Epidemic Waves and Exact Solutions of a Sequence of Nonlinear Differential Equations Connected to the SIR Model of Epidemics.

The SIR model of epidemic spreading can be reduced to a nonlinear differential equation with an exponential nonlinearity. This differential equation can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. The equations from the obtained sequence are treated by the Simple Equations Method (SEsM). This allows us to obtain exact solutions to some of these equations. We discuss several of these solutions. Some (but not all) of the obtained exact solutions can be used for the description of the evolution of epidemic waves. We discuss this connection. In addition, we use two of the obtained solutions to study the evolution of two of the COVID-19 epidemic waves in Bulgaria by a comparison of the solutions with the available data for the infected individuals.

News Waves: Hard News, Soft News, Fake News, Rumors, News Wavetrains.

We discuss the spread of a piece of news in a population. This is modeled by SIR model of epidemic spread. The model can be reduced to a nonlinear differential equation for the number of people affected by the news of interest. The differential equation has an exponential nonlinearity and it can be approximated by a sequence of nonlinear differential equations with polynomial nonlinearities. Exact solutions to these equations can be obtained by the Simple Equations Method (SEsM). Some of these exact solutions can be used to model a class of waves associated with the spread of the news in a population. The presence of exact solutions allow to study in detail the dependence of the amplitude and the time horizon of the news waves on the wave parameters, such as the size of the population, initial number of spreaders of the piece of the news, transmission rate, and recovery rate. This allows for recommendations about the change of wave parameters in order to achieve a large amplitude or appropriate time horizon of the news wave. We discuss five types of news waves on the basis of the values of the transmission rate and recovery rate-types A, B, C, D, and E of news waves. In addition, we discuss the possibility of building wavetrains by news waves. There are three possible kinds of wavetrains with respect of the amplitude of the wave: increasing wavetrain, decreasing wavetrain, and mixed wavetrain. The increasing wavetrain is especially interesting, as it is connected to an increasing amplitude of the news wave with respect to the amplitude of the previous wave of the wavetrain. It can find applications in advertising, propaganda, etc.

Simple Equations Method (SEsM): An Effective Algorithm for Obtaining Exact Solutions of Nonlinear Differential Equations.

Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf-Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers-Huxley, generalized equation of Camassa-Holm, generalized equation of Swift-Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.

Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity.

We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example.

On the Possibility of Chaos in a Generalized Model of Three Interacting Sectors.

In this study we extend a model, proposed by Dendrinos, which describes dynamics of change of influence in a social system containing a public sector and a private sector. The novelty is that we reconfigure the system and consider a system consisting of a public sector, a private sector, and a non-governmental organizations (NGO) sector. The additional sector changes the model's system of equations with an additional equation, and additional interactions must be taken into account. We show that for selected values of the parameters of the model's system of equations, chaos of Shilnikov kind can exist. We illustrate the arising of the corresponding chaotic attractor and discuss the obtained results from the point of view of interaction between the three sectors.

Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods.

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

Statistical Characteristics of Stationary Flow of Substance in a Network Channel Containing Arbitrary Number of Arms.

We study flow of substance in a channel of network which consists of nodes of network and edges which connect these nodes and form ways for motion of substance. The channel can have arbitrary number of arms and each arm can contain arbitrary number of nodes. The flow of substance is modeled by a system of ordinary differential equations. We discuss first a model for a channel which arms contain infinite number of nodes each. For stationary regime of motion of substance in such a channel we obtain probability distributions connected to distribution of substance in any of channel's arms and in entire channel. Obtained distributions are not discussed by other authors and can be connected to Waring distribution. Next, we discuss a model for flow of substance in a channel which arms contain finite number of nodes each. We obtain probability distributions connected to distribution of substance in the nodes of the channel for stationary regime of flow of substance. These distributions are also new and we calculate corresponding information measure and Shannon information measure for studied kind of flow of substance.

On the Motion of Substance in a Channel of a Network: Extended Model and New Classes of Probability Distributions.

We discuss the motion of substance in a channel containing nodes of a network. Each node of the channel can exchange substance with: (i) neighboring nodes of the channel, (ii) network nodes which do not belong to the channel, and (iii) environment of the network. The new point in this study is that we assume possibility for exchange of substance among flows of substance between nodes of the channel and: (i) nodes that belong to the network but do not belong to the channel and (ii) environment of the network. This leads to an extension of the model of motion of substance and the extended model contains previous models as particular cases. We use a discrete-time model of motion of substance and consider a stationary regime of motion of substance in a channel containing a finite number of nodes. As results of the study, we obtain a class of probability distributions connected to the amount of substance in nodes of the channel. We prove that the obtained class of distributions contains all truncated discrete probability distributions of discrete random variable ω which can take values 0,1,⋯,N. Theory for the case of a channel containing infinite number of nodes is presented in Appendix A. The continuous version of the discussed discrete probability distributions is described in Appendix B. The discussed extended model and obtained results can be used for the study of phenomena that can be modeled by flows in networks: motion of resources, traffic flows, motion of migrants, etc.

Release of carbon nanoparticles of different size and shape from nanocomposite poly(lactic) acid film into food simulants.

Poly(lactic) acid (PLA) film with 2 wt% mixed carbon nanofillers of graphene nanoplates (GNPs) and multiwall carbon nanotubes (MWCNTs) in a weight ratio of 1:1 with impurities of fullerene and carbon black (CB) was produced by layer-to-layer deposition and hot pressing. The release of carbon nanoparticles from the film was studied at varying time-temperature conditions and simulants. Migrants in simulant solvents were examined with laser diffraction analysis and transmission electron microscopy (TEM). Film integrity and the presence of migrants on the film surfaces were visualised by scanning electron microscopy (SEM). The partial dissolution of PLA polymer in the solvents was confirmed by swelling tests and differential scanning calorimetry (DSC). Nanoparticle migrants were not detected in the simulants (at the LOD 0.020 µm of the laser diffraction analysis) after migration testing at 40°C for 10 days. However, high-temperature migration testing at 90°C for 4 h provoked a release of GNPs from the film into ethanol, acetic acid and oil-based food simulants. Short carbon nanotubes were observed rarely to release in the most aggressive acetic acid solvent. Obviously, the enhanced molecular mobility at temperatures above the glass transition and partial dissolution of PLA polymer by the food simulant facilitate the diffusion processes. Moreover, shape, size and concentration of nanoparticles play a significant role. Flexible naked GNPs (lateral size 100-1000 nm) easily migrate when the polymer molecules exhibit enhanced mobility, while fibrous MWCNTs (> 1 µm length) formed entangled networks on the film surfaces as the PLA polymer is partly dissolved, preventing their release into food simulants. The impurities of fullerenes and CB (5-30 nm) were of minor concentration in the polymer, therefore their migration is low or undetectable. The total amount of released migrants is below overall migration limits.

Role of surfactants on the approaching velocity of two small emulsion drops.

Here we present the exact solution of two approaching spherical droplets problem, at small Reynolds and Peclet numbers, taking into account surface shear and dilatational viscosities, Gibbs elasticity, surface and bulk diffusivities due to the presence of surfactant in both disperse and continuous phases. For large interparticle distances, the drag force coefficient, f, increases only about 50% from fully mobile to tangentially immobile interfaces, while at small distances, f can differ several orders of magnitude. There is significant influence of the degree of surface coverage, θ, on hydrodynamic resistance ß for insoluble surfactant monolayers. When the surfactant is soluble only in the continuous phase the bulk diffusion suppresses the Marangoni effect only for very low values of θ, while in reverse situation, the bulk diffusion from the drop phase is more efficient and the hydrodynamic resistance is lower. Surfactants with low value of the critical micelle concentration (CMC) make the interfaces tangentially immobile, while large CMC surfactants cannot suppress interfacial mobility, which lowers the hydrodynamic resistance between drops. For micron-sized droplets the interfacial viscosities practically block the surface mobility and they approach each other as solid spheres with high values of the drag coefficient.

Transition of chaotic motion to a limit cycle by intervention of economic policy: an empirical analysis in agriculture.

This paper investigates the transition of dynamics observed in an actual real agricultural economic dataset. Lyapunov spectrum analysis is conducted on the data to distinguish deterministic chaos and the limit cycle. Chaotic and periodic oscillation were identified before and after the second oil crisis, respectively. The statitonarity of the time series is investigated using recurrence plots. This shows that government intervention might reduce market instability by removing a chaotic market's long-term unpredictability.

Chaotic pairwise competition.

We investigate a kind of competition possible in a system of at least three populations competing for the same limited resource. As a model we use generalised Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations. Because of the nonconstant values of the last quantities the system could be repelled from the state of cyclic pairwise competition described by May and Leonard (SIAM J. Appl. Math. 29 (1975) 243.). We investigate the competition in a chaotic regime of evolution of the number of members of populations. We show that the nonconstant competition coefficients can lead to a regularisation of the time intervals of domination of each population and the non-constant growth rates can lead to decreasing length of the time intervals of domination as well as to chaotisation of the occurrence of these intervals. A quantity characterising the time intervals between the successive maxima of the number of the populations individuals is discussed. By means of the wavelet transform modulus maxima method we calculate the tau(q)-spectrum and the Hölder exponent for the time series of this quantity. The results of the theory are illustrated by an example of competition among the three main political parties in Bulgaria and we discuss qualitative aspects of the dynamics of change of preferences of voters.

Convective heat transport in a rotating fluid layer of infinite Prandtl number: optimum fields and upper bounds on Nusselt number.

By means of the Howard-Busse method of the optimum theory of turbulence we investigate numerically upper bounds on convective heat transport for the case of infinite fluid layer with stress-free vertical boundaries rotating about a vertical axis. We discuss the case of infinite Prandtl number, 1-alpha solution of the obtained variational problem and optimum fields possessing internal, intermediate, and boundary layers. We investigate regions of Rayleigh and Taylor numbers R and Ta, where no analytical bounds can be derived, and compare the analytical and numerical bounds for these regions of R and Ta where such comparison is possible. The increasing rotation has a different influence on the rescaled optimum fields of velocity w(1), temperature theta(1) and the vertical component of the vorticity f(1). The increasing Ta for fixed R leads to vanishing of the boundary layers of w(1) and theta(1). Opposite to this, the increasing Ta leads first to a formation of boundary layers of the field f(1) but further increasing the rotation causes vanishing of these boundary layers. We obtain optimum profiles of the horizontal averaged total temperature field which could be used as hints for construction of the background fields when applying Doering-Constantin method to the problems of rotating convection. The wave number alpha(1) corresponding to the optimum fields follows the asymptotic relationship alpha(1)=(R/5)(1/4) for intermediate Rayleigh numbers. However, when R becomes large with respect to Ta, after a transition region, the power law for alpha(1) becomes close to the power law for the case without rotation. The Nusselt number Nu is close to the nonrotational bound 0.32R(1/3) for the case of large R and small Ta. Nu decreases with increasing Taylor number. Thus, the upper bounds reflect the tendency of inhibiting thermal convection by increasing rotation for a fixed Rayleigh number. For the regions of Rayleigh and Taylor numbers where the numerical and asymptotic bounds on Nu can be compared, the numerical bounds are about 70% lower than the asymptotic bounds.

Low-dimensional chaos in zero-Prandtl-number Bénard-Marangoni convection.

Three-dimensional surface-tension-driven Bénard convection at zero Prandtl number is computed in the smallest possible doubly periodic rectangular domain that is compatible with the hexagonal flow structure at the linear stability threshold of the quiescent state. Upon increasing the Marangoni number beyond this threshold, the initially stationary flow becomes quickly time dependent. We investigate the transition to chaos for the case of a free-slip bottom wall by means of an analysis of the kinetic energy time series. We observe a period-doubling scenario for the transition to chaos of the energy attractor, intermittent behavior of a component of the mean velocity field, three characteristic energy levels, and two frequencies that contain a considerable amount of the power spectral density connected with the kinetic energy time series.