# American Institute of Mathematical Sciences

December  2011, 15(3): 707-738. doi: 10.3934/dcdsb.2011.15.707

## An epidemiological approach to the spread of political third parties

 1 Center for Applied Mathematics, Cornell University, Ithaca, NY 14853, United States 2 University of Texas at Arlington, Box 19408, Arlington, TX 76019-0408, United States 3 Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, AZ 85287-1904, United States 4 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, United States

Received  January 2010 Revised  May 2010 Published  February 2011

Third political parties are influential in shaping American politics. In this work we study the spread of a third party ideology in a voting population where we assume that party members/activists are more influential in recruiting new third party voters than non-member third party voters. The study uses an epidemiological metaphor to develop a theoretical model with nonlinear ordinary differential equations as applied to a case study, the Green Party. Considering long-term behavior, we identify three threshold parameters in our model that describe the different possible scenarios for the political party and its spread. We also apply the model to the study of the Green Party's growth using voting and registration data in six states and the District of Columbia to identify and explain trends over the past decade. Our system produces a backward bifurcation that helps identify conditions under which a sufficiently dedicated activist core can enable a third party to thrive, under conditions which would not normally allow it to arise. Our results explain the critical role activists play in sustaining grassroots movements under adverse conditions.
Citation: Daniel M. Romero, Christopher M. Kribs-Zaleta, Anuj Mubayi, Clara Orbe. An epidemiological approach to the spread of political third parties. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 707-738. doi: 10.3934/dcdsb.2011.15.707
##### References:
 [1] L. 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, Physica D, 364 (2006), 513-536. doi: 10.1016/j.physa.2005.08.083. [2] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,'' Springer-Verlag, New York, 2001. [3] California Secretary of State's Office, Statewide election results, Online at http://www.sos.ca.gov/elections/elections_elections.htm, accessed 2009 July 1. [4] California Secretary of State's Office, Voter registration and participation statistics, Online at http://www.sos.ca.gov/elections/elections_u.htm, accessed 2009 July 1. [5] C. Castillo-Chávez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability, in "Mathematical Approaches For Emerging and Reemerging Infectious Diseases: An Introduction,'' IMA Vol. 125 (ed. C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu), Springer-Verlag, New York, (2002), 224-250. [6] C. Castillo-Chávez and B. Song, Models for the transmission dynamics of fanatic behaviors, in "Bioterrorism: Mathematical Modeling Applications in Homeland Security,'' SIAM Frontiers in Applied Mathematics Series, No. 28 (ed. H.T. Banks and C. Castillo-Chávez), SIAM, Philadelphia, 2003, 155-172. [7] N. M. Crisosto, C. M. Kribs-Zaleta, C. Castillo-Chávez and S. Wirkus, Community resilience in collaborative learning, Discrete Continuous Dynam. Systems - B, 14 (2010), 17-40. [8] District of Columbia Board of Elections and Ethics, Election results, Online at http://www.dcboee.org/election_info/election_results/index.asp, accessed 2009 July 1. [9] District of Columbia Board of Elections and Ethics, Voter registration statistics, Online at http://www.dcboee.org/voter_stats/voter_reg/voter.asp, accessed 2009 July 1. [10] M. Gladwell, "The Tipping Point,'' Little, Brown and Company, New York, 2000. [11] B. González, E. Huerta-Sánchez, C. Kribs-Zaleta, A. Ortiz-Nieves and T. Vázquez-Alvarez, Am I too fat? Bulimia as an epidemic, J. Math. Psychol., 47 (2003), 515-526. doi: 10.1016/j.jmp.2003.08.002. [12] M. Granovetter, Threshold models of collective behavior, Amer. J. Sociol., 83 (1978), 1420-1443. doi: 10.1086/226707. [13] R. Huckfeldt and J. Sprague, "Citizens, Politics, and Social Communication: Information and Influence in an Election Campaign,'' Cambridge University Press, New York, 1995. doi: 10.1017/CBO9780511664113. [14] D. Kowalewski, How movements move: The dynamics of an ecoprotest campaign, The Social Science Journal, 32 (1995), 49-67. doi: 10.1016/0362-3319(95)90019-5. [15] M. W. Macy, Threshold effects in collective action, Amer. Sociological Review, 56 (1991), 730-747. doi: 10.2307/2096252. [16] Maine Bureau of Corporations, Elections & Commissions, Election results, Online at http://maine.gov/sos/cec/elec/prior1st.htm, accessed 2009 July 1. [17] Maine Bureau of Corporations, Elections & Commissions, Voter registration, Online at http://www.maine.gov/sos/cec/elec/votreg.htm, accessed 2009 July 1. [18] J. Marks, A historical look at Green structure: 1984 to 1992, Synthesis/Regeneration, 14 (1997), online at http://www.greens.org/s-r/14/14-03.html, accessed 2007 July 1. [19] Maryland State Board of Elections, Elections by year, Online at http://elections.state.md.us/elections/index.html, accessed 2009 July 1. [20] Maryland State Board of Elections, Monthly voter registration activity reports, Online at http://elections.state.md.us/voter_registration/monthly.html, accessed 2009 July 1. [21] New York State Board of Elections, Enrollment by county, Online at http://www.elections.state.ny.us/EnrollmentCounty.html, accessed 2009 July 1. [22] New York State Board of Elections, Election results, Online at http://www.elections.state.ny.us/ElectionResults.html, accessed 2009 July 1. [23] D. W. Nickerson, Is voting contagious? Evidence from two field experiments, Amer. Political Science Review, 102 (2008), 49-57. doi: 10.1017/S0003055408080039. [24] P. E. Oliver and G. Marwell, The paradox of group size in collective action: A theory of critical mass, II, Amer. Sociological Review, 53 (1988), 1-8. doi: 10.2307/2095728. [25] Oregon Secretary of State Election Division, Election registration and participation history, Online at http://www.sos.state.or.us/elections/votreg/regpart.htm, accessed 2009 July 1. [26] Oregon Secretary of State Election Division, Elections history, Online at http://www.sos.state.or.us/elections/other.info/stelec.htm, accessed 2009 July 1. [27] Pennsylvania Bureau of Commissions, Elections & Legislation, Voter registration statistics archives, Online at http://www.dos.state.pa.us/elections/cwp/view.asp?a=1310 &q=447072, accessed 2009 July 1. [28] Pennsylvania Bureau of Commissions, Elections & Legislation, Elections information, Online at http://www.electionreturns.state.pa.us/ElectionsInformation.aspx?Function ID=0, accessed 2009 July 1. [29] The Pew Research Center for the People & the Press, Trends in political values and core attitudes: 1987-2007. Political landscape more favorable to Democrats, Author, Washington, DC, 2007 March 22. Online at http://people-press.org/reports/pdf/312.pdf, accessed 2009 July 1. [30] Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat, World population prospects: The 2008 revision. Online at http://esa.un.org/unpp, accessed 2009 July 1. [31] J. P. Robinson, Interpersonal influence in election campaigns: Two step-flow hypotheses, Public Opinion Quarterly, 40 (1976), 304-319. doi: 10.1086/268307. [32] L. B. Shrestha, Life expectancy in the United States, (updated August 16, 2006). Washington, DC: Congressional Research Service, Library of Congress, 2006. Online at http://aging.senate.gov/crs/aging1.pdf, accessed 2009 July 1. [33] B. Song, M. Castillo-Garsow, K. R. Ríos-Soto, M. Mejran, L. Henson and C. Castillo-Chávez, Raves, clubs and ecstasy: the impact of peer pressure, Math. Biosci. Eng., 3 (2006), 249-266. [34] P. L. Southwell, The politics of alienation: Nonvoting and support of third-party candidates among 18-30-year-olds, The Social Science Journal, 40 (2003), 99-107. doi: 10.1016/S0362-3319(02)00261-6. [35] P. L. Southwell, Nader voters in the 2000 Presidential Election: What would they have done without him?, The Social Science Journal, 41 (2004), 423-431. doi: 10.1016/j.soscij.2004.04.009. [36] R. J. Timpone, Ties that bind: Measurement, demographics, and social connectedness, Political Behavior, 20 (1998), 53-77. doi: 10.1023/A:1024895116980. [37] E. Warren, "U.S. Supreme Court Plurality Opinion, Sweezy v. New Hampshire," 354 U.S. 234, 250-251, 1957. [38] J. Wong, The effects of age and political exposure on the development of party identification among Asian American and Latino immigrants in the United States, Political Behavior, 22 (2000), 341-371. doi: 10.1023/A:1010630130895.

show all references

##### References:
 [1] L. 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, Physica D, 364 (2006), 513-536. doi: 10.1016/j.physa.2005.08.083. [2] F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,'' Springer-Verlag, New York, 2001. [3] California Secretary of State's Office, Statewide election results, Online at http://www.sos.ca.gov/elections/elections_elections.htm, accessed 2009 July 1. [4] California Secretary of State's Office, Voter registration and participation statistics, Online at http://www.sos.ca.gov/elections/elections_u.htm, accessed 2009 July 1. [5] C. Castillo-Chávez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability, in "Mathematical Approaches For Emerging and Reemerging Infectious Diseases: An Introduction,'' IMA Vol. 125 (ed. C. Castillo-Chavez, S. Blower, P. van den Driessche, D. Kirschner and A.-A. Yakubu), Springer-Verlag, New York, (2002), 224-250. [6] C. Castillo-Chávez and B. Song, Models for the transmission dynamics of fanatic behaviors, in "Bioterrorism: Mathematical Modeling Applications in Homeland Security,'' SIAM Frontiers in Applied Mathematics Series, No. 28 (ed. H.T. Banks and C. Castillo-Chávez), SIAM, Philadelphia, 2003, 155-172. [7] N. M. Crisosto, C. M. Kribs-Zaleta, C. Castillo-Chávez and S. Wirkus, Community resilience in collaborative learning, Discrete Continuous Dynam. Systems - B, 14 (2010), 17-40. [8] District of Columbia Board of Elections and Ethics, Election results, Online at http://www.dcboee.org/election_info/election_results/index.asp, accessed 2009 July 1. [9] District of Columbia Board of Elections and Ethics, Voter registration statistics, Online at http://www.dcboee.org/voter_stats/voter_reg/voter.asp, accessed 2009 July 1. [10] M. Gladwell, "The Tipping Point,'' Little, Brown and Company, New York, 2000. [11] B. González, E. Huerta-Sánchez, C. Kribs-Zaleta, A. Ortiz-Nieves and T. Vázquez-Alvarez, Am I too fat? Bulimia as an epidemic, J. Math. Psychol., 47 (2003), 515-526. doi: 10.1016/j.jmp.2003.08.002. [12] M. Granovetter, Threshold models of collective behavior, Amer. J. Sociol., 83 (1978), 1420-1443. doi: 10.1086/226707. [13] R. Huckfeldt and J. Sprague, "Citizens, Politics, and Social Communication: Information and Influence in an Election Campaign,'' Cambridge University Press, New York, 1995. doi: 10.1017/CBO9780511664113. [14] D. Kowalewski, How movements move: The dynamics of an ecoprotest campaign, The Social Science Journal, 32 (1995), 49-67. doi: 10.1016/0362-3319(95)90019-5. [15] M. W. Macy, Threshold effects in collective action, Amer. Sociological Review, 56 (1991), 730-747. doi: 10.2307/2096252. [16] Maine Bureau of Corporations, Elections & Commissions, Election results, Online at http://maine.gov/sos/cec/elec/prior1st.htm, accessed 2009 July 1. [17] Maine Bureau of Corporations, Elections & Commissions, Voter registration, Online at http://www.maine.gov/sos/cec/elec/votreg.htm, accessed 2009 July 1. [18] J. Marks, A historical look at Green structure: 1984 to 1992, Synthesis/Regeneration, 14 (1997), online at http://www.greens.org/s-r/14/14-03.html, accessed 2007 July 1. [19] Maryland State Board of Elections, Elections by year, Online at http://elections.state.md.us/elections/index.html, accessed 2009 July 1. [20] Maryland State Board of Elections, Monthly voter registration activity reports, Online at http://elections.state.md.us/voter_registration/monthly.html, accessed 2009 July 1. [21] New York State Board of Elections, Enrollment by county, Online at http://www.elections.state.ny.us/EnrollmentCounty.html, accessed 2009 July 1. [22] New York State Board of Elections, Election results, Online at http://www.elections.state.ny.us/ElectionResults.html, accessed 2009 July 1. [23] D. W. Nickerson, Is voting contagious? Evidence from two field experiments, Amer. Political Science Review, 102 (2008), 49-57. doi: 10.1017/S0003055408080039. [24] P. E. Oliver and G. Marwell, The paradox of group size in collective action: A theory of critical mass, II, Amer. Sociological Review, 53 (1988), 1-8. doi: 10.2307/2095728. [25] Oregon Secretary of State Election Division, Election registration and participation history, Online at http://www.sos.state.or.us/elections/votreg/regpart.htm, accessed 2009 July 1. [26] Oregon Secretary of State Election Division, Elections history, Online at http://www.sos.state.or.us/elections/other.info/stelec.htm, accessed 2009 July 1. [27] Pennsylvania Bureau of Commissions, Elections & Legislation, Voter registration statistics archives, Online at http://www.dos.state.pa.us/elections/cwp/view.asp?a=1310 &q=447072, accessed 2009 July 1. [28] Pennsylvania Bureau of Commissions, Elections & Legislation, Elections information, Online at http://www.electionreturns.state.pa.us/ElectionsInformation.aspx?Function ID=0, accessed 2009 July 1. [29] The Pew Research Center for the People & the Press, Trends in political values and core attitudes: 1987-2007. Political landscape more favorable to Democrats, Author, Washington, DC, 2007 March 22. Online at http://people-press.org/reports/pdf/312.pdf, accessed 2009 July 1. [30] Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat, World population prospects: The 2008 revision. Online at http://esa.un.org/unpp, accessed 2009 July 1. [31] J. P. Robinson, Interpersonal influence in election campaigns: Two step-flow hypotheses, Public Opinion Quarterly, 40 (1976), 304-319. doi: 10.1086/268307. [32] L. B. Shrestha, Life expectancy in the United States, (updated August 16, 2006). Washington, DC: Congressional Research Service, Library of Congress, 2006. Online at http://aging.senate.gov/crs/aging1.pdf, accessed 2009 July 1. [33] B. Song, M. Castillo-Garsow, K. R. Ríos-Soto, M. Mejran, L. Henson and C. Castillo-Chávez, Raves, clubs and ecstasy: the impact of peer pressure, Math. Biosci. Eng., 3 (2006), 249-266. [34] P. L. Southwell, The politics of alienation: Nonvoting and support of third-party candidates among 18-30-year-olds, The Social Science Journal, 40 (2003), 99-107. doi: 10.1016/S0362-3319(02)00261-6. [35] P. L. Southwell, Nader voters in the 2000 Presidential Election: What would they have done without him?, The Social Science Journal, 41 (2004), 423-431. doi: 10.1016/j.soscij.2004.04.009. [36] R. J. Timpone, Ties that bind: Measurement, demographics, and social connectedness, Political Behavior, 20 (1998), 53-77. doi: 10.1023/A:1024895116980. [37] E. Warren, "U.S. Supreme Court Plurality Opinion, Sweezy v. New Hampshire," 354 U.S. 234, 250-251, 1957. [38] J. Wong, The effects of age and political exposure on the development of party identification among Asian American and Latino immigrants in the United States, Political Behavior, 22 (2000), 341-371. doi: 10.1023/A:1010630130895.
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