Population modelling, systems of differential equations, non-linear analysis

  • Anderson D. Estimates of the future membership of the Church of Jesus Christ of Latter-day Saints.
    An attempt to fit church data to the logistic model – thus using the concept of carrying capacity and limited growth.
  • Ausloos M. (2012). Econophysics of a religious cult: The Antoinists in Belgium [1920-2000]. Physica A: Statistical Mechanics and its Applications, 391(11), 3190-3197.
  • Hayward J. (1995). Mathematical Modelling of Church Growth. Technical Report, UG-M-95-3, 1995.
    The first report on the Limited Enthusiasm Model.
  • Hayward J. (1999). Mathematical Modeling of Church Growth. Journal of Mathematical Sociology. 23(4), 255-292.
    An updated version of the paper in light of more recent work in System Dynamics. Original in the journal.
  • Hayward J. (2005). A General Model of Church Growth and Decline. Journal of Mathematical Sociology, 29(3), 177-207.
    The definitive version of the Limited Enthusiasm Model.
  • Lucero JC. (2021). Subexponential growth of early Christianity. arXiv:2111.08833.
  • Madubueze C.E. & Nwaokolo M.A. (2014). A Mathematical Model to study the effect of Renewal and Reversion of Inactive Christians on Church Growth. International Journal of Mathematics Trends and Technology – Volume 14(2).
    Extension of the model by Ochoche & Gweryina.
  • McCartney M. & Glass D. H. (2014). A three-state dynamical model for religious affiliation. Physica A: Statistical Mechanics and its Applications.
    Elements of the limited enthusiasm model of church growth are embedded in these three state models.
  • Ochoche J.M. & Gweryina R.I. (2013). Evaluating the Impact of Active Members on Church Growth. International Journal of Science and Technology 2(11): 784-791.
    A differential equation model of church growth, based on Hayward’s limited enthusiasm model, and the 80/20 principle of economics. The latter claims that 20% of the people do 80% of the work and vice versa. Thus the model has two categories of believers spreading the faith in different degrees. It provides a viable alternative church growth model to the limited enthusiasm one. Also supported by the results of Medcalfe and Sharp.
  • Nyabadza F. (2008). A deterministic model for church growth with internal revival. Journal of Interdisciplinary Mathematics, 11(1), 11-27.
  • Vitanov N.K., Dimitrova Z.I. & Ausloos M.R. (2011). Verhulst-Lotka-Volterra (VLV) model of ideological struggles. Physica A: Statistical Mechanics and its Applications, 389(21), 4970-4980.
    A differential equation model of ideological spread with competition, relevant to competing churches.
  • Wieder T. (2011). A Simple differential equation system for the description of competition among religions, International Mathematical Forum, 6:35, 1713-1723.

System Dynamics

MIT System Dynamics, invented by J.W. Forrester. Simulation with Stella, Vensim and AnyLogic.

  • Acuua M.N., Cuevas, R. C., Ospina, Y. C., & Valencia, J. P. (2001). Caleb: Microworld of the Christian Church’s Membership Dynamics. Paper presented at The 19th International Conference of the System Dynamics Society, Atlanta, Georgia.
    Uses system dynamics to model the growth of a specific denomination in Latin America. Organisational rather than diffusion.
  • Bullock J.L. (1999). The Parish Learning Laboratory: A Computer-Based Simulation for Exploring the Long-Term Outcome of Policies and Planning. PhD. Thesis. Berkeley, California, Church Divinity School of the Pacific.
    Uses system dynamics to model the growth and finances of a parish.
  • Gaynor A.K., Morrow J. & Georgiou S.N. (1991). Aging, contraction, and cohesion in a religious order: A policy analysis. System Dynamics Review, 7(1): 1-19.
    Systems dynamics model of a religious order. Organisational rather than diffusion.
  • Hayward J. (2000). Growth and Decline of Religious and Sub-cultural Groups. Presented at the 18th International System Dynamics Society, Bergen, Norway, July 2000.
  • Hayward J. (2002). A Dynamical Model of Church Growth and its Application to Contemporary Revivals. Review of Religious Research, 43(3), 218-241.
  • Hayward J. (2002). A Dynamical Model of Strictness and its Effect on Church Growth. Presented at “The Annual Meeting of the Religious Research Association / Scientific Study of Religion”, Salt Lake City, Utah 1-3 November 2002.
  • Hayward J. (2010). Church Growth via Enthusiasts and Renewal. Presented at the 28th International Conference of the System Dynamics Society, Seoul, South Korea, July 2010.
    A model where enthusiasts are generated through renewing existing believers in addition to conversion. Model displays accelerated growth and critical mass.
  • Hayward J. (2013). System Dynamics: A Tool for Investigating Church and Religious Growth. Presented at “The Annual Meeting of the Scientific Study of Religion”, Boston, USA, November 2013. Powerpoint slides.
    How to build a church growth model in system dynamics; applications to Church of England and Southern Baptist Convention data; modelling the effects of church strictness.
  • Hayward J. (2014). The Growth of Islam in the UK: A System Dynamics Perspective. Presented at “The Annual Meeting of the Scientific Study of Religion”, Indianapolis, USA, November 2014. (With Ahmed N and Hassan A.)
    An application of the Limited Enthusiasm Model of Church Growth to Islam in England & Wales.
  • Howells L. (2010). The Dynamics of Religious Revival and Church Growth, 5th Research Student Workshop, University of Glamorgan, March 2010.
  • Jo, C., Kim, D.H. & Lee, J.W. (2021). Sustainability of religious communities. PloS one, 16(5), p.e0250718.
    An alternative approach to passive and active believers. Applied to the Presbyterian churches of South Korea.


Stochastic mathematics, cellular automata, networks, computer simulation, Net Logo

  • Carvalho J.P. (2009). A Theory of the Islamic Revival, Economics Series Working Papers, University of Oxford, Department of Economics.
    Not strictly about church growth, the model uses agent-based ideas to model the spread of religion with economic style assumptions.
  • Iannaccone L.R. & Makowsky M.D. (2007). Accidental Atheists? Agent-Based Explanations for the Persistence of Religious Regionalism. Journal for the Scientific Study of Religion, 46(1), 1-16.
  • Ormerod P. & Roach A.P. (2004). The medieval inquisition: Scale-free networks and the suppression of heresy. Physica A, 339: 645-652.
    Not church growth as such, but does model the spread of religious belief which would be an important part of a more sociological approach to church growth within a given social context.


Sociophysics, statistical physics.

  • Ausloos M. & Petroni F. (2007). Statistical dynamics of religions and adherents. Europhysics Letters, 77: 38002-38006.
  • Ausloos M. & Petroni F. (2009). Statistical dynamics of religion evolutions. Physica A: Statistical Mechanics and its Applications, 388(20): 4438-4444.
  • Ausloos M. and Petroni F. (2010). On World Religion Adherence Distribution Evolution, appears in “Econophysics Approaches to Large-Scale Business Data and Financial Crisis: Proceedings of the Tokyo Tech-Hitotsubashi Interdisciplinary Conference + APFA7, eds. Takayasu M., Watanabe T. and Takayasu H., 289-312, Springer.
  • Logan P.F. & Dye T.W. (1984). Physics for Anthropologists, Search, 15(1-2), 30-32.
    Modelling the growth of a church in a developing country using diffusion ideas from gas dynamics. Uses the logistic model and the concept of positive feedback.
  • Mir T.A. (2014). The Benford law behavior of the religious activity data. Physica A: Statistical Mechanics and its Applications, 408, 1-9.

Statistical Modelling

  • Loomis R.D. (2002). Church Growth and the Latter Day Saints. Presented at The Association for the Sociology of Religion August 15-17, 2002 Chicago, IL.
    Uses statistical modelling to examine the future growth of the Mormon Church. Also, see details.
  • Mangeloja E. (2007). Preaching to the choir? Economic analysis of church growth. European Network on the Economics of Religion Online, paper 07/03, [Accessed 01/03/2010].
    A Statistical model of church growth that supports growth through the activity of the most committed.
  • Penrose L.S. (1952). On The Objective Study of Crowd Behaviour.
    This is the first reference I found that refers to the use of epidemiology to explain the spread of religious behaviour. In addition, the book has applications to politics, fashion and panic reactions. The presentation is more descriptive than mathematical. There is no quantitative model of the spread of religion. However, it does contain a demonstration of the famous Penrose Theorem that is the basis of the square root voting rule of organisations such as the UN. Lionel Penrose was the father of the famous relativist Professor Roger Penrose, who kindly pointed me in the direction of his father’s work.


  • Levy G. & Razin R. (2010). Religious Organizations, Fondazione Eni Enrico Mattei Working Papers.
    Model using game theory.
  • Schell M. (nd). Using computers to support total church growth. Chapter 18 of Church Growth—State of the Art, edited by Peter Wagner, Win Arn, and Elmer L. Towns.
    Although this is computer applications rather than mathematics, Mel Schell produced an early attempt to model growing and successful churches. Many of these ideas would translate into a system dynamics model
  • Shy O. (2005). Dynamic Models of Religious Conformity and Conversion: Theory and Calibration. Discussion paper, Social Science Research Centre, Berlin. SP II 2005 – 12.
    Uses economic modelling to explain changes in numbers of non-believers by differing both rates with believers, and differing ratios of nonconformity in each group. .