 $\alpha$ becomes smaller or $\dot{M}_{\rm CO}$ becomes higher, a new regime is attained where the CO mass reaches orders of magnitude higher values which can explain the typical CO masses observed with ALMA for shielded discs. The final CO mass in this regime
plateaus because it is then dominated by the viscous evolution since the photodissociation timescale becomes greater than the viscous timescale. In this shielded regime, the C$^0$ mass plateaus because the CO mass input rate at $R_0$ becomes lower owing to CO spreading and thus
less C$^0$ can be produced per unit of time. 

Therefore, this figure shows that there is clearly two regimes of secondary gas production with a sharp transition between gas discs of low CO and C$^0$ masses (e.g. $\beta$ Pic) and high CO and C$^0$ masses (e.g. HD 131835) depending on $\alpha$ and $\dot{M}_{\rm CO}$. However, we notice
that if the gas production only started recently (i.e., $\sim$1 Myr, lowermost thin lines on Fig.~\ref{figcoc1alpha}) and for low values of $\dot{M}_{\rm CO}$, 
a disc that is on its way to becoming shielded could be mistaken for an unshielded low-mass secondary gas disc. This is because before becoming a high
mass gas disc, there is a build-up process, the timescale of which is fixed by the time to reach a C$^0$ mass high enough to start shielding CO as shown in Fig.~\ref{figcoc1evol}. Also, we note that the gas input rate will decrease over time because the dust mass loss rate will 
become smaller owing to collisions that are depleting the belt mass \citep[e.g.][]{2007ApJ...663..365W,2008ApJ...673.1123L}. Thus, a shielded disc can become unshielded again after a certain period of time.

\subsubsection{Application of the refined semi-analytic model to HD 131835}\label{app}

\begin{figure*}
   \centering
   \includegraphics[width=18cm]{Figure_Chi2_predicted_CI_mass_v4.pdf}
   \caption{\label{figco13} Best-fit values to get $M_{\rm CO}=0.04$ M$_\oplus$ as observed with ALMA for HD 131835, in the parameter space $\dot{M}_{\rm CO}$ VS $\alpha$. These results are based on the model presented in Sec.~\ref{refined}. The three columns correspond to different evolution time from when the secondary production of gas has started. 
{\it Top:} The colourscale is showing the CO mass $M_{\rm CO}$ and the contours are plotted for 1, 3 and 5$\sigma$ from the observed value. The yellow region is where C$^0$ does not reach densities that are high enough to shield CO. The blue dashed line is the prediction from the analytical model presented in Sec.~\ref{an}. The green dashed line shows the expected CO production rate $\dot{M}_{\rm CO}$ from the stellar and 
disc parameters \citep[see][]{2017ApJ...842....9M}. 
{\it Bottom:} Total C$^0$ mass needed to produce $M_{\rm CO}=0.04$ M$_\oplus$ in different parts of the parameter space. The red and purple lines are for C$^0$ masses of $2.7 \times 10^{-3}$ and $1.2 \times 10^{-2}$ M$_\oplus$ (predicted range from observations), respectively. We overlay the CO contours from the top plots to check where the best-fit models lie in terms of C$^0$ masses.}
\end{figure*}

We now apply the model presented in the previous section \ref{refined} to the specific case of HD 131835 and try to explain the high observed CO mass of 0.04 M$_\oplus$ in this system. We ran a grid of 1000 models for different $\dot{M}_{\rm CO}$ and $\alpha$ values and in Fig.~\ref{figco13} (top), we compute the predicted CO mass shown in colour scale and plot contours that show where the predicted CO mass is within 1, 3, and 5$\sigma_{\rm CO}$ from the mass inferred from the observations, where the uncertainty $\sigma_{\rm CO}$ is assumed to be 0.5 dex from the observed mass \citep{2017ApJ...849..123M}.
As the models are not always at steady state, we ran simulations for different ages. HD 131835 is $\sim$15 Myr old but the secondary production started after the protoplanetary disc dispersed, i.e., likely after a few Myr. We plot the best-fit models for 3, 10 and 30 Myr.
10 Myr is likely to be the most realistic unless the gas production is linked to a recent event (a few Myr old) and we also show 30 Myr when most of the parameter space is then at steady state. The horizontal part of the best-fit region (i.e., where the contours are horizontal for small $\alpha$) is for models that have not reached 
steady state yet. The yellow region shows where the C$^0$ mass produced through the
secondary evolution is not high enough to shield CO from photodissociating (i.e., it is not the shielded disc regime). In the rest of the parameter space, CO is shielded owing to C$^0$ and the contours show where the CO mass is closest to 0.04 M$_\oplus$. The green dashed line shows the CO production rate predicted from the stellar and dust disc parameters \citep[see][noting that uncertainties are up to a factor 10 on that value]{2017ApJ...842....9M}. 
From the 10 Myr plot, we conclude that $10^{-3}<\alpha<10^{-2}$ can explain the observed CO mass, which is similar to what was found with the more simple analytical model presented in Sec.~\ref{an} as can be seen from the blue dashed line (on top plots of Fig.~\ref{figco13}) representing the analytical model developed in Sec.~\ref{an}.

We then compute the C$^0$ mass that is needed to explain that CO mass on the bottom row of Fig.~\ref{figco13} (the red and purple lines are for $2.7 \times 10^{-3}$ and $1.2 \times 10^{-2}$ M$_\oplus$, respectively, corresponding to the range of plausible masses derived from observations). 
We find that by overplotting the best-fit contours for CO mass from the top plot on the bottom one, then for 10 Myr, the C$^0$ mass needed to explain the observed CO mass is between $5 \times 10^{-3}$ and $10^{-2}$ M$_\oplus$,
which is similar to what we deduced from the [CI] observations in Sec~\ref{gasfit} but is slightly narrower. We find that the best-fit is not necessarily at steady state. If $\alpha \sim 10^{-3}$, the disc surface density is still evolving with time as can be inferred from the fact that the contours are horizontal in this range.
We note that even for larger $\alpha$, the inner and outer regions are not necessarily at steady state (even though it is not on the horizontal part anymore) because the global viscous timescale can be $\sim 10$ times longer than the local viscous timescale for which steady state is 
reached at $R_0$ only \citep{1981ARA&A..19..137P}, i.e., the gas disc can be at steady state in the parent belt but not yet in the inner/outer regions.
As noted in Sec.~\ref{gasfit}, as the outer radial profile of the gas density is very steep, it could be that the gas disc has not reached steady state and thus the global viscous timescale should be $<15$ Myr (the age of the system), which translates as $\alpha<6 \times 10^{-3}$ as already described in Sec.~\ref{an}.
Using Fig.~\ref{figco13} and fixing $\alpha=6 \times 10^{-3}$, we therefore find a maximum $\dot{M}_{\rm CO}$ of 0.3 M$_\oplus$/Myr to still produce a best-fit (at the 99.7\% uncertainty level) in the case when steady state is not yet reached. 



We note that when higher resolution images become available, the present model should be extended to a 1D code that viscously evolves CO and carbon together, since that would allow to compare the predicted radial profiles to that observed.

\subsection{Cometary composition}\label{comcomp}
 
Using Eq.~1 in \citet{2017MNRAS.469..521K}, we find that the dust production rate in this belt is $\sim 1.7$ M$_\oplus$/Myr, where we assumed a ${\rm d}r/r$ of $\sim$0.85 as derived in Sec.~\ref{dustfit}, and the fiducial values 
listed in \citet{2017MNRAS.469..521K}. This in turns leads to a prediction of the CO-to-solid mass ratio of $<20\%$ using the maximum $\dot{M}_{\rm CO}$ of 0.3 M$_\oplus$/Myr that we have just derived in