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

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

* Corresponding author

Received  April 2018 Revised  December 2018 Published  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.

Citation: Geofferey Jiyun Kim, Jerim Kim, Bara Kim. A stochastic model of contagion with different individual types. Journal of Industrial & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Clancy, Strong approximations for mobile population epidemic models, Ann. Appl. Probab., 6 (1996), 883-895.  doi: 10.1214/aoap/1034968231.  Google Scholar

[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.  Google Scholar

[7]

A. EconomouA. 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.  Google Scholar

[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.  Google Scholar

[9]

L. F. GordilloS. A. MarionA. 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.  Google Scholar

[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.  Google Scholar

[11]

I. Z. KissC. G. MorrisF. SélleyP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. W. Nickerson, Is voting contagious? Evidence from two field experiments, American Political Science Review, 102 (2008), 49-57.  doi: 10.1017/S0003055408080039.  Google Scholar

[15]

A. SaniD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

D. Clancy, Strong approximations for mobile population epidemic models, Ann. Appl. Probab., 6 (1996), 883-895.  doi: 10.1214/aoap/1034968231.  Google Scholar

[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.  Google Scholar

[7]

A. EconomouA. 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.  Google Scholar

[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.  Google Scholar

[9]

L. F. GordilloS. A. MarionA. 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.  Google Scholar

[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.  Google Scholar

[11]

I. Z. KissC. G. MorrisF. SélleyP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

D. W. Nickerson, Is voting contagious? Evidence from two field experiments, American Political Science Review, 102 (2008), 49-57.  doi: 10.1017/S0003055408080039.  Google Scholar

[15]

A. SaniD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 1
Figure 2.  The probability mass functions of the number of individual afflictions in Application 1
Figure 3.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 2
Figure 4.  The probability mass functions of the number of individual afflictions in Application 2
Figure 5.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 3
Figure 6.  The probability mass functions of the number of individual afflictions in Application 3
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
$ \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
$ \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
$ \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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