There are a growing number of papers that attempt to use mathematics to examine church growth. In 2014 there were papers by McCartney & Glass in Physica A, and Madubueze & Nwaokolo in the International Journal of Mathematics.
For someone new to the field Eddie Gibbs "I Believe in Church Growth" remains a good introduction, even though many years old. Rick Warren's "The Purpose Driven Church", is the classic church growth book and hugely influential.
Revival is a term commonly used within the Christian Church to describe an infusion of spiritual life into those who believe, resulting in the substantial growth of the church. To understand the nature of revival in terms of its biblical and historical contexts read "Revival: A People Saturated by God" by Brian Edwards, and "Revival" by Dr Martyn Lloyd-Jones. Among the older authors, the works by Jonathan Edwards are unbeatable for spiritual value.
System dynamics was pioneered in the 1960s by Jay W Forrester whose books still provide penetrating insights. "Business Dynamics" by John Sterman is a good technical introduction. "The Fifth Discipline" by Peter Senge is a good introductory account to systems thinking suitable for a lay audience.
The sociology of religion had a paradigm shift in the second half of the 20th century. The older view championed secularization theory which claimed the religion's importance would wither away as society becomes more developed. The newer view is that religion not only survives but flourishes in modern society. The newer works provide many insights into the state of the church and its growth from a sociologist's viewpoint. Christians would find Kelley's book: "Why Conservative Churches are Growing" a profitable read, even though many years old.
There are other forms of diffusion in society apart from the spread of a disease. Diffusion models are used extensively to in marketing, spread of technological innovation, and social phenomena such as riots, binge drinking, bulimia and rumours.
An epidemic occurs when an infectious disease spreads rapidly through a population. Epidemiology, the study of the spread of diseases, is a well developed academic discipline with a high mathematical content. The original mathematical paper was by Kermack and McKendrick in 1927, which the basis for the Limited Enthusiasm model of church growth. The book by Anderson and May, is the subject's definitive work.
References used in the Church Growth Modelling project, split into subject area