An epidemic model with nonlocal diffusion on networks

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December 2016, 11(4): 693-719. doi: 10.3934/nhm.2016014

An epidemic model with nonlocal diffusion on networks

Elisabeth Logak ^1, and Isabelle Passat ^2,

1.	University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, Cergy-Pontoise, F-95000
2.	University of Cergy-Pontoise, Department of Mathematics, UMR CNRS 8088, F-95000 Cergy-Pontoise, France

Received November 2015 Revised February 2016 Published October 2016

Abstract
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We consider a SIS system with nonlocal diffusion which is the continuous version of a discrete model for the propagation of epidemics on a metapopulation network. Under the assumption of limited transmission, we prove the global existence of a unique solution for any diffusion coefficients. We investigate the existence of an endemic equilibrium and prove its linear stability, which corresponds to the loss of stability of the disease-free equilibrium. In the case of equal diffusion coefficients, we reduce the system to a Fisher-type equation with nonlocal diffusion, which allows us to study the large time behaviour of the solutions. We show large time convergence to either the disease-free or the endemic equilibrium.

Keywords: Fisher equation, Epidemic models, complex networks., nonlocal diffusion, patchy environments.

Mathematics Subject Classification: 35B35, 35B40, 35B51, 45J05, 47G20, 92C42, 92D3.

Citation: Elisabeth Logak, Isabelle Passat. An epidemic model with nonlocal diffusion on networks. Networks & Heterogeneous Media, 2016, 11 (4) : 693-719. doi: 10.3934/nhm.2016014

References:

[1]	V. Colizza, R. Pastor-Sattoras and A. Vespignani, Reaction-diffusion processes and meta-population models in heterogeneous networks,, Nat. Phys., 3 (2007), 276.
[2]	V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks,, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.148701.
[3]	V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations,, J. Theor. Biol., 251* (2008), 450. doi: 10.1016/j.jtbi.2007.11.028.*
[4]	O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, Wiley, (2000).
[5]	S. N. Dorogotsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach,, Phys. Rev. E, 63 (2001).
[6]	S. N. Dorogotsev and J. F. F. Mendes, Evolution of networks,, Adv. Phys., 51 (2002).
[7]	I. Hanski, A practical model of metapopulation dynamics,, J. Animal Ecology, 63 (1994), 151. doi: 10.2307/5591.
[8]	I. Hanski, Metapopulation dynamics,, Metapopulation Biology, (1997), 69. doi: 10.1016/B978-012323445-2/50007-9.
[9]	H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.
[10]	D. Juher, J. Ripoll and J. Saldana, Analysis and monte carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulation,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.041920.
[11]	M. J. Keeling and K. T. D. Eames, Networks and epidemic models,, J. R. Soc. Interface, 2 (2005), 295. doi: 10.1098/rsif.2005.0051.
[12]	R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control,, Bull. Entomology Soc. of America, 71 (1969), 237. doi: 10.1093/besa/15.3.237.
[13]	E. Logak and I. Passat, A nonlocal model for epidemics on networks in the case of nonlimited transmission,, preprint., ().
[14]	M. E. J. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167. doi: 10.1137/S003614450342480.
[15]	R. Pastor-Sattoras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks,, Rev. Mod. Phys., 87 (2015), 925. doi: 10.1103/RevModPhys.87.925.
[16]	R. Pastor-Sattoras and A. Vespignani, Epidemic spreading in scale-free networks,, Phys. Rev. Lett., 86 (2001). doi: 10.1103/PhysRevLett.86.3200.
[17]	J. Saldana, Continous-time formulation of reaction-diffusion processes on heterogeneous metapopulations,, Phys. Rev. E, 78 (2008).
[18]	J. Saldana, Analysis and Monte-Carlo simulations of a model for spread of infectious diseases in heterogeneous metapopulations,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.041920.
[19]	J. Saldana, Modelling the spread of infectious diseases in complex metapopulations,, Math. Model. Nat. Phenom., 5 (2010), 22. doi: 10.1051/mmnp/20105602.

show all references

References:

[1]	V. Colizza, R. Pastor-Sattoras and A. Vespignani, Reaction-diffusion processes and meta-population models in heterogeneous networks,, Nat. Phys., 3 (2007), 276.
[2]	V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks,, Phys. Rev. Lett., 99 (2007). doi: 10.1103/PhysRevLett.99.148701.
[3]	V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations,, J. Theor. Biol., 251* (2008), 450. doi: 10.1016/j.jtbi.2007.11.028.*
[4]	O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,, Wiley, (2000).
[5]	S. N. Dorogotsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach,, Phys. Rev. E, 63 (2001).
[6]	S. N. Dorogotsev and J. F. F. Mendes, Evolution of networks,, Adv. Phys., 51 (2002).
[7]	I. Hanski, A practical model of metapopulation dynamics,, J. Animal Ecology, 63 (1994), 151. doi: 10.2307/5591.
[8]	I. Hanski, Metapopulation dynamics,, Metapopulation Biology, (1997), 69. doi: 10.1016/B978-012323445-2/50007-9.
[9]	H. W. Hethcote, The mathematics of infectious diseases,, SIAM Review, 42 (2000), 599. doi: 10.1137/S0036144500371907.
[10]	D. Juher, J. Ripoll and J. Saldana, Analysis and monte carlo simulations of a model for the spread of infectious diseases in heterogeneous metapopulation,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.041920.
[11]	M. J. Keeling and K. T. D. Eames, Networks and epidemic models,, J. R. Soc. Interface, 2 (2005), 295. doi: 10.1098/rsif.2005.0051.
[12]	R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control,, Bull. Entomology Soc. of America, 71 (1969), 237. doi: 10.1093/besa/15.3.237.
[13]	E. Logak and I. Passat, A nonlocal model for epidemics on networks in the case of nonlimited transmission,, preprint., ().
[14]	M. E. J. Newman, The structure and function of complex networks,, SIAM Rev., 45 (2003), 167. doi: 10.1137/S003614450342480.
[15]	R. Pastor-Sattoras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks,, Rev. Mod. Phys., 87 (2015), 925. doi: 10.1103/RevModPhys.87.925.
[16]	R. Pastor-Sattoras and A. Vespignani, Epidemic spreading in scale-free networks,, Phys. Rev. Lett., 86 (2001). doi: 10.1103/PhysRevLett.86.3200.
[17]	J. Saldana, Continous-time formulation of reaction-diffusion processes on heterogeneous metapopulations,, Phys. Rev. E, 78 (2008).
[18]	J. Saldana, Analysis and Monte-Carlo simulations of a model for spread of infectious diseases in heterogeneous metapopulations,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.041920.
[19]	J. Saldana, Modelling the spread of infectious diseases in complex metapopulations,, Math. Model. Nat. Phenom., 5 (2010), 22. doi: 10.1051/mmnp/20105602.

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