American Institute of Mathematical Sciences

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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

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), 276-282. Google Scholar

[2]

V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks, Phys. Rev. Lett., 99 (2007), 148701. doi: 10.1103/PhysRevLett.99.148701.  Google Scholar

[3]

V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, J. Theor. Biol., 251 (2008), 450-467. doi: 10.1016/j.jtbi.2007.11.028.  Google Scholar

[4]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000.  Google Scholar

[5]

S. N. Dorogotsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach, Phys. Rev. E, 63 (2001), 056125. Google Scholar

[6]

S. N. Dorogotsev and J. F. F. Mendes, Evolution of networks, Adv. Phys., 51 (2002), 1079. Google Scholar

[7]

I. Hanski, A practical model of metapopulation dynamics, J. Animal Ecology, 63 (1994), 151-162. doi: 10.2307/5591.  Google Scholar

[8]

I. Hanski, Metapopulation dynamics, Metapopulation Biology, (1997), 69-91. doi: 10.1016/B978-012323445-2/50007-9.  Google Scholar

[9]

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

[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), 041920, 9pp. doi: 10.1103/PhysRevE.80.041920.  Google Scholar

[11]

M. J. Keeling and K. T. D. Eames, Networks and epidemic models, J. R. Soc. Interface, 2 (2005), 295-307. doi: 10.1098/rsif.2005.0051.  Google Scholar

[12]

R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomology Soc. of America, 71 (1969), 237-240. doi: 10.1093/besa/15.3.237.  Google Scholar

[13]

E. Logak and I. Passat, A nonlocal model for epidemics on networks in the case of nonlimited transmission,, preprint., ().   Google Scholar

[14]

M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

[15]

R. Pastor-Sattoras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys., 87 (2015), 925-979. doi: 10.1103/RevModPhys.87.925.  Google Scholar

[16]

R. Pastor-Sattoras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200.  Google Scholar

[17]

J. Saldana, Continous-time formulation of reaction-diffusion processes on heterogeneous metapopulations, Phys. Rev. E, 78 (2008), 012902. Google Scholar

[18]

J. Saldana, Analysis and Monte-Carlo simulations of a model for spread of infectious diseases in heterogeneous metapopulations, Phys. Rev. E, 80 (2009), 041920, 9pp. doi: 10.1103/PhysRevE.80.041920.  Google Scholar

[19]

J. Saldana, Modelling the spread of infectious diseases in complex metapopulations, Math. Model. Nat. Phenom., 5 (2010), 22-37. doi: 10.1051/mmnp/20105602.  Google Scholar

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), 276-282. Google Scholar

[2]

V. Colizza and A. Vespignani, Invasion threshold in heterogeneous metapopulation networks, Phys. Rev. Lett., 99 (2007), 148701. doi: 10.1103/PhysRevLett.99.148701.  Google Scholar

[3]

V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, J. Theor. Biol., 251 (2008), 450-467. doi: 10.1016/j.jtbi.2007.11.028.  Google Scholar

[4]

O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000.  Google Scholar

[5]

S. N. Dorogotsev and J. F. F. Mendes, Scaling properties of scale-free evolving networks: Continuous approach, Phys. Rev. E, 63 (2001), 056125. Google Scholar

[6]

S. N. Dorogotsev and J. F. F. Mendes, Evolution of networks, Adv. Phys., 51 (2002), 1079. Google Scholar

[7]

I. Hanski, A practical model of metapopulation dynamics, J. Animal Ecology, 63 (1994), 151-162. doi: 10.2307/5591.  Google Scholar

[8]

I. Hanski, Metapopulation dynamics, Metapopulation Biology, (1997), 69-91. doi: 10.1016/B978-012323445-2/50007-9.  Google Scholar

[9]

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

[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), 041920, 9pp. doi: 10.1103/PhysRevE.80.041920.  Google Scholar

[11]

M. J. Keeling and K. T. D. Eames, Networks and epidemic models, J. R. Soc. Interface, 2 (2005), 295-307. doi: 10.1098/rsif.2005.0051.  Google Scholar

[12]

R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bull. Entomology Soc. of America, 71 (1969), 237-240. doi: 10.1093/besa/15.3.237.  Google Scholar

[13]

E. Logak and I. Passat, A nonlocal model for epidemics on networks in the case of nonlimited transmission,, preprint., ().   Google Scholar

[14]

M. E. J. Newman, The structure and function of complex networks, SIAM Rev., 45 (2003), 167-256. doi: 10.1137/S003614450342480.  Google Scholar

[15]

R. Pastor-Sattoras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys., 87 (2015), 925-979. doi: 10.1103/RevModPhys.87.925.  Google Scholar

[16]

R. Pastor-Sattoras and A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett., 86 (2001), 3200. doi: 10.1103/PhysRevLett.86.3200.  Google Scholar

[17]

J. Saldana, Continous-time formulation of reaction-diffusion processes on heterogeneous metapopulations, Phys. Rev. E, 78 (2008), 012902. Google Scholar

[18]

J. Saldana, Analysis and Monte-Carlo simulations of a model for spread of infectious diseases in heterogeneous metapopulations, Phys. Rev. E, 80 (2009), 041920, 9pp. doi: 10.1103/PhysRevE.80.041920.  Google Scholar

[19]

J. Saldana, Modelling the spread of infectious diseases in complex metapopulations, Math. Model. Nat. Phenom., 5 (2010), 22-37. doi: 10.1051/mmnp/20105602.  Google Scholar

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