# American Institute of Mathematical Sciences

June  2022, 17(3): 401-425. doi: 10.3934/nhm.2022013

## Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties

 1 Istituto Nazionale di Alta Matematica "Francesco Severi" (INdAM), 00185 Roma, Italy 2 Department of Mathematics and Computer Science, University of Ferrara, 44121 Ferrara, Italy 3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong 4 Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

*Corresponding author: Lorenzo Pareschi

Received  October 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

Uncertainty in data is certainly one of the main problems in epidemiology, as shown by the recent COVID-19 pandemic. The need for efficient methods capable of quantifying uncertainty in the mathematical model is essential in order to produce realistic scenarios of the spread of infection. In this paper, we introduce a bi-fidelity approach to quantify uncertainty in spatially dependent epidemic models. The approach is based on evaluating a high-fidelity model on a small number of samples properly selected from a large number of evaluations of a low-fidelity model. In particular, we will consider the class of multiscale transport models recently introduced in [13,7] as the high-fidelity reference and use simple two-velocity discrete models for low-fidelity evaluations. Both models share the same diffusive behavior and are solved with ad-hoc asymptotic-preserving numerical discretizations. A series of numerical experiments confirm the validity of the approach.

Citation: Giulia Bertaglia, Liu Liu, Lorenzo Pareschi, Xueyu Zhu. Bi-fidelity stochastic collocation methods for epidemic transport models with uncertainties. Networks and Heterogeneous Media, 2022, 17 (3) : 401-425. doi: 10.3934/nhm.2022013
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##### References:
Test 1 (a): SIR model in diffusive regime. First row: expectation (left) and standard deviation (right) obtained at $t = 5$ for the variable $I$ with the three methodologies, by using $n = 8$ points for the bi-fidelity approximation. Second row: relative $L^2$ errors of the bi-fidelity approximation for the mean (left) and standard deviation (right) of density $I$ with respect to the number of "important" points $n$ used in the bi-fidelity algorithm, compared with low-fidelity errors
Test 1 (b): SIR model in hyperbolic regime. First row: expectation (left) and standard deviation (right) obtained at $t = 5$ for the variable $I$ with the three methodologies, by using $n = 14$ points for the bi-fidelity approximation. Second row: relative $L^2$ errors of the bi-fidelity approximation for the mean (left) and standard deviation (right) of density $I$ with respect to the number of "important" points $n$ used in the bi-fidelity algorithm, compared with low-fidelity errors
Test 2 (a): SEIAR model in intermediate regime. The baseline temporal and spatial evolution of compartments $S$ (first row, left), $E$ (first row, right), $I$ (second row, left) and $A$ (second row, right) in the high-fidelity model
Test 2 (a): SEIAR model in intermediate regime. Expectation (left) and standard deviation (right) of densities $E$ (first row), $I$ (second row) and $A$ (third row) at time $t = 5$, obtained with the three methodologies, using $n = 6$ for the bi-fidelity solution
Test 2 (a): SEIAR model in intermediate regime. Relative $L^2$ error decay of the bi-fidelity approximation of expectation (left) and standard deviation (right) for the density $A$ with respect to the number of selected "important" points $n$, compared with low-fidelity errors
Test 2 (b): SEIAR model in hyperbolic regime. Baseline temporal and spatial evolution of compartments $S$ (first row, left), $E$ (first row, right), $I$ (second row, left) and $A$ (second row, right) in the high-fidelity model
Test 2 (b): SEIAR model in hyperbolic regime. Expectation (left) and standard deviation (right) of densities $E$ (first row), $I$ (second row) and $A$ (third row) at time $t = 5$, obtained with the three methodologies, using $n = 7$ for the bi-fidelity solution
Test 2 (b): SEIAR model in hyperbolic regime. Relative $L^2$ error decay of the bi-fidelity approximation of expectation (left) and standard deviation (right) for the density $A$ with respect to the number of selected "important" points $n$, compared with low-fidelity errors
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