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We derive a novel hybrid approach, a combination of neural networks and inverse problem, in order to forecast COVID-19 cases, and more generally any infectious disease. For this purpose, we extract a second order nonlinear differential equation for the total confirmed cases from a SIR-like model. The latter is the cornerstone of the present study which allows to rigorously simplify the construction of the time-dependent epidemiological parameters without solving a system of differential equations. The neural network and inverse problems are used to compute the trial functions for total cases and the model parameters, respectively. The number of suspected and infected individuals can be found using the trial function of total confirmed cases. We divide the time domain into two parts, training interval (first 365/395 days) and test interval (first 366 to 395/ 396 to 450 days), and train the neural networks on the preassigned training zones. To examine the efficiency and effectiveness, we apply the proposed method to Canada, and use the Canadian publicly available database to estimate the parameters of the trial function involved with total cases. The trial functions of model parameters show that the basic reproduction number was closed to unity over a wide range, the first 100 to 365 days of the current pandemic in Canada. The proposed prediction models, based on the influence of previous COVID-19 cases and social economic policy, show excellent agreement with the data. The test results reveal that the single path prediction can forecast a period of 30 days, and the prediction using previous social and economical situation can forecast a range of 55 days.

This paper is devoted to the study of numerical methods for solving large, sparse or full fractional linear algebraic systems (FLAS). The intent is to provide relevant and fair accuracy and efficiency comparisons of several solvers for this type of linear systems, typically involved in the approximation of fractional partial differential equations.

International audience; This paper is dedicated to the derivation and analysis of a Schwarz waveform relaxation domain decomposition method for solving time-dependent linear/nonlinear space fractional Schrödinger and heat equations. Along with the details of the derivation of the method and some mathematical properties, we also propose some illustrating numerical experiments and conjectures on the rate of convergence of the method.

International audience; This paper is devoted to the numerical computation of fractional linear systems. The proposed approach is based on an efficient computation of Cauchy integrals allowing to estimate the real power of a (sparse) matrix A. A first preconditioner M is used to reduce the length of the Cauchy integral contour enclosing the spectrum of M A, hence allowing for a large reduction of the number of quadrature nodes along the integral contour. Next, ILU-factorizations are used to efficiently solve the linear systems involved in the computation of approximate Cauchy integrals. Numerical examples related to stationary (deterministic or stochastic) fractional Poisson-like equations are finally proposed to illustrate the methodology.