A fractional mathematical model with nonlinear partial differential equations for transmission dynamics of severe acute respiratory syndrome coronavirus 2 infection.

This study presents a fractional mathematical model based on nonlinear Partial Differential Equations (PDEs) of fractional variable-order derivatives for the host populations experiencing transmission and evolution of the Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) pandemic. Five host population groups have been considered, the Susceptible, Exposed, Infected, Recovered, and Deceased (SEIRD). The new model, not introduced before in its current formulation, is governed by nonlinear PDEs with fractional variable-order derivatives. As a result, the proposed model is not compared with other models or real scenarios. The advantage of the proposed fractional partial derivatives of variable orders is that they can model the rate of change of subpopulation for the proposed model. As an efficient tool to obtain the solution of the proposed model, a modified analytical technique based on the homotopy and Adomian decomposition methods is introduced. Then again, the present study is general and is applicable to a host population in any country.

A multiconsistent computational methodology to resolve a diffusive epidemiological system with effects of migration, vaccination and quarantine.

BACKGROUND: We provide a compartmental model for the transmission of some contagious illnesses in a population. The model is based on partial differential equations, and takes into account seven sub-populations which are, concretely, susceptible, exposed, infected (asymptomatic or symptomatic), quarantined, recovered and vaccinated individuals along with migration. The goal is to propose and analyze an efficient computer method which resembles the dynamical properties of the epidemiological model. MATERIALS AND METHODS: A non-local approach is utilized for finding approximate solutions for the mathematical model. To that end, a non-standard finite-difference technique is introduced. The finite-difference scheme is a linearly implicit model which may be rewritten using a suitable matrix. Under suitable circumstances, the matrices representing the methodology are M-matrices. RESULTS: Analytically, the local asymptotic stability of the constant solutions is investigated and the next generation matrix technique is employed to calculate the reproduction number. Computationally, the dynamical consistency of the method and the numerical efficiency are investigated rigorously. The method is thoroughly examined for its convergence, stability, and consistency. CONCLUSIONS: The theoretical analysis of the method shows that it is able to maintain the positivity of its solutions and identify equilibria. The method's local asymptotic stability properties are similar to those of the continuous system. The analysis concludes that the numerical model is convergent, stable and consistent, with linear order of convergence in the temporal domain and quadratic order of convergence in the spatial variables. A computer implementation is used to confirm the mathematical properties, and it confirms the ability in our scheme to preserve positivity, and identify equilibrium solutions and their local asymptotic stability.

On a reaction–diffusion system modelling infectious diseases without lifetime immunity

In this paper, we study a mathematical model for an infectious disease caused by a virus such as Cholera without lifetime immunity. Due to the different mobility for susceptible, infected human and recovered human hosts, the diffusion coefficients are assumed to be different. The resulting system is governed by a strongly coupled reaction–diffusion system with different diffusion coefficients. Global existence and uniqueness are established under certain assumptions on known data. Moreover, global asymptotic behaviour of the solution is obtained when some parameters satisfy certain conditions. These results extend the existing results in the literature. The main tool used in this paper comes from the delicate theory of elliptic and parabolic equations. Moreover, the energy method and Sobolev embedding are used in deriving a priori estimates. The analysis developed in this paper can be employed to study other epidemic models in biological, ecological and health sciences.

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.

Total Value Adjustment of Multi-Asset Derivatives under Multivariate CGMY Processes

Counterparty credit risk (CCR) is a significant risk factor that financial institutions have to consider in today's context, and the COVID-19 pandemic and military conflicts worldwide have heightened concerns about potential default risk. In this work, we investigate the changes in the value of financial derivatives due to counterparty default risk, i.e., total value adjustment (XVA). We perform the XVA for multi-asset option based on the multivariate Carr–Geman–Madan–Yor (CGMY) processes, which can be applied to a wider range of financial derivatives, such as basket options, rainbow options, and index options. For the numerical methods, we use the Monte Carlo method in combination with the alternating direction implicit method (MC-ADI) and the two-dimensional Fourier cosine expansion method (MC-CC) to find the risk exposure and make value adjustments for multi-asset derivatives.

Well-posedness for a diffusion–reaction compartmental model simulating the spread of COVID-19

This paper is concerned with the well-posedness of a diffusion–reaction system for a susceptible-exposed-infected-recovered (SEIR) mathematical model. This model is written in terms of four nonlinear partial differential equations with nonlinear diffusions, depending on the total amount of the SEIR populations. The model aims at describing the spatio-temporal spread of the COVID-19 pandemic and is a variation of the one recently introduced, discussed, and tested in a paper by Viguerie et al (2020). Here, we deal with the mathematical analysis of the resulting Cauchy–Neumann problem: The existence of solutions is proved in a rather general setting, and a suitable time discretization procedure is employed. It is worth mentioning that the uniform boundedness of the discrete solution is shown by carefully exploiting the structure of the system. Uniform estimates and passage to the limit with respect to the time step allow to complete the existence proof. Then, two uniqueness theorems are offered, one in the case of a constant diffusion coefficient and the other for more regular data, in combination with a regularity result for the solutions. © 2023 The Authors. Mathematical Methods in the Applied Sciences published by John Wiley & Sons, Ltd.

TTRISK: Tensor train decomposition algorithm for risk averse optimization

This article develops a new algorithm named TTRISK to solve high‐dimensional risk‐averse optimization problems governed by differential equations (ODEs and/or partial differential equations [PDEs]) under uncertainty. As an example, we focus on the so‐called Conditional Value at Risk (CVaR), but the approach is equally applicable to other coherent risk measures. Both the full and reduced space formulations are considered. The algorithm is based on low rank tensor approximations of random fields discretized using stochastic collocation. To avoid nonsmoothness of the objective function underpinning the CVaR, we propose an adaptive strategy to select the width parameter of the smoothed CVaR to balance the smoothing and tensor approximation errors. Moreover, unbiased Monte Carlo CVaR estimate can be computed by using the smoothed CVaR as a control variate. To accelerate the computations, we introduce an efficient preconditioner for the Karush–Kuhn–Tucker (KKT) system in the full space formulation.The numerical experiments demonstrate that the proposed method enables accurate CVaR optimization constrained by large‐scale discretized systems. In particular, the first example consists of an elliptic PDE with random coefficients as constraints. The second example is motivated by a realistic application to devise a lockdown plan for United Kingdom under COVID‐19. The results indicate that the risk‐averse framework is feasible with the tensor approximations under tens of random variables.

Analysis on Infection Wave and Decay of Transmission Risk Based on Spatial SIR Model - Taking COVID Epidemic as an Example

Starting with the spatial SIR model, this paper gives the strict boundary conditions, and obtains two theorems in the process of infectious disease transmission through theoretical analysis. After that, the partial differential equations are transformed into ordinary differential equations by the method of traveling wave solution, and the solutions of infectious wave velocity and hypergeometric function are further derived. Beside local diffusion operator model, the paper also developed global transmission risk functions as convolution kernels and discovered their properties. The solution of the spatial infectious disease model is visualized by programming, and the influence of parameter changes on the solution is discussed. Finally, some variants of the model in special cases are given. This paper proves that under generalized assumption the three population densities of the spatial SIR model results at the origin cannot take extreme values at the same time, and when the infected density takes extreme values at the origin, the higher-order derivative of the infected density to the space is zero. The hypergeometric function method verifies the solution at infinity of the equations, and the above solution can be used to approximate when the distance from the infection source radius is large. In this paper, the discussion on the impact of the changes of several infectious disease parameters can inspire the methods of epidemic prevention and control.

The Stability Analysis and Transmission Dynamics of the SIR Model with Nonlinear Recovery and Incidence Rates

In the present paper, the SIR model with nonlinear recovery and Monod type equation as incidence rates is proposed and analyzed. The expression for basic reproduction number is obtained which plays a main role in the stability of disease-free and endemic equilibria. The nonstandard finite difference (NSFD) scheme is constructed for the model and the denominator function is chosen such that the suggested scheme ensures solutions boundedness. It is shown that the NSFD scheme does not depend on the step size and gives better results in all respects. To prove the local stability of disease-free equilibrium point, the Jacobean method is used;however, Schur–Cohn conditions are applied to discuss the local stability of the endemic equilibrium point for the discrete NSFD scheme. The Enatsu criterion and Lyapunov function are employed to prove the global stability of disease-free and endemic equilibria. Numerical simulations are also presented to discuss the advantages of NSFD scheme as well as to strengthen the theoretical results. Numerical simulations specify that the NSFD scheme preserves the important properties of the continuous model. Consequently, they can produce estimates which are entirely according to the solutions of the model.

Modeling nonlocal behavior in epidemics via a reaction-diffusion system incorporating population movement along a network.

The outbreak of COVID-19, beginning in 2019 and continuing through the time of writing, has led to renewed interest in the mathematical modeling of infectious disease. Recent works have focused on partial differential equation (PDE) models, particularly reaction-diffusion models, able to describe the progression of an epidemic in both space and time. These studies have shown generally promising results in describing and predicting COVID-19 progression. However, people often travel long distances in short periods of time, leading to nonlocal transmission of the disease. Such contagion dynamics are not well-represented by diffusion alone. In contrast, ordinary differential equation (ODE) models may easily account for this behavior by considering disparate regions as nodes in a network, with the edges defining nonlocal transmission. In this work, we attempt to combine these modeling paradigms via the introduction of a network structure within a reaction-diffusion PDE system. This is achieved through the definition of a population-transfer operator, which couples disjoint and potentially distant geographic regions, facilitating nonlocal population movement between them. We provide analytical results demonstrating that this operator does not disrupt the physical consistency or mathematical well-posedness of the system, and verify these results through numerical experiments. We then use this technique to simulate the COVID-19 epidemic in the Brazilian region of Rio de Janeiro, showcasing its ability to capture important nonlocal behaviors, while maintaining the advantages of a reaction-diffusion model for describing local dynamics.

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.

Analysis for the spread of COVID-19 in China based on the impulsive partial differential equation with diffusion

At the beginning of 2020, COVID-19 broke out in Wuhan and quickly swept the world. At present, the global epidemic prevention and control is still facing severe challenges. Scientific and effective measures of the epidemic is crucial to epidemic prevention and control. In this paper, a COVID-19 diffusion prediction model is established based on the impulsive partial differential equation and traditional infectious disease model, which can describe the spatial diffusion of viruses. This is also a lack of other models. The model divides the total population into seven groups: susceptible, quarantine, exposed, asymptomatic, infected, diagnosed and recovered, while considering the influence of time and space on the spread of the virus. In order to test the model, we take Jiangsu Province in China as an example, compare the calculated results with the actual data, and verify the effectiveness of the model through numerical calculation. © COPYRIGHT SPIE.

Discretization Fractional-Order Biological Model with Optimal Harvesting

In this paper, a discretization of a three-dimensional fractional-order prey-predator model has been investigated with Holling type III functional response. All its fixed points are determined;also, their local stability is investigated. We extend the discretized system to an optimal control problem to get the optimal harvesting amount. For this, the discrete-time Pontryagin’s maximum principle is used. Finally, numerical simulation results are given to confirm the theoretical outputs as well as to solve the optimality problem.

Finite Time Stability of 2D Fractional Hyperbolic System with Time Delay

In this work, a class of two-dimensional fractional hyperbolic differential linear system (2D-FHDLS) with time delay is investigated. By using generalized Gronwall’s inequality, sufficient conditions for the finite time stability (FTS) of two-dimensional fractional hyperbolic differential system with time delay are given. Numerical examples are also given to illustrate the stability result.

Electro-fluid-mechanics of the heart

This article presents an overview of the dynamics of the human heart and the main goal is the discussion of its fluid mechanic features. We will see, however, that the flow in the heart can not be fully described without considering its electrophysiology and elastomechanics as well as the interaction with the systemic and pulmonary circulations with which it is strongly connected. Biologically, the human heart is similar to that of all warm-blooded mammals and it satisfies the same allometric laws. Since the Paleolithic Age, however, humans have improved their living conditions, have modified the environment to satisfy their needs and, more recently, have developed advanced medical knowledge which has allowed triple the number of heartbeats with respect to other mammals. In the last century, effective diagnostic tools, reliable surgical procedures and prosthetic devices have been developed and refined leading to substantial progress in cardiology and heart surgery with routine clinical practice which nowadays cures many disorders, once lethal. Pulse duplicators have been built to reproduce the pulsatile flow and ‘blood analogues’, have been realized. Heart phantoms, can attain deformations similar to the real heart although the active contraction and the tissue anisotropy still can not be replicated. Numerical models have also become a viable alternative for cardiovascular research: they do not suffer from limitations of material properties and device technologies, thus making possible the realization of truly digital twins. Unfortunately, a high-fidelity model for the whole heart consists of a system of coupled, nonlinear partial differential equations with a number of degrees of freedom of the order of a billion and computational costs become the bottleneck. An additional challenge comes from the inherent human variability and the uncertainty of the heart parameters whose statistical assessment requires a campaign of simulations rather than a single deterministic calculation;reduced and surrogate models can be employed to alleviate the huge computational burden and all possibilities are currently being pursued. In the era of big data and artificial intelligence, cardiovascular research is also advancing by exploiting the latest technologies: equation-based augmented reality, virtual surgery and computational prediction of disease progression are just a few examples among many that will become standard practice in the forthcoming years.

Application of Machine Learning to Predict the Performance of an EMIPG Reactor Using Data from Numerical Simulations

Microwave-driven plasma gasification technology has the potential to produce clean energy from municipal and industrial solid wastes. It can generate temperatures above 2000 K (as high as 30,000 K) in a reactor, leading to complete combustion and reduction of toxic byproducts. Characterizing complex processes inside such a system is however challenging. In previous studies, simulations using computational fluid dynamics (CFD) produced reproducible results, but the simulations are tedious and involve assumptions. In this study, we propose machine-learning models that can be used in tandem with CFD, to accelerate high-fidelity fluid simulation, improve turbulence modeling, and enhance reduced-order models. A two-dimensional microwave-driven plasma gasification reactor was developed in ANSYS (Ansys, Canonsburg, PA, USA) Fluent (a CFD tool), to create 644 (geometry and temperature) datasets for training six machine-learning (ML) models. When fed with just geometry datasets, these ML models were able to predict the proportion of the reactor area with temperature above 2000 K. This temperature level is considered a benchmark to prevent formation of undesirable byproducts. The ML model that achieved highest prediction accuracy was the feed forward neural network;the mean absolute error was 0.011. This novel machine-learning model can enable future optimization of experimental microwave plasma gasification systems for application in waste-to-energy.

Generalized Laplace-Type Transform Method for Solving Multilayer Diffusion Problems

Multilayer diffusion problems have found significant importance that they arise in many medical, environmental, and industrial applications of heat and mass transfer. In this article, we study the solvability of a one-dimensional nonhomogeneous multilayer diffusion problem. A new generalized Laplace-type integral transform is used, namely, the Mρ,m-transform. First, we reduce the nonhomogeneous multilayer diffusion problem into a sequence of one-layer diffusion problems including time-varying given functions, followed by solving a general nonhomogeneous one-layer diffusion problem via the Mρ,m-transform. Hence, by means of general interface conditions, a renewal equations’ system is determined. Finally, the Mρ,m-transform and its analytic inverse are used to obtain an explicit solution to the renewal equations’ system. Our results are of general attractiveness and comprise a number of previous works as special cases.

Mathematical Modeling the Time-Delay Interactions between Tumor Viruses and the Immune System with the Effects of Chemotherapy and Autoimmune Diseases

The immune system is the body’s defense against pathogens, which are complex living organisms found in many parts in the body including organs, tissues, cells, molecules, and proteins. When the immune system works properly, it can recognize and kill the abnormal cells and the infected cells. Otherwise, it can attack the body’s healthy cells even if there is no invader. Many researchers have developed immunotherapy (or cancer vaccines) and have used chemotherapy for cancer treatment that can kill fast-growing cancer cells or at least slow down tumor growth. However, chemotherapy drugs travel throughout the body and tend to kill both healthy cells and cancer cells. In this study, we consider the fact that chemotherapy can kill tumor cells and that the loss of the immune cells may at the same time stir up cancer growth. We present a dynamic time-delay tumor-immune model with the effects of chemotherapy drugs and autoimmune disease. The modeling results can be used to determine the progression of tumor cells in the human body with the effect of chemotherapy, autoimmune diseases, and time delays based on partial differential equations. It can also be used to predict when the tumor viruses’ free state can be reached as time progresses, as well as the state of the body’s healthy cells as time progresses. We also present a few numerical cases that illustrate that the model can be used to monitor the effects of chemotherapy drug treatment and the growth rate of tumor virus-infected cells and the autoimmune disease.

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.