%% formulation
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\begin{align*}
\min_{I(t),t_f} \quad & t_f \\
\text{subject to:} \quad & \dot{S} = - \frac{S}{\tau_p} + \Gamma G(N,S) S + \beta B N\left[ N + P_0 \right] \\
& \dot{N} = \frac{I}{q} - R(N) - \Gamma G(N,S) S \\
& (S,N)(0) = (S_0,N_0) \\
& (S,N)(t_f) = (S_f,N_f) \\
& I_{\min} \leq I(t) \leq I_{\max} \\
\text{where:} \quad & G(N,S) = G_p \left[ N - N_{\mathrm{tr}} \right] \left[ 1 - \epsilon  S \right] \\
& R(N) = AN + BN \left[ N + P_0 \right] + CN \left[ N + P_0 \right]^2
\end{align*}
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%% solution