# Optimal Workplace Occupancy Strategies during the COVID-19 Pandemic

Mansoor Davoodi<EMAIL_ADDRESS>Abhishek Senapati
<EMAIL_ADDRESS>Adam Mertel<EMAIL_ADDRESS>Weronika Schlechte-Welnicz
<EMAIL_ADDRESS>Justin M. Calabrese<EMAIL_ADDRESS>Center
for Advanced Systems Understanding (CASUS), G$\ddot{o}$rlitz, Germany
Helmholtz-Zentrum Dresden-Rossendorf (HZDR), Dresden, Germany  Helmholtz
Centre for Environmental Research-UFZ, Leipzig, Germany  Dept. of Biology,
University of Maryland, College Park, MD, USA

###### Abstract

During the COVID-19 pandemic, many organizations (e.g. businesses, companies,
government facilities, etc.) have attempted to reduce infection risk by
employing partial home office strategies. However, working from home can also
reduce productivity for certain types of work and particular employees. Over
the long term, many organizations therefore face a need to balance infection
risk against productivity. Motivated by this trade-off, we model this
situation as a bi-objective optimization problem and propose a practical
approach to find trade-off (Pareto optimal) solutions. We present a new
probabilistic framework to compute the expected number of infected employees
as a function of key parameters including: the incidence level in the
neighborhood of the organization, the COVID-19 transmission rate, the number
of employees, the percentage of vaccinated employees, the testing frequency,
and the contact rate among employees. We implement the model and the
optimization algorithm and perform several numerical experiments with
different parameter settings. Furthermore, we provide an online application
based on the models and algorithms developed in this paper, which can be used
to identify the optimal workplace occupancy rate for real-world organizations.

###### keywords:

COVID-19, Pandemic, Optimal Presence Strategy, Productivity, Infection

††journal: arXiv

##  1 Introduction

The COVID-19 pandemic has caused worldwide social and economic disruptions
over the past two years. One consequence has been that companies,
organizations, and government facilities (hereafter “organizations”) have
suffered reductions in the quality and quantity of the services they provide
[1, 2, 3, 4, 5]. This decreased efficiency is a direct consequence of the need
to manage infection risk in the workplace. As COVID-19 spreads mostly through
direct/close contacts between individuals, the risk of infection tends to
increase with workplace occupancy rate. To manage infection risk, many
organizations have implemented a range of strategies including: (i) adopting
recommended personal hygiene measures (e.g., wearing a mask, social
distancing, etc.), (ii) reducing the number of employees present in the
workplace via teleworking, (iii) screening employees via periodic testing, and
(iv) encouraging employees to get vaccinated [6, 7, 8]. Though these
strategies have worked quite well to manage workplace infection risk, they
also come with certain costs and limitations. For example, telework has the
potential to completely eliminate workplace infection risk, but it may also
decrease productivity for particular tasks or individuals, and may not
possible at all for certain types of work [1, 9]. It is therefore evident that
a trade-off between productivity and infection risk exists, and an important
question that arises here is “what is the optimal presence rate in the
workplace during the pandemic?”.

Motivated by these circumstances, we develop a model of disease transmission
within organizations that allows us to balance the trade-off between
productivity and infection risk. Our model incorporates the local incidence
level and inherent transmission rate of COVID-19 as well as key within-
organization factors such as the testing frequency, rate of vaccination, and
contact rate among the employees. Additionally, our model accounts for
differences in efficiency between working remotely and being present in the
workplace. This factor varies from organization to organization depending on
the type of work they do and their inputs, processes and outputs. Hereafter,
we refer to the ratio of efficiency at home to efficiency at office as the
productivity for simplicity.

We employ a probabilistic framework that accounts for the aforementioned
influential factors and allows us to compute the expected number of infected
employees at any time after an infection is introduced into the facility. This
framework has the advantages of being simple, fast, and practical, while still
being customizable for specific organizations. Moreover, our model can be
efficiently used for any organization size. Based on this model, we also
develop an optimization application to find Pareto optimal solutions of the
twin objectives of maximizing productivity and minimizing infection risk in
organizations. The application allows the user to adjust all input parameters
and immediately visualize the results of any change.

This paper is organized in five sections. In the second section, we review
related studies. In the third section, we introduce the core model parameters
and formally state the problem. In addition, we propose the probabilistic
model and an algorithm to compute infection risk and the Pareto optimal
solutions. In the fourth section, we perform several numerical experiments and
test the ability of the model and algorithm to identify Pareto optimal
solutions. In the fifth and final section, we draw conclusions and outline
future directions.

##  2 Related Work

Managing infection risk within organizations has been recognized as a key
theme in responding to and living with the pandemic. Consequently, many
mathematical models have been developed specifically to investigate several
aspects of transmission mechanisms and control strategies for COVID-19 in
organizations [10, 11, 12, 13, 14]. In addition, several quantitative studies
have also been carried out to gain insight on the outcomes of strategies
proposed for different organization types[15, 16, 17, 18]. Most of these
studies focused on implementing strategies for organizations like hospitals,
health care facilities, nursing homes, offices, and schools. For example, [19,
20] have focused on identifying the optimal frequency of testing in healthcare
environments and nursing homes, while others have demonstrated that immediate
testing of symptomatic individuals and quarantining of their primary contacts
via contact tracing can potentially reduce disease transmission among health
workers and other high risk groups [21]. Still other surveillance-focused
studies have explored the performance of non-adaptive combinatorial group
testing [22], and the importance of self-isolation and contact tracing
measures [23]. Finally, [24] proposed an optimal testing strategy to minimize
the presence of pre-symptomatic and asymptomatic employees in the workplace.

Some studies have investigated how scheduling personnel in the workplace can
minimize the consequences of infection. In this context, a desynchronization
strategy has been proposed in which the workers in the healthcare system are
divided into two non-overlapping teams who will be working in alternating
weeks [25]. Similarly, [26] studied the effectiveness of regular testing and a
desynchronization protocol in preventing COVID-19 infection in hospitals while
accounting for both internal and external sources of infection. The effect of
a cyclic 4-day work and 10-day lockdown strategy in the workplace has been
investigated by [27]. In nursing homes and long-term care facilities, [28]
constructed a mathematical model based on bipartite networks consisting of
health care workers and facility residents to investigate how restructuring
interactions via the concept of shied immunity can change outbreak dynamics.
Specifically, shield immunity refers to the practice of proportionally
increasing the interactions between recovered and susceptible individuals to
protect the susceptible from infection. Additionally, [29] developed a
decision support system for different employment sectors like agriculture,
manufacturing, construction, and human health in Italy and measured the
occupational risk of infection in workplace for each sector based on type of
work-activity, involvement of third parties in the work processes, and risk of
social aggregation.

Several recommendations based on mathematical models have been proposed in
reopening different activities in workplaces. For instance, some studies have
explored how enhanced levels of testing, contact-tracing, adoption of facial
masks, and home quarantine can facilitate relaxation of social distancing
interventions [30, 31]. Similarly, [32] studied how designing an optimal
employee screening strategy can reduce workplace infections, as well as how
such a strategy affects the feasibility of return to work policies.

From the above discussion, it is clear that most of these studies have focused
on modeling the impact of implementing different infection reduction measures
in the workplace. Additionally, some studies have recommended strategies for
personnel scheduling and reopening different activities. However, no study to
date has modeled the basic mechanics of workplace disease transmission while
accounting for testing frequency, the contact rate, the vaccination rate, and
the differing efficiencies of home vs office work.

##  3 Modeling and Solution Approach for the Workplace Presence Problem

While teleworking has advantages and disadvantages [33, 34], the COVID-19
pandemic has forced many organizations to massively increase the amount of
remote work they allow. The realized productivity of employees working from
home depends strongly on the type of tasks they perform [35]. However, for
many job types (e.g., hospitality, banking, information technology)
productivity increases with the physical presence of employees in the
workplace [36]. So, while working from home clearly reduces infection risk, we
conclude that it will also decrease productivity for many types of work [37,
38]. Therefore, a trade-off, mediated by workplace presence rate, will often
exist between productivity and infection risk.

In this section, we formulate this trade-off as a bi-objective optimization
problem to find the optimal rate of workplace presence while also considering
the risk of infection at the facility. Notably, if the employee productivity
ratio of some organization is not decreased by telework (e.g., due to a 100%
digital workflow), then clearly no trade-off between workplace infection risk
and productivity exists and the optimal (trivial) strategy is minimum
workplace occupancy. While some chance of infection for employees working from
home still exists, organizations do not have direct control over this, and
there is no disease spread in the workplace. Thus, in this paper, we only
consider organizations for which total productivity increases with increasing
workplace occupancy.

Table 1 defines the model’s key parameters, variables, and assumptions.

Table 1: The parameters and variables used in the proposed model. Notation | Definition
---|---
$n$ | The total number of employees
$n_{v}$ | The number of vaccinated employees
$n_{I}$ | The number of infected employees
$\beta_{u}$ | The COVID-19 transmission rate for unvaccinated individuals
$\beta_{v}$ | The COVID-19 transmission rate for vaccinated individuals
$prod$ | (home) Productivity ratio
$\tau$ | The average interval between tests of the same employee
$\rho$ | Probability of an infection arriving at the facility
$\kappa$ | The average number of contacts per employee per day
$occup$ | The workplace occupancy rate

Parameters, Variables, and Assumptions of Model

  * 1.

$n$: The total number of employees who regularly work in the facility

  * 2.

$n_{v}$: The number of employees who are (fully) vaccinated with one of the
available COVID-19 vaccines. We count partially vaccinated individuals (e.g.,
those who have only had one dose) as non-vaccinated.

  * 3.

$n_{I}$: The expected number of infected employees that can spread the disease
in the facility. As soon as they test positive or the disease’s symptoms
appear, they are quarantined and no longer counted as an active employee.

  * 4.

$\beta_{u}$ and $\beta_{v}$: The per contact probability of virus transmission
from one infected individual to one unvaccinated ($\beta_{u}$) or one
vaccinated ($\beta_{v}$) individual.

  * 5.

$prod$: The productivity ratio is the productivity of working at home compared
to working at the office. This factor varies from facility to facility and
from employee to employee, based on their mission and the type of services
they provide. Managers can determine this factor based on their experience in
working under the COVID-19 pandemic and comparing it with normal situations.
Precisely,
$prod=\frac{Productivity~{}of~{}working~{}from~{}Home}{Productivity~{}of~{}working~{}at~{}Office}$.
So, $prod=1$ means that there is no difference between productivity for
working at home and workplace, and $prod=0$ means the employees has no
productivity when they work from home.

  * 6.

$\tau$: The time interval between two sequential tests of the same employee,
averaged over all employees. As the symptoms of COVID-19 appear within two
weeks [39, 40], we assume $\tau$ varies from one day to 14 days (incubation
period). In order to keep the model as simple and parameter sparse as
possible, we assume perfect tests with no false positives or negatives. This
would not qualitatively change the results, and we could easily incorporate
the testing error as a coefficient in computing the expected time to detect
the infection.

  * 7.

$\rho$: The per day probability that an infection arrives at the facility.
This can be computed by considering the number of employees in the facility
and the incidence level in the neighborhood of the facility. Specifically, we
estimate this probability based on the reported number of infections over the
last 7 days in the focal region (e.g., see
https://www.coronavirus.sachsen.de).

  * 8.

$\kappa$: The average number of contacts per employee per day. We assume the
employees and clients mostly wear masks and keep a distance of at least 1.5
meters.

  * 9.

$occup$: The rate (percentage) of employees present in the workplace. This is
the only decision variable of the model, and should be determined optimally.

As previously mentioned, all variables and parameters are given except the
decision variable $occup$, which is the output of the model. Furthermore, we
divide the employees into three groups: ($i$) those who work at the facility,
($ii$) those who work from home, and ($iii$) those who are infected. The
productivity ratio of the first group is one, and those of second and third
groups are $prod$ and zero, respectively. So, $n-n_{I}$ is the number of
healthy and active employees and of them $occup$ percent are at the workplace
with full productivity, and the remaining $1-occup$ percent work from home
with productivity $prod$. Therefore, the total productivity of an organization
can be determined by the following formula111It is possible to customize this
formula and define it based on the outcomes in a company if a clear definition
is available. In this paper, we skip such details and only focus on this basic
and general definition of the total productivity.

$Total~{}Productivity=occup\times(n-\mathbb{E}(n_{I}))+prod\times(1-occup)\times(n-\mathbb{E}(n_{I}))$
(1)

Therefore, one objective function of the model is maximizing the total
productivity, and the other one is minimizing the expected number of
infections, $\mathbb{E}(n_{I})$. There is a clear conflict between these two
objectives. Note that, $prod$ is the productivity ratio determined by the
decision-maker. Without loss of generality, we consider $prod\in[0,1)$. Note
that, for $prod\geq 1$, there is a trivial optimal solution $occup=0$, that is
working from home is strictly preferred to working at the office, and it
results in minimizing the number of infected employees. The most important
part of the objective functions is computing the expected number of infected
employees, $\mathbb{E}(n_{I})$, after the arrival of an infection. Indeed, we
must compute it as a function of time and of the influential parameters
mentioned in Table 1, which are explained in detail below.

###  3.1 Probabilistic framework for computing the number of infected
employees

We start with a full contact network of unvaccinated employees to explain our
approach. We then extend it to accommodate both vaccinated and unvaccinated
employees. All possible pairs of employees are equally likely to contact each
other, but the average number of contacts per employee per day is bounded by
$\kappa$. Suppose an infection arrives at the facility at time (day) $t=0$. We
then compute the probability of an arbitrary individual being infected after
$\Delta t$ days, $P_{I}(\Delta t,n,\beta_{u},\kappa)$, when the disease
transmission rate is $\beta_{u}$ and the number of contacts per day is
$\kappa$. This probability function allows one to easily compute the expected
number of infected individuals as

$\mathbb{E}\left[(n_{I}(\Delta t,n,\beta_{u},\kappa)\right]=1+(n-1)\times
P_{I}(\Delta t,n,\beta_{u},\kappa).$ (2)

For simplicity, we hereafter use $P_{I}(\Delta t)$ to refer to the probability
function $P_{I}(\Delta t,n,\beta_{u},\kappa)$ for known values of $n$ and
$\kappa$. Thus, computing the probability of infection per employee leads
directly to the expected number of infected employees. Let us denote the
source of infection at $t=0$ by $s$, and let $u$ be an employee who stays
healthy until $\Delta t-1$. There are then two ways $u$ can be infected on day
$\Delta t$; via some direct contact with $s$, or via contact with one of the
$n-2$ other employees. The transmission probability for each of the contacts
is $\beta_{u}$, and the probability of infection for the source is one, while
the probability of infection for the other employees is $P_{I}(\Delta t-1)$
(Figure 1). This means that the probability that an arbitrary employee like
$u$ stays healthy after $c$ contacts with $s$ is $(1-1\times\beta_{u})^{c}$.
Similarly, the probability that $u$ remains healthy after $c$ contacts with
the other $n-2$ employees on day $\Delta t$ is $(1-P_{I}(\Delta
t-1)\times\beta_{u})^{c}$. These relationships imply that the infection
probability can be calculated as a recursive function. Since we assume
$\kappa$ contacts for each employee per day and all of them have an equal
chance to occur, this function can be written as

$P_{I}(\Delta t)=1-(1-P_{I}(\Delta
t-1))\times\left[(1-\beta_{u})^{\kappa\frac{1}{n-1}}\times(1-P_{I}(\Delta
t-1)\beta_{u})^{\kappa(1-\frac{1}{n-1})}\right],$ (3)

where $\kappa\frac{1}{n-1}$ is the expected number of contacts between
employee $u$ and the infected source employee $s$, and
$\kappa(1-\frac{1}{n-1})$ is the expected number of contacts between employee
$u$ and the other employees except $s$. Note that, $(1-P_{I}(\Delta t-1))$ is
the probability that the employee stays healthy until day $\Delta t-1$.

Figure 1: Possible infection pathways in $n$ employees of which one of them
($s$) is infected.

Eq. 3 computes the probability that an arbitrary employee gets infected after
$t=\Delta t$ days when the employees have $\kappa$ contacts per day with full
workplace occupancy (i.e., $occup=1$). For $0\leq occup\leq 1$, all employees
(including $s$) stay at home (and healthy) with probability $1-occup$. So the
probability of one contact between $u$ and $s$ occurring is $occup^{2}$.
Therefore, the infection probability formula for known $n$, $\kappa$ and
$occup$ can be extended as follows

$P_{I}(\Delta t)=1-(1-P_{I}(\Delta
t-1))\left[(1-\beta_{u})^{occup^{2}\times\kappa\frac{1}{n-1}}\times(1-P_{I}(\Delta
t-1)\beta_{u})^{occup\times\kappa(1-\frac{1}{n-1})}\right].$ (4)

Also, we set $P_{I}(0)=0$ without loss of generality as the scenario we assume
here. After computing the probability of infection, the expected number of
infected employees can be straightforwardly computed by applying Eq. 2.

Figure 2: Infection possibilities of $n$ employees of which one of them ($s$)
is infected and there are two groups of vaccinated and unvaccinated employees.
The left subfigure shows the scenario for an unvaccinated employee $u$ and the
right one shows the scenario for a vaccinated employee $v$.

Now, let us extend the above computation to two different groups, namely
vaccinated and unvaccinated employees. We assume the disease transmission
probability for the vaccinated group is $\beta_{v}~{}(\beta_{v}\ll\beta_{u})$.
Without loss of generality, we consider the source of infection $s$ as an
unvaccinated employee and compute the probability of infection in two cases:
(i) for an unvaccinated employee $u$, and (ii) for a vaccinated employee $v$
(see Fig. 2). There are three ways of infecting $u$: (i) via a direct contact
with $s$, (ii) via contact with one of the $n_{v}$ vaccinated employees, or
(iii) via contact with one of the $n-n_{v}-2$ unvaccinated employees. The
transmission probability for each of these contacts is $\beta_{u}$ because $u$
is an unvaccinated employee, however, the transmission probability for the
vaccinated group is $\beta_{v}$. Let $P^{u}_{I}(\Delta t-1)$ and
$P^{v}_{I}(\Delta t-1)$ be the probability of infection for an unvaccinated
employee and a vaccinated employee, respectively. Thus, the recursive
equations for each group can be written as

$P^{u}_{I}(\Delta t)=1-(1-P^{u}_{I}(\Delta
t-1))\left[(1-\beta_{u})^{occup^{2}\times\kappa\frac{1}{n-1}}\times\left(1-P^{u}_{I}(\Delta
t-1)\beta_{u}\right)^{occup\times\frac{n-n_{v}-1}{n-1}\times\kappa(1-\frac{1}{n-1})}\times\left(1-P^{v}_{I}(\Delta
t-1)\beta_{v}\right)^{occup\times\frac{n_{v}}{n-1}\times\kappa(1-\frac{1}{n-1})}\right],$
(5)

and

$P^{v}_{I}(\Delta t)=1-(1-P^{v}_{I}(\Delta
t-1))\left[(1-\beta_{v})^{occup^{2}\times\kappa\frac{1}{n-1}}\times\left(1-P^{u}_{I}(\Delta
t-1)\beta_{v}\right)^{occup\times\frac{n-n_{v}-1}{n-1}\times\kappa(1-\frac{1}{n-1})}\times\left(1-P^{v}_{I}(\Delta
t-1)\beta_{v}\right)^{occup\times\frac{n_{v}}{n-1}\times\kappa(1-\frac{1}{n-1})}\right],$
(6)

where $P^{u}_{I}(0)=P^{v}_{I}(0)=0$. One advantage of these recursive formulas
is that $P^{u}_{I}(\Delta t)$ and $P^{v}_{I}(\Delta t)$ can be efficiently
computed in linear time to $\Delta t$ using a simple bottom-up approach.
Finally, the expected number of infected employees $\Delta t$ days after an
infection is introduced can be computed as

$\mathbb{E}\left[n_{I}(\Delta
t,n,n_{v},\beta_{u},\beta_{v},\kappa,occup)\right]=1+(n-n_{v}-1)\times
P^{u}_{I}(\Delta t)+n_{v}\times P^{v}_{I}(\Delta t).$ (7)

Thus, Eq. 7 provides a recursive formula to compute the number of infections
over time. An implementation of this equation and comparison with simulation
results is presented in the Appendix.

###  3.2 Computing optimal presence rate

As explained, the expected number of infected employees $\Delta t$ days after
an infection arrives at the facility can be computed using Eq.7. For
simplicity, we denote this expected number by $\mathbb{E}\left[n_{I}(\Delta
t)\right]$. Let $\rho$ be the per day probability that an infection arrives at
the facility. This probability can be determined using two straightforward
approaches. First, using the (recent) historical data of incidence in the
company, and second, using the local incidences and applying the ratio of the
vaccination rate in the neighborhood of the facility to the vaccination rate
in the company. Therefore, the cumulative number of infected employees for a
time interval $T$ days can be computed as follows

$\mathbb{E}(n_{I})=Accum_{I}(T,\rho)=\rho\times Z(T),$ (8)

where $Z(T)$ is defined

$Z(T)=\left\\{\begin{array}[]{rcl}\mathbb{E}\left[n_{I}(0)\right]+\mathbb{E}\left[n_{I}(1)\right]-\frac{\mathbb{E}\left[n_{I}(1)\right]\times\mathbb{E}\left[n_{I}(0)\right]}{n}&\mbox{,
if}&T=1\\\
\mathbb{E}\left[n_{I}(T)\right]+Z(T-1)-\frac{\mathbb{E}\left[n_{I}(1)\right]\times
Z(T-1)}{n}&\mbox{, if }&T\geq 2\end{array}\right.$ (9)

Note that $Z(T)$ is a linear recursive formula to compute the cumulative
number of infections for $t=0,1,\dots,T$ by removing the expected overlapping
infected employees over time.

Eq. 8 computes the total number of infected employees over a time interval $T$
days. If employees are tested every $\tau$ days on average, and they uniformly
distribute in the test interval (having $k=\frac{n}{\tau}$ tests per day on
average), the expected time to detect the infection can be computed as

$\overline{\tau}=\left[1-{(1-Pr(t))}^{k}\right]+\sum_{t=2}^{\tau}\left[t\times(1-{(1-Pr(t))}^{k})\times\prod_{t^{\prime}=1}^{t-1}{(1-Pr(t^{\prime}))}^{k}\right],$
(10)

where $Pr(t)=\frac{\mathbb{E}\left[n_{I}(t)\right]}{n}$ is the probability of
infection per employee $t$ days after an infection arrives at the facility,
so, ${1-(1-Pr(t))}^{k}$ is the probability of detecting at least one infected
employee by testing a subgroup of $k$ after $t$ days. For detecting the
infection $t>1$ days after of arrival, we must compute the probability that it
has been not detected in the days before $t$, i.e. $t^{\prime}<t$, which is
computed using $\prod_{t^{\prime}=1}^{t-1}{(1-Pr(t^{\prime}))}^{k}$.

$\overline{\tau}$ in Eq.10 is the expected time to detect an infection if the
employees are tested every $\tau$ days on average. Therefore, the expected
number of infected employees before detecting an infection is obtained by
$\mathbb{E}\left[n_{I}(\tau,n,n_{v},\beta_{u},\beta_{v},\kappa,occup)\right]$.
So, the total productivity (see Eq.1) of a facility can be determined for any
given $occup$ in $[0,1]$, and as a function of the other parameters.

###  3.3 A practical multi-objective solution approach

As previously described, the optimal workplace presence strategy is formulated
in the framework of a bi-objective optimization problem as follows

$\displaystyle Minimize~{}~{}~{}\mathbb{E}(n_{I}),$ (11) $\displaystyle
Maximize~{}~{}~{}Total~{}Productivity,$
$\displaystyle~{}~{}~{}~{}Subject~{}to:$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}occup\geq
An~{}Occupancy~{}Threshold.$

The first objective is minimizing the expected number of infected employees in
Eq. (8), and the second objective is maximizing the total productivity of the
organization via Eq. (1). The only constraint that we considered in the model
is related to the minimum possible workplace occupancy, the Occupancy
Threshold, which is the minimum number of employees that must be present at
the facility to perform tasks for which physical presence is required. It is
also possible to include additional lower or upper bound constraints on the
occupancy variable. The model has two conflicting objectives. Clearly, by
increasing workplace occupancy, the expected number of infected employees will
increase as well. This is, however, not the case for the second objective,
because by increasing occupancy, the number of infected employees also
increases, but total productivity will eventually decrease as the productivity
of infected employees is zero. Figure 3 shows the effect of occupancy on the
first and second objective separately for the scenarios of one test per week
(blue curve) and one test per two weeks (red curve). As expected, the number
of infections is a strictly increasing function of occupancy rate, while, the
productivity function is increasing for the low occupancy, and after reaching
the maximum productivity (e.g., for $occup\approx 88\%$ in the first scenario
and for $occup\approx 65\%$ in the second one), then becomes a decreasing
function of occupancy.

Figure 3: Effect of occupancy on the expected number of infections (left
panel) and on the total productivity (right panel) for one test per week and
one test per two weeks. In these simulations, we assumed an organization with
100 employees of which half are vaccinated, with 15% contact rate, and the
home productivity ratio 0.90.

The outcome of the optimization problem presented in LABEL:bi_obj_model is a
set of trade-off, or Pareto optimal solutions, which are defined as solutions
that are not improved for an objective unless it sacrifices the other
objective [41]. There are different approaches to solve multi-objective
optimization problems, such as the weighted-sum method [42], the lexicographic
method [43], the $\epsilon-constraint$ approach [44], goal programming [45],
or evolutionary algorithms [46, 41]. All of these approaches have their
advantages and disadvantages. Some of them have high time complexity, others
work only for differentiable functions or convex models, while some provide no
guarantee of finding all the Pareto-optimal solutions. We therefore propose a
quick and practical method to compute all Pareto optimal solutions of the
model LABEL:bi_obj_model efficiently.

The method utilizes some observations to solve the model. First, $occup$ is a
continuous variable, but in reality, there are at most $n$ possible choices
for the number of employees present in the facility. So initially, for any
$occup\in\\{0,\frac{1}{n},\frac{2}{n},\dots,\frac{n}{n}\\}$ which satisfy the
occupancy threshold, we compute the expected detection time based on Eq. (10),
and the expected number of infected employees and its corresponding total
productivity based on Eq. (7) and Eq.(1), respectively. All of these steps can
be handled in $O(n\tau)$ time using the presented linear recursive equations.
We denote this solution set by $S$. Now, we can find the non-dominated
solutions of $S$ and report them as the Pareto-optimal solutions of the
problem. A solution $\bar{s}\in S$ is a non-dominated solution if there is no
other solution in $S$ such that it is better than $\bar{s}$ in both
objectives, i.e., its expected number of infections is less than the expected
number of infections of $\bar{s}$, and simultaneously, its productivity is
more than the productivity of $\bar{s}$. Since the solutions of $S$ are
constructed one by one, from the minimum expected infections to the maximum
one, the Pareto-optimal solutions of the problem can be computed in linear
time using a sweep-line approach [47]. The pseudocode of the proposed
algorithm is presented as follows.

Algorithm 1 Computing Pareto Optimal Workplace Presence Strategies

Input: Organizations’s Parameters (the notation mentioned in Table 1)
Output: All Pareto Optimal Strategies

$max\\_prod\leftarrow-1$

$i\leftarrow 0$

while $i\leq n$ do

$occup\leftarrow\frac{i}{n}$

if $occup\geq Occupancy\\_Threshold$ then

Compute $\overline{\tau}$ using Eq. (8)

Compute expected number of infections using Eq. (7)

Compute the total productivity using Eq. (1) and denote it by $TP$

if $TP>max\\_prod$ then

Report the current strategy as a Pareto optimal strategy

$max\\_prod\leftarrow TP$

end if

end if

$i\leftarrow i+1$

end while

##  4 Simulation Results and Discussions

In this section, we show some numerical results from the proposed model and
method for computing Pareto optimal solutions that minimize the expected
number of infections while maximizing productivity. Since there are several
influential input parameters in the proposed model, there are too many
possible combinations of them to consider exhaustively. We therefore briefly
display the output of the model for some diverse set of possible inputs, and
instead, we provide an online optimization tool based on the model to allow
interested readers to consider all possible combinations of the model
parameters as input and observe the output. A user-friendly version of this
tool is available at where2test.de/optimization.

In our simulations, we consider a mid-sized organization with $n=100$
employees and suppose the 7-day incidence rate per 100,000 population is 500.
Thus, the average probability of infection per employee in a week will be
$\frac{500}{100,000}$, and consequently, the probability that an infection
arrives at the facility is $\rho=1-(1-\frac{500}{100000})^{n}$. We
specifically consider two scenarios based on the disease transmission rate for
unvaccinated employees ($\beta_{u}$). First, we choose the baseline value as
$\beta_{u}=0.04$ from a possible range of values reported in a previous study
[48]. This baseline scenario can be regarded as the situation where Delta is
the dominant SARS-CoV-2 variant. We then consider the situation where the
transmissibility increases 2.5-fold (i.e. $\beta_{u}=0.1$), which is
consistent with the recently identified Omicron variant [49, 50, 51].
Additionally, we set the transmission rate of vaccinated employees to
$\beta_{v}=(1-0.80)\beta_{u}$, which implies 80% immunity for vaccinated
individuals [52, 53, 54, 55].

Furthermore, we assume two different values each for the home productivity
ratio ($prod=0.6$ and $prod=0.9$), the vaccination rate ($0.4$ and $0.8$), the
test interval ($\tau=7$ and$\tau=14$ days), and the number of contacts (low:
$\kappa=5+0.10\times occup\times n$, and high: $\kappa=5+0.20\times
occup\times n$). For $\kappa$, the low contact scenario assumes every employee
in the workplace has at least 5 contacts plus additional contacts totalling
10% of the employees present. Likewise, the high rate scenario assumes the
number of contacts is at least 5 contacts plus additional contacts totalling
20% of the employees present. Figure 4 displays the result of these 15
scenarios for different setting of input parameters, whereas Table 2 details
the 15 input settings.

Each subfigure in Figure 4 shows Pareto optimal solutions and their objective
values. The horizontal axis shows the possible occupancy values. The blue
diagram and the left vertical axis illustrate the resulting total
productivity, and the red diagram and the right vertical axis illustrate the
expected number of infected employees per week. Note that the Pareto optimal
solutions are displayed only for the occupancy values which result in such
solutions, and the dominated solutions are not displayed. These results
together provide useful information for a decision-maker who may consider a
maximum infection risk threshold and try to find the maximum productivity
possible without exceeding that threshold by changing parameters such as the
vaccination rate, test interval, or contact rate.

For example, when the 7-day incidence is 500 individuals per 100,000
population (this is almost the average number of incidences in Saxony, Germany
from January first till mid February 2022, e.g., see https://www.
where2test.de/saxony), the background risk of infection per employee is
$\frac{5}{1000}$ per week, and for the whole organization it is
$1-(1-\frac{500}{100000})^{100}\approx 0.4$. The background risk can be
interpreted as the risk of infection if the employees do not come to the
office and maintain normal social contacts by, e.g., going to restaurants and
shopping. When the number of incidences is intermediate, decision-makers can
use the background risk as a reference point in determining a proper threshold
for the risk of infection in the company. The horizontal green line in each of
the subfigures displays the corresponding background risk.

For example, if a decision-maker would like to follow a workplace presence
strategy where within-organization risk never exceeds the background risk,
he/she can consider the occupancy corresponding to the intersection point
between the background risk and the computed infection risk, i.e., the green
dashed line and the red diagram. For instance, such occupancy in the first
scenario in Figure 4($a$)
($\tau=7,prod=0.6,vaccination~{}rate=0.5,~{}Low~{}contact~{}rate$ and
$\beta_{u}=0.04$) corresponds with occupancy 64%, and it will result in 85%
productivity, while in the second scenario, Figure
4($o$)($\tau=14,prod=0.9,vaccination~{}rate=0.8,~{}High~{}contact~{}rate$ and
$\beta_{u}=0.10$), it is 46% with more than 94% productivity.

In a reverse usage of the Pareto optimal solutions, a decision-maker may wish
to know what is the risk of infection (e.g., compared to the background risk)
if he/she would like to achieve a particular level of productivity in the
company. For instance, in the second scenario,
4($b$)($\tau=7,prod=0.6,vaccination~{}rate=0.8,~{}Low~{}contact~{}rate$ and
$\beta_{u}=0.04$), a level of 70% productivity can be achieved with only half
of the background risk, i.e., 0.2, resulting in only 25% of employees present
in the workplace. To compute this, we first find the intersection point of 70%
productivity with the productivity curve (blue). In the second scenario, it
results in almost 25% occupancy. Then find the infection risk on the red curve
which corresponds with such occupancy, which in this case is almost 0.2.

Finally, decision makers can use several curves together to get a sense of the
productivity or infection risk associated with changing key parameters, such
as test interval and/or vaccination rate among employees. For instance, if the
current situation in an organization is similar to the $13^{th}$ scenario (see
Figure 4($l$)), then 54% occupancy will result in infection risk equal to the
background risk with 81% productivity. Now, if the test interval among the
employees rises to $\tau=14$ days, (see 15-th scenario, Figure4($l$)), and the
decision maker still would like the risk of infection in the organization to
not exceed the background risk, he/she has to decrease occupancy to almost
46%, which results 78% productivity. Similar analyses can be performed to
investigate the effects of other input parameters. As prevously mentioned,
because of the extremely large number of possible combinations of the input
parameters, it is not possible to concisely discuss all sensitivity analyses
and observe changes in the objectives across the full parameter space.
Instead, we suggest using the online application we provide for this purpose.

Table 2: 15 different settings of input parameters. The corresponding Pareto optimal solutions are depicted in Figure 4. Figure | $\tau$ | prod | vaccine rate | contact rate | $\beta_{u}$
---|---|---|---|---|---
Fig. 4(a) | 7 | 0.6 | 0.5 | Low | 0.04
Fig. 4(b) | 7 | 0.6 | 0.8 | Low | 0.04
Fig. 4(c) | 14 | 0.9 | 0.5 | Low | 0.04
Fig. 4(d) | 14 | 0.6 | 0.8 | Low | 0.04
Fig. 4(e) | 7 | 0.9 | 0.5 | High | 0.04
Fig. 4(f) | 14 | 0.6 | 0.5 | High | 0.04
Fig. 4(g) | 14 | 0.9 | 0.5 | High | 0.04
Fig. 4(h) | 7 | 0.9 | 0.5 | Low | 0.1
Fig. 4(i) | 7 | 0.9 | 0.8 | Low | 0.1
Fig. 4(j) | 14 | 0.6 | 0.5 | Low | 0.1
Fig. 4(k) | 14 | 0.9 | 0.8 | Low | 0.1
Fig. 4(l) | 7 | 0.6 | 0.8 | High | 0.1
Fig. 4(m) | 14 | 0.9 | 0.5 | High | 0.1
Fig. 4(n) | 14 | 0.6 | 0.8 | High | 0.1
Fig. 4(o) | 14 | 0.9 | 0.8 | High | 0.1

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

Figure 4: Pareto optimal solutions for 15 different setting of input
parameters. See Table 2. The horizontal axis shows the possible occupancy. The
blue curve and the left vertical axis illustrate the obtained total
productivity, and the red curve and the right vertical axis illustrate the
expected number of infected employees per week.

##  5 Conclusion and Future Directions

In this study, we developed a model to compute the optimal trade-off solutions
for minimizing the risk of infection and maximizing the productivity in
organizations during the COVID-19 pandemic. First, we proposed a probabilistic
approach to compute the expected number of infected employees over the time by
incorporating basic influential parameters such as the local incidence level,
number of contacts among the employees, the average test interval, and the
vaccination rate. This basic probabilistic model can easily be extended to
compute the number of infected individuals and probability of infection over
time for groups of people differing in terms of their probabilities of
infection. Though our focus here has been on COVID-19, our model and methods
could easily be adapted to optimize workplace occupancy in the context of
other infectious diseases.

We assumed two groups of employees with different infection probabilities
(vaccinated and unvaccinated), and presented a practical approach to compute
the Pareto optimal presence rate of employees to reach the maximum
productivity and minimum infected risk. In addition to the incidence level,
the key parameters that influence the maximum productivity are the home
productivity ratio of the employees, the contact rate between the employees,
the average test interval among the employees and the vaccination rate. We
designed the model to be as simple as possible while still being able to cover
and interpret the effects of all influential parameters. Our approach is
linear in terms of time complexity, and it can be simply extended to consider
the sensitivity of COVID-19 diagnostic tests. Furthermore, the model can be
extended to more than two groups of employees, e.g, different age groups. A
basic implementation of this model and the optimization algorithms are
available online at https://test.where2test.de. Future extensions of this
study may consider the following subjects

  * 1.

In addition to employees, visitors (i.e., clients and customers) who have
direct contact with the employees can be considered for computing the
probability of an infection arrival at the facility.

  * 2.

Heterogeneous groups of employees with different contact rates, physical
networks, productivity ratios and test frequencies can be considered.

  * 3.

For simplicity, we defined the total productivity of the organization based on
the individual productivity, that is, the sum of all employees’ productivity.
However, it is possible for some organizations to define their total
productivity based on the type of the tasks and services in different sections
of the company. To this end, the physical graph of contacts and more details
on the outcomes would be required as input.

## Acknowledgements

This work was partially funded by the Center of Advanced Systems Understanding
(CASUS), which is financed by Germany’s Federal Ministry of Education and
Research (BMBF) and by the Saxon Ministry for Science, Culture, and Tourism
(SMWK) with tax funds on the basis of the budget approved by the Saxon State
Parliament.

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## Appendix: Evaluation of the proposed probabilistic approach

However, we explained the theory behind the probabilistic approach, in this
section, we implement the proposed probabilistic approach for computing the
expected number of infected employees as well. To this end, we assume
different settings of input parameters, and compute the expected number of
infections along the time. Indeed, we suppose one infection at time $t=0$, and
compute the expected infections for one month, i.e., $t<30$ using Eq. 7. Fig.
5 illustrates the results for different values of $\beta_{u}$, $\kappa$,
$occup$ and $n_{v}$. In these results, we assumed $n=150$ employees and
$\beta_{v}=0.15\beta_{u}$. In each subfigure, three cases of transmission
rates, $\beta_{u}=0.05$, $\beta_{u}=0.10$ and $\beta_{u}=0.15$, are shown.
Also, for comparison, we show simulation results (the dots), that is,
simulating the companies using a set of $n=150$ agents (employees) based on
the parameter’s setting. To reduce the effect of random number generators, we
ran the simulations for 100 times and report the average number of infected
agents. Table 3 shows the mean absolute percentage error for each figure and
the average of them. As it can be seen, the average (the last row of the
Table) percentage of difference between the results is 5.6%, 3.8% and 2.7% for
transmission rates $\beta_{u}=0.05$, $\beta_{u}=0.10$ and $\beta_{u}=0.15$,
respectively. The results show the probabilistic analysis estimates the number
of infected individuals with high accuracy, while it is fast and flexible
approach to apply for heterogeneous group of people with different
transmission rate and number of contacts.

Table 3: Mean absolute percentage error between the simulation results (average of 100 iterations) and the proposed probabilistic analysis for estimating the number of infected employees for six different settings of the influential parameters. |  $\beta_{u}=0.05$ | $\beta_{u}=0.10$ | $\beta_{u}=0.15$
---|---|---|---
Fig. 5(a) | 0.055 | 0.042 | 0.041
Fig. 5(b) | 0.057 | 0.024 | 0.026
Fig. 5(c) | 0.028 | 0.017 | 0.014
Fig. 5(d) | 0.078 | 0.057 | 0.040
Fig. 5(e) | 0.051 | 0.053 | 0.024
Fig. 5(f) | 0.064 | 0.036 | 0.014
Average | 0.056 | 0.038 | 0.027

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The simulation results and the probabilistic analysis of the
expected number of infected employees for a facility with $n=150$ employees
along time, $\Delta t=1,2,\dots,29$ days, after the infection arrives. Six
different settings of the influential parameters are shown. The computations
are based on Eq. 7.