Extinction and quasistationarity in the stochastic logistic SIS model

Lecture Notes in Mathematics

Volumn 2022, Issue , 2011, Pages 1-212

  Nåsell, Ingemar ^a  

^a ROYAL INSTITUTE OF TECHNOLOGY   (Sweden)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79960908138     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-20530-9_1     Document Type: Article

References (78)

1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965).
2. L. J. S. Allen, An Introduction to Stochastic Processes with Applications to Biology, Pearson Education Inc., Upper Saddle River, N.J., 2003.
3. H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics 151, Springer Verlag, New York, 2000.
4. H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic, J. Appl. Prob. 35 (1998), 662-670. (Pubitemid 128356773)
5. T.M. Apostol, An elementary view of Euler's summation formula, American Mathematical Monthly, 106, no 5 (1999), 409-418.
6. M. Assaf and B. Meerson, Spectral theory of metastability and extinction in birth-death systems, Physical Review Letters, 97 (2006).
7. N. T. J. Bailey, The Mathematical Theory of Infectious Diseases, Charles Griffin & Company Ltd, London and High Wycombe, 1975.
8. A. D. Barbour, Quasi-stationary distributions in Markov population processes, Adv. Appl. Prob. 8 (1976), 296-314.
9. D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Math. Sociol. 4 (1976), 187-199.
10. M. S. Bartlett, Deterministic and stochastic models for recurrent epidemics, In: Proceedings of the Third Berkeley Symposium on Mathematics, Statistics and Probability, vol 4, University of Calofornia Press, Berkeley, (1956), 81-109.
11. M. S. Bartlett, On theoretical models for competitive and predatory biological systems, Biometrika 44 (1957), 27-42.
12. M. S. Bartlett, The critical community soze fopr measles in the United States, J. Roy. Stat. Soc. Ser. A 123 (1960), 37-44.
13. M. S. Bartlett, J. C. Gower, and P. H. Leslie, A comparison of theoretical and empirical results for some stochastic population models, Biometrika 47 (1960), 1-11.
14. A. T. Bharucha-Reid, Elements of the Theory of Markov Processes, New York: McGraw-Hill, 1960.
15. B. J. Cairns, J. V. Ross, and T. Taimre, A comparison of models for predicting population persistence, Ecological Modelling 201 (2007), 19-26. (Pubitemid 46054344)
16. J. A. Cavender, Quasi-stationary distributions of birth-and-death processes, Adv. Appl. Prob. 10 (1978), 570-586.
17. M. F. Chen, Eigenvalues, Inequalities, and Ergodic Theorry, Springer, London, 2005.
18. D. Clancy and P. K. Pollett, A note on quasi-stationary distributions of birth-death processes and the SIS logistic epidemic, J. Appl. Prob. 40 (2003), 821-825. (Pubitemid 37374299)
19. D. Clancy and S. T. Mendy, Approximating the quasi-stationary distribution of the SIS model for endemic infection, Methodol. Comput. Appl. Probab. (2010).
20. D. R. Cox, The continuity correction, Biometrika, 57 No 1 (1970), 217-219.
21. J.N. Darroch and E. Sneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl Prob. 4 (1967), 192-196.
22. N.G. de Bruijn, Asymptotic Methods in Analysis, Wiley, New York, 1957.
23. R. Dickman and R. Vidigal, Quasi-stationary distributions for stochastic processes with an absorbing state, J. Phys. A: Math. Gen. 35 (2002), 1147-1166. (Pubitemid 37026735)
24. C.R. Doering, K.V. Sargsyan and L.M. Sander, Extinction times for birth-death processes: Exact results, continuum asymptotics, and the failure of the Fokker-Planck approximation, SIAM J. Multiscale Modeling and Sim. 3 (2005), 283-299. (Pubitemid 40951973)
25. R.G. Dolgoarshinnykh and S.P. Lalley, Critical scaling for the SIS stochastic epidemic, J. Appl. Prob., 43 (2006), 892-898.
26. W. Feller, Die Grundlagen des Volterraschen Theorie des Kampfes ums Dasein in wahrscheinlichkeitstheoretischer Behandlung, Acta Biotheoretica, 5 (1939), 11-40.
27. P. Ferrari, H. Kesten, S. Martínez, and P. Picco, Existence of quasi-stationary distributions. A renewal dynamic approach, Ann. Probab. 23 (1995), 501-521.
28. P. Flajolet, P. J. Grabner, P. Kirschenhofer, and H. Prodinger On Ramanujan's Q-function J. Comp. Appl. Math. 58 (1995), 103-116.
29. J.-P. Gabriel, F. Saucy, and L.-F. Bersier, Paradoxes in the logistic equation?, Ecological Modelling, 85 (2005), 147-151. (Pubitemid 40483347)
30. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Springer Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo, Hong Kong, Barcelona, Budapest, 1985.
31. J. Grasman, Stochastic epidemics: The expected duration of the endemic period in higher dimensional models, Math. Biosci. 152 (1998), 13-27. (Pubitemid 28396972)
32. T. E. Harris, Contact interactions on a lattice, Ann. Prob. 2, 969-988, 1974.
33. M. Iosifescu and P. Tautu, Stochastic Processes and Applications in Biology and Medicine, Vol. II. Berlin: Springer Verlag, 1973.
34. S. Karlin and H.M. Taylor, A First Course in Stochastic Processes, Academic Press, New York, 1975.
35. D. G. Kendall, Stochastic processes and population growth, J. Roy. Stat. Soc. Series B 11, 230-264, 1949.
36. M. Kijima, Markov Processes for Stochastic Modeling, Chapman & Hall, London, 1997.
37. M. Kijima and E. Seneta, Some Results for Quasi-Stationary Distributions of Birth-Death Processes, J. Appl. Prob. 28 No. 3 (1991), 503-511.
38. D. E. Knuth, The Art of Computer Programming. Volume 1: Fundamental Algorithms, Addison-Wesley, Reaading, MA, 1973.
39. R. J. Kryscio and C. Lefèvre, On the extinction of the SIS stochastic logistic epidemic, J. Appl. Prob. 27 (1989), 685-694.
40. J.H. Matis and T.R. Kiffe, On approximating the moments of the equilibrium distribution of a stochastic logistic model, Biometrics 52 No 3 (1996), 980-991. (Pubitemid 26296075)
41. R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, N.J. 1974.
42. I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Math. Biosci. 107 No 2 (1991), 187-207. (Pubitemid 23396929)
43. I. Nåsell, The threshold concept in stochastic epidemic and endemic models, In Epidemic Models: Their Structure and Relation to Data (D. Mollison, ed.), Publications of the Newton Institute, Cambridge University Press, Cambridge, 1995, 71-83.
44. I. Nåsell, The quasi-stationary distribution of the closed endemic SIS model, Adv. Appl. Prob. 28 (1996), 895-932. (Pubitemid 126499161)
45. I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic,Math. Biosci. 156 (1999a), 21-40. (Pubitemid 29157684)
46. I. Nåsell, On the time to extinction in recurrent epidemics, J. Roy. Statist. Soc. Series B 61 (1999b), 309-330.
47. I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, J. Theor. Biol. 211 (2001), 11-27. (Pubitemid 32619945)
48. I. Nåsell, Endemicity, persistence, and quasi-stationarity, Mathematical Approaches for Emerging and Reemerging Infectious Diseases, (C Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner, A. Yakubu, eds.), The IMA Volumes in Mathematics and its Applications, Volume 125, Springer Verlag, New York, 2002, 199-227.
49. I. Nåsell, Moment closure and the stochastic logistic model, Theor. Pop. Biol. 63 (2003) 159-168. (Pubitemid 38042776)
50. I. Nåsell, A new look at the critical community size for childhood infections, Theor. Pop. Biol. 67 (2005) 203-216. (Pubitemid 40451539)
51. T.J. Newman and J-B Ferdy and C Quince, Extinction times and moment closure in the stochastic logistic process, Theor. Pop. Biol. 65 (2004), 115-126. (Pubitemid 38215110)
52. R.M. Nisbet andW. S. C. Gurney, Modelling Fluctuating Populations,Wiley, New York, 1982.
53. R. H. Norden, On the distribution of the time to extinction in the stochastic logistic ppopulation model, Adv. Appl. Prob. 14 (1982), 687-708.
54. F. W. J. Olver, Asymptotics and Special Functions, Academic Press, New York, San Fransisco, London, 1974.
55. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, eds., NIST Handbook of Mathematical Functions, http://dlmf.nist.gov, 2010.
56. I. Oppenheim, K. E. Shuler, and G. H. Weiss, Stochastic theory of nonlinear rate processes with multiple stationary states, Physica 88A (1977), 191-214.
57. O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J.Appl. Prob., 38 (2001), 898-907. (Pubitemid 33390105)
58. L. Pearl and L. J. Reed, On the rate of growth of the population of the United States since 1790 and its mathematical representation, Proc. Natl. Acad. of Sci. USA 6, 275-288, 1920.
59. P. K. Pollett, Reversibility, invariance, and μ-invariance, Adv. Appl. Prob. 20 (1988), 600-621.
60. P. K. Pollett, The generalized Kolmogorov criterion, Stoch. Proc Appl. 33 (1989), 29-44.
61. P. K. Pollett, Quasi-Stationary Distributions: A Bibliography, http://www.maths.uq.edu.au/̃pkp/papers/qsds/qsds.pdf, 2010.
62. B. J. Prendiville, Discussion on symposium on stochastic processes, J. Roy. Stat. Soc. Ser. B 11, 273, 1949.
63. S. Ramanujan, Question 294, J. Indian Math. Soc. 3 (1911), 128.
64. E. Renshaw, Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge, 1991.
65. L. M. Ricciardi, Stochastic population theory: birth and death processes, Mathematical Ecology, an Introduction (Hallam & Levin, eds.) 17, 155-190, 1980, Berlin: Springer Verlag.
66. M. Schmitz, Quasi-stationarität in einem epidemiologischen Modell, http://wwwmath.uni-muenster.de/statistik/alsmeyer /Diplomarbeiten/Schmitz- Manuela.pdf, 2006.
67. E. Seneta, Quasi-stationary behavior in the random walk with continuous time, Austr. J. Statist. 8 (1966), 92-98.
68. A. Singh and J.P. Hespanha, A derivative matching approach to moment closure for the stochastic logistic model, Bull. Math. Biol. 69 no6 (2007), 1909-1925. (Pubitemid 47222244)
69. D. Sirl, H. Zhang, and P. Pollett, Computable bounds for the decay parameter of a birth-death process, J. Appl. Prob. 44 (2007), 476-491.
70. M. Takashima, Note on evolutionary processes, Bull. Math. Stat. 7, 18-24, 1957.
71. H. M. Taylor and S. Karlin, An Introduction to Stochastic Modeling, Academic Press, San Diego, 1993.
72. A. Tsoularis and J. Wallace, Analysis of logistic growth models, Math. Biosci. 179 (2002), 21-55. (Pubitemid 34686426)
73. E. A. van Doorn, Quasi-stationary distributions and convergence to quasi-stationarity of birthdeath processes, Adv. Appl. Prob. 23 (1991), 683-700.
74. P. F. Verhulst, Notice sur la loi que la population suit dans son accroisement, Corr. Math. Phys. X, 113-121, 1838.
75. P. F. Verhulst, Recherches mathématiques sur la loi d'accroissement de la population, Noveau Mémoires de l'Academie Royale des Sciences et Belles-Lettres de Bruxelles, 18, 1-41, 1845.
76. G. H. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci. 11 (1971), 261-265.
77. P. Whittle, On the use of the normal approximation in the treatment of stochastic processes, J. Roy. Stat. Soc. Ser B 19, 268-281, 1957.
78. A. M. Yaglom, Certain limit theorems of the theory of branching processes (in Russian), Dokl. Akad. Nauk SSSR 56 (1947), 795-798.