Spatial epidemics: Critical behavior in one dimension

Probability Theory and Related Fields

Volumn 144, Issue 3-4, 2009, Pages 429-469

  Lalley, Steven P ^a  

^a UNIVERSITY OF CHICAGO   (United States)

Author keywords

Branching random walk; Critical scaling; Dawson Watanabe process; Spatial epidemic

Indexed keywords

EID: 64749091340     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-008-0151-0     Document Type: Article

References (28)

1. D. Aldous 1997 Brownian excursions, critical random graphs and the multiplicative coalescent Ann. Probab. 25 2 812 854
2. Athreya, K.B., Ney, P.E.: Branching Processes. Springer, New York (1972). Die Grundlehren der mathematischen Wissenschaften, Band 196
3. L. Belhadji N. Lanchier 2006 Individual versus cluster recoveries within a spatially structured population Ann. Appl. Probab. 16 1 403 422
4. Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. Wiley-Interscience, New York (1999)
5. J.T. Cox R. Durrett E.A. Perkins 2000 Rescaled voter models converge to super-Brownian motion Ann. Probab. 28 1 185 234
6. D.A. Dawson 1978 Geostochastic calculus Canad. J. Statist. 6 2 143 168
7. D.A. Dawson K.J. Hochberg 1979 The carrying dimension of a stochastic measure diffusion Ann. Probab. 7 4 693 703
8. Dawson, D.A., Perkins, E.A.: Measure-valued processes and renormalization of branching particle systems. In: Stochastic Partial Differential Equations: six perspectives, Math. Surveys Monogr., vol. 64, pp. 45-106. American Mathematical Society, Providence, RI (1999)
9. R. Dolgoarshinnykh S.P. Lalley 2006 Critical scaling for the sis stochastic epidemic J. Appl. Probab. 43 892 898
10. R. Durrett 1999 Stochastic spatial models SIAM Rev. 41 4 677 718 (electronic)
11. R. Durrett L. Mytnik E. Perkins 2005 Competing super-Brownian motions as limits of interacting particle systems Electron. J. Probab. 10 35 1147 1220 (electronic)
12. R. Durrett E.A. Perkins 1999 Rescaled contact processes converge to super-Brownian motion in two or more dimensions Probab. Theory Related Fields 114 3 309 399
13. Etheridge, A.M.: An introduction to superprocesses. University Lecture Series, vol. 20. American Mathematical Society, Providence, RI (2000)
14. S.N. Evans E.A. Perkins 1995 Explicit stochastic integral representations for historical functionals Ann. Probab. 23 4 1772 1815
15. K. Fleischmann 1988 Critical behavior of some measure-valued processes Math. Nachr. 135 131 147
16. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus. Graduate Texts in Mathematics, vol. 113. Springer, New York (1991)
17. H. Kesten 1995 Branching random walk with a critical branching part J. Theoret. Probab. 8 4 921 962
18. N. Konno T. Shiga 1988 Stochastic partial differential equations for some measure-valued diffusions Probab. Theory Related Fields 79 2 201 225
19. A. Martin-Löf 1998 The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier J. Appl. Probab. 35 3 671 682
20. D. Mollison 1977 Spatial contact models for ecological and epidemic spread J. R. Statist. Soc. Ser. B 39 3 283 326
21. C. Müller R. Tribe 1995 Stochastic p.d.e.'s arising from the long range contact and long range voter processes Probab. Theory Related Fields 102 4 519 545
22. R. Schinazi 2002 On the role of social clusters in the transmission of infectious diseases Theor. Pop. Biol. 61 163 169
23. T. Shiga 1994 Two contrasting properties of solutions for one-dimensional stochastic partial differential equations Can. J. Math. 46 2 415 437
24. S. Sugitani 1989 Some properties for the measure-valued branching diffusion processes J. Math. Soc. Jpn 41 3 437 462
25. Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d'été de probabilités de Saint-Flour, XIV-1984. Lecture Notes in Mathematics, vol. 1180, pp. 265-439. Springer, Berlin (1986)
26. S. Watanabe 1968 A limit theorem of branching processes and continuous state branching processes J. Math. Kyoto Univ. 8 141 167
27. Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1996. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions, Reprint of the fourth edition (1927)
28. Zheng, X.: Spatial epidemics in higher dimensions. Ph.D. thesis, University of Chicago (in progress)