Observed biological growth curves generally are sigmoid in appearance. It is common practice to fit such data with either a Verhulst logistic or a Gompertz curve. This paper critically considers the conceptual bases underlying these descriptive models.The logistic model was developed by Verhulst to accommodate the common sense observation that populations cannot keep growing indefinitely. A justification for using the same equation to describe the growth of individuals, based on considerations from chemical kinetics (autocatalysis of a monomolecular reaction), (...) was put forward by Richardson, but met with heavy criticism as a result of his erroneous derivation of the basic equation (Snell, 1929). It errs on the side of over-simplicity (Priestley & Pearsall, 1922). Von Bertalanffy (1957) subsequently based a justification on the assumption that, as a first approximation, the rates of catabolism and anabolism may be assumed to be proportional to weight and power (still vacant places, between the maximal possible and the already accumulated population sizes. This point of view was fiercely challenged by Nicholson (1933), Milne (1962), Smith (1954) and Rubinov (1973). And indeed, what is meant by vacant places has never become entirely clear. Finally Lotka (1925) devised a third leading approach by just truncating a Taylor expansion around zero of the differential law for autonomous growth after the second degree term. (shrink)

Sigmoid functions have been applied in many areas to model self limited population growth. The most popular functions; General Logistic (GL), General von Bertalanffy (GV), and Gompertz (G), comprise a family of functions called Theta Logistic ( $$ \Uptheta $$ L ). Previously, we introduced a simple model of tumor cell population dynamics which provided a unifying foundation for these functions. In the model the total population ( N ) is divided into reproducing ( P ) and non-reproducing/quiescent ( (...) Q ) sub-populations. The modes of the rate of change of ratio P / N was shown to produce GL, GV or G growth. We now generalize the population dynamics model and extend the possible modes of the P / N rate of change. We produce a new family of sigmoid growth functions, Trans-General Logistic (TGL), Trans-General von Bertalanffy (TGV) and Trans-Gompertz (TG)), which as a group we have named Trans-Theta Logistic ( T $$ \Uptheta $$ L ) since they exist when the $$ \Uptheta $$ L are translated from a two parameter into a three parameter phase space. Additionally, the model produces a new trigonometric based sigmoid ( TS ). The $$ \Uptheta $$ L sigmoids have an inflection point size fixed by a single parameter and an inflection age fixed by both of the defining parameters. T $$ \Uptheta $$ L and TS sigmoids have an inflection point size defined by two parameters in bounding relationships and inflection point age defined by three parameters (two bounded). While the Theta Logistic sigmoids provided flexibility in defining the inflection point size, the Trans-Theta Logistic sigmoids provide flexibility in defining the inflection point size and age. By matching the slopes at the inflection points we compare the range of values of inflection point age for T $$ \Uptheta $$ L versus $$ \Uptheta $$ L for model growth curves. (shrink)

When consciousness is examined using subjective ratings, the extent to which processing is conscious or unconscious is often estimated by calculating task performance at the subjective threshold or by calculating the correlation between accuracy and awareness. However, both these methods have certain limitations. In the present article, we propose describing task accuracy and awareness as functions of stimulus intensity as suggested by Koch and Preuschoff . The estimated lag between the curves describes how much stimulus intensity must increase for (...) awareness to change proportionally as much as accuracy and the slopes of the curves are used to assess how fast accuracy and awareness increases and whether awareness is dichotomous. The method is successfully employed to assess consciousness characteristics on data from four different awareness scales. (shrink)

The filamentous and branched thallus of Antithamnion plumula is constitued of two different kinds of branches with apical growth: the cladomial axes with a continuous or indefinite growth, and the pleuridia with a limited growth. The size of the pleuridia depends on their position with respect to the lateral cladomial axes.The growth kinetics of 35 pleuridia were analysed using Nelder's generalized logistics. Each sigmoidal curve, which was divided into four growth stages from the instantaneous acceleration variations, was thus characterized by (...) ten kinetic parameters: lengths at the time of the changes in growth stage, durations of the growth stages and maximum growth rate. (shrink)

Humans regularly pursue activities characterized by dramatic success or failure outcomes where, critically, the chances of success depend on the time invested working toward it. How should people allocate time between such make‐or‐break challenges and safe alternatives, where rewards are more predictable (e.g., linear) functions of performance? We present a formal framework for studying time allocation between these two types of activities, and we explore optimal behavior in both one‐shot and dynamic versions of the problem. In the one‐shot version, we (...) illustrate striking discontinuities in the optimal time allocation policy as we gradually change the parameters of the decision‐making problem. In the dynamic version, we formulate the optimal strategy—defined by a giving‐up threshold—which adaptively dictates when people should stop pursuing the make‐or‐break goal. We then show that this strategy is computationally inaccessible for humans, and we explore boundedly rational alternatives. We compare the performance of the optimal model against (a) a myopic giving‐up threshold that is easier to compute, and even simpler heuristic strategies that either (b) only decide whether or not to start pursuing the goal and never give up or (c) consider giving up at a fixed number of control points. Comparing strategies across environments, we investigate the cost and behavioral implications of sidestepping the computational burden of full rationality. (shrink)

The growth activity of an organ (variable y) is defined simultaneously by the instantaneous absolute ratedy/dt and its variationd 2y/dt2. The use of these two descriptors allows a sigmoidal (i.e. continuous and non periodical, as observed for the logistic function) growth curve to be discretized into a series of 5 growth states or phases which are delimited by the following singular values: max, Vmax (=0), max, adult stage. The (V, ) plot, termedgrowth trajectory, visualizes, e.g. in the case of Richards-Nelder's (...) generalized logistics, the unequal duration of these phases and allows the dissymmetry property of the growth function to be characterized precisely.At a more integrated organization level, such anorganic series (e. g. the set of the leaves borne by a same branch), the above concepts allow two kinds of analyses to be carried out: (i) in thekinetic approach, the growth trajectory of the series can characterizes, under certain conditions, the emergence of a new property: theoccurrence of growth activity rhythms which are independent of meristem dormancy processes; (ii) thedemographic approach shows thegrowth structure of the phytomer population (a particular form of age structure) and the associated dimensional structure (plant partition into zones or regions in relation to the growth state of the various phytomers). (shrink)

The fructose 6-phosphate (Fru-6P) saturation curve for phosphofructokinase (PFK) from E. coli is sigmoidal in the presence of saturating MgATP levels, while the corresponding curve for B. stearothermophilus PFK is essentially hyperbolic. Sigmoidality can be due to apparent cooperativity arising from the kinetic mechanism of an enzyme. We have determined the kinetic mechanism of B. stearothermophilus PFK (BsPFK). BsPFK was found to obey a non rapid-equilibrium random mechanism similar to the one E. coli PFK (EcPFK) follows. Substrate inhibition by MgATP (...) was observed. We propose that substrates bind to BsPFK through two alternative pathways, one of which is slower. The substrate inhibition arises in part from reaction flux through the slower pathway. Although EcPFK and BsPFK obey similar kinetic mechanisms, they are inhibited differently by MgATP: EcPFK is profoundly inhibited, BsPFK only weakly. The structural basis for this difference could be closure of the active site via a conformational transition that occurs in EcPFK, but not BsPFK. To investigate the importance of this transition for MgATP inhibition of EcPFK, we have constructed a chimeric enzyme (ChiPFK) that contains the "rigid" ATP-binding domain of BsPFK grafted onto the remainder of the EcPFK subunit. Our results indicate that ChiPFK is locked in an "open" structure resembling the activated form of EcPFK. It is insensitive to heterotropic regulation. Nevertheless, ChiPFK exhibits residual cooperativity. Possible explanations for the cooperativity are discussed. The 6F loop is proposed to be important in PFK allosteric behavior. Residues along the loop are largely conserved between BsPFK and EcPFK, except for 161, which is a glutamate in BsPFK, and a glutamine in EcPFK. Using site-directed mutagenesis, Glu 161 of BsPFK has been changed to glutamine and alanine. Similarly, Gin 161 of EcPFK has been changed to glutamate, arginine and alanine. Of the five mutants, one, QA161, was particularly interesting. Though activated normally by GDP, it was completely insensitive to PEP inhibition. This indicates that the hydrogen-bonding ability of residue 161 is critical for PEP inhibition of EcPFK, and that GDP activation and PEP inhibition follow different structural pathways. (shrink)

This paper investigates the following question: when can one reliably infer the existence of an intersection point from a diagram presenting crossing curves or lines? Two cases are considered, one from Euclid's geometry and the other from basic real analysis. I argue for the acceptability of such an inference in the geometric case but against in the analytic case. Though this question is somewhat specific, the investigation is intended to contribute to the more general question of the extent and (...) limits of reliable diagrammatic reasoning in mathematics. (shrink)

In this article, we analyse the evidential value of the corpus of experimental philosophy. While experimental philosophers claim that their studies provide insight into philosophical problems, some philosophers and psychologists have expressed concerns that the findings from these studies lack evidential value. Barriers to evidential value include selection bias and p-hacking. To find out whether the significant findings in x-phi papers result from selection bias or p-hacking, we applied a p-curve analysis to a corpus of 365 x-phi chapters and articles. (...) Our results suggest that this corpus has evidential value, although there are hints of p-hacking in a few parts of the x-phi corpus. (shrink)

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that cannot be captured within the orthodox Bayesian framework.

I propose three new curved spacetime versions of the Dirac Equation. These equations have been developed mainly to try and account in a natural way for the observed anomalous gyromagnetic ratio of Fermions. The derived equations suggest that particles including the Electron which is thought to be a point particle do have a finite spatial size which is the reason for the observed anomalous gyromagnetic ratio. A serendipitous result of the theory, is that, to of the equation exhibits an asymmetry (...) in their positive and negative energy solutions the first suggestion of which is clear that a solution to the problem as to why the Electron and Moun—despite their acute similarities—exhibit an asymmetry in their mass is possible. The Moun is often thought as an Electron in a higher energy state. Another of the consequences of three equations emanating from the asymmetric serendipity of the energy solutions of two of these equations, is that, an explanation as to why Leptons exhibit a three stage mass hierarchy is possible. (shrink)

Bayesians often assume, suppose, or conjecture that for any reasonable explication of the notion of simplicity a prior can be designed that will enforce a preference for hypotheses simpler in just that sense. Further, it is often claimed that the Bayesian framework automatically implements Occam's razor—that conditionalizing on data consistent with both a simple theory and a complex theory more or less inevitably favours the simpler theory. But it is shown here that there are simplicity-driven approaches to curve-fitting problems that (...) cannot be captured within the orthodox Bayesian framework and that the automatic razor does not function for such problems. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit, pull in opposite directions. To this problem, we propose a solution that strikes a balance between simplicity and goodness-of-fit. Using Bayes' theorem we argue that the notion of prior probability represents a measurement of simplicity of a theory, whereas the notion of likelihood represents the theory's goodness-of-fit. We justify the use of prior probability and show how to calculate the likelihood of a family of curves. We diagnose the (...) relationship between simplicity of a theory and its predictive accuracy. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit pull in opposite directions. To solve this problem, two proposals, the first one based on Bayes's theorem criterion (BTC) and the second one advocated by Forster and Sober based on Akaike's Information Criterion (AIC) are discussed. We show that AIC, which is frequentist in spirit, is logically equivalent to BTC, provided that a suitable choice of priors is made. We evaluate the charges against Bayesianism and contend that AIC approach (...) has shortcomings. We also discuss the relationship between Schwarz's Bayesian Information Criterion and BTC. (shrink)

The main aim of this paper is to revisit the curve fitting problem using the reliability of inductive inference as a primary criterion for the ‘fittest' curve. Viewed from this perspective, it is argued that a crucial concern with the current framework for addressing the curve fitting problem is, on the one hand, the undue influence of the mathematical approximation perspective, and on the other, the insufficient attention paid to the statistical modeling aspects of the problem. Using goodness-of-fit as the (...) primary criterion for ‘best', the mathematical approximation perspective undermines the reliability of inference objective by giving rise to selection rules which pay insufficient attention to ‘accounting for the regularities in the data'. A more appropriate framework is offered by the error-statistical approach, where (i) statistical adequacy provides the criterion for assessing when a curve captures the regularities in the data adequately, and (ii) the relevant error probabilities can be used to assess the reliability of inductive inference. Broadly speaking, the fittest curve (statistically adequate) is not determined by the smallness if its residuals, tempered by simplicity or other pragmatic criteria, but by the nonsystematic (e.g. white noise) nature of its residuals. The advocated error-statistical arguments are illustrated by comparing the Kepler and Ptolemaic models on empirical grounds. ‡I am grateful to Deborah Mayo and Clark Glymour for many valuable suggestions and comments on an earlier draft of the paper; estimating the Ptolemaic model was the result of Glymour's prompting and encouragement. †To contact the author, please write to: Department of Economics, Virginia Tech 3019 Pamplin Hall (0316), Blacksburg, VA 24061; e-mail: [email protected]. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit, pull in opposite directions. To this problem, we propose a solution that strikes a balance between simplicity and goodness-of-fit. Using Bayes' theorem we argue that the notion of prior probability represents a measurement of simplicity of a theory, whereas the notion of likelihood represents the theory's goodness-of-fit. We justify the use of prior probability and show how to calculate the likelihood of a family of curves. We diagnose the (...) relationship between simplicity of a theory and its predictive accuracy. (shrink)

In the curve fitting problem two conflicting desiderata, simplicity and goodness-of-fit, pull in opposite directions. To this problem, we propose a solution that strikes a balance between simplicity and goodness-of-fit. Using Bayes’ theorem we argue that the notion of prior probability represents a measurement of simplicity of a theory, whereas the notion of likelihood represents the theory’s goodness-of-fit. We justify the use of prior probability and show how to calculate the likelihood of a family of curves. We diagnose the (...) relationship between simplicity of a theory and its predictive accuracy. (shrink)

Much of scientific inference involves fitting numerical data with a curve, or functional relation. The received view is that the fittest curve is the curve which best balances the conflicting demands of simplicity and accuracy, where simplicity is measured by the number ofparameters in the curve. The problem with this view is that there is no commonly accepted justification for desiring simplicity. This paper presents a measure of the stability of equations. It is argued that the fittest curve is the (...) curve which best balances stability and accuracy. The received view is defended with a proof that simplicity corresponds to stability, for linear regression equations. 1This paper is based on part of my doctoral dissertation. My thanks go to my thesis supervisor Professor Alasdair Urquhart for his encouragement, constructive criticism, and for directing me to several relevant articles: to my advisor Professor Ian Hacking for reminding me to concentrate on results that might have some application in the real world; and to my friend Wendy Brandts for sharing her ideas on a closely related problem. My thanks also to an anonymous referee of The British Journal of the Philosophy of Science for several helpful comments, to my friends and family for unfailing support, and to the Social Sciences and Humanities Research Council (awards 452-86-5885 and 453-87-0513) and the University of Toronto for financial assistance. (shrink)

Model simplicity in curve fitting is the fewness of parameters estimated. I use a vector model of least squares estimation to show that degrees of freedom, the difference between the number of observed parameters fit by the model and the number of explanatory parameters estimated, are the number of potential dimensions in which data are free to differ from a model and indicate the disconfirmability of the model. Though often thought to control for parameter estimation, the AIC and similar indices (...) do not do so for all model applications, while goodness of fit indices like chi-square, which explicitly take into account degrees of freedom, do. Hypothesis testing with prespecified values for parameters is based on a metaphoric regulative subject/object schema taken from object perception and has as its goal the accumulation of objective knowledge. (shrink)

According to general relativity, we live in a four-dimensional curved universe. Since the human mind cannot visualize those four dimensions, a popular analogy compares the universe to a two-dimensional rubber sheet distorted by massive objects. This analogy is often used when teaching GR to upper secondary and undergraduate physics students. However, physicists and physics educators criticize the analogy for being inaccurate and for introducing conceptual conflicts. Addressing these criticisms, we analyze the rubber sheet analogy through systematic metaphor analysis of textbooks (...) and research literature, and present an empirical analysis of upper secondary school students’ use and understanding of the analogy. Taking a theoretical perspective of embodied cognition allows us to account for the relationship between the experiential and sensory aspects of the metaphor in relation to the abstract nature of spacetime. We employ methods of metaphor and thematic analysis to study written accounts of small groups of 97 students who worked with a collaborative online learning environment as part of their regular physics lessons in five classes in Norway. Students generated conceptual metaphors found in the literature as well as novel ones that led to different conceptions of gravity than those held by experts in the field. Even though most students showed awareness of some limitations of the analogy, we observed a conflict between students’ embodied understanding of gravity and the abstract description of GR. This conflict might add to the common perception of GR being counterintuitive. In making explicit strengths and weaknesses of the rubber sheet analogy and learners’ conceptual difficulties, our results offer guidance for teaching GR. More generally, these findings contribute to the epistemological implications of employing specific scientific metaphors in classrooms. (shrink)

In this book, Francis Diebold and Glenn Rudebusch propose two extensions of the classic yield curve model of Nelson and Siegel that are both theoretically rigorous and empirically successful.

In this paper, we first define the vector product in Minkowski space $\mathbb{R}_{4}^{7}$, which is identified with the space of spatial split-octonions. Next, we derive the $G_{2}-$ frame formulae for a seven dimensional Minkowski curve by using the spatial split-octonions and the vector product. We show that Frenet–Serret formulas are satisfied for a spatial split octonionic curve. We obtain the congruence of two spatial split octonionic curves and give relationship between the $G_{2}-$ frame and Frenet–Serret frame. Furthermore, we present (...) the Frenet–Serret frame with split octonions in $\mathbb{R}_{4}^{8}$. Finally, we give illustrative examples with Matlab codes. (shrink)

Reason Curve, Jury Competence, and the English Criminal Justice System, a cross-jurisdictional and cross-disciplinary book, seeks to stimulate discussion and extend the debate in the area of criminal trials in light of the absence of an articulated explanation for a verdict. The book traces the history and development of the jury, from the Carolingian kings, its advancement in the English Courts following papal intervention, the impact of the Magna Carta, to its general use, current curtailment in England and Wales, and (...) re-emergence in Continental Europe. Central to the book's submission is the dictum that the jurors' franchise to deliver a cryptic verdict is 'a matter between them and their conscience... (shrink)

Four empirically equivalent versions of general relativity, namely standard GR, Lorentz-invariant gravitational theory,and the gravitational gauge theories of the Lorentz and translation groups, are investigated in the form of a case study for theory underdetermination. The various ontological indeterminacies (both underdetermination and inscrutability of reference) inherent in gravitational theories are analyzed in a detailed comparative study. The concept of practical underdetermination is proposed, followed by a discussion of its adequacy to describe scientific progress.

Much of scientific inference involves fitting numerical data with a curve, or functional relation. The received view is that the fittest curve is the curve which best balances the conflicting demands of simplicity and accuracy, where simplicity is measured by the number ofparameters in the curve. The problem with this view is that there is no commonly accepted justification for desiring simplicity.This paper presents a measure of the stability of equations. It is argued that the fittest curve is the curve (...) which best balances stability and accuracy. The received view is defended with a proof that simplicity corresponds to stability, for linear regression equations. (shrink)

In a recent paper published in the Journal of Medical Ethics, Le Morvan and Stock claim that the kantian ideal of treating people always as ends in themselves and never merely as a means is in direct and insurmountable conflict with the current medical practice of allowing practitioners at the bottom of their “learning curve” to “practise their skills” on patients. In this response, I take up the challenge they issue is and try to reconcile this conflict. The kantian ideal (...) offered in the paper is an incomplete characterisation of Kant’s moral philosophy, and the formula of humanity is considered in isolation without taking into account other salient kantian principles. I also suggest that their argument based on “necessary for the patient” assumes too narrow a reading of “necessary”. This reply is intended as an extension to, rather than a criticism of, their work. (shrink)

I propose an account of the metaphysics of the expressions of a mathematical language which brings together the structuralist construal of a mathematical object as a place in a structure, the semantic notion of indexicality and Kit Fine's ontological theory of qua objects. By contrasting this indexical qua objects account with several other accounts of the metaphysics of mathematical expressions, I show that it does justice both to the abstractness that mathematical expressions have because they are mathematical objects and to (...) the element of concreteness that they have because they are also used as signs. In a concluding section, I comment on the pragmatic element that has entered ontology by way of the notion of indexicality and use it to give an answer to a question Stewart Shapiro has recently posed about the status of meta-mathematics in the structuralist philosophy of mathematics. (shrink)

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifolds. The fields are pairs of \,\varpi )\), where \\) is a 1-form with coefficients in the Lie algebra (...) of the Poincaré group and \ is an 8-form with coefficients in the dual of this Lie algebra. The dynamical equations derive from a simple variational principle and imply that the 10-dimensional manifold looks locally like the total space of a fiber bundle over a 4-dimensional base manifold. Moreover this base manifold inherits a metric and a connection which are solutions of a system of Einstein–Cartan equations. (shrink)

In this paper, we show within ${\mathsf{RCA}_0}$ that both the Jordan curve theorem and the Schönflies theorem are equivalent to weak König’s lemma. Within ${\mathsf {WKL}_0}$ , we prove the Jordan curve theorem using an argument of non-standard analysis based on the fact that every countable non-standard model of ${\mathsf {WKL}_0}$ has a proper initial part that is isomorphic to itself (Tanaka in Math Logic Q 43:396–400, 1997).

Mechanisms for QoS provisioning in communication networks range from flow-based resource reservation schemes, providing QoS guarantees, through QoS differentiation based on reservation aggregation techniques to adaptation of applications, compensating for incomplete reservations. Scalable, aggregation-based reservations can also be combined with adaptations for a more flexible and robust overall QoS provisioning. Adaptation is particularly important in wireless networks, where reservations schemes are more difficult to realize. It is widely accepted that usability of Cellular or Mobile IP can be largely improved if (...) adaptation is incorporated. Adaptation, in turn, depends on coding techniques involved and on user requirements. Those impose certain preferences and ordering relations on QoS hierarchies or degradation paths to be followed along when adaptation is performed. With the advent of MPEG4 coding techniques, more sophisticated adaptation schemes are more likely to be supported in the future. (shrink)

ObjectivesTo simulate hemodynamic changes after extraluminal compression in pulsatile tinnitus patients with a prominent transverse-sigmoid sinus junction.MethodsOne patient-specific case was reconstructed based on computed tomography venography images of a PT patient. The compression degree served as a new index in this study. Cases with 10, 20, 30, 40, 50, 60, 70, 80, and 90% of the compression degree of the control subject were constructed. Steady-state computational fluid dynamics were assessed. The wall pressure distribution, wall maximum pressure and flow pattern (...) of the PTSJ were calculated to evaluate hemodynamic differences among all cases.ResultsWith increasing compression, the wall pressure at the compression point and downstream of the PTSJ decreased but increased upstream. When the compression degree exceeded 70%, the upstream pressure increased significantly. Above 50% compression, the blood flow pattern downstream of the sigmoid sinus tended to spiral, especially after 80% compression. Beyond 60% compression, the blood flow pattern under the compression axis became more medial.ConclusionMechanical compression of PTSJ changes wall pressure and blood flow patterns. The degree of compression should be carefully observed to avoid possible complications or reoccurrence. (shrink)

Since the February 2020 publication of the article ‘Flattening the curve’ in The Economist, political leaders worldwide have used this expression to legitimize the introduction of social distancing measures in fighting Covid-19. In fact, this expression represents a complex combination of three components: the shape of the epidemic curve, the social distancing measures and the reproduction number \. Each component has its own history, each with a different history of control. Presenting the control of the epidemic as flattening the curve (...) is in fact flattening the underlying natural-social complexity. The curve that needs to be flattened is presented as a bell-shaped curve, implicitly suggesting that the pathogen’s spread is subject only to natural laws. The \ value, however, is, fundamentally, a metric of how a pathogen behaves within a social context, namely its numerical value is affected by sociopolitical influences. The jagged and erratic empirical curve of Covid-19 illustrates this. Although the virus has most likely infected only a small portion of the total susceptible population, it is clear its shape has changed drastically. This changing shape is largely due to sociopolitical factors. These include shifting formal laws and policies, shifting individual behaviors as well as shifting various other social norms and practices. This makes the course of Covid-19 curve both erratic and unpredictable. (shrink)