Stability analysis of nonstandard finite difference discretization of sir epidemic model with non-monotone incidence rate

A conservative nonstandard finite difference (CNSFD) scheme for a SIR epidemic model with non-monotonic incidence rate is constructed by approximating the first order derivative with the generalization of forward scheme and applying a nonlocal approximation for other terms. The denominator function for the generalization of forward scheme is derived using the exact conservation law of SIR epidemic model. It is shown analytically and numerically that the proposed CNSFD scheme preserves the conservation law, positivity condition and the same equilibrium points and their stability properties as those of the associated continuous SIR model, irrespective of step sizes.