Mathematical modeling of optimized SIRS epidemic model and some dynamical behaviors of the solution

In this paper, a generalized mathematical model of spread of infectious disease as sirs epidemicmodel is considered as a nonlinear system of differential equations. we prove that for positive initial conditions the resulting equivalence system has positive solution and under some hypotheses, this system with initial positive condition, has a positive t-periodic solution which is globally asymp- totically stable. for numerical simulations the fourth order runge-kutta method is applied to the nonlinear system of differential equations.