Nonstandard numerical methods for a mathematical model for influenza disease

In this paper we construct and develop two competitive implicit finite difference scheme for a deterministic mathematical model associated with the evolution of influenza A disease in human population. Qualitative dynamics of the model is determined by the basic reproduction number, R0. Numerical schemes developed here are based on nonstandard finite difference methods. Our aim is to transfer essential properties of the continuous model to the discrete schemes and to obtain unconditional stable schemes. The proposed numerical schemes have two fixed points which are identical to the critical points of the continuous model and it is shown that they have the same stability properties. Numerical simulations with different initial conditions, parameters values and time step sizes are developed for different values of parameter R0, convergence to the disease free equilibrium point when R0<1 and to the endemic equilibrium point when R0>1 are obtained independent of the time step size. These numerical integration schemes are useful since can reproduce the dynamics of original differential equations.