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
References:
[1]

P. S. Abdul SalamW. BockA. Klar and S. Tiwari, Disease contagion models coupled to crowd motion and mesh-free simulation, Math. Models Methods Appl. Sci., 31 (2021), 1277-1295.  doi: 10.1142/S0218202521400066.

[2]

G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Kinetic modelling of epidemic dynamics: Social contacts, control with uncertain data, and multiscale spatial dynamics, in press in Predicting Pandemics in a Globally Connected World, Springer-Nature, (2022).

[3]

G. Albi, L. Pareschi and M. Zanella, Control with uncertain data of socially structured compartmental epidemic models, J. Math. Biol., 82 (2021), 41pp. doi: 10.1007/s00285-021-01617-y.

[4]

G. AlbiL. Pareschi and M. Zanella, Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty, Math. Biosci. Eng., 18 (2021), 7161-7190.  doi: 10.3934/mbe.2021355.

[5]

E. Barbera, G. Consolo and G. Valenti, Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model, Phys. Rev. E, 88 (2013), 13pp. doi: 10.1103/PhysRevE.88.052719.

[6]

N. BellomoR. BinghamM. A. J. ChaplainG. DosiG. Forni and et al., A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591-1651.  doi: 10.1142/S0218202520500323.

[7]

G. BertagliaW. BoscheriG. Dimarco and L. Pareschi, Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty, Math. Biosci. Eng., 18 (2021), 7028-7059.  doi: 10.3934/mbe.2021350.

[8]

G. Bertaglia, V. Caleffi, L. Pareschi and A. Valiani, Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model, J. Comput. Phys., 430 (2021), 20pp. doi: 10.1016/j.jcp.2020.110102.

[9]

G. Bertaglia and L. Pareschi, Hyperbolic compartmental models for epidemic spread on networks with uncertain data: Application to the emergence of COVID-19 in Italy, Math. Models Methods Appl. Sci., 31 (2021), 2495-2531.  doi: 10.1142/S0218202521500548.

[10]

G. Bertaglia and L. Pareschi, Hyperbolic models for the spread of epidemics on networks: Kinetic description and numerical methods, ESAIM Math. Model. Numer. Anal., 55 (2021), 381-407.  doi: 10.1051/m2an/2020082.

[11]

S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35 (2013), A22–A51. doi: 10.1137/110842855.

[12]

S. BoscarinoL. Pareschi and G. Russo, A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation, SIAM J. Numer. Anal., 55 (2017), 2085-2109.  doi: 10.1137/M1111449.

[13]

W. BoscheriG. Dimarco and L. Pareschi, Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations, Math. Models Methods Appl. Sci., 31 (2021), 1059-1097.  doi: 10.1142/S0218202521400017.

[14]

B. Buonomo and R. Della Marca, Effects of information-induced behavioural changes during the COVID-19 lockdowns: The case of Italy, R. Soc. Open Sci., 7 (2020). doi: 10.1098/rsos.201635.

[15]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[16]

R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the Covid-19 pandemic, J. Math. Ind., 10 (2020), 20pp. doi: 10.1186/s13362-020-00090-4.

[17]

G. Dimarco, L. Liu, L. Pareschi and X. Zhu, Multi-fidelity methods for uncertainty propagation in kinetic equations, preprint, arXiv: 2112.00932.

[18]

G. Dimarco and L. Pareschi, Multi-scale control variate methods for uncertainty quantification in kinetic equations, J. Comput. Phys., 388 (2019), 63-89.  doi: 10.1016/j.jcp.2019.03.002.

[19]

G. Dimarco and L. Pareschi, Multiscale variance reduction methods based on multiple control variates for kinetic equations with uncertainties, Multiscale Model. Simul., 18 (2020), 351-382.  doi: 10.1137/18M1231985.

[20]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.

[21]

G. Dimarco, B. Perthame, G. Toscani and M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, J. Math. Biol., 83 (2021), 32pp. doi: 10.1007/s00285-021-01630-1.

[22]

E. Franco, A feedback SIR (fSIR) model highlights advantages and limitations of infection-based social distancing, preprint, arXiv: 2004.13216.

[23]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceed. Nat. Acad. Sci., 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[24]

F. GolseS. Jin and C. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.  doi: 10.1137/S0036142997315986.

[25]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[26]

T. Hillen and A. Swan, The diffusion limit of transport equations in biology, in Mathematical Models and Methods for Living Systems, Lecture Notes in Math., 2167, Fond. CIME/CIME Found. Subser., Springer, Cham, 2016, 73–129. doi: 10.1007/978-3-319-42679-2_2.

[27]

S. Jin, H. Lu and L. Pareschi, Efficient stochastic asymptotic-preserving implicit-explicit methods for transport equations with diffusive scalings and random inputs, SIAM J. Sci. Comput., 40 (2018), A671–A696. doi: 10.1137/17M1120518.

[28]

S. JinL. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.

[29]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Bio. J. IMA, 22 (2005), 113-128.  doi: 10.1093/imammb/dqi001.

[30]

L. Liu, L. Pareschi and X. Zhu, A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs, preprint, arXiv: 2107.09250.

[31]

L. Liu and X. Zhu, A bi-fidelity method for the multiscale Boltzmann equation with random parameters, J. Comput. Phys., 402 (2020), 23pp. doi: 10.1016/j.jcp.2019.108914.

[32]

N. Loy and A. Tosin, A viral load-based model for epidemic spread on spatial networks, Math. Biosci. Eng., 18 (2021), 5635-5663.  doi: 10.3934/mbe.2021285.

[33]

C. Lu and X. Zhu, Bifidelity data-assisted neural networks in nonintrusive reduced-order modeling, J. Sci. Comput., 87 (2021), 30pp. doi: 10.1007/s10915-020-01403-w.

[34]

P. MagalG. F. Webb and Y. Wu, Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2185-2202.  doi: 10.3934/dcdsb.2019223.

[35]

A. Narayan, C. Gittelson and D. Xiu, A stochastic collocation algorithm with multifidelity models, SIAM J. Sci. Comput., 36 (2014), A495–A521. doi: 10.1137/130929461.

[36]

M. Peirlinck, K. Linka, F. Sahli Costabal, J. Bhattacharya, E. Bendavid, J. P. A. Ioannidis and E. Kuhl, Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19, Comput. Methods Appl. Mech. Engrg., 372 (2020), 22pp. doi: 10.1101/2020.05.23.20111419.

[37]

M. Pulvirenti and S. Simonella, A kinetic model for epidemic spread, Math. Mech. Complex Syst., 8 (2020), 249-260.  doi: 10.2140/memocs.2020.8.249.

[38]

F. Riccardo, M. Ajelli, X. D. Andrianou, A. Bella and M. Del Manso, et al., Epidemiological characteristics of COVID-19 cases and estimates of the reproductive numbers 1 month into the epidemic, Italy, 28 January to 31 March 2020, Euro Surveill., 25 (2020). doi: 10.2807/1560-7917.ES.2020.25.49.2000790.

[39]

L. Roques, O. Bonnefon, V. Baudrot, S. Soubeyrand and H. Berestycki, A parsimonious approach for spatial transmission and heterogeneity in the COVID-19 propagation, R. Soc. Open Sci., 7 (2020). doi: 10.1098/rsos.201382.

[40]

G.-Q. Sun, Pattern formation of an epidemic model with diffusion, Nonlinear Dynam., 69 (2012), 1097-1104.  doi: 10.1007/s11071-012-0330-5.

[41]

B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020). doi: 10.3390/jcm9020462.

[42]

A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli and T. J. R. Hughes, et al., Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 9pp. doi: 10.1016/j.aml.2020.106617.

[43]

A. ViguerieA. VenezianiG. LorenzoD. BaroliN. Aretz-Nellesen and et al., Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study, Comput. Mech., 66 (2020), 1131-1152.  doi: 10.1007/s00466-020-01888-0.

[44]

J. Wang, F. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 20pp. doi: 10.1016/j.cnsns.2019.104951.

[45]

G. F. Webb, A reaction-diffusion model for a deterministic diffusion epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6.

[46] D. Xiu, Numerical Methods for Stochastic Computations. A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010. 
[47]

X. ZhuE. M. Linebarger and D. Xiu, Multi-fidelity stochastic collocation method for computation of statistical moments, J. Comput. Phys., 341 (2017), 386-396.  doi: 10.1016/j.jcp.2017.04.022.

[48]

X. ZhuA. Narayan and D. Xiu, Computational aspects of stochastic collocation with multifidelity models, SIAM/ASA J. Uncertain. Quantif., 2 (2014), 444-463.  doi: 10.1137/130949154.

show all references

References:
[1]

P. S. Abdul SalamW. BockA. Klar and S. Tiwari, Disease contagion models coupled to crowd motion and mesh-free simulation, Math. Models Methods Appl. Sci., 31 (2021), 1277-1295.  doi: 10.1142/S0218202521400066.

[2]

G. Albi, G. Bertaglia, W. Boscheri, G. Dimarco, L. Pareschi, G. Toscani and M. Zanella, Kinetic modelling of epidemic dynamics: Social contacts, control with uncertain data, and multiscale spatial dynamics, in press in Predicting Pandemics in a Globally Connected World, Springer-Nature, (2022).

[3]

G. Albi, L. Pareschi and M. Zanella, Control with uncertain data of socially structured compartmental epidemic models, J. Math. Biol., 82 (2021), 41pp. doi: 10.1007/s00285-021-01617-y.

[4]

G. AlbiL. Pareschi and M. Zanella, Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty, Math. Biosci. Eng., 18 (2021), 7161-7190.  doi: 10.3934/mbe.2021355.

[5]

E. Barbera, G. Consolo and G. Valenti, Spread of infectious diseases in a hyperbolic reaction-diffusion susceptible-infected-removed model, Phys. Rev. E, 88 (2013), 13pp. doi: 10.1103/PhysRevE.88.052719.

[6]

N. BellomoR. BinghamM. A. J. ChaplainG. DosiG. Forni and et al., A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world, Math. Models Methods Appl. Sci., 30 (2020), 1591-1651.  doi: 10.1142/S0218202520500323.

[7]

G. BertagliaW. BoscheriG. Dimarco and L. Pareschi, Spatial spread of COVID-19 outbreak in Italy using multiscale kinetic transport equations with uncertainty, Math. Biosci. Eng., 18 (2021), 7028-7059.  doi: 10.3934/mbe.2021350.

[8]

G. Bertaglia, V. Caleffi, L. Pareschi and A. Valiani, Uncertainty quantification of viscoelastic parameters in arterial hemodynamics with the a-FSI blood flow model, J. Comput. Phys., 430 (2021), 20pp. doi: 10.1016/j.jcp.2020.110102.

[9]

G. Bertaglia and L. Pareschi, Hyperbolic compartmental models for epidemic spread on networks with uncertain data: Application to the emergence of COVID-19 in Italy, Math. Models Methods Appl. Sci., 31 (2021), 2495-2531.  doi: 10.1142/S0218202521500548.

[10]

G. Bertaglia and L. Pareschi, Hyperbolic models for the spread of epidemics on networks: Kinetic description and numerical methods, ESAIM Math. Model. Numer. Anal., 55 (2021), 381-407.  doi: 10.1051/m2an/2020082.

[11]

S. Boscarino, L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35 (2013), A22–A51. doi: 10.1137/110842855.

[12]

S. BoscarinoL. Pareschi and G. Russo, A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation, SIAM J. Numer. Anal., 55 (2017), 2085-2109.  doi: 10.1137/M1111449.

[13]

W. BoscheriG. Dimarco and L. Pareschi, Modeling and simulating the spatial spread of an epidemic through multiscale kinetic transport equations, Math. Models Methods Appl. Sci., 31 (2021), 1059-1097.  doi: 10.1142/S0218202521400017.

[14]

B. Buonomo and R. Della Marca, Effects of information-induced behavioural changes during the COVID-19 lockdowns: The case of Italy, R. Soc. Open Sci., 7 (2020). doi: 10.1098/rsos.201635.

[15]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.

[16]

R. M. Colombo, M. Garavello, F. Marcellini and E. Rossi, An age and space structured SIR model describing the Covid-19 pandemic, J. Math. Ind., 10 (2020), 20pp. doi: 10.1186/s13362-020-00090-4.

[17]

G. Dimarco, L. Liu, L. Pareschi and X. Zhu, Multi-fidelity methods for uncertainty propagation in kinetic equations, preprint, arXiv: 2112.00932.

[18]

G. Dimarco and L. Pareschi, Multi-scale control variate methods for uncertainty quantification in kinetic equations, J. Comput. Phys., 388 (2019), 63-89.  doi: 10.1016/j.jcp.2019.03.002.

[19]

G. Dimarco and L. Pareschi, Multiscale variance reduction methods based on multiple control variates for kinetic equations with uncertainties, Multiscale Model. Simul., 18 (2020), 351-382.  doi: 10.1137/18M1231985.

[20]

G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numer., 23 (2014), 369-520.  doi: 10.1017/S0962492914000063.

[21]

G. Dimarco, B. Perthame, G. Toscani and M. Zanella, Kinetic models for epidemic dynamics with social heterogeneity, J. Math. Biol., 83 (2021), 32pp. doi: 10.1007/s00285-021-01630-1.

[22]

E. Franco, A feedback SIR (fSIR) model highlights advantages and limitations of infection-based social distancing, preprint, arXiv: 2004.13216.

[23]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures, Proceed. Nat. Acad. Sci., 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[24]

F. GolseS. Jin and C. Levermore, The convergence of numerical transfer schemes in diffusive regimes. I. Discrete-ordinate method, SIAM J. Numer. Anal., 36 (1999), 1333-1369.  doi: 10.1137/S0036142997315986.

[25]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.

[26]

T. Hillen and A. Swan, The diffusion limit of transport equations in biology, in Mathematical Models and Methods for Living Systems, Lecture Notes in Math., 2167, Fond. CIME/CIME Found. Subser., Springer, Cham, 2016, 73–129. doi: 10.1007/978-3-319-42679-2_2.

[27]

S. Jin, H. Lu and L. Pareschi, Efficient stochastic asymptotic-preserving implicit-explicit methods for transport equations with diffusive scalings and random inputs, SIAM J. Sci. Comput., 40 (2018), A671–A696. doi: 10.1137/17M1120518.

[28]

S. JinL. Pareschi and G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations, SIAM J. Numer. Anal., 38 (2000), 913-936.  doi: 10.1137/S0036142998347978.

[29]

A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models, Math. Med. Bio. J. IMA, 22 (2005), 113-128.  doi: 10.1093/imammb/dqi001.

[30]

L. Liu, L. Pareschi and X. Zhu, A bi-fidelity stochastic collocation method for transport equations with diffusive scaling and multi-dimensional random inputs, preprint, arXiv: 2107.09250.

[31]

L. Liu and X. Zhu, A bi-fidelity method for the multiscale Boltzmann equation with random parameters, J. Comput. Phys., 402 (2020), 23pp. doi: 10.1016/j.jcp.2019.108914.

[32]

N. Loy and A. Tosin, A viral load-based model for epidemic spread on spatial networks, Math. Biosci. Eng., 18 (2021), 5635-5663.  doi: 10.3934/mbe.2021285.

[33]

C. Lu and X. Zhu, Bifidelity data-assisted neural networks in nonintrusive reduced-order modeling, J. Sci. Comput., 87 (2021), 30pp. doi: 10.1007/s10915-020-01403-w.

[34]

P. MagalG. F. Webb and Y. Wu, Spatial spread of epidemic diseases in geographical settings: Seasonal influenza epidemics in Puerto Rico, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 2185-2202.  doi: 10.3934/dcdsb.2019223.

[35]

A. Narayan, C. Gittelson and D. Xiu, A stochastic collocation algorithm with multifidelity models, SIAM J. Sci. Comput., 36 (2014), A495–A521. doi: 10.1137/130929461.

[36]

M. Peirlinck, K. Linka, F. Sahli Costabal, J. Bhattacharya, E. Bendavid, J. P. A. Ioannidis and E. Kuhl, Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19, Comput. Methods Appl. Mech. Engrg., 372 (2020), 22pp. doi: 10.1101/2020.05.23.20111419.

[37]

M. Pulvirenti and S. Simonella, A kinetic model for epidemic spread, Math. Mech. Complex Syst., 8 (2020), 249-260.  doi: 10.2140/memocs.2020.8.249.

[38]

F. Riccardo, M. Ajelli, X. D. Andrianou, A. Bella and M. Del Manso, et al., Epidemiological characteristics of COVID-19 cases and estimates of the reproductive numbers 1 month into the epidemic, Italy, 28 January to 31 March 2020, Euro Surveill., 25 (2020). doi: 10.2807/1560-7917.ES.2020.25.49.2000790.

[39]

L. Roques, O. Bonnefon, V. Baudrot, S. Soubeyrand and H. Berestycki, A parsimonious approach for spatial transmission and heterogeneity in the COVID-19 propagation, R. Soc. Open Sci., 7 (2020). doi: 10.1098/rsos.201382.

[40]

G.-Q. Sun, Pattern formation of an epidemic model with diffusion, Nonlinear Dynam., 69 (2012), 1097-1104.  doi: 10.1007/s11071-012-0330-5.

[41]

B. Tang, X. Wang, Q. Li, N. L. Bragazzi, S. Tang, Y. Xiao and J. Wu, Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions, J. Clin. Med., 9 (2020). doi: 10.3390/jcm9020462.

[42]

A. Viguerie, G. Lorenzo, F. Auricchio, D. Baroli and T. J. R. Hughes, et al., Simulating the spread of COVID-19 via a spatially-resolved susceptible-exposed-infected-recovered-deceased (SEIRD) model with heterogeneous diffusion, Appl. Math. Lett., 111 (2021), 9pp. doi: 10.1016/j.aml.2020.106617.

[43]

A. ViguerieA. VenezianiG. LorenzoD. BaroliN. Aretz-Nellesen and et al., Diffusion-reaction compartmental models formulated in a continuum mechanics framework: Application to COVID-19, mathematical analysis, and numerical study, Comput. Mech., 66 (2020), 1131-1152.  doi: 10.1007/s00466-020-01888-0.

[44]

J. Wang, F. Xie and T. Kuniya, Analysis of a reaction-diffusion cholera epidemic model in a spatially heterogeneous environment, Commun. Nonlinear Sci. Numer. Simul., 80 (2020), 20pp. doi: 10.1016/j.cnsns.2019.104951.

[45]

G. F. Webb, A reaction-diffusion model for a deterministic diffusion epidemic, J. Math. Anal. Appl., 84 (1981), 150-161.  doi: 10.1016/0022-247X(81)90156-6.

[46] D. Xiu, Numerical Methods for Stochastic Computations. A Spectral Method Approach, Princeton University Press, Princeton, NJ, 2010. 
[47]

X. ZhuE. M. Linebarger and D. Xiu, Multi-fidelity stochastic collocation method for computation of statistical moments, J. Comput. Phys., 341 (2017), 386-396.  doi: 10.1016/j.jcp.2017.04.022.

[48]

X. ZhuA. Narayan and D. Xiu, Computational aspects of stochastic collocation with multifidelity models, SIAM/ASA J. Uncertain. Quantif., 2 (2014), 444-463.  doi: 10.1137/130949154.

Figure 1.  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
Figure 2.  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
Figure 3.  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
Figure 4.  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
Figure 5.  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
Figure 6.  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
Figure 7.  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
Figure 8.  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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