# American Institute of Mathematical Sciences

2013, 10(5&6): 1587-1607. doi: 10.3934/mbe.2013.10.1587

## Sociological phenomena as multiple nonlinearities: MTBI's new metaphor for complex human interactions

 1 Mathematics Department, University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States

Received  August 2012 Revised  March 2013 Published  August 2013

Mathematical models are well-established as metaphors for biological and epidemiological systems. The framework of epidemic modeling has also been applied to sociological phenomena driven by peer pressure, notably in two dozen dynamical systems research projects developed through the Mathematical and Theoretical Biology Institute, and popularized by authors such as Gladwell (2000). This article reviews these studies and their common structures, and identifies a new mathematical metaphor which uses multiple nonlinearities to describe the multiple thresholds governing the persistence of hierarchical phenomena, including the situation termed a backward bifurcation'' in mathematical epidemiology, where established phenomena can persist in circumstances under which the phenomena could not initially emerge.
Citation: Christopher M. Kribs-Zaleta. Sociological phenomena as multiple nonlinearities: MTBI's new metaphor for complex human interactions. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1587-1607. doi: 10.3934/mbe.2013.10.1587
##### References:
 [1] Carlos A. Acevedo-Estefania, Christina Gonzalez, Karen R. Rios-Soto, Eric D. Summerville, Baojun Song and Carlos Castillo-Chavez, A mathematical model for lung cancer: The effects of second-hand smoke and education,, Biometrics Unit Technical Report BU-1525-M, (2000). Google Scholar [2] A. A. Alchian, Uncertainty, evolution and economic theory,, J. Political Economy, 58 (1950), 211. doi: 10.1086/256940. Google Scholar [3] Joshua Austin, Emma Smith, Sowmya Srinivasan and Fabio Sanchez, Social dynamics of gang involvement: A mathematical approach,, MCMSC Technical Report MTBI-08-08M, (2011), 08. Google Scholar [4] Luís M. A. Bettencourt, Ariel Cintrón-Arias, David I. Kaiser and Carlos Castillo-Chavez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models,, Physica A, 364 (2006), 513. doi: 10.2172/990668. Google Scholar [5] Corvina D. H. Boyd, Alison Castro, Nicolás Crisosto, Arlene Morales Evangelista, Carlos Castillo-Chavez and Christopher Kribs-Zaleta, A socially transmitted disease: Teacher qualifications and dropout rates,, Studies in Theoretical Biology: A Collection of Undergraduate Research, 1 (2000), 549. Google Scholar [6] Erika Camacho, Julio Villarreal and Monica F. Yichoy, Delinquency dynamics,, Biometrics Unit Technical Report BU-1504-M, (1997). Google Scholar [7] Carlos Castillo-Garsow, Guarionex Jordán-Salivia and Ariel Rodriguez-Herrera, Mathematical models for the dynamics of tobacco use, recovery, and relapse,, Biometrics Unit Technical Report BU-1505-M, (1997). Google Scholar [8] Jonathan Crane, The epidemic theory of ghettos and neighborhood effects on dropping out and teenage childbearing,, Amer. J. Soc., 95 (1989), 1226. doi: 10.1086/229654. Google Scholar [9] Nicolás Crisosto, Christopher Kribs-Zaleta, Carlos Castillo-Chavez and Stephen Wirkus, Community resilience in collaborative learning,, Discrete and Continuous Dynamical Systems, 14 (2010), 17. doi: 10.3934/dcdsb.2010.14.17. Google Scholar [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population,, Journal of Mathematical Biology, 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [11] Jennifer L. Dillon, Natalia Baeza, Mary Cristina Ruales and Baojun Song, A mathematical model of depression in young women as a function of the pressure to be "beautiful,'', Biometrics Unit Technical Report BU-1616-M, (2002). Google Scholar [12] Arlene Morales Evangelista, Angela R. Ortiz, Karen R. Ríos-Soto and Alicia Urdapilleta, USA the fast food nation: Obesity as an epidemic,, MCMSC Technical Report MTBI-01-3M, (2004), 01. Google Scholar [13] Malcolm Gladwell, "The Tipping Point,'', Little, (2000). Google Scholar [14] Natalie S. Glance and Bernardo A. Huberman, The outbreak of cooperation,, J. Math. Soc., 17 (1993), 281. doi: 10.1080/0022250X.1993.9990112. Google Scholar [15] Beverly González, Emilia Huerta-Sánchez, Angela Ortiz-Nieves, Terannie Vázquez-Álvarez and Christopher Kribs-Zaleta, Am I too fat? Bulimia as an epidemic,, Journal of Mathematical Psychology, 47 (2003), 515. doi: 10.1016/j.jmp.2003.08.002. Google Scholar [16] Mark Granovetter, Threshold models of collective behavior,, Am. J. Soc., 83 (1978), 1420. doi: 10.1086/226707. Google Scholar [17] Mark Granovetter and R. Soong, Threshold models of diffusion and collective behavior,, J. Math. Soc., 9 (1983), 165. doi: 10.1080/0022250X.1983.9989941. Google Scholar [18] Howard E. Gruber, Ensembles of metaphors in creative scientific thinking,, in, (2005), 259. Google Scholar [19] Karl P. Hadeler and Carlos Castillo-Chávez, A core group model for disease transmission,, Math. Biosciences, 128 (1995), 41. doi: 10.1016/0025-5564(94)00066-9. Google Scholar [20] J. A. P. Heesterbeek, A brief history of $R_0$ and a recipe for its calculation,, Acta Biotheoretica, 50 (2002), 189. doi: 10.1023/A:1016599411804. Google Scholar [21] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Part I,, Proc. Royal Soc. Edin. A, 115 (1927), 700. Google Scholar [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proc. Royal Soc. Edin. A, 138 (1932), 55. Google Scholar [23] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity,, Proc. Royal Soc. Edin. A, 141 (1933), 94. Google Scholar [24] Christopher M. Kribs-Zaleta, Alternative transmission modes for Trypanosoma cruzi,, Math. Biosci. Eng., 7 (2010), 657. doi: 10.3934/mbe.2010.7.657. Google Scholar [25] Christopher M. Kribs-Zaleta and Jorge X. Velasco-Hernández, A simple vaccination model with multiple endemic states,, Math. Biosciences, 164 (2000), 183. doi: 10.1016/S0025-5564(00)00003-1. Google Scholar [26] Anuj Mubayi, Priscilla E. Greenwood, Carlos Castillo-Chavez, Paul Gruenewald and Dennis M. Gorman, Impact of relative residence times on the distribution of heavy drinkers in highly distinct environments,, Socio-Economic Planning Sciences, 43 (2010), 1. doi: 10.1016/j.seps.2009.02.002. Google Scholar [27] R. R. Nelson and S. G. Winter, "An Evolutionary Theory of Economic Change,'', Harvard Univ. Press, (1982). Google Scholar [28] E. T. Penrose, Biological analogies in the theory of the firm,, Amer. Economic Review, 42 (1952), 804. Google Scholar [29] Tönu Puu, "Nonlinear Economic Dynamics,'', Springer-Verlag, (1992). Google Scholar [30] Daniel M. Romero, Christopher M. Kribs-Zaleta, Anuj Mubayi and Clara Orbe, An epidemiological approach to the spread of political third parties,, Discrete and Continuous Dynamical Systems, 15 (2011), 707. doi: 10.3934/dcdsb.2011.15.707. Google Scholar [31] Ronald Ross, "The Prevention of Malaria,'', Second edition, (1911). Google Scholar [32] Fabio Sánchez, Xiaohong Wang, Carlos Castillo-Chavez, Dennis M. Gorman and Paul J. Gruenewald, Drinking as an epidemic: A simple mathematical model with recovery and relapse,, in, (2007), 353. Google Scholar [33] Thomas Schelling, Dynamic models of segregation,, J. Math. Soc., 1 (1971), 143. doi: 10.1080/0022250X.1971.9989794. Google Scholar [34] Susan Seal, William Z. Rayfield, Carl Ballard II, Holden Tran, Christopher Kribs-Zaleta and Edgar Díaz, A dynamical interpretation of the three-strikes law,, MCMSC Technical Report MTBI-04-07M, (2007), 04. Google Scholar [35] Baojun Song, Melissa Castillo-Garsow, Karen R. Ríos-Soto, Marcin Mejran, Leilani Henson and Carlos Castillo-Chavez, Raves, clubs, and ecstasy: The impact of peer pressure,, Mathematical Biosciences and Engineering, 3 (2006), 249. Google Scholar [36] Pauline Van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [37] Wolfgang Weidlich, Synergetic modelling concepts for sociodynamics with application to collective political opinion formation,, J. Math. Soc., 18 (1994), 267. doi: 10.1080/0022250X.1994.9990129. Google Scholar [38] Wolfgang Weidlich, "Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences,'', Harwood Academic Publishers, (2000). Google Scholar

show all references

##### References:
 [1] Carlos A. Acevedo-Estefania, Christina Gonzalez, Karen R. Rios-Soto, Eric D. Summerville, Baojun Song and Carlos Castillo-Chavez, A mathematical model for lung cancer: The effects of second-hand smoke and education,, Biometrics Unit Technical Report BU-1525-M, (2000). Google Scholar [2] A. A. Alchian, Uncertainty, evolution and economic theory,, J. Political Economy, 58 (1950), 211. doi: 10.1086/256940. Google Scholar [3] Joshua Austin, Emma Smith, Sowmya Srinivasan and Fabio Sanchez, Social dynamics of gang involvement: A mathematical approach,, MCMSC Technical Report MTBI-08-08M, (2011), 08. Google Scholar [4] Luís M. A. Bettencourt, Ariel Cintrón-Arias, David I. Kaiser and Carlos Castillo-Chavez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models,, Physica A, 364 (2006), 513. doi: 10.2172/990668. Google Scholar [5] Corvina D. H. Boyd, Alison Castro, Nicolás Crisosto, Arlene Morales Evangelista, Carlos Castillo-Chavez and Christopher Kribs-Zaleta, A socially transmitted disease: Teacher qualifications and dropout rates,, Studies in Theoretical Biology: A Collection of Undergraduate Research, 1 (2000), 549. Google Scholar [6] Erika Camacho, Julio Villarreal and Monica F. Yichoy, Delinquency dynamics,, Biometrics Unit Technical Report BU-1504-M, (1997). Google Scholar [7] Carlos Castillo-Garsow, Guarionex Jordán-Salivia and Ariel Rodriguez-Herrera, Mathematical models for the dynamics of tobacco use, recovery, and relapse,, Biometrics Unit Technical Report BU-1505-M, (1997). Google Scholar [8] Jonathan Crane, The epidemic theory of ghettos and neighborhood effects on dropping out and teenage childbearing,, Amer. J. Soc., 95 (1989), 1226. doi: 10.1086/229654. Google Scholar [9] Nicolás Crisosto, Christopher Kribs-Zaleta, Carlos Castillo-Chavez and Stephen Wirkus, Community resilience in collaborative learning,, Discrete and Continuous Dynamical Systems, 14 (2010), 17. doi: 10.3934/dcdsb.2010.14.17. Google Scholar [10] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population,, Journal of Mathematical Biology, 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [11] Jennifer L. Dillon, Natalia Baeza, Mary Cristina Ruales and Baojun Song, A mathematical model of depression in young women as a function of the pressure to be "beautiful,'', Biometrics Unit Technical Report BU-1616-M, (2002). Google Scholar [12] Arlene Morales Evangelista, Angela R. Ortiz, Karen R. Ríos-Soto and Alicia Urdapilleta, USA the fast food nation: Obesity as an epidemic,, MCMSC Technical Report MTBI-01-3M, (2004), 01. Google Scholar [13] Malcolm Gladwell, "The Tipping Point,'', Little, (2000). Google Scholar [14] Natalie S. Glance and Bernardo A. Huberman, The outbreak of cooperation,, J. Math. Soc., 17 (1993), 281. doi: 10.1080/0022250X.1993.9990112. Google Scholar [15] Beverly González, Emilia Huerta-Sánchez, Angela Ortiz-Nieves, Terannie Vázquez-Álvarez and Christopher Kribs-Zaleta, Am I too fat? Bulimia as an epidemic,, Journal of Mathematical Psychology, 47 (2003), 515. doi: 10.1016/j.jmp.2003.08.002. Google Scholar [16] Mark Granovetter, Threshold models of collective behavior,, Am. J. Soc., 83 (1978), 1420. doi: 10.1086/226707. Google Scholar [17] Mark Granovetter and R. Soong, Threshold models of diffusion and collective behavior,, J. Math. Soc., 9 (1983), 165. doi: 10.1080/0022250X.1983.9989941. Google Scholar [18] Howard E. Gruber, Ensembles of metaphors in creative scientific thinking,, in, (2005), 259. Google Scholar [19] Karl P. Hadeler and Carlos Castillo-Chávez, A core group model for disease transmission,, Math. Biosciences, 128 (1995), 41. doi: 10.1016/0025-5564(94)00066-9. Google Scholar [20] J. A. P. Heesterbeek, A brief history of $R_0$ and a recipe for its calculation,, Acta Biotheoretica, 50 (2002), 189. doi: 10.1023/A:1016599411804. Google Scholar [21] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. Part I,, Proc. Royal Soc. Edin. A, 115 (1927), 700. Google Scholar [22] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proc. Royal Soc. Edin. A, 138 (1932), 55. Google Scholar [23] W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity,, Proc. Royal Soc. Edin. A, 141 (1933), 94. Google Scholar [24] Christopher M. Kribs-Zaleta, Alternative transmission modes for Trypanosoma cruzi,, Math. Biosci. Eng., 7 (2010), 657. doi: 10.3934/mbe.2010.7.657. Google Scholar [25] Christopher M. Kribs-Zaleta and Jorge X. Velasco-Hernández, A simple vaccination model with multiple endemic states,, Math. Biosciences, 164 (2000), 183. doi: 10.1016/S0025-5564(00)00003-1. Google Scholar [26] Anuj Mubayi, Priscilla E. Greenwood, Carlos Castillo-Chavez, Paul Gruenewald and Dennis M. Gorman, Impact of relative residence times on the distribution of heavy drinkers in highly distinct environments,, Socio-Economic Planning Sciences, 43 (2010), 1. doi: 10.1016/j.seps.2009.02.002. Google Scholar [27] R. R. Nelson and S. G. Winter, "An Evolutionary Theory of Economic Change,'', Harvard Univ. Press, (1982). Google Scholar [28] E. T. Penrose, Biological analogies in the theory of the firm,, Amer. Economic Review, 42 (1952), 804. Google Scholar [29] Tönu Puu, "Nonlinear Economic Dynamics,'', Springer-Verlag, (1992). Google Scholar [30] Daniel M. Romero, Christopher M. Kribs-Zaleta, Anuj Mubayi and Clara Orbe, An epidemiological approach to the spread of political third parties,, Discrete and Continuous Dynamical Systems, 15 (2011), 707. doi: 10.3934/dcdsb.2011.15.707. Google Scholar [31] Ronald Ross, "The Prevention of Malaria,'', Second edition, (1911). Google Scholar [32] Fabio Sánchez, Xiaohong Wang, Carlos Castillo-Chavez, Dennis M. Gorman and Paul J. Gruenewald, Drinking as an epidemic: A simple mathematical model with recovery and relapse,, in, (2007), 353. Google Scholar [33] Thomas Schelling, Dynamic models of segregation,, J. Math. Soc., 1 (1971), 143. doi: 10.1080/0022250X.1971.9989794. Google Scholar [34] Susan Seal, William Z. Rayfield, Carl Ballard II, Holden Tran, Christopher Kribs-Zaleta and Edgar Díaz, A dynamical interpretation of the three-strikes law,, MCMSC Technical Report MTBI-04-07M, (2007), 04. Google Scholar [35] Baojun Song, Melissa Castillo-Garsow, Karen R. Ríos-Soto, Marcin Mejran, Leilani Henson and Carlos Castillo-Chavez, Raves, clubs, and ecstasy: The impact of peer pressure,, Mathematical Biosciences and Engineering, 3 (2006), 249. Google Scholar [36] Pauline Van den Driessche and James Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Mathematical Biosciences, 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [37] Wolfgang Weidlich, Synergetic modelling concepts for sociodynamics with application to collective political opinion formation,, J. Math. Soc., 18 (1994), 267. doi: 10.1080/0022250X.1994.9990129. Google Scholar [38] Wolfgang Weidlich, "Sociodynamics. A Systematic Approach to Mathematical Modelling in the Social Sciences,'', Harwood Academic Publishers, (2000). Google Scholar
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