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September  2020, 16(5): 2175-2193. doi: 10.3934/jimo.2019049

A stochastic model of contagion with different individual types

Geofferey Jiyun Kim 1, , Jerim Kim 2,, and  Bara Kim 3,

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

* Corresponding author

Received  April 2018 Revised  December 2018 Published  September 2020 Early access  May 2019

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.

Keywords: SIS stochastic models, contagion, epidemic.
Mathematics Subject Classification: 92D30, 91B70.
Citation: Geofferey Jiyun Kim, Jerim Kim, Bara Kim. A stochastic model of contagion with different individual types. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2175-2193. doi: 10.3934/jimo.2019049
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.

Figure 1.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 1
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Figure 2.  The probability mass functions of the number of individual afflictions in Application 1
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Figure 3.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 2
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Figure 4.  The probability mass functions of the number of individual afflictions in Application 2
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Figure 5.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 3
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Figure 6.  The probability mass functions of the number of individual afflictions in Application 3
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Table 1.  Parameter values for Application 1
$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(ⅰ) 0.2 0.1313 2.625 1.25 1 1
(ⅱ) 0.15 0.1313 2.625 2.25 1 1
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$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(ⅰ) 0.2 0.1313 2.625 1.25 1 1
(ⅱ) 0.15 0.1313 2.625 2.25 1 1
Table 2.  Parameter values of Application 2
$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(ⅰ) 0.25 0.1313 2.625 0.25 1 1
(ⅱ) 0.25 0.1313 2.625 0.25 1.02 0.6
Table Options

Download as excel

$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(ⅰ) 0.25 0.1313 2.625 0.25 1 1
(ⅱ) 0.25 0.1313 2.625 0.25 1.02 0.6
Table 3.  Parameter values of Application 3
$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(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
Table Options

Download as excel

$ \beta_{11} $ $ \beta_{12} $ $ \beta_{21} $ $ \beta_{22} $ $ \gamma_{1} $ $ \gamma_{2} $
(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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