) was obtained from the linear regression output. Here the R2 value was equally used to assess how well the linear regression model fitted the observed data [Step 3. Using the epidemic growth rate and its 95% confidence interval estimated in step two above, the average epidemic doubling time was estimated using the relation between the epidemic doubling time and the epidemic growth rate given by the expression below [Therefore, a linear regression model was run for this plot and the 95% confidence interval for the already estimated epidemic growth rate ved data . Step 3.on below :Td=ln2αTd) were therefore determined by the interval for the estimated epidemic growth rate (α).The 95% confidence interval lower and upper limits for the epidemic doubling time . All calculations were done in Microsoft Excel Software.For this primary method, the duration of infectiousness for bubonic plague infection was the only parameter that was obtained from literature. Despite estimations greatly varying in the literature, this parameter was simply estimated to range between three and five days. This simply represents the most central estimate of the average period that the infectious individual may pass the infection to susceptible human ectoparasites vectors which transmit the infection to other susceptible individuals before these vectors die from the infection themselves; assuming an early phase transmission cycle in the vectors ,42,43. T0 from the model, the model in this study was developed without the inclusion of disease interventions in order to simulate the transmission dynamics of bubonic plague infection for the period of initial growth of the Nyimba bubonic plague outbreak when the infection was thought to spread naturally in the community. For the purpose of this model, the following main assumptions were made about the disease transmission dynamics for the study outbreak:For the second method, a deterministic compartmental mathematical model was created for the Nyimba bubonic plague outbreak and this model was used to estimate the basic reproduction number. As the infection was assumed to be spread through vectors, the Ross Macdonald Susceptible-Infectious-Removed-Susceptible-Infectious model, denoted as SIR-SI, was initially created as a first step in the modelling process . This moThe human population was initially completely susceptible to the infection.Individuals in the population were considered to be homogenously mixing.It was assumed that during the outbreak the infection was transmitted from human to human through various human ectoparasites vectors ,30–35,47The human population was constant owing to the short duration of the Nyimba bubonic plague outbreak.The infection is immunizing ,23.The human ectoparasites vectors do not recover from the infection.Sh), infectious (Ih), and removed (Rh) humans. The constant total human population Nh is given as Nh = (Sh) + (Ih) + (Rh) and this was estimated to be about 350 people for the village of the outbreak based on estimates by local health experts in Nyimba. Susceptible people in the population (Sh) acquire the infection from the infectious people (Ih) at the rate βc for a period α−1. This transmission rate βc is called the composite human to human transmission rate and it already accounts for the infection transmission dynamics in the intermediate human ectoparasites vectors [α to enter the removed compartment. It is assumed that all infectious individuals eventually either die from the infection at the end of this period α−1 or they may naturally recover from the infection but are all considered no longer able to transmit the infection [In the derived SIR model, the transmission of bubonic plague by the infectious humans through various human ectoparasites vectors is modeled by three equations as given in  vectors . The infnfection ,42. The nfection ,7.The ability of the derived SIR model to reproduce the general disease incidence pattern of the Nyimba bubonic plague outbreak for the period of initial growth of the outbreak up to its peak was assessed. This was done to optimize and validate the model through model simulation against the actual outbreak data. The model simulation was done using the systems dynamics modeling software Vensim—personal learning edition (PLE) version 7 for windows –50. Fig Nh) and at the start the number of susceptible people is given as initial population minus one (the index case). This initial population is equal to the total population for the model system  at any given time (t) during the outbreak: Initial population = total population = susceptible + infectious + removed. The infection term in the diagram describes the flow of individuals from the susceptible compartment to the infectious compartment and it is given as the quantity αIh(t) in Nh = 350, Sh(0) = Nh− 1, Ih(0) = 1, Rh(0) = 0 were used. Fixed parameter values used for the model were as given in βc. Note that βc is comprised of the parameters βf, βh and μ of which βf, the transmission rate of bubonic plague from infectious human ectoparasites vectors to susceptible humans per unit time, was relatively the more uncertain of these sub-parameters in literature. Therefore, a narrow range of about 0.05–0.123 was considered for βf as compared to the fixed value of 0.05 used for this parameter in a similar mathematical model created for bubonic plague infection by [βf was because currently there are limited studies that have investigated the potency of these human ectoparasites (such as the human fleas and body lice) as vectors for bubonic plague infection [βf in this study. Consequently, for the model simulation the fixed values for the other parameters βh and μ were maintained as given in βf were used from the possible range of values given for this parameter. Sixteen model simulations were therefore run by appropriately altering the value of βf which gave different values of βc per run until a value of βc that produced the best model fit to the targeted Nyimba bubonic plague outbreak data was found. This was statistically confirmed using the chi-square goodness of fit test at a significance level of 0.05. The value of βc which produced better model fit to the outbreak data was therefore used in the estimation of the basic reproduction number (R0) for bubonic plague. The model was simulated over a 45-day time period which was approximately the duration of the Nyimba bubonic plague outbreak [rd of March 2015 even though the date of onset of symptoms for the first recorded case during the actual outbreak was the 26th of March 2015. This was done so as to accommodate the brief preclinical stage of the infection for the first case before this patient became symptomatic and recorded on the 26th of March 2016.In the diagram , the iniction by . The reanfection ,8,32. Innfection  was specoutbreak . In the Sh(0), Ih(0), > 0, and Rh(0) = 0, the basic reproduction number (R0) for bubonic plague was estimated directly from the derived SIR model system of 0 was derived as follows:With the initial conditions given as; f R0=βcα ,44,51. TIf the infection is spreading in the population such as during an outbreak, then we expect the number of new human infections to be increasing in the population. This means that:(t) notation for convenience, this statement means that from inequality 1.Removing the 0, inequality 1 is analyzed at the starting point of the outbreak where the entire population can still be considered to be completely susceptible except for the one infectious individual at Ih(0) introduced in the population which is the index case. At this point, Nh ≈ Sh(0). The appropriate notation for inequality 1 at this point can therefore be given as:Nh ≈ Sh(0) when the index case is introduced in the population, it means that inequality 1 simplifies as:As per definition of RFrom this we have that pulation .No ethical issues were encountered as no human or animal subjects were used in this study and cases were anonymous.2 value of 0.9502 was accepted as significant for the fitted exponential curve [th of March, 2015 which corresponded to day 4 of the outbreak in the graph while the 8th of April, when the outbreak was at its peak, corresponded to day 17.For the primary method, al curve . In NyimThe fitted exponential model predicted an average initial epidemic growth rate of about 0.1866 cases/day with the 95% confidence interval estimated to be between 0.1273 and 0.2459 cases/day from the linear regression analysis of the log-transformation of 0 using 0. The range in the possible values of R0 is due to the estimated duration of infectiousness for bubonic plague which was used.The basic reproduction number for bubonic plague in this study was therefore estimated to range between 1.5599–1.9332 with an average estimate of 1.7465 as shown in βc used per simulation after altering the values of βf. The shaded area represents the period of initial growth of the outbreak up to its peak while the un-shaded area shows the model predicted growth of the outbreak without interventions over the projected 45-day time period.For the secondary approach, th of March, 30th March, 5th April, 6th April, 7th April and 8th April 2015 [βf on these runs of the model simulation were taken to be the most optimal estimates for this parameter based on the available outbreak data for the Nyimba bubonic plague outbreak. The corresponding values of βc for these estimated values of βf therefore gave the estimates of R0 for bubonic plague as R0 = 1.4 for run 14 and R0 = 1.5 for run 15 based on the formula The optimal configuration of the model was determined by analysing the outbreak data for the period of initial growth of the outbreak up to its peak (shaded area). This is because it was expected in this study that the disease transmission dynamics occurred naturally during this period of the outbreak; an assumption implicit in the model design. Therefore, the model predicted cumulative number of cases over time for each simulation for this period was compared to the actual numbers recorded during the Nyimba outbreak for the same period. This was done for all dates on which new cases of bubonic plague were recorded within this period which are the 26ril 2015 . These d0) for the most recent outbreak of bubonic plague which occurred in 2015 in Nyimba district of Eastern province in Zambia was estimated. To estimate this R0, a simplified robust mathematical method that estimates R0 directly from epidemic incidence data was used (primary method) in light of the limited epidemiological data for the study outbreak [0 for bubonic plague was estimated to be effectively above unity ranging from 1.5599–1.9332 with an average of 1.7465. The Nyimba plague outbreak reached its peak as early as around the 7th– 8th of April 2015 after which it was generally brought under control with the last case recorded on the 1st of May 2015 [In this study, the basic reproduction number  [0 for the infection in this study [The estimated value of R country ,14,18,21rch 2015 . After t period) . It is tis study . This ma0 was slightly lower but still fell well within the 95% confidence interval of the estimated range for R0 using the primary method. This outcome was taken to validate the results of this study to some extent since there was a general consistency in the estimates of R0 for the same bubonic plague outbreak despite using two very different methods of estimation. This observed “agreement” in the two results  is despite the uncertainties associated with estimating R0 using various parameters of a compartmental model. For example, in this model used, all the parameters (βc and α) on which R0 depends generally had to be estimated from literature which can introduce unavoidable uncertainty in the R0 estimate because these parameter values also greatly vary in literature. For this study, this was of great significance because changes in the composite parameter βc (which comprised of various parameters estimated from literature) had the greatest effect on changes in both the model predicted number of cases over time and the estimated value of R0 for bubonic plague. Other parameter values were kept constant for simplification of the model.A simple SIR deterministic compartmental mathematical model for bubonic plague infection was also developed in this study. As seen in 0 for bubonic plague in this study was also found to be essentially within the same range as that estimated for previous rapid spreading bubonic plague outbreaks of the 14th century Black Death plague in Europe for which R0 was estimated to range between 1.5 and 1.9 [th to 18th century, such as the well-known Black Death, were documented to have been more devastating, with higher attack rates than modern day third pandemic outbreaks [0, but different magnitude of outbreaks between our study outbreak and previous pandemics is most likely due to similar transmission patterns but present day treatment and containment measures reducing the susceptible population fast [0 for the infection. The result of this study therefore puts forward some scientific evidence that bubonic plague outbreaks today are still inherently capable of widespread transmission with devastating effects if left unchecked. In the case of Zambia, this raises particular concern considering that starting from about 1997 there has been a sudden occurrence of large plague epidemics than ever recorded before in the country [0, this current study was able to attach a quantitative aspect to describing and understanding the possible transmission dynamics of bubonic plague in a local setup in Zambia. With this new added knowledge of R0, we can now use this estimate to predict the size and progression of future outbreaks and quantitatively assess the effects of various control measures required to prevent the spread of infection through further mathematical modelling techniques [0 is that it is used to determine the minimum mass treatment or vaccination coverage rate required to prevent spread of the infectious disease in a population [0 estimated in this study, a minimum vaccination (or prophylactic treatment) coverage rate of about 43% would be enough to prevent the spread of bubonic plague infection in an identified risk population [0 in the secondary method, that the disease will die out if βc < α, i.e., if the recovery rate is greater than the composite human to human transmission rate, as expected. Finally, similar to the conclusions by [0, the study shows that the plague outbreak in Nyimba followed an exponential growth rate suggestive of human to human transmission. This implies that a human to human plague transmission pathway via ectoparasites has to be considered under the present conditions. With socio-cultural and ecological changes in plague endemic developing countries, this transmission pathway could become more prominent and adequate control methods should be employed.The estimated value of R and 1.9 . As is gutbreaks ,31,53. Tion fast ,31,54. T country . By estichniques –28,37,55pulation ,37. Therpulation ,18,21,37sions by , the res0 may vary from one area to another depending on population dynamics and other factors. Additionally, it is also acknowledged that the methods used here for estimating R0 are still subject to certain weaknesses most important of which is the completeness and quality of the reported epidemic incidence data used for the analysis. For example, 29 out of all the 111 patients suspected to have bubonic plague during the Nyimba outbreak were not available for physical examination [0 to be underestimated in this study as it is generally possible that some cases during the outbreak may have been misdiagnosed, never reported or even died before they went to the hospital. This could have particularly affected the result from the model as it is possible that the model used in this study may have been fit to data that was incomplete. Another limitation of the study is the small sample size available for the analysis despite use of a robust method of estimating R0 as well as the limited background data about the affected population. More similar studies with larger sample sizes will need to be conducted. In spite of this, it is the researcher’s recommendation that the estimate of R0 for bubonic plague obtained in this study can still be used as a primary reference value in the analysis of future outbreaks of the disease in similar populations in the country especially in emergency situations. Alternatively, this study has demonstrated the practical use of a quick and robust method for estimating R0 directly from outbreak incidence data for an infectious disease such as bubonic plague. Therefore, this method can still be employed in future to obtain an updated estimate of R0 during the early stages of an outbreak of bubonic plague or other infectious diseases using the outbreak incidence data.It is noted however that the result of this study may not be readily representative for all recent bubonic plague outbreaks seen in Zambia since the value of R0 which is above unity is a quantitative indicator that indeed bubonic plague is capable of epidemic spread in Zambia as was seen in Nyimba. The value of R0 estimated here also suggests that there is a present potential threat of the occurrence of large bubonic plague outbreaks in Zambia if conditions become favorable.The basic reproduction number for the 2015 Nyimba district bubonic plague outbreak was estimated to range between 1.5599 and 1.9332 with an average of 1.7465. This value of R