American Institute of Mathematical Sciences

Advanced
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  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 & 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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The probability mass functions of the number of individual afflictions in Application 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  The probability mass functions of the number of individual afflictions in Application 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  The probability density functions and the complementary cumulative distribution functions of the contagion duration in Application 3
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  The probability mass functions of the number of individual afflictions in Application 3
Figure Options

Download full-size image

Download as PowerPoint slide

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
Table Options

Download as excel

$ \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
[1]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201
[2]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175
[3]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223
[4]

Cicely K. Macnamara, Mark A. J. Chaplain. Spatio-temporal models of synthetic genetic oscillators. Mathematical Biosciences & Engineering, 2017, 14 (1) : 249-262. doi: 10.3934/mbe.2017016
[5]

Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053
[6]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817
[7]

Shanshan Chen, Junping Shi, Guohong Zhang. Spatial pattern formation in activator-inhibitor models with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 1843-1866. doi: 10.3934/dcdsb.2020042
[8]

Alina Chertock, Alexander Kurganov, Mária Lukáčová-Medvi${\rm{\check{d}}}$ová, Șeyma Nur Özcan. An asymptotic preserving scheme for kinetic chemotaxis models in two space dimensions. Kinetic & Related Models, 2019, 12 (1) : 195-216. doi: 10.3934/krm.2019009
[9]

J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008
[10]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[11]

Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208
[12]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[13]

Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023
[14]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192
[15]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027
[16]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637
[17]

Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004
[18]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[19]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[20]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (144)
  • HTML views (647)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences