Xdot == m a b Y (1-X) - r X
Ydot == a c x (Exp[-g n] -Y)- g Y

We now construct the matrices that we need in order to calculate (Subscript[R, 0]).  The first is the matrix that keeps track of new infections.

F1 = {m a b Y,  a c X (Exp[-g n] )}

F=Outer[D,F1,{X,Y}]

MatrixForm[F]

Now do the same thing for V, the matrix of transitions.

V1 = {-r X,-g Y}

V = Outer[D,V1,{X,Y}]

MatrixForm[V]

Generate the next generation matrix G = (FV^(-1)).

G = Dot[F, Inverse[V]]

MatrixForm[G]

RR = Eigenvalues[G]

By inspection, we can see that the second element of RR is the dominant eigenvalue (since it is positive).

R0 = RR[[2]]