American Institute of Mathematical Sciences

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2010, 7(4): 809-823. doi: 10.3934/mbe.2010.7.809

An application of queuing theory to SIS and SEIS epidemic models

Carlos M. Hernández-Suárez 1, , Carlos Castillo-Chavez 2, , Osval Montesinos López 1, and  Karla Hernández-Cuevas 1,

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

Received  February 2010 Revised  May 2010 Published  October 2010

In this work we consider every individual of a population to be a server whose state can be either busy (infected) or idle (susceptible). This server approach allows to consider a general distribution for the duration of the infectious state, instead of being restricted to exponential distributions. In order to achieve this we first derive new approximations to quasistationary distribution (QSD) of SIS (Susceptible- Infected- Susceptible) and SEIS (Susceptible- Latent- Infected- Susceptible) stochastic epidemic models. We give an expression that relates the basic reproductive number, $R_0$ and the server utilization, $\rho$.
Keywords: SIS; SEIS; Queuing theory; $R_{0}$; basic reproductive number; stochastic epidemic models..
Mathematics Subject Classification: Primary: 92B05; Secondary: 62J2.
Citation: Carlos M. Hernández-Suárez, Carlos Castillo-Chavez, Osval Montesinos López, Karla Hernández-Cuevas. An application of queuing theory to SIS and SEIS epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (4) : 809-823. doi: 10.3934/mbe.2010.7.809
References:
[1]

S. Ross, "Introduction to Probability Models,", Academic Press, (2007).   Google Scholar

[2]

D. Kendall, Some problems in the theory of queues,, Journal of the Royal Statistical Society Series B (Methodological), 13 (1951), 151.   Google Scholar

[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.  doi: doi:10.1214/aoms/1177728975.  Google Scholar

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T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic,, J. Appl. Probab., 20 (1983), 390.  doi: doi:10.2307/3213811.  Google Scholar

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F. Ball and P. Donnelly, Strong approximations for epidemic models,, Stochastic Processes and Their Applications, 55 (1995), 1.  doi: doi:10.1016/0304-4149(94)00034-Q.  Google Scholar

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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.  doi: doi:10.1016/j.mbs.2009.02.001.  Google Scholar

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H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic,, J. Appl. Probab., 35 (1998), 662.   Google Scholar

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H. Andersson and T. Britton, Stochastic epidemics in dynamic populations: Quasi-stationarity and extinction,, J. Math. Biol., 41 (2000), 559.  doi: doi:10.1007/s002850000060.  Google Scholar

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J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains,, J. Appl. Probability, 4 (1967), 192.  doi: doi:10.2307/3212311.  Google Scholar

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J. Cavender, Quasi-stationary distributions of birth-and-death processes,, Advances in Applied Probability, 10 (1978), 570.  doi: doi:10.2307/1426635.  Google Scholar

[13]

R. J. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic,, J. Appl. Probab., 26 (1989), 685.  doi: doi:10.2307/3214374.  Google Scholar

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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.   Google Scholar

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R. H. Norden, On the distribution of the time to extinction in the stochastic logistic population model,, Adv. in Appl. Probab., 14 (1982), 687.  doi: doi:10.2307/1427019.  Google Scholar

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I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic,, Math. Biosci., 156 (1999), 21.  doi: doi:10.1016/S0025-5564(98)10059-7.  Google Scholar

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G. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic,, Math. Biosci., 11 (1971), 261.  doi: doi:10.1016/0025-5564(71)90087-3.  Google Scholar

[18]

D. J. Bartholomew, Continuous time diffusion models with random duration of interest,, J. Mathematical Sociology, 4 (1976), 187.  doi: doi:10.1080/0022250X.1976.9989853.  Google Scholar

[19]

O. Ovaskainen, The quasistationary distribution of the stochastic logistic model,, J. Appl. Probab., 38 (2001), 898.  doi: doi:10.1239/jap/1011994180.  Google Scholar

[20]

I. Nåsell, On the quasi-stationary distribution of the Ross malaria model,, Mathematical Biosciences, 107 (1991).  doi: doi:10.1016/0025-5564(91)90004-3.  Google Scholar

[21]

I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: doi:10.1006/jtbi.2001.2328.  Google Scholar

[22]

I. Nåsell, Stochastic models of some endemic infections,, Math. Biosci., 179 (2002), 1.  doi: doi:10.1016/S0025-5564(02)00098-6.  Google Scholar

[23]

C. Hernández-Suárez and C. Castillo-Chavez, A basic result on the integral for birth-death Markov processes,, Mathematical Biosciences, 161 (1999), 95.  doi: doi:10.1016/S0025-5564(99)00034-6.  Google Scholar

[24]

V. T. Stefanov and S. Wang, A note on integrals for birth-death processes,, Math. Biosci., 168 (2000), 161.  doi: doi:10.1016/S0025-5564(00)00046-8.  Google Scholar

[25]

F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes,, J. Appl. Probab., 38 (2001), 995.  doi: doi:10.1239/jap/1011994187.  Google Scholar

[26]

M. VanHoorn, Algorithms and approximations for queueing systems,, CWI Tract No. 8, (1984).   Google Scholar

[27]

D. Cox, "Renewal Theory,", Monographs on Applied Probability and Statistics, (1962).   Google Scholar

show all references

References:
[1]

S. Ross, "Introduction to Probability Models,", Academic Press, (2007).   Google Scholar

[2]

D. Kendall, Some problems in the theory of queues,, Journal of the Royal Statistical Society Series B (Methodological), 13 (1951), 151.   Google Scholar

[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.  doi: doi:10.1214/aoms/1177728975.  Google Scholar

[4]

M. Kitaev, The M/G/1 processor-sharing model: Transient behavior,, Queueing Systems, 14 (1993), 239.  doi: doi:10.1007/BF01158868.  Google Scholar

[5]

H. Andersson and T. Britton, "Stochastic Epidemic Models and Their Statistical Analysis,", Springer Verlag, (2000).   Google Scholar

[6]

T. Sellke, On the asymptotic distribution of the size of a stochastic epidemic,, J. Appl. Probab., 20 (1983), 390.  doi: doi:10.2307/3213811.  Google Scholar

[7]

F. Ball and P. Donnelly, Strong approximations for epidemic models,, Stochastic Processes and Their Applications, 55 (1995), 1.  doi: doi:10.1016/0304-4149(94)00034-Q.  Google Scholar

[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.  doi: doi:10.1016/j.mbs.2009.02.001.  Google Scholar

[9]

H. Andersson and B. Djehiche, A threshold limit theorem for the stochastic logistic epidemic,, J. Appl. Probab., 35 (1998), 662.   Google Scholar

[10]

H. Andersson and T. Britton, Stochastic epidemics in dynamic populations: Quasi-stationarity and extinction,, J. Math. Biol., 41 (2000), 559.  doi: doi:10.1007/s002850000060.  Google Scholar

[11]

J. N. Darroch and E. Seneta, On quasi-stationary distributions in absorbing continuous-time finite Markov chains,, J. Appl. Probability, 4 (1967), 192.  doi: doi:10.2307/3212311.  Google Scholar

[12]

J. Cavender, Quasi-stationary distributions of birth-and-death processes,, Advances in Applied Probability, 10 (1978), 570.  doi: doi:10.2307/1426635.  Google Scholar

[13]

R. J. Kryscio and C. Lefèvre, On the extinction of the S-I-S stochastic logistic epidemic,, J. Appl. Probab., 26 (1989), 685.  doi: doi:10.2307/3214374.  Google Scholar

[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.   Google Scholar

[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.  doi: doi:10.2307/1427019.  Google Scholar

[16]

I. Nåsell, On the quasi-stationary distribution of the stochastic logistic epidemic,, Math. Biosci., 156 (1999), 21.  doi: doi:10.1016/S0025-5564(98)10059-7.  Google Scholar

[17]

G. Weiss and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic,, Math. Biosci., 11 (1971), 261.  doi: doi:10.1016/0025-5564(71)90087-3.  Google Scholar

[18]

D. J. Bartholomew, Continuous time diffusion models with random duration of interest,, J. Mathematical Sociology, 4 (1976), 187.  doi: doi:10.1080/0022250X.1976.9989853.  Google Scholar

[19]

O. Ovaskainen, The quasistationary distribution of the stochastic logistic model,, J. Appl. Probab., 38 (2001), 898.  doi: doi:10.1239/jap/1011994180.  Google Scholar

[20]

I. Nåsell, On the quasi-stationary distribution of the Ross malaria model,, Mathematical Biosciences, 107 (1991).  doi: doi:10.1016/0025-5564(91)90004-3.  Google Scholar

[21]

I. Nåsell, Extinction and quasi-stationarity in the Verhulst logistic model,, Journal of Theoretical Biology, 211 (2001), 11.  doi: doi:10.1006/jtbi.2001.2328.  Google Scholar

[22]

I. Nåsell, Stochastic models of some endemic infections,, Math. Biosci., 179 (2002), 1.  doi: doi:10.1016/S0025-5564(02)00098-6.  Google Scholar

[23]

C. Hernández-Suárez and C. Castillo-Chavez, A basic result on the integral for birth-death Markov processes,, Mathematical Biosciences, 161 (1999), 95.  doi: doi:10.1016/S0025-5564(99)00034-6.  Google Scholar

[24]

V. T. Stefanov and S. Wang, A note on integrals for birth-death processes,, Math. Biosci., 168 (2000), 161.  doi: doi:10.1016/S0025-5564(00)00046-8.  Google Scholar

[25]

F. Ball and V. T. Stefanov, Further approaches to computing fundamental characteristics of birth-death processes,, J. Appl. Probab., 38 (2001), 995.  doi: doi:10.1239/jap/1011994187.  Google Scholar

[26]

M. VanHoorn, Algorithms and approximations for queueing systems,, CWI Tract No. 8, (1984).   Google Scholar

[27]

D. Cox, "Renewal Theory,", Monographs on Applied Probability and Statistics, (1962).   Google Scholar

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