THREE-DIMENSIONAL ANALYTICAL SOLUTION OF THE ADVECTION-DIFFUSION EQUATION FOR AIR POLLUTION DISPERSION

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.

The Analytic Solutions of the Fractional-Order Model for the Spatial Epidemiology of the COVID-19 Infection

This paper provides a mathematical fractional-order model that accounts for the mindset of patients in the transmission of COVID-19 disease, the continuous inflow of foreigners into the country, immunization of population subjects, and temporary loss of immunity by recovered individuals. The analytic solutions, which are given as series solutions, are derived using the fractional power series method (FPSM) and the residual power series method (RPSM). In comparison, the series solution for the number of susceptible members, using the FPSM, is proportional to the series solution, using the RPSM for the first two terms, with a proportional constant of ψΓnα+1, where ψ is the natural birth rate of the baby into the susceptible population, Γ is the gamma function, n is the nth term of the series, and α is the fractional order as the initial number of susceptible individuals approaches the population size of Ghana. However, the variation in the two series solutions of the number of members who are susceptible to the COVID-19 disease begins at the third term and continues through the remaining terms. This is brought on by the nonlinear function present in the equation for the susceptible subgroup. The similar finding is made in the series solution of the number of exposed individuals. The series solutions for the number of deviant people, the number of nondeviant people, the number of people quarantined, and the number of people recovered using the FPSM are unquestionably almost identical to the series solutions for same subgroups using the RPSM, with the exception that these series solutions have initial conditions of the subgroup of the population size. It is observed that, in this paper, the series solutions of the nonlinear system of fractional partial differential equations (PDEs) provided by the RPSM are more in line with the field data than the series solutions provided by the FPSM.

Interactive Python Notebooks for Physical Chemistry

Chemistry simulations using interactive graphic user interfaces (GUIs) represent uniquely effective and safe tools to support multidimensional learning. Computer literacy and coding skills have become increasingly important in the chemical sciences. In response to both of these facts, a series of Jupyter notebooks hosted on Google Colaboratory were developed for undergraduate students enrolled in physical chemistry. These modules were developed for use during the COVID-19 pandemic when Millsaps College courses were virtual and only virtual or online laboratories could be used. These interactive exercises employ the Python programming language to explore a variety of chemical problems related to kinetics, the Maxwell–Boltzmann distribution, numerical versus analytical solutions, and real-world application of concepts. All of the modules are available for download from GitHub (https://github.com/Abravene/Python-Notebooks-for-Physical-Chemistry). Accessibility was prioritized, and students were assumed to have no prior programming experience;the notebooks are cost-free and browser-based. Students were guided to use widgets to build interactive GUIs that provide dynamic representations, immediate access to multiple investigations, and interaction with key variables. To evaluate the perceived effectiveness of this introduction to Python programming, participants were surveyed at the beginning and end of the course to gauge their interest in pursuing programming and data analysis skills and how they viewed the importance of programming and data analysis for their future careers. Student reactions were generally positive and showed increased interest in programming and its importance in their futures, so these notebooks will be incorporated into the in-person laboratory in the future.

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.

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.

A Hybrid Interpolation Method for Fractional PDEs and Its Applications to Fractional Diffusion and Buckmaster Equations

This study presents a novel numerical method to solve PDEs with the fractional Caputo operator. In this method, we apply the Newton interpolation numerical scheme in Laplace space, and then, the solution is returned to real space through the inverse Laplace transform. The Newton polynomial provides good results as compared to the Lagrangian polynomial, which is used to construct the Adams–Bashforth method. This procedure is used to solve fractional Buckmaster and diffusion equations. Finally, a few numerical simulations are presented, ensuring that this strategy is highly stable and quickly converges to an exact solution.

Closed-form pricing formulas for variance swaps in the Heston model with stochastic long-run mean of variance

The Heston model is a popular stochastic volatility model in mathematical finance and it has been extended or modified in several ways by researchers to overcome the shortcomings of the model in the context of pricing derivatives. However, the extended models usually do not lead to a closed-form formula for the derivative prices. This paper is focused on a stochastic extension of the constant long-run mean of variance in the Heston model for the pricing of variance swaps. The extension is given by a positive function perturbed by an amplitude-modulated Brownian motion or Ito integral. We obtain two closed-form formulas for the fair strike prices of a variance swap under two corresponding underlying models. The formulas are explicitly given by elementary functions without any integral terms involved. Further, the two models show better performance than the Heston model when the market implied volatility has a concave-down pattern as shown in an unstable market circumstance caused by the COVID-19 pandemic.

A Modified Iterative Algorithm for Numerical Investigation of HIV Infection Dynamics

The human immunodeficiency virus (HIV) mainly attacks CD4+ T cells in the host. Chronic HIV infection gradually depletes the CD4+ T cell pool, compromising the host’s immunological reaction to invasive infections and ultimately leading to acquired immunodeficiency syndrome (AIDS). The goal of this study is not to provide a qualitative description of the rich dynamic characteristics of the HIV infection model of CD4+ T cells, but to produce accurate analytical solutions to the model using the modified iterative approach. In this research, a new efficient method using the new iterative method (NIM), the coupling of the standard NIM and Laplace transform, called the modified new iterative method (MNIM), has been introduced to resolve the HIV infection model as a class of system of ordinary differential equations (ODEs). A nonlinear HIV infection dynamics model is adopted as an instance to elucidate the identification process and the solution process of MNIM, only two iterations lead to ideal results. In addition, the model has also been solved using NIM and the fourth order Runge–Kutta (RK4) method. The results indicate that the solutions by MNIM match with those of RK4 method to a minimum of eight decimal places, whereas NIM solutions are not accurate enough. Numerical comparisons between the MNIM, NIM, the classical RK4 and other methods reveal that the modified technique has potential as a tool for the nonlinear systems of ODEs.

The “Hygiene Package”: Analysis of Fraud Rates in Italy in the Period before and after Its Entry into Force

In violation of EU legislation, fraudulent activities in agri-food chains seek to make economic profits at the expense of consumers. Food frauds (FFs) often constitute a public health risk as well as a risk to animal and plant health, animal welfare and the environment. To analyze FFs in Italy during 1997–2020 with the aim of gaining observational insights into the effectiveness of the legislation in force and consequently of inspection activities, FFs were determined from official food inspections carried out by the Central Inspectorate of Quality Protection and Fraud Repression of Agri-food Products in 1997–2020. Inspected sectors were wine, oils and fats, milk and dairy products, fruit and vegetables, meat, eggs, honey, feeds and supplements, and seeds. Data show that the inspection activities have significantly improved in terms of sampling and fraud detection. However, a higher incidence of fraud involving the meat sector was observed. The obtained results demonstrate that there has not been a clear change of direction after the so-called “hygiene package” (food hygiene rules in the EU) came into force. Thus, more effective measures are needed to manage risk as well as new analytical solutions to increase the deterrence against meat adulteration and the rapid detection of fraud.

Stability Results of Some Fractional Neutral Integrodifferential Equations with Delay

Differential equations with fractional derivative are being extensively used in the modelling of the transmission of many infective diseases like HIV, Ebola, and COVID-19. Analytical solutions are unreachable for a wide range of such kind of equations. Stability theory in the sense of Ulam is essential as it provides approximate analytical solutions. In this article, we utilize some fixed point theorem (FPT) to investigate the stability of fractional neutral integrodifferential equations with delay in the sense of Ulam-Hyers-Rassias (UHR). This work is a generalized version of recent interesting works. Finally, two examples are given to prove the applicability of our results.

The Chinese system of government relief supplies: from the perspective of knowledge management

Purpose>Public crisis often generates new knowledge that should be incorporated into a government’s macro-control to ensure the relief supply. From the perspective of public crisis knowledge management, the Chinese system of Government relief supplies can be considered as a special case of the knowledge system. This paper aims to investigate the supply and production mechanism of relief goods and explore the advantages of the Chinese system when a sudden public crisis occurs.Design/methodology/approach>Under the Chinese system, the authors construct a relief supply chain model consisting of the Chinese Government, one manufacturer and one supplier, where the supplier has no capital constraints. Given the demand for relief goods, the government purchases from the manufacturer with a guide price. Then, the manufacturer decides on its order quantity and offers a wholesale price to the supplier. The supplier has a random capacity and decides on the level of knowledge acquisition to improve its capacity.Findings>The authors first obtain the analytical solution for the manufacturer to motivate a high level of knowledge acquisition from the supplier. Specifically, the manufacturer’s optimal order quantity is equal to the demand and the optimal wholesale price has a cost-plus form that reimburses the supplier for its production cost and knowledge-acquisition cost. Next, the authors derive the optimal guide price for the government, which should be set to subsidize the manufacturer with a proportion of the sourcing cost. Finally, the authors compare the Chinese system with the market mechanism where the supplier has capital constraints and confirm that the Chinese system is more beneficial to both the manufacturer and the government.Originality/value>Quantitative research on the Chinese system of Government relief supplies is difficult to be conducted. This paper provides feasible and practical methods to quantify the benefits of the Chinese system. The results reveal that the Chinese system is an effective mechanism of public crisis knowledge management, which can be helpful to the government’s policy-making in practice.

Ulam-Hyers-Rassias Stability of Nonlinear Differential Equations with Riemann-Liouville Fractional Derivative

Fractional derivatives are used to model the transmission of many real world problems like COVID-19. It is always hard to find analytical solutions for such models. Thus, approximate solutions are of interest in many interesting applications. Stability theory introduces such approximate solutions using some conditions. This article is devoted to the investigation of the stability of nonlinear differential equations with Riemann-Liouville fractional derivative. We employed a version of Banach fixed point theory to study the stability in the sense of Ulam-Hyers-Rassias (UHR). In the end, we provide a couple of examples to illustrate our results. In this way, we extend several earlier outcomes.

Application of a Kinetic Model for Studying the Spatial Spread of COVID-19

A one-dimensional model based on a kinetic-type equation is proposed for studying the dynamic distribution density of virus carriers in time and space while taking into account their distribution from a dedicated center. This model is new and fundamentally different from known models of the diffusion–reaction type. The analytical solution is built;for obtaining a series of calculations, numerical methods are also used. The model and real data from Italy, Russia, and Chile are compared. In addition to the rate of infection, the “rate of recovery” is considered. When the wave of recovery passes through a territory with the greater part of the commonwealth, a conclusion is made about the onset of global recovery, which corresponds to real data. The predictions are proved to have been accurate also for the second wave of the pandemic in Russia. The model is expected to be able also to describe adequately subsequent epidemics instead of only the development of COVID-19.

Generalized Mittag-Leffler Kernel Form Solutions of Free Convection Heat and Mass Transfer Flow of Maxwell Fluid with Newtonian Heating: Prabhakar Fractional Derivative Approach

In this article, the effects of Newtonian heating along with wall slip condition on temperature is critically examined on unsteady magnetohydrodynamic (MHD) flows of Prabhakar-like non integer Maxwell fluid near an infinitely vertical plate under constant concentration. For the sake of generalized memory effects, a new mathematical fractional model is formulated based on a newly introduced Prabhakar fractional operator with generalized Fourier’s law and Fick’s law. This fractional model has been solved analytically and exact solutions for dimensionless velocity, concentration, and energy equations are calculated in terms of Mittag-Leffler functions by employing the Laplace transformation method. Physical impacts of different parameters such as α, Pr, β, Sc, Gr, γ, and Gm are studied and demonstrated graphically by Mathcad software. Furthermore, to validate our current results, some limiting models such as classical Maxwell model, classical Newtonian model, and fractional Newtonian model are recovered from Prabhakar fractional Maxwell fluid. Moreover, we compare the results between Maxwell and Newtonian fluids for both fractional and classical cases with and without slip conditions, showing that the movement of the Maxwell fluid is faster than viscous fluid. Additionally, it is visualized that both classical Maxwell and viscous fluid have relatively higher velocity as compared to fractional Maxwell and viscous fluid.

UN World Food Programme: Toward Zero Hunger with Analytics

Each year, the United Nations World Food Programme (WFP) provides food assistance to around 100 million people in more than 80 countries. Significant investments over the last decade have put planning and optimization at the forefront of tackling emergencies at WFP. A data-driven approach to managing operations has gradually become the norm and has culminated in the creation of a supply chain planning unit and savings of more than USD 150 million-enough to support two million food-insecure people for an entire year. In this paper, we describe three analytical solutions in detail: the Supply Chain Management Dashboard, which uses descriptive and predictive analytics to bring end-to-end visibility and anticipate operational issues;Optimus, which uses a mixed-integer programming model to simultaneously optimize food basket composition and supply chain planning;and DOTS, which is a data integration platform that helps WFP automate and synchronize complex data flows. Three impact studies for Iraq, South Sudan, and COVID-19 show how these tools have changed the way WFP manages its most complex operations. Through analytics, decision makers are now equipped with the insights they need to manage their operations in the best way, thereby saving and changing the lives of millions and bringing the world one step closer to zero hunger.

Numerical simulation of particle transport in the urban boundary layer with implications for SARS-CoV-2 virion distribution

This paper presents a description, verification, and application of a model simulating the transport of aerosol particles with different properties in the urban boundary layer by using a Lagrangian approach. The model takes input fields of air flow characteristics from arbitrary external models or analytical solutions and allows estimating the movement, sedimentation, and decay of particles. In this paper, the accuracy of the model is successfully estimated on the basis of exact analytical solutions. Simulations are made for a series of urban canyons under different conditions of stratification and wind speed to assess the effects of these meteorological parameters on particle transport in urban areas. Under similar conditions, the transport of particles simulating SARS-CoV-2 coronavirus particles is calculated.

Discrete and continuum models of COVID-19 virus, formal solutions, stability and comparison with real data.

Very recently, various mathematical models, for the dynamics of COVID-19 with main contribution of suspected-exposed-infected-recovered people have been proposed. Some models that account for the deceased, quarantined or social distancing functions were also presented. However, in any local space the real data reveals that the effects of lock-down and traveling are significant in decreasing and increasing the impact of this virus respectively. Here, discrete and continuum models for the dynamics of this virus are suggested. The continuum dynamical model is studied in detail. The present model deals with exposed, infected, recovered and deceased individuals (EIRD), which accounts for the health isolation and travelers (HIT) effects. Up to now no exact solutions of the parametric-dependent, nonlinear dynamical system NLDS were found. In this work, our objective is to find the exact solutions of a NLDS. To this issue, a novel approach is presented where a NLDS is recast to a linear dynamical system LDS. This is done by implementing the unified method (UM), with auxiliary equations, which are taken coupled linear ODE's (LDS). Numerical results of the exact solutions are evaluated, which can be applied to data in a local space (or anywhere) when the initial data for the IRD are known. Here, as an example, initial conditions for the components in the model equation of COVID-19, are taken from the real data in Egypt. The results of susceptible, infected, recovered and deceased people are computed. The comparison between the computed results and the real data shows an agreement up to a relative error 1 0 - 3 . On the other hand it is remarked that locking-down plays a dominant role in decreasing the number of infected people. The equilibrium states are determined and it is found that they are stable. This reveals a relevant result that the COVID-19 can be endemic in the case of a disturbance in the number of the exposed people. A disturbance in the form of an increase in the exposed number, leads to an increase in the number of infected people. This result is, globally, valid. Furthermore, initial states control is analyzed, where region of initial conditions for infected and exposed is determined. We developed a software tool to interact with the model and facilitate applying various data of different local spaces.

Analysis of transmission dynamics of COVID-19 via closed-form solutions of a susceptible-infectious-quarantined-diseased model with a quarantine-adjusted incidence function.

We analyze the disease control and prevention strategies in a susceptible-infectious-quarantined-diseased (SIQD) model with a quarantine-adjusted incidence function. We have established the closed-form solutions for all the variables of SIQD model with a quarantine-adjusted incidence function provided ß ≠ γ + α by utilizing the classical techniques of solving ordinary differential equations (ODEs). The epidemic peak and time required to attain this peak are provided in closed form. We have provided closed-form expressions for force of infection and rate at which susceptible becomes infected. The management of epidemic perceptive using control and prevention strategies is explained as well. The epidemic starts when ρ 0 > 1, the peak of epidemic appears when number of infected attains peak value when ρ 0 = 1 , and the disease dies out ρ 0 < 1. We have provided the comparison of estimated and actual epidemic peak of COVID-19 in Pakistan. The forecast of epidemic peak for the United states, Brazil, India, and the Syrian Arab Republic is given as well.

Exactly solvable SIR models, their extensions and their application to sensitive pandemic forecasting.

The classic SIR model of epidemic dynamics is solved completely by quadratures, including a time integral transform expanded in a series of incomplete gamma functions. The model is also generalized to arbitrary time-dependent infection rates and solved explicitly when the control parameter depends on the accumulated infections at time t. Numerical results are presented by way of comparison. Autonomous and non-autonomous generalizations of SIR for interacting regions are also considered, including non-separability for two or more interacting regions. A reduction of simple SIR models to one variable leads us to a generalized logistic model, Richards model, which we use to fit Mexico's COVID-19 data up to day number 134. Forecasting scenarios resulting from various fittings are discussed. A critique to the applicability of these models to current pandemic outbreaks in terms of robustness is provided. Finally, we obtain the bifurcation diagram for a discretized version of Richards model, displaying period doubling bifurcation to chaos.