Action for $N$ D0-Branes Invariant Under Gauged Galilean Transformations

J. Klusoň, 111Email addresses<EMAIL_ADDRESS>(J.Klusoň),

Department of Theoretical Physics and Astrophysics, Faculty of Science,
Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

In this short note we formulate an action for $N$ D0-branes that is manifestly
invariant under gauged Galilean transformations. We also find its canonical
form and determine first class constraints that are generators of gauge
transformations.

##  1 Introduction and Summary

Relational mechanics is formulation of dynamics of particles that is closely
related to Mach’s idea that claims that dynamics of $N$ particles should be a
theory of relations about these quantities without any reference to external
non-material entities. The question is how to make these ideas more concrete.
One such a possibility was proposal that the Lagrangian should be invariant
under gauged Galilean group [1] 222See also [2, 3].. Such Lagrangian can be
found when its measure (kinetic energy term) is replaced with a measure that
is defined in the space of orbits, where orbits correspond to a set of
configurations which are equivalent under gauge transformations. Then it was
shown that the solutions of this gauge invariant dynamics correspond to the
solutions of the original Lagrangian with vanishing total momentum and angular
momentum. An alternative proposal how to construct relational mechanics was
presented in [4]. This construction starts with the original Lagrangian
invariant under rigid Galilean transformation. As the next step we add to it
specific counterterms that compensate changes in the kinetic energy term under
time dependent Galilean transformations. As a result Lagrangian invariant
under time dependent Galilean transformation was derived in [4] and it was
also shown that corresponding equations of motion are valid in any frame
making concrete implementation of the Mach’s principle.

In more details, Newton’s mechanics is invariant under time independent
translations, space translations and rotations of particle’s positions
$\mathbf{x}_{i}$

$\displaystyle t^{\prime}=t+\epsilon\ ,\quad\epsilon=\mathrm{const}\ ,$
$\displaystyle\mathbf{x}^{\prime}_{i}=\mathbf{x}_{i}+\xi\
,\quad\xi=\mathrm{const}\ ,$
$\displaystyle\mathbf{x}^{\prime}_{i}=\mathbf{A}\mathbf{x}_{i}\ ,$

where $\mathbf{A}$ is orthogonal matrix, $i=1,\dots,N$ where $i$ labels
particles in ansamble. Further, Newton’s laws are invariant under Galilean
transformations

$\mathbf{x}^{\prime}_{i}=\mathbf{x}_{i}+\mathbf{V}t\ ,$ (2)

which are special case of local time dependent translations when we identify
$\xi=\mathbf{V}t$. The transformations (1) and (2) represent Galilean group of
Newtonian mechanics. In fact, there is a privileged set of inertial frames and
clocks in Newtonian mechanics which are related each other through the
transformation of the Galilean group. Then we leave an idea of these
privileged frames when the Galilean group of transformation becomes gauge
group with time dependent parameters. We can also eliminate the absolute time
when we gauge time translation as $t^{\prime}=t+\epsilon(t)$. In such a
formulation of mechanics there are no privileged frames and clocks and it
becomes purely relational.

Detailed analysis of formulation of relational mechanics was performed in [4]
where systems of $N$ non-relativistic particles was studied. The Lagrangian
that is invariant under time dependent Galilean group was found there together
with corresponding Hamiltonian formulation. It was also shown there that
Relational Mechanics contain frames where Newtonian mechanics is valid and
these frames are determined by mass distribution of all particles which is an
essence of the Mach’s principle.

In this article we would like to follow the procedure used in [4] and in [5]
in case of one very interesting non-relativistic system which is low energy
Lagrangian for $N$ D0-branes in string theory [6, 7]. It is well known that
the low-energy Lagrangian for a system of many type IIA D0-branes is the
matrix quantum mechanics Lagrangian arising from the dimensional reduction to
$0+1$ dimensions of the $10D$ super Yang-Mills Lagrangian, for review see for
example [11, 12, 13]. An importance of this action is that it is the key point
in the formulation of Matrix theory [8] which is strictly speaking defined for
infinite number of D0-branes even if there is a version of Matrix theory that
is correct for finite $N$ as well [9, 10]. Now we show that it is possible to
formulate Lagrangian for $N$ D0-branes that is manifestly invariant under
gauged Galilean transformations and hence, according to the extended
discussion presented in [4] corresponds to relational mechanics for $N$
D0-branes. On the other hand it is not possible to write it in manifestly
relational form where the Lagrangian depends on relative velocities and
distances of particles in the general case due to the fact that fundamental
objects in matrix theory are matrices rather than coordinates of individual
D0-branes. On the other hand this can be easily done in the approximation when
off-diagonal components of matrices are small with respect to the diagonal
ones so that we can neglect them and we show that in this case the Lagrangian
takes purely relational form.

As the next step in our research we find canonical form of this theory and we
identify two sets of the generators of the first class constraints. We also
show that by appropriate fixing the gauge symmetry this theory reduces to the
ordinary finite Matrix theory.

We mean that this is nice and interesting result that should be developed
further. In fact, the natural and very important question is to include
fermionic terms in the constructions of relational mechanics for $N$
D0-branes. The second question is whether it is possible to generalize this
construction of relational mechanics to the case of the full non-linear
version of the action as it is represented by non-abelian Dirac-Born-Infeld
action for $N$ D0-branes [14, 15]. We hope to return to these problems in near
future.

The structure of this paper is as follows. In the next section (2) we find an
action for $N$ D0-branes that is invariant under gauged Galilean
transformations. Then in section (3) we determine its canonical form and
identify constraints structure of the theory. Finally in section (4) we show
how it is possible to make this theory manifestly reparametrization invariant.

##  2 Relational Formulation of Action for $N$ D0-Branes

In this section we find the form of low energy effective action for $N$
D0-branes that is invariant under gauged Galilean transformations. We start
with the Lagrangian for $N$ D0-branes that at the leading order corresponds to
$U(N)$ Super-Yang-Mills mechanical system that has the form

$S=\int dtL\ ,\quad
L=\frac{1}{2gl_{s}}\mathrm{Tr}\left[\dot{\Phi}^{I}\dot{\Phi}^{I}+\frac{1}{2}[\Phi^{I},\Phi^{J}][\Phi^{I},\Phi^{J}]+\mathrm{fermions}\right]\
,$ (3)

where $\Phi^{I}_{ij}$ are $N\times N$ Hermitean matrices where
$I,J=1,2,\dots,9$ and where $i,j,\dots=1,\dots,N$. Further, $g_{s}$ is string
coupling constant and $l_{s}$ is the string length. Note that the transverse
space is nine-dimensional Euclidean space and repeated indices mean summation
over them. Finally this Lagrangian contains terms with fermions that makes the
Lagrangian $N=16$ Super-Yang-Mills mechanics. In more details, the Lagrangian
(3) can be defined by dimensional reduction of $10$ dimensional
$\mathcal{N}=1$ super-Yang-Mills theory with gauge group $U(N)$ to the $0+1$
dimensions. In what follows we restrict ourselves to the bosonic terms only
leaving its extension to the fully supersymmetric invariant case in the near
future.

The action (3) is invariant under rigid translation

$\Phi^{\prime I}(t)=\Phi^{I}(t)+\xi^{I}\mathbf{I}_{N\times N}\ ,$ (4)

where $\xi^{I}$ is constant and where $\mathbf{I}_{N\times N}$ is unit
$N\times N$ matrix. Further, the Lagrangian is invariant under rigid rotation

$\Phi^{I}(t)=\Lambda^{I}_{\ J}\Phi^{J}(t)\ ,$ (5)

where $\Lambda^{I}_{\ J}$ obey the relations

$\Lambda^{I}_{\ K}\delta_{IJ}\Lambda^{J}_{\ L}=\delta_{KL}\ .$ (6)

Let us now construct Lagrangian that is invariant under time dependent
translation

$\Phi^{\prime I}(t)=\Phi^{I}(t)+\xi^{I}(t)\mathbf{I}_{N\times N}\ .$ (7)

Clearly the potential term is invariant under this transformation while the
kinetic term transforms as

$\displaystyle\delta\left(\frac{1}{2g_{s}l_{s}}\mathrm{Tr}[\dot{\Phi}^{I}\dot{\Phi}^{I}]\right)=\frac{1}{g_{s}l_{s}}\mathrm{Tr}\delta\dot{\Phi}^{I}\dot{\Phi}^{I}=\frac{1}{g_{s}l_{s}}\dot{\xi}^{I}\mathrm{Tr}\dot{\Phi}^{I}\
.$

In order to compensate this transformation we consider following variation

$\displaystyle\frac{1}{2Ng_{s}l_{s}}\delta(\mathrm{Tr}\dot{\Phi}^{I}\mathrm{Tr}\dot{\Phi}^{I})=\frac{1}{gl_{s}}\dot{\xi}^{I}\mathrm{Tr}\dot{\Phi}^{I}$

so that following combination

$\frac{1}{2g_{s}l_{s}}\mathrm{Tr}[\dot{\Phi}^{I}\dot{\Phi}_{I}]-\frac{1}{2Ng_{s}l_{s}}\mathrm{Tr}\dot{\Phi}^{I}\mathrm{Tr}\dot{\Phi}^{I}$
(10)

is invariant under the gauge symmetry (7). As the next step we proceed to the
analysis of the invariance of the Lagrangian under time dependent rotation.
Time Dependent Rotation

As we have argued above the action (3) is invariant also under rigid rotation
(5) where $\Lambda^{I}_{\ J}$ obey the relation (6). It is convenient to write
it in infinitesimal form when we define

$\Lambda^{I}_{\ J}=\delta^{I}_{\ I}+\omega^{I}_{\ J}\ ,\quad\omega^{I}_{\
J}\ll\delta^{I}_{J}\ .$ (11)

Then (6) implies

$\delta_{KJ}\omega^{J}_{\ L}+\delta_{LI}\omega^{I}_{\
K}=0\Rightarrow\omega_{KL}+\omega_{LK}=0\ .$ (12)

Let us now presume that $\omega^{I}_{\ J}$ depend on $t$ so that (11) gives

$\delta\Phi^{I}=\Phi^{\prime I}-\Phi^{I}=\omega^{I}_{\ J}\Phi^{J}\
,\quad\delta\dot{\Phi}^{I}\equiv\dot{\omega}^{I}_{\ J}\Phi^{J}+\omega^{I}_{\
J}\dot{\Phi}^{J}\ .$ (13)

Then the variation of the kinetic term is equal to

$\displaystyle\frac{1}{2g_{s}l_{s}}\delta\mathrm{Tr}(\dot{\Phi}^{I}\delta_{IJ}\dot{\Phi}^{J})=\frac{1}{g_{s}l_{s}}\dot{\omega}^{I}_{\
K}\mathrm{Tr}(\Phi^{K}\delta_{IJ}\dot{\Phi}^{J})+\frac{1}{g_{s}l_{s}}\mathrm{Tr}(\dot{\Phi}^{J}\omega_{JK}\dot{\Phi}^{K})$
$\displaystyle=\frac{1}{g_{s}l_{s}}\dot{\omega}^{I}_{\
K}\mathrm{Tr}(\Phi^{K}\delta_{IJ}\dot{\Phi}^{J})\ ,$

where the last term on the first line vanishes due to the anti-symmetry of
$\omega_{IJ}$. As the next step we should calculate the variation of new
compensating term and we find

$\displaystyle-\delta\left(\frac{1}{2Ng_{s}l_{s}}\mathrm{Tr}\dot{\Phi}^{I}\mathrm{Tr}\dot{\Phi}^{I}\right)=-\dot{\omega}^{I}_{\
K}\frac{1}{Ng_{s}l_{s}}\mathrm{Tr}\Phi^{K}\delta_{IJ}\mathrm{Tr}\dot{\Phi}^{J}-\mathrm{Tr}\frac{1}{Ng_{s}l_{s}}\omega_{KJ}\mathrm{Tr}\dot{\Phi}^{K}\mathrm{Tr}\dot{\Phi}^{J}=$
$\displaystyle=-\dot{\omega}^{I}_{\
K}\frac{1}{Ng_{s}l_{s}}\mathrm{Tr}\Phi^{K}\delta_{IJ}\mathrm{Tr}\dot{\Phi}^{J}\
,$

where again the last term on the first line vanishes due to the anti-symmetry
of $\omega_{IJ}$. In other words the variation of the kinetic term is equal to

$\displaystyle\delta\left(\frac{1}{2g_{s}l_{s}}\mathrm{Tr}(\dot{\Phi}^{I}\delta_{IJ}\dot{\Phi}^{J})-\frac{1}{2Ng_{s}l_{s}}\mathrm{Tr}\dot{\Phi}^{I}\mathrm{Tr}\dot{\Phi}^{I}\right)=\dot{\omega}_{JK}\mathbf{J}^{KJ}\
,$
$\displaystyle\mathbf{J}^{IJ}=\frac{1}{2g_{s}l_{s}}\mathrm{Tr}(\Phi^{I}\dot{\Phi}^{J}-\Phi^{J}\dot{\Phi}^{I})-\frac{1}{2g_{s}l_{s}N}(\mathrm{Tr}\Phi^{I}\mathrm{Tr}\dot{\Phi}^{J}-\mathrm{Tr}\Phi^{J}\mathrm{Tr}\dot{\Phi}^{I})=-\mathbf{J}^{JI}\
.$

Note that $\mathbf{J}^{IJ}$ transforms under time dependent rotation as

$\displaystyle\delta\mathbf{J}^{IJ}=\omega^{I}_{\
K}\mathbf{J}^{KJ}+\mathbf{J}^{IK}\omega_{K}^{\ J}-\dot{\omega}^{I}_{\
K}I^{KJ}-I^{IK}\dot{\omega}_{K}^{\ J}\ ,$

where

$I^{KJ}=\frac{1}{2g_{s}l_{s}}\mathrm{Tr}(\Phi^{K}\Phi^{J})-\frac{1}{2g_{s}l_{s}N}\mathrm{Tr}\Phi^{K}\mathrm{Tr}\Phi^{J}\
.$ (18)

Our goal is to add new additional term to the Lagrangian to make it invariant
under time dependent rotation. We propose that such term has a form

$-\frac{1}{2}\mathbf{J}^{IJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}\ ,$ (19)

where the matrix $\mathbf{M}_{IJ,KL}$ is matrix in $I,J,K,L$ indices while it
is scalar with respect to the $U(N)$ structure.

Now we proceed to the construction of the object $\mathbf{M}_{IJ,KL}$. First
of all we demand that it does not depend on time derivative of $\Phi$ so that
it transforms as ordinary tensor under time dependent rotation. Then under
time dependent rotation the new term (19) transforms as

$\displaystyle\delta(\frac{1}{2}\mathbf{J}^{IJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL})=-\dot{\omega}^{I}_{\
M}I^{MJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}-I^{IM}\dot{\omega}_{M}^{\
J}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}=$ $\displaystyle=-\dot{\omega}^{I}_{\
M}I^{MJ}(M_{IJ,KL}-M_{JI,KL})\mathbf{J}^{KL}=2\dot{\omega}^{I}_{\
M}I^{MJ}\mathbf{M}_{JI,KL}\mathbf{J}^{KL}\ ,$

where we presume that $\mathbf{M}_{IJ,KL}=-\mathbf{M}_{JI,KL}$ as a
consequence of the fact that $\mathbf{J}^{IJ}=-\mathbf{J}^{JI}$. In order to
cancel variation of the kinetic term we should require that

$2I^{MJ}\mathbf{M}_{JI,KL}=\delta_{IL}\delta_{K}^{M}\ .$ (21)

To proceed further we introduce $I^{-1}_{IJ}$ as matrix inverse to $I^{JK}$ so
that $I^{IJ}I^{-1}_{JK}=\delta^{I}_{K}$. Then if we multiply (21) with
$(I^{-1})_{RM}$ we get

$\mathbf{M}_{RI,KL}=\frac{1}{2}\delta_{IL}I^{-1}_{RK}\ .$ (22)

We see that it is natural to define matrix $\mathbf{M}_{IJ,KL}$ as

$\displaystyle\mathbf{M}_{IJ,KL}=\frac{1}{2}\delta_{JL}I^{-1}_{IK}\
,\quad\mathbf{M}_{JI,KL}=-\frac{1}{2}\delta_{JL}I^{-1}_{IK}\ ,$
$\displaystyle\mathbf{M}_{KL,IJ}=\frac{1}{2}\delta_{LJ}I^{-1}_{KI}\
,\quad\mathbf{M}_{IJ,LK}=-\frac{1}{2}\delta_{JL}I^{-1}_{IK}\ .$

In summary, we have found Lagrangian for $N$ D0-branes that is invariant under
local translation and rotation and that has the form

$L=\frac{1}{2gl_{s}}\mathrm{Tr}\left[\dot{\Phi}^{I}\dot{\Phi}_{I}+\frac{1}{2}[\Phi^{I},\Phi^{J}][\Phi^{I},\Phi^{J}]\right]-\frac{1}{2Ng_{s}l_{s}}\mathrm{Tr}\dot{\Phi}^{I}\dot{\Phi}^{I}-\frac{1}{2}\mathbf{J}^{IJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}\
.$ (24)

This is final form of the Lagrangian for $N$ D0-branes that is invariant under
time dependent Galilean transformation so that this Lagrangian is valid in any
frame. In fact, following [4] we can interpret is as relational formulation of
D0-brane mechanics. To see this more clearly let us consider situation when
the matrices $\Phi^{I}$ are diagonal, or say alternatively, situation when we
can neglect all off diagonal terms with respect to diagonal ones. Then the
matrices $\Phi^{I}$ have the form

$\Phi^{I}_{ij}=x^{I}_{i}\delta_{ij}\ ,$ (25)

where $x^{I}_{i}$ are coordinates of individual $i-$th D0-brane. With such a
configuration we find that the potential term vanishes while the kinetic term
has the form

$\frac{1}{2g_{s}l_{s}}(\sum_{i}v_{i}^{I}v_{i}^{I}-\frac{1}{N}\sum_{i}v_{i}^{I}\sum_{j}v_{j}^{I})\
,\quad v^{I}_{i}=\frac{dx^{I}_{i}}{dt}$ (26)

that can be written in an alternative form

$\frac{1}{4Ng_{s}l_{s}}\sum_{i,j}(v_{i}^{I}-v_{j}^{I})(v_{i}^{I}-v_{j}^{I})$
(27)

which nicely demonstrate the relational form of this Lagrangian. Further,
matrix $I^{IJ}$ has the form

$I^{IJ}=\frac{1}{2g_{s}l_{s}}(\sum_{i}x^{I}_{i}x^{J}_{i}-\sum_{i}x^{I}_{i}\sum_{j}x^{J}_{j})=\frac{1}{4g_{s}l_{s}N}\sum_{i,j}(x_{i}^{I}-x_{j}^{I})(x^{J}_{i}-x^{J}_{j})\
.$ (28)

In the same way we proceed with $\mathbf{J}^{IJ}$ and we get

$\displaystyle\mathbf{J}^{IJ}=\frac{1}{2g_{s}l_{s}}\sum_{i}(x^{I}_{i}v^{J}_{i}-x^{J}_{i}v^{I}_{i})-\frac{1}{2g_{s}l_{s}N}(\sum_{i}x^{I}_{i}\sum_{j}v^{J}-\sum_{i}x^{J}_{i}\sum_{j}v^{I}_{j})=$
$\displaystyle=\frac{1}{4g_{s}l_{s}N}\sum_{i}\sum_{j}\left((x^{I}_{i}-x^{I}_{j})(v^{J}_{i}-v^{J}_{j})-(x^{J}_{i}-x^{J}_{j})(v^{I}_{i}-v^{I}_{j})\right)\
$

which again depends on relative distances and velocities. In summary we obtain
Lagrangian

$\displaystyle
L=\frac{1}{4Ng_{s}l_{s}}\sum_{i,j}(v_{i}^{I}-v_{j}^{I})(v_{i}^{I}-v_{j}^{I})-\frac{1}{2}\mathbf{J}^{IJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}\
$

that has manifestly form of relational mechanics as follows from (28) and (2).

##  3 Hamiltonian Formalism

In this section we find Hamiltonian from Lagrangian (24). In the first step we
introduce conjugate momenta to the matrix elements $\Phi^{I}_{ij}$ where we
will tread matrix elements as independent keeping in mind that we have
$\Phi_{ij}=\Phi_{ji}^{*}$. Then from (24) we obtain

$\displaystyle(\Pi_{I})_{ij}=\frac{\delta
L}{\delta\dot{\Phi}^{I}_{ij}}=\frac{1}{g_{s}l_{s}}(\dot{\Phi}_{I})_{ji}-\frac{1}{Ng_{s}l_{s}}\delta_{ji}\mathrm{Tr}{\dot{\Phi}_{I}}-$
$\displaystyle\frac{1}{2g_{s}l_{s}}(\Phi^{K}_{ji}\delta^{L}_{I}-\Phi^{L}_{ji}\delta^{K}_{I})-\frac{1}{N}(\mathrm{Tr}\Phi^{K}\delta_{ji}\delta^{L}_{I}-\mathrm{Tr}\Phi^{L}\delta_{ji}\delta^{K}_{I}))\mathbf{M}_{KL,MN}\mathbf{J}^{MN}\
,$

using

$\frac{\delta\mathbf{J}^{KL}}{\delta\dot{\Phi}^{I}_{ij}}=\frac{1}{2g_{s}l_{s}}(\Phi^{K}_{ji}\delta^{L}_{I}-\Phi^{L}_{ji}\delta^{K}_{I})-\frac{1}{2g_{s}l_{s}N}(\mathrm{Tr}\Phi^{K}\delta_{ji}\delta^{L}_{I}-\mathrm{Tr}\Phi^{L}\delta_{ji}\delta^{K}_{I})\
.$ (32)

As the next step we define Hamiltonian in the standard way

$\displaystyle H=(\Pi_{I})_{ij}\dot{\Phi}^{I}_{ij}-L$
$\displaystyle=\frac{1}{2g_{s}l_{s}}\dot{\Phi}^{I}_{ij}\dot{\Phi}^{I}_{ji}-\frac{1}{2g_{s}l_{s}N}\mathrm{Tr}\dot{\Phi}^{I}\mathrm{Tr}\dot{\Phi}^{I}-\frac{1}{2}\mathbf{J}^{IJ}\mathbf{M}_{IJ,KL}\mathbf{J}^{KL}-\frac{1}{4g_{s}l_{s}}\mathrm{Tr}[\Phi^{I},\Phi^{J}][\Phi^{I},\Phi^{J}]\
.$

To proceed further note that (3) implies

$\displaystyle\mathbf{P}_{I}\equiv\mathrm{Tr}\Pi_{I}=(\Pi_{I})_{ij}\delta_{ji}=0\
,$

so that $\mathbf{P}_{I}\approx 0$ is primary constraint of the theory.
Further, from (3) we also get

$\displaystyle\Phi^{I}_{ij}(\Pi_{J})_{ij}-\Phi^{J}_{ij}(\Pi_{I})_{ij}=0$

so that there is second set of primary constraints

$\mathbf{J}^{IJ}=\Phi^{I}_{ij}\Pi^{J}_{ij}-\Phi^{J}_{ij}\Pi^{I}_{ij}\approx 0\
,$ (36)

where again repeated indices mean summation over them.

Now we should check that they are the first class constraints. To do this we
introduce canonical Poisson brackets

$\left\\{\Phi^{I}_{ij},(\Pi_{J})_{kl}\right\\}=\delta^{I}_{J}\delta_{ik}\delta_{jl}\
.$ (37)

Clearly we have

$\displaystyle\left\\{\mathbf{P}_{I},\mathbf{P}_{J}\right\\}=0$ (38)

and also

$\displaystyle\left\\{\mathbf{J}^{IJ},\mathbf{P}_{K}\right\\}=\delta^{I}_{K}\mathrm{Tr}\Pi^{J}-\delta^{J}_{K}\mathrm{Tr}\Pi^{I}=\delta^{I}_{K}\mathbf{P}^{J}-\delta^{J}_{K}\mathbf{P}^{I}\approx
0\ .$

As the last Poisson bracket we calculate
$\left\\{\mathbf{J}^{IJ},\mathbf{J}^{KL}\right\\}$ and we obtain

$\displaystyle\left\\{\mathbf{J}^{IJ},\mathbf{J}^{KL}\right\\}=\delta^{IL}\Pi^{J}_{ij}\Phi^{K}_{ij}-\delta^{JK}\Phi^{I}_{mn}\Pi^{L}_{mn}-\delta^{JL}\Pi^{I}_{mn}\Phi^{K}_{mn}+\delta^{IK}\Phi_{mn}^{J}\Pi^{L}_{mn}-$
$\displaystyle-\delta^{IK}\Phi_{mn}^{L}\Pi_{mn}^{J}+\delta^{JL}\Phi^{I}_{mn}\Pi^{K}_{mn}+\delta^{JK}\Pi^{I}_{mn}\Phi^{L}_{mn}-\delta^{IL}\Phi^{J}_{mn}\Pi^{K}_{mn}=$
$\displaystyle=\delta^{IK}\mathbf{J}^{JL}-\delta^{IL}\mathbf{J}^{JK}-\delta^{JK}\mathbf{J}^{IL}+\delta^{JL}\mathbf{J}^{IK}\approx
0\ .$

In summary we find that $\mathbf{P}_{I}\approx 0\ ,\mathbf{J}^{IJ}\approx 0$
are first class constraints. We will discuss their properties below.

Finally we return to the Hamiltonian and express it in the form of canonical
variables. Using (3) we obtain

$\displaystyle(\Pi_{I})_{ij}(\Pi_{I})_{ji}=\frac{1}{g_{s}^{2}l_{s}^{2}}\mathrm{Tr}\dot{\Phi}^{I}\dot{\Phi}_{I}-\frac{1}{Ng_{s}^{2}l_{s}^{2}}\mathrm{Tr}\dot{\Phi}_{I}\mathrm{Tr}\dot{\Phi}_{I}-\frac{2}{g_{s}l_{s}}\mathbf{J}^{KL}\mathbf{M}_{KL,MN}\mathbf{J}^{MN}+\frac{1}{g_{s}l_{s}}\mathbf{J}^{KL}\mathbf{M}_{KL,MN}\mathbf{J}^{MN}$

and we find that the bare Hamiltonian is equal to

$\displaystyle
H_{B}=\frac{g_{s}l_{s}}{2}\mathrm{Tr}\Pi_{I}\Pi_{I}-\frac{1}{4g_{s}l_{s}}\mathrm{Tr}[\Phi^{I},\Phi^{J}][\Phi^{I},\Phi^{J}]\
.$ (42)

Then it is easy to see that total Hamiltonian that is given as linear
combination of the bare Hamiltonian with the first class constraints has the
form

$\displaystyle
H_{T}=\frac{g_{s}l_{s}}{2}\mathrm{Tr}\Pi_{I}\Pi_{I}-\frac{1}{4g_{s}l_{s}}\mathrm{Tr}[\Phi^{I},\Phi^{J}][\Phi^{I},\Phi^{J}]+\lambda_{I}\mathbf{P}_{I}+\lambda^{IJ}\mathbf{J}_{IJ}\
.$

Since $\mathbf{P}_{I}$ and $\mathbf{J}_{IJ}$ are first class constraints the
standard procedure is to fix them. For example, we can impose the gauge fixing
condition that says that the center of mass coordinates are equal to zero. In
other words we define gauge fixing functions $\mathcal{G}_{I}$ as

$\mathcal{G}_{I}\equiv\mathrm{Tr}\Phi_{I}\approx 0\ .$ (44)

Then we have

$\left\\{\mathcal{G}_{I},\mathbf{P}_{J}\right\\}=\left\\{\Phi^{I}_{ij},(\Pi_{J})_{kl}\right\\}\delta_{ji}\delta_{lk}=\delta^{I}_{J}\delta_{ik}\delta_{jl}\delta_{ji}\delta_{lk}=\delta^{I}_{J}\delta_{il}\delta_{li}=N\delta^{I}_{J}\
.$ (45)

In other words $\mathcal{G}_{I}$ and $\mathbf{P}_{J}$ are set of second class
constraints that now strongly vanish. We further fix generators
$\mathbf{J}_{IJ}\approx 0$ by imposing conditions that off-diagonal components
of the matrix $I^{IJ}$ are zero

$\mathcal{G}^{IJ}\equiv I^{IJ}\approx 0\ ,I\neq J\ ,$ (46)

where

$I^{IJ}=\frac{1}{2g_{s}l_{s}}\mathrm{Tr}(\Phi^{I}\Phi^{J})-\frac{1}{2g_{s}Nl_{s}}\mathrm{Tr}(\Phi^{I})\mathrm{Tr}(\Phi^{J})\
.$ (47)

Note that we have following Poisson brackets

$\displaystyle\left\\{\mathcal{G}^{IJ},\mathbf{P}_{K}\right\\}=0$ (48)

together with

$\displaystyle\left\\{\mathcal{G}^{IJ},\mathbf{J}^{KL}\right\\}=\frac{1}{2g_{s}l_{s}}(\delta^{IL}\Phi^{J}_{nm}\Phi_{mn}^{K}-\delta^{IK}\Phi^{J}_{ji}\Phi^{L}_{ij}+\delta^{JL}\Phi^{K}_{mn}\Phi^{I}_{nm}-\delta^{JK}\Phi^{I}_{mn}\Phi^{L}_{nm})-$
$\displaystyle-\frac{1}{2g_{s}N}(\delta^{IL}\Phi^{K}_{ij}\delta_{ji}-\delta^{IK}\Phi_{ij}^{L}\delta_{ji})\mathrm{Tr}\Phi^{J}-\frac{1}{2g_{s}N}(\delta^{JL}\Phi^{K}_{ij}\delta_{ji}-\delta^{JK}\Phi_{ij}^{L}\delta_{ji})\mathrm{Tr}\Phi^{I}=$
$\displaystyle=\delta^{IL}I^{JK}-\delta^{IK}I^{JL}+\delta^{JL}I^{KI}-\delta^{JK}I^{IL}\
.$

For $I=L$ and $J=K$ we obtain non-zero result

$\left\\{\mathcal{G}^{LK},\mathbf{J}^{KL}\right\\}=I^{KK}\ .$ (50)

Since $I^{KK}\neq 0$ by definition we find that $\mathcal{G}^{LK}$ is gauge
fixing function for $\mathbf{J}^{KL}$ and they form collection of the second
class constraints.

It is important to stress that gauge fixed theory with
$\mathbf{P}_{I}=\mathbf{J}^{IJ}=0$ corresponds to the original Hamiltonian for
$N$ D0-branes and we can interpret these frames with vanishing total momentum
and angular momentum as Newtonian frames, for more details we recommend
discussion presented in [4].

##  4 Gauging Time translation

In order to find an action invariant under arbitrary time dependent
translation $t^{\prime}=t+\epsilon(t)$ we follow the standard procedure of
parametrized systems, see for example [16]. We begin with the canonical form
of the action

$S=\int dt((\Pi_{I})_{ij}\dot{\Phi}_{ij}^{I}-H_{T})\ ,$ (51)

where $H_{T}$ is given in (3). As the next step we introduce variable $t$ and
conjugate momenta $p_{t}$ and rewrite the action into the form

$S=\int
d\tau(p_{t}\frac{d}{d\tau}t+(\Pi_{I})_{ij}\frac{d}{d\tau}\Phi_{ij}^{I}-N(p_{t}+H_{T}))\
.$ (52)

In order to see equivalence between (52) and (51) let us consider equations of
motion for $N$ and $p_{t}$ that give

$\frac{d}{d\tau}t-N=0\ ,\quad p_{t}+H_{T}=0\ $ (53)

that inserting back to the action (52) we obtain

$S=\int d\tau\frac{dt}{d\tau}((\Pi_{I})_{ij}\frac{d}{dt}\Phi_{ij}^{I}-H_{T})\
,$ (54)

where we presumed that the first relation in (53) can be inverted. Then it is
easy to see that (54) is equivalent do (51). The action (52) is manifestly
reparametrization invariant under transformation

$\tau=f(\tau^{\prime})\ ,\quad t^{\prime}(\tau^{\prime})=t(\tau)\ ,\quad
N(\tau)=N^{\prime}(\tau^{\prime})\frac{1}{\frac{df}{d\tau^{\prime}}}\ $ (55)

and

$(\Pi^{\prime}_{I})_{ij}(\tau^{\prime})=(\Pi_{I})_{ij}(\tau)\
,\quad\Phi^{\prime I}_{ij}(\tau^{\prime})=\Phi_{ij}^{I}(\tau)\ .$ (56)

In summary we got the action (52) that is invariant under time dependent
Galilean transformation together with arbitrary redefinition of the time
$\tau$. Clearly this construction is generally correct even in our specific
case of $N$ D0-branes. On the other hand the question is physical
interpretation of the coordinate $t$ and how it should be interpreted in the
context of non-abelian nature of D0-brane action. In other words this
construction cannot be interpreted as covariant form of the action for $N$
D0-brane which is very difficult to construct, see for example [17, 18]. For
that reason we mean that this construction has only formal meaning.

Acknowledgement:

The work of J.K. was supported by the Grant Agency of the Czech Republic under
the grant P201/12/G028.

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††thanks: Corresponding author

# On modeling airborne infection risk

Yannis Drossinos<EMAIL_ADDRESS>Thermal Hydraulics & Multiphase
Flow Laboratory, Institute of Nuclear & Radiological Sciences and Technology
and Technology, Energy & Safety, National Centre for Scientific Research
“Demokritos”, 15314 Agia Paraskevi, Greece  Nikolaos I. Stilianakis
<EMAIL_ADDRESS>Joint Research Centre (JRC), European
Commission, 21027 Ispra (VA), Italy  Department of Biometry and Epidemiology,
University of Erlangen-Nuremberg, Erlangen, Germany

###### Abstract

Airborne infection risk analysis is usually performed for enclosed spaces
where susceptible individuals are exposed to infectious airborne respiratory
droplets by inhalation. It is usually based on exponential, dose-response
models of which a widely used variant is the Wells-Riley (WR) model. We employ
a population-based Susceptible-Exposed-Droplet-Infected-Recovered (SEDIR)
model to revisit the infection-risk estimate at the population level during an
epidemic. We demonstrate the link between epidemiological models and the WR
model, including its Gammaitoni-Nucci (GN) generalization. This connection
shows how infection quanta are related to the number of infectious airborne
droplets. For long latent periods, the SEDIR model reduces to the GN model
with parameters that depend on biological properties of the pathogen (size-
dependent pathogen droplet concentration, infection probability of a deposited
infectious droplet), physical droplet properties (lung-deposition
probability), and individual behavioral properties (exposure time). In two
scenarios we calculate the probability of infection during the epidemic. The
WR and GN limits of the SEDIR model reproduce accurately the SEDIR-calculated
infection risk.

infectious diseases, SARS-CoV-2 transmission, aerosol, airborne, infection
risk, Wells-Riley infections risk model, Gammaitoni-Nucci infection risk model

##  I Introduction

The determination of the risk of infection during an epidemic is an important
quantitative indicator that, among others, influences decisions of public
health authorities on intervention strategies and their implementation,
including vaccine administration. It contributes, also, to individual
decisions on accepting recommended social distancing, implementing proper
wearing of face masks, and adhering to mobility restrictions. Estimates of the
risk associated with airborne respiratory-pathogen infection have become
numerous since the beginning of the coronavirus disease 2019 (COVID-19)
pandemic.

Most airborne infection-risk analyses during the COVID-19 pandemic
concentrated on risk calculations in small, enclosed spaces within which
susceptible individuals are exposed by inhalation to infectious airborne
respiratory droplets for a brief period. For example, the probability of
infection due to the (severe acute respiratory syndrome coronavirus 2 (SARS-
CoV-2) has been estimated in numerous micro-environments, such as in an office
as a function of number of occupants and their exposure time, in a pharmacy, a
supermarket, a restaurant, a post office and a bank [1]; in a hospital room,
conference room and auditorium [2]; in shared indoor space [3]; in public
spaces like a shopping mall corridor [4] or small shops [5]; in a ski cabin
[6]; in a university office [7]. The majority of these risk analyses were
based on the exponential, dose-response Wells-Riley (WR) model or its
variants, see, for example, the recent generalization to poly-pathogen
aerosols and the validity of the Poisson-distribution assumption [8] or the
use of a double Poisson model [9].

The Wells-Riley [10, 11, 12] model is a deterministic exposure model, based on
the probabilistic airborne infection model proposed by Wells [13]. Wells
introduced the quantum of airborne infection 111We suspect that the choice of
the word quantum refers to the discrete nature of the carriers of the airborne
infection, i.e., the infectious respiratory droplets. Wells might have been
paying homage to the great successes of quantum mechanics, universally
accepted by the time he introduced the notion of a quantum of infection. as a
discrete entity of the infectious dose that would give [8] a $63.21\%$
probability of infection infected is, according to the Poisson distribution,
or, in modern terminology, as the Infectious Dose ID63.21. Riley et al. [10],
expanding on Riley (1974) [14] and using Wells’ quantum of infection,
introduced the average number of quanta inhaled during an individual’s
exposure to an airborne pathogen in an exponential dose-response model to
obtain a model for the probability of airborne infections in a indoor
environment. They assumed that the micro-environment is homogeneous, and hence
infection quanta were uniformly distributed (well mixed approximation), and
that the quantum concentration was at steady state, as well as the ventilation
conditions (steady-state outdoor air supply). The resulting steady-state model
is commonly referred to as the Wells-Riley model. Moreover, they took $I$, the
number of infectors, constant during exposure, but not so the number of
susceptibles $S$, assuming that the viral latent period, the time of infection
to the time of becoming infectious, is much longer than the exposure time,
namely the time interval individuals are exposed to the pathogen. An important
generalization of the WR model was proposed by Gammaitoni and Nucci [15] (GN).
They removed the steady-state quantum concentration assumption to generalize
the airborne infection-risk model to time-dependent quanta concentrations
within the confined environment.

One of the characteristics of the WR model is that it uses input from aerosol
dynamics in, e.g., estimates of the generation rate of the quanta of
infection, and their removal rate via e.g., gravitational settling or indoor-
air ventilation, to estimate viral transmissibility. Human behavior, however,
is naively modeled by the lumped parameter of exposure time. The model, being
an individual-level model and in contrast to compartmental epidemiological
models, does not consider the total population $N$. Instead, the enclosed-
space volume $V$ determines the spatial scale. This is required to render the
exponent in the risk expression an intensive variable, that is a density which
is independent of the scale of the system.

Infection risk estimates in larger, including closed or semi-closed,
populations and at longer, but intermediate, spatial and temporal time scales
than those investigated by micro-environmental models are equally important.
Envisioned intermediate spatial scales are those encountered in, e.g.,
hospitals, prisons, cruise and military ships, boarding schools, nursing
homes, military camps. Mesoscopic epidemiological models address these scales.
The Susceptible-Droplet-Infected-Recovered (SDIR) model [7, 4] is one such
model. It has two distinguishing features: it retains the structure of
compartmental epidemiological models, and it incorporates explicitly the
dynamics of the pathogen-carrying agent. In the case of SARS-CoV-2 where the
pathogen-carrying agent is the infectious respiratory droplets, the SDIR model
retains the necessary information on the dynamics of the infectious droplets,
in addition to incorporating biological aspects of the virus, and behavioral
aspects of the individuals. Contrary to micro-environmental models the SDIR
model is a population-level model.

Macroscopic models, on the other hand, address much larger populations and
much longer temporal and spatial scales, for example country-wide and province
scales [18, 19, 20] or regional scales [21]. At such scales, the models do not
consider explicitly micro-environmental dynamics. Instead, the intricate
dynamics of the respiratory droplets and other micro-environmental dynamics
are implicitly incorporated via effective transmission rates or parameters,
via a procedure akin to coarse-grained descriptions of physical systems [4].

Noakes et al. [12] presented an early attempt to reconcile the WR expression
with a standard Susceptible-Infected-Recovered (SIR) compartmental
epidemiological model. Their derivation was reconsidered and amplified by
Bazant and Bush [22] who included explicitly the exposed population
compartment and considered both short and long latent periods, as we do in
this work. We use an extended version of the SDIR model to revisit the
derivation and to estimate what we shall refer to as the epidemic airborne
infection risk, the infection risk during an epidemic. In doing so, we
elucidate and establish firmly the connection between compartmental
epidemiological models and micro-environmental risk models, like the Wells-
Riley model and its Gammaitoni-Nucci generalization, and the relevance of
respiratory droplet dynamics. One of the essential observations is that
neither the GN nor the WR model considers the time-dependent changes of
infected population.

##  II Infection probability in compartmental epidemiological models

The epidemic infection risk $P(t_{0},\delta
t;\langle\tau_{\textrm{exp}}\rangle)$ is the probability of infection at time
$t_{0}$ from the beginning of an epidemic within a prediction interval $\delta
t$. Expressed in terms of the number of susceptible individuals $S$, it is
their relative change [23, 12] in the period $[t_{0},t_{0}+\delta t]$,

$P(t_{0},\delta
t;\langle\tau_{\textrm{exp}}\rangle)=\frac{S(t_{0};\langle\tau_{\textrm{exp}}\rangle)-S(t_{0}+\delta
t;\langle\tau_{\textrm{exp}}\rangle)}{S(t_{0};\langle\tau_{\textrm{exp}}\rangle)}.$
(1)

The decreasing time series $S(t)$ depends on the average daily exposure time
$\langle\tau_{\textrm{exp}}\rangle$. This dependence may be explicit, as in
the SDIR model, or implicit via, e.g., the daily number of contacts between
susceptible and infected individuals in the standard SIR model. The time from
the start of the epidemic $t_{0}$ determines the initial conditions: for
example, in a Susceptible-Exposed-Infected-Recovered (SEIR) model it specifies
the initial number of infected $I(t_{0})=I_{0}$, of susceptibles $S_{0}$ and
exposed $E_{0}$. Additionally, in models that include respiratory-droplet
dynamics, $t_{0}$ specifies the initial number of airborne $D_{0}$ and settled
$C_{0}$ droplets. The probability of infection may also be expressed in terms
of $\textrm{Cases}(t_{0})$, the number of new infectious cases at time
$t_{0}$, since $\textrm{Cases}(t_{0}+\delta
t;\langle\tau_{\textrm{exp}}\rangle)=S(t_{0};\langle\tau_{\textrm{exp}}\rangle)-S(t_{0}+\delta
t;\langle\tau_{\textrm{exp}}\rangle)$. Bazant and Bush [22] used the secondary
attack rate, new cases relative to $S_{0}$, to obtain the infection
probability in a form as reported here.

Given Eq. (1) any epidemiological model that calculates $S(t)$ can be used to
calculate the probability of infection. It provides the connection between
epidemiological compartmental models and infection-risk models.

##  III  Droplet models

###  III.1 Susceptible-Droplet-Infected-Recovered (SDIR) model

The SDIR model of infectious disease transmission via infectious respiratory
droplets [16] extends the standard SIR model by coupling the population
dynamics of susceptibles and infected to the dynamics of a population of
infectious respiratory droplets. It is a population epidemiological model in
that the population under consideration is divided into compartments and
individuals can move between the compartments. Its particularity is that it
adduces the population of airborne and settled infectious respiratory
droplets. As the temporal and spatial scales associated with infectious
droplets are relatively short, effective removal time scales of infectious
droplets are less than a day, the SDIR model is a mesoscopic model [17]. It
provides a natural extension of micro-environmental models in that it
considers intermediate scales where the properties and the dynamics of the
pathogen-carrying droplets are explicitly calculated and incorporated in the
model.

According to the initial formulation, respiratory droplets are partitioned
into two compartments: airborne ($D$) and settled ($S$). As in the standard
SIR model individuals are divided into three population compartments:
susceptibles $S$, infected $I$, and recovered $R$. Infection does not occur
via direct $I\leftrightarrow S$ interaction: instead, this interaction is
mediated by the infectious droplets, be they airborne or settled. The model
allows for a distribution of droplets characterized by their diameter, be it
pre- or post-evaporation [24, 25].

###  III.2 Susceptible-Exposed-Droplet-Infected-Recovered (SEDIR) model

As other respiratory viruses SARS-CoV-2 virus exhibits a latent period. During
the latent period $\tau_{\textrm{lat}}$ exposed individuals are infected but
not infectious. Accordingly, we generalize the SDIR model by adding an exposed
population compartment $E$. We, thus, introduce the latent period
$\tau_{\textrm{lat}}=1/\sigma$, a time scale to be contrasted to the average
time a susceptible individual is exposed to the pathogen
$\langle\tau_{\textrm{exp}}\rangle$ that, as we will argue, is embedded in the
transmission rates $\beta$s. As previously mentioned, Bazant and Bush [22]
also included the exposed population to connect the SEIR model to the WR-
model. They, as we do herein, also considered cases of short and long latent
periods. The SEDIR model is defined by the following set of coupled ordinary
differential equations (ODEs)

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle-\sum_{i=1}^{i=i_{\textrm{max}}}\Big{(}\frac{\beta_{i}^{d}}{N}D_{i}S+\frac{\beta_{i}^{c}}{N}C_{i}S\Big{)},$
(2a) $\displaystyle\frac{dE}{dt}$ $\displaystyle=$
$\displaystyle-\frac{dS}{dt}-\sigma E,$ (2b) $\displaystyle\frac{dI}{dt}$
$\displaystyle=$ $\displaystyle\sigma E-\mu_{I}I,$ (2c)
$\displaystyle\frac{dD_{i}}{dt}$ $\displaystyle=$
$\displaystyle\kappa_{i}^{d}I-\alpha_{i}^{d}D_{i},\quad\mbox{for}\quad
i=1,2\ldots i_{\textrm{max}},$ (2d) $\displaystyle\frac{dC_{i}}{dt}$
$\displaystyle=$
$\displaystyle\kappa_{i}^{c}D-\alpha_{i}^{c}C_{i},\quad\mbox{for}\quad
i=1,2\ldots i_{\textrm{max}}.$ (2e)

We do not show the equation for the recovered compartment $R$ since the total
population $S+I+R=N$ is constant representing a closed population. A schematic
diagram of the model is shown in Fig. 1.

Figure 1: Schematic diagram of the Susceptible-Exposed-Droplet-Infected-
Recovered (SEDIR) model (based on a figure of Ref. [17]). Droplet compartments
are denoted by $D_{i}$, airborne droplets, and $C_{i}$, settled droplets.
Superscripts ($d,c$) denote (airborne, settled) droplets, the subscript $i$
refers to droplets with post evaporation diameter $d_{i}^{\textrm{post}}$.
Infection transmission rates are denoted by $\beta_{i}^{d,c}$, droplet
generation rates by $\kappa_{i}^{d,c}$, and removal rates by
$\alpha_{i}^{d,c}$. The latent period is $\tau_{\textrm{lat}}=1/\sigma$ and
the infection recovery rate $\mu_{I}$.

The number of infectious airborne droplets of post-evaporation diameter
$d_{i}^{\textrm{post}}$ is denoted by $D_{i}$ (number), and that of settled
droplets by $C_{i}$ (number), cf. the Supporting Information Appendix for a
discussion of droplet evaporation and associated droplet diameters. The number
of droplet classes is $i_{\textrm{max}}$. The rate of transition from the
exposed compartment $E$ to the infected compartment $I$ is denoted by
$\sigma$, whose inverse is the virus latent period
$\tau_{\textrm{lat}}=1/\sigma$. The infection recovery rate, the rate at which
$I\rightarrow R$, is $\mu_{I}$. Superscripts denote airborne droplet ($d$) and
settled ($c$), and the subscript $i$ denotes the droplet class specified by
the post-evaporation diameter $d_{i}^{\textrm{post}}$. The transmission rate
per infectious, airborne respiratory droplet that has been inhaled and
deposited in the respiratory tract of a susceptible is denoted by
$\beta_{i}^{d}$ (inverse time), whereas that of an infectious settled droplet
transferred to facial membranes is denoted by $\beta_{i}^{c}$ (inverse time).

The airborne droplet generation rate per infected individual (by normal oro-
nasal activities-e.g., speaking, laughing, breathing- or by violent expiratory
events - sneezing, coughing-) is $\kappa_{i}^{d}$ (number/time) and the
corresponding airborne droplet removal rate is $\alpha_{i}^{d}$ (number/time),
the later including droplet removal by ventilation (if present). Settled
droplets may be generated either via direct generation by an infected
individual and deposition on facial mucous tissues or via deposition of
airborne droplets. Direct deposition would introduce an additional generation
term in Eq. (2e) proportional to the number of infected individuals, similar
to the generation term in the airborne-droplets equation, Eq. (2d). In this
version of the model we neglect this mechanism. Instead, settled droplets are
generated via deposition of airborne droplets, and specifically solely by
gravitational settling. Hence the generation rate
$\kappa_{i}^{c}=\theta_{i}(d_{i}^{\textrm{post}})$ (number/time) with $\theta$
the gravitational settling rate in still air. The corresponding settled
droplet removal rate is $\alpha_{i}^{c}$ (number/time).

We present explicit expressions for the transmission $\beta_{i}^{c,d}$ and
removal $\alpha_{i}^{c,d}$ rates, along with justifications for our choices,
in the Appendix. We remark that the transmission and removal rates are
_derived_ quantities. In addition, both transmission rates depend (linearly as
we argue in the Appendix) on the average exposure time
$\langle\tau_{\textrm{exp}}\rangle$, i.e.,
$\beta=\beta(\langle\tau_{\textrm{exp}}\rangle)$. The SDIR basic reproduction
number is [16, 17]

$R_{0}^{\textrm{SDIR}}=\sum_{i=1}^{i=i_{\textrm{max}}}\Big{(}\frac{\beta_{i}^{d}\kappa_{i}^{d}}{\alpha_{i}^{d}\mu_{I}}+\frac{\beta_{i}^{c}\kappa_{i}^{c}}{\alpha_{i}^{c}\mu_{I}}\Big{)}.$
(3)

Equation (3) also gives the SEDIR basic reproduction number, see, for example,
Ref. [26].

###  III.3 Gammaitoni-Nucci (GN) limit

We limit the droplet classes in the SEDIR model to a single airborne droplet
class $D_{1}$, as the original GN model considered only one droplet diameter.
It is easily shown, for example by integrating the linear ODE for the infected
population Eq. (2c), that if $\sigma\delta t\ll 1$, latent period much greater
than the time infected individuals generate infectious droplets within the
enclosed space, and $\mu_{I}\delta t\ll 1$, infectiousness period much greater
than the prediction time $\delta t$, then $dI/dt|_{t_{0}}=0$, the number of
infected is constant at $t_{0}$, and denoted as $I_{0}$. If, in addition, we
disregard the equation for the exposed population Eq. (2b), which is
irrelevant over the prediction time for the evolution of the infection (the
number of $E$ increases, but not that of $I$), the SEDIR model reduces to

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\beta_{1}^{d}}{N}D_{1}S,$ (4a)
$\displaystyle\frac{dD_{1}}{dt}$ $\displaystyle=$
$\displaystyle\kappa_{1}^{d}I_{0}-\alpha_{1}^{d}D_{1}.$ (4b)

The system of Eqs. (4a,4b) can be compared to the GN equations (5) for the
rate of change of the number of susceptibles and total number of quanta of
infection $Q$ in the enclosed space, which expressed in our notation read

$\displaystyle\frac{dS}{dt}$ $\displaystyle=$ $\displaystyle-\frac{B}{V}QS,$
(5a) $\displaystyle\frac{dQ}{dt}$ $\displaystyle=$ $\displaystyle
qI_{0}-\lambda_{\textrm{air}}Q,$ (5b)

where $q$ is the quantum generation rate per infectious individual
(quanta/sec), see also [12], $B$ is the pulmonary ventilation rate (also
referred to as breathing rate, m3/sec), and $V$ is the space volume (m3). We
explicitly denote the number of infected individuals during exposure as
$I_{0}$, Eq. (5b), to stress that their number is constant. The parameter
$\lambda_{\textrm{air}}$ that determines the quantum-removal rate is the
ventilation rate in air exchanges per hour (referred to as the disinfection
rate expressed as the number of effective, or equivalent, air exchanges, in
the original reference [15]). Since the initial formulation of the model, the
quanta removal rate has been expanded to include the rate of pathogen
inactivation, droplet surface deposition, inactivation due to UV irradiation,
filter penetration, mask efficiency, etc. (see also the droplet removal rates
$\alpha_{1}^{d}$ used in this work and summarized in the Appendix).

The analytical solution of Eq. (5b) is

$Q(t)=\frac{qI_{0}}{\lambda_{\textrm{air}}}+\left(Q_{0}-\frac{qI_{0}}{\lambda_{\textrm{air}}}\right)\exp(-\lambda_{\textrm{air}}\delta
t),$ (6)

where $Q_{0}$ is the initial (at time $t=t_{0}$) total concentration of the
infection quanta in the enclosed space.

The comparison of Eqs. (4) and (5) provides insights on the differences and
formal similarities of the SEDIR and GN models. Let the number of quanta be
proportional to the number of infectious respiratory droplets, $Q=\xi D_{1}$,
and the transmission rate proportional to the breathing rate,
$\beta_{1}^{d}=B\tilde{\beta}_{1}^{d}$, as argued in the Appendix. Moreover,
consider indoor-air ventilation the only droplet or quantum removal process,
$a_{1}^{d}=\lambda_{\textrm{air}}$. Their substitution into Eqs. (4), and a
mapping of the resulting equations to Eqs. (5) determines the conversion
factor $\xi$ to be

$\xi=\frac{\beta_{1}^{d}}{B}\,\frac{V}{N}\equiv\tilde{\beta}_{1}^{d}\rho_{\textrm{scale}},$
(7)

where the last equation defines the scaling density
$\rho_{\textrm{scale}}=V/N$. Hence, in this model infection quanta, up to an
overall scaling factor, are infectious respiratory droplets modified by
$\tilde{\beta}_{1}^{d}$, a parameter that includes the probability of
infection of a lung-deposited pathogen, number of pathogens in a droplet,
lung-deposition probability, and average exposure time, cf. the Appendix. The
combination of these factors converts the infectious airborne droplets to
infection quanta. Their generation $q$ is similarly related to the respiratory
droplet generation rate via $q=\kappa_{1}^{d}\xi$. The mapping of the two
models also manifests the different inherent scales: the extensive variable,
namely the variable that scales linearly with the size of the system, is the
volume of the enclosed space in the GN model, whereas it becomes the total
population $N$ in the SEDIR model. The scaling factor $\rho_{\textrm{scale}}$
implements the transition from a microscopic models which depends on the
enclosed-space volume $V$, to a mesoscopic epidemiological model, which
depends on the total population $N$. This scaling is reminiscent of the
scaling proposed in Ref. [18] to transition from an ODE to a PDE (Partial
Differential Equations) epidemiological model. Care should be exercised in
interpreting $\rho_{\textrm{scale}}$: if $V$ is taken to refer to a mesoscopic
volume then the GN model is essentially extended to much greater scales. If
$N$ is taken to be the number of occupants in an enclosed, micro-environment
the SEDIR model is restricted to smaller scales; however, in that case it may
not be considered a proper compartmental epidemiological model. These
consideration have important repercussion on the choice of model parameters
and prediction times in micro or mesoscale models.

The GN limit of the SEDIR-calculated infection risk may be calculated by
solving Eqs. (4) to obtain $S(t)$ and subsequently the infection risk
according to Eq. (1). In fact, the droplet equation Eq. (4b) may be solved
analytically to obtain an equation formally identical to Eq. (6).Obviously,
given an analytical solution of Eq. (4b), the susceptibles equation Eq. (4a)
may also be integrated. In the numerical simulations we used the analytical
solutions.

###  III.4 Wells-Riley (WR) limit

Reference [27] considered analytically the very common limit where the
duration of the infectiousness of an infected individual $T_{I}=1/\mu_{I}$ is
significantly longer than the lifespan of the airborne pathogen
$T_{p}=1/\alpha_{1}^{d}$, i.e., when
$\rho_{1}\equiv\mu_{I}/\alpha_{1}^{d}=T_{p}/T_{I}\ll 1$. For appropriately
chosen non-dimensional variables [27] the quasi steady-state limit is defined
as $\rho_{1}d\tilde{D}_{1}/d\tilde{t}=0$ which implies
$\tilde{D}_{1,\textrm{qss}}=\tilde{I}$, or in terms of of the original
variables $D_{1,\textrm{qss}}=(\kappa_{1}^{d}/\alpha_{1}^{d})I$. Note that the
quasi steady-state condition does _not_ imply that the number of infected
individuals is constant, $dI/dt|_{\textrm{qss}}\neq 0$, that is
$I_{\textrm{qss}}$ is _time dependent_.

The substitution of the steady-state ($I,D_{1}$) relationship in the original
equations Eqs. (2) gives the quasi steady-state limit of SEDIR,

$\displaystyle\frac{dS_{\textrm{qss}}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\beta_{1}^{d}\kappa_{1}^{d}}{\alpha_{1}^{d}N}I_{\textrm{qss}}S_{\textrm{qss}},$
(8a) $\displaystyle\frac{dE_{\textrm{qss}}}{dt}$ $\displaystyle=$
$\displaystyle\frac{\beta_{1}^{d}\kappa_{1}^{d}}{\alpha_{1}^{d}N}I_{\textrm{qss}}S_{\textrm{qss}}-\sigma
E_{\textrm{qss}},$ (8b) $\displaystyle\frac{dI_{\textrm{qss}}}{dt}$
$\displaystyle=$ $\displaystyle\sigma
E_{\textrm{qss}}-\mu_{I}I_{\textrm{qss}},$ (8c) $\displaystyle\mbox{with}\quad
D_{\textrm{qss}}(t)$ $\displaystyle=$
$\displaystyle\frac{\kappa_{1}^{d}}{\alpha_{1}^{d}}I_{\textrm{qss}}(t).$ (8d)

In the quasi steady-state limit the dependence on the number of infectious
droplets $D_{1}(t)$ disappears.

As before, in the previously considered double limit, $\sigma\delta
t,\mu_{I}\delta t\ll 1$, we can neglect the equation for the exposed
population, Eq. (8b), and take the number of infected individuals constant,
$I_{\textrm{qss}}=I_{0}$. The model equations become

$\displaystyle\frac{dS_{\textrm{qss}}}{dt}$ $\displaystyle=$
$\displaystyle-\frac{\beta_{1}^{d}\kappa_{1}^{d}}{\alpha_{1}^{d}N}I_{0}S_{\textrm{qss}},$
(9a) $\displaystyle I_{\textrm{qss}}$ $\displaystyle=$ $\displaystyle
I_{0},\quad\mbox{{and}}\quad
D_{\textrm{qss}}=\frac{\kappa_{1}^{d}}{\alpha_{1}^{d}}I_{0}.$ (9b)

The analytical solution of Eq. (9a) leads directly to the WR limit of the
SEDIR model as follows

$P_{\textrm{WR}}^{\textrm{SEDIR}}(t_{0},\delta
t;\langle\tau_{\textrm{exp}}\rangle)=1-\exp\Big{(}-\frac{\beta_{1}^{d}\kappa_{1}^{d}}{\alpha_{1}^{d}N}I_{0}\delta
t\Big{)}.$ (10)

For completeness, we also present the WR equation as usually written

$P_{\textrm{WR}}(\delta
t)=1-\exp\Big{(}-\frac{Bq}{\lambda_{\textrm{air}}V}I_{0}\delta t\Big{)},$ (11)

where the variables were defined after Eq. (5). Hence, the WR equation is
obtained from the quasi steady-state SEDIR equations in the triple limit of
latent period and infectiousness period longer than the time scale of
observation and $\rho_{1}\ll 1$. As before, if we let $k_{1}^{d}=q/\xi$ and
$\alpha_{1}^{d}=\lambda_{\textrm{air}}$ in the SEDIR expression
$P_{\textrm{WR}}^{\textrm{SEDIR}}$, Eq. (10), we obtain the WR infection
probability $P_{\textrm{WR}}$, Eq. (11).

Of course, the WR limit of the GN model may be easily obtained by setting
$dQ/dt=0$ in Eq. (5b). The steady-state quantum concentration, then, becomes
$Q_{ss}=qI_{0}/(\lambda_{\textrm{air}})$, leading via Eq. (5a) to the number
of susceptibles

$S(t)=S_{0}\exp\Big{(}-\frac{Bq}{\lambda_{\textrm{air}}V}I_{0}\delta
t\Big{)},$ (12)

and, thus, to the WR risk-model expression Eq. (11). However, the alternative
derivation for the WR limit we presented in term of the quasi steady-state
solution of the SEDIR model specifies under which conditions this limit is
valid, instead of arbitrarily setting $dQ/dt=0$.

##  IV Numerical results

We performed numerical simulations of the SEDIR model, Eqs. (2), to
investigate the effect of the prediction interval $\delta t$ and the latent
period $\tau_{\textrm{lat}}$ on epidemic risk. We also investigate numerically
and analytically the validity of the GN, Eqs. (4), and WR, Eq. (10),
approximations to the SEDIR model-predictions for the calculated risk.

Figure 2: Left: Epidemic infection probability according to the SEDIR model.
Curves were calculated for two prediction intervals ($\delta t=1,7$ days) and
two latent periods ($\tau_{\textrm{lat}}=0.1,6$ days). Two airborne-droplet
classes were considered, ($d_{i}^{\textrm{post}}=2.05,82.13$ $\mu$m
($i=1,2$)), susceptible-infectious droplet encounters per day were taken to be
$c=18$, exposure time for each $S\leftrightarrow D_{i}$ ($i=1,2$), encounter
$\tau_{d_{1}}=25$ min and $\tau_{d_{2}}=1$ min, leading to a total daily
susceptible-infectious droplet average exposure time of
$\langle\tau_{\textrm{exp}}\rangle=7.8$ hours (per day). The ventilation rate
was taken to be $\lambda_{\textrm{air}}=0.2$ air exchanges per hour, a typical
value for an Italian building [1]. Total population $N=1000$. Right:
Corresponding dynamics of the two epidemics. The left curves (filled symbols)
correspond to the short latent period, $\tau_{\textrm{lat}}=0.10$ days, with
$I$ peaking at $t\approx 43$ days, and no discernible exposed population. The
right curves (lines, no symbols) show the epidemic for the long latent period
$\tau_{\textrm{lat}}=6$ days, with $I$ peaking at $t\approx 96$ days, and an
appreciable exposed population.

For the simulations we used parameters related to the COVID-19 pandemic, e.g.,
individual behavior characteristics in addition to physico-chemical and
biological properties of the SARS-CoV-2 virus, e.g., viral load. We note,
though, that we do not attempt to reproduce a COVID-19 scenario as in our
attempt to present the minimal model that reduces to the GN or WR models we do
not consider the asymptomatic stage of the disease.

We used two airborne droplet classes of post-evaporation diameter
$d_{i}^{\textrm{post}}=2.05,82.13$ $\mu$m ($i=1,2$). As generally accepted
[28], the pathogen concentration was taken to be droplet-size dependent. We
opted to limit the airborne droplet classes to two and not to simulate settled
droplets to render easier the interpretation of our results: either
restriction may be easily removed. The evaporation factor [17],
$d_{i}^{\textrm{post}}=z_{\textrm{evap}}d_{i}^{\textrm{pre}}$, was set to
$z_{\textrm{evap}}=0.40$. Airborne droplet generation rates were taken to
correspond to speaking. A complete list of model parameters is presented in
the Appendix.

Individual behavior determines a number of model parameters. We considered the
contact rate, the number of susceptible-infected individual encounters, to be
$c=18$ per day [29]. The duration of an encounter of a susceptible with an
infectious droplet,i.e., the breathing time during a $S\leftrightarrow I$
encounter, was taken to depend on the droplet size: $\tau_{d_{1}}=25$ min and
$\tau_{d_{2}}=1$ min. Thus, the average exposure time per day of a single
susceptible is $c\times(\tau_{d_{1}}+\tau_{d_{2}})=7.8$ hours per day.

Figure 2 summarizes the main results of four simulations to determine epidemic
airborne infection risk. We used two latent periods $\tau_{\textrm{lat}}=0.1$
days (short) and $\tau_{\textrm{lat}}=6.0$ days (long), along with a short
$\delta t=1.0$ day and a long prediction interval $\delta t=7$ days. The left
panel shows the calculated infection probabilities for each scenario. Two
groups of curves may be identified: for the short latent period the infection
probability peaks at about $t_{\textrm{peak}}\approx 43$ days, whereas for the
long latent period the peak occurs at $t_{\textrm{peak}}\approx 96$. Within
each group of curves, infection risk increases with increasing prediction
interval, as would have been expected.

The qualitative behavior of infection risk may be understood by considering
the dynamics of the epidemic, described by the time-dependent number of
$S,E,I$ and $R$ shown in the right panel of Fig. 2. The four curves on the
left (filled symbols) correspond to the short latent period, whereas those on
the right (no symbols) to the long latent period. We also present the maximum
number of infected individuals for each epidemic. We note that, as expected,
infection risk follows the time-dependent behavior of the infected
individuals. For the short-latent period epidemic, the number of exposed
individuals is very small, not discernible on the figure, whereas for the
long-latent period epidemic the number of exposed individuals is comparable to
the number of infected. In fact, before the $I$ maximum, $E>I$, whereas
afterwards $I>E$. Even though not discernible, the number of exposed
individuals $E$ peaks earlier than the number of infected $I$.

The validity of the GN and WR approximations is investigated numerically in
Fig. 3. Four groups of curves are shown, each corresponding to the ordered
pair ($\delta t,\tau_{\textrm{lat}})$. For each pair choice, we plot the SEDIR
infection risk, $P^{\textrm{SEDIR}}$ as determined via the numerical solutions
of Eqs. (2) (filled blue diamonds), $P_{\textrm{GN}}^{\textrm{SEDIR}}$ as
described in Section III.3 (square, unfilled symbols), and
$P_{\textrm{WR}}^{\textrm{SEDIR}}$ as calculated via Eq. (10) (cross,
continuous line).

Figure 3: Epidemic risk calculated via the SEDIR model and its GN and WR
limits, Eq. (10). The four ordered pairs associated with each graph triplet
are specified by ($\delta t,\tau_{\textrm{lat}})$. For all the simulations the
GN and WR limits were identical: they differed from the SEDIR model
predictions only for the long prediction interval ($\delta t=7$ days). See the
main text for the explanation.

Two observations are in order. For the epidemics considered, the GN and WR
limits are identical. Whether the two limits would differ depends on the
airborne droplet removal rate $\alpha_{1}^{d}$ (and hence on the dimensionless
parameter $\rho_{1}=\mu_{I}/\alpha_{1}^{d}$). We restrict the analysis to a
single airborne droplet class, for simplicity: the arguments are easily
generalized. The importance of the removal rate is apparent from the
analytical solution of the droplet equation Eq. (4b), which as noted before is
formally identical to its time-dependent part, which determines the difference
between the steady-state and non steady-state model, vanishes as
$\alpha_{1}^{d}\delta t\gg 1$, a condition satisfied for all cases considered.
The same observation holds for the GN-WR comparison whereby the time-dependent
part of the quantum concentration in Eq. (6) vanishes as the droplet removal
term tends to infinity. If ventilation is the dominant aerosol removal
process, for $\lambda_{\textrm{air}}\delta t\gg 1$ the two models become
identical. Hence, for high ventilation rates the difference between the steady
and non steady-state quantum concentration models decreases or even vanishes.
The opposite limit $\alpha_{1}^{d}\delta t\ll 1$ is a bit more subtle as it
depends, additionally, on the number of infectious droplets $D_{1}(t_{0})$ or
infection quanta $Q_{0}$ at the beginning of the infection-risk calculation.

The other observation is that for the short prediction interval $\delta t=1$
day all three calculations predict the same infection risk, irrespective of
the viral latent period. The calculations start to differ for the long
prediction interval, the difference decreasing with the latent period
increasing, i.e., as $\sigma=1/\tau_{\textrm{lat}}\rightarrow 0$. This is
expected as both the GN and WR models assume that the latent period is much
longer than the exposure time. Hence, the number of susceptibles may decrease
due to infection, but the number of infected remains constant since infected
susceptibles move to the infected/non-infectious compartment of the exposed.
If we analyze the time-dependence of the infection risk we note that as the
number of $I$ increase, i.e., before the maximum of the number of infected,
$P^{\textrm{SEDIR}}>P_{\textrm{GN,WR}}^{\textrm{SEDIR}}$, whereas the opposite
holds when $dI/dt<0$. Again this is expected as the GN model considers that
$I=I_{0}$ is constant, whereas the SEDIR does not: when $I$ increase more
infectious droplets are generated than predicted for a constant $I_{0}$
leading to a larger infection probability, and vice versa.

In an attempt to investigate the GN-WR difference we considered an extreme
case of the SEDIR model by shortening the model time scales from days to
hours. The calculated infection risk, not shown, behaved as described in the
previous paragraphs, confirming the initial estimate that even for short
prediction times, e.g., $\delta t=12$ hours, the condition
$\alpha_{1}^{d}\delta t\gg 1$ remained valid. We note that the GN and WR
models may, however, differ in micro-environmental simulations if the
necessary conditions, e.g., $\alpha_{1}^{d}\delta t\ll 1$ are satisfied.

##  V Discussion

We presented a model to calculate epidemic infection risk due to infectious
respiratory pathogens, be they airborne or settled. The model, which is based
on the compartmental epidemiological SEDIR model, may be considered an
epidemiological generalization of the GN [15] model of airborne infection in
an enclosed space. It is valid for arbitrary virus latent period in contrast
to the GN model that assumes a long latent period, and hence it neglects
changes in the number of infected individuals during exposure to the pathogen.
In addition, the SEDIR, being an epidemiological model, provides a connection
between SIR-like epidemiological models and infection-risk models based on
Wells-Riley [10], (WR)-like models. We emphasize the importance of system
scales, since both the GN and WR models, as initially conceived, are
individual-level models that describe infection risk in enclosed micro-
environments. SIR-like models are population level model: in particular, the
SEDIR model is a mesoscopic model that includes explicitly the dynamics of the
pathogen-carrying agents, i.e., the infectious respiratory droplets in the
case or airborne infections such as of COVID-19 or influenza.

We argued that for long virus latent periods the SEDIR reduces to a set of
equations that are reminiscent of the GN equations. Their mapping identified
infection quanta as infectious respiratory particles modified by a scaling
density and, more importantly, by a combination of parameters that include
biological properties of the pathogen (size-dependent pathogen droplet
concentration, probability of infection due to a deposited infectious
droplet), physical properties (lung-deposition probability), and behavioral
properties (exposure time). We noted that the SEDIR model, being an
epidemiological model, depends on the total population $N$, whereas both the
WR and GN models consider much smaller scales in terms of the enclosed volume
$V$. We identified the scaling density as the factor to transition from one
class of models to the other, and we discusses how this density allows an
extension of these micro-environmental models.

We performed numerical simulations of two scenarios for an epidemic specified
by a short and a long virus latent period and driven by two classes of
airborne infectious droplets. Model parameter were based on properties of the
SARS-CoV-2 virus, even though we do not claim to model specifically the SARS-
CoV-2 transmission dynamics with all of its characteristics. However, the
SARS-CoV-2 transmission dynamics reflect those of a range of airborne
infections such as influenza. We used the dynamics of the epidemic,
specifically the time series of the number of susceptible individuals, to
calculate the probability of infection during the epidemic, what we referred
to as epidemic airborne risk. We found that the WR and GN limits of the SEDIR-
model reproduced accurately infection risk as determined from the numerical
solution of the model. Differences arose for large prediction intervals
($\delta t=7$ days), increasing with decreasing virus latent period.

We remark that the WR and GN limiting forms of the epidemic infection risk
were almost identical for all our simulations. This is a consequence of the
droplet removal rate $\alpha_{1}^{d}$ being much greater than the inverse
prediction interval, i.e., $\alpha_{1}^{d}\delta t\gg 1$. In fact, this is a
general result suggesting that with increasing droplet removal rates, for
example via increased ventilation rate, the WR-calculated airborne infection
risk with a steady-state quantum concentration provides an excellent
approximation to the GN-calculated infection risk with non-steady-state
quantum concentrations.

The comparative analysis presented here bridges the gap and provides the
missing links in the mathematical relationship between individual infection
risk models and associated population based models. The corresponding insights
allow for a more nuanced epidemiological interpretation of infectious disease
outbreaks.

###### Acknowledgements.

YD would like to thank the PEACoG (Physical Epidemiology Amherst Covid Group)
members for their many insightful discussions and helpful comments over the
last years. We thank Marguerite Robinson for discussions during the initial
stages of our work, and Vladimir M. Veliov for comments on the connection
between the SDIR and WR models. The views expressed are purely those of the
authors and may not in any circumstances be regarded as stating an official
position of the European Commission.

## Data availability statement

All data supporting the findings of this study are available within the paper
and its Supplementary Information.

## Data availability statement

All data supporting the findings of this study are available within the paper
and its Supplementary Information.

## Author contributions

Y.D. designed research, performed research, analyzed the results and wrote the
article; N.I.S. designed research, analyzed the results, and wrote the
article.

## Competing interest

The authors declare no competing interest

## Funding

The research was performed with institutional support only.

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##  Appendix A Supporting Information

###  A.1 Susceptible-Exposed-Droplet-Infected-Recovered (SEDIR) model
parameters

The droplet population compartments $D_{i}$, number of airborne droplets, and
$C_{i}$, number of settled droplets, are identified by the droplet diameter.
Respiratory droplets are generated in the respiratory tract under conditions
of 100% relative humidity and approximately 37o C degrees. Upon expulsion,
they equilibrate quickly to the local temperature and relative humidity
conditions by water evaporation. As evaporation is a molecular process,
droplet shrinking occurs very rapidly, see, for example Refs. [1, 2, 3], and
the droplet diameter after equilibration is the droplet diameter most often
experimentally accessible. We refer to the droplet diameter at generation as
the pre-evaporation diameter, $d_{i}^{\textrm{pre}}$, and that after
equilibration as the post-evaporation diameter, $d_{i}^{\textrm{post}}$. Their
ratio defines the evaporation factor [4] $\zeta_{\textrm{evap}}$

$d_{i}^{\textrm{post}}=\zeta_{\textrm{evap}}d_{i}^{\textrm{pre}}.$ (13)

The pre-evaporation diameter, via $\rho_{p}$, the pathogen concentration at
the location of droplet generation, e.g., oral region, determines the number
of pathogens $N_{\textrm{path}}^{(i)}$, within a $d_{i}^{\textrm{pre}}$
droplet,

$N_{\textrm{path}}^{(i)}=\rho_{p}(d_{i}^{\textrm{pre}})\times\frac{\pi}{6}\big{(}d_{i}^{\textrm{pre}}\big{)}^{3}=\rho_{p}(d_{i}^{\textrm{pre}})\times\frac{\pi}{6}\big{(}d_{i}^{\textrm{post}}/\zeta_{\textrm{evap}}\big{)}^{3}.$
(14)

The post-evaporation diameter determines the physical properties of the
droplet like the removal rate $\lambda_{\textrm{dep}}$, via gravitational
settling or other surface-deposition processes, and droplet transport
processes. We also consider that it determines their lung-deposition
probability $q_{d_{i}}$. These observation confirm the importance of the
evaporation factor $\zeta_{\textrm{evap}}$, a factor that depends strongly on
the ambient relative humidity. Not only does it determine
$N_{\textrm{path}}^{(i)}$ and the droplet deposition and transport properties,
it also influences viral infectivity, and eventually the viral inactivation
rate $\mu_{d,c}$ in that changes in the droplet diameter lead to changes of
the concentration of the within-droplet species [4]. These concenentration
changes may have important consequences since, for example, increased
concentration of salts, proteins, organics, and acid may damage the pathogen
and modify its infectivity [5, 6].

###  A.2 Transmission rates

The infection transmission rates depend on numerous parameters that may be
categorized as biological, behavioral, or physical. In Ref. [7] we showed that
the transmission rate associated with a $d_{i}^{\textrm{post}}$ droplet, be it
airborne $\beta_{i}^{d}$ or settled $\beta_{i}^{c}$, may be expressed as

$\displaystyle\beta_{i}^{d}$ $\displaystyle=$ $\displaystyle
c\tau_{d_{i}}\times\frac{B}{V_{cl}}q_{d_{i}}\times
p_{d}\times\rho_{p}^{(i)}(d_{i})\times\frac{\pi}{6}\big{(}d_{i}^{\textrm{post}}/\zeta_{\textrm{evap}}\big{)}^{3}\times\epsilon_{i}^{d},\quad
i=1,\ldots,i_{\textrm{max}},\quad{\mbox{{airborne droplets,}}}$ (15a)
$\displaystyle\beta_{i}^{c}$ $\displaystyle=$ $\displaystyle
c\tau_{c_{i}}\times\eta_{c}q_{c_{i}}\times
p_{c}\times\rho_{p}^{(i)}(d_{i})\times\frac{\pi}{6}\big{(}d_{i}^{\textrm{post}}/\zeta_{\textrm{evap}}\big{)}^{3}\times\epsilon_{i}^{c},\quad
i=1,\ldots,i_{\textrm{max}},\quad{\mbox{{settled droplets,}}}$ (15b)

where $i_{\textrm{max}}$ is the total number of droplet compartments as
specified by their post-evaporation diameter. In the main text, we also argued
that the breathing rate $B$ (m3/day) may be factored out in Eq. (15a) to
define

$\tilde{\beta}_{i}^{d}\equiv\frac{\beta_{i}^{d}}{B}=c\tau_{d_{i}}\times\frac{1}{V_{cl}}q_{d_{i}}\times
p_{d}\times\rho_{p}^{(i)}(d_{i})\times\frac{\pi}{6}\big{(}d_{i}^{\textrm{post}}/\zeta_{\textrm{evap}}\big{)}^{3}\times\epsilon_{i}^{d},$
(16)

a parameter that converts infectious respiratory droplets to infection quanta.

The parameters in Eqs. (15) that depend on biological properties are: the
pathogen concentration $\rho_{p}^{(i)}$ at the generation location of droplet
$d_{i}^{\textrm{pre}}$ (number per volume) which we take to be droplet-size
dependent; the probability of infection $p_{d}$ due to a lung-deposited
airborne droplet per pathogen (dimensionless); and the probability of
infection $p_{c}$ due to a settled droplet that has been transferred from a
surface to a susceptible individual facial membranes per pathogen. The
breathing rate $B$ may also be consider a biological parameter, but we prefer
to consider it a physical parameter (see Table 1). Lastly, the infection
recovery rate $\mu_{I}$ (number per day), not present in Eqs. (15), is also a
biologically-determined parameter.

We consider the lung-deposition probability $q_{d_{i}}$ of a
$d_{i}^{\textrm{post}}$ droplet to be a physically determined parameter. The
characteristic personal-cloud volume, the volume surrounding an individual, is
denoted by $V_{cl}$. Recently, Xenakis (2023) [8] referred to the personal-
cloud volume as the “breathing zone volume, i.e., the air volume surrounding a
susceptible occupant and determining their epidemiological status”.

The transmission-rate parameters that depend on an individual’s behavior
include the individual-infectious person average contact rate $c$ (number per
day), and the transfer rate of settled droplets to facial mucus membranes
$\eta_{c}$ (number per day). During each infectious-susceptible encounter, the
susceptible individual is exposed to airborne infectious droplets for a
droplet-depending breathing time $\tau_{d_{i}}$ (days), and to settled
infectious droplets for the duration of a hands-face exposure time
$\tau_{c_{i}}$ (days). The combination of these average exposure times per
contact leads to an average total exposure time to infectious droplets per day
of
$\langle\tau_{\textrm{exp}}\rangle=c\times\sum_{i}^{i_{\textrm{max}}}(\tau_{d_{i}}+\tau_{c_{i}})$.
The parameters $\epsilon_{i}^{d,c}$ include other effects that could modify
the transmission rates and not initially considered in Ref. [7]. For example,
the filtration efficiency of personal protective equipment or face masks is an
important factor that should be included in $\epsilon_{i}^{d}$.

####  A.2.1 Removal rates

The droplet removal rates are _effective_ removal rates of infectious droplets
in that they include virus inactivation in addition to more traditional
removal rates like surface deposition or removal induced by indoor air
ventilation. The removal rates of airborne $\alpha_{i}^{d}$ and settled
$\alpha_{i}^{c}$ droplets of post-evaporation diameter $d_{i}^{\textrm{post}}$
are

$\displaystyle\alpha_{i}^{d}$ $\displaystyle=$
$\displaystyle\big{(}1+c\tau_{d_{i}}\big{)}\frac{B}{V_{cl}}q_{d_{i}}+\mu_{d}+\lambda_{\textrm{dep}}^{i}(d_{i}^{\textrm{post}})+\lambda_{\textrm{air}}+\phi_{i}^{d},\quad
i=1,\ldots,i_{\textrm{max}},\quad\mbox{{airborne droplets,}}$ (17)
$\displaystyle\alpha_{i}^{c}$ $\displaystyle=$
$\displaystyle\big{(}1+c\tau_{c_{i}}\big{)}\eta_{c}q_{c_{i}}+\mu_{c}+\phi_{i}^{c},\quad
i=1,\ldots,i_{\textrm{max}},\quad{\mbox{{settled droplets}}}.$ (18)

Similarly to the infection transmission rates, droplet removal mechanisms may
be associated with behavioral, biological, or physical processes. The first
term in both Eq. (17) and (18) is a self removal term: in the case of airborne
droplets it models removal by inhalation by the susceptible (shown to be
negligible for influenza-related parameters [9]), in the case of settled
droplets is self-transfer of a deposited droplet to facial membranes.The viral
inactivation rate in airborne droplets is denoted by $\mu_{d}$ (number per
day), and that of settled droplets by $\mu_{c}$ (number per day). They are
determined by the properties of the virus under ambient conditions, and hence
a strong function of the relative humidity [4]. The ventilation rate is
denoted by $\lambda_{\textrm{air}}$ (number of air exchanges per day), whereas
the surface deposition rate is denoted by $\lambda_{\textrm{dep}}$ (number of
droplets per day.) In our simulations we considered that the only physical
processes that leads to droplet deposition is gravitational settling,
$\lambda_{\textrm{dep}}^{i}=\theta(d_{i}^{\textrm{post}})$. The parameters
$\phi_{i}^{d,c}$ denote any other process that might induce particle removal:
for example, UV radiation would be an additional viral inactivation mechanism
that would modify $\mu_{d,c}$. Another possible inactivation mechanism would
be indoor spraying of nonhazardous levels of an acid, e.g., nitric, to
decrease droplet pH [5] or spraying a basic solution to increase indoor micro-
environmental conditions to basic [6].

####  A.2.2 Droplet generation rates

Normal oro-nasal activities, like breathing, talking, laughing, singing, and
more violent expiratory events, like sneezing and coughing, produce a
distribution of respiratory droplet sizes. As we try to retain features of the
spread of SARS-CoV-2 we opted to limit the estimate of the droplet generation
rates to normal oro-nasal activities. In addition, we neglect super-spreaders,
or super-emitters, [10]. The generation rates we analyzed are based on
measurements reported by Johnson et al. (2011) [11], see, also, de Oliveira et
al. (2021) [3]. We used the first two distributions [11], B (bronchiolar
droplet generation mode) and L (laryngeal droplet generation mode), to
determine the concentration-weighed droplet diameter $d_{1}^{\textrm{post}}$.
Their emission rate was determined from the reported data for
$\textrm{Cn}_{i}$, the droplet number concentration (number of droplet per
cm3). The droplet concentration was converted to droplet number per second via
the flow rate of the Aerodynamic Particle Sizer (APS) of $5$ lt/min. The
emitted respiratory droplet per second was converted to number of expelled
droplets per day by assuming 1.5 hours of speaking per day (hence the explicit
1.5 in Table 1).

Since the APS measures aerosol particles in the size range $0.50\leq d_{p}\leq
20$, we decided to use the data of Ref. [11] only for the the smaller diameter
$d_{1}^{\textrm{post}}$. The emission rate of the $d_{2}^{\textrm{post}}$
droplets was based on the data of Loudon and Roberts (1967) [12], as described
in Ref. [7], and preserving the total volume of the expelled oral fluid.

Table 1: Simulation parameters: two airborne droplet classes. Parameter | Description | Estimate | Reference
---|---|---|---
Biological Parameters
$\rho_{p}^{(1)}$ | pathogen |  $7.0\times 10^{7}$ #/cm3 | Stadnytskyi et al. (2020) [13]
| concentration ($d_{1}^{\textrm{post}}$) | (viral copies /cm3) |
$\rho_{p}^{(2)}$ | pathogen |  $3.50\times 10^{6}$ #/cm3 | ibid.
| concentration ($d_{2}^{\textrm{post}}$) | (viral copies /cm3) |
$\mu_{I}$ | infection | $1/6=0.1677$ | Kevrekidis et al. (2021) [14]
| recovery rate | (per day) |
$\mu_{d}$ | inactivation | $15.13$ | van Doremalen et al. (2020) [15],
| rate (airborne) | (per day) | Buonanno et al. (2020) [16]
$p_{d}$ | probability of | 0.052 | Drossinos and Stilianakis (2010) [7]
| infection (airborne) | (-) |
$1/\sigma$ | latent period | 0.1 or 6 days | Scenario parameter
Behavioural Parameters
$c$ | contact rate | 18 #/day | Sypsa et al. (2021) [17]
| per day |  |
$\tau_{d_{1}}$ | characteristic breathing | 25 min | Based on
| time ($d_{1}^{\textrm{post}}$) |  | Drossinos and Stilianakis (2010) [7]
$\tau_{d_{2}}$ | characteristic breathing | 1 min | ibid.
| time $(d_{2}^{\textrm{post}}$) |  |
Physical and physiological parameters
$d_{1}^{\textrm{post}}$ | small-droplet diameter | 2.05 $\mu$m | Johnson et al. (2011) [11]
| speaking |  |
$d_{2}^{\textrm{post}}$ | large-droplet diameter | 82.13 $\mu$m | Loudon and Roberts (1967) [12]
| speaking |  |
$\zeta_{\textrm{evap}}$ | evaporation factor | 0.40 (-) | Lieber et al. (2021) [18]
$B$ | breathing rate | 12 m3/day | Drossinos and Housiadas (2006 [19]
$V_{cl}$ | volume personal cloud | 8 m3 | Drossinos and Stilianakis (2010) [7]
$q_{d_{1}}$ | inhaled-droplet | 0.88 (-) | Drossinos and Housiadas (2010) [19]
| deposition probability ($d_{1}^{\textrm{post}}$) |  |
$q_{d_{2}}$ | inhaled-droplet | 1.00 (-) | ibid.
| deposition probability ($d_{2}^{\textrm{post}}$) |  |
$\kappa_{1}^{d}$ | airborne droplet generation rate | $1.5\times 51,182=$ | Johnson et al. (2011) [11]
| speaking (droplets/day)($d_{1}^{\textrm{post}}$) |  $76,773$ #/day |
$\kappa_{2}^{d}$ | airborne droplet generation rate | $0.8\times 47,160=$ | Loudon and Roberts (1976) [12]
| speaking (droplets/day) ($d_{2}^{\textrm{post}}$) |  $37,728$ #/day |
$\lambda_{\textrm{dep}}^{1}=\theta_{1}$ | airbone droplet deposition rate |  $7.8$ #/day | Drossinos and Housiadas (2006) [19]
| still-air gravitational settling ($d_{1}^{\textrm{post}}$) |  |
$\lambda_{\textrm{dep}}^{2}=\theta_{2}$ | settled droplet generation rate |  $10,558$ #/day | ibid.
| still-air gravitational settling ($d_{2}^{\textrm{post}}$) |  |
$\lambda_{\textrm{air}}$ | air exchange rate (AER) | 4.8 exchanges/day | Typical value for Italian buildings
|  |  | Buonanno et al. (2020) [16]
Infection-risk parameter
$\delta t$ | prediction time | 1 or 7 days | estimate

####  A.2.3 Other parameters

All simulation parameters, along with the associated references, are reported
in Table 1. We note that observations [20] and simulations [4] suggest the
importance of the ventilation rate. We chose to use a characteristic value for
typical Italian buildings as reported in Ref. [16], namely
$\lambda_{\textrm{air}}=0.2$ air exchanges per hour.

The evaporation factor $\zeta_{\textrm{evap}}$ was chosen to be $0.40$, an
intermediate value between the recent estimate [18] of $0.20$ and our initial
estimate [7] of $0.50$.

The viral inactivation rate in airborne droplets was based on the early
measurements of van Doremalen et al. (2020) [15]. It is frequently quoted [16,
21] as the removal rate in terms of the viral half-life $t_{1/2}$ as
$\lambda_{\textrm{inact}}=\ln(2)/t_{1/2}$: both references used
$\lambda_{\textrm{inact}}=0.63$ per hour gives $\mu_{d}=38$ per day. Our
estimate is slightly smaller.

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