How to find the basic reproductive number $R_0$ of a non-autonomous dynamical system? For instance given a dynamical system
$X'=-\delta\frac{X(I+E)}{N}+\epsilon M+\mu N-\mu X$
$I'=\delta\frac{X(I+E)}{N}-(1-sinBt)\alpha_1 I+(1+sin Bt)\alpha_2 E-\gamma I-\mu I$
$E'=(1-sinBt)\alpha_1 I-(1+sin Bt)\alpha_2 E-\mu E$
$M'=\gamma I-\epsilon M-\mu M$.
Can someone show me how to solve the $R_0$ of this dynamical system? or give any references that would lead in solving the $R_0$ of this system.