# Vaccine Discourse on Twitter During the COVID-19 Pandemic

Gabriel Lindelöf
Department of Computer Science
Aalto University
02150 Espoo, Finland
<EMAIL_ADDRESS>
Talayeh Aledavood
Department of Computer Science
Aalto University
02150 Espoo, Finland
<EMAIL_ADDRESS>
Barbara Keller
Department of Computer Science
Aalto University
02150 Espoo, Finland
<EMAIL_ADDRESS>

###### Abstract

Since the onset of the COVID-19 pandemic, vaccines have been an important
topic in public discourse. The discussions around vaccines are polarized as
some see them as an important measure to end the pandemic, and others are
hesitant or find them harmful. A significant portion of these discussions take
place openly on social media platforms. This allows us to closely monitor the
opinions of different groups and their changes over time. This study
investigates posts related to COVID-19 vaccines on Twitter and focuses on
those which have a negative stance toward vaccines. We look into the evolution
of the percentage of negative tweets over time. We also examine the different
topics discussed in these tweets, in order to understand the concerns and
discussion points of those holding a negative stance toward the vaccines. A
dataset of 16,713,238 English tweets related to COVID-19 vaccines was
collected covering the period from March 1, 2020, to July 31, 2021. We used
the Scikit-learn Python library to apply a support vector machine (SVM)
classifier to identify the tweets with a negative stance toward the COVID-19
vaccines. A total of 5,163 tweets were used to train the classifier, out of
which a subset of 2,484 tweets were manually annotated by us and made publicly
available. We used the BERTtopic model to extract and investigate the topics
discussed within the negative tweets and how they changed over time. We show
that the negativity with respect to COVID-19 vaccines has decreased over time
along with the vaccine roll-outs. We identify 37 topics of discussion and
present their respective importance over time. We show that popular topics
consist of conspiratorial discussions such as 5G towers and microchips, but
also contain legitimate concerns around vaccination safety and side effects as
well as concerns about policies. The most prevalent topic among vaccine-
hesitant tweets is related to the use of mRNA and fears about speculated
negative effects on our DNA. Hesitancy toward vaccines existed prior to
COVID-19. However, given the dimension and circumstances surrounding the
COVID-19 pandemic, some new areas of hesitancy and negativity toward the
COVID-19 vaccines have arisen, for example, whether there has been enough time
for them to be properly tested. There is also an unprecedented amount of
conspiracy theories associated with them. Our study shows that even unpopular
opinions or conspiracy theories can become widespread when paired with a
widely popular discussion topic such as COVID-19 vaccines. Understanding the
concerns and the discussed topics and how they change over time is essential
for policymakers and public health authorities to provide better and in-time
information and policies, to facilitate vaccination of the population in
future similar crises.

_Keywords_ COVID-19 vaccines $\cdot$ SARS-COV-2 $\cdot$ vaccine hesitancy
$\cdot$ social media $\cdot$ twitter $\cdot$ natural language processing
$\cdot$ machine learning $\cdot$ stance detection $\cdot$ topic modeling

##  1 Introduction

Since the emergence of the COVID-19 pandemic, vaccines against the SARS-COVID
virus have become a highly salient topic in public discourse. While most agree
that the pandemic should end as soon as possible, the opinions on how and
through which mechanisms or policies to get there, differ greatly. As a main
point of disagreement, many see vaccination (and re-vaccination) of the
majority of the world’s population as the only way to have the pandemic fully
under control, while another group of people is hesitant or completely opposed
to the idea of getting vaccinated. The importance and sensitivity of the topic
of vaccination leads to a large amount of discourse, commonly expressed on
social media platforms, which are often highly polarized. Under circumstances
such as social distancing and remote work, imposed under the pandemic, social
media platforms have taken an even more central role in people’s lives [16].
In these online social spheres, people openly discuss and share their opinions
on vaccines with each other globally. A better understanding of how these
conversations have developed over time and more insight into the topics
discussed within them, can help us better understand those hesitant to the
vaccines or those with low confidence in the surrounding processes.

Many attempts have been made to define the concept of vaccine hesitancy [18].
One such attempt was made by the SAGE working group on vaccine hesitancy who
proposed the following definition “Vaccine hesitancy refers to delay in
acceptance or refusal of vaccines despite availability of vaccine services.
Vaccine hesitancy is complex and context-specific, varying across time, place,
and vaccines. It is influenced by factors such as complacency, convenience and
confidence” [2]. A word commonly used in relation to hesitancy is confidence,
which highlights the aspect of trust in vaccines and trust in the actors
involved in the process, such as health care workers, researchers,
governments, and pharmaceutical companies [18]. The current study focuses on
tweets expressing a negative stance toward the COVID-19 vaccines. This
includes those tweets expressing a hesitancy toward taking available vaccines,
a negative attitude toward policies promoting vaccination, or distrust in
actors involved in the vaccination process. We explore how much space these
voices take on social media and what the most prevalent topics of discussion
are.

A better understanding of individuals with low trust in vaccines or a
hesitancy toward taking a vaccine can help shape interventions to improve
trust. This will be important to get the current COVID-19 pandemic fully under
control, but can also contribute valuable insights how to shape communication
in similar situations in the future. Although previous studies have looked at
the discourse around COVID-19 vaccines on Twitter [15, 35], the current study
is one of the first to cover the entire period from the WHO declaration of the
pandemic until the summer of 2021. The study will also provide temporal
insights into how these discussions have changed over time.

Our main contributions are as follows:

  1. 1.

A classifier capable of identifying tweets that express a negative stance
toward COVID-19 vaccines.

  2. 2.

A timeline of the development of the percentage of tweets expressing a
negative stance toward COVID-19 vaccines for the first 18 months of the
COVID-2019 pandemic.

  3. 3.

An overview of the major topics discussed by those with a negative stance
during this time, their development, and what events co-occurred with changes
in the discourse.

  4. 4.

A dataset of 2,484 tweets manually labeled with their stance toward the
COVID-19 vaccines, along with the codebook used in the annotation process [3].

##  2 Background

###  2.1 Sentiment Analysis

A number of approaches have been taken to analyze sentiments around infectious
diseases on social media [4]. Sentiment mining on social media has proven to
be a valuable resource to understand people’s opinions about ongoing events,
and could potentially help with controlling pandemics [29]. Previous studies
investigating sentiments around infectious diseases can broadly be grouped
into three categories; lexicon-based, machine learning, and hybrids between
the former two [4]. Before the dawn of the COVID-19 pandemic Du et al. [9]
used a machine learning approach to investigate the stance toward HPV vaccines
on Twitter. Using an annotated dataset of 6,000 tweets they were able to train
an SVM classifier that could classify tweets as positive, negative, or neutral
with satisfactory performance.

At the beginning of the COVID-19 pandemic Medford et al. [20] investigated
sentiments discussed on Twitter. They found that approximately 50% of COVID-19
related tweets could be classified as showing fear, while 30% were expressing
surprise. Dominating topics discussed were economical and political impacts of
the virus, quarantine efforts, as well as transmission of the virus and how to
prevent it.

###  2.2 Stance Detection

Stance detection analyzes textual input from people and identifies whether
someone is favorable, against, or neutral toward a certain topic. Stance
detection is related to sentiment analysis but is known to be a more difficult
task [30]. Sentiment analysis aims to identify the opinion of a person and
determine whether the text bears positive, negative, or neutral sentiments. A
negative sentiment of a text does not always imply an unfavorable stance
toward the topic. As an example, Skeppstedt et al. [30] uses the phrase “The
diseases that vaccination can protect you from are horrible.”. This phrase is
favorable toward the pre-chosen topic of vaccination but contains negative
sentiments toward an identified topic which is “diseases”. However, in
previous literature, there is not always a clear distinction between stance
detection and sentiment analysis. For instance, some of the work that study
the stance of social media users toward vaccination does not use the term
stance [13].

Stance detection has also been used by Cotfas et al. [7] to study hesitancy
during the month following the UK vaccination roll-out. Their cleaned dataset
contained 1.2 million tweets and approximately 7 out of 10 were classified as
neutral, and 2 out of 10 as having a negative stance. Some of the larger peaks
in volume of negative tweets correlated with the roll-out of the Pfizer
vaccine (Dec 8, 2020), FDA approval of Moderna (Dec 17, 2020) and the
vaccination dry-run in India (Jan 2, 2021) [7]. Some of the most discussed
topics were labeled as being about mistrust, scam, side effects, and the
hiding of relevant information. The period before or after the month of the
roll-out was not studied, nor was it investigated how the popularity of found
topics developed over time.

###  2.3 Vaccination Discourse on Social Media

The widespread use of social media gives us the opportunity to gain insights
about opinions of a broad population. A topic of particular interest with
respect to public health is the discourse around vaccination. Understanding
concerns and insecurities about vaccinations can help policymakers in finding
an appropriate way to address them. In 2011 Salathé and Khandelwal [26]
assessed the sentiments expressed in tweets toward vaccination in the fall
wave of the H1N1 (swine flu) pandemic. The authors could establish a
correlation between the expressed sentiments on Twitter and the corresponding
estimated vaccination rate (gained via phone surveys) in the same area. On the
other hand, another study on vaccination discourse on Facebook, focusing on
the common flu vaccination, found an asymmetric participation of vaccination
defenders and critics [12]. While the defenders were able to reach 24% of the
investigated network, the vaccination critics were only able to reach 8%.

With the onset of the COVID-19 pandemic, extensive work was done on the area
of COVID-19 vaccinations. Some studies focus on the demographics of the
vaccination hesitant population [19, 33] while others try to understand what
leads to this hesitancy [6, 25], finding similar to our study, that common
concerns are potential side effects and distrust toward the pharmaceutical
industry. Melton et al. [21] used a similar approach as us but focused on the
social media platform Reddit. They investigated topics of vaccine discussion
for the period of December 1, 2020, to May 15, 2021 and found that a majority
of posts were using a positive tone. Lyu et al. [17] focused on tweets made
about COVID-19 from the declaration of the pandemic until February 2021. Their
results showed that opinions and emotions around vaccines was the most tweeted
topic, and that sentiment became more positive over time.

##  3 Materials and Methods

###  3.1 Dataset

Tweets were collected using the academic research track of the Twitter
Application Programming Interface (API). Using the full-archive search we were
able to retroactively collect Tweets from each day for the period of March 1,
2020, to July 31, 2021. We queried English-speaking tweets containing synonyms
of the words “COVID-19” and “vaccine” excluding any retweets. A Python script
utilizing Twarc2 [32] was used to send requests and collect results.

Since random sampling is not possible when gathering historical tweets, tweets
were collected at six different time points of each day, corresponding to noon
in six major time zones with a large population of English speakers: AEST,
IST, CET, EST and PST. This means we get a more balanced dataset collected
during high activity hours for different parts of the world.

At each time point, 30 twitter-pages of tweets were collected. Since pages do
not necessarily correspond to a fixed number of tweets, there are some
variations in the collected volume from one day to another, as can be observed
in Figure 1. For the first day of each month, as well as all days in the month
of July 2021, the API unexpectedly returned a lower number of tweets per page.
Given the large number of collected tweets for each day we did not consider
this an issue for the analysis.

The collected dataset contained a total of 16,713,238 tweets, with an average
of 32,203 for each day in the period. To put these numbers into perspective we
also investigated how many tweets matched our search query in total. Using the
Counts endpoint of the Twitter API we can quickly retrieve how many tweets
match our keywords, without having to go through the lengthy process of
collecting them all. During the investigated period a total of 85 million were
made matching the query, meaning our sample represents roughly 20% of relevant
tweets.

As we were interested in the discourse led by people, tweets posted by the
bots active on Twitter needed to be excluded. To investigate their prevalence
in the dataset a random sample of 300 tweets were run through the OSoMe
Botometer API [28]. Botometer returns a complete automation probability (CAP)
that indicates the likelihood of an account being automated. 26% of tweets
were made from accounts with a CAP above .80, and out of these accounts 87%
were sharing links. Shared links pose their own challenges in classification
tasks. On one hand, it is difficult to judge if the link shared does indeed
reflect the stance of the sharer and on the other hand retrieving the
information following the link can be difficult especially as the information
can change over time. Based on these challenges and the fact that many bots
are sharing links, we decided to remove all tweets containing links. After
this, 7,292,705 tweets remained, corresponding to 44% of the dataset. Tweets
with identical text content to other tweets from the same author were also
removed in order to reduce the amount of spam, making the new total 5,966,905.
Finally, to reduce over-representation of individual users, tweets made from
users with more than 1000 tweets contained in this dataset were removed. An
example of such cases were bots that continuously announced free vaccination
appointments. This removed 108,749 tweets, making the total number of tweets
in the cleaned dataset 5,858,156.

Figure 1: 7-day moving average of tweets before and after the cleaning steps
described in 3.1.

###  3.2 Pre-Processing

The text data of the collected tweets were extracted and transformed using a
few common pre-processing steps. Text was lower-cased, special characters were
removed, “&” symbols were replaced with “and” and letters occurring more than
twice in a row were removed (helllo becomes hello). Text was also separated
into units of words using the NLTK TweetTokenizer. Stop-words contributing
little meaning to the text such as “the”, “a”, and “in” were removed using the
NLTK English stop word list. Since the terms used in the twitter search query
occur at least once in every tweet, they add bias and contribute only little
additional information, and therefore were also removed. The same pre-
processing steps were taken on the text data for both the classifier and the
topic model. Pre-processing is not generally recommended for BERTopic (the
topic model used in our analysis, see section 3.5), unless the data contains a
lot of noise such as HTML tags. In this case, however, pre-processing led to a
better topic model. Perhaps the large volume of data analyzed in this study
made the small loss of information due to the cleaning process negligible.

###  3.3 Annotation

A first step to computationally categorizing documents (in our case tweets)
into different groups (e.g. negative and not negative stance toward COVID-19
vaccines) is to manually annotate a subsample of the documents with the
correct labels as determined by the annotators. A codebook is often used to
have a common framework between different annotators. The codebook sets out
the rules that help decide which label a tweet receives. Although the
classification in this study is limited to a binary negative/not negative, we
used three classes for our codebook. This made the annotation more intuitive,
and also enables future studies to make a distinction between the neural and
positive stances when required. This dataset which was annotated using the
codebook in Table 1, is published along with this article, so that other
researchers can benefit from it. In our codebook a tweet can have a positive,
negative or neutral/unclear stance toward the COVID-19 vaccines. The
explanations of these categories were inspired by a previous codebook
classifying tweets about HPV vaccines [9].

Table 1: The codebook developed for the purpose of this study with example tweets for each category. Stance |  Definition |  Example tweet
---|---|---
Positive |  Showing positive opinion toward the COVID-19 vaccines. Prompting uptake of the vaccines. Expressing the intention of taking or having taken the vaccine. |  “Im getting vaccinated tomorrow yay”
Negative |  Expressing concerns around safety, efficacy, injuries, cost, resistance to COVID-19 vaccines due to cultural or emotional matters. Discouraging uptake of the vaccines. Expressing intention of not taking or refusing the vaccines. Questioning motives behind vaccine deployment, for example, those of scientists, pharmaceuticals companies or governments. |  “No mask and no COVID19 vaccine for me!”
Neutral |  Contain no stance, or stance is unclear. Expresses both pro and anti-stance at the same time. Expresses stance of someone else without own added input. Not related to COVID-19 vaccine topic. Discussion on other medical treatments without relating it to vaccines. Discussion on vaccines for other diseases. Unclear what the person is trying to say. |  “Would you take a COVID vaccine?”

Tweets to be annotated were randomly sampled from the cleaned dataset. The
first 485 tweets were coded by the authors while the codebook was still being
refined. Thereafter two university students were hired to annotate 1999 more
tweets using the developed codebook. Out of these, a total of 543 tweets were
labeled positive, 471 negative, and the remaining 1,450 as neutral/unclear.
All inter-annotator disagreements were discussed in an attempt to find
agreement on a final label. If no agreement could be made, the label was set
to neutral/unclear. The majority of disagreements were between neutral/unclear
and the two other labels. There were only a total of 22 disagreements were one
annotator said positive and the other negative.

###  3.4 Classifier

To investigate the discourse of those with a negative stance toward the
vaccines, we had to find a way to identify tweets belonging to this category.
Considering the size of the dataset we opted for a machine learning approach.
While there are many viable classification algorithms available, previous
research classifying stance toward vaccines on Twitter have shown success
using support vector machines (SVM) [9, 15, 7]. Initial exploration comparing
multinomial naive bayes, random tree and an SVM classifier also showed most
promising results for the SVM, leading us to choose this as our classification
method.

SVM is a type of classification algorithm that can be used to automatically
divide documents into categories [24]. In this study, each tweet is a document
that needs to be labeled with an either _negative_ or _not negative_ stance.
Using thousands of tweets annotated by hand, the SVM can learn from these
examples (training dataset) to automatically label the remaining millions of
tweets. The pre-processed tweets were vectorized using term frequency-inverse
document frequency (TF-IDF), giving each word a weight indicating how
important it is in the text. The tweets that had been manually annotated were
used to train the SVM to recognize tweets with a negative stance. The training
data consisted of 2,484 tweets annotated for the purpose in this study as well
as 2,679 annotated tweets made available by Cotfas et al. [7]. Randomized
search with cross-validation was used to find optimized parameters for the
vectorizer and the SVM. Since the focus was on negative tweets, it was a
priority to minimize the number of tweets incorrectly labeled to this
category. By prioritizing the precision on negative tweets we can decrease the
risk of themes extracted not corresponding to negative tweets [15]. We
therefore opted for a classifier with the highest precision on the negative
class.

###  3.5 Topic Modeling

Topic modeling is a technique that is used to extract themes from a set of
text documents. The current study aims to investigate topics of discourse with
a negative stance toward the COVID-19 vaccines. Manually categorizing all
296,321 negative tweets into topics would have been incredibly time-consuming,
but topic modeling allows this to be done automatically in a matter of hours.
We used BERTopic to train a model and label each tweet with a topic. BERTopic
is a topic modeling technique that utilizes a complex language model (BERT) to
cluster documents based on semantic similarity. For this work we used the
Python implementation of this model available as a package [14]. A BERT model
pre-trained on a large set of texts can be fine-tuned to be used in a wide
variety of language recognition tasks [8]. We used the BERTopic default
embeddings model that has been fine-tuned on 1B English sentence pairs.

The model was fitted on all tweets classified by the SVM as negative. BERTopic
allows the user to specify the desired number of topics, as well as the
minimum number of documents that should constitute a topic. For the purpose of
keeping the size of the model manageable, the minimum topic size was set to
500 documents, and the number of topics to “auto”. Words with a term frequency
lower than 0.0001 were also removed. The resulting model contained 37 distinct
topics, presented in the results section.

To explore changes in negative discourse over time, a dynamic topic model was
also developed using BERTopic’s topics over time function. BERTopic has a
parameter that controls the number of topic representations each topic should
have in the timeline. A higher value gives more timestamps in the graph, but
risks decreasing the quality of the topic representations. This value was set
to 35 as this number led to graphs with an appropriate level of granularity
for our analysis. To allow for easier comparisons between topics of different
sizes the frequencies were normalized to values between 0 and 1. A topic
having a higher peak than another in the graphs should therefore not be
interpreted as it being more popular, but rather as relative popularity for
that particular topic.

##  4 Results

In the following section we present the performance of the classifier and the
findings based on categorization of all the tweets in the clean dataset using
the classifier. We show how negativity toward the COVID-19 vaccines developed
over the course of the pandemic, which topics made up this negative discourse,
and how these individual topics evolved over time.

###  4.1 Classifier Performance

The best classifier found using randomized search is presented in Table 2. A
macro F1-Score of 0.67 was achieved with a precision on the negative class of
0.8. This classifier used a RBF kernel and a vectorizer with 3000 features of
unigrams and bigrams. The classifier was used to label the 5,858,156 tweets in
the cleaned dataset as having a negative or not negative stance toward the
COVID-19 vaccines. A total of 296,321 tweets were classified as negative,
corresponding to 5.1% of all clean tweets.

Table 2: Performance of SVM classifier. | Precision | Recall | F1-Score | Support
---|---|---|---|---
Negative | 0.80 | 0.33 | 0.46 | 131
Other | 0.81 | 0.97 | 0.88 | 386
Accuracy |  |  | 0.81 | 517
Macro avg. | 0.80 | 0.65 | 0.67 | 517
Weighted avg. | 0.81 | 0.81 | 0.78 | 517

###  4.2 Percentage of Negative Tweets Over Time

A timeline of the percentage of tweets with a negative stance toward the
COVID-19 vaccines during the period March 1, 2020 to July 31, 2021 can be seen
in Figure 2. The average percentage of negativity is 5.1%. This number is
slightly lower than what Cotfas et al. [7] found for the period 8 December,
2020 – 7 January, 2021, where the percentage of tweets against the COVID-19
vaccines was 6.78%. For this same period, our estimate is 4.7%. The
discrepancy could likely be explained by the design choice of our study to
employ a conservative classifier that heavily prioritized not having false
positives for the negative class, with the cost of more false negatives.
Figure 2 shows the percentage of negative tweets per day for the entire time
period. Around the WHO declaration of the pandemic (March 11, 2020), the
amount of negativity was quite stable at 4%, to then rise in April 2020 and
remain relatively high for the rest of the year. Co-occurring with the
vaccination roll-out in December is a decline in the share of negative tweets,
and the following period showed similar negativity to that observed before the
declaration of the pandemic.

Figure 2: 7-day moving average of percentage tweets classified as negative.

###  4.3 Topic model of Negative Tweets

The produced topic model contains 37 different topics with sizes ranging from
531 to 119,196 tweets. The topics are presented in Table 3 along with the top
keywords that were calculated to be the most representative of each topic
using BERTopic’s modified version of TF-IDF. A Dynamic Topic Model was also
created representing the popularity of each topic at 35 different points in
time of the investigated period. In addition to the graphs included in the
results section, the timelines of all topics are available in interactive form
on the companion website [1].

Table 3: Topics of the model ordered by size: The Frequency column contains the number of documents predicted to belong to each topic. Top-words contains the top-scoring words for each topic. The first topic UA contains unassigned documents that did not fit into any of the other topics. Topic 27 contained only usernames as top-words, now censored. Topic | Frequency | Top-words
---|---|---
UA* | 103953 | gates, take, dont, bill
0 | 119196 | virus, take, flu, dont
1 | 10415 | dna, mrna, rna, gene
2 | 9431 | pharma, trump, big, trust
3 | 9378 | mask, masks, wear, wearing
4 | 3232 | passport, passports, id, travel
5 | 3058 | africa, africans, black, african
6 | 3037 | children, kids, school, risk
7 | 2889 | china, chinese, trust, virus
8 | 2820 | guinea, pigs, pig, first
9 | 2159 | exam, india, indian, oxygen
10 | 2048 | pfizer, blood, clots, astrazeneca
11 | 1931 | virus, coronaviruses, years, take
12 | 1787 | 5g, bill, gates, us
13 | 1749 | bill, microchip, chip, tracking
14 | 1549 | jab, jabs, experimental, jabbed
15 | 1363 | hydroxychloroquine, zinc, gates, chloroquine
16 | 1289 | test, testing, untested, rushed
17 | 1184 | polio, measles, smallpox, pox
18 | 1089 | russian, russia, putin, trust
19 | 1028 | hiv, aids, years, 40
20 | 969 | sars, sarscov2, years, mers
21 | 907 | rushed, first, take, im
22 | 874 | poison, poisonous, body, take
23 | 872 | trust, dont, im, wouldnt
24 | 827 | experimental, take, taking, experiment
25 | 794 | antibodies, antibody, test, natural
26 | 772 | boris, brexit, eu, borisjohnson
27 | 737 | username, username, username, username*
28 | 729 | liability, manufacturers, sue, liable
29 | 676 | cure, treatment, want, cures
30 | 629 | science, scientists, trust, dont
31 | 624 | recovery, rate, 99, need
32 | 614 | lockdown, lockdowns, want, dont
33 | 611 | hcq, pharma, big, gates
34 | 570 | injected, injection, inject, body
35 | 531 | travel, fly, flying, airlines

Under this section, a selection of topics from the model will be discussed
using example tweets and graphs of their development over time. The topics
will be referred to with an index that can be used to locate them in Table 3.
The dendrogram in Figure 3 shows how closely related topics are, according to
our topic model. In the following discussion we have chosen to group some
topics together when discussing them, these groupings are based on their
closeness in the dendrogram as well as qualitative similarities seen by the
authors.

Figure 3: Hierarchical structure of the topics labeled with topic index and
the highest scoring word.

### Travel, COVID-passports and territories (topics 4, 7, 9, 18, and 35)

This group contains topics related to traveling and COVID-passports, as well
as discourse around the countries China, Russia, India, and, the continent of
Africa.

Topic 4 is a cluster of documents discussing COVID-passports and certificates
of different forms as a requirement for travel and other activities. The words
“need” and “want” are highly scored in the representation, with many tweets
arguing that they don’t want or need a vaccine nor a passport. The word
mandatory is also prevalent as a result of many discussing the enforcement of
such passports.

Figure 4 shows that this topic first gained popularity in the negative tweets
by the beginning of 2021 as vaccines were being rolled out. This trend is true
also for the general search interest as indicated by the blue line showing the
Google Trends popularity for the query “COVID passport”. The topic shows major
peaks in both the topic model as well as Google trends, May and July of 2021.
The first peak in April has especially high search volume from Great Britain,
as indicated when isolating this period in Google Trends. The go-ahead from
Boris Johnson for COVID-passports could likely be the event sparking this
discussion. Negative tweets from this period discuss COVID-passports in
general, but many also speak to Boris Johnson directly. “@BorisJohnson You
must ensure GB does not push vaccine passports. It is intolerable to think
Pharma and Gates Foundation should have any influence here. You are elected
and accountable-they are not. The vaccinated can still pass on COVID. Stop
penalising hard pressed medics to comply. Disgrace.”. Topic 35 about travel
contains similar discussion but with a focus on travel, airlines and going
abroad, and seems to have had a more constant popularity of discussion
throughout the entire period.

The peak in Figure 4 for July is likely related to the European Union’s
introduction of the EU Digital COVID Certificate (EUDCC). This was introduced
to be used in travel within the Schengen area to prove vaccination, recovery
or recent negative test. Many tweets in this topic argue the passports are
unnecessary and unfair since vaccinated people still can transmit the virus.
Others discuss the forging of COVID passports, “The black market should have a
fake vaccine passport or whatever they are calling it.. This will be my
approach”, “Anybody selling a fake vaccine passport yet? Or am I abit too
early?”. More conspiratorial voices also discussed how the vaccine passports
were a part of a bigger plan: “No thanks. Vaccine passports where in plan
before COVID-19”.

China was also a major topic in the model with 2889 negative tweets being
classified to belong to this cluster. Major keywords for this topic is
“chinese”, “trust”, and “fake”. Some tweets in this topic express distrust in
China, even claiming the virus was “manufactured” there: “I pondered why China
has yet to come up with a vaccine for COVID since it began there. Then I
remembered that any vaccine would be manufactured in China so why would they
hurry? Why would they want to alert the WHO that this virus had to be taken
seriously? China is a threat?”. The topic saw an increase in popularity in
mid-march 2020 (see Figure 4), when the Chinese vaccine was approved for human
testing. Some negative tweets from this period expressed unwillingness to take
a Chinese vaccine, while others discussed conspiracies of a China-developed
virus. This topic was highly salient in the negative discussion at the
beginning of the pandemic and then slowly lost traction over time.

Russia was also a topic of negative discussion with a majority of tweets
between July and November 2020, with a major peak in August, see Figure 4.
This peak is likely due to the Russian vaccine Sputnik V being registered
August 11. Some of these tweets are discussing whether one should trust a
Russian vaccine: “I wouldn’t want a covid vaccine from putin would you?” It
should be noted that our study only investigates tweets in English and the
topics about Russia and China are therefore mainly talking about these
countries from a foreign perspective.

India is also a country prominent in the negative discussion, with a focus on
their exams. Most of the discussion in this topic concerns mandatory in-person
exams during the pandemic. It should be noted that tweets in this topic may
have posed a difficult challenge for the machine learning classifier. That is,
some tweets that express negativity toward in-person exams have incorrectly
been classified as having a negative stance toward the vaccines. For example,
many tweets demanded a shift to online exams, until a vaccine has been made
available: “Shift exams to online mode or no continuous exams. Everyone
demanding for Delay in Compliance due dates. Think Same way, students are not
punching bag for everyone. We are also human, we are not corona immune. We
have not taken corona vaccine yet???”.

The topic about Africa contains several different themes of discussion. A
major theme is claims of racist motives behind the vaccines. Users argue that
vaccines are being sent to African countries part of a secret plot by Bill
Gates to test the side-effects of the vaccines or for the purpose of
population control. Both “‘bill”, “guinea” and “pigs” are highly scoring words
in the topic representation. Moreover the topic also contain similar claims of
racism against African-Americans. This discourse seems to have been sparked by
a statement by Melinda Gates where she proposed that the black population
should receive priority for the vaccines since they face disproportionate
effects from the virus. Another theme of this topic is indicated by the
keyword “madagascar” in the representation, and concerns a tea that is being
used by the Madagascan population with claimed benefits against COVID-19,
which some argue makes a vaccine redundant.

Figure 4: COVID-passports, China, and Russia topic over time. Google Trends
normalized search frequency for “COVID passport” on the axis to the right.

### Pharma and Alternatives to Vaccines (topics 2, 15, 25, 28, 31, and 33)

Under this heading, we have grouped topics that concern discussion around
pharmaceutical companies, their profit motives, as well proposed alternatives
to vaccines. Topic 2 has “big”, “pharma”, “trust” and “rushed’ as highly
scoring words, with many users questioning whether the vaccine is pushed
primarily to make money: “Big Pharma can’t make as much money with a pill
versus a year long process for a vaccine.”. Many users voice their concerns
about the vaccines directly to the then-sitting president, tagging
“@realDonaldTrump” in their tweets. Another commonly raised critique in the
discourse concerns the fact that vaccine manufacturers can not be easily sued,
seen in topic 28: “Who would want a vaccine when Drug companies can’t be sued
if something goes wrong there has not been enough testing of vaccines to make
sure they are safe.”. Two topics (15 & 33) mostly discuss hydroxychloroquine
as a better measure than vaccines against COVID-19. Topic 25 is a topic where
people discuss the protection antibodies give against the virus. A commonly
raised argument is that the vaccines should not be necessary for those who
already had the virus. Topic 31, a topic with 624 tweets, is dominated by
tweets citing a high recovery rate from COVID-19 as an argument against the
vaccines. An example of a very typical tweet from this category is: “If there
is a 99% recovery rate, why would we need a vaccine?”. The topic grew in
popularity during the second half of 2020, reaching a peak in the months
leading up to the vaccine roll-outs, see Figure 5.

Figure 5: Hydroxychloroquine, pharma, and recovery topics over time.

### Popular Conspiracies (topics 12 and 13)

Topic 12 has a very strong peak in April (see Figure 6) which coincides with
the first 5G network tower to be put on fire on the second of April in
Liverpool, UK [27]. Apparently a rumor started that 5G towers are partially to
be blamed for the spread of COVID-19. These rumors were combined with the
sentiment that the 5G towers are used to control people’s minds using
microchips that are supposedly inserted through the Corona vaccination. The
Major of Liverpool declared these rumors as false, which seems to have
triggered some people to take action into their own hands. This incident
triggered discussions among the vaccine-hesitant population, as can be seen in
our timeline, where the words “5G”, “tower”, “chips”, “microchips” are highly
weighted.

Topic 13 is closely related to the topic of 5G (topic 12) but with a stronger
focus on the microchip rumor. The peak and curve behaviors are very similar
for both topics (as they both share microchip as an important word). They peak
in the beginning of April and after that lose in importance. The peak is co-
occurring with the following events that may be contributing to its peak. On
March 18, Bill Gates logged onto Reddit and answered questions. While doing
so, he predicted, that one day we would all carry a digital passport of our
health records. He did not suggest a microchip for that but some kind of
e-vaccine card. On March 19, a Swedish site picked that up and put the
following headline to their article. “Bill Gates will use microchip implants
to fight coronavirus.”. With that the conspiracy theory was born. The
discussion it started among vaccine-hesitant people is reflected in our data.
Simultaneously the non-governmental institution ID2020 (Digital Identity
Alliance) was brought into this as they are advocating for a digital ID for
undocumented people like refugees. Vaccine hesitant individuals drew the
conclusion that ID2020 is involved in putting microchips into people to reach
their goal. Topic 13 has words like “gates”, “implant”, and “tacking” highly
rated with many talking about how the pandemic was planned by gates: “It’s
already planned by Gates. Thank Trump for not allowing ANY Gates vaccine!”.
The topic did not closely correlate with Google search volume for Gates,
indicating that this could be a topic with specific popularity on Twitter.

Figure 6: Bill Gates, 5G, and microchip topics over time.

### Testing and Guinea pigs (topic 8 and 16)

This topic mostly contains tweets talking about the vaccine trials with a
critical tone. The terms “pig” and “pigs” score highly in this topic because
of their appearance in the phrase “Guinea pigs”, used to refer to those
choosing to take the vaccine. One user said: “I WILL NOT BE A GUINEA PIG with
this vaccine! Why don’t you and your family be the test dummies !”. The topic
had most popularity during the vaccine trials of 2020, and has since lost
traction during the roll-out of the vaccines.

Closely related is the topic of testing (topic 16) that also follows a similar
trend over time. Important keywords for this topic like “rushed”, “tested” and
“wouldn’t” reflect the stance expressed that the development process is being
“rushed” and an unwillingness to take an “untested” vaccine. “I have to agree
with you on not taking it. This is an example of why vaccines should not be
rushed with much needed testing not done before it’s approval.”.

### Pfizer and AstraZeneca (topic 10)

This topic peaks in November 2020, see Figure 7. Compared to Google Trends,
there is a clear peak with the search term for Pfizer and Corona on November
9. This is the day that Pfizer announced a successful trial 3 phase of their
corona vaccine with an effectiveness of 90%. On the same day “Nature” and “BBC
News” reported on it. This announcement likely lead to an increase of critical
voices concerned about the safety of the vaccine and triggered discussions
among vaccine hesitant groups. We also see peaks of discourse during the month
of March 2021, with the key term “aztrazeneca” and “blood clots” being used
especially often during this period. At this time concerns surrounding a
potential side effect of blood clots started circulating, causing many
countries to pause their use of the AstraZeneca vaccine. These news seem to
have caused a lot of negative discourse around the vaccine, such as this user
voicing their distrust in the process: “Denmark, Norway, Iceland and Bulgaria
halt use of AstraZeneca’s COVID-19 vaccine over reports of blood clots. Has
any research been done on effects on various blood types and regional DNA
variances in countries?? Thought NOT. Don’t trust Chinese “run” WHO!”.

Figure 7: Pfizer, mRNA, and mask topic over time.

### DNA and mRNA (topic 1)

This topic has two peaks in the investigated time-line. The first peak can be
seen in August 2020 (see Figure 7). Pfizer published its results for phase 1
and phase 2 clinical trials for its mRNA-based vaccine in the journal Nature
on August 12 [23]. Around this time, many news outlets started to describe the
promising vaccine candidates, with a special focus on the new technique to use
mRNA, as done by Pfizer and Moderna. This new technique to use the body’s own
protein generating mechanism to produce the spike protein of the coronavirus
instead of using a dead or a weakened virus, gave rise to concerns about the
vaccines’ safety. Some people were concerned that it could have additional,
adverse effects to a human’s DNA and permanently alter it. The second peak of
the mRNA topic in December 2020 lies close to a peak also seen for the Pfizer
topic, likely due to the publication of Pfizer’s phase 3 trails utilizing the
mRNA-technique.

### Preventative Measures (topic 3, 6, and 32)

One of the largest topics found in the dataset with 8,984 tweets concerns face
masks as a protection measure against the virus. The topic picked up pace in
June 2020 (see Figure 7), and remained popular through the entire period of
study. Top-words for this topic include “mask”, “wearing”, “distancing”,
“protect” and “mandatory”. Since the analysis is limited to tweets with
negative stance toward vaccines, much of the discourse consists of critiques
against mask mandates, calling them discriminatory. One user said: “Yet if I
don’t take the covid “vaccine” or wear a mask you want me to suffer total
discrimination.”. Other users expressed support of masks seeing it as an
alternative to vaccines: “Definitely masks but no to the vaccine. I don’t
trust it because djt pushed them too fast.”.

In addition to masks, lockdowns is another measure that has been widely used
in an attempt to limit the spread of the virus. Many users in topic 3 are
questioning policies being set in place to limit the spread of the virus.
Tweets from early in the pandemic are often arguing that lockdowns are
unfeasible with no end in sight for the pandemic. As the vaccines were
deployed, this topic seems to have shifted arguments that the lockdowns have
been put in place in order to persuade people to take what they perceive as an
unnecessary vaccine.

The eighth-largest topic of the model concerns the vaccination of children
(topic 6). A commonly expressed view in this topic is that the vaccine is too
experimental to use on children and that it is unethical to make vaccines
mandatory for attending school. Many parents claim that they will opt for
homeschooling rather than a vaccine. An argument often used in this topic is
that kids have strong immune systems, and should therefore not require a
vaccine. Another common sentiment in this topic is that the fear of COVID-19
is overblown and that schools should be re-opened.

### Parallels to other disease (topics 17, 19 and 20)

Three topics make comparisons between viral infections and COVID-19. The three
most common in our model are Polio, HIV and earlier strains of SARS. Many
raise the argument that it is implausible that an effective vaccine has
already been found for COVID-19, since many years of research never lead to a
successful vaccine for HIV or SARS. Many also question mRNA technology,
arguing that it is not a “real vaccine”. Some users bring up the Polio
vaccines as an example of vaccines that actually worked, since this disease
has been eradicated in most places, something that cannot be said for
COVID-19.

##  5 Discussion

###  5.1 Principal Findings

In this research we investigated the discourse about COVID-19 vaccines that
took place in English on Twitter between March 1, 2020 to July 31, 2021. We
developed a classifier to identify tweets with a negative stance toward
vaccines, and then modeled the topics of the negative tweets. Looking at the
topics found, we see that a number of different conspiracy theories play a
large role in the negative discourse. Some topics have an obvious link to
popular conspiracy theories, such as those about 5G towers, microchips, and
Bill Gates. However, in topics discussing other concerns such as COVID-
passports, pharmaceutical companies, and racism, references to grand
conspiracies are also commonplace. Another overarching theme concern negative
perceptions of how restrictions and guidelines impact daily life. Distrust in
pharmaceutical companies also seems to fuel much of the hesitancy, and a
discussion of alternatives to vaccines.

The percentage of negative tweets, as well as these more conspiratorial
topics, saw a noteworthy increase in the month following the declaration of
the pandemic. It appears that the attention put on the pandemic gave fuel to
these conspiracies, that previously held a more fringe position. Previous
research has indicated that anxious people are more likely to believe in and
share misinformation [11, 5]. It could therefore be hypothesized that an
increase in anxiety provoked by the news of a pandemic could explain the
increase of hesitant and conspiratorial tweets in the following months.
Furthermore, partaking in misinformation sharing has also been shown to
contribute further to anxiety [34]. It could therefore be posed that a vicious
circle could be formed where anxiousness leads people to share more
misinformation, leading to even more anxiety. Another possible explanation is
that these negative ideas were able to spread to a higher degree as the
network of people involved in the discussion grew.

Similar to Lyu et al. [17] we saw a general decline in the percentage of
negative tweets co-occurring with the vaccine roll-out. Furthermore, beyond
the time-frame investigated by Lyu et al. [17], we could show that negativity
stayed on this lower level during the following seven months.

A decreased popularity was also observed for many of the more conspiratorial
topics such as 5G towers, microchips, and the Gates foundation. This could be
indicative that communication by governments and health authorities around the
issue was successful at combating some of the negative perceptions people had
about the vaccines. However, another explanation could be that the deployment
of vaccines attracted more public attention, shifting the composition of users
involved in the discourse. It can also not be ruled out that Twitter started
enforcing their crisis misinformation policy more strictly as the vaccination
process began, possibly contributing to a decrease in conspiratorial and
vaccine-hesitant tweets.

###  5.2 Limitations

It is important to note that this study is limited in its scope to only
English-speaking Twitter. Although Sloan et al. [31] have found that there is
widespread Twitter use among the general population, the conversations
observed in this study cannot be considered representative of the entirety of
the general public discourse. Furthermore, the demographic using Twitter in
English likely also differs in many respects to the overall user-base. The
analysis of topics was also limited to negative discourse. Future studies
could benefit from investigating how the positive discourse has changed over
the course of the pandemic. This could also help answer questions about which
positive discussions took place as the vaccines rolled out, taking up space
previously held by negativity. As the pandemic has continued into 2022, new
studies could also include an even longer time period to get a more
comprehensive picture of the evolution of topics over the course of the
pandemic

###  5.3 Conclusion

Although the negativity toward vaccinations on Twitter decreased as the
vaccines rolled out, many countries still faced difficulties with vaccination
willingness within their populations. In our analysis of negative tweets over
the course of the pandemic, we saw different indications for vaccine
hesitation such as low trust toward authorities, strong insistence on personal
freedom with respect to following guidelines such as the wearing of masks or
uptake of vaccines, as well as a non-negligible influence of conspiracy
theories around COVID-19 itself as well as the vaccinations. While the
percentage of negative tweets and conspiratorial topics reduced as the
vaccines rolled out, it is still possible that the negative discourse in the
preceding months had already swayed some users into a more hesitant stance. As
unvaccinated people face a much higher rate of hospitalization and death [10],
hesitancy could have severely negative outcomes on society. In a country where
non-pharmaceutical interventions are relaxed, it has been estimated that
hesitancy could lead to a more than 7 times higher mortality [22]. In order to
improve the handling of the current pandemic as well as being prepared for
future pandemics, a successful communication strategy should address concerns
circulating on social media in an early stage, preventing negative perceptions
from taking hold. The fact that such bizarre conspiracies as mind-controlling
5G antennas were widely circulated must be seen as a failure of communicative
and educational efforts and take up valuable space where constructive
conversations around the pros and cons of vaccination could be had.

### Data Availability

The dataset of tweets annotated for the purpose of this study has been made
available on GitHub [3]. Plots can be viewed interactively on the companion
website [1].

### Acknowledgments

We acknowledge the computational resources provided by the Aalto Science-IT
project. We also acknowledge the input from Leila Gharavi, the feedback from
Dr. Koustuv Saha, as well as the work of the two annotators.

### Author contributions

Conceptualization, G.L., T.A. and B.K.; methodology, G.L., T.A. and B.K;
software G.L.; validation, G.L., T.A. and B.K; formal analysis, G.L., T.A. and
B.K.; data curation, G.L.; writing—review and editing, G.L., T.A. and B.K. All
authors have read and agreed to the final version of the manuscript.

### Funding

This research received no external funding.

### Conflicts of Interest

The authors declare no conflict of interest.

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# Proving dualities for $q$MZVs with connected sums

Benjamin Brindle The author has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 101001179).

Abstract. This paper gives a new application of so-called connected sums,
introduced recently by Seki and Yamamoto [SY]. Special about our approach is
that it proves a duality for the Schlesinger– Zudilin and the Bradley–Zhao
model of qMZVs simultaneously. The latter implies the duality for MZVs and the
former can be used to prove the shuffle product formula for MZVs. Furthermore,
the $q$-Ohno relations, a generalization of Bradley–Zhao duality, is also
obtained.
2010 Mathematics Subject Classification. 11M32, 05A30.
Key words: Multiple zeta values; $q$-multiple zeta values; duality; connected
sums.

##  1 Notation and definitions

For an _admissible index_ $\mathbf{k}=(k_{1},\dots,k_{r})\in\mathbb{N}^{r}$,
i.e., $r\geq 0$ and $k_{1}\geq 2$, its _multiple zeta value_ (MZV) is

$\displaystyle\zeta(\mathbf{k}):=\sum\limits_{m_{1}>\dots>m_{r}>0}\frac{1}{m_{1}^{k_{1}}}\cdots\frac{1}{m_{r}^{k_{r}}}.$

To understand the algebraic structure of MZVs better on the one hand and to
get connections to holomorphic functions, in particular, modular forms (see
[GKZ], [Bac]), on the other hand, it is useful to introduce $q$-analogs of
MZVs. There are several models of $q$-analogs. We focus in this paper on the
one by Schlesinger, and Zudilin and Bradley and Zhao. For further details on
these and other models, we refer to [Zha], [Bri]. In this note $q$ will be a
formal variable or a real number with $0<q<1$.

The Bradley–Zhao model is defined as follows: Set
$\zeta_{q}^{\mathrm{BZ}}(\emptyset):=1$ and for
$\mathbf{k}=(k_{1},\dots,k_{r})$ an admissible index define

$\displaystyle\zeta_{q}^{\mathrm{BZ}}(\mathbf{k}):=\sum\limits_{m_{1}>\dots>m_{r}>0}\frac{q^{m_{1}(k_{1}-1)}}{(1-q^{m_{1}})^{k_{1}}}\cdots\frac{q^{m_{r}(k_{r}-1)}}{(1-q^{m_{r}})^{k_{r}}}.$

Similar, but different, we define Schlesinger–Zudilin $q$MZVs via
$\zeta_{q}^{\mathrm{SZ}}(\emptyset):=1$ and for every _SZ-admissible index_
$\mathbf{k}$, i.e., $\mathbf{k}\in\mathbb{N}_{0}^{r}$ for some $r\geq 0$ with
$k_{1}\geq 1$, we set

$\displaystyle\zeta_{q}^{\mathrm{SZ}}(\mathbf{k}):=\sum\limits_{m_{1}>\dots>m_{r}>0}\frac{q^{m_{1}k_{1}}}{(1-q^{m_{1}})^{k_{1}}}\cdots\frac{q^{m_{r}k_{r}}}{(1-q^{m_{r}})^{k_{r}}}.$

##  2 Dualities

Write an admissible index $\mathbf{k}$ in the shape
$\mathbf{k}=\left(k_{1}+1,\\{1\\}^{d_{1}-1},\dots,k_{r}+1,\\{1\\}^{d_{r}-1}\right)$
with $k_{j},\,d_{j}\geq 1$ unique ($\\{1\\}^{d}$ means that $1$ is repeated
$d$-times). For the next two theorems, we need the _dual index_ ,

$\displaystyle\mathbf{k}^{\vee}:=\left(d_{r}+1,\\{1\\}^{k_{r}-1},\dots,d_{1}+1,\\{1\\}^{k_{1}-1}\right).$

######  Theorem 1 (MZV-Duality, [Zag, §9]).

For every admissible index $\mathbf{k}$, we have
$\zeta\left(\mathbf{k}\right)=\zeta\left(\mathbf{k}^{\vee}\right)$.

The next theorem can be seen as a $q$-analogue of MZV-duality since MZV-
duality follows immediately from it (cf. the proof):

######  Theorem 2 (BZ-Duality, [Bra, Thm. 5]).

For every admissible index $\mathbf{k}$, we have
$\zeta_{q}^{\mathrm{BZ}}\left(\mathbf{k}\right)=\zeta_{q}^{\mathrm{BZ}}\left(\mathbf{k}^{\vee}\right)$.

A generalization of BZ-duality are the so-called $q$-Ohno relations, of which
BZ-duality is the special case $c=0$:

######  Theorem 3 ($q$-Ohno relation, [OT, Thm. 1]).

For any admissible index ${\mathbf{k}}=(k_{1},\dots,k_{r})$ and any
$c\in\mathbb{N}_{0}$ we have

$\displaystyle\sum\limits_{|{\mathbf{c}}|=c}\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}+{\mathbf{c}}\right)=\sum\limits_{|{\mathbf{c}}|=c}\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}^{\vee}+{\mathbf{c}}\right),$

where we sum over all ${\mathbf{c}}=(c_{1},\dots,c_{r})\in\mathbb{N}_{0}^{r}$
with $|{\mathbf{c}}|:=c_{1}+\dots+c_{r}=c$.

For the SZ-model, we write an SZ-admissible index in the shape
$\mathbf{k}=\left(k_{1}+1,\\{0\\}^{d_{1}},\dots,k_{r}+1,\\{0\\}^{d_{r}}\right)$
with $k_{j},\,d_{j}\geq 0$ unique and define the _SZ-dual index_ ,

$\displaystyle\mathbf{k}^{\dagger}:=\left(d_{r}+1,\\{0\\}^{k_{r}},\dots,d_{1}+1,\\{0\\}^{k_{1}}\right).$

######  Theorem 4 (SZ-Duality, [Zha, Thm. 8.3]).

For $\mathbf{k}$ SZ-admissible, we have
$\zeta_{q}^{\mathrm{SZ}}\left(\mathbf{k}\right)=\zeta_{q}^{\mathrm{SZ}}\left(\mathbf{k}^{\dagger}\right)$.

Note that BZ- and SZ-duality on algebraic level look the same, both can be
obtained by the same anti-automorphism on the non-commutative free algebra in
two variables (see, e.g., [Bri, Thm. 3.5, Thm. 3.16]). But they imply
different things. BZ-duality gives direct duality for MZVs, while SZ-duality
does not. However, SZ-duality implies another important result in the theory
of MZVs, namely the shuffle product formula (cf. [EMS], [Sin], for details
[Bri, Thm. 3.46]).

For some calculations in the next section we need the connection between
admissible and SZ-admissible index: An index $\mathbf{k}$ is admissible if and
only if $\mathbf{k}-{\mathbf{1}}$ is SZ-admissible ($\mathbf{k}+{\mathbf{1}}$
is the index, which is $\mathbf{k}$ with every entry increased by $1$; similar
for $\mathbf{k}-{\mathbf{1}}$). Furthermore, we have for $\mathbf{k}$
admissible

$\displaystyle(\mathbf{k}-{\mathbf{1}})^{\dagger}=\mathbf{k}^{\vee}-{\mathbf{1}}.$
(2.1)

##  3 Connected sums & proof of dualities

As a new tool for proving identities among ($q$-)multiple zeta values, Seki
and Yamamoto introduced the concept of so-called connected sums (this notion
is independent of connected sums in topology). With connected sums, they have
proven, e.g., the duality of MZVs, Hoffman’s identity, and the $q$-analogue of
Ohno’s relation, cf. [Sek] or [SY].

Using connected sums, we give a new proof of the duality of
Schlesinger–Zudilin $q$MZVs, the duality of Bradley–Zhao $q$MZVs and the usual
duality of MZVs. It turns out that the connected sum defined below has the
power to prove all three statements at once. As a by-product, we also get a
proof for the $q$-Ohno relation. The proof is inspired by the one of Seki and
Yamamoto ([SY]), where the authors proved $q$-Ohno’s relation for non-modified
Bradley–Zhao $q$MZVs. We work with modified $q$MZVs, which will be here the
reason that we can prove also Schlesinger–Zudilin duality at the same time.

######  Definition 5 (Connected sum).

Let be $r,\,s\geq 0$, $\mathbf{k}=(k_{1},\dots,k_{r})\in\mathbb{N}_{0}^{r},\
\boldsymbol{\ell}=(\ell_{1},\dots,\ell_{s})\in\mathbb{N}_{0}^{s}$ and $x$ real
with $|x|<1$. Define the connected sum as

$\displaystyle
Z_{q}\left(\mathbf{k};\boldsymbol{\ell};x\right):=\sum\limits_{\begin{subarray}{c}m_{1}>\dots>m_{r}>m_{r+1}=0\\\
n_{1}>\dots>n_{s}>n_{s+1}=0\end{subarray}}$
$\displaystyle\prod\limits_{i=1}^{r}\frac{q^{m_{i}k_{i}}}{(1-q^{m_{i}}x)(1-q^{m_{i}})^{k_{i}}}\prod\limits_{j=1}^{s}\frac{q^{n_{j}\ell_{j}}}{(1-q^{n_{j}}x)(1-q^{n_{j}})^{\ell_{j}}}$
$\displaystyle\times\frac{q^{m_{1}n_{1}}f_{q}(m_{1};x)f_{q}(n_{1};x)}{f_{q}(m_{1}+n_{1};x)},$

where $f_{q}(m;x):=\prod\limits_{h=1}^{m}(1-q^{h}x)$.

######  Remark 6.

  1. (i)

The connected sum $Z_{q}$ is symmetric in ${\mathbf{k}}$ and
${\boldsymbol{\ell}}$ by definition.

  2. (ii)

Notice that the connected sum is well-defined in the sense that it is a series
over positive real numbers and hence either a positive real number (if
convergent) or $+\infty$ (if not convergent).

  3. (iii)

If $k_{1}\geq 1$, then
$Z_{q}\left({\mathbf{k}};\emptyset;0\right)=\zeta_{q}^{\mathrm{SZ}}\left(\mathbf{k}\right)$.

  4. (iv)

If $k_{1}\geq 1$, then $\lim\limits_{x\rightarrow
1}Z_{q}\left({\mathbf{k}};\emptyset;x\right)=\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k+1}}\right)$.

######  Proposition 7 (Boundary conditions).

If $k_{1}\geq 1,\,0<q<1$ and $x\in\mathbb{R}$ with $|x|<1$, then
$Z_{q}\left({\mathbf{k}};\emptyset;x\right)$ is a well-defined real number.

###### Proof.

One has

$\displaystyle Z_{q}\left({\mathbf{k}};\emptyset;x\right)=$
$\displaystyle\,\sum\limits_{m_{1}>\dots>m_{r+1}:=0}\prod\limits_{i=1}^{r}\frac{q^{m_{i}k_{i}}}{(1-q^{m_{i}}x)(1-q^{m_{i}})^{k_{i}}}$
$\displaystyle\leq$
$\displaystyle\,\frac{1}{(1-qx)^{r}}\sum\limits_{m_{1}>\dots>m_{r+1}:=0}\prod\limits_{i=1}^{r}\frac{q^{m_{i}k_{i}}}{(1-q^{m_{i}})^{k_{i}}}=\,\frac{1}{(1-qx)^{r}}\,\zeta_{q}^{\mathrm{SZ}}(k_{1},\dots,k_{r}),$

which is well-defined since $k_{1}\geq 1$, i.e., $(k_{1},\dots,k_{r})$ SZ-
admissible. ∎

After we have checked well-definednes of $Z_{q}$, we state and prove now
distinguished relations among our connected sums.

######  Theorem 8 (Transport relations).

Let be $r,s\geq 0$ and $k_{1},\dots,k_{r},\,\ell_{1},\dots,\ell_{s}\geq 0$. If
$s>0$,

$\displaystyle
Z_{q}\left((0,k_{1},\dots,k_{r});(\ell_{1},\dots,\ell_{s});x\right)=Z_{q}\left((k_{1},\dots,k_{r});(\ell_{1}+1,\ell_{2},\dots,\ell_{s});x\right)$
(3.1)

and if $r>0$,

$\displaystyle
Z_{q}\left((k_{1}+1,k_{2},\dots,k_{r});(\ell_{1},\dots,\ell_{s});x\right)=Z_{q}\left((k_{1},\dots,k_{r});(0,\ell_{1},\ell_{2},\dots,\ell_{s});x\right).$
(3.2)

###### Proof.

The second equality follows from the first by symmetry and the first one is
obtained from

$\displaystyle\sum\limits_{a>m}\frac{1}{1-q^{a}x}\frac{q^{an}f_{q}(a;x)f_{q}(n;x)}{f_{q}(a+n;x)}$
$\displaystyle=\,$
$\displaystyle\,\frac{q^{n}}{1-q^{n}}\sum\limits_{a>m}\left(\frac{q^{(a-1)n}f_{q}(a-1;x)f_{q}(n;x)}{f_{q}(a+n-1;x)}-\frac{q^{an}f_{q}(a;x)f_{q}(n;x)}{f_{q}(a+n;x)}\right)$
$\displaystyle=\,$
$\displaystyle\,\frac{q^{n}}{1-q^{n}}\frac{q^{mn}f_{q}(m;x)f_{q}(n;x)}{f_{q}(m+n;x)}$

and setting $m=m_{1},\,n=n_{1},\,a=m_{0}$. ∎

This theorem is the key of proving Theorems 1–4. Especially, the following
corollary will be needed, together with the connection of $Z_{q}$ with
$\zeta_{q}^{\mathrm{BZ}}$ resp. $\zeta_{q}^{\mathrm{SZ}}$ (Remark 6).

######  Corollary 9.

For every SZ-admissible index ${\mathbf{k}}$ and real $x$ with $|x|<1$ we have

$\displaystyle
Z_{q}\left({\mathbf{k}};\emptyset;x\right)=Z_{q}\left(\emptyset;{\mathbf{k}}^{\dagger};x\right).$

###### Proof.

For all indices ${\mathbf{k}}$ and ${\boldsymbol{\ell}}$ and $k\geq 1,\,d\geq
0$ we obtain (by $\left(k,\\{0\\}^{d},{\mathbf{k}}\right)$ we mean the
concatination of the indices $\left(k,\\{0\\}^{d}\right)$ and ${\mathbf{k}}$)
by applying $k$-times (3.2) first and then $(d+1)$-times (3.1)

$\displaystyle
Z_{q}\left(\left(k,\\{0\\}^{d},{\mathbf{k}}\right);{\boldsymbol{\ell}};x\right)=Z_{q}\left(\left(\\{0\\}^{d+1},{\mathbf{k}}\right);\left(\\{0\\}^{k},{\boldsymbol{\ell}}\right);x\right)=Z_{q}\left({\mathbf{k}},\left(d+1,\\{0\\}^{k-1};{\boldsymbol{\ell}}\right);x\right).$
(3.3)

Now, set ${\boldsymbol{\ell}}=\emptyset$ and write an SZ-admissible index
${\mathbf{k}}$ in the form

$\displaystyle{\mathbf{k}}=\left(k_{1},\\{0\\}^{d_{1}},\dots,k_{r},\\{0\\}^{d_{r}}\right).$

Then we obtain the corollary by induction on $r$ and using (3.3) in the
induction step.∎

With the connection of $Z_{q}$ and $\zeta_{q}^{\mathrm{SZ}}$ (Rem. 6 (iii)),
SZ-duality follows directly:

###### Proof of Theorem 4.

Take some SZ-admissible index ${\mathbf{k}}$. Using the symmetry of $Z_{q}$
and setting $x=0$, the claim follows by Corollary 9:

$\displaystyle\zeta_{q}^{\mathrm{SZ}}\left({\mathbf{k}}\right)=Z_{q}\left({\mathbf{k}};\emptyset;0\right)=Z_{q}\left(\emptyset;{\mathbf{k}}^{\dagger};0\right)=Z_{q}\left({\mathbf{k}}^{\dagger};\emptyset;0\right)=\zeta_{q}^{\mathrm{SZ}}\left({\mathbf{k}}^{\dagger}\right).$
∎

Analogously, we are able to prove BZ-duality:

###### Proof of Theorem 2.

For an admissible index ${\mathbf{k}}$ we have, using Remark 6 and Corollary
9,

$\displaystyle\zeta_{q}^{\mathrm{BZ}}({\mathbf{k}})$
$\displaystyle=\lim\limits_{x\rightarrow
1}Z_{q}\left({\mathbf{k}}-{\mathbf{1}};\emptyset;x\right)=\lim\limits_{x\rightarrow
1}Z_{q}\left(\emptyset;({\mathbf{k}}-{\mathbf{1}})^{\dagger};x\right)$
$\displaystyle=\lim\limits_{x\rightarrow
1}Z_{q}\left(({\mathbf{k}}-{\mathbf{1}})^{\dagger};\emptyset;x\right)=\zeta_{q}^{\mathrm{BZ}}\left(({\mathbf{k}}-{\mathbf{1}})^{\dagger}+{\mathbf{1}}\right)=\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}^{\vee}\right).$
∎

######  Example 10.

We give a concrete example of applying transport relations step by step to
make clear what happens:

$\displaystyle\,Z_{q}\left((1,0);\emptyset;x\right)=Z_{q}\left((0,0);(0);x\right)$
$\displaystyle=\,$
$\displaystyle\,Z_{q}\left((0);(1);x\right)=Z_{q}\left(\emptyset;(2);x\right)$

By Remark 6 (iii) resp. (iv), we obtain
$\zeta_{q}^{SZ}(1,0)=\zeta_{q}^{SZ}(2)$ resp.
$\zeta_{q}^{BZ}(2,1)=\zeta_{q}^{BZ}(3)$. We have $(1,0)^{\dagger}=(2)$ and
$(2,1)^{\vee}=(3)$, why these results indeed correspond to SZ- resp. BZ-
duality.

We derive in the following the proof of MZV-duality, Theorem 1, from BZ-
duality:

###### Proof of Theorem 1.

Let ${\mathbf{k}}$ be any admissible index. Denote by
$\operatorname{wt}\left(\mathbf{k}\right):=k_{1}+\dots+k_{r}$ the sum of all
entries, the _weight_ of $\mathbf{k}$. Obviously, one has
$\operatorname{wt}\left(\mathbf{k}\right)=\operatorname{wt}\left(\mathbf{k}^{\vee}\right)$.
We have

$\displaystyle\zeta\left({\mathbf{k}}\right)=\lim\limits_{q\rightarrow
1}(1-q)^{\operatorname{wt}\left(\mathbf{k}\right)}\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}\right)=\lim\limits_{q\rightarrow
1}(1-q)^{\operatorname{wt}\left(\mathbf{k}^{\vee}\right)}\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}^{\vee}\right)=\zeta\left({\mathbf{k}}^{\vee}\right).$
∎

Consider for the proof of Theorem 3 connected sums of the form
$Z_{q}\left({\mathbf{k}};\emptyset;x\right)$ and the related one of the form
$Z_{q}\left(\emptyset;{\boldsymbol{\ell}};x\right)$ using transport relations.
In both, we will develop all occurring terms as a Taylor series at $x=1$,
mainly we use that for all $m\in\mathbb{N}$, we have

$\displaystyle\frac{1}{1-q^{m}x}=\,\frac{1}{1-q^{m}}\frac{1}{1-\frac{q^{m}}{1-q^{m}}(x-1)}=\frac{1}{1-q^{m}}\sum\limits_{c\geq
0}\left(\frac{q^{m}}{1-q^{m}}\right)^{c}(x-1)^{c}=\,\sum\limits_{c\geq
0}\frac{q^{mc}}{(1-q^{m})^{c+1}}(x-1)^{c}.$

###### Proof of Theorem 3.

Let ${\mathbf{k}}=(k_{1},\dots,k_{r})$ be an admissible index. Then we have

$\displaystyle Z_{q}\left({\mathbf{k}}-{\mathbf{1}};\emptyset;x\right)=\,$
$\displaystyle\sum\limits_{m_{1}>\dots>m_{r}>0}\prod\limits_{j=1}^{r}\frac{1}{1-q^{m_{j}}x}\frac{q^{m_{j}(k_{j}-1)}}{(1-q^{m_{j}})^{k_{j}-1}}$
$\displaystyle=\,$
$\displaystyle\sum\limits_{m_{1}>\dots>m_{r}>0}\prod\limits_{j=1}^{r}\left(\sum\limits_{c_{j}\geq
0}\frac{q^{m_{j}c_{j}}}{(1-q^{m_{j}})^{c_{j}+1}}(x-1)^{c_{j}}\frac{q^{m_{j}(k_{j}-1)}}{(1-q^{m_{j}})^{k_{j}-1}}\right)$
$\displaystyle=\,$ $\displaystyle\sum\limits_{c_{1},\dots,c_{r}\geq
0}\sum\limits_{m_{1}>\dots>m_{r}>0}\left(\prod\limits_{j=1}^{r}\frac{q^{m_{j}(k_{j}+c_{j}-1)}}{(1-q^{m_{j}})^{k_{j}+c_{j}}}\right)(x-1)^{c_{1}+\dots+c_{r}}$
$\displaystyle=\,$ $\displaystyle\sum\limits_{c_{1},\dots,c_{r}\geq
0}\zeta_{q}^{\mathrm{BZ}}({\mathbf{k}}+{\mathbf{c}})(x-1)^{|{\mathbf{c}}|}.$

Since $\mathbf{k}$ was an arbitrary admissible index and $\mathbf{k}^{\vee}$
is admissible too, we get

$\displaystyle
Z_{q}\left(\emptyset;{\mathbf{k}}^{\vee}-{\mathbf{1}};x\right)=\sum\limits_{c_{1},\dots,c_{r}\geq
0}\zeta_{q}^{\mathrm{BZ}}\left({\mathbf{k}}^{\vee}+{\mathbf{c}}\right)(x-1)^{|{\mathbf{c}}|}.$

Now, since
$Z_{q}\left({\mathbf{k}}-{\mathbf{1}};\emptyset;x\right)=Z_{q}\left(\emptyset;{\mathbf{k}}^{\vee}-{\mathbf{1}};x\right)$
for every admissible index by using the transport relations, the result
follows by comparing the coefficient of $(x-1)^{c}$ on both sides.∎

In the same way, we can consider
$Z_{q}\left({\mathbf{k}}-{\mathbf{1}};\emptyset;x\right)$ when developing
$\frac{1}{1-q^{m}x}$ around some $a\in\mathbb{R}$, i.e.,

$\displaystyle\frac{1}{1-q^{m}x}=\frac{1}{1-aq^{m}-q^{m}(x-a)}=\frac{1}{1-aq^{m}}\frac{1}{1-\frac{q^{m}}{1-aq^{m}}(x-a)}=\sum\limits_{c\geq
0}\frac{q^{mc}}{(1-aq^{m})^{c+1}}(x-a)^{c}.$

Then it is

$\displaystyle Z_{q}\left({\mathbf{k}};\emptyset;x\right)=\,$
$\displaystyle\sum\limits_{m_{1}>\dots>m_{r}>0}\prod\limits_{j=1}^{r}\frac{1}{1-q^{m_{j}}x}\frac{q^{m_{j}k_{j}}}{(1-q^{m_{j}})^{k_{j}}}$
$\displaystyle=\,$
$\displaystyle\sum\limits_{m_{1}>\dots>m_{r}>0}\prod\limits_{j=1}^{r}\sum\limits_{c_{j}\geq
0}\frac{q^{m_{j}c_{j}}}{(1-aq^{m_{j}})^{c_{j}+1}}\frac{q^{m_{j}k_{j}}}{(1-q^{m_{j}})^{k_{j}}}(x-a)^{c_{j}}.$

######  Remark 11.

The series

$\displaystyle\sum\limits_{m_{1}>\dots>m_{r}>0}\prod\limits_{j=1}^{r}\frac{q^{m_{j}c_{j}}}{(1-aq^{m_{j}})^{c_{j}+1}}\frac{q^{m_{j}k_{j}}}{(1-q^{m_{j}})^{k_{j}}}$

for $c_{1},\dots,c_{r}\geq 0$, $k_{1}\geq 2,\,k_{2},\dots,k_{r}\geq 1$ and
$a\in[0,1]$ can be seen as $q$-analogue of MZVs: For $a=1$ we have seen
already by proving the $q$-Ohno relation, how this works. For arbitrary $a$,
it is not clear so far, whether we can prove more identities among $q$MZVs
with this shape of the connected sum. This could be interesting for the
future.

## Acknowledgements

I would like to thank to thank Kathrin Bringmann for her helpful comments on
the paper. Furthermore, I thank Henrik Bachmann and Ulf Kühn for fruitful
discussions and lots of comments while supervising my master thesis, of which
this paper is part of.

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University of Cologne, Department of Mathematics and Computer Sciences

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