# Surveillance Testing for Rapid Detection of Outbreaks in Facilities

Yanyue Ding
Graduate Program in Operations Research & Industrial Engineering (ORIE)
University of Texas at Austin
Austin, Texas 78712
Sudesh K. Agrawal
Graduate Program in ORIE
University of Texas at Austin
Austin, Texas 78712
&Jincheng Cao
Department of Mathematics
University of Texas at Austin
Austin, Texas 78712
Lauren Meyers
Graduate Program in ORIE, and
Department of Integrative Biology
University of Texas at Austin
Austin, Texas 78712
&John J. Hasenbein
Graduate Program in ORIE
University of Texas at Austin
Austin, Texas 78712
<EMAIL_ADDRESS>

###### Abstract

This paper develops an agent-based disease spread model on a contact network
in an effort to guide efforts at surveillance testing in small to moderate
facilities such as nursing homes and meat-packing plants. The model employs
Monte Carlo simulations of viral spread sample paths in the contact network.
The original motivation was to detect COVID-19 outbreaks quickly in such
facilities, but the model can be applied to any communicable disease. In
particular, the model provides guidance on how many test to administer each
day and on the importance of the testing order among staff or workers.

Keywords: COVID-19, surveillance testing, agent-based model, contact networks,
nursing homes

##  1 Introduction

In this paper, we develop an agent-based disease spread model on a contact
network to optimize surveillance testing in facilities. Generally speaking,
the model’s scope involves facilities of small to moderate size, say, with
dozens to hundreds of employees. The original motivation derives from the need
to quickly detect outbreaks of COVID-19 in facilities such as nursing homes,
factories, and meat-packing plants, in which close contact is nearly
unavoidable. Our focus is on _surveillance testing_ or proactive testing of
staff. In other words, the premise is that there is not currently an outbreak
in the facility, and the manager wishes to proactively test asymptomatic staff
in order to quickly detect, and prevent, an outbreak. Given a contact network,
along with other modeling parameters detailed in the sequel, the primary goal
is to answer the following:

  1. 1.

How many employees should be tested each day in order to detect an outbreak
with a given probability?

  2. 2.

What is the optimal sequence in which to test employees?

We propose a relatively simple Monte Carlo simulation model that is tailored
to answering the two questions. Our model is similar to other agent-based SEIR
models in the literature. However, the novelty is in our focus on optimization
and decision making, and how it relates to the structure of the contact
network. In the first part of the paper, we examine generic contact networks
that may be of use when there is little data on the interactions in a
facility. In the second part of the paper, we present more nuanced contact
models, that are derived from the staff and interaction structure in nursing
homes. Such models are preferred when there is detailed data on a facility’s
operations.

##  2 Literature Review

COVID-19 is a critical health issue worldwide and is likely to continue to be
of concern for many years. Even with vaccine development, it is imperative to
detect outbreaks at an early stage to develop effective mitigation measures.
According to some studies, 38%–80% of cases are pre-symptomatic or
asymptomatic in Long-Term-Care Centers (LTCs) [5, 35]. Moreover, 56%–95.5% of
disease transmissions are caused by asymptomatic cases [14, 17]. Early in the
COVID-19 pandemic, in many facilities, only individuals with COVID-like
symptoms were tested, due to limited testing resources. However, several
studies show that screening symptomatic cases only is not enough to control
outbreaks [33, 16, 17]. Starting in March 2020, the Centers for Disease
Control and Prevention (CDC) and the Centers for Medicare & Medicaid Services
(CMS) offered various COVID-19 surveillance testing strategies to hospitals,
nursing homes, schools, and other facilities, as the pandemic progressed. Now,
surveillance testing is a widely applied strategy to contain the spread of the
virus, especially for asymptomatic cases [4, 12]. Apart from suggestions
provided by the government, managers and researchers in various facilities
developed their own surveillance testing strategies, combined with contact
interventions, based on the facility’s contact patterns and healthcare
resources. In LTCs for example, the strategies include daily individual health
screening via surveys or front desk checks [3, 16], daily temperature and
oxygen level checks [33], serological anti-body checks [31], and reverse
transcription-polymerase chain reaction (RT-PCR) tests.

In terms of evaluation of testing strategies, some strategies are evaluated by
outbreak case studies or analysis of publicly recorded data [1, 8, 21, 23,
32]. In addition, strategies may be evaluated using variations of the
stochastic susceptible-exposed-infected-recovered/removed (SEIR) model. Some
researchers have made modifications to the traditional SEIR model, to account
for nuances of COVID-19. These include adding, removing, or subdividing the
standard stages in the network. For example, the exposed stage $E$ is
sometimes removed, and the remaining SIR model is used [11, 15]. Saha [26]
sub-divides the infected stage $I$ into detected and undetected infected
stages. Chapman et al. [7] assume all infected individuals experience a pre-
symptomatic stage, after which the corresponding node in the model transitions
to an asymptomatic, mild symptomatic or severe symptomatic stage. Another
innovation is to allow the parameters in the differential equations defining
the SEIR model to be non-linear. Chen and Wei [9] and Shu et al. [28] assume
that the parameters in the differential equations can be non-linear functions
of time. Some studies modify parameters and the contact network structure over
time. These adaptations are primarily used in comparing and finding the best
surveillance screening and intervention strategies. Hou et al. [20] and Shaikh
et al. [27] estimate the parameter settings under various surveillance testing
and intervention efficacy assumptions and perform a parametric analysis.
Ciaperon et al. [11] study the dynamic SEIR model by temporarily removing the
edges in the contact network due to quarantine intervention for individuals
who have COVID-like symptoms or tested positive.

One key assumption in the differential-equation-based SEIR models is that the
population is well-mixed and features are homogeneous among individuals [7, 9,
10]. The standard SEIR model might be suitable for predicting outbreak
progression in a large well-mixed population while it might over-simplify
dynamics in special networks [10]. Broadly speaking, two ways to address these
issues have been developed in the literature. One is to classify individuals
who share similar characteristics into subgroups and use specific differential
parameters in each subgroup. For example, Besse and Faye [6] divide the
population by spatial features and use a standard SIR model with heterogeneous
parameters in each spatial group. The disease transmission between groups is
estimated by heat equations. Shaikh et al. [27] divide the population in an
LTC by the occupations, and use different transmission probabilities for each
group. A second solution, requiring significant computational effort, is to
use network-based stochastic models that model each individual. Ames et al.
[2] believe human social networks tend to be clustered and heterogeneous. In
particular, they study disease transmission features in networks with various
coefficient and clustering properties. Their results show that the structure
of the network has a profound influence on disease progression. Similarly,
Meyers et al. [24] test intervention strategies for severe acute respiratory
syndrome (SARS) in various networks. Their analysis indicates that contact
patterns in the network impact the size of epidemic significantly, even if the
basic reproduction rates are the same among different networks. Herrera et al.
[18] study surveillance strategies in different networks and find that the
best surveillance strategies are based on network structures.

Rennert et al. [25] modify the SIR model by distinguishing symptomatic and
pre-symptomatic stages and adding an isolation stage. Their results show that
voluntary and random testing is not enough to protect students and faculty on
a campus. They recommend performing rigorous testing during the semester. They
also provide a novel solution by continuing to test the target group if a
positive case in random surveillance occurs, which can be more effective in
controlling disease spread. Chapman et al. [7] applied the SEIR model to
simulate COVID-19 transmission in a congregated shelter population. The model
was tested in shelters in San Francisco, Boston, and Seattle. The results
showed that a combined strategy that involves daily screening, twice-weekly
PCR testing and contact intervention is the best strategy to detect and avert
an outbreak at an early stage. Smith et al. [29, 30] used an individual-based
SEIR model to simulate the transmission of COVID-19 based on detailed contact
data in an LTC in order to find the best surveillance strategy. They compared
the strategies when daily testing capacities of 1, 2, 4, 8, 16, and 32 are
available.

Apart from COVID-19-based models, other researchers have discussed the
importance of considering the structure of contact networks when designing
surveillance strategies. For example, Herrera-Diestra et al. [19] study
contact networks modeling the spread of foot-and-mouth disease among cattle
farms in Turkey, and note that such networks have typical “small-world”
properties. Although they suggest some general surveillance principles based
on their observations, they do not provide detailed suggestions. Mastin et al.
[22] examine the spread of plant pathogens. Their paper is close in spirit to
our work, in that they focus of optimization of surveillance nodes by building
a stochastic optimization model, using Monte Carlo samples. However, their
model applies to a “static” surveillance strategy in which a fixed number of
nodes are selected to aid detection. In this paper we investigate dynamic
testing, in which different nodes are tested each day.

Overall, we note that most previous research focuses on the effectiveness of
various surveillance strategies to detect an outbreak of COVID-19, but few
papers provide guidance on specific testing strategies, especially when
testing resources are in shortage. The model in this paper is predicated on
the belief that decisions makers have different risk tolerances, which in turn
determine the amount of testing that is performed. We also assume the contacts
of individuals in a facility can be heterogeneous. We simulate disease
progression on a network and find the probability of detecting an outbreak for
various levels of testing capacities and time thresholds, as described in the
next section.

##  3 Model Description

The first component of our model is a contact graph $G=(V,E)$ on a set $V$ of
nodes and a set $E$ of undirected edges. We call two nodes _neighbors_ if they
are connected by an edge. Each node in the graph corresponds to a single staff
member, and there is an edge connecting them if they have close contact during
a day, i.e., there is a non-zero probability of COVID-19 passing between them
if one of them is infectious. Apart from the graph structure just described,
the model is characterized by the following parameters:

  * •

$K:=|V|$, the number of staff in the facility

  * •

$p$: daily probability of external infection

  * •

$\ell$: latency, the number of days to move from the exposed state to the
infected state

  * •

$d$: degree of each node in the symmetric case, indicating the density of
contact graph

  * •

$r$: daily probability of infection between neighbors, a representation of the
force of infection

  * •

$f$: false negative rate for a disease test, after a node is infectious

  * •

$t$: outbreak threshold tolerance, in days.

We now describe the progression of the disease through the network. The spread
model is a discrete-time Markov chain, with time index $n=0,1,2,\ldots.$ At
each time point the nodes can be in one of three states: susceptible ($S$),
exposed ($E$), and infected ($I$). We do not model recovered nodes since our
focus is on detecting initial outbreaks, generally in less than 10 days from
the first infected staff member entering the facility. Nodes may be exposed
either from external interactions, or internal interactions. We also have
simplified assumptions regarding the exposed and infected states. In the
exposed state a node cannot spread the disease to other nodes and cannot be
detected via a test. After exactly $\ell$ days, an exposed node becomes
infected. When a node is infected, each neighboring susceptible node enters
the exposed state with probability $r$ each day. The sequence of events
corresponding to infected nodes infecting susceptible nodes is assumed to be
mutually independent, both among nodes and across days. In addition, to model
external infections, each susceptible node can move directly to the infected
state, each day, with probability $p$. The reason we assume this direct move
is that an exposed node cannot be detected by a test, and cannot infect other
nodes, until it is in the “infected” state. Hence, from a dynamic systems
point of view, there is no need to track or model nodes that are just
“exposed” externally. The sequence of events corresponding to external
infections are also mutually independent. Finally, an external infection
resulting in a change to the infected state obviously supersedes a pending
move to “infected” if the same node has already been exposed internally.

In terms of disease testing, a decision maker can administer $k$ tests per day
to try to detect an outbreak. For simplicity, we assume that such a test is
administered in the “morning” to each of the $k$ individuals chosen that day,
and that it is a rapid test, whose results are available before the staff
member interacts with the rest of the staff. From the point of view of our
model, this means that the tested individuals are identified immediately if:
(a) they are tested on a given day, (b) are in the infected state on the same
day, and (c) the test does not yield a false negative result on this day. At
this point, we stop the process, since a detection has occurred. We do not
assume that the test is perfect. Specifically, there is a false negative rate
given by $f$. This rate applies to a test given to an individual that is
already in the infected state. We assume the false negative rate is constant
throughout the period a node is infected. This is obviously a simplification
of the case for most diseases, such as COVID-19, in which the false negative
rate fluctuates as an individual moves through the progression of the disease.
However, recall that our model is generally concerned with rapid detection,
within the first several days of infection (after the “incubation” period
given by $\ell$).

Up until Section 5, we assume that the contact network is homogeneous in
several ways. First, all nodes have an equal probability of external infection
(characterized by $p$). Second, all nodes have equal degree and are vertex
transitive (see Section 3.1). Third, each edge has an equal “weight,” in that
the infection force $r$ is the same for every edge. In Section 5, we relax
some of these assumptions to model heterogeneous contact networks, based on
more realistic contact data. Part of the reason to first study homogeneous
networks is to gain insights into different principles, such as the effect of
the structure of the graph on testing thresholds and efficient testing
protocols. To the best of our knowledge, these issues have not been studied
before in the literature on surveillance testing.

###  3.1 Graph Structure

In our study of so-called homogeneous graphs, we confine our analysis to
$d$-regular graphs, in order to preserve some sense of internal symmetry. In
addition, this allows us to systematically explore the effect of internal
interaction density, as embodied by the single parameter $d$. Of course, there
are numerous other ways to quantify graph structure, even for $d$-regular
graphs, but we do not explore those other avenues in this paper.

First, we recall that in a $d$-regular graph each node has exactly $d$
neighbors. A complete graph is a $d$-regular graph in which each node is
connected to every other node in the graph. Although there are many types of
$d$-regular graphs for a given $d$, in this paper we only study the class of
_circulant graphs_ , which are always $d$-regular. A circulant graph is a
graph whose adjacency matrix is a circulant matrix. More specifically, a
circulant graph can be characterized by a vector $(a_{1},a_{2},\ldots,a_{m})$
where $a_{1}<a_{2}<\cdots<a_{m-1}<a_{m}$ and $a_{m}<(K+1)/2$. For simplicity,
we assume that $a_{m}<K/2$. In this case, the corresponding circulant graph is
$d$-regular with $d=2m$. This allows us to uniquely define $m$, for a
specified even $d$ (which we vary to study the effect of graph density on
disease spread). The vector $(a_{1},a_{2},\ldots,a_{m})$ is sometimes called
the _jump sequence_ , and it characterizes the connections in the graph as
follows. Suppose we label the nodes in the graph $0,1,2,\ldots,K-1$. Then each
node $i$ has nodes $(i\pm a_{1})\mod{K}$, $(i\pm a_{2})\mod{K}$, …, $(i\pm
a_{m})\mod{K}$, as neighbors.

Circulant graphs are useful for our study as they have a number of desirable
properties. As mentioned above, they are always $d$-regular, with a degree
directly related to the parameter $m$. We also design our graphs so that they
are always connected. Further, circulant graphs are _vertex transitive_ so
that, roughly speaking, every vertex looks similar. Note that circulant graphs
need not be edge transitive and hence are not symmetric in the graph-theoretic
sense. In studying the effect of graph structure and density on surveillance
protocols, we use three classes of circulant graphs: complete graphs,
neighboring graphs, and crossing graphs. The first class is standard. The
latter two terms were created for this study. A _neighboring graph_ is one in
which the jump sequence is given by $(1,2,\ldots,m)$. In other words, each
node is connected to its $m$ closest neighbors, if one envisions the nodes
being arranged, as labeled, in a circle. A crossing graph is one in which the
jump sequence is given by $(\lfloor K/2-1\rfloor-m+1,\ldots,\lfloor
K/2-1\rfloor)$. In this case, each is node is connected to its $m$ farthest
neighbors. In Figure 1 we display representatives of each graph class for a
small network.

Figure 1: Some examples of $d$-regular graphs: (a) a complete 11-regular
graph, (b) a neighboring 4-regular graph, (c) a crossing 3-regular graph.

###  3.2 Monte Carlo Simulation

In order to answer the questions posed in the introduction, we perform Monte
Carlo simulations of disease spread, and detection, on networks of varying
structures and sizes. To keep the insights manageable, the parameters
$\ell=3$, $r=0.05$, and $f=0.21$ are kept constant throughout the numerical
studies. Otherwise, we vary network size, density, external infection rate,
and the outbreak threshold tolerance. Given a particular network and testing
protocol we typically simulate 50,000 outbreaks in a facility. An outbreak is
initiated by at least one node becoming infectious at time 0, and the
resulting nodes that are exposed and infected, until the outbreak threshold
tolerance $t$, at which point the simulation is stopped. For each sample
outbreak, we record whether or not there was a successful detection. An
outbreak may not be detected due to an infected node not being tested, or due
to a false negative result. These simulations then produce point estimates and
confidence intervals on the true detection probability for a particular
testing protocol. In general, we found 50,000 sample paths sufficient to
distinguish among policies.

In our numerical results, the primary performance criterion is the probability
of successfully detecting an outbreak within $t$ days, given that an outbreak
occurs. It is important to note that this is what one might call a conditional
outbreak probability. We believe that it is more useful than computing the
unconditional probability of detecting an outbreak, as this type of
performance metric essentially penalizes a protocol for failing to detect an
outbreak that does not exist. As we shall see below, this needs to be kept in
mind, as some results under this metric might appear counterintuitive. For
example, the probability of successful detection actually increases with the
community infection rate of $p$. To see why, imagine the extreme case in which
everyone walks in the door on day 1 infected with a disease and the false
negative rate is 0. In this case, it does not matter who is tested, as
everyone will set off the detection alarm. In particular you only need one
test to detect an outbreak when $p=1$ and $f=0$. However, when the community
infection rate is very low, this means that it is very likely that one, and
only one, person is the genesis of an outbreak. This makes the outbreak more
difficult to detect with a small number of tests.

##  4 Computational Results for Testing Protocols

###  4.1 Parameter Comparisons

In our first set of experiments, we investigate the effect of graph density,
embodied by the parameter $d$, on the number of tests per day needed to detect
an outbreak. In our model, $d$ represents the number of other staff members a
particular staff member is in contact with each day. For example, when $K=100$
and $d=99$, the resulting contact network is a complete graph, and the graph
implies that all staff members come into contact with all other staff members
each day. Obviously, this is an extreme situation. As such, we also examine
lower density networks. In Figure 2, we examine the case of 100 staff members
and an external community infection probability of $p=0.0001$ (per day). We
set $t=6$, indicating that our goal is to detect an outbreak within 6 days of
the introduction of the disease into the facility. Keeping these parameters
fixed, we plot the point estimates for the probability of detection versus the
number of tests per day, and plot this curve for various values of $d$. The
first thing to note is that there appear to be diminishing marginal benefits
of administering more tests, as one might expect. From the graph we also see
that the number of tests required to meet a certain benchmark probability of
detection can vary significantly with $d$. For example, if our threshold
probability is 0.7, and $d=99$ then roughly 4 tests per day are needed.
However, if $d=20$ then about 13 tests per day are needed. This may seem
counterintuitive but it is related to our observation above relating to the
community infection rate. The more quickly a disease spreads in a facility,
the more quickly it can be detected. We also observe that if we wish to
enforce a very high (say 0.95) detection probability, then a large number of
staff need to be tested daily. This is unlikely to be palatable for management
and staff.

$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$d=20$$d=40$$d=60$$d=80$$d=99$ Figure 2: Number of tests versus
detection probability for circulant neighboring graphs of varying degrees.
Fixed graph parameters are $K=100$, $p=0.0001$, $r=0.05$, $f=0.21$, $\ell=3$,
$t=6$. The results were obtained by sampling 50,000 virus sample paths.

In the next set of experiments, we vary the outbreak threshold tolerance $t$
while keeping other parameters fixed. We choose a neighboring graph with
$d=60$ as the results for this graph show the contrast among different
tolerance levels well. The results are shown in Figure 3. As must be the case,
the detection probability curves are nested, with larger tolerance curves
dominating lower tolerance curves. Again we notice that the number of tests
required to detect an outbreak at various tolerance levels varies widely. If
the probability threshold is again 0.7, and $t=8$ then just one test per day
is needed. However, if $t=4$ then around 10 tests per day are needed, which
implies that all staff are tested every 10 days.

$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$ Figure 3: Number of tests versus
detection probability for circulant neighboring graphs of tolerance levels.
Fixed graph parameters are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$,
$d=60$. The results were obtained by sampling 50,000 virus sample paths.

Finally, we examine the effect of the external infection probability on the
detection probability curves. The networks tested in Figure 4 have the same
parameters as the networks tested in Figure 3, with one exception: we change
the value of $p$ from 0.01 to $0.0001$. For each tolerance level $t$ we note
that the curve corresponding to $p=0.01$ dominates the curve for $p=0.0001$.
For example when the probability threshold is 0.7, $t=4$, and $p=0.01$, around
10 tests per day are needed, as observed above. However, when we examine the
case with $p=0.0001$ the required number of tests per day is approximately 17.
At first glance, this may seem counterintuitive: we require fewer tests when a
disease is more prevalent in the community. However, the result is
mathematically sound. For example, suppose we examine the extreme case when
$p$ is very close to 1. In that case, a large number of staff members would
arrive to the facility each day being infectious. As such, we only need to
test a small fraction of the staff to detect an outbreak. In contrast, when
the community infection rate is low, it is likely that only 1 staff member out
of, say 100, would initiate an outbreak. This one seed of an outbreak is
harder to catch with a low number of tests.

$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$ Figure 4: Number of tests versus
detection probability for circulant neighboring graphs of varying tolerance
levels. Fixed graph parameters are $K=100$, $p=0.0001$, $r=0.05$, $f=0.21$,
$\ell=3$, $d=60$. The results were obtained by sampling 50,000 virus sample
paths.

###  4.2 The Effect of Testing Order

In the experiments of Section 4.1, the nodes of the graph were tested in
“standard order.” What this means is that we envision the nodes with a fixed
numbering, 0 to $K-1$, and arranged in a circle according to the numbering.
This numbering is used to characterize various types of circulant graphs, as
described already. In addition, the standard testing order is to begin by
testing nodes 0 through $k-1$ on day 1, then testing nodes $k$ through $2k-1$
on day 2, etc. We continue testing in this manner, returning to node 1 when
all nodes have been tested. Of course, for some values of $t$ and $K$ not all
nodes will be tested. With respect to our simulation results, it does not
matter with which node we start since outbreaks are simulated in a uniform
manner across all nodes, and the graphs are vertex transitive. However, it is
not clear that testing in this standard order, for any given initial starting
node, is optimal. Obtaining the optimal testing order is computationally
prohibitive, as the underlying optimization problem is a stochastic integer
program, with a very large number of feasible solutions. In this paper, we
make no attempt to prove optimality bounds on testing orders, or to develop
complex heuristics. Rather, we examine a few different testing algorithms on
circulant graphs. Our results indicate that the testing order has little
effect on the detection probability, at least for circulant graphs. This is
good news in that it indicates that optimizing the testing order likely is not
a high priority in tuning test protocols in contact networks.

In our next set of experiments, we examine the effect of changing the testing
order, while keeping all the network parameters fixed. In Figure 5 we examine
three possible testing protocols. The first protocol, labeled “circle” is the
standard order described in the previous paragraph. The second protocol,
labeled “new” works as follows. The first node to be tested is chosen
uniformly at random. Call this node $A$. The second node that is tested is the
node that has the maximum distance from $A$, with ties broken randomly. Recall
that the distance between two nodes in a graph is the number of edges in the
shortest path connecting the nodes. To continue the selection process after
some set of nodes has already been selected, the next node chosen is one whose
sum of distances from all previously selected nodes is largest. To break ties
among equal distance sums, we select a node for which these distances have the
smallest sample standard deviation (if there is still a tie, it is broken
randomly). We repeat the selection process until all nodes have been selected,
to create an ordered list that defines the testing order. In our simulation
analysis, we test the nodes in the given order and then return to the
beginning of the list once all nodes have been tested once. The third protocol
is based on a randomly selected permutation of the numbers 0 through $K-1$.

For the neighboring graph testing in Figure 5, we see that the distance
algorithm does slightly outperform the circle protocol for testing levels of
about 5 to 10 nodes per day. This also implies a slight difference in the
required number of tests at the mid-level probability thresholds. For example,
for a probability threshold of 0.9 the difference in the required tests per
day appears to be about 2. Outside this middle zone, the difference is
essentially negligible. Note that we do expect the circle protocol to have a
slightly lower performance in a neighboring graph. When the virus is most
likely to spread among close neighbors, when the nodes are arranged in a
circle, then there is some redundancy in testing nodes in this same circular
order. It is better to hop across the circle for subsequent tests, which is
essentially what the “new” protocol does.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 5: Number of tests versus detection probability
for various testing protocols in a neighboring graph. Fixed graph parameters
are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$. The
results were obtained by sampling 10,000 virus sample paths.

We next test the same three testing protocols in a crossing graph. In Figure
6, we see that there is almost no difference in the three protocols, across a
range of testing levels. This is also expected in this case. In a crossing
graph, the testing protocols circle and new are quite similar, because nodes
that are “neighbors” when arranged on a circle are actually rather distant as
measured by the number of hops required to travel between such nodes. Again,
these results indicate that testing protocol does not have a large effect on
detection probability, at least for circulant graphs.

Finally, in Figure 7, we test the same protocols on a randomly generated
$d$-regular graph. As expected, there are relatively small differences among
protocols. We also tested protocols for different graph densities, and the
results are similar to those presented herein. We conclude that the difference
among testing orders is usually insignificant, with some minor differences as
shown in Figure 5. Since the “new” protocol performs well in all cases, we
recommend this protocol, assuming that the decision maker has concrete
knowledge of the graph structure. Otherwise, nodes can likely be tested in an
arbitrary manner, with little loss of performance versus a good heuristic.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 6: Number of tests versus detection probability
for various testing protocols in a crossing graph. Fixed graph parameters are
$K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$. The results
were obtained by sampling 50,000 virus sample paths.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 7: Number of tests versus detection probability
for various testing protocols in a random $d$-regular graph. Fixed graph
parameters are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$.
The results were obtained by sampling 10,000 virus sample paths.

##  5 Heterogeneous Graphs and Real-world Test Cases

###  5.1 Real-world Test Cases

####  5.1.1 Model adjustment

In previous sections, we considered simplified contact networks in order to
gain insight into the relationship between the parameters of the graph, and
the number of tests needed to detect an outbreak. In the remaining sections,
we consider more general graphs that are inspired by real-world situations and
data. A primary difference is that we allow the parameters $r$ and $p$ to be
heterogeneous, i.e., they can vary by node. The outside probability of
infection $p$ may vary since each staff member has different social
interactions and risk outside of work. Also, in real facilities staff in
different categories can have very different contact patterns. Hence, the
corresponding contact graph is no longer “symmetric,” and the internal
infection probability $r$ can vary by node. Our assumptions regarding the
latency $l$, the false negative rate $f$, and disease testing procedures are
the same as in Section 3.

In many facilities, the staff rosters differ each day as do the contact
patterns. In theory, it might be possible to develop even more nuanced models
in a particular facility. For example, Duval et al. [13] studied the contact
patterns in one nursing home by asking the staff and patients to use wearable
sensors and recorded the contact duration, distances, and frequencies.
However, for legal and ethical reasons, there are obviously significant
barriers to collecting such detailed data in most facilities. Hence, we do not
attempt to model a representative facility at this level of detail. Instead,
we collect general data on staff levels, categories and interactions. We
believe that this level of modeling still provides important insight into
appropriate testing levels for outbreak detection.

Let $c$ be the number of different staff categories in a facility. The staff
categories are given by the $C=\\{1,2,...,c\\}$. Since we have heterogeneous
internal infection parameters, we now use $r_{c_{1},c_{2}}$ to denote the
daily probability of infection between staff members in categories $c_{1}$ and
$c_{2}$, assuming that they are connected by an edge in the contact graph. Of
course, we allow $c_{1}=c_{2}$ to reflect interactions between staff in the
same category. Staff in a facility can be categorized by their job titles,
rosters, offices, working units, etc. In our real-world case study, we
categorize the staff in a nursing home by job title, and we assume the staff
with the same job titles share the same contact patterns.

####  5.1.2 Parameter fitting

We estimate the parameter $r_{c_{1},c_{2}}$ for different choices of $c_{1}$
and $c_{2}$ based on the contact patterns among staff. Inspired by Smith et
al. [29] and Temime et al. [34], we use the contact frequencies, average
duration of each contact, and the rate of infection with close contact to
estimate the probability of internal infection for staff in various
categories. Let $p_{0}$ be the probability of infection per minute between two
individuals with close contact, $d_{0}$ the average contact duration (in
minutes), and $n_{0}$ the average contact frequency. Then we can estimate the
internal probability of infection as $r=p_{0}\cdot d_{0}\cdot n_{0}$. We
collected data from a nursing home in the northern US by collecting
information from the facility manager. The details of this contact data are
shown in Table 1. Next, for each staff category, we estimated the number of
other staff they come into contact with daily, for all other categories. This
data is shown in Table 2. For a job category in row $i$ of the table, the
$i,j$th entry in the upper part of the table is the average number of contacts
for a staff member of category $i$ with staff members in category $j$. Note
that the table would not be symmetric if the lower half was filled in, but the
data in the upper portion is sufficient to estimate contacts between all pairs
of staff categories.

Job title |  Staff # |  # of daily
contact
with residents |  # of daily
contact
with staff |  Avg. contact
distance
(ft) |  Avg. contact
duration
(min)
---|---|---|---|---|---
Nurses and nurse aids | 69 | 30 | 20 |  $<$ 6 |  $>$ 15
House keeping | 16 | 10 | 10 |  $>$ 6 |  $<$ 15
Rehabilitation | 4 | 10 | 10 |  $<$ 6 |  $>$ 15
Activity lifestyle | 12 | 20 | 10 |  $<$ 6 | 15
Salon | 2 | 10 | 10 |  $<$ 6 | 15
Maintenance | 12 | 10 | 10 |  $<$ 6 |  $<$ 15
Reception | 6 | 10 | 40 |  $<$ 6 |  $>$ 15
Dining | 6 | 30 | 20 |  $<$ 6 |  $<$ 15
Social | 5 | 10 | 10 |  $<$ 6 |  $<$ 15
Finance | 14 | 0 | 10 |  $<$ 6 |  $>$ 15
Logistics | 3 | 0 | 20 |  $<$ 6 |  $<$ 15
Driver | 4 | 20 | 10 |  $<$ 6 |  $>$ 15
Table 1: Staff data for the nursing home case study Job title | NR | HK | RH | MS | SL | MT | RC | DN | SC | AD | LG | DR
---|---|---|---|---|---|---|---|---|---|---|---|---
NR | 15 | 1.5 | 0.2 | 1 | 0.1 | 0.1 | 2 | 0.5 | 1 | 1 | 0.5 | 0.22
HK | $-$ | 4 | 0.5 | 0.5 | 0.1 | 1 | 1 | 0.2 | 0.2 | 1 | 0.1 | 0.25
RH | $-$ | $-$ | 1.5 | 2 | 0.1 | 0.1 | 0.5 | 0.2 | 0.2 | 1 | 0.5 | 0.2
MS | $-$ | $-$ | $-$ | 3 | 0.1 | 0.1 | 0.5 | 0.2 | 0.2 | 1 | 0.5 | 0.2
SL | $-$ | $-$ | $-$ | $-$ | 1 | 0.1 | 0.5 | 0.1 | 0.1 | 1.2 | 0.5 | 0.2
MT | $-$ | $-$ | $-$ | $-$ | $-$ | 4 | 0.5 | 0.1 | 0.1 | 1.2 | 0.5 | 0.3
RC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2 | 0.1 | 0.1 | 3 | 1 | 0.4
DN | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 4 | 1 | 1 | 0.1 | 0.1
SC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 3 | 1 | 0.5 | 0.2
AD | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 5 | 1 | 0.5
LG | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.5 | 0.4
DR | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.5
Table 2: Estimated daily contacts between staff of different categories,
rounded to two decimals places

Based on this data, we are now able to generate a representative contact graph
for the case-study nursing home. First, we label all the nodes in the graph
with their corresponding staff category. Then, we calculate the probability of
generating an edge between two nodes by using the data in Tables 1 and 2. For
example, if a node is labeled as a nurse, the probability that a contact is
triggered with another nurse node is $15/68\approx 0.22$, as each nurse has an
average of 15 contacts with the other 68 nurses during a shift. Then, for
every pair of nurses, we generate a Bernoulli(0.22) random variable and assign
an edge to this pair if the outcome is a 1. Otherwise, there is no edge
connecting the pair. When generating edges between nodes in different staffing
categories, we use similar logic. For example, the probability that there is
an edge connecting a nurse with a member of the housekeeping staff is 1.5/16,
since, using Table 2, each nurse has an average of 1.5 contacts per day with a
member of the housekeeping staff.  Figure 8 shows an instance of the contact
network induced by the procedure just described. Our estimates of these
$r_{i,j}$ values appear in Table 3. For example, if in the contact network
there exists an edge between a nurse and a housekeeper, then the probability
of transmission (per day) between these two staff members is 0.01. If there is
no edge between two staff members, then the transmission probability is 0. We
only provide data in the upper part of the table, as this table is symmetric,
by our assumptions.

Job title | NR | HK | RH | MS | SL | MT | RC | DN | SC | AD | LG | DR
---|---|---|---|---|---|---|---|---|---|---|---|---
NR | 0.08 | 0.01 | 0.06 | 0.015 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
HK | $-$ | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.04
RH | $-$ | $-$ | 0.08 | 0.015 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
MS | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
SL | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
MT | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.04
RC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.02 | 0.02 | 0.01 | 0.02 | 0.08
DN | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.02 | 0.04 | 0.02 | 0.01 | 0.08
SC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.02 | 0.03 | 0.02 | 0.08
AD | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.03 | 0.08
LG | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.08
DR | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08

  * •

Abbreviations: NR, nurse; HK, house-keeper; MS, mission; SL, salon; RC,
receptionist; DN, dining; SC, social; AD, administration; LG, logistic; DR,
driver.

Table 3: Estimated probability of internal infection among staff in different
categories Figure 8: A real-world LTC network with $s=152,c=12$

The probability of external infection is a dynamic quantity that depends
highly on the current community prevalence of COVID-19. In this study, we
assume that all staff have the same value of $p$. For a given area, such as a
state or a county, we can estimate the daily probability of infection for an
individual as

$p=\frac{\text{Number of new infections over the past 7
days}}{7\cdot\text{Population size}}.$

Of course, the official count new infections in most cases is underestimating
the true number of cases. Therefore, in conducting sensitivity analysis, we
multiply the estimate above by $3$, $5$, and $10$.

####  5.1.3 Computational experiments

In this section, we present results using the network generated from the data
in Tables 1, 2, and 3. Similar to Section 4.1, we vary some basic model
parameters to perform comparisons. In the first set of experiments, we fix the
probability of external infection to $p=0.003$, and vary the outbreak
threshold tolerance $t$. As shown in Figure 9, the results align with the
experiments we did for the circulant graph. Obviously, as before, a larger
tolerance value requires a smaller number of tests per day, for a fixed
conditional probability target. For example, if the facility manager targets a
0.8 probability of detecting an outbreak, with $t=3$ then approximately 44
tests per day are required. However, if the threshold of tolerance is set to
4, 5 or even 10 days, then the required number of tests per day is 29, 19 and
3, respectively. We also note for $t=3$ the graph appears to be piecewise
linear. This makes intuitive sense, as we do not benefit from the “network”
effect of virus spread in this case.

$0$$10$$20$$30$$40$$50$$60$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per
day ($k$)Conditional probability of detecting
outbreak$t=3$$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$$t=11$$t=12$$t=13$$t=14$
Figure 9: Estimated probabilities of outbreak detection for $K=152$, $c=12$,
and $p=0.003$ for various values of $t$

Next, we examine the effect of changing $p$, the probability of external
infection. Table 10 shows the results of our simulations. As argued in Section
4.1, although it is counterintuitive that higher values of $p$ require fewer
tests per day to meet a given probability target, this is actually a logical
outcome of the model.

$0$$10$$20$$30$$40$$50$$60$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per
day ($k$)Conditional probability of detecting
outbreak$p=0.001$$p=0.003$$p=0.006$$p=0.01$$p=0.015$$p=0.03$ Figure 10:
Estimated probabilities of outbreak detection in a nursing home case study
with $K=152$, $c=12$, $t=7$ and varying values of $p$

Keeping this observation in mind, we note the a facility manager may want to
examine additional metrics when setting the tolerance level $t$ and the
detection probability target. For one parameter setting in our case-study
network, we provide in Figure 11 various statistics on the spread on
infection, as reflected by 50,000 simulation runs. In Figure 11, the blue line
represents the cumulative number of infectious staff, the red line represents
the number of infected staff, and the yellow line represents the number of
newly infected staff. The shaded areas around the lines represent 95% of
confidence intervals around the point estimates for these quantities, various
days after the outbreak. As expected, the blue line point estimate is higher
than the red line, because infection occurs three days before an individual
becomes infectious. Examining the graph, if a facility manager wishes to
detect an outbreak before 10% of the staff becomes infected then she would set
$t=5$ for the network depicted in Figure 11.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$20$$40$$60$$80$$100$$120$$140$Number
of days after outbreakNumber of staffcumulative infected staffcumulative
infectious staffnewly infected staffconfidence interval = 95% Figure 11:
Infection statistics in a case-study model with $K=152$, $c=124$, $p=0.003$,
$l=3$, and $f=0.21$

Finally, in Figure 12 we show the proportion of infected staff in various
categories over 20 days, again averaged over 50,000 simulated outbreaks. Note
that receptionists and nurses are the categories with the highest infection
proportions. In contrast, salon workers have the lowest infection rates in
this example. Our discussions with nursing home staff, indicate that their
perception is that nurses are quite vulnerable to outbreaks, but receptionists
are not. However, if we scrutinize Table 1 we note that receptionists and
nurses both have frequent, close, and prolonged contact with other staff, so
the model does seem to be reflecting the data provided.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
days after outbreakProportion of staff infected within each
categoryNRHKRHMSSLMTRCDNSCADLGDR Figure 12: Proportion of infected staff in
different job categories with $s=152$, $c=12$, $p=0.003$, $l=3$, and $f=0.21$

###  5.2 Effect of Testing Order

In Section 4.2, we examined the effects of testing order in what we called
homogeneous graphs, i.e., graphs with a significant amount symmetry and
homogeneity among the nodes. There, we found small, but mostly insignificant
differences among testing protocols. In contact networks arising from the real
world, the associated graphs are not expected to have such a regular
structure. Due to the less regular structure, there are also a larger variety
of heuristic testing protocols that can be tested. The most intuitive
protocols are based on testing “central” or important nodes first.

In this section, we test three such heuristics of varying levels of
complexity. In each heuristic we need to produce an ordered list (permutation)
of the nodes that determines the testing protocol. The first heuristic, which
we call _degree rank_ , we simply order the nodes from highest degree to
lowest degree, breaking ties in an arbitrary manner. Nodes are then tested in
this order. The idea, of course, is a higher degree node is more likely to
“detect” an outbreak since it has connections to lots of other nodes. The
second heuristic, which is called _PageRank_ , is based on the well-known node
ranking system first developed by the founders of Google. We compute the
PageRank for all nodes, and test in the order of highest rank to lowest.
Again, the idea of this ranking is that higher ranked nodes would be visited
more often by a virus surfing the contact network. A potential drawback of
this notion, as applied to surveillance testing, is that PageRank is
essentially derived from the steady-state distribution of the virus surfing
process alluded to above. However, in our virus spread and detection model, we
are really concerned with the _transient_ behavior of the virus. In
particular, what is most important for early detection is the initial
trajectory of the virus over the network. This observation inspires us to
consider another algorithm which we call _simulation rank_. The idea is as
follows: if our outbreak tolerance is $t$ days, then we are interested in
testing the nodes that are most likely to be involved in an outbreak during
the $t$ days. In particular, we should rank the nodes according to the
probability that they are part of a $t$-day outbreak. In theory, one could
compute this probability exactly, but it is clearly an intractable computation
for networks of even moderate size. Instead, we simulate many virus sample
paths to estimate the aforementioned probability for each node, and then use
these estimates to create the ranking. It is clear that the degree rank
ordering requires very little computational effort, as it just involves
counting edges. PageRank is more difficult, but there are efficient algorithms
to estimate the PageRank, even for large networks. Finally, simulation rank
requires a considerable amount of effort, as potentially thousands of
simulation samples are required to get a good estimate.

In Figures 13 and 14, we show the results of testing the three ranking
algorithms for test protocols. In each case, we determine the test order and
then evaluate the protocol using the network depicted in Figure 8. As in our
previous computations, we using 50,000 virus sample paths to perform the
evaluation. Figure 13 displays the results for a tolerance of $t=4$ days.
Clearly, the difference among the algorithms is imperceptible in this figure.
We also ran statistical tests confirming that there no statistically
significant difference among the algorithms. Figure 14 displays the results
for a tolerance of $t=7$ days. Again, the differences are almost negligible,
although PageRank appears to slightly outperform the other algorithms when 7
to 10 tests per days are performed. Still, these differences do not appear to
be statistically significant. We performed similar tests on a small set of
other networks, with similar results. Although the tests are not
comprehensive, they seem to indicate that a relatively simple algorithm such
as degree rank works just as well as more sophisticated algorithms.

$0$$7$$14$$21$$28$$35$$42$$49$$56$$63$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting outbreakdegree
rankPageRanksimulation rank Figure 13: Performance of heuristic testing
protocols for the LTC network with $s=152$, $c=12$, $p=0.006$, $l=3$, $t=4$
and $f=0.21$
$0$$7$$14$$21$$28$$35$$42$$49$$56$$63$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting outbreakdegree
rankPageRanksimulation rank Figure 14: Performance of heuristic testing
protocols for the LTC network with $s=152$, $c=12$, $p=0.006$, $l=3$, $t=7$
and $f=0.21$

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# Jacob’s ladders and some new consequences from A. Selberg’s formula

Jan Moser  Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.

It is proved in this paper that the Jacob’s ladders together with the A.
Selberg’s classical formula (1942) lead to a new kind of formulae for some
short trigonometric sums. These formulae cannot be obtained in the classical
theory of A. Selberg, and all the less, in the theories of Balasubramanian,
Heath-Brown and Ivic.

###### Key words and phrases:

Riemann zeta-function

##  1\. The A. Selberg’s formula

A. Selberg has proved in 1942 the following formula

(1.1) $\int_{T}^{T+U}X^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\sqrt{\frac{\pi}{2}}\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$

(see [19], p. 55), where

(1.2) $\begin{split}&X(t)=\frac{1}{2}t^{1/4}e^{\frac{1}{4}\pi
t}\pi^{-\frac{s}{2}}\zeta(s),\ s=\frac{1}{2}+it,\\\ &U=T^{1/2+\epsilon},\
\xi=\left(\frac{T}{2\pi}\right)^{\epsilon/10},\ \epsilon\leq\frac{1}{10},\
P=\sqrt{\frac{T}{2\pi}}\\\ &n_{1},n_{2}\in\mathbb{N},(n_{1},n_{2})=1,\
n_{1},n_{2}\leq\xi,\end{split}$

(comp. [19], pp. 10, 18, $a=1/2+\epsilon,\ \epsilon>0$) and $c$ is the Euler’s
constant. Since (see [19], p. 10, [20], p. 79)

$Z^{2}(t)=\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}=\sqrt{\frac{2}{\pi}}X^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right),$

i.e.

(1.3)
$X^{2}(t)=\sqrt{\frac{2}{\pi}}Z^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right)$

where

(1.4)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{1}{2}t\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}(\frac{1}{t})\end{split}$

is the signal defined by the Riemann zeta-function $\zeta(s)$. Following eqs.
(1.1) and (1.3) we obtain

(1.5) $\int_{T}^{T+U}Z^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$

######  Remark 1.

If $n_{1}=n_{2}=1$ then the Hardy-Littlewood-Ingham formula

$\int_{T}^{T+U}Z^{2}(t){\rm
d}t=U\ln\frac{T}{2\pi}+2cU+\mathcal{O}(T^{1/2}\xi^{5})$

follows from the A. Selberg’s formula (1.5) (comp. [20], p. 120).

######  Remark 2.

Let us remind that the A. Selberg’s formula (1.5) played the main role in
proving the fundamental Selberg’s result

$N_{0}(T+U)-N_{0}(T)>A(\epsilon)U\ln T$

where $N_{0}$ stands for the number of zeroes of the function $\zeta(1/2+it),\
t\in(0,T]$.

In this paper it is proved that the Jacob’s ladders together with the A.
Selberg’s classical formula lead to a new kind of results for some short
trigonometric sums.

This paper is a continuation of the series of works [3] \- [18].

##  2\. The result

### 2.1.

Let us remind some notions. First of all

(2.1) $\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t),$

where

(2.2)
$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$

(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)) and $\varphi(t)$ is the
Jacob’s ladder, i.e. the solution of the following nonlinear integral equation

$\int_{0}^{\mu[x(T)]}Z^{2}(t)e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}Z^{2}(t){\rm d}t$

that was introduced in our paper [3]. Next, we have (see [1], comp. [18])

(2.3) $\begin{split}&G_{3}(x)=G_{3}(x;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu}\leq
T+U}\\{t:\ g_{2\nu}(-x)\leq t\leq g_{2\nu}(x)\\},\ 0<x\leq\frac{\pi}{2},\\\
&G_{4}(y)=G_{4}(y;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu+1}\leq T+U}\\{t:\
g_{2\nu+1}(-y)\leq t\leq g_{2\nu+1}(y)\\},\ 0<y\leq\frac{\pi}{2},\end{split}$

and the collection of sequences $\\{g_{\nu}(\tau)\\},\ \tau\in[-\pi,\pi],\
\nu=1,2,\dots$ is defined by the equation (see [1], [18], (6))

$\vartheta_{1}[g_{\nu}(\tau)]=\frac{\pi}{2}\nu+\frac{\tau}{2};\
g_{\nu}(0)=g_{\nu}$

where (comp. (1.4))

$\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}.$

### 2.2.

In this paper we obtain some new integrals containing the following short
trigonometric sums

$\begin{split}&\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos(t\ln p),\
\sum_{2\leq n\leq\xi}\frac{1}{\sqrt{n}}\cos(t\ln n),\\\ &\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(t\ln n)\end{split}$

where $p$ is the prime, $n\in\mathbb{N}$ and $d(n)$ is the number of divisors
of $n$. In this direction, the following theorem holds true.

######  Theorem.

Let

(2.4) $G_{3}(x)=\varphi_{1}(\mathring{G}_{3}(x)),\
G_{4}(y)=\varphi_{1}(\mathring{G}_{4}(y)).$

Then we have

(2.5) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2x}{\pi}U\ln P\ln\ln P,\ x\in(0,\pi/2],\\\
&\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2y}{\pi}U\ln P\ln\ln P,\ y\in(0,\pi/2],\end{split}$

(2.6) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)x+\frac{\epsilon^{2}}{50}\sin
x\right\\}U\ln^{2}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)y-\frac{\epsilon^{2}}{50}\sin
y\right\\}U\ln^{2}P,\end{split}$

(2.7) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &\frac{\sin
x}{2500\pi^{3}}U\ln^{4}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &-\frac{\sin
y}{2500\pi^{3}}U\ln^{2}P,\end{split}$

where

(2.8) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\ t\to\infty,$

and $\pi(t)$ is the prime-counting function.

######  Remark 3.

Let $T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, (comp.
(2.4)). Then from (2.8), similarly to [14], (1.8), we obtain

$\rho\\{[T,T+U];[\mathring{T},\widering{T+U}]\\}\sim(1-c)\pi(T);\
T+U<\mathring{T},$

where $\rho$ stands for the distance of the corresponding segments.

######  Remark 4.

The formulae (2.5) - (2.7) cannot be obtained in the classical theory of A.
Selberg, and, all the less, in the theories of Balasubramanian, Heath-Brown
and Ivic.

##  3\. New asymptotic formulae for the short trigonometric sums: their
dependence on $|\zeta\left(\frac{1}{2}+it\right)|^{2}$

We obtain, putting $x=y=\pi/2$ in (2.5)

$\begin{split}&\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &2U\ln
P\ln\ln P.\end{split}$

Using successively the mean-value theorem (since
$\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$ is a segment), we have

(3.1) $\begin{split}&\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}Z^{2}\\{\varphi_{1}(t)\\}Z^{2}(t){\rm
d}t=\\\ &=\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}Z^{2}\\{\varphi_{1}(\alpha_{2})\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}(t){\rm
d}t\sim\\\ &2U\ln P\ln\ln P,\
\alpha_{1},\alpha_{2}\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2);\
\alpha_{1}=\alpha_{1}(T,U)=\alpha_{1}(T,\epsilon),\dots.\end{split}$

Since

(3.2)
$\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}{\rm
d}t=\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|$

(comp. Remark 8), and

(3.3) $m\\{\mathring{G}_{3}(x)\\}\sim\frac{x}{\pi}U,\
m\\{\mathring{G}_{4}(y)\\}\sim\frac{y}{\pi}U\ \Rightarrow\
\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|\sim U$

(see [2], (13), $m$ stands for the measure) then we obtain from (2.5) (see
(3.1) - (3.3)) the following

######  Corollary 1.

For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{1}(T),\alpha_{2}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that

(3.4) $\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1}(T))\ln
p\\}\sim\frac{2\ln P\ln\ln
P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{2}(T))\right)\right|^{2}},\
T\to\infty$

where
$\varphi_{1}(\alpha_{1}(T)),\varphi_{1}(\alpha_{2}(T))\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$.

Similarly, we obtain from (2.6)

######  Corollary 2.

For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{3}(T),\alpha_{4}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that

(3.5) $\begin{split}&\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{\varphi_{1}(\alpha_{3}(T))\ln n\\}\sim\\\
&\sim\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)\frac{\ln^{2}P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{4}(T))\right)\right|^{2}},\
T\to\infty\end{split}$

where $\varphi_{1}(\alpha_{3}(T)),\varphi_{1}(\alpha_{4}(T))\in
G_{3}(\pi/2)\cup G_{4}(\pi/2)$.

######  Remark 5.

From the asymptotic formulae (3.4), (3.5) it follows that the values of
mentioned short trigonometric sums are connected with the values of the
Riemann zeta-function $\zeta\left(\frac{1}{2}+it\right)$ for some infinite
subset of $t$.

##  4\. New asymptotic formulae on two collections of disconnected sets
$G_{3}(x),G_{4}(y)$

From (2.7), similarly to p. 3, we obtain

######  Corollary 3.

(4.1) $\begin{split}&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{3}(x)}\sim\\\
&\sim\frac{1}{2500\pi^{2}}\frac{\sin x}{x}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{3}(x)}}\\\
&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{4}(y)}\sim\\\
&\sim-\frac{1}{2500\pi^{2}}\frac{\sin y}{y}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{4}(y)}},\
T\to\infty\end{split}$

where $\langle(\dots)\rangle|_{\mathring{G}_{3}(x)},\dots$ denote the mean-
value of $(\dots)$ on $\mathring{G}_{3}(x),\dots$ .

######  Remark 6.

It follows from (4.1) that the short trigonometric sum

$\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\},\ t\geq
T_{0}[\varphi_{1}]$

has an infinitely many zeroes of the odd order.

##  5\. Law of the asymptotic equality of areas

Let

$\begin{split}&\mathring{G}_{3}^{+}(x)=\left\\{t:\ t\in\mathring{G}_{3}(x),\
\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}>0\right\\},\\\ &\vdots\\\ &\mathring{G}_{4}^{-}(x)=\left\\{t:\
t\in\mathring{G}_{4}(x),\ \sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}<0\right\\}.\end{split}$

Then we obtain from (2.7), (comp. Corollary 3 in [14])

######  Corollary 4.

(5.1)
$\begin{split}&\int_{\mathring{G}_{3}^{+}(x)\cup\mathring{G}_{4}^{+}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\sim-\int_{\mathring{G}_{3}^{-}(x)\cup\mathring{G}_{4}^{-}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t.\end{split}$

######  Remark 7.

The formula (5.1) represents the law of the asymptotic equality of the areas
(measures) of complicated figures corresponding to the positive part and the
negative part, respectively, of the graph of the function

(5.2) $\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t),\
t\in\mathring{G}_{3}(x)\cup\mathring{G}_{4}(x),$

where $x\in(0,\pi/2]$. This is one of the laws governing the _chaotic_
behaviour of the positive and negative values of the signal (5.2). This signal
is created by the complicated modulation of the fundamental signal
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$, (comp. (1.4),
(2.2)).

##  6\. Proof of the Theorem

### 6.1.

Let us remind that the following lemma holds true (see [8], (2.5); [9],
(3.3)): for every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ we have

(6.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\ U\in(0,T/\ln T],$

where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$. In the case (comp. (2.4))
$T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, we obtain
from (6.1) the following equality

(6.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$

### 6.2.

First of all, we have from (6.2), for example,

$\int_{\mathring{g}_{2\nu}(-x)}^{\mathring{g}_{2\nu}(x)}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{g_{2\nu}(-x)}^{g_{2\nu}(x)}f(t){\rm d}t,$

(see (2.3). Next, in the case

$f(t)=\left(\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}$

we have the following $\tilde{Z}^{2}$-transformation

(6.3) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t.\end{split}$

Let us remind that we have proved (see [2], (13) and Corollary 7) the
following formulae

(6.4) $\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2x}{\pi}\ln P\ln\ln P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2y}{\pi}\ln P\ln\ln P.\end{split}$

Now, our formulae (2.5) follow from (6.3), (6.4).

### 6.3.

Similarly, from the formulae

$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{x}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}+\frac{\epsilon^{2}}{50}\frac{\sin
x}{x}\right)U\ln^{2}P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{y}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}-\frac{\epsilon^{2}}{50}\frac{\sin
y}{y}\right)U\ln^{2}P,\end{split}$

and

$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm
d}t\sim\frac{\sin x}{2500\pi^{3}}U\ln^{4}P,\\\
&\int_{G_{4}(y)}\left(\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln
n\\}\right)Z^{2}(t){\rm d}t\sim-\frac{\sin
y}{2500\pi^{3}}U\ln^{4}P\end{split}$

(see [2], (13) and Corollaries 8 and 9) we obtain (2.6) and (2.7),
respectively.

######  Remark 8.

The formulae of type (3.2) can be obtained from (6.2) putting $f(t)\equiv 1$.

I would like to thank Michal Demetrian for helping me with the electronic
version of this work.

## References

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  * [2] J. Moser, ‘The structure of the A. Selberg’s formula in the theory of the Riemann zeta-function‘, Acta Math. Univ. Comen., 48-49, (1986), 93-121, (in russian).
  * [3] J. Moser, ‘Jacob’s ladders and the almost exact asymptotic representation of the Hardy-Littlewood integral’, (2008), arXiv:0901.3973.
  * [4] J. Moser, ‘Jacob’s ladders and the tangent law for short parts of the Hardy-Littlewood integral’, (2009), arXiv:0906.0659.
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