The book by Coleman became the definitive volume in mathematical sociology. He put forward models of innovation diffusion which have been extensively investigated since. It is particularly relevant for the church growth models as he proposed the use of the epidemic equations for modelling social diffusion, i.e. the time period of those spreading the phenomena is limited. Bartholomew gives stochastic models of such diffusion.
- Bartholomew D.J. (1982), Stochastic Models for Social Processes. Wiley NY. The stochastic approach to dynamical modelling as opposed to the deterministic approach used in the Church Growth Models.
- Coleman J.S. (1964), Introduction to Mathematical Sociology. The Free Press of Glencoe NY. The foundational work on dynamical models in sociology. Applies diffusion type ideas to the adoption of medical innovations.
- Gilbert N. & Troitzsch (1999), Simulation for the Social Scientist. The Open University PA. An overview of a number of simulation approaches to sociological dynamics. Includes a section on system dynamics.
- Weidlich W. (2000), Sociodynamics - A Systematic Approach to Mathematical Modelling in the Social Sciences. Dover. A stochastic style of modelling for a range of social phenomena.
Spread of Languages
- Abrams D.M. & Strogatz S.H. (2003). Modelling the dynamics of language death. Nature, 424:6951, 900. Heavily cited paper and popular model. The deterministic version is too simple for spread of religion, but it has a very powerful agent-based version with phase transitions.
Ausloos, M. & Petroni, F. (2009). Statistical dynamics of religion evolutions. Physica A: Statistical Mechanics and its Applications, 388(20), 4438-4444.
Ausloos, M. (2010). On religion and language evolutions seen through mathematical and agent based models. In Proc. First Interdisciplinary CHESS Interactions Conf , pp. 157-182.
- Baggs I. & Freedman H.I. (1990). A Mathematical Model for the Dynamics of Interactions between a Unilingual and a Bilingual population: Persistence versus Extinction. Journal of Mathematical Sociology, 16(1), 51-75.
- Wyburn J. & Hayward J. (2008). The Future of Bilingualism: An Application of the Baggs and Freedman Model. Journal of Mathematical Sociology 32(4) Pages: 267-284.
- Wyburn J. & Hayward J. (2009). OR and Language Planning: Modelling The Interaction Between Unilingual and Bilingual Populations. Journal of the Operational Research Society, 60 (5), 626-636.
- Wyburn J. & Hayward J. (2010). A Model of language-group interaction and evolution including language acquisition planning. Journal of Mathematical Sociology, 34:3, 167-200.
- Jeffs R.A., Hayward J., Roach P.A. & Wyburn J. (2016). Activist Model of Political Party Growth. Physica A: Statistical Mechanics and its Applications, 442, 359-372. DOI: 10.1016/j.physa.2015.09.002; arXiv:1509.07805. Physica A.
Romero D.M., Kribs-Zaleta C.M., Mubayi A., & Orbe C. (2011). An Epidemiological Approach to the Spread of Political Third
Parties. Discrete and Continuous Dynamical Systems-Series B (DCDS-B), 15(3):707–738, 2011.
- Burbeck S.L., Raine W.J. & Stark M.J.A. (1979). The Dynamics of Riot Growth: An Epidemiological Approach. Journal of Mathematical Sociology, 6, 1-22.
- Castillo-Chavez C. & Song B. (2003). Models for the Transmission Dynamics of Fanatic Behaviors. Bioterrorism-mathematical modeling applications in homeland security. Philadelphia: SIAM, 155-72.
- Crane J., Boccara N. & Higdon K. (2000). The Dynamics of Street Gang Growth and Policy Response. Journal of Policy Modelling, 22(1) pp 1-25.
- Hayward J., Jeffs R.A., Howells L. & Evans K.S. (2014). Model Building with Soft Variables: A Case Study on Riots. Presented at the 32nd International Conference of the System Dynamics Society, Delft, Netherlands, July 2014.
- Nizamani S., Memon N. & Galam S. (2014). From Public Outrage to the Burst of Public Violence: An Epidemic-Like Model. Physica A: Statistical Mechanics and its Applications, 416, pp 620-630.
- Bettencourt L.M.A., Cintron-Arias A., Kaiser D.I. &Castillo-Chavez C. (2006). The Power of a good idea: Quantitative modelling of the spread of ideas from epidemiological models. Physica A, 364, pp 513-536.
- Gonzalez B., Huerta-Sanchez E., Ortiz-Nieves A., Vazquez-Alvarez T. & Kribs-Zaleta C. (2003). Am I too fat? Bulimia as an epidemic. Journal of Mathematical Psychology, 47(5-6): 515-526.
- Manthey J.L., Aidoo A.Y. & Ward K.Y. (2008).
Campus drinking: an epidemiological model. Journal of
Biological Dynamics, 2(3): 346-356.
- Rodgers J.L. & Rowe D.C. (1993). Social contagion and adolescent sexual behavior: A developmental EMOSA model. Psychological Review, 100(3): 479-510.
- Rowe D., Chassin C., Presson C., Edwards D. & Sherman S. (1992). An epidemic model of adolescent cigarette smoking. Journal of Applied Social Psychology, 22: 261-285.
- Sanchez F., Wang X., Castillo-Chavez C., Gruenwald P. & Gorman D. (2006). Drinking as an epidemic - a simple mathematical model with recovery and relapse. Appears in Therapist's Guide to Evidence-Based Relapse Prevention, ed. K.A. Witkiewitz and G.A. Marlatt. Elsevier.
- Santonja F.J., Villanueva R.J., Jodar L. & Gonzalez-Parra G. (2010). Mathematical modelling of social obesity epidemic in the region of Valencia, Spain, Mathematical and Computer Modelling of Dynamical Systems, 16:1, 23-34.
- Crane J. (1991). The Epidemic Theory of Ghetto's and Neighbourhood Effects on Dropping Out and Teenage Childbearing. The American Journal of Sociology, 96(5), pp 1226-1259.
- Davidoff L., Sutton K., Toutain G.Y., Sanchez F., Kribs-Zaleta C. &
Castillo-Chavez C. (2006). Mathematical modeling of the sex
worker industry as a supply and demand system. Technical
report of the Mathematical and Theoretical Biology Institute,
Arizona State University, MTBI-03-06M.
- Kawachi K. (2008). Deterministic models for rumor transmission. Nonlinear Analysis: Real World Applications, 9: 1989-2028.
- McCartney M., & Glass D.H. (2015). The dynamics of coupled logistic social groups. Physica A: Statistical Mechanics and its Applications, 427, 141-154.
- Lane D.C. & Husemann E. (2004). Movie marketing strategy formation with system dynamics: towards a multidisciplinary adoption/diffusion theory of cinema-going. In: Maier F (ed.) Komplexitat und dynamik als herausforderung fur das management. DUV, pp. 179-222.
- Mao S., Vassileva J. & Grassmann W. (2007). A System Dynamics Approach to Study Virtual Communities. 40th Annual Hawaii International Conference on System Sciences, HICSS 2007.
- Santonja F.J., Garcia I. & Jodar L. (2010). Modelling the dynamic of addictive buying. Appears in Modelling for Addictive Behaviour, Medicine and Engineering, Instituto de Matematica Multidisciplinar Universidad Politecnica de Valencia, Spain.
Product Diffusion & Marketing
- Bass F. (1969). A New Product Growth Model for Consumer Durables. Management Science, 15(5) (January), pp 215-227. The original paper of the famous Bass model.
- Fisher J. and Pry R. (1971). A Simple Substitution Model of Technological Change. Technological Forecasting and Social Change, 3, pp 75-88. The original paper of the famous Fisher Pry model.
- Kumar V. & Kumar U. (1992). Innovation Diffusion: Some New Technological Substitution Models. Journal of Mathematical Sociology, 17(2-3), 175-194. A review of product diffusion focusing on the mathematics.
- Mahajan V., Muller E. & Bass F.M. (1990). New Product Diffusion models in Marketing. Journal of Marketing, 54, 1-26. A review of product diffusion and marketing models focusing on the applications
- Rodrigues H.S. & Fonseca M.J. (2015). Viral marketing as epidemiological model. arXiv preprint arXiv:1507.06986.
Theory of Social Diffusion
- Granovetter M. & Song R. (1983). Threshold Models of Diffusion and Collective Behaviour. Journal of Mathematical Sociology, 9, 165-179. Has the thesis that in a population people have varying rates of susceptibility to adoption of an idea, behaviour etc.
- Granovetter M. (1973). The Strength of Weak Ties. American Journal of Sociology 78 (6): 1360–1380. Presents the thesis that weak ties between groups are more influential in social diffusion than the strong ties within groups, because, for example, there is less redundancy in weak ties which open up new groups of susceptible people.
Although not directly connected with social diffusion, like the mathematics of epidemics, a number of concepts are transferable in the field.
- Freedman H.I. (1980). Deterministic Mathematical Models in Population
Ecology, Dekker, New York. Includes the mathematical derivation of Holling term, in the context of predation. Similar arguments apply to word of mouth social transmission and are used in church growth models.
- Holling C.S. (1959). The Components of Predation as Revealed by a Study of
Small-Mammal Predation of the European Pine
Sawfly. The Canadian Entomologist, 91, 293-320; Some Characteristics of Simple Types of Predation and
Parasitism, The Canadian Entomologist}, 91(7).
385-398 The original papers quantifying density effects in population dynamics which gave rise to the name "Holling term". Used in the Renewal Model of church growth.
- Holling C.S. (1965). Some characteristics of simple types of predation and
parasitism. Memoirs of the Entomological Society of Canada, 45, 5-60; A more mathematical treatment of density effects in population dynamics.
Ludwig, D., Jones, D.D., & Holling, C.S. (1978). Qualitative analysis of insect outbreak systems: the spruce budworm and forest. The Journal of Animal Ecology, 315-332.
Differential equation model with Holling term.