Front dynamics in fractional-order epidemic models

Journal of Theoretical Biology

Volumn 279, Issue 1, 2011, Pages 9-16

  Hanert, Emmanuel ^a   Schumacher, Eva ^a   Deleersnijder, Eric ^a  

^a UNIVERSITÉ CATHOLIQUE DE LOUVAIN   (Belgium)

Author keywords

Epidemics spatial spread; Fractional order diffusion; Front dynamics; L vy flights

Indexed keywords

DIFFUSION; EPIDEMIC; NUMERICAL MODEL; POWER LAW; THEORETICAL STUDY;

ARTICLE; CONTROLLED STUDY; DIFFUSION; DISEASE TRANSMISSION; EPIDEMIC; HIGH RISK POPULATION; INFECTION RISK; INFECTION SENSITIVITY; MATHEMATICAL MODEL; MOLECULAR DYNAMICS; NORMAL DISTRIBUTION; PREDICTION; PRIORITY JOURNAL; PROCESS DEVELOPMENT; PROCESS MODEL;

ANIMALS; COMMUNICABLE DISEASES; DIFFUSION; EPIDEMICS; HUMANS; MODELS, BIOLOGICAL; TIME FACTORS;

ANIMALIA;

EID: 79953170542     PISSN: 00225193     EISSN: 10958541     Source Type: Journal    
DOI: 10.1016/j.jtbi.2011.03.012     Document Type: Article

References (40)

1. Brockmann D. Human mobility and spatial disease dynamics. Reviews of Nonlinear Dynamics and Complexity 2009, 1-24. Wiley-VCH. H.G. Schuster (Ed.).
2. Brockmann D., Hufnagel L., Geisel T. The scaling laws of human travel. Nature 2006, 439:462-465.
3. Buchanan M. Ecological modelling: the mathematical mirror to animal nature. Nature 2008, 453:714-716.
4. Cartea A., del Castillo Negrete D. Fractional diffusion models of option prices in markets with jumps. Physica A: Statistical Mechanics and its Applications 2007, 374:749-763.
5. Cencini M., Lopez C., Vergni D. Reaction-diffusion systems: front propagation and spatial structures. The Kolmogorov Legacy in Physics 2003, 187-209. Springer, Berlin. A. Vulpiani, R. Livi (Eds.).
6. Chaves A.S. A fractional diffusion equation to describe Lévy flights. Physics Letters A 1998, 239:13-16.
7. del Castillo Negrete D. Truncation effects in superdiffusive front propagation with Lévy flights. Physical Review E 2009, 79:1-10.
8. del Castillo Negrete D., Carreras B.A., Lynch V.E. Front dynamics in reaction-diffusion systems with Lévy flights: a fractional diffusion approach. Physical Review Letters 2003, 91.
9. del Castillo Negrete D., Carreras B.A., Lynch V.E. Fractional diffusion in plasma turbulence. Physics of Plasmas 2004, 11:3854-3864.
10. del Castillo Negrete D., Carreras B.A., Lynch V.E. Nondiffusive transport in plasma turbulence: a fractional diffusion approach. Physical Review Letters 2005, 94.
11. Deng Z.Q., de Lima J.L.M.P., de Lima M.I.P., Singh V.P. A fractional dispersion model for overland solute transport. Water Resources Research 2006, 42:W03416.
12. Ding Y., Ye H. A fractional-order differential equation model of HIV infection of CD4+ T-cells. Mathematical and Computer Modelling 2009, 50:386-392.
13. Djordjević V.D., Jarić J., Fabry B., Fredberg J.J., Stamenović D. Fractional derivatives embody essential features of cell rheological behavior. Annals of Biomedical Engineering 2003, 31:692-699.
14. Ebert U., van Saarloos W. Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D 2000, 146:1-99.
15. Edwards A.M., Phillips R.A., Watkins N.W., Freeman M.P., Murphy E.J., Afanasyev V., Buldyrev S.V., Da Luz M.G.E., Raposo E.P., Stanley H.E., Viswanathan G.M. Revisiting Lévy flight search patterns of wandering albatrosses, bumblebees and deer. Nature 2007, 449:1044-1048.
16. Feller W. An Introduction to Probability Theory and its Applications 1971, vol. II. Wiley.
17. Gnedenko B., Kolmogorov A. Limit Distributions for Sums of Independent Random Variables 1954, Addison-Wesley.
18. González M.C., Hidalgo C.A., Barabási A.-L. Understanding individual human mobility patterns. Nature 2008, 453:779-782.
19. Gorenflo R., Mainardi F., Scalas E., Raberto R. Fractional calculus and continuous-time finance III. The diffusion limit. Mathematical Finance 2010, 171-180. Birkhauser. M. Kohlmann, S. Tang (Eds.).
20. Hanert E. A comparison of three Eulerian numerical methods for fractional-order transport models. Environmental Fluid Mechanics 2010, 10:7-20. doi:10.1007/s10652-009-9145-4.
21. Hanert, E. On the numerical solution of space-time fractional diffusion models. Computers and Fluids, in press. doi:. http://10.1016/j.compfluid.2010.08.010.
22. Hufnagel L., Brockmann D., Geisel T. Forecast and control of epidemics in a globalized world. Proceedings of the National Academy of Sciences of the United States of America 2004, 101:15124-15129.
23. Humphries N.E., Queiroz N., Dyer J.R.M., Pade N.G., Musyl M.K., Schaefer K.M., Fuller D.W., Brunnschweiler J.M., Doyle T.K., Houghton J.D.R., Hays G.C., Jones C.S., Noble L.R., Wearmouth V.J., Southall E.J., Sims D.W. Environmental context explains Lévy and Brownian movement patterns of marine predators. Nature 2010, 465:1066-1069.
24. Kim S., Kavvas M.L. Generalized Fick's law and fractional ADE for pollution transport in a river: detailed derivation. Journal of Hydrologic Engineering 2006, 11:80-83.
25. Lévy, P., 1954. Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars.
26. Mainardi F., Luchko Y., Pagnini G. The fundamental solution of the space-time fractional diffusion equation. Fractional Calculus and Applied Analysis 2001, 4:153-192.
27. Mainardi F., Raberto M., Gorenflo R., Scalas E. Fractional calculus and continuous-time finance II: the waiting-time distribution. Physica A: Statistical Mechanics and its Applications 2000, 287:468-481.
28. Metzler R., Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics Reports 2000, 339:1-77.
29. Murray J. Mathematical Biology. II: Spatial Models and Biomedical Applications 2002, Springer-Verlag, New York.
30. Noble J.V. Geographic and temporal development of plagues. Nature 1974, 250:726-729.
31. Oldham K., Spagnier J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order 1974, Academic Press.
32. Pachepsky Y., Timlin D., Rawls W. Generalized Richards' equation to simulate water transport in unsaturated soils. Journal of Hydrology 2003, 272:3-13.
33. Podlubny I. Fractional Differential Equations. Mathematics in Science and Engineering 1999, vol. 198. Academic Press.
34. Podlubny I., Chechkin A., Skovranek T., Chen Y., Jara B.M.V. Matrix approach to discrete fractional calculus II: partial fractional differential equations. Journal of Computational Physics 2009, 228:3137-3153.
35. Scalas E., Gorenflo R., Mainardi F. Fractional calculus and continuous-time finance. Physica A: Statistical Mechanics and its Applications 2000, 284:376-384.
36. Sims D.W., Southall E.J., Humphries N.E., Hays G.C., Bradshaw C.J.A., Pitchford J.W., James A., Ahmed M.Z., Brierley A.S., Hindell M.A., Morritt D., Musyl M.K., Righton D., Shepard E.L.C., Wearmouth V.J., Wilson R.P., Witt M.J., Metcalfe J.D. Scaling laws of marine predator search behaviour. Nature 2008, 451:1098-1102.
37. Small M., Walker D.M., Tse C.K. Scale-free distribution of avian influenza outbreaks. Physical Review Letters 2007, 99.
38. van Saarloos W. Front propagation into unstable states. Physics Reports 2003, 386:29-222.
39. Viswanathan G.M., Afanasyev V., Buldyrev S.V., Murphy E.J., Prince P.A., Stanley H.E. Lévy flight search patterns of wandering albatrosses. Nature 1996, 381:413-415.
40. Viswanathan G.M., Buldyrev S.V., Havlin S., Da Luz M.G.E., Raposo E.P., Stanley H.E. Optimizing the success of random searches. Nature 1999, 401:911-914.