A multiscale approach for spatially inhomogeneous disease dynamics

Markus Schmidtchen; Oliver T.C. Tse; Stephan Wackerle

doi:10.5614/cbms.2018.1.2.1

Authors

Markus Schmidtchen Department of Mathematics, Imperial College London, London SW7 2AZ
Oliver T.C. Tse Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600MB Eindhoven
Stephan Wackerle Fraunhofer Institute for Industrial Mathematics, Fraunhofer-Platz 1, 67663 Kaiserslautern

DOI:

https://doi.org/10.5614/cbms.2018.1.2.1

Keywords:

Epidemiology, disease dynamics, agent-based models, multiscale modeling, stochastic dynamics, mean-field limit

Abstract

In this paper we introduce an agent-based epidemiological model that generalizes the classical SIR model by Kermack and McKendrick. We further provide a multiscale approach to the derivation of a macroscopic counterpart via the mean-field limit. The chain of equations acquired via the multiscale approach is investigated, analytically as well as numerically. The outcome of these results provides strong evidence of the models' robustness and justifies their practicality in describing disease dynamics, in particularly when mobility is involved. The numerical results provide further insights into the applicability of the different scaling limits.

References

L. J. S. Allen, F. Brauer, P. Van den Driessche, and J. Wu. Mathematical epidemiology. Springer, 2008.

N. Bellomo. Modeling complex living systems: a kinetic theory and stochastic game approach. Springer Science & Business Media, 2008.

W. Bock, T. Fattler, I. Rodiah, and O. Tse. An analytic method for agent-based modeling of spatially inhomogeneous disease dynamics. AIP Conference Proceedings, 1871(1):020008, 2017.

F. Bolley, J. A. Canizo, and J. A. Carrillo. Stochastic mean-field limit: non-lipschitz forces and swarming. Mathematical Models and Methods in Applied Sciences, 21(11):2179-2210, 2011.

F. Brauer and C. Castillo-Chavez. Mathematical models in population biology and epidemiology, volume 40. Springer, 2001.

W. Braun and K. Hepp. The vlasov dynamics and its fluctuations in the 1/n limit of interacting classical particles. Communications in mathematical physics, 56(2):101-113, 1977.

D. Brockmann, V. David, and A. M. Gallardo. Human mobility and spatial disease dynamics. Reviews of nonlinear dynamics and complexity, 2:1-24, 2009.

J. A. Carrillo, M. Fornasier, G. Toscani, and F. Vecil. Particle, kinetic, and hydrodynamic models of swarming. In Mathematical modeling of collective behavior in socio-economic and life sciences, pages 297-336. Springer, 2010.

J. A. Carrillo, A. Klar, S. Martin, and S. Tiwari. Self-propelled interacting particle systems with roosting force. Mathematical Models and Methods in Applied Sciences, 20:1533-1552, 2010.

G. Crippa and M. L'ecureux-Mercier. Existence and uniqueness of measure solutions for a system of continuity equations with non-local flow. Nonlinear Differential Equations and Applications NoDEA, 20(3):523-537, 2013.

S. De Lillo, M. Delitala, and M. C. Salvatori. Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles. Mathematical Models and Methods in Applied Sciences, 19(supp01):1405-1425, 2009.

M. Delitala. Generalized kinetic theory approach to modeling spread-and evolution of epidemics. Mathematical and computer modelling, 39(1):1-12, 2004.

R. L. Dobrushin. Vlasov equations. Functional Analysis and Its Applications, 13(2):115-123, 1979.

R. Durrett. Stochastic calculus: a practical introduction, volume 6. CRC press, 1996.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for stochastic dynamics of continuous systems. Journal of Statistical Physics, 141(1):158-178, 2010.

D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for the glauber dynamics in continuum. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(04):537-569, 2011.

R. A. Fisher. The wave of advance of advantageous genes. Annals of eugenics, 7(4):355-369, 1937.

P. Hanggi, P. Talkner, and M. Borkovec. Reaction-rate theory: fifty years after kramers. Reviews of modern physics, 62(2):251, 1990.

JAP Heesterbeek. Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, volume 5. John Wiley & Sons, 2000.

H. W. Hethcote. The mathematics of infectious diseases. SIAM review, 42(4):599-653, 2000.

H. Hu, K. Nigmatulina, and P. Eckhoff. The scaling of contact rates with population density for the infectious disease models. Mathematical biosciences, 244(2):125-134, 2013.

W. O. Kermack and A. G. McKendrick. A contribution to the mathematical theory of epidemics. In Proceedings of the Royal Society of London A: mathematical, physical and engineering sciences, volume 115, pages 700-721. The Royal Society, 1927.

A. Klar, F. Schneider, and O. Tse. Approximate models for stochastic dynamic systems with velocities on the sphere and associated fokker-planck equations. Kinetic and Related Models, 7(3):509-529, 2014.

P. E. Kloeden and E. Platen. Numerical Solution of Stochastic Differential Equations. Stochastic Modelling and Applied Probability. Springer Berlin Heidelberg, 2011.

M. A. Lewis, P. K. Maini, and S. V. Petrovskii. Dispersal, individual movement and spatial ecology. Lecture Notes in Mathematics (Mathematics Bioscience Series), 2071, 2013.

T. Liggett. Interacting particle systems, volume 276. Springer Science & Business Media, 2012.

P. L. Lions and A. S. Sznitman. Stochastic differential equations with reflecting boundary conditions. Communications on Pure and Applied Mathematics, 37(4):511-537, 1984.

H. P. McKean. Propagation of chaos for a class of non-linear parabolic equations. Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pages 41-57, 1967.

D. Morale. Modeling and simulating animal grouping: individual-based models. Future Generation Computer Systems, 17(7):883-891, 2001.

D. Morale, V. Capasso, and K. Oelschlager. An interacting particle system modelling aggregation behavior: from individuals to populations. Journal of mathematical biology, 50(1):49-66, 2005.

J. D. Murray. Mathematical Biology I: An Introduction, volume 17. Springer-Verlag, New York, 2002.

J. D. Murray. Mathematical Biology II: Spatial Models and Biomedical Applications, volume 18. Springer-Verlag New York, 2003.

H. Spohn. Large scale dynamics of interacting particles. Springer Science & Business Media, 2012.

A.-S. Sznitman. Topics in propagation of chaos. In Ecole d''et'e de probabilit'es de Saint-Flour XIX1989, pages 165-251. Springer, 1991.

E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, chapter The HLL and HLLC Riemann Solvers, pages 315-344. Springer Berlin Heidelberg, Berlin, Heidelberg, 2009.

C. Villani. Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.

E. Zeidler. Nonlinear Analysis and Its Applications I: Fixed-Point Theorems. Springer-Verlag, New York, 1993.