text:
\newcommand{\cZ}{\mathcal{Z}}
\newcommand{\cl}{\operatorname{cl}}
\newcommand{\sA}{\mathrsfs{A}}

\section{Introduction}

This article is motivated by the following question:  is Langlands functoriality in the case
of cohomological automorphic forms
on Shimura varieties
induced by algebraic cycle classes?
When the forms in question contribute to $H^1$, this follows from
Faltings' theorem \cite{faltings-mordell} on the Tate conjecture for divisors on abelian varieties,
 but for higher $H^i$ it seems completely open even in the simplest of cases.
Since constructing algebraic cycle classes seems extremely difficult, one can ask for the next
best thing, namely to construct either {\it absolute Hodge classes}
\cite{dmos} or better still, {\it
  motivated cycles} \cite{andre}.
We study this problem in the most classical example of functoriality, namely the Jacquet-Langlands correspondence
for $\GL_2$ and its inner forms.

\subsection{The main theorem}
Let $F$ be a totally real field, $[F:\Q]=n$.
Denote by $\Sigma_\infty$  the set of infinite places of $F$, and
for $v\in \Sigma_\infty$, let $\sigma_v: F \hookrightarrow \R\subset
\C $ denote
the corresponding embedding of $F$ in $\C$. Let
$F^c\subset \Qbar \subset \C$ be the compositum of $\sigma_v (F)$ as $v$ varies over
$\Sigma_\infty$. Thus $F^c$ is the Galois closure of the image of
$\sigma(F)$ for any $\sigma \in \Sigma_\infty$.

Let $\pi=\otimes_v \pi_v $ be an automorphic representation of
$\GL_2(\A_F)$ corresponding to a
(cohomological) holomorphic Hilbert modular newform of weight $(\uk,r)$ where $\uk=(k_1,\ldots, k_n)$ and
$k_1\equiv k_2 \equiv \cdots \equiv k_n \equiv r \bmod 2$.
For simplicity, we will assume that $\pi$ has trivial Nebentypus
character so that it is self-dual up to a (Tate) twist. (See Remark
\ref{sec:non-self-dual} for the non self-dual case.)
Moreover, in the introduction alone, we assume that $\pi$ has parallel weight two and that
 the Hecke eigenvalues $a_v (\pi)$ (suitably normalized) are rational; thus $\pi$ (at least conjecturally) corresponds to an
elliptic curve $A/F$.  In any case, it is known that to such a $\pi$ and every rational prime $\ell$ one can attach
an $\ell$-adic
Galois representation $ \rho_{\pi,\ell} $
of the Galois group $\Gal (\Qbar/F)$. The representations
$\rho_{\pi,\ell}$ (for varying $\ell$) form a compatible system in the sense that
for all finite primes $v$ of $F$ not dividing $\fN$, we have
\[
\tr \rho_{\pi,\ell} (\Frob_v) = a_v (\pi),
\]
where $\Frob_v$ denotes a {\it geometric} Frobenius element attached to $v$;
in particular, this trace is independent of $\ell$.

Let $B_1$ and $B_2$ be two (non-isomorphic) quaternion algebras over $F$ such that
$\pi$ admits Jacquet-Langlands transfers to the algebraic groups $G_1
=\Res_{F/\Q} B_1^\times$ and $G_2=\Res_{F/\Q} B_2^\times$; we denote
the corresponding automorphic representations of $G_1(\A)$ and
$G_2(\A)$ by $\pi_1$ and $\pi_2$ respectively.
We assume that
the set of infinite places of $F$ where
$B_1$ is split agrees with the set of infinite places where $B_2$ is
split, and denote this common set of infinite places by
$\Sigma \subseteq \Sigma_\infty$.
Let $F_\Sigma$ be the subfield of $\C$ given by:
\begin{align*}
F_\Sigma &:= \Qbar^{\{ \sigma \in \Gal(\Qbar/\Q), \sigma (\Sigma)=\Sigma\}} =
           (F^c)^{\{\sigma \in \Gal(F^c/\Q),\sigma \Sigma=\Sigma\}}.
\end{align*}
Then $F_\Sigma$ is also characterized as the subfield of $\Qbar$ generated (over $\Q$)
by the elements
\[
\sum_{v\in \Sigma} \sigma_v(x), \quad x\in F
\]
and is called the reflex field of the pair $(F, \Sigma)$.

Let $X_{1}$ and $X_{2}$ denote the {\it quaternionic Shimura varieties}
associated with $G_1$ and $G_2$.
Then $X_1$ and $X_2$ are of dimension $d:=|\Sigma|$ and have canonical models over the {\it same} reflex field $F_\Sigma\subset \Qbar \subset \C$.
The Langlands-Kottwitz method can be used to study the $\ell$-adic cohomology of the varieties $X_{1}$ and $X_{2}$.
Following the work of several authors  (\cite{langlands72}, \cite{brylinski-labesse}, \cite{carayol-hmf}, \cite{reimann}, \cite{nekovar}), we have the following theorem: for $i=1,2$, the $\pi_i$-isotypic part of $H^*_{\et} (X_{i,\Qbar}, \Q_\ell)$ is concentrated entirely in the middle degree $d$ and moreover is isomorphic to the {\it tensor induction}
\begin{equation}
\label{eqn:intro-ten-ind}
 {\bigotimes_{v\in \Sigma}}' \ \rho_{\pi,\ell}^v
\end{equation}
where $\rho_{\pi,\ell}^v$ denotes the representation of $\Gal (\Qbar/ \sigma_v (F))$ given by $g\mapsto  \rho_{\pi,\ell} (\sigma_v g \sigma_v^{-1})$.
As a consequence, for all rational primes $\ell$, we have isomorphisms
\begin{equation}
\label{eqn:hd-ell-isom}
H^d (X_1, \Q_\ell)_\pi \simeq H^d (X_2, \Q_\ell)_\pi
\end{equation}
as representations of $\Gal(\Qbar/F_\Sigma)$. Here and henceforth we write $H^*(X,\Q_\ell)$ for the $\Q_\ell$-vector space
$H^*_{\et} (X_{\Qbar}, \Q_\ell)$.

The isomorphisms \eqref{eqn:hd-ell-isom}
above may be viewed as giving a collection of
Tate classes in
\[
H^{2d} (X_1\times X_2, \Q_\ell(d)),
\]
and it is natural to ask if there is a single algebraic cycle $\mathcal{Z} \in \CH^{d} (X_1 \times X_2)$ that gives rise to this collection of Tate classes. If $p_1$ and $p_2$ are
the two projections below,
\[
\xymatrix{
& X_1 \times X_2 \ar[ld]_{p_1} \ar[rd]^{p_2} & \\
X_1 & & X_2
}
\]
the class of such a putative algebraic cycle $\cZ$ gives rise to a map
\[
\mathrm{cl} (\mathcal{Z})^*: H^d(X_1) \rightarrow H^d(X_2), \quad x\mapsto p_{2,*} (\cl(\cZ) \cup p_1^*(x))
\]
for any Weil cohomology theory, which induces isomorphisms
\begin{equation}
\label{eqn:hweil-isom}
H^d(X_1)_{\pi} \simeq H^d(X_2)_{\pi_2}.
\end{equation}
Moreover, these isomorphisms for different Weil cohomology theories would be compatible with the usual comparison theorems.

With this motivation, we state our main theorem:

\begin{mtheorem}
\label{thm:intro-main-full}
Suppose that there is at least one infinite place of $F$ at which $B_1$ and $B_2$ are ramified. Then
there is a non-zero {\it Hodge} class
\[
\xi \in H^{2d} (X_1 \times X_2, \Q)_{\pi_1 \boxtimes \pi_2}
\]
such that for all rational primes $\ell$, the image $\xi_\ell(d)$ of
(the Tate twist) $\xi(d)$ in the $\ell$-adic etale realization
\[
H^{2d} (X_1 \times X_2, \Q_\ell)_{\pi_1 \boxtimes \pi_2} (d)
\]
 is $\Gal(\Qbar/{F_\Sigma})$ invariant. Moreover,
\begin{enumerate}
\item The induced map

\begin{equation}
\label{eqn:intro-xi-Hodge}
\xi(d)^*: H^d (X_1,\Q)_{\pi_1} \rightarrow H^d (X_2,\Q)_{\pi_2}, \quad x\mapsto p_{2,*} (\xi (d) \cup p_1^*(x))
\end{equation}
is an isomorphism of $\Q$-Hodge structures. (i.e., is an isomorphism of $\Q$-vector spaces, that after extending scalars to $\C$, preserves the Hodge filtration.)

\item The induced map
\begin{equation}
\label{eqn:intro-xi-ell}
\xi(d)^*_\ell: H^d (X_1,\Q_\ell)_{\pi_1} \simeq H^d (X_2,\Q_\ell)_{\pi_2},\quad x\mapsto p_{2,*} (\xi (d) \cup p_1^*(x))
\end{equation}
is an isomorphism of $\Gal(\Qbar/{F_\Sigma})$-modules. (Here we view $\xi(d)$ as an etale class via the Betti-etale comparison theorems.)
\end{enumerate}

\end{mtheorem}

\begin{mremark}Before outlining the proof, we make some remarks on the statement of the theorem.
\label{rem:mainremark-intro}

\noindent (i) The assumption that $B_1$ and $B_2$ are ramified at some infinite place is made for technical reasons; it ensures that
a certain auxiliary Shimura variety used in the proof is compact. We
believe that with some extra work (e.g., working with intersection cohomology), this assumption could be
relaxed.

\noindent (ii) Our proof of Theorem \ref{thm:intro-main-full} is actually conditional on some ongoing developments in the theory of automorphic forms. Namely, we need

\begin{enumerate}[label=(\alph*)]
\item \label{cond:classification} The classification of {\it non-tempered} automorphic representations on unitary groups (associated to
hermitian spaces over a CM field)
in terms of local and global $A$-packets.
The expected results that we need are stated carefully in \S
\ref{sec:classification-global} and \S \ref{ss:local-A-packets}, and are the subject of work in progress of
Kaletha, Minguez, Shin and White \cite{kmsw}.

\item \label{cond:characterization-of-galrep} The characterization of Galois representations associated with {\it non-tempered} automorphic representations on unitary groups (associated to
hermitian spaces over a CM field) in terms of $A$-parameters. The expected results here are stated in
\S \ref{sec:gal-rep-on-H2}.
The proofs of these are expected to be fairly standard generalizations of existing results in the tempered case.
These will appear elsewhere since the methods are quite different from those in the main argument of this paper and
since they also depend on ongoing work \cite{ksz} extending the work of Kottwitz (\cite{kot}, \cite{kot-jams}, \cite{kot-invent}) to the case of Shimura varieties of abelian type. (See also \S \ref{sec:motivated} below.)
\end{enumerate}
The cautious reader may therefore take the main theorem as tentative until these results appear in print.

\noindent (iii) While we have stated the main result for trivial coefficients, it works equally well
for local systems. In the main text, this more general case is treated.

\noindent (iv) Our proof does not use the previously known isomorphisms \eqref{eqn:hd-ell-isom}. Rather, it provides an alternative
verification of these isomorphisms which may be of independent interest.

\noindent (v) The assumption that $\pi$ has trivial central character forces
  $\pi$ to be self-dual, and is just made for simplicity.
The non-self-dual case can also be treated similarly. See \S \ref{sec:non-self-dual}
below.

\noindent (vi) The main theorem above is close to saying that the class
$\xi$ is an {\it absolute Hodge class} in the sense of Deligne \cite{dmos}. However, what is missing is the
de Rham piece of the story, i.e., in order to show that $\xi$ is absolutely Hodge, we
would need to show in addition to the above that it is also de Rham rational and that for every embedding $\tau$ of $F_\Sigma$ in
$\C$, the class $\tau(\xi)$ is a Hodge class, whose image in $\ell$-adic cohomology is
Galois invariant for all $\ell$. While it seems difficult to show this directly, it seems likely that we
can build on the results of this paper to
 show something even stronger than being absolutely Hodge, namely that $\xi$ is a {\it motivated cycle} in the sense of Andr\'{e}.
 See \S \ref{sec:motivated} and \S \ref{sec:jl-for-unitary} below.

\noindent(vii)
Finally, it seems possible to
give an independent proof of the existence of an isomorphism
\eqref{eqn:intro-xi-Hodge} of Hodge structures,
for instance by relating the periods of modular forms on $B_1^\times$ and $B_2^\times$
to the fourier coefficients of half-integral weight modular
forms. (See \cite{oda-book}, \cite{oda-hs}, \cite{mur-ram}.)
However, it seems very unlikely that such methods can show that the
isomorphism is also Galois equivariant.

\end{mremark}

\subsection{Outline of the proof}
We now explain the strategy of the proof of Theorem \ref{thm:intro-main-full}.
 In fact, the proof in the general case is very similar to that for $F=\Q$, $n=d=1$, and
so we first describe this case, even though formally speaking this case is excluded from the theorem
on account of the assumption that $B_1$ and $B_2$ be ramified at at
least one infinite place. To be precise, one should work with
intersection cohomology in this case, but for simplicity we just use
usual cohomology with the understanding that the proof given below is
only
correct once generalized to the setting where $F$ is a totally real
field and there is some infinite place where $B_1$ and $B_2$ are both ramified.

 The basic idea of the proof
is to embed $X_1 \times X_2$ in a larger Shimura variety $X$, construct a
Hodge class $\xi$ on $X$ and then show that its pullback to $X_1 \times X_2$ has the right property.
The implementation of this idea is a bit involved and breaks up as follows:

\smallskip

\noindent (i) {\bf Unitary Shimura varieties.}
 We first replace $X_1$ and $X_2$ by closely related unitary Shimura varieties.
Pick an imaginary quadratic field $E$ that embeds in both
$B_1$ and $B_2$.
Let $\VV_1 = B_1$ and $\VV_2 = B_2$, viewed as (right) $E$-vector spaces. These are equipped with
natural hermitian forms that are of signature $(1,1)$ at the infinite place.
The corresponding unitary similitude groups are given by
\begin{equation}
\label{eqn:gubv12}
\GU(\VV_1) \simeq  (B_1^\times  \times E^\times)/ F^\times, \quad \GU(\VV_2) \simeq (B_2^\times  \times E^\times)/ F^\times.
\end{equation}
Let $\VV = \VV_1 \oplus \VV_2$. Thus $\VV$ has signature $(2,2)$ at the infinite place. Consider the maps of
algebraic groups
\begin{equation}
\label{eqn:guv-embed}
B_1^\times \times B_2^\times \rightarrow \PB_1^\times \times \PB_2^\times \leftarrow \G (\U_E(\VV_1)\times \U_E(\VV_2)) /E^\times \rightarrow \GU_E(\VV)/E^\times.
\end{equation}
These induce maps of Shimura varieties
\[
X_1 \times X_2 \rightarrow \tilde{X}_1 \times \tilde{X}_2 \leftarrow Y \rightarrow X
\]
which
may be viewed as giving a correspondence on $(X_1 \times X_2) \times X$. This correspondence induces a map
on cohomology:
\[
 \iota^*: H^*(X) \rightarrow H^*(X_1 \times X_2).
\]
As such, since the kernel of the map
\[
\G (\U_E(\VV_1)\times \U_E(\VV_2)) /E^\times \rightarrow  \PB_1^\times \times \PB_2^\times
 \]
 is isomorphic to
 \[
 \G (E^\times \times E^\times)/E^\times \simeq E^{(1)} \simeq E^\times/F^\times,
 \]
 one can introduce a character $\eta$ of $E^\times/F^\times$ in the construction of the correspondence; this gives a map
 \[
 \iota_\eta^*: H^*(X) \rightarrow H^*(X_1 \times X_2).
\]
that depends on the choice of $\eta$.

\smallskip

\noindent (ii) {\bf Cohomological representations and Vogan-Zuckerman theory.}
Since the cohomology of $X$ is given by automorphic forms \cite{borel-wallach},  it is natural to first look for a {\it non-tempered}
automorphic representation $\Pi$ of $\GU(\VV)$ (or say of $\U(\VV)$ for simplicity) which contributes
to $H^2(X)$ but only to the $(1,1)$-part.
The paper of Vogan-Zuckerman \cite{vz} classifies cohomological representations; one finds
that there
is a unique non-trivial (non-tempered) representation $\Pi_\infty^1$ of $\U (\VV_\R) =\U (2,2)_\R$ with the property that
\[
H^{1,1} (\fg, K; \Pi_\infty^1) \neq 0.
\]
The representation $\Pi_\infty^1$ can be realized as a cohomologically induced representation $A_\fq$ where
$\fq$ is a $\theta$-stable parabolic subalgebra of $\fg$ with Levi component $\mathfrak{u}(1,1) \oplus \mathfrak{u}(1,1)$.
In order to construct $\Pi$, it is first natural to look for an explicit construction of $\Pi_\infty^1$ which is
what is accomplished in the next step.

\smallskip

\noindent (iii) {\bf An exceptional isogeny: archimedean theta correspondence and Kudla-Millson theory.}
The representation $\Pi_\infty^1$ can be constructed as a theta lift of the trivial representation of $\U(1,1)$ with
appropriate choices of splitting characters. However, for rather subtle reasons, this fact does not seem to be
useful in our construction. Instead,
 we use the fact that there
is an exceptional isogeny
\begin{equation}
\label{eqn:isogen-intro}
\SU(2,2)_\R \rightarrow \SO(4,2)_\R.
\end{equation}
Ignoring for the moment the difference between $\U$ and $\SU$, and between $\O$ and $\SO$, we may view
$\Pi_\infty^1$ as a representation of $\O(4,2)_\R$, and viewed this way, the representation
$\Pi_\infty^1$ is in fact a theta lift from $\SL_2$.  This fact may appear somewhat
familiar to connoisseurs of Kudla-Millson theory. Indeed, Kudla-Millson theory studies certain explicit closed forms that are Poincare dual to
geodesic cycles coming from embedded $\O(3,2)$s in $\O(4,2)$,
and shows that the corresponding
automorphic representations of $\O(4,2)$ (which contribute to $H^{1,1}$)
can be constructed as theta lifts of forms of weight $3$ on $\SL_2$.

\smallskip

\noindent (iv) {\bf Inner forms.}
For our purposes we need {\it inner form} versions both of the isogeny \eqref{eqn:isogen-intro}
and of the theta lift. Moreover, we need to work with similitude groups rather than isometry groups.
First the theta lift: let $B$ be the
quaternion algebra given by $B=B_1 \cdot B_2$ in the Brauer group of $\Q$.
Since $B_1$ and $B_2$ are assumed to be non-isomorphic, $B$ is a non-split quaternion algebra.
Then there is a theta lift
\[
\Theta: \sA(\GU_B(W)) \longrightarrow \sA( \GU_B(\tilde{V})^0)
\]
where $\tilde{V}=V \oplus V_0$ (with $V_0$ a one dimensional skew-hermitian $B$-space) is a certain three dimensional $B$-vector space
equipped with a $B$-skew-hermitian form, $W$ is a one dimensional
$B$-vector space equipped with a $B$-hermitian form and $\sA(G)$
denotes the space of automorphic forms on $G$.
(To be precise, the theta lift
depends on a choice of Schwartz function.) Note that the groups
$\U_B(W)$ and $\U_B(\tilde{V})$ are respectively inner forms of $\SL_2$ and $\O(4,2)$.
As for the isogeny, we construct (in \S \ref{sec:global-ex-isom}) an explicit
{\it isomorphism}
\begin{equation}
\label{eqn:intro-delta}
\delta: \PGU_E(\VV) \xrightarrow{\simeq} \PGU_B(\tilde{V})^0
\end{equation}
which is an inner form version of \eqref{eqn:isogen-intro} above for (projectivized) similitude groups.

\smallskip

\noindent (v) {\bf The global theta lift: Schwartz forms.}
With this preparation, we can describe the construction of a $(1,1)$-class on $X$.
Let $h$ be a modular form of weight $3$ and central character $\xi_E$, the quadratic
character associated with the extension $E/\Q$, chosen such that it admits a Jacquet-Langlands transfer
to $B^\times$.
Let
$\tilde{\tau}_h$ be the corresponding representation of $\GL_2 (\A)$. Let $\JL$ denote the
Jacquet-Langlands correspondence. Consider the composite maps of automorphic forms
\[
\sA(\GL_2) \xrightarrow{\JL} \sA (B^\times) = \sA (\GU_B (W)) \xrightarrow{\Theta} \sA (\GU_B(\tilde{V})^0 )
\]
and
\[
 \sA (\PGU_B(\tilde{V})^0 ) \xrightarrow{\delta^{-1}} \sA (\PGU_E(\VV))
 \rightarrow \sA (\GU_E(\VV)).
\]
We show that $\Theta \circ \JL (\tilde{\tau}_h)$ has trivial central
character, and so may be viewed as an
automorphic representation of the group $\PGU_B(\tilde{V})^0 $. Thus
we can consider the composite
\[
\Pi := \delta^{-1} \circ \Theta \circ \JL (\tilde{\tau}_h),
\]
which we may view as an automorphic representation of the group $\GU_E (\VV) (\A)$.
This representation has the property that $\Pi_\infty \simeq \tilde{\Pi}_\infty^1$, where
$\tilde{\Pi}_\infty^1$ denotes the unique representation of $\GU(2,2)_\R$ with trivial central character whose restriction to
$\U(2,2)_\R$ is isomorphic to $\Pi_\infty^1$. Further, one can check that:
\[
\dim H^{p,q} (\fg, K; \Pi_\infty) = \begin{cases}
1 & \text{if } (p,q)=(1,1) \text{ or } (3,3); \\
2 & \text{if } (p,q)=(2,2).
\end{cases}
\]
Explicity, we construct following the ideas of Kudla-Millson (and Millson-Funke in the higher weight case),
a {\it Schwartz form} $\varphi_\infty$ (rather than a Schwartz function)
such that with
$\varphi=\varphi_{\fin} \otimes \varphi_\infty$ for any choice of
a Schwartz funtion $\varphi_{\fin}$, the theta lift
\[
\theta_\varphi (\phi)
\]
may be viewed as giving a $(1,1)$-class on $X$, for $\phi$ in the
space of $\JL(\tilde{\tau}_h)$. To be precise, the construction only
depends on the restriction of $\JL(\tilde{\tau}_h)$ to the subgroup
$\GL_2(\A)^+$ (consisting of elements in $\GL_2(\A)$ with positive determinant at infinity) and the vector $\phi$ must be chosen to lie in the {\it
  antiholomorphic} component of this restriction.

\smallskip

\noindent (vi) {\bf Nonvanishing of the restriction.}
Next we show that for suitable choice of $\eta$, $h$, $\varphi_{\fin}$ and
$\phi$, the $(1,1)$-form $\iota_\eta^*(\theta_\varphi (\phi))$ is
non-vanishing, when projected to the $\pi_1\boxtimes \pi_2$-isotypic
component.
Let us now explain the main idea to prove this non-vanishing.
Let $\omega_{f_{B_1}} $ and $\omega_{f_{B_2}}$ denote holomorphic one-forms in
$H^1 (X_{B_1}, \C)$ and $H^1 (X_{B_2}, \C)$ respectively. The strategy is to compute the integral
\[
\int_{X_{B_1} \times X_{B_2}} \iota_{\eta}^* \theta_{\varphi}(\phi) \cdot (p_1^* \omega_{f_{B_1}} \wedge \overline{p_2^* \omega_{f_{B_2}}})
\]
and show it is non-zero.
Using the isomorphism $\delta$ from \eqref{eqn:intro-delta}, we can reduce the integral to a period on the left-hand side of the seesaw diagram below, which
again involves quaternionic unitary groups:
\[
 \xymatrix{
 \GU_B(\tilde{V})  \ar@{-}[dr] \ar@{-}[d] & \G(\U_B(W) \times \U_B (W))
   \ar@{-}[dl] \ar@{-}[d]  \\
 \G(\U_B(V) \times \U_B(V_0)) & \GU_B(W).   }
\]
The seesaw then implies that the period can be computed on the right where it becomes
a triple product period of the form
\[
\int_{[\GU_B(W)]=[B^\times]}\phi \cdot \overline{ \theta(\eta) f_B}.
\]
We then show that
$\eta$,  $h$, $\varphi_{\fin}$ and $\phi$ can be chosen so as to make this triple product integral nonzero.
A similar argument also shows that
\[
\int_{X_{B_1} \times X_{B_2}} \iota_{\eta}^* \theta_{\varphi}(\phi) \cdot (\overline{p_1^* \omega_{f_{B_1}} }\wedge p_2^* \omega_{f_{B_2}})
\]
is nonzero, and in fact the induced map
\[
\iota_{\eta}^* \theta_{\varphi}(\phi) : H^1 (X_{B_1}, \C) \rightarrow H^1 (X_{B_2}, \C)
\]
is an isomorphism.

\smallskip

\noindent (vii) {\bf Hodge classes.}
As yet we do not know that $\theta_\varphi(\phi)$ is a Hodge class. In fact, strictly speaking it is not likely to be a rational cohomology class, but
we show that it lies in the $\C$-span of the Hodge classes in $H^2(X)$.
The key point here is that the (expected)
classification of automorphic representations implies that any
automorphic representation that is nearly equivalent to $\Pi$ must have
archimedean component lying in the (unique) $A$-packet containing $\tilde{\Pi}_\infty^1$.
Moreover, this archimedean $A$-packet consists of two representations
$\tilde{\Pi}_\infty^1, \tilde{\Pi}_\infty^2$ and the latter contributes only to $H^4(X)$ and not $H^2(X)$.
From this we  deduce that $H^2(X,\C)[\Pi_\fin] $ is entirely of type
$(1,1)$.  (The notation $H^2(X,\C)[\Pi_\fin] $ stands for the
subspace of $H^2(X,\C)$ on which the unramified Hecke algebra at some
finite level acts
by the same Hecke eigenvalues as on $\Pi_\fin$.)

Suppose for the moment that $\Pi$ has coefficients in $\Q$. Then
\[
H^2(X,\Q)[\Pi_\fin] \otimes_{\Q} \C = H^2(X,\C)[\Pi_\fin],
\]
hence $H^2(X,\Q)[\Pi_\fin]$
is a rational Hodge structure, pure of type $(1,1)$.
Since $\theta_\varphi(\phi)$ lies in $H^2(X,\C)[\Pi_\fin]$, we see that it lies in the
$\C$-span of $H^2(X,\Q)[\Pi_\fin] $ and in particular is a $\C$-linear
combination of Hodge classes $\xi$.
We have already seen that $\theta_\varphi(\phi)\neq 0$ and moreover that its restriction to the
$\pi_1 \boxtimes \pi_2$-component of
to $X_1 \times X_2$ is nonzero.
From this and a simple continuity argument one deduces that there is a Hodge class $\xi \in H^2(X,\Q)[\Pi_{\fin}]$ such that the induced map
\[
\iota_\eta ( \xi (1))^* : H^1 (X_{B_1}, \Q)_{\pi_1} \rightarrow H^1 (X_{B_2}, \Q)_{\pi_2}
\]
given by
\[
x \mapsto p_{2,*} (p_1^* (x) \cdot \iota_{\eta}^* \xi (1))
\]
is an isomorphism of rational Hodge structures.

\smallskip

\noindent (viii) {\bf Galois representations.}
For this we need to understand the Galois representation on $H^* (X)$
associated to the $A$-packet containing $\Pi$.
Again, for simplicity let us suppose $F=\Q$, $d=1$, the general case being similar.
Then the expected relation between the Galois representation and the
$A$-parameter gives:
\begin{align*}
H^2(X,\Q_\ell)_{\Pi} &= \Q_\ell (-1),  \\
H^4(X, \Q_\ell)_{\Pi} &= \Q_\ell(-2) \oplus \Sym^2 \rho_{h,\ell}, \\
H^6 (X,\Q_\ell)_{\Pi} &= \Q_\ell (-3),
\end{align*}
where $\rho_{h,\ell}$ is the two dimensional $\ell$-adic representation
attached to $h$.
From this one deduces that as a Galois module, $H^2
(X,\Q_\ell)[\Pi_\fin]\simeq \Q_\ell (-1)^{m} $ for some integer $m$.
For every rational prime $\ell$, the action of $\Gal(\Qbar/\Q)$ on $\xi(1)$ is then trivial
and thus $\xi(1)^*$ (viewed as acting on $\ell$-adic cohomology via the
Betti-etale comparison) is a Galois equivariant isomorphism.

\smallskip

\noindent (ix) {\bf Descending coefficients.}
The argument above needs a bit more care, since $\Pi$ may not have
coefficients in $\Q$. Thus one needs some care to ensure that the
Hodge class constructed has coefficients in $\Q$. This argument needed
to achieve this is
explained in detail in \S
\ref{sec:proof-of-main-thm}.
Roughly, the point is to replace
$H^2 (X,\Q)[\Pi_\fin]$ by
$H^2 (X,\Q)[\I]$,
 where $\I$ is the kernel of the action of the unramified Hecke algebra (with $\Q$-coefficients) on
$\Pi_\fin$. Another possible source of extra coefficients is the character
$\eta$, and this needs to be handled separately.

\smallskip

This completes the outline of the proof of Theorem \ref{thm:intro-main-full} in the case
$F=\Q$, $n=d=1$, $k=2$. The general case (assuming still that
$\uk=(2,\ldots,2)$) is only slightly more complicated.
In general we have
\[
\U_E(\VV) (\R) \simeq \U(2,2)^d \times \U(4)^{n-d},
\]
where the $\U(2,2)$ factors correspond to the places in $\Sigma$ and
the $\U(4)$ factors to the infinite places not in $\Sigma$.
At the infinite places in $\Sigma$, i.e., where $B_1$ and $B_2$ are both split, we
just imitate the constructions above. However, we need to deal as well
with the infinite places where $B_1$ and $B_2$ are both ramified.
At such places the representation $\Pi_\infty$ is trivial and the
local $A$-packet is a singleton, consisting of just the trivial representation.
This is consistent with the fact that at such places $v$, we have
\[ \U_E(\VV)_v\simeq \U(4), \quad \U_B(\tilde{V})_v \simeq \O(6), \quad \U_B(W)_v \simeq \SL_2
\]
and the theta lift of the weight $3$ {\it holomorphic} discrete series representation on $\SL_2$ is the
trivial representation of $\O(6)$.
The conclusion then is that $H^{2d} (X,\C)[\Pi_\fin]$ consists entirely
of $(d,d)$-classes and one can find a Hodge class $\xi \in H^{2d}(X,\Q)[\Pi_{\fin}]$ such that the induced map
\[
 \xi (d)^* : H^d (X_{B_1}, \Q) \rightarrow H^d (X_{B_2}, \Q)
\]
given by
\[
x \mapsto p_{2,*} (p_1^* (x) \cdot \iota_{\eta}^* \xi (d))
\]
is an isomorphism of rational Hodge structures, that is also Galois
invariant.

\begin{rem}
We note the following conceptual reason why we work with the group $\U_B(\tilde{V})$ which at archimedean places looks like a product $\O(4,2)^d \times \O(0,6)^{n-d}$.
After all, in principle, one could also construct Kudla-Millson classes directly on the group $\U_B(V)$,  which at archimedean places looks like a product $\O(2,2)^d \times \O(0,4)^{n-d}$, by taking a lift of a form of parallel weight two. However, the issue is that on this smaller group, the Hodge classes are
mixed up with other classes of the same degree, and therefore it is difficult to see that the Kudla-Millson class is in the $\C$-span of the Hodge classes, except in the ``trivial" situation when $B_1=B_2$; in that case the group $\U_B(V)$ is quasi-split and there are obvious ``diagonal" cycles in the correct degree. On the larger group however, the Hodge classes in degree $(d,d)$ can be separated out
using Hecke operators.
\end{rem}

\begin{rem} (Local systems and normalizations)
We briefly mention the numerology in the case of general local
systems. Suppose that the form $\pi$ has weights $\uk=(k_1,\ldots,
k_n)$. Then (in the classical normalization) the Hodge structure of $H^*(X_i)_{\pi_i}$ is a tensor product
over the places $v$ in $\Sigma$ of a Hodge structure of type
\[
(k_v-1,0) + (0,k_v-1).
\]
Thus $H^*(X_1)_{\pi_1} \otimes H^*(X_2)_{\pi_2}$ is a tensor product
over the places $v$ in $\Sigma$ of a Hodge structure of type
\begin{equation}
\label{eqn:hs-of-tensor}
(2k_v-2,0) + 2 (k_v-1,k_v-1) + (0,2k_v-2).
\end{equation}
The Hodge class in $H^*(X_1)_{\pi_1} \otimes H^*(X_2)_{\pi_2}$  should come from the tensor product over the places
$v$ in $\Sigma$ of a class of type $(k_v-1,k_v-1)$. In our
construction, we pick an auxiliary form $\tilde{\tau}$ of weights
$\uk+\1 = (k_1+1, \ldots, k_n+1)$. Then $\JL( \tilde{\tau})$  corresponds to
a Hodge structure which is a tensor
 product
over the places $v$ in $\Sigma$ of a Hodge structure of type
\[
(k_v,0) + (0,k_v).
\]
Its lift $\Pi$ to $\U_B(\tilde{V})$ contributes to different cohomological
degrees; so there is an associated Hodge diamond which is the tensor
 product
over the places $v$ in $\Sigma$ of a Hodge diamond of the form:

\begin{equation}
\label{eqn:hs-of-unitary}
\xymatrix{
& (k_v+1,k_v+1) & \\
(2k_v,0) &2 (k_v,k_v) & (0,2k_v) \\
& (k_v-1,k_v-1) &
}
\end{equation}

The Hodge class in $H^*(X)_\Pi $  comes from the tensor product over the places
$v$ in $\Sigma$ of the class of type $(k_v-1,k_v-1)$. The ``rest'' of
the Hodge structure at any place $v$ consists of Tate twists
of this $(k_v-1,k_v-1)$ class and $\Sym^2$ of the Hodge structure
attached to $\tilde{\tau}$.

In the main text, we use the ``automorphic normalization'' instead of
the classical normalization. This amounts to twisting the Hodge
structures in \eqref{eqn:hs-of-tensor} and \eqref{eqn:hs-of-unitary} above by
$(2-k_v,2-k_v)$. This twist is therefore not visible in parallel
weight $2$.
\end{rem}

In the next section, we discuss some possible extensions and
generalizations of the main theorem.

\subsection{Extensions and generalizations}

\subsubsection{The non-self-dual case}
\label{sec:non-self-dual}
Let us denote the contragredient of $\pi$ by $\pi^\vee$ and let $\chi$ be the central character of $\pi$ so that
$\pi \simeq \pi^\vee \otimes \chi$.
The method described above extends to this case, except that we must
choose $\eta$ such that $\eta|_{\A^\times_F} = \chi^{-1} $ to
compensate for the central character of $\pi$.
We remark on one unusual feature. Namely, it seems that the method outlined here naturally produces a Hodge class $\xi \in H^{2d}( X, \Q(\chi^2))_\Pi $ such that:
\begin{enumerate}
\item The induced map
\[
\xi(d)^*: H^d(X_{B_1}, \Q(\chi^{-1}))_{\pi_1^\vee} \rightarrow H^d(X_{B_2}, \Q(\chi))_{\pi_2}
\]
is an isomorphism of $\Q(\chi)=\Q(\chi^{-1})$-vector spaces and preserves the Hodge filtration (on tensoring with $\C$).

\item The Galois module $H^{2d}(X,\Q_\ell (\chi^2))$ is isomorphic to (a sum of copies of) $\Q_\ell (-d) (\chi^2)$ and the induced map
\[
\xi(d) ^*: H^d(X_{B_1}, \Q_\ell(\chi^{-1}))_{\pi_1^\vee} \rightarrow H^d(X_{B_2}, \Q_\ell(\chi))_{\pi_2}
\]
satisfies the following Galois equivariance:
\[
\sigma (\xi^*(x)) = \chi^2 (\sigma) \cdot \xi^* (\sigma (x)).
\]
We note that $\Q(\chi^{-1})=\Q(\chi)$ and $\Q_\ell (\chi^{-1})=\Q_\ell (\chi)$.
\end{enumerate}
The reason this is unusual is that one might expect to have a natural construction producing a {\it rational} Hodge class in
$H^*(X_{B_1}, \Q(\chi))_{\pi_1^\vee} \otimes H^*(X_{B_2}, \Q(\chi))_{\pi_2}$, since
after all the Galois representation $H^*(X_{B_1}, \Q_\ell (\chi))_{\pi_1^\vee} \otimes H^*(X_{B_2}, \Q_\ell (\chi))_{\pi_2}$ always
contains the {\it trivial} representation as a direct summand. Instead, our construction naturally produces a Hodge class (with coefficients in a number field) in
$H^* (X_{B_1}, \Q(\chi))_{\pi_1} \otimes  H^*(X_{B_2}, \Q(\chi))_{\pi_2}$
and then one has to ``untwist'' it to produce the rational Hodge class
that one expects to exist.

\subsubsection{Motivated cycles}
\label{sec:motivated}
It is natural to ask if the Hodge class we have constructed on $X$ may
be
obtained by a simple modification of
the class of an algebraic cycle of an
embedded Shimura variety.
This is unlikely in general, since
any cycle constructed in such a fashion would have a dense set of CM
points,
and then by the Andr\'{e}-Oort conjecture (see \cite{klingler-yafaev},
for example), the pull-back of this cycle
to
$X_1 \times X_2$ would be a sub-Shimura variety. However, if $B_1
\neq B_2$, the product variety $X_1 \times X_2$ has no interesting
sub-Shimura varieties.
On the other hand,
it seems likely that one can build on the methods of this paper to
show that the Hodge class we have constructed on $X_1 \times X_2$ is the class of a
{\it motivated cycle} in the sense of Andr\'{e} \cite{andre}, i.e., it
is
the class of an algebraic cycle up to inverting the Lefschetz operator $L$
on some auxiliary variety.
In our case, we expect the auxiliary variety is simply the Shimura
variety $X$. We have constructed in this paper a Hodge class $\xi$ on
$X$ of type $(d,d)$, and $(L^{2d})^{-1} (\xi)$ is a Hodge class of type
$(3d,3d)$.
The basic idea is to show that this Hodge class is the class of a
rather explicit
algebraic cycle, namely coming from an embedded unitary Shimura
variety associated to a group $H$ such that $H(\R) = \U(1,1)^d \times
\U(2)^{n-d}$. This extension will be treated in a forthcoming paper
\cite{mot-cyc}. It seems likely that this extension will obviate the need for
the results on Galois representations in Remark \ref{rem:mainremark-intro},(ii)(b),
namely the characterization of the Galois representations associated to non-tempered forms on the cohomology of
unitary Shimura varieties.

\subsubsection{Functoriality for unitary groups}
\label{sec:jl-for-unitary}
We expect that the picture described above can be generalized to other instances of
functoriality,
especially when the varieties $X_1$ and $X_2$ have the same dimension
and the automorphic representation in question contributes to the same
(middle) cohomological degree. Namely,
there should exist a Shimura variety $X$ of dimension $m$ say, a subvariety $Y \subset X$,
 \[
 \xymatrix{
 & X & \\
  X_1 \times X_2 \ar[ur]^{i} & & Y \ar[ul]_{j}
  }
  \]
with $\dim(Y) = \dim(X_1) = \dim (X_2)=d$,  such that
\[
 i^* \circ (L^{m-2d})^{-1} (\cl_X (Y))
 \]
is a (motivated) cycle which realizes the Jacquet-Langlands
correspondence (for a fixed automorphic representation).
For example, suppose that $X_1$ and $X_2$ correspond to $\U(\VV_1)$
and $\U(\VV_2)$ respectively,
where $\VV_i$ is a hermitian space over an imaginary quadratic extension
of $\Q$, with
$\U(\VV_i) (\R) \simeq \U(p,q)$. One might speculate that in this case
one could take
$X$ attached to the unitary group of $\VV_1\oplus \VV_2$ and $Y$ attached to $\U(\WW)$
where $\WW \subset \VV$ is an auxiliary hermitian space of signature
$(p,q)$; this corresponds to the following picture of real groups:
 \[
 \xymatrix{
 & \U(2p,2q) & \\
  \U(p,q) \times \U(p,q)  \ar[ur] & & \U(p,q) \ar[ul]
  }
  \]
with $d=pq$ and $m=4pq$. There are several issues that need to be
resolved to determine if this does indeed realize functoriality,
the most subtle of which seems to be establishing  nonvanishing of the
relevant periods. The prediction above should be viewed as complementary to
the Tate conjecture, which by itself gives no indication of where to
find the relevant cycle; moreover,
knowing that the cohomology class realizing functoriality is related
in an explicit fashion via the Lefschetz operator to the class
of a Shimura sub-variety on an auxiliary Shimura variety may have
useful applications.
 It would also be interesting to compare this
with the recent work on Tate cycles in finite characteristic, e.g.,
\cite{tian-xiao-gorenoort}, \cite{tian-xiao-tate-quat},
\cite{helm-tian-xiao}, \cite{xiao-zhu}.

{\bf Acknowledgements.} During the preparation
of this article, A.I. was partially supported by JSPS KAKENHI Grant Numbers 26287003,
and K.P. was partially supported by NSF grants DMS 1160720, DMS 1600494 and a grant from the
Simons Foundation (\# 305784). He is also thankful to the Institute for Advanced Study
for its support in 2014-15 as a von Neumann fellow, when the work on this paper was initiated.


TITLE: What is an indexed set?
QUESTION [1 upvotes]: I stuck on something. I am trying to understand what is said in the image I linked above.
"A function I from a set Λ onto a set a is said to index the set a by Λ. The set Λ is called the index and a is the indexed set. If I(λ) = a, then we write aλ for I(λ)."
There's no examples and further explanations in the textbook. As far as I understood, Λ can be a set {1,2,3,4...} or, say {a,b,c,d..} and the set a can be {tree, fox, grass}. So after set a got indexed by the set Λ = {1,2,3}, we get a = {tree(1), fox(2), grass(3)} (Sorry, I don't know how to type so that "1" appear under the word "tree")
Am I right? And by the way, what is it supposed to mean: "If I(λ) = a, then we write aλ for I(λ)." I swear there's no λ mentioned before that line in the textbook. Is λ a variable that might represent one of those elements of {1,2,3...}? Say I could say that λ = 1 or λ = 15, right?
Explain please what is this all about. Thank you in advance.

REPLY [4 votes]: In many situations it's handy to have a way for referring to elements of a set in a “uniform way”.
Think to this “indexing” as a way to name the elements of the set: it's quite similar to identifying a car with its plate number.
Suppose the set is $a=\{\mathrm{tree},\mathrm{fox},\mathrm{grass}\}$. The idea is to pick a surjective function from some set to $a$. The set could be $\{1,2,3\}$ or $\{\mathrm{Larry},\mathrm{Moe},\mathrm{Curly}\}$, but there's no restriction to the size of $I$, so long as the function we pick is surjective.
Remember that the set $a$ can also be described as $a=\{\mathrm{grass},\mathrm{fox},\mathrm{tree}\}$, so even if we choose $I=\{1,2,3\}$, there's no “canonical” indexing.
Both functions
$$
\begin{cases}
1\mapsto\mathrm{tree}\\
2\mapsto\mathrm{fox}\\
3\mapsto\mathrm{grass}
\end{cases}
\qquad\text{and}\qquad
\begin{cases}
1\mapsto\mathrm{grass}\\
2\mapsto\mathrm{fox}\\
3\mapsto\mathrm{tree}
\end{cases}
$$
are valid as an indexing of $a$. Also the function $\{1,2,3,4\}\to a$ defined by
$$
\begin{cases}
1\mapsto\mathrm{tree}\\
2\mapsto\mathrm{fox}\\
3\mapsto\mathrm{grass}\\
4\mapsto\mathrm{fox}
\end{cases}
$$
would be a valid indexing.
The indexing function is usually “unnamed”, so it's denoted by $\lambda\mapsto a_\lambda$, because its exact definition is mostly irrelevant. The important thing is that, for every $x\in a$, there exists $\lambda\in I$ such that $x=a_\lambda$, so that $x$ has a “plate number” (maybe more than one).


text:
section \<open>Integral and Bounded Matrices and Vectors\<close>

text \<open>We define notions of integral vectors and matrices and bounded vectors and matrices
  and prove some preservation lemmas. Moreover, we prove a bound on determinants.\<close>
theory Integral_Bounded_Vectors
  imports
    Missing_VS_Connect
    Sum_Vec_Set
    LLL_Basis_Reduction.Gram_Schmidt_2 (* for some simp-rules *)
begin

(* TODO: move into theory Norms *)
lemma sq_norm_unit_vec[simp]: assumes i: "i < n"
  shows "\<parallel>unit_vec n i\<parallel>\<^sup>2 = (1 :: 'a :: {comm_ring_1,conjugatable_ring})"
proof -
  from i have id: "[0..<n] = [0..<i] @ [i] @ [Suc i ..< n]"
    by (metis append_Cons append_Nil diff_zero length_upt list_trisect)
  show ?thesis unfolding sq_norm_vec_def unit_vec_def
    by (auto simp: o_def id, subst (1 2) sum_list_0, auto)
qed

definition Ints_vec ("\<int>\<^sub>v") where
  "\<int>\<^sub>v = {x. \<forall> i < dim_vec x. x $ i \<in> \<int>}"

definition indexed_Ints_vec  where
  "indexed_Ints_vec I = {x. \<forall> i < dim_vec x. i \<in> I \<longrightarrow> x $ i \<in> \<int>}"

lemma indexed_Ints_vec_UNIV: "\<int>\<^sub>v = indexed_Ints_vec UNIV"
  unfolding Ints_vec_def indexed_Ints_vec_def by auto

lemma indexed_Ints_vec_subset: "\<int>\<^sub>v \<subseteq> indexed_Ints_vec I"
  unfolding Ints_vec_def indexed_Ints_vec_def by auto

lemma Ints_vec_vec_set: "v \<in> \<int>\<^sub>v = (vec_set v \<subseteq> \<int>)"
  unfolding Ints_vec_def vec_set_def by auto

definition Ints_mat ("\<int>\<^sub>m") where
  "\<int>\<^sub>m = {A. \<forall> i < dim_row A. \<forall> j < dim_col A. A $$ (i,j) \<in> \<int>}"

lemma Ints_mat_elements_mat: "A \<in> \<int>\<^sub>m = (elements_mat A \<subseteq> \<int>)"
  unfolding Ints_mat_def elements_mat_def by force

lemma minus_in_Ints_vec_iff[simp]: "(-x) \<in> \<int>\<^sub>v \<longleftrightarrow> (x :: 'a :: ring_1 vec) \<in> \<int>\<^sub>v"
  unfolding Ints_vec_vec_set by (auto simp: minus_in_Ints_iff)

lemma minus_in_Ints_mat_iff[simp]: "(-A) \<in> \<int>\<^sub>m \<longleftrightarrow> (A :: 'a :: ring_1 mat) \<in> \<int>\<^sub>m"
  unfolding Ints_mat_elements_mat by (auto simp: minus_in_Ints_iff)

lemma Ints_vec_rows_Ints_mat[simp]: "set (rows A) \<subseteq> \<int>\<^sub>v \<longleftrightarrow> A \<in> \<int>\<^sub>m"
  unfolding rows_def Ints_vec_def Ints_mat_def by force

lemma unit_vec_integral[simp,intro]: "unit_vec n i \<in> \<int>\<^sub>v"
  unfolding Ints_vec_def by (auto simp: unit_vec_def)

lemma diff_indexed_Ints_vec:
  "x \<in> carrier_vec n \<Longrightarrow> y \<in> carrier_vec n \<Longrightarrow> x \<in> indexed_Ints_vec I \<Longrightarrow> y \<in> indexed_Ints_vec I
  \<Longrightarrow> x - y \<in> indexed_Ints_vec I"
  unfolding indexed_Ints_vec_def by auto

lemma smult_indexed_Ints_vec: "x \<in> \<int> \<Longrightarrow> v \<in> indexed_Ints_vec I \<Longrightarrow> x \<cdot>\<^sub>v v \<in> indexed_Ints_vec I"
  unfolding indexed_Ints_vec_def smult_vec_def by simp

lemma add_indexed_Ints_vec:
  "x \<in> carrier_vec n \<Longrightarrow> y \<in> carrier_vec n \<Longrightarrow> x \<in> indexed_Ints_vec I \<Longrightarrow> y \<in> indexed_Ints_vec I
  \<Longrightarrow> x + y \<in> indexed_Ints_vec I"
  unfolding indexed_Ints_vec_def by auto

lemma (in vec_space) lincomb_indexed_Ints_vec: assumes cI: "\<And> x. x \<in> C \<Longrightarrow> c x \<in> \<int>"
  and C: "C \<subseteq> carrier_vec n"
  and CI: "C \<subseteq> indexed_Ints_vec I"
shows "lincomb c C \<in> indexed_Ints_vec I"
proof -
  from C have id: "dim_vec (lincomb c C) = n" by auto
  show ?thesis unfolding indexed_Ints_vec_def mem_Collect_eq id
  proof (intro allI impI)
    fix i
    assume i: "i < n" and iI: "i \<in> I"
    have "lincomb c C $ i = (\<Sum>x\<in>C. c x * x $ i)"
      by (rule lincomb_index[OF i C])
    also have "\<dots> \<in> \<int>"
      by (intro Ints_sum Ints_mult cI, insert i iI CI[unfolded indexed_Ints_vec_def] C, force+)
    finally show "lincomb c C $ i \<in> \<int>" .
  qed
qed

definition "Bounded_vec (b :: 'a :: linordered_idom) = {x . \<forall> i < dim_vec x . abs (x $ i) \<le> b}"

lemma Bounded_vec_vec_set: "v \<in> Bounded_vec b \<longleftrightarrow> (\<forall> x \<in> vec_set v. abs x \<le> b)"
  unfolding Bounded_vec_def vec_set_def by auto

definition "Bounded_mat (b :: 'a :: linordered_idom) =
  {A . (\<forall> i < dim_row A . \<forall> j < dim_col A. abs (A $$ (i,j)) \<le> b)}"

lemma Bounded_mat_elements_mat: "A \<in> Bounded_mat b \<longleftrightarrow> (\<forall> x \<in> elements_mat A. abs x \<le> b)"
  unfolding Bounded_mat_def elements_mat_def by auto

lemma Bounded_vec_rows_Bounded_mat[simp]: "set (rows A) \<subseteq> Bounded_vec B \<longleftrightarrow> A \<in> Bounded_mat B"
  unfolding rows_def Bounded_vec_def Bounded_mat_def by force

lemma unit_vec_Bounded_vec[simp,intro]: "unit_vec n i \<in> Bounded_vec (max 1 Bnd)"
  unfolding Bounded_vec_def unit_vec_def by auto

lemma Bounded_matD: assumes "A \<in> Bounded_mat b"
  "A \<in> carrier_mat nr nc"
shows "i < nr \<Longrightarrow> j < nc \<Longrightarrow> abs (A $$ (i,j)) \<le> b"
  using assms unfolding Bounded_mat_def by auto

lemma Bounded_vec_mono: "b \<le> B \<Longrightarrow> Bounded_vec b \<subseteq> Bounded_vec B"
  unfolding Bounded_vec_def by auto

lemma Bounded_mat_mono: "b \<le> B \<Longrightarrow> Bounded_mat b \<subseteq> Bounded_mat B"
  unfolding Bounded_mat_def by force

lemma finite_Bounded_vec_Max:
  assumes A: "A \<subseteq> carrier_vec n"
    and fin: "finite A"
  shows "A \<subseteq> Bounded_vec (Max { abs (a $ i) | a i. a \<in> A \<and> i < n})"
proof
  let ?B = "{ abs (a $ i) | a i. a \<in> A \<and> i < n}"
  have fin: "finite ?B"
    by (rule finite_subset[of _ "(\<lambda> (a,i). abs (a $ i)) ` (A \<times> {0 ..< n})"], insert fin, auto)
  fix a
  assume a: "a \<in> A"
  show "a \<in> Bounded_vec (Max ?B)"
    unfolding Bounded_vec_def
    by (standard, intro allI impI Max_ge[OF fin], insert a A, force)
qed

definition det_bound :: "nat \<Rightarrow> 'a :: linordered_idom \<Rightarrow> 'a" where
  "det_bound n x = fact n * (x^n)"

lemma det_bound: assumes A: "A \<in> carrier_mat n n"
  and x: "A \<in> Bounded_mat x"
shows "abs (det A) \<le> det_bound n x"
proof -
  have "abs (det A) = abs (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>i = 0..<n. A $$ (i, p i)))"
    unfolding det_def'[OF A] ..
  also have "\<dots> \<le> (\<Sum>p | p permutes {0..<n}. abs (signof p * (\<Prod>i = 0..<n. A $$ (i, p i))))"
    by (rule sum_abs)
  also have "\<dots> = (\<Sum>p | p permutes {0..<n}. (\<Prod>i = 0..<n. abs (A $$ (i, p i))))"
    by (rule sum.cong[OF refl], auto simp: abs_mult abs_prod sign_def simp flip: of_int_abs)
  also have "\<dots> \<le> (\<Sum>p | p permutes {0..<n}. (\<Prod>i = 0..<n. x))"
    by (intro sum_mono prod_mono conjI Bounded_matD[OF x A], auto)
  also have "\<dots> = fact n * x^n" by (auto simp add: card_permutations)
  finally show "abs (det A) \<le> det_bound n x" unfolding det_bound_def by auto
qed

lemma minus_in_Bounded_vec[simp]:
  "(-x) \<in> Bounded_vec b \<longleftrightarrow> x \<in> Bounded_vec b"
  unfolding Bounded_vec_def by auto

lemma sum_in_Bounded_vecI[intro]: assumes
  xB: "x \<in> Bounded_vec B1" and
  yB: "y \<in> Bounded_vec B2" and
  x: "x \<in> carrier_vec n" and
  y: "y \<in> carrier_vec n"
shows "x + y \<in> Bounded_vec (B1 + B2)"
proof -
  from x y have id: "dim_vec (x + y) = n" by auto
  show ?thesis unfolding Bounded_vec_def mem_Collect_eq id
  proof (intro allI impI)
    fix i
    assume i: "i < n"
    with x y xB yB have *: "abs (x $ i) \<le> B1" "abs (y $ i) \<le> B2"
      unfolding Bounded_vec_def by auto
    thus "\<bar>(x + y) $ i\<bar> \<le> B1 + B2" using i x y by simp
  qed
qed

lemma (in gram_schmidt) lincomb_card_bound: assumes XBnd: "X \<subseteq> Bounded_vec Bnd"
  and X: "X \<subseteq> carrier_vec n"
  and Bnd: "Bnd \<ge> 0"
  and c: "\<And> x. x \<in> X \<Longrightarrow> abs (c x) \<le> 1"
  and card: "card X \<le> k"
shows "lincomb c X \<in> Bounded_vec (of_nat k * Bnd)"
proof -
  from X have dim: "dim_vec (lincomb c X) = n" by auto
  show ?thesis unfolding Bounded_vec_def mem_Collect_eq dim
  proof (intro allI impI)
    fix i
    assume i: "i < n"
    have "abs (lincomb c X $ i) = abs (\<Sum>x\<in>X. c x * x $ i)"
      by (subst lincomb_index[OF i X], auto)
    also have "\<dots> \<le> (\<Sum>x\<in>X. abs (c x * x $ i))" by auto
    also have "\<dots> = (\<Sum>x\<in>X. abs (c x) * abs (x $ i))" by (auto simp: abs_mult)
    also have "\<dots> \<le> (\<Sum>x\<in>X. 1 * abs (x $ i))"
      by (rule sum_mono[OF mult_right_mono], insert c, auto)
    also have "\<dots> = (\<Sum>x\<in>X. abs (x $ i))" by simp
    also have "\<dots> \<le> (\<Sum>x\<in>X. Bnd)"
      by (rule sum_mono, insert i XBnd[unfolded Bounded_vec_def] X, force)
    also have "\<dots> = of_nat (card X) * Bnd" by simp
    also have "\<dots> \<le> of_nat k * Bnd"
      by (rule mult_right_mono[OF _ Bnd], insert card, auto)
    finally show "abs (lincomb c X $ i) \<le> of_nat k * Bnd" by auto
  qed
qed

lemma bounded_vecset_sum:
  assumes Acarr: "A \<subseteq> carrier_vec n"
    and Bcarr: "B \<subseteq> carrier_vec n"
    and sum: "C = A + B"
    and Cbnd: "\<exists> bndC. C \<subseteq> Bounded_vec bndC"
  shows "A \<noteq> {} \<Longrightarrow> (\<exists> bndB. B \<subseteq> Bounded_vec bndB)"
    and "B \<noteq> {} \<Longrightarrow> (\<exists> bndA. A \<subseteq> Bounded_vec bndA)"
proof -
  {
    fix A B :: "'a vec set"
    assume Acarr: "A \<subseteq> carrier_vec n"
    assume Bcarr: "B \<subseteq> carrier_vec n"
    assume sum: "C = A + B"
    assume Ane: "A \<noteq> {}"
    have "\<exists> bndB. B \<subseteq> Bounded_vec bndB"
    proof(cases "B = {}")
      case Bne: False
      from Cbnd obtain bndC where bndC: "C \<subseteq> Bounded_vec bndC" by auto
      from Ane obtain a where aA: "a \<in> A" and acarr: "a \<in> carrier_vec n" using Acarr by auto
      let ?M = "{abs (a $ i) | i. i < n}"
      have finM: "finite ?M" by simp
      define nb where "nb = abs bndC + Max ?M"
      {
        fix b
        assume bB: "b \<in> B" and bcarr: "b \<in> carrier_vec n"
        have ab: "a + b \<in> Bounded_vec bndC" using aA bB bndC sum by auto
        {
          fix i
          assume i_lt_n: "i < n"
          hence ai_le_max: "abs(a $ i) \<le> Max ?M" using acarr finM Max_ge by blast
          hence "abs(a $ i + b $ i) \<le> abs bndC"
            using ab bcarr acarr index_add_vec(1) i_lt_n unfolding Bounded_vec_def by auto
          hence "abs(b $ i) \<le> abs bndC + abs(a $ i)" by simp
          hence "abs(b $ i) \<le> nb" using i_lt_n bcarr ai_le_max unfolding nb_def by simp
        }
        hence "b \<in> Bounded_vec nb" unfolding Bounded_vec_def using bcarr carrier_vecD by blast
      }
      hence "B \<subseteq> Bounded_vec nb" unfolding Bounded_vec_def using Bcarr by auto
      thus ?thesis by auto
    qed auto
  } note theor = this
  show "A \<noteq> {} \<Longrightarrow> (\<exists> bndB. B \<subseteq> Bounded_vec bndB)" using theor[OF Acarr Bcarr sum] by simp
  have CBA: "C = B + A" unfolding sum by (rule comm_add_vecset[OF Acarr Bcarr])
  show "B \<noteq> {} \<Longrightarrow> \<exists> bndA. A \<subseteq> Bounded_vec bndA" using theor[OF Bcarr Acarr CBA] by simp
qed

end

subset_name: curated


text:
\begin{document}

\begin{center}
{\Large{\bf On the dynamics of the Coronavirus epidemic and the unreported cases:
the Chilean case
}}\\
\vspace{0.5cm}
{\large Andr\'es Navas\footnote{Supported by MICITEC Chile.}
\,\,\& \,\,
Gast\'on Vergara-Hermosilla\footnote{Supported by  the European Union’s Horizon 2020 research and
innovation programme under the Marie Sklodowska-Curie grant agreement No 765579.}
}
\end{center}

\vspace{0.3cm}

\noindent{\bf \Large{Introduction}}

\vspace{0.3cm}

One of the main problems faced in the mathematical modeling of the coronavirus epidemic has been
the lack of quality data. In particular, it is estimated that a large number of cases have been unreported,
especially those of asymptomatic patients. This is mostly due to the strong demand of tests required by
a relatively complete report of the infected cases. Although the countries/regions that have managed to
control the epidemic have been precisely those that have been able to develop a great capacity of testing,
this has not been achieved in most of the situations.

In general, the number of unreported patients has been estimated by extrapolating data from the reported cases
assuming that both numbers vary proportionally. However, this view can be criticized (al least) in two directions:

\noindent -- It does not consider the dynamical role of the unreported cases in the evolution of the epidemic and,
by extension, in the number of reported cases;

\noindent -- it does not allow a possible variation of the proportion between the numbers of reported
and unreported cases when the conditions of social distancing remain unchanged.

In this work, we address these points incorporating the unreported cases into the modeling. In a first introductory
section, we discuss from a mathematical perspective what happens when the testing capacity is very low. Using
a very simple argument we show that, in this context, the dynamics of the disease is actually governed by the growth
of the number of unreported cases, and that of the reported patients becomes much smaller (in a progressive way) than
that of the total cases. Although the discussion is presented in a context of extreme (hence ``ideal'') conditions, it certainly
illustrates how essential is to consider the role of the unreported cases to analyze the global evolution of the epidemic.

Several mathematical models have been proposed to deal with the unreported cases and their role in the progression of the
epidemic. In the second section of this work we address one of them, with the acronym SIRU, recently proposed by
Zhihua Liu, Pierre Magal, Ousmane Seydi and Glen Webb \cite{1,2,3}. After a brief presentation of the model and
an explanation of why it is more adjusted to the current epidemic, we proceed to establish a series of structural results.
Although this does not completely close the study of the qualitative properties of the underlying differential equations,
we can already glimpse an analogy with those of the classical SIR model. However, we stress an important difference:
in the SIRU model, the curves that appear do not necessarily have a single peak. Specifically, we exhibit a simple
method to detect parameters that give rise to curves with at least two peaks.

The model of Liu, Magal, Seydi and Webb has already been used to describe the evolution of the epidemic in various countries
(China, South Korea, the United Kingdom, Italy, France and Spain). In the last section of this work, we implement this modeling
to the global Chilean scenario using the official COVID-19 data provided by the Chilean government \cite {8}. However, unlike
\cite{1,2,3}, our implementation is novel in that it uses a variable transmission rate for the disease, which is more pertinent
according to the local epidemiological evolution.

This work concludes with a section of general conclusions in which some future lines of research are also described.

\section{On the dynamics with low testing capacity}

\subsection{The context}

We denote by $U$  the number of positive cases that one would expect to detect within a certain unit of time
in some region using the maximal
capacity of testing that is available.\footnote{We note that this quantity shouldn't be equal to the total number of tests, but to a
fraction of this. Indeed, it is expected (and international statistics confirm this) that, for each positive test, a certain fixed
amount of tests will have negative outputs. This percentage depends on numerous factors, in particular, on the bias to
preferably test symptomatic patients.} Suppose that the epidemic has evolved to a point where the number
of detected cases in this unit time is systematically very close to this threshold $ U $. The reported
contagion curve is then taking a plateau shape, but a question immediately arises: has the contagion process
entered into an stationary phase -with a reproduction rate $ R $ equal to 1 or slightly higher-, or is a
significant number of positive cases being indetected?

It is impossible to answer {\em a priori} to this question. However, it is very unlikely that
the process enters a stationary regime exactly when the threshold is reached. We argue below that,
if the entry into an stationary regime has not occurred, then not only a fixed ``proportion'' of the number of cases is being
undetected, but this is the case of a much larger amount; more precisely, the latter follows an exponential rate growth.

\subsection{A mathematical argument}

To simplify the discussion, our unit time will be equal to the total period of the disease.
For an instant $ i $, we denote by $ c_i $ the number of reported positive cases, and
by $ C_i $ that of total cases. We assume that, while $ c_i $ remains below the threshold,
it assumes values very close to it in the future evolution. We write this as follows:
$$ U-v \leq c_i \leq U, $$
where $ v $ is relatively small compared to $ U $.

We will also assume that we are not in a stationary regime (that is, the apparent stationarity is actually
a consequence of a default of testing). Therefore, the value of $ C_i $ is substantially greater than the
threshold and, consequently, than $ c_i $. At the instant $i$, the number of cases
that are not being detected is $C_i - c_i$.
These individuals will have a higher social activity than those detected as infected (since the latter will be quarantined).
Therefore, the average number of new infected individuals by these undetected agents will be a value
$ \hat{\mathcal{R}} $ strictly larger than $  \mathcal{R} $.

Thus, on average, the detected infected individuals of time $i$ generate $ \mathcal{R} $ new infections at
time $ i + 1 $, while those undetected at time $ i $ (whose number is $ C_i - c_i $) each generate
$ \hat{\mathcal{R}}$ new infections. Therefore, the following inequality holds:
$$C_{i+1} \geq  \mathcal{R} c_i + \hat{\mathcal{R}} \, [C_i - c_i].$$
Since $ \mathcal{R} \geq 1 $, this implies
$$C_{i+1} \geq c_i + \hat{\mathcal{R}} \, [C_i-c_i],$$
hence
$$C_{i+1} - c_{i+1} \geq c_i - c_{i+1} + \hat{\mathcal{R}}  \, [C_i - c_i ] .$$
Since $ c_i \geq U - v $ and $ c_{i + 1} \leq U $, the above implies
$$C_{i+1} - c_{i+1} \geq \hat{\mathcal{R}} \, [C_i - c_i ] - v.$$
By simple recurrence, for an initial time $ i_0 $ and all $ n \geq 1 $, this gives
\begin{eqnarray*}
C_{i_0 + n} - c_{i_0 + n}
&\geq& \hat{\mathcal{R}} \, [C_{i_0+n-1} - c_{i_0+n-1}] - v\\
&\geq& \hat{\mathcal{R}} \, [ \hat{\mathcal{R}} \, [C_{i_0+n-2} - c_{i_0+n-2}] - v] - v
\,\, = \,\, \hat{\mathcal{R}}^2 \, [C_{i_0+n-2} - c_{i_0+n-2}] - v\, [1 + \hat{\mathcal{R}} ] \\
&\vdots& \\
&\geq& \hat{\mathcal{R}}^n \, [C_{i_0} - c_{i_0}] - v \, [1+\hat{\mathcal{R}} + \hat{\mathcal{R}}^2 + \ldots + \hat{\mathcal{R}}^{n-1}]
\,\, =  \,\, \hat{\mathcal{R}}^n \, [C_{i_0} - c_{i_0}] - v \, \left[ \frac{\hat{R}^n - 1}{\hat{\mathcal{R}} - 1} \right]\\
&\geq& \hat{\mathcal{R}}^n \left[ C_{i_0} - c_{i_0} - \frac{v}{\hat{\mathcal{R}} - 1} \right].
\end{eqnarray*}
In other words,
$$C_{i_0 + n}  \geq  c_{i_0 + n} + \hat{\mathcal{R}}^n \left[ C_{i_0} - c_{i_0} - \frac{v}{\hat{\mathcal{R}} - 1} \right].$$
It is natural to expect that the term $ v / (\hat{R} -1) $ is (very)
small with respect to $ C_ {i_0} - c_ {i_0} $. Indeed, on the one hand, $ \hat{R} $ does not approximate indefinitely
to $ 1 $ (since the undetected infected individuals do not significantly reduce their social activity); on the other hand, our analysis deals
with times for which $ C_i $ is very (although, {\em a priori}, not exponentially) greater than $ c_i $. (A more robust argument
consists in choosing not only a single initial time $ i_0 $, but to implement the previous inequality along a sequence of
consecutive times and finally average along these inequalities.)

Assuming all of the above, the conclusion is clear: the number of infected people $ C_ {i_0 + n} $ is much higher than the number $ c_ {i_0 + n} $
of reported infected individuals. Indeed, the difference between the two is bounded from below by an exponential of ratio $ \hat{\mathcal {R}} $, while
that the evolution of the detected cases is governed by the rate $ \mathcal{R} $, which is strictly less than $ \hat {\mathcal{R}} $. So, the curve that
we see (that of the values $ c_{i + n} $) is not only very far from the real one (that of $ C_{i + n} $), but the difference between them has an
exponential growth that we are not perceiving.

\subsection{A more theoretical discussion}

\vspace{0.2cm}

The ``toy argument'' above shows something evident: if we are not able to follow the evolution of an epidemic through appropriate mass testing, then we
lose track of the infectious curve. In more sophisticated terms, what it reveals is that as long we are not aware that the reproduction rate $ \mathcal{R} $ is
{\em strictly smaller} than $ 1 $, the dynamical system of the epidemic moves in a regime of either slight exponential growth or, at least, of {\em instability}.
In such a regime, small variations of the initial conditions can lead to exponential explosion. Now, to a large extent, these initial conditions are provided by
official data. However, if these move around the maximum of what the system can detect, then we can hardly know how accurate they are and, therefore,
whether we really are in a stationary situation or whether we have advanced to an exponential explosion of cases that we are not perceiving. For this reason, for
each instance in which the number of positive detected cases becomes nearly constant (that is, when the curve of reported cases begins to acquire a plateau
shape), it seems reasonable to apply a substantive increase in the number of tests (both in quantity and spectrum). If this leads to a significant increase
in the number of positive cases, then most likely this would mean that, actually, the regime was not stationary, but was simply exceeding a threshold
above which a significant amount of infections cannot be reported. We will return to this point in the general conclusions of this work.

\section{About the model of Liu, Magal, Seydi and Webb}

\subsection{Presentation}

The key argument in the preceding section is that unreported patients have a greater dynamic role than those  that are reported (since the former do not
enter into quarantine), and therefore they contribute more importantly to the epidemic. However, in the traditional SIR model, both types of patients are
part of the same compartment. In \cite{1}, Liu, Magal, Seydi and Webb solve this problem by separating them into two compartments. Denoting respectively
by $ S, I, R$ and $U $ the susceptible individuals, infected individuals who do not yet have symptoms (and are at incubation stage), reported infected
individuals,  and unreported (either asymptomatic or low symptomatic) infected individuals, they consider the following diagram flux:
\begin{figure}[H]
\centering
    \includegraphics[width=0.9\textwidth]{Untitled2.png}
    \caption{Diagram flux}
    \label{fig:mesh1}
\end{figure}

The differential equations attached to the diagram above and that govern the dynamics of the epidemic are the following:

\begin{equation}\label{modelo}
 \arraycolsep=1.5pt\def\arraystretch{1.5}
\left\{\begin{array}{l}S^{\prime}(t)=-\tau S(t)[I(t)+U(t)] \\ I^{\prime}(t)=\tau S(t)[I(t)+U(t)]-\nu I(t) \\ R^{\prime}(t)=\nu_{1} I(t)-\eta R(t) \\ U^{\prime}(t)=\nu_{2} I(t)-\eta U(t),\end{array}\right.
\end{equation}
where $ t \geq t_0 $ corresponds to time, with $ t_0 $ being the starting date for the study (as in \cite{1,2,3},
in the implementation, we will consider the time $ t_0 $ corresponding to the beginning of the epidemic). Although
this system of differential equations makes perfect sense when prescribing any initial condition, in
epidemiological modeling one is naturally lead to use data of the following type:
$$
S(t_0)=S_0>0,\quad I(t_0)=I_0>0,\quad R(t_0) \geq 0\quad \mbox{and} \quad U(t_0)=U_0\geq 0.
$$
The parameters used in the model are described in the Table below.
In particular, note that $ \nu = \nu_1 + \nu_2 $. In addition, all the parameters
that are considered $ \tau, \nu, \nu_1, \nu_2, \eta $ are positive.
\begin{table}[H]
\begin{center}
 \arraycolsep=1.4pt\def\arraystretch{1.2}
\begin{tabular}{ p{2.3cm} p{11.8cm}  }
\hline \hline
Symbol & Interpretation\\
\hline \hline
$t_0$ & Initial time.  \\
$S_0$ & Number of individuals susceptible to the disease at time $ t_0 $.  \\
$I_0$ & Number of infected individuals (in incubation period)
without symptoms at time $ t_0 $.  \\
$R_0$ & Number of reported infected individuals at time $ t_0 $. \\
$U_0$ & Number of unreported infected individuals at time $ t_0 $. \\
$\tau$ & Transmission rate of the disease.  \\
$1/\nu$ & Average time during which the infectious asymptomatic individuals remain in incubation.\\
$f$ & Fraction of asymptomatic infected individuals that become reported infected. \\
$\nu_{1}=f \nu$ & Rate at which asymptomatic infected cases become reported.  \\
$\nu_{2}=(1-f) \nu$ & Rate at which asymptomatic infected become unreported infected individuals (asymptomatic or mildly symptomatic). \\
$1/\eta$ & Average time during which an infected individual presents symptoms. \\
\hline \hline
\end{tabular}
\caption{Parameters and initial conditions of the model.}
\label{tabla:sencilla0}
\end{center}
\end{table}
\vspace{-0.5cm}

Note that in the first of the equations of (\ref{modelo}), namely
$$S' (t) = - \tau S(t) \, [I(t) + U(t)],$$
the role of $ I $ and $ U $ in the spread of the infection is the same. This is the essential point of the model: it
gives the same dynamic role to those who are infected and do not have symptoms as to those who are not reported,
because the latter do not go into quarantine and, in fact, have a similar social activity. Certainly, the model can be refined in many
directions; for example, one could give different though still dynamical roles to $ I $ and $ U $ by attaching to them different
positive parameters $ \tau_1 \neq \tau_2 $ (this could be justified by that $ U $ includes individuals with low
symptomaticity who can practice self-care). However, it is already worth to visualize some of the main
properties of this model and to implement it in specific situations following this general  format.

\subsection{Some basic qualitative properties}

We next consider the system of equations (\ref{modelo}) in more detail.
We will assume an epidemic situation, which is summarized by the condition
\begin{equation}\label{epidemia}
\tau S_0 - \nu > 0.
\end{equation}
We will also consider the initial conditions of Liu, Magal, Seydi and Webb:
\begin{equation}\label{inicio}
S_0 > 0, \,\, I_0 > 0, \,\, R_0 = 0 \,\, \mbox{ and } \,\, U_0 = \frac{\nu_2 \, I_0}{\eta + \chi_2}
\end{equation}
for a certain $ \chi_2> 0 $. (The precise value of the parameter $ \chi_2 $ will be given in the next section;
here we just retain the fact that it is positive.) For simplicity, our starting time will be  $ t_0: = 0 $.

\begin{thm}
{\em In an epidemic situation} (\ref{epidemia}) {\em and starting with the initial conditions} (\ref{inicio}),
{\em the following three properties are fulfilled:}
\begin{enumerate}
\item[(i)] {\em For all time $t>0$, the values of  $ S (t) $, $ U (t), I (t) $ and $ R (t) $ exist, are positive and strictly
smaller than $ P: = S_0 + I_0 + U_0 $ (the total population);}
\item[(ii)]  {\em $ S (t) $ converges to a certain positive limit value $ S_{\infty} $ as $ t \to \infty $,
while $ I (t), R (t), U (t)$ converge to $ 0 $;}
\item[(iii)] {\em $ R (t) / U (t) $ converges to $ \nu_1 / \nu_2 $ from below.}
\end{enumerate}
\end{thm}

\begin{proof} We first note that, due to (\ref{epidemia}) and (\ref{inicio}), we have
$$S' (0) = -\tau S_0 I_0 < 0,$$
$$I' (0) = (\tau S_0 - \nu) I_0 + \tau S_0 U_0 > 0,$$
$$R' (0) = \nu_1 I_0 > 0,$$
$$U' (0) = \nu_2 I_0 - \eta U_0 = \nu_2 I_0 \left( 1 - \frac{\eta}{\eta + \chi_2} \right) > 0.$$
Therefore, there exists $ \varepsilon> 0 $ such that $ S (t), I (t), R (t) $ and $ U (t) $ (are defined in $ [0, \varepsilon) $ and) are
strictly positive. The arguments that follow are inspired by an observation contained in the classical book of Vladimir Arnold \cite{arnold}.

\noindent (i) Suppose $ S $ vanishes, and let $ T > 0$ be the first time this occurs. Let $C> 0$ be such that $I(t) + U(t) \leq C$
for all $t \in [0,T]$.  Then $S' (t) \geq - \tau C S(t)$, and hence, for $t \in [0,T)$, we have
$$\frac{S' (t)}{S(t)} \geq -\tau C .$$
Integrating between  $0$ and $s < T$, this gives
$$\log (S(s)) - \log (S_0) \geq -\tau C s,$$
and then,
$$S(s) \geq S_0 \, e^{-\tau C s}.$$
Letting $ s $ go to $ T $, this contradicts the assumption $ S (T) = 0 $.

Suppose now that $ U $ vanishes, and let $ T > 0$ be the first time this occurs. If $ I $ has not vanished until this
time, then $U' (t) = \nu_2 I(t) - \eta U(t) \geq - \eta U(t)$ for all $t \in [0,T]$. By integration, this gives
$$U(T) \geq U_0 \, e^{-\eta T},$$
which contradicts our assumption.  Hence, $I$ should have vanished in $ [0, T] $.

The same argument above shows that if $ R $ vanishes at a time $ T > 0$, then $ I $ must have vanished at some time in $ [0, T] $.

Finally, suppose that $ I $ vanishes, and let $ T $ be the first moment this occurs. Then, $U(t) \geq 0$
for all  $t \in [0,T]$, and therefore
$$I' (t) = \tau S(t) \, [I(t)+U(t)] - \nu I(t) \geq -\nu I(t).$$
However, by integration, this again gives  a contradiction, namely $ I (T) \geq I_0 e^{- \nu T} $.

We next show that neither $ I $ nor $ U $ explode (that is, none of them tends to infinity along an increasing sequence
of times tending to a finite time $ T $). Indeed, if anyone does it in time $ T $ then, from the above,
$ P \geq S (t) \geq 0 $ on $ [0, T) $, and $ U, I $ are positive on this interval. Therefore, on $ [0, T) $,
$$(I+U)'  = \tau P \, [I+U] - \nu_1 I - \eta U \leq \tau P  \, [I + U],$$
which implies by integration that $ (I + U) (t) \leq (I_0 + U_0) \, e^{\tau P t} $ for $ t \in [0, T) $.
Letting $ t $ go to $ T $, this contradicts the explosion.

To see that $ R $ does not explode, we proceed again by contradiction: if this occurs at time $ T $, then from
$ R = \nu_1 I - \eta R \leq \nu_1 I $ we deduce $ R (t) \leq R_0 \, e^{\nu_1 C} $ for all $ t \in [0, T) $ and $C:= \max_{t \in [0,T]} I(t)$.

Finally, $ S $ does not explode because it is decreasing and positive.

In conclusion, $ S, I, R, U $ are all positive and do not explode. To prove that they are bounded from above by $ P $,
we introduce the equation of the {\em deletted} (removed) individuals from the system:
$$D' (t) = \eta \, [R (t) + U (t)], \qquad D(0) = 0.$$
We have a constant population $ P = S (t) + I (t) + R (t) + U (t) + D (t) $, and since $ D '> 0 $, we have that $ D ( t)> 0 $ for all $ t> 0 $.
Now, since $ S, I, R, U $ are positive for $ t> 0 $, we conclude that each of them must be strictly smaller than $ P $.

\noindent (ii) First we show that $ S $ converges towards a positive limit. Let $ S _ {\infty} $ be the limit of $ S $ (which
exists because $ S $ is decreasing). Denoting $c := \min \{\eta, \nu\}$, we have
$$(I+U)' = \tau S \, [I+U] - \nu_1 I - \eta U \leq (\tau S - c) \, (I+U).$$
Since $S' = - \tau S \, [I+U]$, this implies
$$(I+U)' \leq \frac{(\tau S - c) \, S'}{- \tau S} = - S' + \frac{c}{\tau} \frac{S'}{S}.$$
Integrating between $0$ and $t > 0$, this gives
$$(I+U+S)(t) - (I_0+U_0+S_0) \leq \frac{c}{\tau} \log \left( \frac{S(t)}{S_0} \right).$$
Since $I_0+U_0+S_0 = P$,  this implies
$$-P \leq \frac{c}{\tau} \log \left( \frac{S(t)}{S_0} \right).$$
Thus, for all $t > 0$, we have \,
$S(t) \geq S_0 e^{-\frac{\tau}{c} P},$ \,
and therefore, \, $S_{\infty} \geq S_0 e^{-\frac{\tau}{c} P} > 0.$

Next, we simultaneously prove that $ I $ and $ U $ converge to $ 0 $ (the convergence of $ R $ to $0$ will be then a
consequence of the convergence of $ R / U $ to $ \nu_1 / \nu_2 $ proved in (iii) below). To do this, we first note that, since
$$S' = - \tau S \, [I+U],$$
it follows that there is a sequence of times $ t_n \to \infty $ such that $ (I + U) (t_n) \to 0 $. Otherwise, there would
exist  $ c> 0 $ such that $ (I + U) (t) \geq c $ for all $ t > 0$, which implies  $ S '/ S \leq - \tau c $,  and therefore
$S(t) \leq S_0 e^{-c \tau t}$. However, for $ t $ large enough, this contradicts the inequality $S(t) \geq S_{\infty} > 0$.

Now, since $\varepsilon > 0$, we may fix $T$ such that  $(I+U) (T) \leq \varepsilon / 2$ and
$S(t) \leq S_{\infty} + \varepsilon/2$ for all $t \geq T$. We claim that $(I+U) (t) \leq \varepsilon$ for all $t \geq T$.
(Since $\varepsilon > 0$ was arbitrarily chosen, this concludes the proof of the convergence of $ I + U $ towards $ 0 $.)
To prove this, note that, since $(S+I+U)' (t) = -\nu_1 I - \eta U < 0$,  we have $(S+I+U)(t) \leq (S+I+U)(T)$ for all $t \geq T$, hence
$$(I+U)(t) \leq (I+U)(T) + [S(T)-S(t)] \leq \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon,$$
as we wanted to show.

\noindent (iii) To prove that $R/U$ converges to $\nu_1 / \nu_2$, we first remark that
\begin{equation}\label{derivada}
\left( \frac{R}{U} \right)' = \frac{R' U - R U'}{U^2} = \frac{(\nu_1 I - \eta R ) \, U - R \, (\nu_2 I - \eta U)}{U^2}
= \nu_2 \, \frac{I}{U} \left( \frac{\nu_1}{\nu_2} - \frac{R}{U} \right).
\end{equation}
Therefore,
$$\frac{R}{U} > \frac{\nu_1}{\nu_2} \Longrightarrow \left( \frac{R}{U} \right)' (t) < 0,$$
$$\frac{R}{U} < \frac{\nu_1}{\nu_2} \Longrightarrow \left( \frac{R}{U} \right)' (t) > 0.$$
In other words, if $ R / U $ is smaller (resp. greater) than $ \nu_1 / \nu_2 $ at a point
$ t $, then it is increasing (resp. decreasing) around this point.

We first prove that $ R / U $ cannot be equal to $ \nu_1 / \nu_2 $ at any point. To do this,
we note that $ R_0 / U_0 = 0 \neq \nu_1 / \nu_2 $. We define $\varphi := \frac{\nu_1}{\nu_2} - \frac{R}{U}$.
Equality (\ref{derivada}) then becomes
$$\varphi' = - \nu_2 \frac{I}{U} \varphi.$$
If $ T $ were the first instant at which $ R / U = \nu_1 /  \nu_2 $, then $ T $ would be  the first zero of $ \varphi $.
By (i), there exists $C > 0$ such that $\nu_2 I / U \leq C$ on $[0,T]$. Since $R_0 = 0$,
one has $\varphi (t) > 0$ for all positive but small-enough $t$. Choosing such a $t$ smaller than $T$ we obtain, on $[t,T),$
$$\frac{\varphi'}{\varphi} \geq - C.$$
By integration, this gives $ \varphi (T) \geq \varphi (t) \, e^{t-T} $, which contradicts the fact that $ \varphi (T) = 0 $.

To prove the convergence of $ R / U $ towards $ \nu_1 / \nu_2 $, which is equivalent to that of $ \varphi $ towards $ 0 $,
we will use the fact (proven below) that $ I / U $ is bounded from below by a positive constant $ c$. Assuming this, we have
$$\frac{\varphi'}{\varphi} \leq -\nu_2 \, c,$$
and hence,
$$\varphi (T) \leq \varphi (t) \, e^{(t-T) \nu_2 c},$$
which converges to $ 0 $ as $T \to \infty$.

To conclude, we must show that $ I / U $ does not approach zero. For this, we begin by noting that
$$\left( \frac{I}{U} \right)'
= \frac{I'U - IU'}{U^2}
= \frac{(\tau S \, [I+U] - \nu I ) \, U -  I \, (\nu_2 I - \eta U)}{U^2}
= \tau S - \nu_2 \left( \frac{I}{U} \right)^2 + \frac{I}{U} (\tau S + \eta - \nu).$$
Since $ S \geq S _ {\infty} $, if $ I / U $ is very small, then the derivative $ (I / U) '$ becomes positive,
and therefore $ I / U $ grows. As a consequence, $ I / U $ cannot arbitrarily approximate $ 0 $.
\end{proof}

There are several remarks to the proof above.

\begin{rem}\label{first-r}
The proof above applies to more general initial conditions
than (\ref{inicio}): it only requires that the values $ S_0> 0, I _0> 0, R_0 \geq 0 $ and $ U_0 \geq 0 $
are such that $ I '(t_0) $, $ R' (t_0) $ and $ U '(t_0) $ are all positive. However, note that,
at this level of generality, in statement (iii) above the convergence of $ R / U $ towards $ \nu_1 / \nu_2 $ can occur
from above, and even the quotient $ R / U $ can remain constant and equal to $ \nu_1 / \nu_2 $ throughout the whole evolution.
\end{rem}

\begin{rem}
In the classical SIR model, the fact that the population
of susceptibles converges towards a positive limit is often presented as a consequence of the so-called
{\em final size relation} \cite{libro}. In our context, $ S'$ not only depends on $ I $, but also on $ U $. For this reason, it is
hard to expect such a simple relation, and this partly justifies the use of robust estimates in the preceding proof.
\end{rem}

\begin{rem}
The convergence of $ R $ and $ U $ towards $0$ must hold at a speed lower than $ e^{- \eta t} $. Indeed, from
the last two equations of the system one obtains
$$\frac{R' + \eta R}{\nu_1} = I = \frac{U' + \eta U}{\nu_2},$$
which can be rewritten in the form $\nu_2 ( R e^{\eta t})' = \nu_1 (U e^{\eta t})'$. By integration, this gives
$$\nu_2 R e^{\eta t} = \nu_1 U e^{\eta t} + C, \qquad \mbox{where} \quad C := - \nu_1 U_0.$$
Therefore,
$$\frac{R}{U} = \frac{ \nu_1 }{ \nu_2 } + \frac{C}{\nu_2 \, U e^{\eta t}}.$$
Since $ R / U $ converges to $ \nu_1 / \nu_2 $, the product $ U e^{\eta t} $ must tend to infinity,
thus showing our claim for $ U $. The claim for $ R $ immediately follows from this.
\end{rem}

\begin{rem}
Another observation is related to the proportion $ R / U $. Since this number grows throughout the epidemic,
the dynamic variability that we mentioned at the beginning of this work holds. Moreover, the convergence of $ R / U $
towards $ \nu_1 / \nu_2 $ tells us what should be the values of the parameters in the implementation.
We will return to this point for a specific case in the next section.
\end{rem}

\begin{rem}
Finally, we must mention that one of the basic results on the SIR model has not been incorporated into the theorem
above, namely, that of the uniqueness of the peak of the curve of infected cases. In fact, this uniqueness is no longer valid
for the SIRU model. Below we draw an example of a double peak for the reported and unreported cases curves. Note that though this
occurs under initial conditions different from (\ref{inicio}), it holds in a context in which the theorem is still valid, according to Remark \ref{first-r}.
\begin{figure}[H]
\centering
    \includegraphics[width=0.4\textwidth]{doble.png}
    \caption{A double peak for the curves.}
    \label{fig:mesh2}
\end{figure}
Examples of this type can be easily obtained. For the curve $ R $, one starts by imposing the conditions
$ R '(t_p) = 0 $ and $ R''(t_p)> 0 $  for a time $ t_p > 0 $ different from the starting time. These conditions
can be translated into conditions only on the values of $ S, I, R $ and $ U $  at that time.
Then the SIRU equations are implemented in both directions of time around $ t_p $. This
moment will hence correspond to a local minimum of the curve, necessarily
located between two peaks.

The same argument allows to build examples in which the curve of infected cases has two peaks. Naturally, this is due to the presence
of $ U $ in the derivative of $ I $, which is an absent element of the SIR model. In fact, in this one, the condition $ I' = 0 $ necessarily
implies $ I '' <0 $, as a straightforward computation shows. This occurs only at the peak of the curve, which corresponds to the
time when herd immunity is achieved.
\end{rem}

\section{Observations based on official numbers}

\subsection{General implementation of the SIRU model}

Our goal now is to compare the data available on COVID-19 in Chile until May 14 (2020) \cite{8} with what is
predicted by the SIRU model, so that we can estimate the evolution of the number of unreported cases
throughout the period. We point out that a first estimate of unreported cases in (some regions of) Chile using the
data of March 2020 and the SIRU model was carried out by M\'onica Candia and Gast\'on Vergara-Hermosilla in \cite{7}.

To begin with, let us point out that in outbreaks of influenza disease, the parameters $ \tau, \ \nu, \ \nu_1, \ \nu_2, \ \eta $, as well as the
initial values $ S_0, I_0 $ and $ U_0 $, are generally unknown. However, it is possible to identify them from specific time data of
reported symptomatic cases. The {\em cumulative number} of infectious cases reported at a time $ t $, denoted by
$ CR (t) $, is given by
$$CR(t) := \nu_1 \int_{t_0}^t I(s)ds.$$
This is data that is openly available. Likewise, the cumulative  number of unreported cases at a time $ t $ is
$$CU(t) := \nu_2 \int_{t_0}^t I(s)ds.$$
We note that, up to constants, these values coincide respectively with
$$R(t) + \eta \int_{t_0}^t R(s) \, ds \qquad \mbox{ and } \qquad U(t) + \eta \int_{t_0}^t U(s) \, ds.$$

Following the scheme used by Liu, Magal, Seydi and Webb, we will assume that, at an early stage of the disease,
$ CR (t) $ has an (almost) exponential form:
$$
CR(t) =\chi_ 1\exp(\chi_ 2 t)-\chi_ 3.
$$
For simplicity,
we will assume that $ \chi_3 = 1 $. The values of $ \chi_ 1 $ and $ \chi_2$
will then be adjusted to the accumulated data of cases reported in the early phase of the epidemic\footnote{Specifically,
to adjust $\chi_1,\ \chi_2$ we consider the next 28 days since the detection of the first case in Chile (Talca).} using a
classical least squares method (after passing to logarithmic coordinates). According to the above (see \cite{1} for details),
for numerical simulations, the initial time for the beginning of the exponential growth phase is fitted at
$$
t_{0} := -\frac{1}{\chi_ {2}}\cdot\log \left(\chi_ {1}\right).
$$

Again, for simplicity, we will identify the initial value $ S_0 $ to that of the total population of Chile (since there is no prior immunity against
the virus). Once the values of $ \nu $, $ \eta $ and $ f $ are set, the conditions at the beginning of the disease are naturally fitted as
$$
I_{0}:=\frac{\chi_ {2}}{f\left(\nu_{1}+\nu_{2}\right)} = \frac{\chi_2}{\nu_1},
\quad U_{0}:=\left(\frac{(1-f)\left(\nu_{1}+\nu_{2}\right)}{\eta+\chi_{2}}\right) I_{0} = \frac{\nu_2 \, I_0}{\eta + \chi_2},
\quad R_{0}:=0.
$$

We recall that $ 1 / \nu $ corresponds to the average time during which patients are asymptomatic infectious. As in \cite{1,2,3}, we will let this
parameter be equal to 7 (days), and the same value will be used for the average time in which patients are reported or unreported infectious:
$$\tau = \eta = 1/7.$$
Note that although the incubation period has been reported as being slightly smaller, the lag in the delivery of the results of the PCR  tests
in Chile justifies this choice.

\subsection{On the fraction of unreported cases}

To implement the SIRU system we still need to establish a good value for the parameter $ f $ (the fraction of symptomatic cases that are
reported). Once this is fitted, we will have the values of
$$
\nu_1=f\nu\quad \mbox{ y }\quad \nu_2=(1-f)\nu,
$$
and we will just need to fit the value of $ \tau $.

Before continuing, it is worth pointing out that in \cite{1,2,3} there is no major discussion on the  criterion used to establish
the value of $ f $ in the different scenarios. Actually, a value issued by the medical counterpart is assumed as valid. (For example,
$ f =  0.8 $ is considered for China.) In our modeling, we will use the work of Baeza-Yates \cite{6} and that of Castillo and
Past\'en \cite{CP}, who use the case fatality rate of the disease (with the correct correction according to its duration; see
\cite{avner, letal}) to give estimates for the right number of infected individuals.\footnote{They estimate this number
between $ 60 \% $ and $ 70 \% $ higher than the one reported for the period studied.} Since Baeza-Yates' argument is
simpler and is not included in an academic publication, we borrow it below in a language closer to that of the SIRU
model. As we will see, it yields a method to adjusting the value of $ f $ that can be used in almost all contexts.

Since we know that $ R / U $ converges towards $ \nu_1 / \nu_2 $, we  assume for this calculation that $ R / U $
is simply equal to $ \nu_1 / \nu_2 $. In addition, we will argue in discrete units of time (in days). Then we have
$$R(n) = f \, [R(n) + U(n)]$$
If $ d $ is the average time of illness to death and $ M (n) $
the number of deaths in day $n$, then the case fatality rate $ L $ corresponds to
$$L = \frac{M(n)}{[R (n-d) + U(n-d)]}.$$
Moreover, the reported case fatality rate is
$$L_R = \frac{M(n)}{R(n-d)} = \frac{M(n)}{[R(n-d)+U(n-d)]} \cdot \frac{[R(n-d)+U(n-d)]}{R(n-d)} = \frac{L}{f}.$$
This gives
$$f = \frac{L}{L_R}.$$
In the Chilean context, deaths in the period studied occurred within $9.4$ days after the disease was reported.
Adjusting $ d = 9$, the computation of $ L_R $ is made feasible from the data available in \cite{8}.

Finally, regarding the natural case fatality ratio $ L $ of the disease, it is deduced from international studies that, after
adjusting it to the age distribution of the Chilean population, it should vary between $ 0.2 \% $ and $ 1 \% $, with
a very high tendency to be around $ 0.6 \% $. In summary, this gives a parameter $ f $ varying between $ 0.1 $
and $ 0.5 $, with a high tendency to be close to $\frac{1}{3.4} \sim 0.3$.

\subsection{Variations in the transmission rate}

Given the heterogeneity of the safeguard measures taken by the Chilean government, instead of directly applying the SIRU model, it
became more pertinent to us to consider a variable transmission rate $ \tau $ (as a function of time). To analyze the variation of the values
of $ \tau (t) $, we observe how the percentage of the Chilean population subjected to confinement has been changing, which is illustrated below:
\begin{figure}[H]
\centering
    \includegraphics[width=0.65\textwidth]{porc_cuar2.png}
    \caption{Variation of the percentages of the population in quarantine in Chile corresponding to the first 49 days from March 13, 2020.}
    \label{fig:mesh3}
\end{figure}
We hence propose a function $ \tau (t) $ of the form
$$
\tau(t)=\left\lbrace
\begin{array}{ll}
\tau_0 & \mbox{ if } t\in \mathcal{I}_1 = [N_0,N_1],\\
\tau_1(t)=\tau_0 \exp(-\mu_1(t-N_1)) & \mbox{ if } t\in  \mathcal{I}_2 = (N_1,N_2],\\
\vdots& \,\, \vdots\\
\tau_r(t)=\tau_{r-1}(t) \exp(-\mu_{r-1}(t-N_{r-1})) & \mbox{ if } t\in  \mathcal{I}_{r} = (N_{r-1}, N_r],
\end{array}
\right.
$$
where the $ \mathcal{I} _i $'s correspond to successive time intervals. For the Chilean context, according to the graph of quarantines
illustrated above (Figure 3), the chosen time intervals are those described in Table \ref{tabla:sencilla1} below:
\vspace{0.5cm}
\begin{table}[H]
\begin{center}
 \arraycolsep=1.4pt\def\arraystretch{1.1}
\begin{tabular}{ p{2.5cm} p{6cm} p{2.2cm}  }
\hline \hline
Interval   & Time frame & $N_i$\\
 \hline \hline
$\mathcal{I}_{1}$  & from March 3 to 22 & March 22 \\
$\mathcal{I}_{2}$  & from March 23 to April 1 & April 1  \\
$\mathcal{I}_3$  & from April 2 to 11 & April 11  \\
$ \mathcal{I}_{4}$  & from April  12 to 21  &  April 21  \\
$ \mathcal{I}_{5}$  & from April 22 to 31 & April 31  \\
$\mathcal{I}_6$  & from May 1 to 10  & May 10    \\
$\mathcal{I}_7$  & from May 11 to 14  & May 14  \\
\hline \hline
\end{tabular}
\caption{Time intervals used in our numerical simulations.}
\label{tabla:sencilla1}
\end{center}
\end{table}
\vspace{0.14cm}
Following \cite {1,2,3}, the value of the transmission rate in $ \mathcal {I} _1 $ is fitted to
$$
\tau_0 := \left( \frac{\chi_ {2}+\nu}{S_{0}}\right) \left(  \frac{\eta+\chi_ {2}}{\nu_{2}+\eta+\chi_ {2}}\right).
$$
Then, the parameters $ \mu_i $ are chosen in such a way that the reported cumulated cases in the numerical
simulation align with the data of the reported cumulative number of infections at time $ t $ according to \cite{8}.
This is implemented with various values of $ f $, following the guidelines of the preceding subsection (specifically,
we work with $ f =  0.2$, $ f = 0.3$ and $ f =  0.4$). A summary is described in the following Table:
\begin{table}[H]
\begin{center}
 \arraycolsep=1.4pt\def\arraystretch{1.1}
\begin{tabular}{ p{2.5cm} p{2cm}  p{2cm}   p{2cm}   }
\hline \hline
 Parameter   &$  f=0.2$ &$  f=0.3$ & $ f=0.4$  \\
 \hline \hline
$ \mu_{1}$  & 8.3$\cdot 10^{-4}$ & 8.3$\cdot 10^{-4}$  &8.3$\cdot 10^{-4}$  \\
$ \mu_{2}$  & 2.1$\cdot 10^{-6}$ & 2.1$\cdot 10^{-6} $& 2.1$\cdot 10^{-6}$  \\
$\mu_3$  & 5$\cdot 10^{-6}$ & 5$\cdot 10^{-6}$ & 5$\cdot 10^{-6}$   \\
$ \mu_{4}$  & 1.41$\cdot 10^{-6}$ &1.41$\cdot 10^{-6}$ & 1.41$\cdot 10^{-6}$  \\
$ \mu_{5}$  & 5.08$\cdot 10^{-5}$ & 5.08$\cdot 10^{-5}$ & 5.08$\cdot 10^{-5}$  \\
$\mu_6$  & 2.96$\cdot 10^{-4}$ & 2.96$\cdot 10^{-4}$ & 2.96$\cdot 10^{-4}$  \\
\hline \hline
\end{tabular}
\caption{Parameters $ \mu_i $ corresponding to the different values of $ f $ considered in our numerical simulations.}
\label{tabla:sencilla2}
\end{center}
\end{table}
\vspace{-0.5cm}

Using the data of the cumulated reported cases available in \cite{8}, we can finally proceed to the simulations, which
are shown below. They illustrate numerical estimates for the curves of $ CR (t), \ CU (t), \ R (t) $ and $ U (t) $.

\begin{figure}[H]
\centering
    \includegraphics[width=0.71\textwidth]{0,2.png}
    \centering
    \includegraphics[width=0.71\textwidth]{0,3.png}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=0.71\textwidth]{0,4.png}
\vspace{0.8cm}
\caption{Plots of the numerical approximations of the functions $ CR (t), \ CU (t), \ R (t) $ and $ U (t) $ obtained from the numerical
solutions of the model (\ref{modelo}) applied to the Chilean context based on the data of reported cumulated cases up to 14
May 2020 \cite {8}. The plots (A), (B) and (C) were obtained by considering $ f = 0.2 $, $ f = 0.3 $ and $ f = 0.4 $ respectively,
using the parameters $\eta = 1/7$ and $\nu= 1/7$.}
    \label{fig:mesh4}
\end{figure}

\vspace{0.5cm}

\noindent{\Large{\bf Discussion and future work}}

\vspace{0.5cm}

The epidemic outbreak of the new human coronavirus COVID-19 was first detected in Wuhan, China, in late 2019.
In Chile, the first case was reported on March 3, 2020, in Talca (Maule region). Since then, modeling the epidemic
in the country has faced the problem of the low availability of disaggregated data \cite{baeza}.

The first part of this work was inspired by the situation experienced in Chile during the month of April, when the number of reported
cases stabilized around 450 per day. According to most specialists, this was not accurately representing the genuine epidemiological evolution.
Although it is difficult to argue that our reasoning from the first section fully applies to this situation (in particular, because the number of daily
tests during the period was variable), the explosive increase of cases that subsequently occurred should call us to reflection on the point.
Nevetheless, it is clear that this increase was also due in part to the relaxation of the protection measures, which is reflected not only
in the absolute (and exponential) increase in the number of cases, but also in the proportion of positive cases with respect to
tests (the latter despite of the significant increase in the number of tests \cite{art-ciper}).

The first part of the work naturally led us to model the dynamics of the epidemic incorporating a compartment for unreported cases so that we could deal
with their active role in the evolution. To do this, we used the SIRU model recently introduced/implemented by Liu, Magal, Seydi and Webb in \cite{1,2,3}.
Since the qualitative theory of the underlying differential equations has not yet been treated, we developed some essenttial points of it in the second
section, thus rigorously establishing fundamental structure theorems. However, we pointed out an important difference with the SIR model, namely,
the possibility of multiple peaks for the curves of infected, reported and unreported patients. For the future, it would be desirable to complete
the qualitative description of the solutions of the equations with respect to the parameters, with an emphasis on the phenomenon of multiplicity
of peaks. For example, until now we do not know whether there can be more than two peaks and/or whether there is any restriction on the relative
position between them. Without any doubt, this discussion is relevant for the implementation of disease containment policies, since it is directly
related to the recognition of the moment when the curves begin to definitively descend.

In the last section of the work, we implemented an extension of the model to the case of Chile. To do this, we considered
a variable transmission rate, which is more appropriate according to the local reality. Our rate is coupled
to the official statistics provided by the government in \cite{8}, information that also allowed us to make the
parametric afjustement. The last ingredient to launch the simulations was to fit the value of the fraction $f$ of infected
individuals that were reported during the period of study. For this we used the work of Baeza-Yates \cite{6} and
that of Castillo and Past\'en \cite{CP}. Naturally, our modeling is consistent with their estimates. In particular, in Section 2.3 of \cite{CP},
the authors estimate the number of real cases for April 28 \footnote{This corresponds to day 57 from the first case detected in Talca.}
as 43095, which is remarkably close to the number of total cases that can be inferred from the plot (B) in Figure 4 obtained
by considering $ f = 0.3 $. We point out that the prediction of Castillo and Past\'en is recovered
with complete accuracy through the method used in this work
for the value $ f = 0.31791 $. According to our modeling, on April 28, a percentage of $ 31.791 \% $ of the
infections was reported to health agencies, and consequently $ 68.209 \% $ of the infected cases was not reported.

Although the estimates above were obtained {\em a posteriori}, their complementarity puts us in a good position
to use them for modeling the future evolution of the epidemic. However, working in this direction requires great caution.
In particular, it would be necessary to consider a variable parameter $f$, since both the quantity and the criteria of the tests have been
modified during late May. Despite this, we strongly believe that the basis for pursuing the implementation of the SIRU model are fullfilled,
and it would be very useful to advance in a more compartmentalized and georeferenced implementation of it.

\vspace{0.2cm}

\noindent{\bf Acknowledgments.} We would like to thank Mar\'ia Paz Bertoglia, M\'onica Candia, Alicia Dickenstein, Yamileth Granizo,
Rafael Labarca,  Mario Ponce and Marius Tucsnak for their reading, their kind remarks and suggestions of bibliography.

\vspace{-0.3cm}

\begin{small}

config: arxiv


TITLE: Multiple Regression with Categorical Predictor Variables of More than Two Levels
QUESTION [0 upvotes]: I'm planning on running a hierarchical multiple regression in SPSS. In the first step, I would like to enter demographic characteristics, second step continuous predictor variables of interest, and third step interactions between the continuous predictor variables. However, the issue I've run into is that some of the demographic variables are more than two levels and either nominal (e.g., education level: less than BA, BA, graduate degree) or ordinal (e.g., age: 20-30, 31-40, 41-50, 51+). My dependent variable is continuous.
My question is: can I dummy code the nominal/ordinal variables and then run the multiple regression as I normally would? Or, would you recommend handling the data differently? I'm a bit rusty on SPSS, so I appreciate any help/input. Thanks!

REPLY [1 votes]: The most flexible thing you can do is assign binary variables for each individual category and include them all in your model. For example, let $Age_1 = 1$ if age was between 20-30 and $0$ otherwise. Let $Age_2 = 1$ if age was between 31-40 and $0$ otherwise..., etc. The cost of this approach is that it significantly increases the number of parameters you wish to estimate.
At the other end, you can simply code up the existing variables as a variables that ranges from, for example 1-4 where 1 corresponds to ages 21-30, 2 corresponds to ages 31-40 and so forth. This is a crude method, but it is a proxy of age (or education).
There is no right or wrong answer and these are far from the only 2 methods. Any combination would also work. If you have a large sample and computation time is not a concern I would go with method 1.