Mathematical techniques have been widely used to model the infection and progress of communicable diseases, especially since the times of Daniel Bernoulli in early 18th century. Another development in the field of modeling has been to model social beliefs, ideas, addictions, etc. which are communicated from one person to another through social influence and pressure. A common pattern visible in such phenomena is that there are people who are unaffected or neutral, who are moderate, and who are severe by the infection under study. Deviating from the usual dynamic system based models, by dividing the population into the above three classes based on the intensity of the infection, in this paper, we propose a general and much simpler game theory approach to model such social contact spread. Based on the concept of an evolutionarily stable population state and using the associated replicator dynamics, we give three illustrative examples.

game theoretic models, evolutionary games, applications of game theory.

Received: April 13, 2021; Accepted: June 3, 2021; Published: November 10, 2021

How to cite this article: Gigi Thomas, A game theoretic approach to the modeling of social contact contagions, Far East Journal of Applied Mathematics 110(2) (2021), 99-118. DOI: 10.17654/AM110020099

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References

[1] N. T. Bailey, A simple stochastic epidemic, Biometrika 37 (1950), 193-202.
[2] N. T. Bailey, The total size of a general stochastic epidemic, Biometrika 40(1-2) (1953), 177-185.
[3] M. S. Bartlett, Some evolutionary stochastic processes, Journal of the Royal Statistical Society. Series B (Methodological) 11(2) (1949), 211-229.
[4] D. Bernoulli, Essai d’une nouvelle analyse de la mortalité causée par lat petite Vérole, & des avantages de l’inoculation pour la prévenir, 1760.
[5] L. M. A. Bettencourt, A. Cintrón-Arias, D. I. Kaiser and C. Castillo-Chávez, The power of a good idea: quantitative modeling of the spread of ideas from epidemiological models, Phys. A 364 (2006), 513-536. DOI 10.1016/j.physa.2005.08.083.
[6] L. M. A. Bettencourt, D. I. Kaiser, J. Kaur, C. Castillo-Chávez and D. E. Wojick, Population modeling of the emergence and development of scientific fields, Scientometrics 75(5) (2008), 395-518. DOI 10.1016/j.physa.2005.08.083.
[7] P. S. Brachman, Infectious diseases past, present, and future, International Journal of Epidemiology 32(5) (2003), 684-686. DOI 10.1093/ije/dyg282.
[8] F. Brauer, Mathematical epidemiology: past, present, and future, Infectious Disease Modelling 2(2) (2017), 113-127. DOI 10.1016/j.idm.2017.02.001.
[9] M. Broom and J. Rychtar, Game-theoretical Models in Biology, 1st ed., Chapman and Hall/CRC, Boca Raton, 2013.
[10] E. Camacho, The development and interaction of terrorist and fanatic groups, Commun. Nonlinear Sci. Numer. Simul. 18(11) (2013), 3086-3097.
[11] C. Castillo-Chavez and B. Song, Models for the transmission dynamics of fanatic behaviors, Bioterrorism: Mathematical Modeling Applications in Homeland Security, H. T. Banks and C. Castillo-Chvez, eds., SIAM 28 (2003), 155-172.
[12] R. Cressman, Evolutionary Dynamics and Extensive Form Games, 1st ed., The MIT Press, Cambridge, 2003.
[13] E. van Damme, Stability and Perfection of Nash Equilibria, 1st ed., Springer-Verlag, Berlin, Heidelberg, 1991.
[14] O. Diekmann, J. Heesterbeek and J. A. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28(4) (1990), 365-382.
[15] P. D. En’ko, On the course of epidemics of some infectious diseases, International Journal of Epidemiology 18(4) (1989), 749-755.
[16] I. M. Foppa, A historical introduction to mathematical modeling of infectious diseases, Seminal Papers in Epidemiology, Academic Press, London, Vol. 197, 2017.
[17] C. Gan, X. Yang, W. Liu, Q. Zhu and X. Zhang, Propagation of computer virus under human intervention: a dynamical model, Discrete Dyn. Nat. Soc. 2012, Art. ID 106950, pp. 1-8. DOI 10.1155/2012/106950.
[18] W. Goffman and V. Newhill, Generalization of epidemic theory: an application to the transmission of ideas, Nature 204 (1964), 225-228.
[19] B. Gonzalez, E. Huerta-Sánchez, A. Ortiz-Nieves, T. Vázquez-Alvarez and C. Kribs-Zaleta, Am i too fat? Bulimia as an epidemic, J. Math. Psych. 47(5-6) (2003), 515-526. DOI 10.1016/j.jmp.2003.08.002.
[20] W. H. Hamer, The Milroy Lectures on Epidemic Disease in England: The Evidence of Variability and of Persistency of Type, Bedford Press, 1906
[21] J. Hayward, Mathematical modeling of church growth, Journal of Mathematical Sociology 23(4) (1999), 255-292. DOI 10.1080/0022250X.1999.9990223.
[22] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, 1st ed., Cambridge University Press, Cambridge, 1998.
[23] R. Isea and R. Mayo-Garcia, Mathematical analysis of the spreading of a rumor among different subgroups of spreaders, Pure and Applied Mathematics Letters 2015 (2015), 50-54.
[24] R. A. Jeffs, J. Hayward, P. A. Roach and J. Wyburn, Activist model of political party growth, Phys. A 442 (2016), 359-372. DOI 10.1016/j.physa.2015.09.002.
[25] F. Jin, E. Dougherty, P. Saraf, Y. Cao and N. Ramakrishnan, Epidemiological modeling of news and rumors on twitter, Proceedings of the 7th Workshop on Social Network Mining and Analysis, 2013.
[26] M. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Part I, Proceedings of the Royal Society. Series A, Containing Papers of a Mathematical and Physical Character 115(5) (1927), 700-721.
[27] M. Kubo, K. Naruse, H. Sato and T. Matsubara, The possibility of an epidemic meme analogy for web community population analysis, Intelligent Data Engineering and Automated Learning - IDEAL 2007, H. Yin, P. Tino, E. Corchado, W. Byrne and X. Yao, eds., Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, Vol. 4881, 2007.
[28] W. Liu, X. Wu, W. Yang and X. Zhu, Modeling cyber rumor spreading over mobile social networks: a compartment approach, Appl. Math. Comput. 343 (2019), 214-229. DOI 10.1016/j.amc.2018.09.048.
[29] G. Macdonald, The analysis of equilibrium in malaria, Tropical Diseases Bulletin 49 (9) (1952) 813–829
[30] B. K. Mishra and D. K. Saini, SEIRS epidemic model with delay for transmission of malicious objects in computer network, Appl. Math. Comput. 188(2) (2007), 1476-1482. DOI 10.1016/j.amc.2006.11.012.
[31] B. K. Mishra and N. Jha, SEIQRS model for the transmission of malicious objects in computer network, Appl. Math. Model. 34(3) (2010), 710-715.
DOI 10.1016/j.apm.2009.06.011.
[32] F. Nyabadza and S. Hove-Musekwa, From heroin epidemics to methamphetamine epidemics: modelling substance abuse in a South African province, Math. Biosci. 225(2) (2010), 132-140. DOI 10.1016/j.mbs.2010.03.002.
[33] P. Ormerod, C. Mounfield and L. Smith, Non-linear modelling of burglary and violent crime in the UK, Modelling Crime and Offending: Recent Developments in England and Wales, C. Lewis, ed., Research, Development and Statistics Directorate (80), 2001.
[34] J. R. C. Piqueira, Rumor propagation model: an equilibrium study, Math. Probl. Eng. 2010 (2010), 1-7. DOI 10.1155/2010/631357.
[35] J. R. C. Piqueira, B. F. Navarro and L. H. A. Monteiro, Epidemiological models applied to viruses in computer networks, Journal of Computer Science 1(1) (2005), 31-34. DOI 10.3844/jcssp.2005.31.34.
[36] D. M. Romero, C. M. Kribs-Zaleta, A. Mubayi and C. Orbe, An epidemiological approach to the spread of political third parties, Discrete Contin. Dyn. Syst. Ser. B 15(3) (2011), 703-738. DOI 10.3934/dcdsb.2011.15.707.
[37] R. Ross, The Prevention of Malaria, Dutton, 1910.
[38] R. Ross, Some quantitative studies in epidemiology, Nature 87 (1911), 466-467.
[39] L. Samuelson, Evolutionary Games and Equilibrium Selection, 1 ed., The MIT Press, Cambridge, 1998.
[40] S. Sharma and G. P. Samanta, Analysis of a drinking epidemic model, Int. J. Dynam. Control 3(3) (2015), 288-305.
[41] O. Sharomi and A. Gumel, Curtailing smoking dynamics: a mathematical modeling approach, Appl. Math. Comput. 195(2) (2008), 475-499. DOI 10.1016/j.amc.2007.05.012.
[42] J. Smith, The theory of games and the evolution of animal conflicts, J. Theoret. Biol. 47(1) (1974), 209-221. DOI 10.1016/0022-5193(74)90110-6.
[43] J. Smith, Evolution and the Theory of Games, 1st ed., Cambridge University Press, Cambridge, 1982.
[44] J. Smith and G. Price, The logic of animal conflict, Nature 246 (1973), 15-18.
DOI 10.1038/246015a0.
[45] J. Sooknanan, B. Bhatt and D. M. G. Comissiong, Life and death in a gang-a mathematical model of gang membership, J. Math. Res. 4(4) (2012), 10-27.
DOI 10.5539/jmr.v4n4p10.
[46] J. Sooknanan and D. M. G. Comissiong, When behaviour turns contagious: the use of deterministic epidemiological models in modeling social contagion phenomena, Int. J. Dynam. Control 5 (2017), 1046-1050. DOI 10.1007/s40435-016-0271-9.
[47] H. Soper, The interpretation of periodicity in disease prevalence, Journal of the Royal Statistical Society, A 92 (1929), 34-73.
[48] P. Taylor and L. Jonker, Evolutionary stable strategies and game dynamics, Math. Biosci. 40(1-2) (1978), 145-156. DOI 10.1016/0025-5564(78)90077-9.
[49] P. Van den Driessche and J. Watmough, Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180(1) (2002), 29-48.
[50] F. Wang, Y. Zhang, C. Wang, J. Ma and S. Moon, Stability analysis of a SEIQV epidemic model for rapid spreading worms, Computers and Security 29(4) (2010), 410-418. DOI 10.1016/j.cose.2009.10.002.
[51] P. Whittle, The outcome of a stochastic epidemic–a note on Bailey’s paper, Biometrika 42(1-2) (1955), 116-122.
[52] WHO, Global Status Report on Alcohol and Health 2014, World Health Organization, Geneva, 2014.
[53] WHO, Global adult tobacco survey 2 (gats 2) India 2016-17 report, 2018. https://mohfw.gov.in/newshighlights/global-adult-tobacco-survey-2-gats-2-india- 2016-17-report.
[54] J. Woo, J. Son and H.-C. Chen, An SIR model for violent topic diffusion in social media, Proceedings of 2011 IEEE International Conference on Intelligence and Security Informatics, ISI 2011, 2011, pp. 15-19.