A dynamically consistent NSFD scheme for an SIS epidemic model with vaccination and variable population

This study investigates a sequential nonstandard finite difference (NSFD) approximation of an established SIS vaccination model with variable population. Disease-free and endemic stability are classified by the spectral abscissa in four parameter planes. The continuous model does not exhibit backward bifurcation, and the threshold \(R_0=1\) separates disease elimination from the existence of a biologically feasible endemic equilibrium. The proposed sequential NSFD scheme preserves nonnegativity and the equilibrium set and reproduces the disease-free threshold exactly for every positive step size. First-order consistency and finite-time convergence are established, while discrete stability is examined through the spectral radius of the full discrete Jacobian. Numerical experiments near the threshold, with \(R_0=0.99\) and \(R_0=1.01\) , show the expected slow transients and the transition from decay to persistence. Comparisons with the forward Euler and fourth-order Runge-Kutta methods show that these standard explicit schemes may lose positivity or stability for larger step sizes, whereas the NSFD scheme retains the qualitative properties established analytically. These results provide a structure-preserving numerical treatment of the SIS vaccination model.