English

The primary purpose of this report is to model the behavior of seasonal diseases by systems of differential equations. It also deals with  the study of the dynamic properties, such as positivity, periodicity and stability, of  the analytical solutions and  with the construction of numerical schemes for the calculation of approximate solutions of the systems of first order nonlinear differential equations, which model the behavior of seasonal infectious diseases such
as the transmission of the  Respiratory syncytial virus RSV.

Two mathematical models of seasonal diseases are generalized and we show that solutions are periodic using a Jean Mawhin's Theorem of Coincidence. To corroborate the analytical results, numerical schemes are developed using the non-standard finite difference techniques developed by Ronald Mickens and also by the differential transformation method, which allow us to reproduce the dynamic behavior of the analytical solutions, such as positivity and periodicity.

Finally, numerical simulations are performed using the implemented schemes with parameters derived from clinical data
of the Region of Valencia in persons infected with the virus RSV. These results are compared with those produced using the methods of Euler, Runge-Kutta
and the routine of ODE45 of Matlab.  The new methods ensure better approximations using step sizes larger than those normally used by traditional schemes.