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Strategic inventory under suppliers competition
A stochastic model of contagion with different individual types
1. | Underwood International College, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul, 03722, Korea |
2. | Department of Mathematics, University of Seoul, 163 Seoulsiripdae-ro, Dongdaemun-gu, Seoul, 02504, Korea |
3. | Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Korea |
We develop a stochastic model of contagion with two individual types by extending an archetypal SIS CTMC model. Our results include the analyses of the contagion duration and the number of individual afflictions. Numerical applications with the minority and majority types are provided to consider various contagions.
References:
[1] |
L. J. S. Allen,
Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[2] |
L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008,179–189.
doi: 10.1007/978-3-540-78911-6. |
[3] |
F. Ball and P. Neal,
Network epidemic models with two levels of mixing, Math. Biosci., 212 (2008), 69-87.
doi: 10.1016/j.mbs.2008.01.001. |
[4] |
F. Brauer, P. van den Driessche and J. Wu (eds.), Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008.
doi: 10.1007/978-3-540-78911-6. |
[5] |
D. Clancy,
Strong approximations for mobile population epidemic models, Ann. Appl. Probab., 6 (1996), 883-895.
doi: 10.1214/aoap/1034968231. |
[6] |
D. J. D. Earn, A light introduction to modelling recurrent epidemics, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008, 3–17.
doi: 10.1007/978-3-540-78911-6. |
[7] |
A. Economou, A. Gómez-Corral and M. López-García,
A stochastic SIS epidemic model with heterogeneous contacts, Phys. A., 421 (2015), 78-97.
doi: 10.1016/j.physa.2014.10.054. |
[8] |
J. H. Fowler and N. A. Christakis, Dynamic spread of happiness in a large social network: Longitudinal analysis over 20 years in the Framingham heart study, British Medical Journal, 337 (2008), a2338.
doi: 10.1136/bmj.a2338. |
[9] |
L. F. Gordillo, S. A. Marion, A. Martin-Löf and P. E. Greenwood,
Bimodal epidemic size distributions for near-critical SIR with vaccination, Bulletin of Mathematical Biology, 70 (2008), 589-602.
doi: 10.1007/s11538-007-9269-y. |
[10] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London Series A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[11] |
I. Z. Kiss, C. G. Morris, F. Sélley, P. L. Simon and R. R. Wilkinson,
Exact deterministic representation of Markovian SIR epidemics, J. Math. Biol., 70 (2015), 437-464.
doi: 10.1007/s00285-014-0772-0. |
[12] |
G. E. Lahodny Jr. and L. J. S. Allen,
Probability of a disease outbreak in stochastic multipatch epidemic models, Bull. Math. Biol., 75 (2013), 1157-1180.
doi: 10.1007/s11538-013-9848-z. |
[13] |
A. Martin-Löf,
The final size of a nearly critical epidemic and the first passage time of a Wiener process to a parabolic barrier, Journal of the Applied Probability, 35 (1998), 671-682.
doi: 10.1239/jap/1032265215. |
[14] |
D. W. Nickerson,
Is voting contagious? Evidence from two field experiments, American Political Science Review, 102 (2008), 49-57.
doi: 10.1017/S0003055408080039. |
[15] |
A. Sani, D. P. Kroese and P. K. Pollett,
Stochastic models for the spread of HIV in a mobile heterosexual population, Math. Biosci., 208 (2007), 98-124.
doi: 10.1016/j.mbs.2006.09.024. |
[16] |
R. Schiller and J. Pound,
Survey evidence on diffusion of interest and information among investors, Journal of Economic Behavior and Organization, 12 (1989), 47-66.
doi: 10.1016/0167-2681(89)90076-0. |
[17] |
K. J. Sharkey,
Deterministic epidemiological models at the individual level, J. Math. Biol., 57 (2008), 311-331.
doi: 10.1007/s00285-008-0161-7. |
[18] |
A.-A. Yakubu and J. E. Franke,
Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |
show all references
References:
[1] |
L. J. S. Allen,
Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105.
doi: 10.1016/0025-5564(94)90025-6. |
[2] |
L. J. S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008,179–189.
doi: 10.1007/978-3-540-78911-6. |
[3] |
F. Ball and P. Neal,
Network epidemic models with two levels of mixing, Math. Biosci., 212 (2008), 69-87.
doi: 10.1016/j.mbs.2008.01.001. |
[4] |
F. Brauer, P. van den Driessche and J. Wu (eds.), Mathematical Epidemiology, Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008.
doi: 10.1007/978-3-540-78911-6. |
[5] |
D. Clancy,
Strong approximations for mobile population epidemic models, Ann. Appl. Probab., 6 (1996), 883-895.
doi: 10.1214/aoap/1034968231. |
[6] |
D. J. D. Earn, A light introduction to modelling recurrent epidemics, in Mathematical Epidemiology (eds. F. Brauer, P. van den Driessche and J. Wu), Lecture Notes in Mathematics, Mathematical Biosciences Subseries 1945, Springer-Verlag, Heidelberg, 2008, 3–17.
doi: 10.1007/978-3-540-78911-6. |
[7] |
A. Economou, A. Gómez-Corral and M. López-García,
A stochastic SIS epidemic model with heterogeneous contacts, Phys. A., 421 (2015), 78-97.
doi: 10.1016/j.physa.2014.10.054. |
[8] |
J. H. Fowler and N. A. Christakis, Dynamic spread of happiness in a large social network: Longitudinal analysis over 20 years in the Framingham heart study, British Medical Journal, 337 (2008), a2338.
doi: 10.1136/bmj.a2338. |
[9] |
L. F. Gordillo, S. A. Marion, A. Martin-Löf and P. E. Greenwood,
Bimodal epidemic size distributions for near-critical SIR with vaccination, Bulletin of Mathematical Biology, 70 (2008), 589-602.
doi: 10.1007/s11538-007-9269-y. |
[10] |
W. O. Kermack and A. G. McKendrick,
A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London Series A, 115 (1927), 700-721.
doi: 10.1098/rspa.1927.0118. |
[11] |
I. Z. Kiss, C. G. Morris, F. Sélley, P. L. Simon and R. R. Wilkinson,
Exact deterministic representation of Markovian SIR epidemics, J. Math. Biol., 70 (2015), 437-464.
doi: 10.1007/s00285-014-0772-0. |
[12] |
G. E. Lahodny Jr. and L. J. S. Allen,
Probability of a disease outbreak in stochastic multipatch epidemic models, Bull. Math. Biol., 75 (2013), 1157-1180.
doi: 10.1007/s11538-013-9848-z. |
[13] |
A. Martin-Löf,
The final size of a nearly critical epidemic and the first passage time of a Wiener process to a parabolic barrier, Journal of the Applied Probability, 35 (1998), 671-682.
doi: 10.1239/jap/1032265215. |
[14] |
D. W. Nickerson,
Is voting contagious? Evidence from two field experiments, American Political Science Review, 102 (2008), 49-57.
doi: 10.1017/S0003055408080039. |
[15] |
A. Sani, D. P. Kroese and P. K. Pollett,
Stochastic models for the spread of HIV in a mobile heterosexual population, Math. Biosci., 208 (2007), 98-124.
doi: 10.1016/j.mbs.2006.09.024. |
[16] |
R. Schiller and J. Pound,
Survey evidence on diffusion of interest and information among investors, Journal of Economic Behavior and Organization, 12 (1989), 47-66.
doi: 10.1016/0167-2681(89)90076-0. |
[17] |
K. J. Sharkey,
Deterministic epidemiological models at the individual level, J. Math. Biol., 57 (2008), 311-331.
doi: 10.1007/s00285-008-0161-7. |
[18] |
A.-A. Yakubu and J. E. Franke,
Discrete-time SIS epidemic model in a seasonal environment, SIAM J. Appl. Math., 66 (2006), 1563-1587.
doi: 10.1137/050638345. |






(ⅰ) | 0.2 | 0.1313 | 2.625 | 1.25 | 1 | 1 |
(ⅱ) | 0.15 | 0.1313 | 2.625 | 2.25 | 1 | 1 |
(ⅰ) | 0.2 | 0.1313 | 2.625 | 1.25 | 1 | 1 |
(ⅱ) | 0.15 | 0.1313 | 2.625 | 2.25 | 1 | 1 |
(ⅰ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1 | 1 |
(ⅱ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1.02 | 0.6 |
(ⅰ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1 | 1 |
(ⅱ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1.02 | 0.6 |
(ⅰ) | 0.5 | 0 | 0 | 0.5 | 1 | 1 |
(ⅱ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1 | 1 |
(iii) | 0 | 0.2625 | 5.25 | 0 | 1 | 1 |
(ⅰ) | 0.5 | 0 | 0 | 0.5 | 1 | 1 |
(ⅱ) | 0.25 | 0.1313 | 2.625 | 0.25 | 1 | 1 |
(iii) | 0 | 0.2625 | 5.25 | 0 | 1 | 1 |
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