r_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\mathbbm{R}\mathbbm{T}_{t}\bigg{(}-\underbrace{rt_{t+1}+\bigg{(}rt_{t}+\frac{1-m_{i}}{t_{i}}it_{t}+\frac{1-m_{h}}{h_{t}}h_{t}+\tau_{t}\gamma\bigg{(}r_{t}+\frac{1-m_{i}}{t_{i}}i_{t}\bigg{)}\bigg{)}}_{=\Delta
rt_{t}}\bigg{)}+$
$\displaystyle+\sum_{t=1}^{\infty}\mathbbm{H}_{t}\bigg{(}-\underbrace{h_{t+1}+\bigg{(}\bigg{(}1-\frac{1}{t_{h}}\bigg{)}h_{t}+\frac{m_{i}}{t_{i}}(i_{t}+it_{t})\bigg{)}}_{\Delta
h_{t}}\bigg{)},$

where we used our standard shorthand notations: $s_{t}=\sum_{k\leqslant
t}s_{t}^{k}$, $i_{t}=\sum_{k\leqslant t}i_{t}^{k}$, $r_{t}=\sum_{k\leqslant
t}r_{t}^{k}$, $\widehat{s}_{t}=\sum_{k\leqslant t}\alpha_{t}^{k}s_{t}^{k}$,
$\widehat{i}_{t}=\sum_{k\leqslant t}\alpha_{t}^{k}i_{t}^{k}$,
$\widehat{r}_{t}=\sum_{k\leqslant t}\alpha_{t}^{k}r_{t}^{k}$,
$y_{t}=\lambda_{t}(s_{t}+i_{r}+r_{t}+rt_{t})$ and
$d_{t}=1-(s_{t}+i_{t}+it_{t}+r_{t}+rt_{t}+h_{t})$.

The optimal level of social distancing is characterized by the necessary
first-order condition ($FOC-\alpha_{t}^{k}$). For completeness, we copy the
condition:

$\displaystyle
1-\alpha^{k}_{t}+\frac{\kappa}{s_{t}^{k}+i_{t}^{k}+r_{t}^{k}}\bigg{(}\frac{1}{\alpha^{k}_{t}}-\frac{1}{1-\alpha^{k}_{t}}\bigg{)}$
$\displaystyle=\frac{\phi^{+}}{c}\frac{s^{k}_{t}}{s^{k}_{t}+i^{k}_{t}+r^{k}_{t}}\beta_{s}{\widehat{I}_{t}}+\frac{\phi^{-}}{c}\frac{i^{k}_{t}}{s^{k}_{t}+i^{k}_{t}+r^{k}_{t}}\beta_{s}{\widehat{S}_{t}}+\frac{\Delta_{t}^{k}}{c(1-\delta_{A})\delta_{A}^{t-1}q_{t}}\frac{s_{t}^{k}}{s_{t}^{k}+i_{t}^{k}+r_{t}^{k}}\beta_{s}{\widehat{I}_{t}},$

where
$\Delta_{t}^{k}=(1-\tau_{t}\gamma)\mathbbm{S}^{k}_{t}+\tau_{t}\gamma\mathbbm{S}^{t+1}_{t}-(1-\tau_{t})\mathbbm{I}^{k}_{t}-\tau_{t}\mathbbm{I}\mathbbm{T}_{t}$.
It is easy to see that the equation above admits a unique solution for small
enough $\kappa>0$.313131In fact, Equation ($FOC-\alpha_{t}^{k}$) can be
rewritten as
$\alpha_{t}^{k}=A_{t}^{k}+\frac{\kappa}{s_{t}^{k}+i_{t}^{k}+r_{t}^{k}}\bigg{(}\frac{1}{\alpha_{t}^{k}}-\frac{1}{1-\alpha_{t}^{k}}\bigg{)}$
for a certain number $A_{t}^{k}$, which is independent of $\kappa$. The
equation defines a polynomial of degree 3. Then, it is possible to show that
the solution in trigonometric functions can be obtained using the Viete’s
formula, and that $\alpha_{t}^{k}$ converges to
$\max\\{0,\min\\{1,A_{t}^{k}\\}\\}$ as $\kappa\downarrow 0$.

The remaining first-order conditions with respect to the vector of state
variables are as follows:

$\displaystyle\mathbbm{S}^{k}_{t}$
$\displaystyle=(1-\tau_{t+1}\gamma)\mathbbm{S}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{S}^{t+2}_{t+1}-\Delta^{k}_{t+1}\left(\beta_{w}\lambda^{2}_{t+1}I_{t+1}+\beta_{s}\widehat{I}_{t+1}\alpha^{k}_{t+1}\right)+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}+$
$\displaystyle+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\left(1-(1-\alpha^{k}_{t+1})^{2}\right)-\phi^{+}\left(\beta_{w}\lambda^{2}_{t+1}I_{t+1}+\beta_{s}\alpha^{k}_{t+1}\widehat{I}_{t+1}\right)\bigg{)}q_{t+1},\;\;\tiny{k=1,\ldots,t+1}$
($\Delta\mathbbm{S}_{t}^{k}$) $\displaystyle\mathbbm{I}^{k}_{t}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\bigg{(}(1-\tau_{t+1})\mathbbm{I}^{k}_{t+1}+\tau_{t+1}\mathbbm{I}\mathbbm{T}_{t+1}\bigg{)}+\frac{1-m_{i}}{t_{i}}\bigg{(}(1-\tau_{t+1}\gamma)\mathbbm{R}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{R}\mathbbm{T}_{t+1}\bigg{)}+\frac{m_{i}}{t_{i}}\mathbbm{H}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}+$
$\displaystyle+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\left(1-(1-\alpha^{k}_{t+1})^{2}\right)-\phi^{-}\left(\beta_{w}\lambda^{2}_{t+1}S_{t+1}+\beta_{s}\alpha^{k}_{t+1}\widehat{S}_{t+1}\right)\bigg{)}q_{t+1},\;\;\tiny{k=1,\ldots,t}$
($\Delta\mathbbm{I}_{t}^{k}$) $\displaystyle\mathbbm{I}\mathbbm{T}_{t}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\mathbbm{I}\mathbbm{T}_{t+1}+\frac{1-m_{i}}{t_{i}}\mathbbm{R}\mathbbm{T}_{t+1}+\frac{m_{i}}{t_{i}}\mathbbm{H}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t},$
($\Delta\mathbbm{I}\mathbbm{T}_{t}$) $\displaystyle\mathbbm{R}^{k}_{t}$
$\displaystyle=(1-\tau_{t+1}\gamma)\mathbbm{R}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{R}\mathbbm{T}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\bigg{(}1-(1-\alpha^{k}_{t+1})^{2}\bigg{)}\bigg{)}q_{t+1},\;\;\tiny{k=1,\ldots,t}$
($\Delta\mathbbm{R}_{t}^{k}$) $\displaystyle\mathbbm{R}\mathbbm{T}_{t}$
$\displaystyle=\mathbbm{R}\mathbbm{T}_{t+1}+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\bigg{)}q_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t},$
($\Delta\mathbbm{R}\mathbbm{T}_{t}$) $\displaystyle\mathbbm{H}_{t}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}\mathbbm{H}_{t+1}+\frac{1-m_{h}}{t_{h}}\mathbbm{R}\mathbbm{T}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}.$
($\Delta\mathbbm{H}_{t}$)

As discussed in Section 4.2, the system of adjoint equations is backwards with
the following boundary condition:

$\lim_{t\to\infty}\frac{\mathbbm{S}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{I}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{R}^{k}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{I}\mathbbm{T}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{R}\mathbbm{T}_{t}}{p_{t}\delta_{A}^{t-1}}=\lim_{t\to\infty}\frac{\mathbbm{H}_{t}}{p_{t}\delta_{A}^{t-1}}=\delta_{A}\left(1+\frac{c}{2}\right)\;\forall
k.$

As usual, we write
$\Delta\Upsilon_{t}:=\big{(}(\Delta\mathbbm{S}_{t}^{k})_{k\leqslant
t+1},(\Delta\mathbbm{I}_{t}^{k})_{k\leqslant
t},\Delta\mathbbm{I}\mathbbm{T}_{t},(\Delta\mathbbm{R}_{t}^{k})_{k\leqslant
t},\Delta\mathbbm{R}\mathbbm{T}_{t},\Delta\mathbbm{H}_{t}\big{)}$ for the
system of adjoint equations. For the fixed aggregates
$\big{(}S_{t},I_{t},\widehat{S}_{t},\widehat{I}_{t}\big{)}$ and policies
$(\lambda_{t},\tau_{t})$, Equations ($\Delta\pi_{t}$), ($\Delta\Upsilon_{t}$)
and ($FOC-\alpha_{t}^{k}$) determine a representative agent’s best response,
that is her rate of social distancing $(\alpha_{t}^{k})_{k\leqslant t}$ as
well as the individual state and adjoint vectors, $\pi_{t}$ and
$\Upsilon_{t}$, respectively. Since the agents are ex-ante identical, the Law
of large numbers implies that in equilibrium the individual state variables
match exactly the aggregate state variables, that is $\pi_{t}=\Pi_{t}$, in
particular:
$\big{(}s_{t},i_{t},\widehat{s}_{t},\widehat{i}_{t}\big{)}=\big{(}S_{t},I_{t},\widehat{S}_{t},\widehat{I}_{t}\big{)}$.

###  10.2 Derivation of the government’s optimal policy

The government’s optimization problem is described in details in Section 5:
The government chooses the rates of lockdown, social distancing and testing as
well as the aggregate state variables and agent’s adjoint variables to
maximize her own payoff. In this task the government is constrained by the
fact every agent must indeed find it in her interest to follow the designated
social distancing policy, that is the agent’s optimality conditions must be
respected. In addition, the total test count must be exactly equal to the
capacity, i.e., $\overline{X}_{t}$. Thus, we arrive at the following problem:

$\displaystyle\max\limits_{\lambda_{t},(\alpha_{t}^{k})_{k\leqslant
t},\tau_{t},\Pi_{t},\Upsilon_{t}}\quad Y_{G}+\xi\cdot(1-D)\quad\text{subject
to}\quad\text{\hyperref@@ii[eqpi]{$(\Delta\Pi_{t})$}},\;\text{\hyperref@@ii[equps
A]{$(\Delta\Upsilon_{t})$}},\;\text{(\ref{agent_foc})}\;\text{and}\;\text{(\ref{eq
testing capacity})}.$

We now set up the Lagrangian for the problem denoting the undiscounted dual
variables associated with Equations $(\Delta\Pi_{t})$ by
$\overline{\Upsilon}_{t}=\big{(}(\overline{\mathbbm{S}}^{k}_{t})_{k\leqslant
t+1},(\overline{\mathbbm{I}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{IT}}_{t},(\overline{\mathbbm{R}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{RT}}_{t},\overline{\mathbbm{H}}_{t}\big{)}$ and those
associated with Equations $(\Delta\Upsilon_{t})$ by
$\big{(}(\overline{S}_{t}^{k})_{k\leqslant
t},(\overline{I}_{t}^{k})_{k\leqslant
t},\overline{IT}_{t},(\overline{R}_{t}^{k})_{k\leqslant
t},\overline{RT}_{t},\overline{H}_{t}\big{)}$. In addition, we let
$(\eta_{t}^{k})_{k\leqslant t}$ and $\chi_{t}$ be the Lagrange multipliers on
($FOC-\alpha_{t}^{k}$) and ($F-\tau_{t}$), respectively. Taking all pieces
together, the Lagrangian can be written as follows:

$\displaystyle\underbrace{(1-\delta_{G})\sum_{t=1}^{\infty}\delta_{G}^{t-1}\bigg{(}Y_{t}q_{t}+\frac{\delta_{G}+\xi/\delta_{G}^{t-1}}{1-\delta_{G}}(1-D_{t+1})p_{t}\bigg{)}}_{Y_{G}}+\sum_{t=1}^{\infty}\chi_{t}\bigg{(}\underbrace{-\overline{X}_{t}+\tau_{t}\bigg{(}\gamma\bigg{(}S_{t}-\beta_{w}\lambda^{2}_{t}S_{t}I_{t}-\beta_{s}\widehat{S}_{t}\widehat{I}_{t}\bigg{)}+\gamma\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)}+\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}\widehat{S}_{t}\widehat{I}_{t}\bigg{)}}_{F-\tau_{t}}\bigg{)}+$
$\displaystyle+$ $\displaystyle\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\eta_{t}^{k}\underbrace{\bigg{(}(1-\delta_{A})c\delta_{A}^{t-1}\bigg{(}(1-\alpha_{t}^{k})(S_{t}^{k}+I_{t}^{k}+R_{t}^{k})+\kappa\bigg{(}\frac{1}{\alpha^{k}_{t}}-\frac{1}{1-\alpha^{k}_{t}}\bigg{)}\bigg{)}q_{t}-(1-\delta_{A})\delta_{A}^{t-1}\beta_{s}\left(\phi^{+}S_{t}^{k}\widehat{I}_{t}+\phi^{-}I_{t}^{k}\widehat{S}_{t}\right)-\Delta_{t}^{k}\beta_{s}S_{t}^{k}\widehat{I}_{t}\bigg{)}}_{FOC-\alpha_{t}^{k}}+$
$\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\overline{\mathbbm{S}}^{k}_{t}\bigg{(}-\underbrace{S^{k}_{t+1}+(1-\tau_{t}\gamma)\bigg{(}S^{k}_{t}-\beta_{w}\lambda^{2}_{t}S^{k}_{t}I_{t}-\beta_{s}\alpha^{k}_{t}S^{k}_{t}\widehat{I_{t}}\bigg{)}}_{\Delta
S_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\overline{\mathbbm{S}}^{t+1}_{t}\bigg{(}-\underbrace{S^{t+1}_{t+1}+\tau_{t}\gamma\bigg{(}S_{t}-\beta_{w}\lambda^{2}_{t}S_{t}I_{t}-\beta_{s}\widehat{S_{t}}\widehat{I_{t}}\bigg{)}}_{\Delta
S_{t}^{t+1}}\bigg{)}+$ $\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\overline{\mathbbm{I}}^{k}_{t}\bigg{(}-\underbrace{I^{k}_{t+1}+(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I^{k}_{t}+\beta_{w}\lambda^{2}_{t}S^{k}_{t}I_{t}+\beta_{s}\alpha^{k}_{t}S^{k}_{t}\widehat{I_{t}}\bigg{)}}_{\Delta
I_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\overline{\mathbbm{IT}}_{t}\bigg{(}-\underbrace{IT_{t+1}+\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}IT_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}\widehat{S_{t}}\widehat{I_{t}}\bigg{)}\bigg{)}}_{\Delta
IT_{t}}\bigg{)}+$ $\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t}\overline{\mathbbm{R}}^{k}_{t}\bigg{(}-\underbrace{R^{k}_{t+1}+\bigg{(}(1-\tau_{t}\gamma)\bigg{(}R^{k}_{t}+\frac{1-m_{i}}{t_{i}}I^{k}_{t}\bigg{)}\bigg{)}}_{\Delta
R_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\overline{\mathbbm{RT}}_{t}\bigg{(}-\underbrace{RT_{t+1}+\bigg{(}RT_{t}+\frac{1-m_{i}}{t_{i}}IT_{t}+\frac{1-m_{h}}{h_{t}}H_{t}+\tau_{t}\gamma\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)}\bigg{)}}_{=\Delta
RT_{t}}\bigg{)}+$
$\displaystyle+\sum_{t=1}^{\infty}\overline{\mathbbm{H}}_{t}\bigg{(}-\underbrace{H_{t+1}+\bigg{(}\bigg{(}1-\frac{1}{t_{h}}\bigg{)}H_{t}+\frac{m_{i}}{t_{i}}(I_{t}+IT_{t})\bigg{)}}_{\Delta
H_{t}}\bigg{)}+\sum_{t=1}^{\infty}\sum_{k\leqslant
t+1}\overline{S}_{t+1}^{k}\bigg{(}\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{-\mathbbm{S}^{k}_{t}+\bigg{(}(1-\tau_{t+1}\gamma)\mathbbm{S}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{S}^{t+2}_{t+1}-\Delta^{k}_{t+1}\left(\beta_{w}\lambda^{2}_{t+1}I_{t+1}+\beta_{s}\widehat{I}_{t+1}\alpha^{k}_{t+1}\right)+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\bracelu\leaders\vrule\@height\z@\@depth\z@\hfill\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@$\crcr\kern 3.0pt\p@\cr}}}\limits+$
$\displaystyle+\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\left(1-(1-\alpha^{k}_{t+1})^{2}\right)-\phi^{+}\left(\beta_{w}\lambda^{2}_{t+1}I_{t+1}+\beta_{s}\alpha^{k}_{t+1}\widehat{I}_{t+1}\right)\bigg{)}q_{t+1}\bigg{)}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\bracerd\braceld\leaders\vrule\@height\z@\@depth\z@\hfill\braceru$\crcr\kern
3.0pt\p@\cr}}}\limits_{\Delta\mathbbm{S}_{t}^{k}}\bigg{)}+$
$\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t+1}\overline{I}_{t+1}^{k}\bigg{(}\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{-\mathbbm{I}^{k}_{t}+\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\bigg{(}(1-\tau_{t+1})\mathbbm{I}^{k}_{t+1}+\tau_{t+1}\mathbbm{I}\mathbbm{T}_{t+1}\bigg{)}+\frac{1-m_{i}}{t_{i}}\bigg{(}(1-\tau_{t+1}\gamma)\mathbbm{R}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{R}\mathbbm{T}_{t+1}\bigg{)}+\frac{m_{i}}{t_{i}}\mathbbm{H}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\bracelu\leaders\vrule\@height\z@\@depth\z@\hfill\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@$\crcr\kern 3.0pt\p@\cr}}}\limits+$
$\displaystyle+\mathop{\vtop{\m@th\halign{#\cr$\hfil\displaystyle{(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\left(1-(1-\alpha^{k}_{t+1})^{2}\right)-\phi^{-}\left(\beta_{w}\lambda^{2}_{t+1}S_{t+1}+\beta_{s}\alpha^{k}_{t+1}\widehat{S}_{t+1}\right)\bigg{)}q_{t+1}\bigg{)}}\hfil$\crcr\kern
3.0pt\p@\nointerlineskip\cr$\m@th@\hbox{$\braceld$}\vrule\@width\p@\kern
0.0pt\p@\vrule\@width\p@\kern 0.0pt\p@\vrule\@width\p@\kern
0.0pt\p@\bracerd\braceld\leaders\vrule\@height\z@\@depth\z@\hfill\braceru$\crcr\kern
3.0pt\p@\cr}}}\limits_{\Delta\mathbbm{I}_{t}^{k}}\bigg{)}+\sum_{t=1}^{\infty}\overline{IT}_{t+1}\bigg{(}\underbrace{-\mathbbm{I}\mathbbm{T}_{t}+\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\mathbbm{I}\mathbbm{T}_{t+1}+\frac{1-m_{i}}{t_{i}}\mathbbm{R}\mathbbm{T}_{t+1}+\frac{m_{i}}{t_{i}}\mathbbm{H}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}\bigg{)}}_{\Delta\mathbbm{I}\mathbbm{T}_{t}}\bigg{)}+$
$\displaystyle+\sum_{t=1}^{\infty}\sum_{k\leqslant
t+1}\overline{R}_{t+1}^{k}\bigg{(}\underbrace{-\mathbbm{R}^{k}_{t}+\bigg{(}(1-\tau_{t+1}\gamma)\mathbbm{R}^{k}_{t+1}+\tau_{t+1}\gamma\mathbbm{R}\mathbbm{T}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\bigg{(}1-(1-\alpha^{k}_{t+1})^{2}\bigg{)}\bigg{)}q_{t+1}\bigg{)}}_{\Delta\mathbbm{R}_{t}^{k}}\bigg{)}+$
$\displaystyle+\sum_{t=1}^{\infty}\overline{RT}_{t+1}\bigg{(}\underbrace{-\mathbbm{R}\mathbbm{T}_{t}+\bigg{(}\mathbbm{R}\mathbbm{T}_{t+1}+(1-\delta_{A})\delta_{A}^{t}\bigg{(}\lambda_{t+1}+\frac{c}{2}\bigg{)}q_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}\bigg{)}}_{\Delta\mathbbm{R}\mathbbm{T}_{t}}\bigg{)}+\sum_{t=1}^{\infty}\overline{H}_{t+1}\bigg{(}\underbrace{-\mathbbm{H}_{t}+\bigg{(}\bigg{(}1-\frac{1}{t_{h}}\bigg{)}\mathbbm{H}_{t+1}+\frac{1-m_{h}}{t_{h}}\mathbbm{R}\mathbbm{T}_{t+1}+\delta_{A}^{t}\bigg{(}1+\frac{c}{2}\bigg{)}p_{t}\bigg{)}}_{\Delta\mathbbm{H}_{t}}\bigg{)},$

In the problem above the following short notations are used:
$S_{t}=\sum_{k\leqslant t}S_{t}^{k}$, $I_{t}=\sum_{k\leqslant t}I_{t}^{k}$,
$R_{t}=\sum_{k\leqslant t}R_{t}^{k}$, $\widehat{S}_{t}=\sum_{k\leqslant
t}\alpha_{t}^{k}S_{t}^{k}$, $\widehat{I}_{t}=\sum_{k\leqslant
t}\alpha_{t}^{k}I_{t}^{k}$, $\widehat{r}_{t}=\sum_{k\leqslant
t}\alpha_{t}^{k}R_{t}^{k}$, $Y_{t}=\lambda_{t}(S_{t}+I_{r}+R_{t}+RT_{t})$,
$\overline{Y}_{t}=\lambda_{t}(\overline{S}_{t}+\overline{I}_{t}+\overline{R}_{t}+\overline{RT}_{t})$
and $D_{t}=1-(S_{t}+I_{t}+IT_{t}+R_{t}+RT_{t}+H_{t})$. In addition, we let
$\Delta_{t}^{k}=(1-\tau_{t}\gamma)\mathbbm{S}^{k}_{t}+\tau_{t}\gamma\mathbbm{S}^{t+1}_{t}-(1-\tau_{t})\mathbbm{I}^{k}_{t}-\tau_{t}\mathbbm{I}\mathbbm{T}_{t}$
and
$\overline{\Delta}_{t}^{k}=(1-\tau_{t}\gamma)\overline{\mathbbm{S}}^{k}_{t}+\tau_{t}\gamma\overline{\mathbbm{S}}^{t+1}_{t}-(1-\tau_{t})\overline{\mathbbm{I}}^{k}_{t}-\tau_{t}\overline{\mathbbm{IT}}_{t}$.

The optimal lockdown is a unique maximizer of Equation (OPT-$\lambda_{t}$),
which is given by

$\displaystyle\lambda_{t}$
$\displaystyle=\arg\max\limits_{\lambda\in[\overline{\lambda},1]}\;\bigg{\\{}\underbrace{\bigg{(}(1-\delta_{G})\delta_{G}^{t-1}Y_{t}q_{t}+(1-\delta_{A})\delta_{A}^{t-1}\overline{Y}_{t}q_{t}\bigg{)}}_{=:a_{t}}\lambda-$
(OPT-$\lambda_{t}$)
$\displaystyle-\underbrace{\bigg{(}\beta_{w}\sum_{k\leqslant
t}\bigg{(}\overline{\Delta}_{t}^{k}S_{t}^{k}I_{t}+\Delta_{t}^{k}\overline{S}_{t}^{k}I_{t}\bigg{)}+\beta_{w}\chi_{t}\tau_{t}(1-\gamma)S_{t}I_{t}+\beta_{w}(1-\delta_{A})\delta_{A}^{t-1}\bigg{(}\phi^{+}\overline{S}_{t}I_{t}+\phi^{-}S_{t}\overline{I}_{t}\bigg{)}q_{t}\bigg{)}}_{=:b_{t}}\lambda^{2}\bigg{\\}}.$

Denote by $a_{t}$ and $b_{t}$ the first and second coefficients in Equation
(OPT-$\lambda_{t}$), respectively. Then, the optimal value of $\lambda_{t}$
can be succinctly written as

$\lambda_{t}=\begin{cases}1&\text{if}\;b_{t}\leqslant 0,\\\
\min\left\\{1,\max\left\\{0,\overline{\lambda},\frac{a_{t}}{2b_{2}}\right\\}\right\\}&\text{if}\;b_{t}>0.\end{cases}$

As for the other remaining control variables: The first-order conditions with
respect to the testing rate $\tau_{t}$ and social distancing
$(\alpha_{t}^{k})_{k\leqslant t}$ pin down the Lagrange multipliers on the
corresponding constraints, $\chi_{t}$ and $(\eta_{t}^{k})_{k\leqslant t}$,
respectively. The former is given by the following:

$\displaystyle\chi_{t}\bigg{(}\gamma
S_{t}+\bigg{(}1-\frac{1}{t_{i}}\bigg{)}I_{t}+\gamma\bigg{(}R_{t}+\frac{1-m_{i}}{t_{i}}I_{t}\bigg{)}+(1-\gamma)\left(\beta_{w}\lambda^{2}_{t}S_{t}I_{t}+\beta_{s}\widehat{S}_{t}\widehat{I}_{t}\right)\bigg{)}=$
(OPT-$\tau_{t}$) $\displaystyle=\bigg{(}\sum_{k\leqslant
t}\bigg{(}\gamma\left(\overline{\mathbbm{S}}_{t}^{k}-\overline{\mathbbm{S}}_{t}^{t+1}\right)S_{t}^{k}+\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\left(\overline{\mathbbm{I}}_{t}^{k}-\overline{\mathbbm{IT}}_{t}\right)I_{t}^{k}+\gamma\left(\overline{\mathbbm{R}}_{t}^{k}-\overline{\mathbbm{RT}}_{t}\right)\bigg{(}R_{t}^{k}+\bigg{(}1-\frac{1-m_{i}}{t_{i}}\bigg{)}I_{t}^{k}\bigg{)}\bigg{)}+$
$\displaystyle+\sum_{k\leqslant
t}\bigg{(}\gamma\left(\mathbbm{S}_{t}^{k}-\mathbbm{S}_{t}^{t+1}\right)\overline{S}_{t}^{k}+\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\left(\mathbbm{I}_{t}^{k}-\mathbbm{I}\mathbbm{T}_{t}\right)\overline{I}_{t}^{k}+\gamma\left(\mathbbm{R}_{t}^{k}-\mathbbm{R}\mathbbm{T}_{t}\right)\bigg{(}\overline{R}_{t}^{k}+\bigg{(}1-\frac{1-m_{i}}{t_{i}}\bigg{)}\overline{I}_{t}^{k}\bigg{)}\bigg{)}+$
$\displaystyle+\left(\overline{\mathbbm{I}}_{t}^{k}-\overline{\mathbbm{IT}}_{t}-\gamma\left(\overline{\mathbbm{S}}_{t}^{k}-\overline{\mathbbm{S}}_{t}^{t+1}\right)\right)\left(\beta_{w}\lambda^{2}S_{t}^{k}I_{t}+\beta_{s}\alpha_{t}^{k}+\eta_{t}^{k})S_{t}^{k}\widehat{I}_{t}\right)+\left(\mathbbm{I}_{t}^{k}-\mathbbm{I}\mathbbm{T}_{t}-\gamma\left(\mathbbm{S}_{t}^{k}-\mathbbm{S}_{t}^{t+1}\right)\right)\left(\beta_{w}\lambda^{2}\overline{S}_{t}^{k}I_{t}+\beta_{s}(\alpha_{t}^{k}+\eta_{t}^{k})k\overline{S}_{t}^{k}\widehat{I}_{t}\right)\bigg{)}.$

We note that Equation (OPT-$\tau_{t}$) uniquely determines the Lagrange
multiplier $\chi_{t}$ for the interior choice of $\tau_{t}\in(0,1)$. Recall
that the testing rate is determined by Equation ($F-\tau_{t}$) in a way that
exactly $\overline{X}_{t}$ are conducted at date $t$. As a result, as long as
$\overline{X}_{t}$ is positive and not too large, the implied testing rate is
indeed interior.

The first-order condition of the government’s problem with respect to the rate
of social distancing is as follows:

$\displaystyle\eta_{t}^{k}(1-\delta_{A})\delta_{A}^{t-1}c\bigg{(}S_{t}^{k}+I_{t}^{k}+R_{t}^{k}+\kappa\bigg{(}\frac{1}{(\alpha_{t}^{k})^{2}}+\frac{1}{(1-\alpha_{t}^{k})^{2}}\bigg{)}\bigg{)}q_{t}+\sum_{m\leqslant
t}\eta_{t}^{m}\bigg{(}(1-\delta_{A})\delta_{A}^{t-1}\beta_{s}\left(\phi^{+}S_{t}^{m}I_{t}^{k}+\phi^{-}I_{t}^{m}S_{t}^{k}\right)q_{t}-\Delta_{t}^{m}\beta_{s}S_{t}^{m}I_{t}^{k}\bigg{)}=$
$\displaystyle=\chi_{t}\tau_{t}(1-\gamma)\beta_{s}\left(S_{t}^{k}\widehat{I}_{t}+I_{t}^{k}\widehat{S}_{t}\right)-\beta_{s}\left(\overline{\Delta}_{t}^{k}S_{t}^{k}+\Delta_{t}^{k}\overline{S}_{t}^{k}\right)\widehat{I}_{t}-\beta_{s}\sum_{m\leqslant
t}\left(\overline{\Delta}_{t}^{m}S_{t}^{m}+\Delta_{t}^{m}\overline{S}_{t}^{m}\right)\widehat{I}_{t}^{k}+(1-\delta_{A})\delta_{A}^{t-1}c(1-\alpha_{t}^{k})\left(\overline{S}_{t}^{k}+\overline{I}_{t}^{k}+\overline{R}_{t}^{k}\right)q_{t}-$
$\displaystyle-(1-\delta_{A})\delta_{A}^{t-1}\beta_{s}\bigg{(}\phi^{+}\overline{S}_{t}^{k}\widehat{I}_{t}+\phi^{-}\overline{I}_{t}^{k}\widehat{S}_{t}+\phi^{+}\widehat{\overline{S}_{t}}I_{t}^{k}+\phi^{-}\widehat{\overline{I}_{t}}S_{t}^{k}\bigg{)}q_{t}.$
(OPT-$\alpha_{t}^{k}$)

Equation (OPT-$\alpha_{t}^{k}$) can be used to solve for the Lagrange variable
$(\eta_{t}^{k})_{k\leqslant t}$. According to the aforementioned equation,
$(\eta_{t}^{k})_{k\leqslant t}$ must be a solution to the linear system, i.e.,
$\mathcal{A}_{t}\vec{\eta_{t}}=\mathcal{B}_{t}$ for the certain $t\times t$
matrix $\mathcal{A}_{t}$ and $t\times 1$ vector $\mathcal{B}_{t}$. Thus, as
long as the matrix $\mathcal{A}_{t}$ is invertible the solution is unique, and
it is given by

$\vec{\eta_{t}}=\mathcal{A}_{t}^{-1}\mathcal{B}_{t}.$

Moving on, recall that the vector of agent’s adjoints
$\Upsilon_{t}=\big{(}(\mathbbm{S}^{k}_{t})_{k\leqslant
t+1},(\mathbbm{I}^{k}_{t})_{k\leqslant
t},\mathbbm{I}\mathbbm{T}_{t},(\mathbbm{R}^{k}_{t})_{k\leqslant
t},\mathbbm{R}\mathbbm{T}_{t},\mathbbm{H}_{t}\big{)}$ is associated to the
Lagrange variables
$\overline{\Delta\Pi}_{t}=\big{(}(\overline{S}_{t}^{k})_{k\leqslant
t},(\overline{I}_{t}^{k})_{k\leqslant
t},\overline{IT}_{t},(\overline{R}_{t}^{k})_{k\leqslant
t},\overline{RT}_{t},\overline{H}_{t}\big{)}$. The first-order condition with
respect to $\Upsilon_{t}$ specifies the forward system of equations (see
$(\Delta\overline{\Pi}_{t})$) describing the dynamics of $\overline{\Pi}_{t}$.
For completeness, we copy the system from the main text:

$\displaystyle\overline{S}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}\overline{S}^{k}_{t}-\beta_{w}\lambda^{2}_{t}\overline{S}^{k}_{t}I_{t}-\beta_{s}\bigg{(}\alpha^{k}_{t}\overline{S}^{k}_{t}-\eta^{k}_{t}{S}^{k}_{t}\bigg{)}\widehat{I}_{t}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{S}_{t}^{k}$)
$\displaystyle\overline{S}^{t+1}_{t+1}$
$\displaystyle=\tau_{t}\gamma\bigg{(}\overline{S}_{t}-\beta_{w}\lambda^{2}_{t}\overline{S}_{t}I_{t}-\beta_{s}\bigg{(}\widehat{\overline{S}}_{t}-\sum_{k\leqslant
t}\eta_{t}^{k}S_{t}^{k}\bigg{)}\widehat{I}_{t}\bigg{)},$
($\Delta\overline{S}_{t}^{t+1}$) $\displaystyle\overline{I}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t})\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{I}^{k}_{t}+\beta_{w}\lambda^{2}_{t}\overline{S}^{k}_{t}I_{t}+\beta_{s}\bigg{(}\alpha^{k}_{t}\overline{S}^{k}_{t}-\eta^{k}_{t}{S}^{k}_{t}\bigg{)}\widehat{I}_{t}\bigg{)}\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{I}_{t}^{k}$)
$\displaystyle\overline{IT}_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{IT}_{t}+\tau_{t}\bigg{(}\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{I}_{t}+\beta_{w}\lambda^{2}_{t}\overline{S}_{t}I_{t}+\beta_{s}\bigg{(}\widehat{\overline{S}}_{t}-\sum_{k\leqslant
t}\eta_{t}^{k}S_{t}^{k}\bigg{)}\widehat{I}_{t}\bigg{)},$
($\Delta\overline{IT}_{t}$) $\displaystyle\overline{R}^{k}_{t+1}$
$\displaystyle=(1-\tau_{t}\gamma)\bigg{(}\overline{R}^{k}_{t}+\frac{1-m_{i}}{t_{i}}\overline{I}^{k}_{t}\bigg{)},\;\;\;\text{for
}k=1,\ldots,t,$ ($\Delta\overline{R}_{t}^{k}$)
$\displaystyle\overline{RT}_{t+1}$
$\displaystyle=\overline{RT}_{t}+\frac{1-m_{i}}{t_{i}}\overline{IT}_{t}+\frac{1-m_{h}}{h_{t}}\overline{H}_{t}+\tau_{t}\gamma\bigg{(}\overline{R}_{t}+\frac{1-m_{i}}{t_{i}}\overline{I}_{t}\bigg{)},$
($\Delta\overline{RT}_{t}$) $\displaystyle\overline{H}_{t+1}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}\overline{H}_{t}+\frac{m_{i}}{t_{i}}(\overline{I}_{t}+\overline{IT}_{t}),$
($\Delta\overline{H}_{t}$)

and $\overline{I}_{t+1}^{t+1}=\overline{R}_{t+1}^{t+1}$ at all dates. The
initial condition is that $\overline{\Pi}_{1}=0$.

It remains only to identify the first-order condition with respect to the
state variables. The vector of state variables
$\Pi_{t}=\big{(}(S_{t}^{k})_{k\leqslant t},(I_{t}^{k})_{k\leqslant
t},IT_{t},(R_{t}^{k})_{k\leqslant t},RT_{t},H_{t}\big{)}$ is associated to the
vector of Lagrange multipliers $\overline{\Upsilon}_{t}$, which is
$\overline{\Upsilon}_{t}=\big{(}(\overline{\mathbbm{S}}^{k}_{t})_{k\leqslant
t+1},(\overline{\mathbbm{I}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{IT}}_{t},(\overline{\mathbbm{R}}^{k}_{t})_{k\leqslant
t},\overline{\mathbbm{RT}}_{t},\overline{\mathbbm{H}}_{t}\big{)}$. Thus, the
first-order conditions with respect to $\Pi_{t}$ outputs the backward system
of equations that describes the dynamics of $\overline{\Upsilon}_{t}$. The
system is as follows:

$\displaystyle\overline{\mathbbm{S}}_{t}^{k}$
$\displaystyle=\left((1-\tau_{t+1}\gamma)\overline{\mathbbm{S}}_{t+1}^{k}+\tau_{t+1}\gamma\overline{\mathbbm{S}}_{t+1}^{t+2}\right)+\chi_{t+1}\tau_{t+1}\bigg{(}\gamma+(1-\gamma)\left(\beta_{w}\lambda^{2}I_{t+1}+\beta_{s}\widehat{I}_{t+1}\alpha_{t+1}^{k}\right)\bigg{)}+$
($\overline{\mathbbm{S}}_{t}^{k}$)
$\displaystyle+(1-\delta_{G})\delta_{G}^{t}\lambda_{t+1}q_{t+1}+(\delta_{G}^{t}+\xi)p_{t}+\eta_{t+1}^{k}\bigg{(}(1-\delta_{A})\delta_{A}^{t}c(1-\alpha_{t+1}^{k})q_{t+1}-(1-\delta_{A})\delta_{A}^{t}\beta_{s}\phi^{+}\widehat{I}_{t+1}q_{t+1}-\Delta_{t+1}^{k}\beta_{s}\widehat{I}_{t+1}\bigg{)}-$
$\displaystyle-\sum_{m\leqslant
t+1}\eta_{t+1}^{m}(1-\delta_{A})\delta_{A}^{t}\beta_{s}\phi^{-}I_{t+1}^{m}\alpha_{t+1}^{k}-\overline{\Delta}_{t+1}^{k}\bigg{(}\beta_{w}\lambda^{2}_{t+1}I_{t+1}^{k}+\beta\alpha_{t+1}^{m}\widehat{I}_{t+1}\alpha_{t+1}^{k}\bigg{)}-$
$\displaystyle-(1-\delta_{A})\delta_{A}^{t}\sum_{m\leqslant
t+1}\overline{I}_{t+1}^{m}\phi^{-}\left(\beta_{w}\lambda_{t+1}^{2}+\beta_{s}\alpha_{t+1}^{m}\right)q_{t+1},$
$\displaystyle\overline{\mathbbm{I}}_{t}^{k}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\left((1-\tau_{t+1})\overline{\mathbbm{I}}_{t+1}^{k}+\tau_{t+1}\overline{\mathbbm{IT}}_{t+1}\right)+\frac{1-m_{i}}{t_{i}}\left((1-\tau_{t+1}\gamma)\overline{\mathbbm{R}}_{t+1}^{k}+\tau_{t+1}\gamma\overline{\mathbbm{RT}}_{t+1}\right)+\frac{m_{i}}{t_{i}}\overline{\mathbbm{H}}_{t+1}+$
($\overline{\mathbbm{I}}_{t}^{k}$)
$\displaystyle+\chi_{t+1}\tau_{t+1}\bigg{(}\gamma\frac{1-m_{i}}{t_{i}}+1-\frac{1}{t_{i}}+(1-\gamma)\left(\beta_{w}\lambda^{2}S_{t+1}+\beta_{s}\widehat{S}_{t+1}\alpha_{t+1}^{k}\right)\bigg{)}+(1-\delta_{G})\delta_{G}^{t}\lambda_{t+1}q_{t+1}+(\delta_{G}^{t}+\xi)p_{t}+$
$\displaystyle+\eta_{t+1}^{k}(1-\delta_{A})\delta_{A}^{t}\bigg{(}c(1-\alpha_{t+1}^{k})-\beta_{s}\phi^{-}\widehat{S}_{t+1}\bigg{)}q_{t+1}-\sum_{m\leqslant
t+1}\eta_{t+1}^{m}(1-\delta_{A})\delta_{A}^{t}\beta_{s}\bigg{(}\phi^{+}+\Delta^{m}_{t+1}\bigg{)}S^{m}_{t+1}\alpha_{t+1}^{k}-$
$\displaystyle-\sum_{m\leqslant
t+1}\Delta_{t+1}^{m}\bigg{(}\beta_{w}\lambda^{2}_{t+1}\overline{S}_{t+1}^{m}+\beta\alpha_{t+1}^{m}\overline{S}_{t+1}^{m}\alpha_{t+1}^{k}\bigg{)}-\sum_{m\leqslant
t+1}\overline{\Delta}_{t+1}^{m}\bigg{(}\beta_{w}\lambda^{2}_{t+1}S_{t+1}^{m}+\beta\alpha_{t+1}^{m}S_{t+1}^{m}\alpha_{t+1}^{k}\bigg{)}-$
$\displaystyle-(1-\delta_{A})\delta_{A}^{t}\sum_{m\leqslant
t+1}\overline{S}_{t+1}^{m}\phi^{+}\left(\beta_{w}\lambda_{t+1}^{2}+\beta_{s}\alpha_{t+1}^{m}\right)q_{t+1},$
$\displaystyle\overline{\mathbbm{IT}}_{t}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{i}}\bigg{)}\overline{\mathbbm{IT}}_{t+1}+\frac{1-m_{i}}{t_{i}}\overline{\mathbbm{RT}}_{t+1}+\frac{m_{i}}{t_{i}}\overline{\mathbbm{H}}_{t+1}+(\delta_{G}^{t}+\xi)p_{t},$
($\overline{\mathbbm{IT}}_{t}$) $\displaystyle\overline{\mathbbm{R}}^{k}_{t}$
$\displaystyle=(1-\tau_{t+1}\gamma)\overline{\mathbbm{R}}_{t+1}^{k}+\tau_{t+1}\gamma\overline{\mathbbm{RT}}_{t+1}+\chi_{t+1}\tau_{t+1}\gamma+\eta_{t+1}^{k}(1-\delta_{A})c\delta_{A}^{t}(1-\alpha_{t+1}^{k})q_{t+1}+(1-\delta_{G})\delta_{G}^{t}\lambda_{t+1}q_{t+1}+(\delta_{G}^{t}+\xi)p_{t}$
$\displaystyle\overline{\mathbbm{RT}}_{t}$
$\displaystyle=\overline{\mathbbm{RT}}_{t+1}+(1-\delta_{G})\delta_{G}^{t}\lambda_{t+1}q_{t+1}+(\delta_{G}^{t}+\xi)p_{t},$
($\Delta\overline{\mathbbm{RT}}_{t}$)
$\displaystyle\overline{\mathbbm{H}}_{t}$
$\displaystyle=\bigg{(}1-\frac{1}{t_{h}}\bigg{)}\overline{\mathbbm{H}}_{t+1}+\frac{1-m_{h}}{t_{h}}\overline{\mathbbm{RT}}_{t+1}+(\delta_{G}^{t}+\xi)p_{t}.$
($\Delta\overline{\mathbbm{H}}_{t}$)

As already standard, we write
$\overline{\Delta\Upsilon}_{t}:=\big{(}(\Delta\overline{\mathbbm{S}}_{t}^{k})_{k\leqslant
t+1},(\Delta\overline{\mathbbm{I}}_{t}^{k})_{k\leqslant
t},\Delta\overline{\mathbbm{IT}}_{t},(\Delta\overline{\mathbbm{R}}_{t}^{k})_{k\leqslant
t},\Delta\overline{\mathbbm{RT}}_{t},\Delta\overline{\mathbbm{H}}_{t}\big{)}$
for the system of adjoint equations. We note that this system, as one for
$\Upsilon_{t}$, is backwards with the following “initial" condition:

$\displaystyle\lim_{t\to\infty}\frac{\overline{\mathbbm{S}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{I}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{R}}^{k}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{IT}}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{RT}}_{t}}{p_{t}}-\delta_{G}^{t}=\lim_{t\to\infty}\frac{\overline{\mathbbm{H}}_{t}}{p_{t}}-\delta_{G}^{t}=\xi\;\forall
k.$

###  10.3 Calibration

In this section we discuss in details the rationale behind the assumed
government policies and the set of fixed parameters which are used to
calibrate the parameters of interest.

####  10.3.1 Lockdown function

We start with the approximation of actual lockdown (see Figure 3(a)) which is
assumed to take the following parametric form:

$\displaystyle\lambda_{t}$ $\displaystyle=\begin{cases}1&t=1,\ldots,16,\\\
\sqrt{1-\frac{1-\overline{\lambda}}{\big{(}\frac{32}{17}-1\big{)}^{\sigma_{1}}}\big{(}\frac{t}{55}-1\big{)}^{\sigma_{1}}}&t=17,\ldots,32,\\\
\overline{\lambda}&t=33,\ldots,103,\\\
\sqrt{\overline{\lambda}^{2}\sigma_{2}^{t-1-103}+(1-\sigma_{2})^{t-1-103}}&t=104,\ldots,T,\end{cases}$

where $1-\overline{\lambda}$ reflects the maximum level of lockdown that can
be imposed by the government, while $\sigma_{1}$ and $\sigma_{2}$ controls its
severity. The assumed lockdown function has four broad phases: (i) between Feb
29, 2020 ($t=1$) and Mar 15, 2020 ($t=16$), the US government did not impose
any lockdown and the economy worked at full capacity, (ii) severe lockdown
including bans on international travel were imposed between Mar 16 ($t=17$)
and Mar 31, 2020 ($t=32$), reflected by the steep decline in the lockdown
function.323232This was the period in which the US government formally
acknowledged Covid-19 to be a national emergency. Link to Proclamation 9994 by
President Donald J. Trump. The flat line (phase (iii)) between March 31
($t=32$) to June 5 ($t=103$) reflects the severe lockdown imposed in majority
of the states and cities. Moreland et al. (2020) provides a comprehensive
analysis of lockdowns imposed by various states in the US. They show that
about 80% of the states had mandatory stay-at-home orders in some form by
March 31st, 2020. Most of these orders were gradually rescinded or allowed to
exhaust from June 6 ($t=103$) onwards. This is reflected in the gradual rise
in phase (iv) of the assumed lockdown function.

The choice of $\sigma_{1}$ and $\sigma_{2}$ reflects the curvature of the
lockdown policy. After the proclamation of national emergency, there was a
flurry of measures taken to curb down international and interstate travel, as
well as closure of most offices. This is denoted by a steep curvature in phase
(ii), reflected by a high value of $\sigma_{1}=2$ ensuring approximately the
maximum level of lockdown at the start of phase (iii). On the other hand, the
relaxation of lockdown has been much slower, reflected in a gradual increase
in the lockdown function at the fixed rate of $\sigma_{2}$ in phase (iv). We
set $\sigma_{2}=0.9995$, so that $60\%$ of the economy is functioning by Fall,
2021.

####  10.3.2 Test count

As explained above in Section 6, measuring the total number of tests and the
number of positive and negative cases has been a tough task for researchers.
The difficulty is driven by the fact that in the US there is no Federal
standard for reporting testing data, and each individual state can decide to
report a number of tests in terms of a) people tested at least once (unique
people), b) samples collected and c) testing encounters. The "COVID Tracking
Project" aggregates the state level data as follows: for each state, a total
number of tests equals to a number of testing encounters if it is reported,
otherwise it equals to a number of unique people if it is reported, otherwise
it equals to a number of samples collected.

In order to calibrate and estimate the model, we aggregate the testing data by
imputing a number of unique people reported by each state. For each state $s$,
let $\overline{X}_{s,t}$ be the number of unique tested at time $t$ if it is
reported by this state; otherwise, let $\overline{X}_{s,t}$ be the total
number of test in state $s$ at time $t$ multiplied by the average ratio of the
total number of tests to the number of unique people at time $t$ across states
which reported both. Then, we simple set

$\overline{X}_{t}:=\sum_{s\in\text{states}}\overline{X}_{s,t}.$

The main reason for working with unique people is that this metric is reported
by most states in our dataset, thus its imputation is more reliable; in
contrast, testing encounters were used only by a few smaller states. We do
however re-calibrate and re-estimate the model using the aggregated testing
data provided by the "COVID Tracking Project". It turns out that both
approaches yield very similar results. Instead of reporting all the estimation
and optimal policy results again, we simply plot four time series of lockdown,
social distancing, infections and deaths for these two specifications of
deaths—see Figure 13.

(a) Lockdown

(b) Social Distancing

(c) Flow of infections

(d) Cumulative deaths

Figure 13: Optimal policy when the total number of tests is measured
differently

The total number of tests available almost doubles in comparison to the
baseline. However, in order to fit the data, the efficacy of tracing $\gamma$
must become worse. So the gains in the total number of deaths in non-trivial
but not large.

####  10.3.3 Fixed parameters

We now move to the fixed parameters shown in Table 2. First of all, we utilize
the collective wisdom of various medical studies to calibrate the medical
parameters. The average number of days of infection $t_{i}$ is set to $10$
days which includes the incubation and latency period. Conditional on
developing further severe symptoms, the average number of days of
hospitalization (or at home critical care) is assumed to be $t_{h}=7$. Our
main basis for these numbers come from the CDC Planning Scenario dated March
19, 2021. For instance, their estimate for the mean time from exposure of the
disease to the onset of symptoms is around 8 days. Similarly, the population
weighted mean time between the onset of symptoms to hospitalization is around
9 days. The average hospitalization period is calculated similarly.

Since the time series of data on hospitalizations and daily deaths is
available in the United States, and majority of Covid-19 related deaths have
taken place at hospitals, we do a simple OLS regression between the two
variables to calculate the rate at which those who develop severe symptoms are
likely to die, which turns out to be $m_{h}=0.1705$. This basically means that
everything else being equal, $17\%$ of all people who develop a severe case of
Covid-19 will end up dying. Technically this parameter is not fixed because we
estimate it from the data we have, but once estimated it will be fixed for the
grand-estimation procedure.

We calculate the rate at which an infected person can develop severe symptoms
and require extra care, that is the rate of transition from the state $(\bf
I)$ to the state $(\bf H)$, to be $m_{i}=0.0176$ using the following argument:
First, the IFR or the infection fatality rate is taken to be $0.003$. This
means, ceteris paribus, there is a $0.3\%$ chance of death upon contracting
the virus. This corroborates with estimates obtained in some of the medical
literature (see for instance Ioannidis (2021)). Note that the value of IFR is
by no means settled in the medical literature. For instance, the CDC Planning
Scenario of 19th March, 2021 pegs IFR to be around 0.004. Alvarez et al.
(2021) uses a value of 0.0005, while Farboodi et al. (2021) uses an IFR of
0.0062. If anything, our choice of $0.003$ may turn out to be on the higher
side when the dust settles and one widely agreed upon number emerges. Once IFR
is fixed, $m_{i}$ is calculated using the identity $m_{i}\times m_{H}=IFR$.

The arrival of vaccine and more broadly vaccine dissemination that culminates
in the date when the pandemic can be declared to have ended is modelled
through a negative binomial distribution with mean 540 days and variance 180
days (see the entries for $\mathbb{E}T$ and $\mathbb{V}T$ in Table 2). This
ensures that the pandemic will surely not end till around 404 days and it will
almost certainly end in 593 days from the first day of the virus spread. This
roughly squares with the prevailing wisdom, aggregated well in New York Times
(2020), which had pegged the best case scenario for ending the pandemic to be
the summer of 2021. Since the arrival of vaccine in our model coincides with
the end of the pandemic, a mean of 540 days implies that the pandemic is
expected to end in the Fall of 2021 (around August or September, 2021). At the
current pace of vaccination in the US, it is expected that around 80% of the
US population will be vaccinated by September, 2021 (see New York Times
[2020]).

The maximum amount of lockdown the government can impose is capped at 70
percent. So $\overline{\lambda}=0.3$ means that thirty percent of economic
activities need to happen for the basic ingredients of society to keep
functioning. We have in mind health, retail, government, utilities, and food
manufacturing. This is the number used, for example, by Alvarez et al. (2021)
and the uniform lockdown policy used in Acemoglu et al. (2021). Then, as
standard in the literature, the government is forward-looking and discounts
the future in a way that the annual interest rate equals to $5$ percent. Our
model allows for the flexibility of choosing any level of discounting by the
agent, we calibrate the model for $\delta_{A}=\delta_{G}$.

Finally, we note that there is some indeterminacy in jointly identifying the
total prevalence and the initial number of infected agent, see Korolev (2021).
The standard approach in the literature is either to fix both or fix one of
these parameters and calibrate the other. We decided to calibrate the total
prevalence by taking $e_{0}=0.007$ percent, i.e., there were $23000$ infected
people at the end of February (roughly one month since the first positive case
was reported in the US). It must be noted that the choice of the initial seed
varies greatly in the literature, depending on the time frame of the analysis.
For instance, Eichenbaum et al. (2021a) works with an initial seed as high as
328,000 while Farboodi et al. (2021) work with an initial seed of roughly
17,000.

####  10.3.4 Fixing $\delta_{A}=0$

The agents are modeled to be as forward looking as the government. McAdams
(2021) notes that some studies in the literature fix $\delta_{A}=0$, that is
take the agents to be myopic optimizers, both for tractability and as thumb
rule which approximates some reality. To assess robustness of our results and
make them comparable to other studies, we fix $\delta_{A}=0$, re-estimate the
model and then solve for the optimal policy. All the other fixed parameters
are kept the same as before.

(a) Lockdown

(b) Social Distancing

(c) Flow of infections

(d) Cumulative deaths

Figure 14: Results for myopic agents, $\delta_{A}=0$

Since the agents are myopic, they internalize the costs of getting infected to
a lesser extent and social distance less (Figure 14(b)). Expecting the reduced
social distancing from the agents, the government thus locks down more (Figure
14(a)). Both social distancing and lockdown have the similar shape as the
forward looking model — broadly persistent with a gradual decrease during the
course of the pandemic, with a sharp decline towards the end. As a consequence
of the agents’ myopia, the number of infections and deaths is larger (Figures
14(c) and 14(d)). While the greater lockdown picks up some of the slack from
reduced social distancing, the government still prefers not to completely
lockdown at the maximal amount for a prolonged period for that would choke the
economy. As before the optimal policy aims at reducing the effective
reproductive number to one through the course of the pandemic.

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# Discovery of a FR 0 radio galaxy emitting at $\gamma$-ray energies

Paola Grandi1 , Alessandro Capetti2 and Ranieri D. Baldi3
1INAF-IASFBO, Via Gobetti 101, I-40129 Bologna, Italy
2INAF-Osservatorio Astrofisico di Torino, Strada Osservatorio 20, I-10025,
Pino Torinese, Italy
3Department of Physics and Astronomy, The University, Southampton SO17 1BJ, UK
E-mail:grandi@iasfbo.inaf.itcapetti<EMAIL_ADDRESS>
###### Abstract

We present supporting evidence for the first association of a Fermi source,
3FGLJ1330.0-3818, with the FR 0 radio galaxy Tol1326-379. FR 0s represent the
majority of the local radio loud AGN population but their nature is still
unclear. They share the same properties of FR Is from the point of view of the
nuclear and host properties, but they show a large deficit of extended radio
emission. Here we show that FR 0s can emit photons at very high energies.
Tol1326-379 has a GeV luminosity of $L_{>1~{}{\rm GeV}}\sim 2\times 10^{42}$
erg s-1, typical of FR Is, but with a steeper $\gamma$-ray spectrum
($\Gamma=2.78\pm 0.14$). This could be related to the intrinsic jet properties
but also to a different viewing angle.

###### keywords:

galaxies: active-galaxies:radio-galaxies:individual
(Tol1326$-$379)-galaxies:jet - $\gamma$-rays:galaxies

##  1 Introduction

Radio Galaxies (RGs) has been historically divided into faint edge-darkened FR
I and bright edge-brightened FR II (Fanaroff & Riley, 1974) on the basis of
their extended radio morphology with the transition occurring at,
approximately, a radio power of P${}_{\rm 178~{}MHz}\sim$ 1025 W Hz-1 sr-1.
From the optical point of view, radio galaxies are split into Broad Line Radio
Galaxies (BLRGs) and Narrow Line Radio Galaxies (NLRGs). NLRGs can be further
classified as High Excitation Galaxies (HEGs) and Low Excitation Galaxies
(LEGs) based on their optical emission line ratios (Buttiglione et al., 2010).
While HEGs and BLRGs show almost exclusively a FR II morphology, LEGs can
assume both radio morphologies. It is believed that in LEGs, the AGN
luminosity is sustained by hot gas via advection-dominated flow-like/Bondi
accretion (Balmaverde, Baldi & Capetti, 2008), while in BLRGs and HEGs by a
cold geometrically–thin, optically–thick disk (Best & Heckman, 2012; Heckman &
Best, 2014). Steep Spectrum Radio Quasars (SSRQs) are similar to BLRGs but
more distant and luminous. FRIs and FR IIs are considered the parent
population of BL LACs (BLs) and Flat Spectrum Radio Quasars (FSRQs),
respectively (Urry & Padovani, 1995).

Being observed at large angles , the jets of RGs and SSRQs (collectively named
Misaligned AGN: MAGN) do not benefit from the strong Doppler amplification
typical of blazars. Although geometrically disfavored (i.e. less amplified),
RGs and SSRQs have been detected above 100 MeV. The Fermi satellite found 11
MAGN in only 15 months of Large Area Telescope (LAT) survey (Abdo et al.,
2010c) with four of them also detected in the TeV band. Although their number
is a tiny fraction of the total Fermi detected sources, their discovery had a
strong impact on the study of the high energy process in AGN.

A clear link, in at least two BLRGs, 3C 111 and 3C 120, has been established
between the expulsion of bright superluminal knots from the radio core and
intense $\gamma$-ray flares (Grandi, Torresi & Stanghellini, 2012; Casadio et
al., 2015).

The firm detection of GeV emission from the radio lobes in nearby RG Centaurus
A has shown that extranuclear extended regions can be a source of gamma-ray
photons, implying the presence of highly energetic particles at large
distances from the nucleus (Abdo et al., 2010b). Because their radiative
lifetimes ($<1$ to 10 million years) approach plausible electron transport
time scales across the lobes, their presence is difficult to explain unless
successive particle acceleration occurs even at large distances from the black
hole.

Spectral Energy Distribution (SED) studies of FR I radio galaxies showed that
a pure, one-zone homogeneous, Synchrotron Self-Compton (SSC) emitting region
is inadequate in reproducing the radio to TeV data, stimulating the
elaboration of more complex models. Stratified jets with different regions
interacting with each other (Georganopoulos & Kazanas, 2003; Ghisellini,
Tavecchio & Chiaberge, 2005; Böttcher & Dermer, 2010) as well as magnetic
reconnection events along the jet (Giannios, Uzdensky & Begelman, 2010) or in
the vicinity of the black hole (Khiali, de Gouveia Dal Pino & Sol, 2015) have
been suggested as possible sources of $\gamma$-ray photons. Hadronic models
based on proton-photon interaction (see for a review Böttcher 2012) have also
been explored providing possible connections among AGN, Ultra High Energy
Cosmic Rays and neutrinos (Becker & Biermann, 2009)

All these results are based on the study of the brightest MAGN discovered in
the first years of the Fermi-LAT activity, i.e. bright radio sources with flux
density of 2Jy or more. The recent publication of the new Fermi-LAT point
source catalog (3FGL - Acero et al. 2015) and the Fermi-LAT AGN catalogs (3LAC
- Ackermann et al. 2015) collecting 4 years of data now allow us to extend the
study of GeV radio galaxies to lower fluxes. Although the ’standard’ picture,
i.e. the predominance of bright FR Is among MAGN, is confirmed, we can now
enter into the unexplored territory of fainter radio AGN. Indeed we propose
here the first association between a 3LAC gamma-ray source and a FR 0 galaxy.

The cross-match of radio and optical data favored by the advent of large area
surveys has surprisingly revealed that the bulk of the radio-galaxy population
lacks prominent extended radio structures. Best & Heckman (2012) built a
sample of radio-galaxies by cross-correlating the Sloan Digital Sky Survey
(SDSS), the National Radio Astronomy Observatory (NRAO) Very Large Array (VLA)
Sky Survey (NVSS), and the the Faint Images of the Radio Sky at Twenty
centimetres (FIRST) survey datasets. This sample is selected at $F_{\rm
1.4}>5$ mJy and it includes RGs up to $z\sim$ 0.3, covering the range $L_{\rm
1.4}\sim 10^{22}-10^{26}$ W Hz-1. Most of them ($\sim$ 80 %) are unresolved or
barely resolved at the 5′′ FIRST resolution, corresponding to a limit to their
size of $\sim$10 kpc (Baldi & Capetti, 2009). The lack of extended radio
structures, that characterize the morphology of the traditional Fanaroff-Riley
classes (FR I and FR II), suggests defining these objects as FR 0s (see Baldi,
Capetti & Giovannini 2015b).

FR 0s share the same properties of FR Is from the point of view of the nuclear
and host properties. By comparing FR Is and FR 0s of similar AGN power,
estimated from the optical line luminosity, FR 0s show the same radio core
power but a strong deficit of extended radio emission. Most of them ($\sim
70\%$) can be classified as LEGS, the same spectroscopic class to which FR Is
belong. Finally, the hosts of these two classes are effectively
indistinguishable being in both cases red early-type galaxies with central
black hole masses larger than $\sim 10^{8}M_{\odot}$ (Baldi & Capetti, 2009,
2010; Sadler et al., 2014; Baldi, Capetti & Giovannini, 2015a).

Although FR 0s represent the majority of the low luminosity RGs in the local
Universe, they are a puzzling class of AGN completely unexplored at high
energies.

In this paper, we show that the galaxy Tol1326-379, identified as the
counterpart of the $\gamma$-ray source 3FGLJ1330.0-3818 and classified in the
3LAC as a Flat Spectrum Radio Quasar, is actually the first FR 0 radio source
discovered in the GeV sky.

A cosmology with $H_{0}=67$ km s-1 Mpc-1, $\Omega_{m}=0.32$, and
$\Omega_{\Lambda}=0.68$ is assumed.

##  2 3FGLJ1330.0-3818 - Tol1326-379

The $\gamma$-ray source 3FGLJ1330.0-3818 was detected by Fermi with a
significance of $\sim 5\sigma$, a flux above 1 GeV of F${}_{>1~{}{\rm
GeV}}=(3.1\pm 0.8)\times 10^{-10}$ phot cm-2 s-1 and a power law index
$\Gamma=2.78\pm 0.14$ (Acero et al., 2015).

3FGLJ1330.0-3818 is listed in the 3LAC catalog and associated to the early
type galaxy Tol1326-379 at z=0.02843 with a Bayesian probability of
association of 90%. The Bayesian method calculates the posterior probability
that a source from a catalog of candidates is the counterpart of a
$\gamma$-ray source detected by the LAT, evaluating the significance of a
spatial coincidence between the candidate counterpart and the LAT-detected
source. For more details see Abdo et al. (2010a).

In order to strengthen the association proposed by the 3LAC catalog, we
explored the region around 3FGLJ1330.0-3818 within the 95% $\gamma$-ray
position error ellipse (20$\arcmin$ $\times$ 13$\arcmin$) looking for
alternative identifications. We considered all radio sources with a NVSS flux
density larger than 20 mJy ($\sim$3 times fainter than Tol1326-379), finding
three objects spanning the range 70 to 240 mJy (see Figure 1). Their radio
spectral indices, estimated between 0.843 and 1.4 GHz, are steep ($\alpha_{\rm
r}$ $\sim$ 0.7-0.9) and none of them is detected at 4.8 GHz with the PARKES
telescope (Griffith & Wright, 1993), implying that their flux densities are
less than $\sim$35 mJy, suggesting the dominance of the extended radio
emission. Furthermore none of them has a 2MASS counterpart apart from
Tol1326-379 which shows an early-type morphology. This confirms that
Tol1326-379 is the most likely association.

Figure 1: NVSS radio sources in the field of 3FGLJ1330.0-3818. Symbol sizes
are proportional to the flux density at 1.4 GHz. Tol1326-379 is marked with
the red point. Three bright radio sources, NVSS J132911-380918, NVSS
J132952-381251, and NVSS J132857-381703 (cross symbols) are within the Fermi
position error ellipse, but they are rejected as possible counterparts (see
text). This image was produced by using the ASDC tool
http://www.asdc.asi.it/fermi3fgl/

Figure 2: Left: optical spectrum of Tol1326-379 from the 6dF Galaxy Survey
(Jones et al., 2004). Right: radio spectrum of Tol1326-379. Data are from
literature (Healey et al., 2007; Wright et al., 1994; Condon et al., 1998;
Feain et al., 2009; Mauch et al., 2003). The 20 GHz flux density was measured
with the Australian Telescope Compact Array (see text for the details). The
dotted line is the single power fit to the data having a slope of 0.37.

##  3  The galaxy Tol1326-379

###  3.1 Optical properties

The optical spectrum of Tol1326-379 from the 6dF Galaxy Survey (Jones et al.,
2004, 2009) is presented in Fig. 2, left panel. It does not show any evidence
for broad emission lines, arguing against its identification as FSRQ; on the
other hand, the equivalent width (EW) of the emission lines largely exceed the
limit of 5Å commonly adopted for BL Lac objects (Stickel et al., 1991). We
obtained, in particular, EW(H$\alpha$) $\sim$ 45$\pm$4Å and EW([O III]) $\sim$
8$\pm$1Å. The diagnostic line ratios (log [O III]/H$\beta$ =0.10, log [N
II]/H$\alpha$ =-0.03, log [O I]/H$\alpha$ =-0.40, and log [S II]/H$\alpha$
=0.01) are characteristic of a LEG spectrum (Kewley et al., 2006; Buttiglione
et al., 2010). This lends further weight against its association with a FSRQ,
because such sources always show a high ionization optical spectrum (Shaw et
al., 2012).

Unfortunately the 6dFGS spectrum is not flux calibrated. In order to obtain
the emission lines luminosity we need to rely on an indirect estimate. We
derived the flux in the J band Two Micron All Sky Survey (2MASS) image from a
synthetic aperture of 6$\aas@@fstack{\prime\prime}$7, the diameter of the
6dFGS fiber, after having degraded the image resolution to match the seeing
reported in the observing log. We then adopted a V-J color of 2.43 (Mannucci
et al., 2001), typical of early-type galaxies, and obtained of magnitude
V=15.1. From its measured EW we finally obtain an [O III] flux of 3.2$\times
10^{-15}$ erg cm-2 s-1 and a luminosity of 4$\times 10^{40}$ erg s-1.

In order to assess the accuracy of such a procedure, we performed the same
analysis on a group of 7 early-type emission line galaxies in common between
the 6dFGS and the SDSS survey. Our line measurements agree with those provided
by the SDSS database within a factor of 4.

From its 2MASS image, we derived a total K magnitude of 11.22, corresponding
to a luminosity of 1.0$\times 10^{11}$ solar luminosity. The tight correlation
between $M_{\rm BH}$ and the near-infrared bulge luminosity proposed by
Marconi & Hunt (2003) allows us to estimate (within a factor 2) the black hole
mass of $M_{\rm BH}=2\times 10^{8}$ M⊙.

Using the relation Lbol=3500$~{}L_{\rm[O~{}III]}$ measured by Heckman et al.
(2004), we obtain a bolometric luminosity L${}_{\rm bol}=44.1$ erg s-1 (with
an uncertainty of 0.4 dex) for this source. The Eddington-scaled accretion
rate of Tol1326-379 is $\dot{L}=L_{\rm bol}/L_{\rm Edd}\sim 5\times 10^{-3}$,
a value typical of LEGs (Best & Heckman, 2012).

###  3.2 Radio properties

We collected the radio flux density data from the NASA/ IPAC Infrared Science
Archive and Extragalactic Database (NED). To these data we added our own
measurement at 20 GHz from the Australian Telescope Compact Array (ATCA); the
observations were obtained on Oct 27th 2004 (project C1049, PI Ekers) as part
of the AT20G survey (Murphy et al., 2010), centered at 19.9 GHz and with a
bandwidth of 256 MHz and with a duration of 3.6 minutes. We used PKS 1934 –
638 for the primary flux calibration, 1036–52 as bandpass calibrator and PKS
1349-439 as phase calibrator. After deconvolution the source appears as
unresolved (angular resolution of 10$\arcsec$); we perform a Gaussian fit
using ’jmfit’ task of AIPS software111The NRAO Astronomical Image Processing
System (AIPS) is a package to support the reduction and analysis of data taken
with radio telescopes. to estimate its flux density of 23.2 $\pm 0.8$ mJy. The
available radio data cover the frequency range from 843 MHz to 20 GHz (see
Fig. 2, right panel). None of the available radio images shows extended
emission and in particular Tol1326-379 is reported by Healey et al. (2007) as
a point source in their CRATES catalog at 8.4 GHz with an angular resolution
of $\sim 2\arcsec$ provided by the ATCA observation. The overall spectral
slope is $\alpha=0.37$ but the data points show a large scatter. This in part
due to the source variability; indeed its flux density varied from 53.2 to
71.5 mJy in the NVSS and ATCA observations, respectively, two measurements
obtained at the same frequency and with similar spatial resolution. In
addition, the observations have rather different spatial resolution, typically
$\sim 45\arcsec$ for the four lower frequencies and of a few arcsec for the
data at 8.4 and 20 GHz. The radio spectrum measured between these two high
frequency (and high resolution) measurements is even flatter than the overall
value, $\alpha_{8.4-20~{}{\rm GHz}}=0.11$. The core dominance R parameter
defined as the ratio between the 4.85 GHz and 1.4 GHz flux densities is $\log
R=-0.03$.

Summarizing, Tol1326-379 fulfills all the spectro-photometric requirements for
a FR 0 classification listed by Baldi, Capetti & Giovannini (2015a). It is an
early-type galaxy with a black hole mass larger than $10^{8}$ M⊙, a low
excitation optical spectrum, and a high radio core dominance. Furthermore
Tol1326-379 , given its radio luminosity at 1.4 GHz of 2$\times 10^{39}$ erg
s-1 , falls within the region typical of FR 0s in the diagram shown in Fig.3
that compares the total radio and emission line luminosities. It also shows,
as characteristic of this class of sources, a large deficit (a factor of
$\sim$300) of radio emission respect to FR I radio-galaxies with equal
emission line power. Tol1326-379 is also consistent, considering the
relatively large errors related to the line measurement, with the core-[O III]
luminosity relation found for FR Is (Baldi, Capetti & Giovannini, 2015a)
another crucial prerogative of FR 0s.

Figure 3: Logarithm of the radio vs. [O III] luminosities (both in erg s-1
units) for the SDSS/NVSS sub-sample ($0.03<z<0.1$) analyzed by Baldi & Capetti
(2009) and mainly composed of FR 0 sources. The solid line reproduces the
line-radio correlation followed by FR Is of the 3CR sample (green stars). The
dashed line marks the boundary of the location of Seyfert galaxies (e.g.
Whittle 1985). FR 0s show a strong deficit of total radio emission, occupying
the region to the left of the FR Is (Baldi, Capetti & Giovannini, 2015a).
Tol1326-379 (red point) falls into the FR 0s area.

###  3.3 Gamma-ray properties

We explored the position of Tol1326-379 in a plot where the radio luminosity
at 1.4 GHz of all the blazars and MAGN of the 3LAC clean sample (with known
optical classification and redshift) are reported as a function of the gamma-
ray luminosities (see Fig. 4, left panel). We also add 3C 111 and Cen B (not
found in the clean sample because of their low galactic latitudes) and 3C 120
already reported in the 15 month-sample of MAGN and that has recently
undergone strong flares (Casadio et al., 2015). As the origin (jet, lobes) of
the FornaxA $\gamma$-ray emission is unclear, we prefer not to include it.

We derived for all sources in the 3LAC catalog the k-corrected 1.4 GHz rest
frame flux density by assuming $\alpha_{\rm r}=0$ for blazars and 0.8 for
MAGN.222In some cases the 3LAC radio flux density was provided at different
radio frequencies (for example at 20 GHz or at 843 MHz). The above reported
spectral radio slopes were also adopted to convert the listed flux to the 1.4
GHz flux. For Tol1326-379 we assume the observed value of $\alpha_{\rm
r}=0.37$. As expected, different classes lie in different zones of the diagram
with FSRQs at higher luminosities than BL Lacs. We recover the well known
radio $\gamma$-ray correlation for blazars (Ghirlanda et al., 2010; Ackermann
et al., 2011). On average, MAGN are offset from the blazar strip showing a
radio excess with respect to BL Lacs and FSRQs with similar $\gamma$-ray
luminosities. This is due to the additional contribution from extranuclear
radio emission. Tol1326-379 falls into the low luminosity tail of the blazar
strip. It is, together with IC 310, the least powerful radio galaxy with
$\gamma$-ray detection.

To further explore the nature of our source, the $\gamma$-ray spectral slope
($\Gamma$) of blazars and MAGN of the clean 3LAC sample was plotted in Fig. 4,
right panel, as a function of the rest frame isotropic luminosity above 1 GeV
($L_{>1{\rm GeV}}$ in ergs cm-1 s-1). As is well known, different classes of
AGN occupy different locations in the $\Gamma$-$L_{>1{\rm GeV}}$ plane (Abdo
et al., 2010c). FSRQs and SSRQs are in the upper right part of the diagram
(high luminosities and steep spectra), BL Lacs in the bottom right side (low
luminosities and flat spectra), while RGs are all at low luminosities, but
with a large scatter in spectral indeces. It is evident that Tol1326-379 falls
into the general radio-galaxy area and not into the FSRQs region.

Figure 4: Left: the radio luminosity of the 3 LAC FR I radio galaxies (green
circles), FR II radio sources (magenta squares), BL Lacs (open blue
triangles), FSRQs (open black squares), and SSRQs (cyan crosses) is plotted as
a function of the $\gamma$-ray luminosity between 1 GeV and 100 GeV. MAGN,
extended at 1.4 GHz, are more luminous in radio than blazars with similar
$\gamma$-ray luminosity. Tol1326-379 (red open diamond) has a typical FRI
$\gamma$-ray luminosity but falls into the low luminosity tail of the blazar
strip. Right: $\gamma$-ray spectral slope versus 1-100 GeV luminosity.
Tol1326-379 is located into the MAGN region and outside the FSRQs zone. For
clarity, error-bars are reported for non-blazars only. All the luminosities
are in erg s-1.

##  4 Discussion

The 3LAC catalog associates the $\gamma$-ray source 3FGLJ1330.0-3818 to the
galaxy Tol1326-379. We confirm that no other flat radio source brighter than
Tol1326-379 at 5 GHz is present with the 95% error circle position of
3FGLJ1330.0-3818. Although flat in the radio, Tol1326-379 is not a FSRQ but a
FR 0 radio galaxy, the first source of this new radio class with a
$\gamma$-ray counterpart.

Several observations support this conclusion:

  * •

Tol1326-379 is an early-type galaxy at z=0.02843;

  * •

it does not show any evidence for broad emission lines, arguing against its
identification as FSRQs;

  * •

the high values of the line equivalent widths exclude the possibility that it
is a BL Lac;

  * •

its line ratios are typical of LEGs;

  * •

its estimated black hole mass of $M_{\rm BH}=2\times 10^{8}$ M⊙ and accretion
rate $\dot{L}=L_{\rm bol}/L_{\rm Edd}\sim 5\times 10^{-3}$ are characteristic
of radio-loud AGN (Chiaberge & Marconi, 2011) associated with low efficiency
accretion flows;

  * •

it is unresolved in the radio images, showing a high core dominance and a flat
radio spectrum;

  * •

when put in a radio versus [O III] luminosity plot, it falls into the FR 0
region and not in the FR I area.

In Fig. 5, we present the SED of Tol1326-379. Besides the radio and Fermi
observations already discussed, we complemented it with the WISE, Galex, and
ROSAT data.

Tol1326-379 is detected at high significance by the WISE satellite in all four
bands and its colors are W1-W2=0.38 $\pm$ 0.03 and W2-W3=2.58$\pm$0.03,
respectively. These are significantly different from those typical of
elliptical galaxies (Wright et al., 2010). This indicates that the emission
seen, at least, in the W3 band is dominated by the AGN component. Indeed,
Tol1326-379 lies in a region populated by BL Lac objects (Massaro et al.,
2011), although offset from the main blazar strip.

The UV data, from the Galex archive333http://galex.stsci.edu/GR6/, are
corrected for galactic extinction assuming E(B-V)=0.0678. In the X-ray band,
Tol1326-379 was only detected by the PSPC instrument (0.1-2.4 keV) onboard of
the ROSAT satellite. Its count rate reported in the ROSAT All-Sky Survey Faint
Source Catalog444http://www.xray.mpe.mpg.de/rosat/survey/rass-fsc/ is
0.029$\pm 0.013$ c/s. This can be converted to an unabsorbed flux of
F${}_{0.1-2.4~{}keV}=(7.6\pm 3.4)\times 10^{-13}$ erg s-1 cm-2 if a power law
with spectral slope $\Gamma_{x}=2$ and Galactic absorption of $N_{H}=5.5\times
10^{20}$ cm-2 are assumed. Due to the low spatial resolution of the UV and
X-ray images, and the lack of any spectral information, the corresponding
measurements should be considered as upper limits, as they include the host
galaxy emission. In particular, its soft X-ray luminosity is compatible with
that expected from the hot corona of an early-type galaxy with the luminosity
estimated above for our source (Fabbiano, Kim & Trinchieri, 1992).

The SED of Tol1326-379 is compared to those of Centaurus A and M87, the
prototype nearby FR I radio galaxies. Despite of the similar radio luminosity,
M 87 is less luminous than Tol1326-379 by a factor of 30 at 1 GeV and has a
flat SED in the $\gamma$-ray domain. On the contrary, Centaurus A and
Tol1326-379 are quite similar in shape but the former source is about two
orders of magnitude fainter.

Balmaverde & Capetti (2006) found that the SEDs of the FR I nuclei differ from
those of BL Lacs. This is partly related to the frequency shift in the SED due
to the relativistic beaming, but also to differences in the emitting regions.
Indeed, due to the presence of a velocity stratification with the relativistic
jets (Kovalev et al., 2007; Nagai et al., 2014) in BL Lacs we are seeing the
regions of the jet with the highest Doppler factor (the so-called jet spine)
while in FR Is the emission could be dominated by the slower jet layers. The
SED differences might be witnessing the diverse physical conditions in these
two regions (Chiaberge et al., 2000; Ghisellini, Tavecchio & Chiaberge, 2005;
Tavecchio & Ghisellini, 2014) leading to a dependence of their shape on the
viewing angle.

This effect might account for the SED differences between M 87 and Tol1326-379
and, on the other hand, the similarities with Centaurus A. Indeed, M 87 jet is
seen at a rather small angle from the line of sight with $\theta\sim
15-25^{\circ}$ (Acciari et al., 2009), while Cen A lies close to the plane of
the sky, $\theta\sim 50-80^{\circ}$ (Tingay et al., 1998). We argue that the
SED shapes are, similarly to what is derived from the comparison between BL
Lacs and FR Is, mainly driven by the jet viewing angle. Indeed, by excluding
Centaurus A, all the FR Is have flat $\gamma$-ray spectra and are seen at
small angles (see Table 1). The only exception is 3C 120, that has a slope
comparable to that of Cen A (and Tol1326-379), but smaller inclination angle.
3C 120 is, however, a peculiar radio galaxy. Although classified as FR I, it
has optical-UV-X-ray properties typical of FR IIs. It shows broad optical
emission lines (Tadhunter et al., 1993), an UV bump, and an Kα iron line in
the X-ray spectrum (Zdziarski & Grandi, 2001; Ogle et al., 2005; Kataoka et
al., 2007; Chatterjee et al., 2011), all clear signatures of the presence of
an efficient accretion disk. Indeed, in Fig. 4 (right panel) 3C 120 falls
between 3C 111 ($\Gamma$=2.79,Lγ=43.21), and Pictor A
($\Gamma$=2.49,Lγ=42.65), two (FR II) BLRGs detected by Fermi.

Figure 5: SEDs of Tol1326-379 (red symbols) compared to those of Cen A (green)
and M 87 (blue). The dotted lines are polynomial functions connecting the
points and do not represent model fits to data. Data are collected from the
literature. M 87: Biretta, Stern & Harris (1991); Despringre, Fraix-Burnet &
Davoust (1996); Tan et al. (2008); Perlman et al. (2001); Balmaverde, Capetti
& Grandi (2006); Ackermann et al. (2015); Broderick et al. (2015); Aharonian
et al. (2006); Aliu et al. (2012). Cen A: Meisenheimer et al. (2007); Grandi
et al. (2003); Harmon et al. (2004); Ackermann et al. (2015); Aharonian et al.
(2009). The Compton Gamma Ray Observatory fluxes of Cen A, collected during a
multi-frequency campaign in 1995, are provided by Steinle H. at the web page
http://www.mpe.mpg.de/ hcs/Cen-A/ and included in the NASA/IPAC EXTRAGALACTIC
DATABASE.

Although we do not have any direct estimate for the orientation of
Tol1326-379, we speculate that it is oriented at a large angle and this causes
the similarity between its SED and that of Cen A.

Alternatively, if we are seeing the jet of Tol1326-379 at a small angle, its
SED would be intrinsically different from those of the FR Is. This would be
the first indication of a discrepant property between these two classes (other
than the paucity of extended radio emission in FR 0) and might be an important
clue to understand their nature. Interesting enough, the Compton peak in
Tol1326-379 appears to be more prominent than the Synchrotron one. This
generally occurs in FSRQs, where a surplus of seed photons coming from the
accretion disk, the broad line region and the torus, contributes to the high
energy emission by External Compton (Sikora et al., 2009). Incidentally we
note that Tol1326-379 is characterized by a steep $\gamma$-ray spectrum
($\Gamma_{\gamma}=2.78\pm 0.14$), more similar to that generally observed in
FSRQs (and their misaligned population, i.e BLRGs and SSRQs) than in BL Lacs
(and FRIs). It is, however, improbable that an External Compton mechanism is
responsible for the cooling of the jet particles of Tol1326-379. Its nuclear
environment is poor in photons as indicated by the low accretion rate. It is
then more plausible that its $\gamma$-ray luminosity is sustained by different
jet components that mutually interact amplifying the IC emission as suggested
for FR Is. The excess of $\gamma$-ray radiation could then reflect different
physical conditions of the high energy dissipation regions (i.e. of the spine
and/or the layers).

Progress in our understanding of the properties of Tol1326-379 can come from a
better definition of its SED. In particular, the information in the X-rays can
be improved with a firm detection and with a measurement of the spectral slope
in this band. This might be used to test the indication that the Compton peak
in Tol1326-379 is more prominent than the synchrotron one.

Since a large diversity of spectral behavior among this class of sources, as
already observed in blazars, cannot be excluded, other observations of
$\gamma$-rays emitting FR 0s are necessary to consolidate the overall picture.

Table 1: Jet inclination angle from literature (Ref.) , $\gamma$-rays spectral
slope ($\Gamma$) and luminosity $L_{\gamma}$ [$>1$ GeV] (erg s-1) of the FR I
radio galaxies.

Source | $\theta$ | $\Gamma$ | Lγ | Ref.
---|---|---|---|---
3C 78 | $~{}50^{\circ}$ | 2.07$\pm$0.11 | 42.92 | 1
IC 310 |  $<$20∘ | 1.90$\pm$0.14 | 42.70 | 2
NGC1275 | 30-$55^{\circ}$ | 1.95$\pm$0.01 | 44.03 | 3
B20331$+$39 |  $<$45∘ | 2.11$\pm$0.17 | 42.40 | 4
3C 120a | $18-22^{\circ}$ | 2.7$\pm 0.1$ | 42.75 | 5
PKS 0625-45 |  $<$61∘ | 1.87$\pm$0.05 | 44.00 | 6
3C 189 | $~{}27^{\circ}$ | 2.16$\pm$0.16 | 42.86 | 7
3C 264 |  $\sim$50∘ | 1.98$\pm$0.19 | 42.37 | 8
M 87 | 15-$25^{\circ}$ | 2.05$\pm$0.06 | 41.61 | 9
Cen A | 50-$80^{\circ}$ | 2.70$\pm$0.03 | 41.00 | 10
Cen Ba,b | $<80^{\circ}$ | 2.32$\pm$0.09 | 42.63 | 11
NGC 6251 | 10-$40^{\circ}$ | 2.04$\pm$0.08 | 42.87 | 12

a – Cen B and 3C 120 are not in the clean 3LAC sample.
b – Cen B: Jet inclination upper limit estimated using the jet/counterjet flux
ratio $J\sim 2$ at 4.8 GHz provided by Jones, Lloyd & McAdam (2001). Possible
$\gamma$-ray contribution from the lobes (Katsuta et al., 2013).

(1) Kharb et al. (2009), (2) Schulz et al. (2015), (3) Walker, Romney & Benson
(1994), (4) Giovannini et al. (2001), (5) Jorstad et al. (2005), (6) Venturi
et al. (2000), (7) Bondi et al. (2000), (8) Lara et al. (2004), (9) Acciari et
al. (2009), (10) Tingay et al. (1998), (11) Jones, Lloyd & McAdam (2001), (12)
Migliori et al. (2011).

## Acknowledgments

We are grateful to F. Paresce for carefully reading the paper and useful
comments which improved the final version. We thank the referee for
constructive comments/suggestions on the manuscript. This research made use of
the NASA/ IPAC Infrared Science Archive and Extragalactic Database (NED),
which are operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration. Part of this work is based on archival data, software or
online services provided by the Italian Space Agency (ASI) Scientific Data
Center (ASDC).

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