  Over the past several decades there has been a proliferation of
epidemiological models with ordinary derivatives replaced by fractional
derivatives in an an-hoc manner. These models may be mathematically interesting
but their relevance is uncertain. Here we develop an SIR model for an epidemic,
including vital dynamics, from an underlying stochastic process. We show how
fractional differential operators arise naturally in these models whenever the
recovery time from the disease is power law distributed. This can provide a
model for a chronic disease process where individuals who are infected for a
long time are unlikely to recover. The fractional order recovery model is shown
to be consistent with the Kermack-McKendrick age-structured SIR model and it
reduces to the Hethcote-Tudor integral equation SIR model. The derivation from
a stochastic process is extended to discrete time, providing a stable numerical
method for solving the model equations. We have carried out simulations of the
fractional order recovery model showing convergence to equilibrium states. The
number of infecteds in the endemic equilibrium state increases as the
fractional order of the derivative tends to zero.
