An algebraic formula, deep learning and a novel SEIR-type model for the COVID-19 pandemic.

The most extensively used mathematical models in epidemiology are the susceptible-exposed-infectious-recovered (SEIR) type models with constant coefficients. For the first wave of the COVID-19 epidemic, such models predict that at large times equilibrium is reached exponentially. However, epidemiological data from Europe suggest that this approach is algebraic. Indeed, accurate long-term predictions have been obtained via a forecasting model only if it uses an algebraic as opposed to the standard exponential formula. In this work, by allowing those parameters of the SEIR model that reflect behavioural aspects (e.g. spatial distancing) to vary nonlinearly with the extent of the epidemic, we construct a model which exhibits asymptoticly algebraic behaviour. Interestingly, the emerging power law is consistent with the typical dynamics observed in various social settings. In addition, using reliable epidemiological data, we solve in a numerically robust way the inverse problem of determining all model parameters characterizing our novel model. Finally, using deep learning, we demonstrate that the algebraic forecasting model used earlier is optimal.

Integrable nonlinear evolution equations in three spatial dimensions.

There are integrable nonlinear evolution equations in two spatial variables. The solution of the initial value problem of these equations necessitated the introduction of novel mathematical formalisms. Indeed, the classical Riemann-Hilbert problem used for the solution of integrable equations in one spatial variable was replaced by a non-local Riemann-Hilbert problem or, more importantly, by the so-called d -bar formalism. The construction of integrable nonlinear evolution equations in three spatial dimensions has remained the key open problem in the area of integrability. For example, the two versions of the Kadomtsev-Petviashvili (KP) equation constitute two-dimensional generalizations of the celebrated Korteweg-de Vries equation. Are there three-dimensional generalizations of the KP equations? Here, we present such equations. Furthermore, we introduce a novel non-local d -bar formalism for solving the associated initial value problem.

Easing COVID-19 lockdown measures while protecting the older restricts the deaths to the level of the full lockdown.

Guided by a rigorous mathematical result, we have earlier introduced a numerical algorithm, which using as input the cumulative number of deaths caused by COVID-19, can estimate the effect of easing of the lockdown conditions. Applying this algorithm to data from Greece, we extend it to the case of two subpopulations, namely, those consisting of individuals below and above 40 years of age. After supplementing the Greek data for deaths with the data for the number of individuals reported to be infected by SARS-CoV-2, we estimated the effect on deaths and infections in the case that the easing of the lockdown measures is different for these two subpopulations. We found that if the lockdown measures are partially eased only for the young subpopulation, then the effect on deaths and infections is small. However, if the easing is substantial for the older population, this effect may be catastrophic.

A new approach to integrable evolution equations on the circle.

We propose a new approach for the solution of initial value problems for integrable evolution equations in the periodic setting based on the unified transform. Using the nonlinear Schrödinger equation as a model example, we show that the solution of the initial value problem on the circle can be expressed in terms of the solution of a Riemann-Hilbert problem whose formulation involves quantities which are defined in terms of the initial data alone. Our approach provides an effective solution of the problem on the circle which is conceptually analogous to the solution of the problem on the line via the inverse scattering transform.

A quantitative framework for exploring exit strategies from the COVID-19 lockdown.

Following the highly restrictive measures adopted by many countries for combating the current pandemic, the number of individuals infected by SARS-CoV-2 and the associated number of deaths steadily decreased. This fact, together with the impossibility of maintaining the lockdown indefinitely, raises the crucial question of whether it is possible to design an exit strategy based on quantitative analysis. Guided by rigorous mathematical results, we show that this is indeed possible: we present a robust numerical algorithm which can compute the cumulative number of deaths that will occur as a result of increasing the number of contacts by a given multiple, using as input only the most reliable of all data available during the lockdown, namely the cumulative number of deaths.

Mathematical models and deep learning for predicting the number of individuals reported to be infected with SARS-CoV-2.

We introduce a novel methodology for predicting the time evolution of the number of individuals in a given country reported to be infected with SARS-CoV-2. This methodology, which is based on the synergy of explicit mathematical formulae and deep learning networks, yields algorithms whose input is only the existing data in the given country of the accumulative number of individuals who are reported to be infected. The analytical formulae involve several constant parameters that were determined from the available data using an error-minimizing algorithm. The same data were also used for the training of a bidirectional long short-term memory network. We applied the above methodology to the epidemics in Italy, Spain, France, Germany, USA and Sweden. The significance of these results for evaluating the impact of easing the lockdown measures is discussed.

Fokas method for linear boundary value problems involving mixed spatial derivatives.

We obtain solution representation formulae for some linear initial boundary value problems posed on the half space that involve mixed spatial derivative terms via the unified transform method (UTM), also known as the Fokas method. We first implement the method on the second-order parabolic PDEs; in this case one can alternatively eliminate the mixed derivatives by a linear change of variables. Then, we employ the method to biharmonic problems, where it is not possible to eliminate the cross term via a linear change of variables. A basic ingredient of the UTM is the use of certain invariant maps. It is shown here that these maps are well defined provided that certain analyticity issues are appropriately addressed.

A numerical technique for linear elliptic partial differential equations in polygonal domains.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

X-ray, spectroscopic and normal-mode dynamics of calexcitin: structure-function studies of a neuronal calcium-signalling protein.

The protein calexcitin was originally identified in molluscan photoreceptor neurons as a 20 kDa molecule which was up-regulated and phosphorylated following a Pavlovian conditioning protocol. Subsequent studies showed that calexcitin regulates the voltage-dependent potassium channel and the calcium-dependent potassium channel as well as causing the release of calcium ions from the endoplasmic reticulum (ER) by binding to the ryanodine receptor. A crystal structure of calexcitin from the squid Loligo pealei showed that the fold is similar to that of another signalling protein, calmodulin, the N- and C-terminal domains of which are known to separate upon calcium binding, allowing interactions with the target protein. Phosphorylation of calexcitin causes it to translocate to the cell membrane, where its effects on membrane excitability are exerted and, accordingly, L. pealei calexcitin contains two protein kinase C phosphorylation sites (Thr61 and Thr188). Thr-to-Asp mutations which mimic phosphorylation of the protein were introduced and crystal structures of the corresponding single and double mutants were determined, which suggest that the C-terminal phosphorylation site (Thr188) exerts the greatest effects on the protein structure. Extensive NMR studies were also conducted, which demonstrate that the wild-type protein predominantly adopts a more open conformation in solution than the crystallographic studies have indicated and, accordingly, normal-mode dynamic simulations suggest that it has considerably greater capacity for flexible motion than the X-ray studies had suggested. Like calmodulin, calexcitin consists of four EF-hand motifs, although only the first three EF-hands of calexcitin are involved in binding calcium ions; the C-terminal EF-hand lacks the appropriate amino acids. Hence, calexcitin possesses two functional EF-hands in close proximity in its N-terminal domain and one functional calcium site in its C-terminal domain. There is evidence that the protein has two markedly different affinities for calcium ions, the weaker of which is most likely to be associated with binding of calcium ions to the protein during neuronal excitation. In the current study, site-directed mutagenesis has been used to abolish each of the three calcium-binding sites of calexcitin, and these experiments suggest that it is the single calcium-binding site in the C-terminal domain of the protein which is likely to have a sensory role in the neuron.

Generating mechanism for higher-order rogue waves.

We introduce a mechanism for generating higher-order rogue waves (HRWs) of the nonlinear Schrödinger (NLS) equation: the progressive fusion and fission of n degenerate breathers associated with a critical eigenvalue λ(0) creates an order-n HRW. By adjusting the relative phase of the breathers in the interacting area, it is possible to obtain different types of HRWs. The value λ(0) is a zero point of an eigenfunction of the Lax pair of the NLS equation and it corresponds to the limit of the period of the breather tending to infinity. By employing this mechanism we prove two conjectures regarding the total number of peaks, as well as a decomposition rule in the circular pattern of an order-n HRW.

Systematic construction and prediction of the arrangement of the strands of sandwich proteins.

The problem of predicting the three-dimensional structure of a protein starting from its amino acid sequence is regarded as one of the most important open problems in biology. Here, we solve aspects of this problem for the so-called sandwich proteins that constitute a large class of proteins consisting of only beta-strands arranged in two sheets. A breakthrough for this class of proteins was announced in Kister et al. (Kister et al. 2002 Proc. Natl Acad. Sci. USA 99, 14 137-14 141), in which it was shown that sandwich proteins contain a certain invariant substructure called interlock. It was later noted that approximately 90% of the observed sandwich proteins are canonical, namely they are generated by certain geometrical structures. Here, employing a topological investigation, we prove that interlocks and geometrical structures are the direct consequence of certain biologically motivated fundamental principles. Furthermore, we construct all possible canonical motifs involving 6-10 strands. This construction limits dramatically the number of possible motifs. For example, for sandwich proteins with nine strands, the a priori number of possible canonical motifs exceeds 360000, whereas our construction yields only 49 geometrical structures and 625 canonical motifs.

Electro-magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries.

The problem of determining a continuously distributed neuronal current inside the brain under the assumption of a three-shell model is analysed. It is shown that for an arbitrary geometry, electroencephalography (EEG) provides information about one of the three functions specifying the three components of the current, whereas magnetoencephalography (MEG) provides information about a combination of this function and of one of the remaining two functions. Hence, the simultaneous use of EEG and MEG yields information about two of the three functions needed for the reconstruction of the current. In particular, for spherical and ellipsoidal geometries, it is possible to determine the angular parts of these two functions as well as to obtain an explicit constraint satisfied by their radial parts. The complete determination of the radial parts, as well as the determination of the third function, requires some additional a priori assumption about the current. One such assumption involving harmonicity is briefly discussed.

Reconstruction algorithm for single photon emission computed tomography and its numerical implementation.

The modern imaging techniques of positron emission tomography and of single photon emission computed tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so-called attenuated Radon transform, respectively. An exact formula for the inverse Radon transform is well known, whereas that for the inverse attenuated Radon transform was obtained only recently by R. Novikov. The latter formula was constructed by using a method introduced earlier by R. Novikov and the first author in connection with a novel derivation of the inverse Radon transform. Here, we first show that the appropriate use of that earlier result yields immediately an analytic formula for the inverse attenuated Radon transform. We then present an algorithm for the numerical implementation of this analytic formula, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithm is capable of producing accurate reconstruction for realistic phantoms such as the well-known Shepp-Logan phantom.

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

Strict rules determine arrangements of strands in sandwich proteins.

From a computer analysis of the spatial organization of the secondary structures of beta-sandwich proteins, we find certain sets of consecutive strands that are connected by hydrogen bonds, which we call "strandons." The analysis of the arrangements of strandons in 491 protein structures that come from 69 different superfamilies reveals strict regularities in the arrangements of strandons and the formation of what we call "canonical supermotifs." Six such supermotifs account for approximately 90% of all observed structures. Simple geometric rules are described that dictate the formation of these supermotifs.

A geometric construction determines all permissible strand arrangements of sandwich proteins.

For a large class of proteins called sandwich-like proteins (SPs), the secondary structures consist of two beta-sheets packed face-to-face, with each beta-sheet consisting typically of three to five beta-strands. An important step in the prediction of the three-dimensional structure of a SP is the prediction of its supersecondary structure, namely the prediction of the arrangement of the beta-strands in the two beta-sheets. Recently, significant progress in this direction was made, where it was shown that 91% of observed SPs form what we here call "canonical motifs." Here, we show that all canonical motifs can be constructed in a simple manner that is based on thermodynamic considerations and uses certain geometric structures. The number of these structures is much smaller than the number of possible strand arrangements. For instance, whereas for SPs consisting of six strands there exist a priori 900 possible strand arrangements, there exist only five geometric structures. Furthermore, the few motifs that are noncanonial can be constructed from canonical motifs by a simple procedure.

Prediction of the structural motifs of sandwich proteins.

We investigate the supersecondary structure of a large group of proteins, the so-called sandwich proteins. The analysis of a large number of such proteins has led us to propose a set of rules that can be used to predict the possible arrangements of strands in the two beta-sheets forming a given sandwich structure. These rules imply the existence of certain invariant supersecondary substructures common to all sandwich proteins. Furthermore, they dramatically restrict the number of permissible arrangements. For example, whereas for proteins consisting of three strands in each beta-sheet 180 possible strand arrangements exist a priori, our rules imply that only 15 of them are permissible. Five of these predicted arrangements describe all currently known sandwich proteins with six strands.

Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence.

A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0< or =x< or =L-y< or =L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.

Mathematical model of granulocytopoiesis and chronic myelogenous leukemia.

We present a mathematical model of granulocytopoiesis that depends on certain physiologically meaningful parameters. By choosing different values of these parameters, the model describes both the normal process and that in chronic myelogenous leukemia (CML). The model fits all the available experimental data tested. Furthermore, it shows how the CML cells can ultimately outnumber the normal cells and how this process can be very slow. The model provides a quantitative approach to the relationship between proliferation and maturation and resolves the apparent contradiction between decreased proliferation and increased production, by assuming that a greater fraction of CML cells is produced by division rather than by maturation. The model should be helpful in designing experiments to better define the abnormalities of proliferation and maturation in CML and in seeking to define the specific alterations in the cell regulatory networks resulting from the production of the chimeric p210bcr-abl protein characteristic of CML.