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An application of queuing theory to SIS and SEIS epidemic models
1. | Facultad de Ciencias, Universidad de Colima, Apdo. Postal 25, Colima, Colima, Mexico, Mexico, Mexico |
2. | Mathematics, Computational and Modeling Sciences Center, Arizona State University PO Box 871904, Tempe, AZ, 85287, United States |
References:
[1] |
S. Ross, "Introduction to Probability Models," Academic Press, 2007. |
[2] |
D. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society Series B (Methodological), 13 (1951) 151-185. |
[3] |
D. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded markov chain, The Annals of Mathematical Statistics, 24 (1953), 338-354.
doi: doi:10.1214/aoms/1177728975. |
[4] |
M. Kitaev, The M/G/1 processor-sharing model: Transient behavior, Queueing Systems, 14 (1993), 239-273.
doi: doi:10.1007/BF01158868. |
[5] |
H. Andersson and T. Britton, "Stochastic Epidemic Models and Their Statistical Analysis," Springer Verlag, 2000. |
[6] |
T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, J. Appl. Probab., 20 (1983), 390-394.
doi: doi:10.2307/3213811. |
[7] |
F. Ball and P. Donnelly, Strong approximations for epidemic models, Stochastic Processes and Their Applications, 55 (1995), 1-21.
doi: doi:10.1016/0304-4149(94)00034-Q. |
[8] |
P. Trapman and M. Bootsma, A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection, Mathematical Biosciences, 219 (2009), 15-22.
doi: doi:10.1016/j.mbs.2009.02.001. |
[9] |
H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic, J. Appl. Probab., 35 (1998), 662-670, http://projecteuclid.org/getRecord?id=euclid.jap/1032265214. |
[10] |
H. Andersson and T. Britton, Stochastic epidemics in dynamic populations: Quasi-stationarity and extinction, J. Math. Biol., 41 (2000), 559-580.
doi: doi:10.1007/s002850000060. |
[11] |
J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Probability, 4 (1967), 192-196.
doi: doi:10.2307/3212311. |
[12] |
J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, 10 (1978), 570-586.
doi: doi:10.2307/1426635. |
[13] |
R. J. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 26 (1989), 685-694.
doi: doi:10.2307/3214374. |
[14] |
W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics-iii. Further studies of the problem of endemicity, Bulletin of Mathematical Biology, 53 (1991), 89-118. |
[15] |
R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. in Appl. Probab., 14 (1982), 687-708.
doi: doi:10.2307/1427019. |
[16] |
I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156 (1999), 21-40, epidemiology, cellular automata and evolution (Sofia, 1997).
doi: doi:10.1016/S0025-5564(98)10059-7. |
[17] |
G. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265.
doi: doi:10.1016/0025-5564(71)90087-3. |
[18] |
D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Mathematical Sociology, 4 (1976), 187-199.
doi: doi:10.1080/0022250X.1976.9989853. |
[19] |
O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907.
doi: doi:10.1239/jap/1011994180. |
[20] |
I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Mathematical Biosciences, 107 (1991), 187.
doi: doi:10.1016/0025-5564(91)90004-3. |
[21] |
I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27.
doi: doi:10.1006/jtbi.2001.2328. |
[22] |
I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.
doi: doi:10.1016/S0025-5564(02)00098-6. |
[23] |
C. Hernández-Suárez and C. Castillo-Chavez, A basic result on the integral for birth-death Markov processes, Mathematical Biosciences, 161 (1999), 95-104.
doi: doi:10.1016/S0025-5564(99)00034-6. |
[24] |
V. T. Stefanov and S. Wang, A note on integrals for birth-death processes, Math. Biosci., 168 (2000), 161-165.
doi: doi:10.1016/S0025-5564(00)00046-8. |
[25] |
F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes, J. Appl. Probab., 38 (2001), 995-1005.
doi: doi:10.1239/jap/1011994187. |
[26] |
M. VanHoorn, Algorithms and approximations for queueing systems, CWI Tract No. 8, CWI, Amsterdam, 1984. |
[27] |
D. Cox, "Renewal Theory," Monographs on Applied Probability and Statistics, Methuen and Co., 1962. |
show all references
References:
[1] |
S. Ross, "Introduction to Probability Models," Academic Press, 2007. |
[2] |
D. Kendall, Some problems in the theory of queues, Journal of the Royal Statistical Society Series B (Methodological), 13 (1951) 151-185. |
[3] |
D. Kendall, Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded markov chain, The Annals of Mathematical Statistics, 24 (1953), 338-354.
doi: doi:10.1214/aoms/1177728975. |
[4] |
M. Kitaev, The M/G/1 processor-sharing model: Transient behavior, Queueing Systems, 14 (1993), 239-273.
doi: doi:10.1007/BF01158868. |
[5] |
H. Andersson and T. Britton, "Stochastic Epidemic Models and Their Statistical Analysis," Springer Verlag, 2000. |
[6] |
T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic, J. Appl. Probab., 20 (1983), 390-394.
doi: doi:10.2307/3213811. |
[7] |
F. Ball and P. Donnelly, Strong approximations for epidemic models, Stochastic Processes and Their Applications, 55 (1995), 1-21.
doi: doi:10.1016/0304-4149(94)00034-Q. |
[8] |
P. Trapman and M. Bootsma, A useful relationship between epidemiology and queueing theory: The distribution of the number of infectives at the moment of the first detection, Mathematical Biosciences, 219 (2009), 15-22.
doi: doi:10.1016/j.mbs.2009.02.001. |
[9] |
H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic, J. Appl. Probab., 35 (1998), 662-670, http://projecteuclid.org/getRecord?id=euclid.jap/1032265214. |
[10] |
H. Andersson and T. Britton, Stochastic epidemics in dynamic populations: Quasi-stationarity and extinction, J. Math. Biol., 41 (2000), 559-580.
doi: doi:10.1007/s002850000060. |
[11] |
J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains, J. Appl. Probability, 4 (1967), 192-196.
doi: doi:10.2307/3212311. |
[12] |
J. Cavender, Quasi-stationary distributions of birth-and-death processes, Advances in Applied Probability, 10 (1978), 570-586.
doi: doi:10.2307/1426635. |
[13] |
R. J. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic, J. Appl. Probab., 26 (1989), 685-694.
doi: doi:10.2307/3214374. |
[14] |
W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics-iii. Further studies of the problem of endemicity, Bulletin of Mathematical Biology, 53 (1991), 89-118. |
[15] |
R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model, Adv. in Appl. Probab., 14 (1982), 687-708.
doi: doi:10.2307/1427019. |
[16] |
I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic, Math. Biosci., 156 (1999), 21-40, epidemiology, cellular automata and evolution (Sofia, 1997).
doi: doi:10.1016/S0025-5564(98)10059-7. |
[17] |
G. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci., 11 (1971), 261-265.
doi: doi:10.1016/0025-5564(71)90087-3. |
[18] |
D. J. Bartholomew, Continuous time diffusion models with random duration of interest, J. Mathematical Sociology, 4 (1976), 187-199.
doi: doi:10.1080/0022250X.1976.9989853. |
[19] |
O. Ovaskainen, The quasistationary distribution of the stochastic logistic model, J. Appl. Probab., 38 (2001), 898-907.
doi: doi:10.1239/jap/1011994180. |
[20] |
I. Nåsell, On the quasi-stationary distribution of the Ross malaria model, Mathematical Biosciences, 107 (1991), 187.
doi: doi:10.1016/0025-5564(91)90004-3. |
[21] |
I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model, Journal of Theoretical Biology, 211 (2001), 11-27.
doi: doi:10.1006/jtbi.2001.2328. |
[22] |
I. Nåsell, Stochastic models of some endemic infections, Math. Biosci., 179 (2002), 1-19.
doi: doi:10.1016/S0025-5564(02)00098-6. |
[23] |
C. Hernández-Suárez and C. Castillo-Chavez, A basic result on the integral for birth-death Markov processes, Mathematical Biosciences, 161 (1999), 95-104.
doi: doi:10.1016/S0025-5564(99)00034-6. |
[24] |
V. T. Stefanov and S. Wang, A note on integrals for birth-death processes, Math. Biosci., 168 (2000), 161-165.
doi: doi:10.1016/S0025-5564(00)00046-8. |
[25] |
F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes, J. Appl. Probab., 38 (2001), 995-1005.
doi: doi:10.1239/jap/1011994187. |
[26] |
M. VanHoorn, Algorithms and approximations for queueing systems, CWI Tract No. 8, CWI, Amsterdam, 1984. |
[27] |
D. Cox, "Renewal Theory," Monographs on Applied Probability and Statistics, Methuen and Co., 1962. |
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