no. 2
Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 381-407

We consider the development of hyperbolic transport models for the propagation in space of an epidemic phenomenon described by a classical compartmental dynamics. The model is based on a kinetic description at discrete velocities of the spatial movement and interactions of a population of susceptible, infected and recovered individuals. Thanks to this, the unphysical feature of instantaneous diffusive effects, which is typical of parabolic models, is removed. In particular, we formally show how such reaction-diffusion models are recovered in an appropriate diffusive limit. The kinetic transport model is therefore considered within a spatial network, characterizing different places such as villages, cities, countries, etc. The transmission conditions in the nodes are analyzed and defined. Finally, the model is solved numerically on the network through a finite-volume IMEX method able to maintain the consistency with the diffusive limit without restrictions due to the scaling parameters. Several numerical tests for simple epidemic network structures are reported and confirm the ability of the model to correctly describe the spread of an epidemic.

DOI : 10.1051/m2an/2020082
Classification : 65M08, 35L50, 65L04, 35K57, 82C40, 92D30
Keywords: Kinetic equations, hyperbolic systems, spatial epidemic models, SIR model, network models, IMEX Runge–Kutta schemes, diffusive limit 
@article{M2AN_2021__55_2_381_0,
     author = {Bertaglia, Giulia and Pareschi, Lorenzo},
     title = {Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {381--407},
     year = {2021},
     publisher = {EDP-Sciences},
     volume = {55},
     number = {2},
     doi = {10.1051/m2an/2020082},
     mrnumber = {4229196},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/m2an/2020082/}
}
TY  - JOUR
AU  - Bertaglia, Giulia
AU  - Pareschi, Lorenzo
TI  - Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2021
SP  - 381
EP  - 407
VL  - 55
IS  - 2
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/m2an/2020082/
DO  - 10.1051/m2an/2020082
LA  - en
ID  - M2AN_2021__55_2_381_0
ER  - 
%0 Journal Article
%A Bertaglia, Giulia
%A Pareschi, Lorenzo
%T Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2021
%P 381-407
%V 55
%N 2
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2020082/
%R 10.1051/m2an/2020082
%G en
%F M2AN_2021__55_2_381_0
Bertaglia, Giulia; Pareschi, Lorenzo. Hyperbolic models for the spread of epidemics on networks: kinetic description and numerical methods. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 2, pp. 381-407. doi: 10.1051/m2an/2020082

[1] A. Aktay, S. Bavadekar, G. Cossoul, J. Davis, D. Desfontaines, A. Fabrikant, E. Gabrilovich, K. Gadepalli, B. Gipson, M. Guevara, C. Kamath, M. Kansal, A. Lange, C. Mandayam, A. Oplinger, C. Pluntke, T. Roessler, A. Schlosberg, T. Shekel, S. Vispute, M. Vu, G. Wellenius, B. Williams and R. J. Wilson, Google COVID-19 Community Mobility Reports: Anonymization Process Description (version 1.1). Preprint: (2020). | arXiv

[2] G. Albi, M. Zanella and L. Pareschi, Control with uncertain data of socially structured compartmental epidemic models. Preprint: (2020). | arXiv | MR

[3] D. Amadori and G. Guerra, Global weak solutions for systems of balance laws. Appl. Math. Lett. 12 (1999) 123–127. | MR | Zbl | DOI

[4] U. M. Ascher, S. J. Ruuth and R. J. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. App. Numer. Math. 25 (1997) 151–167. | MR | Zbl | DOI

[5] D. Balcan, V. Colizza, B. Gonçalves, H. Hu, J. J. Ramasco and A. Vespignani, Multiscale mobility networks and the spatial spreading of infectious diseases. Proc. Nat. Acad. Sci. United States Am. 106 (2009) 21484–21489. | DOI

[6] 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). | MR | DOI

[7] S. Bargmann, P. M. Jordan and W. Lambert, A second-sound based hyperbolic SIR model for high-diffusivity spread. Phys. Lett. A 375 (2011) 898–907. | Zbl | DOI

[8] N. Bellomo, R. Bingham, M. A. J. Chaplain, G. Dosi, G. Forni, D. A. Knopoff, J. Lowengrub, R. Twarock and M. E. Virgillito, A multi-scale model of virus pandemic: heterogeneous interactive entities in a globally connected world. Math. Models Methods Appl. Sci. 30 (2020) 1591–1651. | MR | DOI

[9] G. Bertaglia, M. Ioriatti, A. Valiani, M. Dumbser and V. Caleffi, Numerical methods for hydraulic transients in visco-elastic pipes. J. Fluids Struct. 81 (2018) 230–254. | DOI

[10] G. Bertaglia, V. Caleffi and A. Valiani, Modeling blood flow in viscoelastic vessels: the 1D augmented fluid-structure interaction system. Comput. Methods Appl. Mech. Eng. 360 (2020). | MR | DOI

[11] S. Boscarino, L. Pareschi and G. Russo, A unified IMEX Runge-Kutta approach for hyperbolic systems with multiscale relaxation. SIAM J. Numer. Anal. 55 (2017) 2085–2109. | MR | DOI

[12] A. Bressan, S. Čanić, M. Garavello, M. Herty and B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1 (2014) 47–111. | MR | Zbl | DOI

[13] G. Bretti, R. Natalini and B. Piccoli, Numerical approximations of a traffic flow model on networks. Networks Heterogen Media 1 (2006) 57–84. | MR | Zbl | DOI

[14] G. Bretti, R. Natalini and M. Ribot, A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: M2AN 48 (2014) 231–258. | MR | Zbl | Numdam | DOI

[15] V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model. Math. Biosci. 42 (1978) 43–61. | MR | Zbl | DOI

[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) 22. | MR | DOI

[17] M. Dumbser and E. F. Toro, On universal Osher-type schemes for general nonlinear hyperbolic conservation laws. Commun. Comput. Phys. 10 (2011) 635–671. | MR | DOI

[18] L. Fermo and A. Tosin, A fully-discrete-state kinetic theory approach to traffic flow on road networks. Math. Models Methods Appl. Sci. 25 (2015) 423–461. | MR | DOI

[19] E. Franco, A feedback SIR (fSIR) model highlights advantages and limitations of infection-based social distancing. Preprint: (2020). | arXiv

[20] K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. 68 (1971) 1686–1688. | MR | Zbl | DOI

[21] M. Gatto, E. Bertuzzo, L. Mari, S. Miccoli, L. Carraro, R. 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

[22] G. Giordano, F. Blanchini, R. Bruno, P. Colaneri, A. Di Filippo, A. Di Matteo and M. Colaneri, Modelling the COVID-19 epidemic and implementation of population-wide interventions in Italy. Nat. Med. 26 (2020) 855–860. | DOI

[23] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Math. Acad. Sci. Paris 334 (2002) 337–342. | MR | Zbl | DOI

[24] H. W. Hethcote, The mathematics of infectious diseases. SIAM Rev. 42 (2000) 599–653. | MR | Zbl | DOI

[25] S. Jin, L. Pareschi and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations. SIAM J. Numer. Anal. 35 (1998) 2405–2439. | MR | Zbl | DOI

[26] D. Koch, R. Illner and J. Ma, Edge removal in random contact networks and the basic reproduction number. J. Math. Bio. 67 (2013) 217–238. | MR | Zbl | DOI

[27] A. Korobeinikov and P. K. Maini, Non-linear incidence and stability of infectious disease models. Math. Med. Biol. J. IMA 22 (2005) 113–128. | Zbl | DOI

[28] M. U. G. Kraemer, C.-H. Yang, B. Gutierrez, C.-H. Wu, B. Klein, D. M. Pigott, L. Du Plessis, N. R. Faria, R. Li, W. P. Hanage, J. S. Brownstein, M. Layan, A. Vespignani, H. Tian, C. Dye, O. G. Pybus and S. V. Scarpino, The effect of human mobility and control measures on the COVID-19 epidemic in China. Science 368 (2020) 493–497. | DOI

[29] P.-L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoamericana 13 (1997) 473–513. | MR | Zbl | DOI

[30] S. Merler and M. Ajelli, Human mobility and population heterogeneity in the spread of an epidemic. Proc. Comput. Sci. 1 (2010) 2237–2244. | DOI

[31] W. Min Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23 (1986) 187–204. | MR | Zbl | DOI

[32] I. Müller and T. Ruggeri, Rational Extended Thermodynamics. Springer, New York (1998). | MR | Zbl | DOI

[33] J. D. Murray, Mathematical Biology I, II. Springer-Verlag, New York (2002–2003). | Zbl | MR

[34] G. Naldi and L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37 (2000) 1246–1270. | MR | Zbl | DOI

[35] L. Pareschi and G. Russo, Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. J. Sci. Comput. 25 (2005) 129–155. | MR | Zbl

[36] L. Pellis, F. Ball, S. Bansal, K. Eames, T. House, V. Isham and P. Trapman, Eight challenges for network epidemic models. Epidemics 10 (2015) 58–62. | DOI

[37] B. Piccoli and M. Garavello, Traffic Flow on Networks. American Institute of Mathematical Sciences (2006). | MR | Zbl

[38] S. Riley, K. Eames, V. Isham, D. Mollison and P. Trapman, Five challenges for spatial epidemic models. Epidemics 10 (2015) 68–71. | DOI

[39] G.-Q. Sun, Pattern formation of an epidemic model with diffusion. Nonlinear Dyn. 69 (2012) 1097–1104. | MR | DOI

[40] E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics, 3rd edition. Springer Verlag (2009). | MR | Zbl

[41] M. A. C. Vollmer, S. Mishra, H. J. T. Unwin, A. Gandy, T. A. Mellan, H. Zhu, H. Coupland, I. Hawryluk, M. Hutchinson, O. Ratmann, P. Walker, C. Whittaker, L. Cattarino, C. Ciavarella, L. Cilloni, M. Baguelin, S. Bhatia, A. Boonyasiri, N. Brazeau, G. Charles, V. Cooper, Z. Cucunuba, G. Cuomo-Dannenburg, A. Dighe, B. Djaafara, J. Eaton, L. V. Elsland, R. Fitzjohn, K. Fraser, K. Gaythorpe, W. Green, S. Hayes, N. Imai, E. Knock, D. Laydon, J. Lees, T. Mangal, A. Mousa, G. Nedjati-Gilani, P. Nouvellet, D. Olivera, K. V. Parag, M. Pickles, H. A. Thompson, R. Verity, H. Wang, Y. Wang, O. J. Watson, L. Whittles, X. Xi and A. Ghani, Report 20: Using mobility to estimate the transmission intensity of COVID-19 in Italy: A subnational analysis with future scenarios. Technical Report May, Imperial College London (2020).

[42] Y. Wang, J. Wang and L. Zhang, Cross diffusion-induced pattern in an SI model. Appl. Math. Comput. 217 (2010) 1965–1970. | MR | Zbl

[43] 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). | MR | DOI

Cité par Sources :