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Mathematical models and methods 545
ability ptj of passing from state i to statej for all values of i andj is independ-
ent of time). In this particular, then, the human sciences are unquestionably
backward in relation, say, to biology: today works on epidemiology for in-
stance, rely almost exclusively on stochastic models and the classical analytical
models prevalent at the time of Volterra’s Theory of the Struggle for Life be-
long to apast age. None of the social or human sciences is as yet, in this partic-
ular, even approaching the stage reached by biology. There are several rea-
sons for this: for like hypotheses, the mathematics of stochastic processes are
much more complex than the mathematics of the corresponding deterministic
models. Putting this another way, it amounts to saying that at any given level of
complexity, a hypothesis is much more difficult to reduce to a stochastic than to
a deterministic model. Secondly, it so happens that although processes with
highly simplified assumptions can be of great help in epidemiology, it is not the
same, say, in economics or sociology, where even elementary phenomena can-
not be represented by models using highly simplified assumptions.
To summarize, for the convenience of the reader, the difference between
deterministic and stochastic models, it is well to present a simple example of
two equivalent models, of which one is a classical analytical model of deter-
ministic type and the other stochastic. For this we can use the ‘geometrical
progression’ to which Tarde attached great importance for the explanation of
social phenomena.
Let us assume that the increased number of those converted to an innovation,
at any arbitrary moment, varies with the number of people who have already
adopted that innovation at that moment. If we call this number x(t), the
model is expressed by the differential equation :
~- dx(t) - kx(t)
dt
Assuming that the number of converts x(o) at time zero is equal to xo, the
integration of the above equation gives:
x(t) = xoekf
This is a deterministic model: it enables us, in the light of parameters xo and Ir,
to predict the exact number of converts at time t. Now let us consider the sto-
chastic equivalent of this model.
If x(t) is the number of converts at time t, we posit that the Probability of
having gained one more convert after a time interval of (f, t+dt) is proportional
to x(t)dt, and additionally that the probability of gaining two or more in the same
interval of time moves towards zero when At moves towards zero. The prob-
ability of finding exactly x converts after the interval of time (f, t+ At) is then
given by the following expression:
(probability of x converts in t) x (probability of no change after the interval
of time) + (probability of there being x- I proselytes at t) x (probability of
a new convert in the interval) + (probability of two proselytes appearing).
Symbolically, this expression is written :
Px(t + At) = P,(t) (I-kxdt) + Px-,(t) k (x-1)At + o(d t)