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    High order discretization methods for spatial-dependent epidemic models
    (Amsterdam [u.a.] : Elsevier Science, 2022) Takács, Bálint; Hadjimichael, Yiannis
    In this paper, an epidemic model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of partial-differential equations with integral terms. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different spatial and temporal discretizations are employed, and step-size restrictions for the discrete model’s positivity, monotonicity preservation, and population conservation are investigated. We provide sufficient conditions under which high-order numerical schemes preserve the stability of the computational process and provide sufficiently accurate numerical approximations. Computational experiments verify the convergence and accuracy of the numerical methods.
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    High order discretization methods for spatial-dependent epidemic models
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Takács, Bálint; Hadjimichael, Yiannis
    In this paper, an SIR model with spatial dependence is studied and results regarding its stability and numerical approximation are presented. We consider a generalization of the original Kermack and McKendrick model in which the size of the populations differs in space. The use of local spatial dependence yields a system of integro-differential equations. The uniqueness and qualitative properties of the continuous model are analyzed. Furthermore, different choices of spatial and temporal discretizations are employed, and step-size restrictions for population conservation, positivity, and monotonicity preservation of the discrete model are investigated. We provide sufficient conditions under which high order numerical schemes preserve the discrete properties of the model. Computational experiments verify the convergence and accuracy of the numerical methods.
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    Positivity preservation of implicit discretizations of the advection equation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2021) Hadjimichael, Yiannis; Ketcheson, David I.; Lóczi, Lajos
    We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semidiscretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary _x0012_θ-method in time (including the forward and backward Euler methods, and a second-order method by choosing _x0012_ θ ∈ [0, 1] suitably). The full discretization generates a two-parameter family of circulant matrices M ∈ ℝ m_x0002_xm , where each matrix entry is a rational function in θ and _x0017_ν . Here, _x0017_ν denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix M---which is equivalent to the positivity preservation of the fully discrete scheme---is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points m is even. However, it turns out that positivity preservation of the fully discrete system is recovered for odd values of m provided that θ ≥ 1/2 and ν are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are not positivity preserving.