I need to plot this system of differential equations but I don't know how.. Can anyone help?

S_(n+1)(t)=S_n(t)-int_0^t[d/dw S_n(w)- mu*N+beta*S_n(w)*J_n(w)+mu*S_n(w)] dw

E_(n+1)(t)=E_n(t)-int_0^t[d/dw E_n(w) - beta*S_n(w)*J_n(w)+mu*E_n(w)+sigma*E_n(w)]dw

J_(n+1)(t)=J_n(t)-int_0^t[d/dw J_n(w)- sigma*E_n(w)+mu*J_n(w)+gamma1*J_n(w)] dw

R_(n+1)(t)=R_n(t)-int_0^t[d/dw R_n(w)-gamma1*J_n(w)+mu*R_n(w)] dw

with initial value

beta=0.00001

mu=0.02

sigma=45.6

gamma1=73

S(0)=12500000

E(0)=50000

J(0)=30000

R(0)=37420000

S_0(t)=S(0) and the same with E_(0). J_(0), and R_(0). And n is the n-th approximation. 

I have problem with the diff and sum inside the integral..

i need to plot this equation but can't find a way to do so. Can anyone help?

beta parameter and initial value of S(k), E(k) is known, but i have problems at the sum and (k+1)..

Error, More than two dependent variables, please indicate the desired scene