Numerical Modeling of the Transmission Dynamics of Influenza

A deterministic mathematical model for the transmission dynamics of Influenza is studied. Although the equilibria of the model could not be expressed in closed form, their existence and threshold conditions for their stability are theoretically investigated. It is shown that the diseasefree equilibrium is locally–asymptotically stable if the basic reproductive number R0 < 1 (thus, Influenza can be eradicated from the community) and unstable if R0 > 1 (leading to the persistence of Influenza within the community). A competitive finite-difference method will be constructed and used for the solution of the resulting system of first-order, non-linear, initial-value problem (IVP). Unlike the fourth-order Runge-Kutta method (RK4), which fails when the discretization parameters exceed certain values, the novel numerical method to be developed in this paper gives convergent results for all parameter values. The introduction of seasonal variation into the model leads to periodic and chaotic dynamics of epidemics which are present in the numerical simulations.