# A Corpus for Sentence-level Subjectivity Detection
on English News Articles

Francesco Antici1 Andrea Galassi1🖂 Federico Ruggeri1🖂 Katerina Korre2
Arianna Muti2 and Alessandra Bardi2 and Alice Fedotova2 and Alberto Barrón-
Cedeño2
1 DISI, University of Bologna, Italy
2 DIT, University of Bologna, Italy
<EMAIL_ADDRESS>
{aikaterini.korre2,arianna.muti2,a.barron<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

We present a novel corpus for subjectivity detection at the sentence level. We
develop new annotation guidelines for the task, which are not limited to
language-specific cues, and apply them to produce a new corpus in English. The
corpus consists of 411 subjective and 638 objective sentences extracted from
ongoing coverage of political affairs from online news outlets. This new
resource paves the way for the development of models for subjectivity
detection in English and across other languages, without relying on language-
specific tools like lexicons or machine translation. We evaluate state-of-the-
art multilingual transformer-based models on the task, both in mono- and
cross-lingual settings, the latter with a similar existing corpus in Italian
language. We observe that enriching our corpus with resources in other
languages improves the results on the task.

##  1 Introduction

Subjectivity Detection (SD) is a task that contributes to several other
Natural Language Processing applications Riloff and Wiebe (2003). However,
spotting subjectivity is a challenging task even for human experts. Indeed,
the perception of subjectivity is subjective in itself: it may derive from
different interpretations of the language, different background knowledge, and
personal biases Chaturvedi et al. (2018). Consequently, the creation of
corpora for SD is notably difficult and costly Riloff and Wiebe (2003).

Standard approaches to overcome creating corpora for SD rely on word spotting
Wiebe and Riloff (2005); Riloff and Wiebe (2003) and existing lexicons Das and
Sagnika (2020); Yu and Hatzivassiloglou (2003); Villena-Román et al. (2015).
Nonetheless, these solutions are known to be limited to domain- and language-
specific assumptions Pang and Lee (2004a), and require external tools like
machine translation for language transfer Benamara et al. (2011). Recently,
few attempts have investigated the use of annotation guidelines to define
general-purpose SD tasks. However, such approaches face demanding issues like
annotation ambiguity Chaturvedi et al. (2018), interpretation bias Geva et al.
(2019), and ambiguous case resolution.

We take inspiration from the latter approach, and we (i) develop a novel set
of annotation guidelines that are language-agnostic, and that can be used for
the creation of new corpora in other languages; (ii) design an annotation
procedure based on the prescriptive paradigm Rottger et al. (2022) to mitigate
annotation conflicts by discussing and resolving controversial cases; (iii)
manually curate a novel corpus for SD concerning controversial topics like
political affairs in news articles. More precisely, the corpus contains around
1,000 sentences, each manually labeled by at least two annotators.

In our experiments, we evaluate a set of machine learning models on our
corpus, including traditional classifiers and transformer-based language
models like BERT Devlin et al. (2019) and SBERT Reimers and Gurevych (2019).
Moreover, we perform multilingual and zero-shot cross-lingual experiments on
an Italian corpus, as cross-lingual transfer learning has shown to be
effective for low-resource languages with little or no labeled training data
de Vries et al. (2022). We observe that transformer-based models achieve the
best performance on our corpus when trained in the multi-lingual setting.
Additionally, we observe that SBERT achieves comparable performance to its
monolingual counterpart in the cross-lingual setting. Our corpus, our
guidelines, and our code are publicly
available.111https://www.dropbox.com/sh/pterfc16inz0h7b/AAB3csoKxYJNa11A-CZZt4xZa.

##  2 Related Work

The definition of subjective is very wide and can include opinions, rants,
allegations, accusations, suspicions, and speculations Riloff and Wiebe
(2003). Previous works have mostly coupled it with well-known tasks like
sentiment analysis Stepinski and Mittal (2007); Chaturvedi et al. (2018), and
bias detection Aleksandrova et al. (2019); Hube and Fetahu (2019). But being
able to automatically capture subjectivity aspects may also impact tasks of
question answering, summarization, information retrieval, claim extraction
Riloff and Wiebe (2003); Banea et al. (2014), and fact-checking Vieira et al.
(2020); Jerônimo et al. (2019). SD has been investigated at several
granularity levels of the detection: at sentence level Riloff and Wiebe
(2003), segment level Benamara et al. (2011) and document level Antici et al.
(2021). Following the results of Vieira et al., which "indicate that the
fragmentation of news allows the identification of subjectivity markers that
cannot be identified when considering the entire documents”, we build our
corpus by annotating subjectivity at the sentence level.

For what concerns the domain, previous work has addressed movie reviews Pang
and Lee (2004b); Benamara et al. (2011), social media content Volkova et al.
(2013); Bosco et al. (2013), and, most notably, news media Wiebe et al.
(1999); Antici et al. (2021). We address the domain of news since it is
particularly of interest for tasks of SD: while news media are generally
expected to be objective, Riloff and Wiebe (2003) observed that newspaper
articles contain a substantial amount of subjective content. Despite the
existence of previous corpora for SD in news, we hypothesize that SD
classifiers trained on corpora of recent news would perform better than models
trained on corpora of older news. Indeed, the rapid advancement of technology
over the last decades has impacted language and its evolution and also the
communication methods used by news media Favilla (2017).

Few contributions on SD have addressed languages other than English, such as
French Benamara et al. (2011) and Italian Antici et al. (2021); Esuli and
Sebastiani (2006). For what concerns multilingual approaches, Banea et al.
(2010) investigated SD on 6 languages using machine translation and found that
multilingual information can be beneficial for SD. Banea et al. (2014)
addressed multilingual cross-lingual SD in Romanian and English documents by
aligning the corresponding version of WordNet Fellbaum (2010). To the best of
our knowledge, we are the first to address multi-lingual sentence-level SD
experimenting over two corpora that have been manually annotated without the
use of machine translation.

##  3 Corpus

We manually select 23 articles from 8 English outlets222The outlets are
tribunemag.co.uk, spectator.co.uk, shtfplan.com, vdare.com, theweek.com,
frontpagemag.com, economist.com, and theguardian.com. about controversial
political affairs, such as law, civil rights, economics, and internal
politics. Our selection was constrained by three aspects: the popularity of
the source, a wide range of coverage standpoints (e.g., left vs right, liberal
vs conservative) and controversial topics.

For the annotation, we recruit 7 annotators with near-to-native English
knowledge and a background in either linguistics or computing science. We
carry out two pilot annotations following a preliminary version of the
annotation guidelines Antici et al. (2021). We define such guidelines by
following the prescriptive paradigm Rottger et al. (2022) to impose a specific
and consistent conceptualization of subjectivity for annotation. In
particular, we conceive SD as a step that can contribute to information-
retrieval-oriented tasks, such as claim extraction and fact-checking. We focus
on detecting cases where the subjectivity of the author influences the
information in the text and therefore impacts the retrieval of information. In
total, annotators label 270 sentences as subjective (SUBJ) or objective (OBJ).
After each stage, all annotators gather together to discuss ambiguous
sentences and analyze edge cases toward the refinement of the guidelines. This
is in line with the prescriptive paradigm, where annotator disagreement is a
call to action to refine annotation guidelines. In this process, we refer to
linguistic resources Finegan (1995); University of Adelaide (2014) to refine
our guidelines concerning edge cases like speculations, biased language,
conclusions backed up by factual evidence, and the speaker’s emotions. In the
first pilot annotation, the inter-annotator agreement (IAA) measured as
Krippendorf’s alpha Krippendorff (2011) is 0.40 (“fair agreement”). During the
second pilot annotation, we test the impact of considering contextual
information when annotating. Half of the annotators label the sentences in the
same order as they appear in their corresponding articles, while half perform
labeling in random order. The first group’s IAA is 0.38, while the second
group’s is 0.53 (“moderate agreement”). Moreover, we observe that contextual
information leads to an increased annotators’ workload. For this reason, we
agree on labeling without context.

A sentence is subjective if its content is based on or influenced by personal
feelings, tastes, or opinions. Otherwise, the sentence is objective.
---
More precisely, a sentence is subjective if one or more of the following
conditions apply:  1. expresses an explicit personal opinion from the author
(e.g., speculations to draw conclusions); 2. includes sarcastic or ironic
expressions; 3. gives exhortations of personal auspices; 4. contains
discriminating or downgrading expressions; 5. contains rhetorical figures that
convey the author’s opinion. The following ambiguous cases in a sentence are
objective: a third-party’s opinion, comments that do not draw conclusions and
leave open questions, and factual conclusions.
Note 1: Quotation marks represent an explicit third-party’s opinion when used
to quote a third person, and, thus, are objective.
Note 2: Personal feelings, emotions or mood of the author, without conveying
opinions on the matter, are considered objective since the author is the most
reliable source for information regarding their own emotions. Emotion-carrying
statements are not excluded since they frequently occur in news articles and
excluding them from the corpus would turn it less useful in real application
scenarios.
Figure 1: Excerpt of the guidelines for manual annotation of subjective and
objective sentences.

We provide the annotators with the guidelines reported in Appendix D and
summarized in Figure 1. The guidelines include a definition of subjectivity,
specific examples, and cases that, during the pilot annotation stages, we
identify as ambiguous. With respect to Antici et al. (2021), we address
quotes, one of the controversial cases highlighted in their discussion,
considering them as a specific case of third-party opinion. Moreover, we label
sentences where the author explicitly states their emotions as objective
instead of subjective. This decision may seem counter-intuitive since the
presence of emotions is widely used as an indicator of subjectivity Mihalcea
et al. (2012); Veronika (2006), especially when the purpose is to capture
subjectivity as an expression of a “private state” of the author Riloff and
Wiebe (2003). However, in these cases, emotions do not influence the message
conveyed in the sentence, but rather are the content of the sentence. Thus,
the sentence is not influenced by emotions, and we label it as objective. We
observe that annotations produced following our guidelines significantly
differ from those obtained using a lexicon, suggesting that a lexicon-based
approach is not suitable for our problem. We elaborate further in Appendix A.

We perform the rest of the annotation process in three stages: (i) two
annotators label their assigned sentences; (ii) each couple of annotators
discusses ambiguous sentences to agree on one common decision; and (iii) in
cases in which the two annotators fail to agree, a third annotator labels the
disputed sentence. We aggregate labels via majority voting for the purpose of
training machine learning models Nguyen et al. (2017). The IAA is 0.51
(“moderate”) after step (i) and 0.83 (“near-perfect”) after step (ii),
requiring step (iii) in less than 10% of cases.333We do not measure the
agreement after step (iii) since each label assigned in such step would
necessarily agree with one annotator and disagree with the other. It is
important to stress that each sentence was annotated by at least two
annotators and that we measured the IAA on the entire corpus rather than on a
subsample.

Our final corpus contains 1,049 sentences, out of which 638 are objective and
411 are subjective. We perform train, validation, and test splitting such that
all the sentences of the same article belong to the same split. We balance the
number of SUBJ and OBJ sentences in the validation and test sets to ensure a
sound model evaluation. Table 1 reports corpus statistics for each split.
Following Basile et al. (2021) and Abercrombie et al. (2022), we flag the 89
sentences for which agreement is not reached during step (ii), hence requiring
the intervention of a third annotator. This process is considered as a measure
of quality assurance Rottger et al. (2022); Teruel et al. (2018); Davani et
al. (2022).

| # Art. | # Sent. | # OBJ | # SUBJ
---|---|---|---|---
Train | 16 | 731 | 487 (12) | 244 (46)
Val | 3 | 99 | 45 (3) | 54 (8)
Test | 4 | 219 | 106 (4) | 113 (16)
Total | 23 | 1,049 | 638 (19) | 411 (70)
Table 1: Statistics of our corpus. The number of disputed sentences for which
a third annotator was necessary is reported between parenthesis.

##  4 Experimental Setting

We formulate SD as a binary classification task where a model classifies a
sentence as subjective (SUBJ) or objective (OBJ). To understand if our
annotations are compatible with corpora developed in other languages, we
perform a preliminary evaluation considering three experimental settings:
monolingual, multilingual, and cross-lingual. In the first one, we test a
model on the same corpus it has been trained on. For the second and third, we
follow the setting of Muti and Barrón-Cedeño (2022) and experiment with
multilingual and zero-shot cross-lingual scenarios. The former consists of
training models on two training corpora in different languages and evaluating
them on one of them. In the latter, models are trained on one language and
tested on another one without any fine-tuning Huang et al. (2021). We
experiment with Italian as the second language and use the SubjectivITA corpus
Antici et al. (2021) for the cross-lingual settings, using the same splits as
defined in the original paper. 444Among the main differences between our and
SubjectivITA’s annotation guidelines there is the labeling of quotes and
explicit emotions. With the purpose of increasing the compatibility between
the two corpora, we partially re-annotate SubjectivITA, labeling sentences
that belong to these categories as objective. Further information is reported
in Appendix C.

As classifiers we consider the following models: a linear kernel Support
Vector Machine (SVM) Cortes and Vapnik (1995) with TF-IDF
representations;555We use the default configuration of scikit-learn. a
Logistic Regressor (LR) with TF-IDF representations; SentenceBERT (M-SBERT)
Reimers and Gurevych (2019), a multilingual version of SBERT embeddings with a
logistic regressor on top;666We use paraphrase-multilingual-MiniLM-L12-v2.
MultilingualBERT (M-BERT) Devlin et al. (2019), a multilingual version of BERT
with a linear classification layer on top.777We use bert-base-multilingual-
cased.

Moreover, we evaluate a random uniform (RND-B) and a majority (MAJ-B)
baselines. We fine-tune transformer-based models for 4 epochs following
standard practice Devlin et al. (2019). For evaluation, we report macro
F1-score and class-specific F1-scores (F1-OBJ and F1-SUBJ) averaged over three
individual seed runs. Additional details are reported in Appendix B.

##  5 Results

Model | F1-Macro | F1-OBJ | F1-SUBJ
---|---|---|---
baselines: en $\rightarrow$ en |  |  |
MAJ-B | 0.33 | 0.65 | 0.00
RND-B | 0.50 | 0.49 | 0.50
mono-lingual: en $\rightarrow$ en |  |  |
SVM | 0.44 | 0.64 | 0.24
LR | 0.55 | 0.63 | 0.48
M-SBERT | 0.69 | 0.70 | 0.69
M-BERT | 0.75 | 0.77 | 0.71
multi-lingual: en+it $\rightarrow$ en |  |  |
SVM | 0.49 | 0.63 | 0.34
LR | 0.64 | 0.63 | 0.65
M-SBERT | 0.71 | 0.67 | 0.76
M-BERT | 0.80 | 0.81 | 0.80
cross-lingual: it $\rightarrow$ en |  |  |
M-SBERT | 0.67 | 0.61 | 0.74
M-BERT | 0.60 | 0.72 | 0.46

Table 2: Classification performance on our test set.

Model | F1-Macro | F1-OBJ | F1-SUBJ
---|---|---|---
baselines: it $\rightarrow$ it |  |  |
MAJ-B | 0.42 | 0.85 | 0.00
RND-B | 0.47 | 0.58 | 0.36
mono-lingual: it $\rightarrow$ it |  |  |
SVM | 0.59 | 0.85 | 0.34
LR | 0.60 | 0.77 | 0.42
M-SBERT | 0.69 | 0.82 | 0.56
M-BERT | 0.74 | 0.88 | 0.59
multi-lingual: en+it $\rightarrow$ it |  |  |
SVM | 0.60 | 0.85 | 0.35
LR | 0.61 | 0.81 | 0.42
M-SBERT | 0.69 | 0.81 | 0.56
M-BERT | 0.77 | 0.88 | 0.66
cross-lingual: en $\rightarrow$ it |  |  |
M-SBERT | 0.66 | 0.83 | 0.49
M-BERT | 0.65 | 0.85 | 0.46

Table 3: Classification performance on SubjectvITA test set.

Table 2 shows the classification performance on our corpus. Regarding the
monolingual setting, we observe that M-BERT and M-SBERT greatly outperform SVM
and LR. M-BERT achieves comparable performance to M-SBERT concerning F1-SUBJ,
while it outperforms M-SBERT on F1-OBJ.

The low scores concerning F1-SUBJ achieved by SVM and LR are possibly due to
the unbalanced class distribution in the training set.

Concerning the multilingual setting, training on both English and Italian
corpora leads to improved performances on our corpus. In particular, M-BERT is
the best-performing model with a clear improvement of 0.05 on F1-macro, while
M-SBERT achieves an improvement of 0.02 on F1-macro. In this setting, we also
evaluate our models on SubjectivITA. Table 3 reports classification
performance. In the monolingual setting, we observe a similar trend where
M-BERT achieves the best performance. When trained with both language training
sets, M-BERT obtains an improvement of 0.07 on F1-SUBJ. In contrast, M-SBERT
shows no relevant performance improvement compared to its monolingual
counterpart. These results indicate that the multilingual setting is
beneficial and we interpret it as a sign that the two corpora are coherent in
their annotation. Otherwise, the use of an additional language would have
provided more noise rather than information, lowering model performance.

Lastly, in the cross-lingual setting, M-BERT is underperforming compared to
its monolingual counterpart, whereas M-SBERT achieves comparable performance
in both corpora. The latter observation is in accordance with the results of
Reimers and Gurevych (2020).

##  6 Conclusion

We presented a corpus for subjectivity detection at the sentence level in news
articles. The corpus was annotated following language-agnostic guidelines and
a prescriptive approach. We experimented with mono-, multi- and cross-lingual
scenarios, obtaining the best performance when training on materials from
newspapers both in English and Italian, suggesting that our approach is
general enough to be applied in other languages. Future research directions
include the annotation of news in other languages, detailed error analysis of
misclassified instances to provide insight about which types of subjective
sentences are hard to detect, and perform model evaluation when the disputed
sentences in the training set are removed.

## Limitations and Risks

The perception of subjectivity is subjective in itself. It may derive from
different interpretations of the language, different background knowledge, and
personal biases Chaturvedi et al. (2018). We have addressed this limitation by
using multiple annotators to label each sentence. While different genders,
nationalities, and areas of expertise were represented among the annotators,
it is impossible to exclude that they may all share similar biases that
influence their perception of subjectivity. The annotators belong to 3
different countries of origin, two of which are European. Three annotators
identify themselves as males and the remaining four as females.

Some of the sources we have used, especially the ones that had many subjective
sentences, may address some topics with more frequency than others. Therefore,
we can not exclude that a model may learn a bias regarding certain topics and
the keywords associated to them. Similar problems have emerged in other NLP
tasks, such as abusive language detection Wiegand et al. (2019); Park et al.
(2018). We remark that our corpus should be used only for research purposes.

## Ethics Statement

Subjectivity, being an indicator of bias in the author’s statement, is of
valuable importance in analyzing and recognizing opinionated content. Our work
could provide useful insight when studying relevant problems like fact-
checking in foreign languages, where is more difficult for researchers to
recognize peculiar patterns which may hide the author’s point of view. Our
work aims at improving the awareness, and consequently the fairness, in
textual resources.

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## Appendix

##  Appendix A Comparison with Lexicon-Based Annotations

We use TextBlob,888https://textblob.readthedocs.io/ a widely-adopted lexicon
for subjectivity detection, to label our corpus. Specifically, we approximate
the score obtained through TextBlob to 0 (OBJ) or 1 (SUBJ). The average IAA
between each human annotator and textblob is -0.21. The negative value means
that it systematically disagrees with human annotators, suggesting that a
lexicon-based approach is not suitable for our problem.

##  Appendix B Reproducibility

###  B.1 Metrics

We used F1 score as the main evaluation metric. It is defined as:

$F_{1}=\frac{2TP}{2TP+FP+FN}$

Where TP are True Positive samples, FP are False Positive, and FN are False
Negative. The macro-averaged F1 is obtained as the average of the F1 on the 2
classes.

###  B.2 Computing Infrastructure

We conducted all of our experiments on a machine equipped with an intel i7
3.7GHz processor, 32Gb of RAM and an NVidia 1080ti 11 GB.

###  B.3 Hyperparameters and Runtime

The models are trained with their default configuration setup. For the M-BERT
model, we iterated over the training set for 4 epochs, using Adam as optimizer
with a learning rate of $1e-5$. The model, which has around 170 millions
parameters, takes an average of 30 seconds per epoch during training while
having an inference time of 2 seconds.

###  B.4 Validation Performances

Table 4 and Table 5 report classification performance on the English and
Italian validation sets, respectively.

###  B.5 Seeds

We perform multiple runs for the experiments, the tables report results
averaged over three seed runs. The seeds are 42, 1000, and 500.

Model | F1-Macro | F1-OBJ | F1-SUBJ
---|---|---|---
mono-lingual: en $\rightarrow$ en |  |  |
SVM | 0.50 | 0.63 | 0.36
LR | 0.56 | 0.62 | 0.49
M-SBERT | 0.63 | 0.63 | 0.62
M-BERT | 0.62 | 0.67 | 0.57
multi-lingual: en+it $\rightarrow$ en |  |  |
SVM | 0.55 | 0.77 | 0.32
LR | 0.60 | 0.72 | 0.47
M-SBERT | 0.71 | 0.69 | 0.74
M-BERT | 0.61 | 0.66 | 0.56
cross-lingual: it $\rightarrow$ en |  |  |
M-SBERT | 0.66 | 0.62 | 0.70
M-BERT | 0.41 | 0.64 | 0.19

Table 4: Classification performance on our corpus validation set.

Model | F1-Macro | F1-OBJ | F1-SUBJ
---|---|---|---
mono-lingual: it $\rightarrow$ it |  |  |
SVM | 0.51 | 0.82 | 0.21
LR | 0.57 | 0.78 | 0.35
M-SBERT | 0.73 | 0.82 | 0.64
M-BERT | 0.71 | 0.87 | 0.55
multi-lingual: en+it $\rightarrow$ it |  |  |
SVM | 0.55 | 0.77 | 0.32
LR | 0.60 | 0.72 | 0.47
M-SBERT | 0.60 | 0.72 | 0.47
M-BERT | 0.74 | 0.87 | 0.62
cross-lingual: en $\rightarrow$ it |  |  |
M-SBERT | 0.70 | 0.84 | 0.57
M-BERT | 0.71 | 0.87 | 0.55

Table 5: Classification performance on SubjectivITA validation set.

##  Appendix C SubjectivITA

The distribution of classes in the SubjectivITA corpus is presented in Table
6.

| # Art. | # Sent. | # OBJ | # SUBJ
---|---|---|---|---
Train | 80 | 1,399 | 1,079 | 320
Val | 13 | 214 | 152 | 62
Test | 10 | 227 | 167 | 60
Total | 103 | 1,840 | 1,398 | 442
Table 6: Statistics of SubjectivITA corpus.

##  Appendix D Guidelines

### Definitions

#### Subjective

A sentence is considered subjective when it is based on —or influenced by—
personal feelings, tastes, or opinions.

Otherwise, the sentence is considered objective.

Table 7 shows an example for each type. Sentence 1 is objective because the
author describes a historical event without giving any opinion or personal
comment. In contrast, in Sentence 2 the author explicitly conveys their
personal emotions, making the sentence subjective.

|  Sentence | Label
---|---|---
1. |  India, who was the bridesmaid at the King’s wedding to Princess Diana in 1981, could not be seen in the footage, but filmed the video as she walked through the grounds of the royal residence. | OBJ
2. |  It is a sad truth that many of the villages in this region of Portugal are dying. | SUBJ
Table 7: Examples of subjective and objective sentences.

### Specific Subjective Cases

#### SUBJ 1

A sentence is subjective if it explicitly reports the personal opinion of its
author. Rhetorical questions are considered as an expression of an opinion as
well; see Ex. c. Additionally, speculations which draw conclusions are
considered as opinions, see Ex. d.

Examples:

  1. (a)

It has everything you could want in a holiday: beautiful sandy beaches and
clear waters, ancient history and culture, delicious food (the Greek salads
are simply on another level), island hopping, nightlife and more.

  2. (b)

After treading vineyard soils and seeing grapes ripening, that merlot becomes
more than just a Wednesday night relaxant.

  3. (c)

Do they really think other nations sprouted up out of the ground?

  4. (d)

But Putin will hope to sow uncertainty in the eyes of policymakers’ meetings
in New York.

#### SUBJ 2

A sentence is subjective if it contains sarcastic or ironic expressions
attributable to its author.
Examples:

  1. (e)

It’s no lie that the USA is one heck of a big country (said in a southern
twang).

  2. (f)

With Land Rover bowdlerising images of the vehicle into little more than a
perfume advertisement on TV[…].

  3. (g)

Especially if you’re more excited at the prospect of sampling rare bottles
from the cellar than snapping vineyard selfies.

#### SUBJ 3

A sentence is subjective if it contains exhortations of personal auspices made
by its author.
Examples:

  1. (h)

The West should arm Ukraine faster.

#### SUBJ 4

A sentence is subjective if it contains discriminating or downgrading
expressions.
Examples:

  1. (i)

And what is even more evident is the perverse role reversal that is taking
place, in which he who sits in Rome has the task of formulating heterodox
principles opposed to Catholic doctrine, and his accomplices in the Dioceses
have the role of scandalously applying them, in an infernal attempt to
undermine the Moral law in order to obey the spirit of the world.

  2. (j)

How did we reach the stage where priests and bishops cowered like frightened
puppies before a common flu, where their predecessors ministered fearlessly
among the lepers, the cripples, and the victims of typhoid, cholera, smallpox,
and Bubonic Plague?

#### SUBJ 5

A sentence is subjective if it contains rhetorical figures, like hyperboles,
explicitly made by its author to convey their opinion.
Examples:

  1. (k)

Barcelona where it all began, Messi was a king in Catalonia and he lived like
one too.

  2. (l)

The churches, and the Catholic Church in particular (which is by far the
largest), had the ability to put an end to the lockdown madness and the COVID-
terror campaign, had they wished to do so.

  3. (m)

So it must be biochemistry that is really what is racist.

### Specific Cases of Objectivity

If a sentence does not meet any subjectivity type listed in the previous
section, it is considered objective. Here we include examples of objective
sentences which may be wrongly interpreted as subjective.

#### Example 1

A sentence is objective when it reports on news or historical facts that are
quoted by the author of the sentence.
Examples:

  1. (a)

President Putin has just reiterated his threat to use nuclear weapons and
announced that Russian-controlled Ukrainian territory will become part of the
Russian Federation.

  2. (b)

In the modern era electroconvulsive therapy, first used in 1938, became a
treatment for some serious forms of depression in the post-war decades.

#### Example 2

A sentence is objective when it describes the personal feelings, emotions or
moods of the writer, without conveying opinions on other matters.
Examples:

  1. (c)

I was definitely surprised at how emotional I felt watching the service.

  2. (d)

The second I saw him, I felt a jolt of connection.

#### Example 3

A sentence is objective if it expresses an opinion, claim, emotion or a point
of view that is explicitly attributable to a third-party (e.g., a person
mentioned in the text).
Examples:

  1. (e)

Frank Drake believed that the universe had to contain other intelligent
beings.

  2. (f)

"You showed callous indifference to Dean’s fate after he had been repeatedly
stabbed" the judge said.

Note: The presence of quotation marks (“ ”), when used to quote a third person
(be it at the beginning of the sentence, at the end, or both), represents an
explicit third-party opinion, even if it is not clearly stated in the
sentence.
Examples:

  1. (g)

“Crosbie is an extremely violent man who has no place in society, and we
welcome the jury’s verdict today.”

  2. (h)

“My children have lost their hero and I have lost my chosen person - the
person I chose to spend my life with.

  3. (i)

For these reasons and out of conviction, I consider myself bound in my
conscience to say no.”

#### Example 4

A sentence is objective if it contains a comment made by the author of the
sentence that does not draw any conclusion. In particular, the author doesn’t
convey their personal interpretation or opinion, leaving the discussion on the
topics of interest open.
Examples:

  1. (j)

It is not clear yet which of the couples from the E4 reality show remain
together and who have now, because the series has not concluded.

  2. (k)

Do car manufacturers know how far their EVs will really go?

  3. (l)

Exact figures are hard to come by, but Ukraine may well have more troops
available than Russia now.

#### Example 5

A sentence is objective if it contains factual conclusions made by the author
of the sentence that do not convey any stance or personal opinion, or are
justified up by a non-personal hypothesis.
Examples:

  1. (m)

In years gone by, travel to Japan was notoriously expensive, but the devaluing
of the yen has made it more accessible.

  2. (n)

The bottom-up approaches which target the molecular, genetic and electrical
fundamentals of the brain can assist top-down approaches to brain disorder
such as talking therapies.

  3. (o)

Based on our experiences and road tests, a good rule of thumb is to expect to
achieve somewhere between 75 and 80 per cent of a car’s WLTP Combined range[…]

#### Example 6

When referring to an individual, any kind of well-known nickname or title is
considered objective.
Examples:

  1. (p)

Things have certainly progressed on the pitch for Spurs this season.

  2. (q)

The Duke of York ‘plotted’ with Diana to ‘push Prince Charles aside’.

#### Example 7

Any kind of common expression or proverb is considered objective.
Examples:

  1. (r)

the adage ‘sticks and stones may break my bones, but words can never hurt me’.

  2. (s)

Home sweet home: George poses in one of the rooms at his sprawling Hampstead
home during a photoshoot in 2002.


A sensitivity analysis of a gonorrhoea dynamics and control model

Louis Omenyi${}^{1,3,\ast,\href https://orcid.org/0000-0002-8628-0298},$
Aloysius Ezaka${}^{2},$ Henry O. Adagba${}^{2},$ Gerald Ozoigbo${}^{1},$
Kafayat Elebute${}^{1},$

1 Department of Mathematics and Statistics,

Alex Ekwueme Federal University, Ndufu-Alike, Nigeria

2Department of Industrial Mathematics and Applied Statistics,

Ebonyi State University, Abakaliki, Nigeria

3Department of Mathematical Sciences,

Loughborough University, Leicestershire, United Kingdom

Corresponding author, email<EMAIL_ADDRESS>
###### Abstract

We formulate and analyse a robust mathematical model of the dynamics of
gonorrhoea incorporating passive immunity and control. Our results show that
the disease-free and endemic equilibria of the model are both locally and
globally asymptotically stable. A sensitivity analysis of the model shows that
the dynamics of the model is variable and dependent on waning rate, control
parameters and interaction of the latent and infected classes. In particular,
the lower the waning rate, the more the exponential decrease in the passive
immunity but the susceptible population increases to the equilibrium and wanes
asymptotically due to the presence of the control parameters and restricted
interaction of the latent and infected classes.

_Keywords:_ gonorrhoea; passive immunity; equilibria; reproduction number;
sensitivity analysis; asymptotic stability.
_2010 AMS Subject Classification:_ 97M10, 97N80.

##  1 Introduction

Gonorrhoea, a sexually transmitted disease, is a major cause of infertility
and other debilitating ailments among couples. Thus, it becomes necessary to
undertake prompt prevention and control activities to tackle the ugly
incidence of this sexually transmitted diseases, [6]. It is caused by a
bacterium called Neisseria gonorrhoeae, [20]. According to Mushayabasa and
Bhunu in [11], neisseria gonorrhoea is characterized by a very short period of
latency, namely, $2-10$ days and is commonly found in the glummer epithelium
such as the urethra and endo-cervix epithelia of the reproductive track, [5].
Gonorrhoea is transmitted to a new born infant from the infected mother
through the birth canal thereby causing inflammations and eye infection such
as conjunctivitis. It is also spread through unprotected sexual intercourse,
[19].

Studies by Usman and Adam [21] and Center for Disease Control Report in [3]
show that male patients of gonorrhoea have pains in the testicles (known as
epididymitis), painful urination due to scaring inside the urethra while in
female patients, the disease may ascend the genital tract and block the
fallopean tube leading to pelvic inflammatary disease (PID) and infertility,
see also [14]. Other complications associated with this epidemic include
arthritis, endocarditis, chronic pelvic pain, meningitis and ectopic
pregnancy, [15].

Gonorrhoea confers temporal immunity on some individuals in the susceptible
class while some others are not immuned, [19]. This immunity through the
immune system plays an important role in protecting the body against the
infection and other foreign substances, [3]. That is why an immuno-compromised
patient has a reduced ability to fight infectious disease such as gonorrhoea
due to certain diseases and genetic disorder, [17]. Such patient may be
particularly vulnerable to opportunistic infection such as gonorrhoea. Hence,
immune reaction can be stimulated by drug-induced immune system such as
Thrombocytopenia, [17]. This helps to reduce the waning rate of passive
immunity in the immune class, [2]. However, if the activity of immune system
is excessive or over-reactive due to lack of cell mediated immunity, a
hypersensitive reaction develops such as auto-immunity and allergy which may
be injurious to the human body or may even cause death [25].

Statistically, gonorrhoea infection has spread worldwide with more than $360$
million new cases witnessed globally in adults aged $15-49$ years, [3]. In
1999, above over $120$ million people in African countries were reported to
have contracted the disease. Over $82$ million people were reported in
Nigeria, [3]. Researches abound on the modelling and control of this epidemic
with various approaches and controls, see e.g. [3, 9, 10, 18, 16, 19, 20] and
mostly recently [1, 24, 13] and [4]. This present study continues the
discussion by investigating the dynamics of the disease by carrying out the
sensitivity analysis of the effective reproduction number of the model.
Besides, the theory of epidemiology signifies the phenomenon of bifurcation at
the equilibria. This is a classical requirement for the model’s effective
reproduction number, $R_{e}.$ Although this is necessary, it is no longer
sufficient to conclude the effective control or elimination of gonorrhoea in a
population, see e.g. [25]. Therefore in this paper, we consider the nature of
the equilibrium solution near the bifurcation point $R_{e}=1$ in the
neighbourhood of the disease-free equilibrium, $E_{0},$ through sensitivity
analysis to determine the most sensitive parameters to target by a way of
intervention strategy.

##  2 Materials and Methods

We formulate a modified SIR model by extending an existing one to incorporate
passive immunity. Let $Q(t)$ be passive immune class, $S(t)$ the susceptible
compartment, $L(t)$ the latent class, $I(t)$ the infectious class, $T(t)$ the
treated class and $R(t)$ be the recovered compartment over time $t.$ Let the
parameters of the model be $\sigma$ as level of recruitment, $\upsilon$ as
waning rate of immunity, $\mu$ as rate of natural mortality, $\lambda$ as
contact rate between the susceptible and the latent classes, $\eta$ as
treatment rate of latent class, $\gamma$ as induced death rate due to the
infection, $\alpha$ as treatment rate of infected compartment, $\beta$ as
infectious rate of latent class, $\omega$ as recovery rate of treated class,
$\delta$ as rate at which recovered class becomes susceptible again, $\theta$
as infectious rate from the susceptible class direct to the infectious class,
$k_{1}$ as control measure given to latent class and $k_{2}$ as control
measure given to infected class.

Next, assume the recruitment into the population is by birth or immigration;
all the parameters of the model are positive; some proportions of new birth
are immunized against the infection; the immunity conferred on the new birth
wanes after sometime; and that the rate of contact of the disease due to
interaction, $\lambda,$ rate is due to the movement of the infected
population. Consequently, the total population at time $t$ is
$N(t)=Q(t)+S(t)+L(t)+I(t)+T(t)+R(t).$ The existing schematic model diagram is
given as Figure 1 and that of the extended model as 2:

Figure 1: The diagram of the model; see [19]. Figure 2: The model with control

So, the system of equations for the model without control is (2.7):

$\displaystyle\left.\begin{array}[]{rcl}\frac{dQ}{dt}&=&f\sigma-\upsilon Q-\mu
Q\\\ \frac{dS}{dt}&=&\upsilon Q+(1-f)\sigma+\delta R-\mu S-\theta S-\theta
SI\\\ \frac{dL}{dt}&=&\theta SI-\eta L-\mu L-\beta L\\\ \frac{dI}{dt}&=&\beta
L+\theta S-\mu I-\gamma I-\alpha I\\\ \frac{dR}{dt}&=&\omega T-\mu R-\delta
R\\\ \frac{dT}{dt}&=&\alpha I+\eta L-\mu T-\omega T\end{array}\right\\}$ (2.7)

and those with control is (2.14):

$\displaystyle\left.\begin{array}[]{rcl}\frac{dQ}{dt}&=&f\sigma-\upsilon Q-\mu
Q\\\ \frac{dS}{dt}&=&\upsilon Q+(1-f)\sigma-\theta S(1-k_{2})+\delta R-\mu
S-\theta SI\\\ \frac{dL}{dt}&=&\theta SI-\beta L-\mu L-\eta(1+k_{1})L\\\
\frac{dI}{dt}&=&\beta L+\theta S(1-k_{2})-((\mu+\gamma)+\alpha(1+k_{2}))I\\\
\frac{dR}{dt}&=&\omega T-\mu R-\delta R\\\
\frac{dT}{dt}&=&\eta(1+k_{1})L+\alpha(1+k_{2})I-\mu T-\omega
T\end{array}\right\\}$ (2.14)

By sensitivity theory, see e.g. [2, 5, 9, 18] and [12], it is expected that a
significant perturbation in the model parameters leads to a change in the
behaviour of the equilibrium solution of the model. We proceed to study the
qualitative properties of the model through the variation of the model
parameters such as in [6] and [8].

##  3 Results and discussions

We firstly prove the positivity of the solution set of the model.

######  Lemma 3.1.

The extended gonorrhoea model equations admit non-negative solution set.

###### Proof.

A direct computation of the model equations (2.14) gives

$\displaystyle Q(t)$ $\displaystyle=$
$\displaystyle\frac{f\sigma}{\mu+\upsilon}+c_{1}e^{-(\mu+\upsilon)t}>0;$
$\displaystyle S(t)$ $\displaystyle=$ $\displaystyle\frac{\upsilon
Q+(1-f)\sigma+\delta R}{\mu+\theta(1-k_{2})+\theta
I}+c_{2}e^{-(\mu+\theta(1-k_{2})+\theta I)t}>0\Rightarrow S(t)>0;$
$\displaystyle L(t)$ $\displaystyle=$ $\displaystyle\frac{\theta
IS}{\mu+\beta+\eta(1+k_{1})}+c_{3}e^{-(\mu+\beta+\eta(1+k_{1}))t}>0\Rightarrow
L(t)>0;$ $\displaystyle I(t)$ $\displaystyle=$ $\displaystyle\frac{\beta
L+\theta
S(1-k_{2})}{\mu+\gamma+\alpha(1+k_{2})}+c_{4}e^{-(\mu+\gamma+\alpha(1+k_{2}))t}>0;$
$\displaystyle R(t)$ $\displaystyle=$ $\displaystyle\frac{\omega
T}{\mu+\delta}+c_{5}e^{-(\mu+\delta)t}>0\Rightarrow
R(t)>0~{}~{}\text{and}~{}~{}$ $\displaystyle T(t)$ $\displaystyle=$
$\displaystyle\frac{\eta(1+k_{1})L+\alpha(1+k_{2})I}{\mu+\omega}+c_{6}e^{-(\mu+\omega)t}>0\Rightarrow
T(t)>0$

for arbitrary positive constants $c_{1},c_{2},\cdots,c_{6}.$ ∎

Next, we show that there exists a feasible solutions region for the model.

######  Proposition 3.2.

Let $\dot{x}=f(x)$ in $D\subset\mathbb{R}_{+}^{n}$ be a system of equations in
the feasible solutions region of the model, (2.14). Then the solutions $x(t)$
are feasible for all $t\geq 0$ if $x(t)\in D\subset\mathbb{R}_{+}^{6}.$

###### Proof.

It suffices to prove that the solution set
$x(t)=\\{(Q(t),S(t),L(t),I(t),T(t),R(t))\\}$ enters the invariant region
$\mathbb{D}\subset\mathbb{R}_{+}^{6}.$ Since $N=Q+S+L+I+R+T,$ it follows that

$\frac{dN}{dt}=\frac{dQ}{dt}+\frac{dS}{dt}+\frac{dL}{dt}+\frac{dI}{dt}+\frac{dR}{dt}+\frac{dT}{dt}.$

This implies that

$\frac{dN}{dt}=\sigma-\mu N-\gamma I\leq\sigma-\mu N\\\
\Rightarrow\frac{dN}{dt}\leq\sigma-\mu N.$

Solving gives

$N\leq\frac{\sigma-qe^{-\mu
t}}{\mu}~{}~{}\text{with}~{}~{}N(0)=N0\leq\frac{\sigma-qe^{-\mu t}}{\mu}$

where $q$ is arbitrary constant and $q=\sigma-\mu N(0).$ Therefore,

$N(t)\leq\frac{\sigma}{\mu}-\frac{(\sigma-\mu N(0))e^{-\mu t}}{\mu}.$

Observe that as $t\rightarrow\infty,$ the population size approaches the
carrying capacity $\frac{\sigma}{\mu}.$ That is,

$0<N\leq\frac{\sigma}{\mu}\Rightarrow N\rightarrow\frac{\sigma}{\mu}.$

Hence, the feasible solution set of the system enters the invariant region
$D\subset\mathbb{R}^{6}_{+}$, and

$S>0,Q>0,L\geq 0,I\geq 0,T\geq 0,R\geq 0.$

Whenever $N\leq\frac{\sigma}{\mu}$, every solution with initial condition in
$D$ remains in that region for $t>0.$ Hence, the region $D$ is positively
invariant or bounded with $\frac{\sigma}{\mu}$ as the upper bound, $x(t)\in
D.$ ∎

######  Lemma 3.3.

The extended model admits disease-free and endemic equilibria.

###### Proof.

Setting the right hand side of the system (2.14) equal to zero supposes there
is no gonorrhoea infection in the population, i.e. $L=I=R=T=0$ which gives the
disease-free equilibria. On the other hand, suppose $L\neq 0,~{}I\neq
0,~{}R\neq 0~{}~{}\text{and}~{}~{}T\neq 0,$ one solves to obtain the endemic
equilibrium state of the model to be

$\displaystyle Q^{\ast}$ $\displaystyle=$
$\displaystyle\frac{f\sigma}{\mu+\upsilon};$ $\displaystyle S^{\ast}$
$\displaystyle=$
$\displaystyle\frac{(\mu+\delta)(\mu+\omega)f\sigma+(\mu+\upsilon)(\mu+\delta)(\mu+\omega)\sigma(1-f)+(\mu+\upsilon)\delta\omega(\alpha+\eta)}{(\mu+\upsilon)(\mu+\delta)(\mu+\omega)};$
$\displaystyle L^{\ast}$ $\displaystyle=$
$\displaystyle\frac{(\lambda)(\mu+\delta)(\mu+\omega)f\sigma+(\mu+\upsilon)(\mu+\delta)(\mu+\omega)\sigma(1-f)+(\mu+\upsilon)\delta\omega(\alpha+\eta)}{(\mu+\beta+\eta)(\mu+\upsilon)(\mu+\delta)(\mu+\omega)};$
$\displaystyle I^{\ast}$ $\displaystyle=$
$\displaystyle\frac{(\mu+\delta)(\mu+\omega)f\sigma+(\mu+\upsilon)(\mu+\delta)(\mu+\omega)\sigma(1-f)+(\mu+\upsilon)\delta\omega(\alpha+\eta)(\beta\lambda+(\mu+\beta+\eta)\theta)}{(\mu+\alpha+\gamma)(\mu+\beta+\eta)(\mu+\upsilon)(\mu+\delta)(\mu+\omega)};$
$\displaystyle R^{\ast}$ $\displaystyle=$
$\displaystyle\frac{\omega(\alpha+\eta)}{(\mu+\delta)(\mu+\omega)}~{}~{}\text{and}~{}~{}T^{\ast}=\frac{\alpha+\eta}{\mu+\omega}.$

∎

Now, we recall that the basic reproduction number $R_{0}$ is the expected
number of secondary infection produced in a completely susceptible population
by a typical or one infected individual. It determines how long an infectious
disease prevails in a given population, see e.g. [22]. When $R_{0}<1,$ it
indicates that with time the disease will die out of the population thereby
giving it a clean health bill. But if $R_{0}$ is greater than one, we expect
the disease will persist in the population. So for the infection to die out of
the population, $R_{0}$ must be less than $1.$ We have the following result.

######  Theorem 3.4.

The basic reproduction number of the extended model $0<R_{0}\ll 1.$

###### Proof.

Following [7], we obtain the basic reproduction number $R_{0}$ from the model
(2.7) without control measures, using next generation matrix method.

So, let

$f_{i}=\begin{bmatrix}\theta IS\\\ \theta S\\\ 0\end{bmatrix}$

so that

$\frac{\partial f_{i}}{\partial x_{j}}=F=\begin{pmatrix}0&\theta S&0\\\
0&0&0\\\ 0&0&0\end{pmatrix};$

and let

$v_{i}=\begin{pmatrix}\beta L+\mu L+\eta L\\\ \mu I+\gamma I+\alpha I-\beta
L-\theta S\\\ \mu T+\omega T-\eta L-\alpha I\end{pmatrix}$

so that

$\frac{\partial v_{i}}{\partial x_{j}}=\begin{pmatrix}(\beta+\mu+\eta)&0&0\\\
-\beta&(\mu+\gamma+\alpha)&0\\\ -\eta&-\alpha&(\mu+\omega)\end{pmatrix}.$

The matrix formed by the co-factors of the determinant is

$V=\begin{pmatrix}(\mu+\gamma+\alpha)(\mu+\omega)&\beta(\mu+\omega)&\alpha\beta+\eta(\mu+\gamma+\alpha)\\\
0&(\beta+\mu+\eta)(\mu+\omega)&\alpha(\beta+\mu+\eta)\\\
0&0&(\beta+\mu+\eta)(\mu+\gamma+\alpha)\end{pmatrix}$

with inverse

$V^{-1}=\begin{pmatrix}\frac{1}{\beta+\mu+\eta}&0&0\\\
\frac{\beta}{(\beta+\mu+\eta)(\mu+\alpha+\gamma)}&\frac{1}{\mu+\gamma+\alpha}&0\\\
\frac{\alpha\beta+\eta(\mu+\alpha+\gamma)}{(\mu+\alpha+\gamma)(\mu+\omega)}&\frac{\alpha}{\mu+\gamma+\alpha)(\mu+\omega)}&\frac{1}{\mu+\omega}\end{pmatrix}.$

Thus,

$|FV^{-1}-\lambda I|=\begin{vmatrix}\frac{\beta\theta
S}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}-\lambda&\frac{\theta
S}{\mu+\gamma+\alpha}&0\\\ 0&0-\lambda&0\\\ 0&0&0-\lambda\end{vmatrix}=0$

implies

$\lambda^{2}(\frac{(\beta\theta
S)}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}-\lambda)=0.$

Therefore, either $\displaystyle{\lambda^{2}=0}$ or
$\displaystyle{\lambda=\frac{(\beta\theta
S)}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}}.$ So for $\lambda\neq 0$ we obtain
the $R_{0}$ to be

$R_{0}=\frac{(\beta\theta
S)}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}=\frac{\sigma\beta\theta(\mu+\upsilon)-\mu
f)}{\mu(\mu+\alpha+\gamma)(\mu+\beta+\eta)(\mu+\upsilon)}.$ (3.1)

Parameter/Variable | $\beta$ | $\theta$ | $\mu$ | $\eta$ | $\gamma$ | $\alpha$ | $\delta$ | $\upsilon$ | $\omega$ | $\sigma$
---|---|---|---|---|---|---|---|---|---|---
Value | $0.01$ | $0.5$ | $0.2$ | $0.1$ | $0.01$ | $0.2$ | $0.8$ | $0.4$ | $0.7$ | $0.4$
Parameter/Variable | $d_{1}=k_{1}$ | $d_{2}=k_{2}$ | $f$ | $S$ | $Q$ | $R$ | $T$ | $L$ | $I$ |
Value | $0.5$ | $0.8$ | $0.91$ | $2000$ | $1000$ | $500$ | $1000$ | $1000$ | $500$ |

Table 1: Parameters/variables and values.

Using the data Table 1, we quickly observe that

$R_{0}=0.1566070960\ll 1.$

When control measure is given to a model, we compute the effective
reproduction number, $R_{e},$ similarly. From (2.14) let

$f_{i}=\begin{bmatrix}\theta IS\\\ \theta(1-k_{2})S\\\ 0\end{bmatrix}$

So that

$\frac{\partial f_{i}}{\partial x_{j}}=\begin{pmatrix}0&\theta S&0\\\ 0&0&0\\\
0&0&0\end{pmatrix}.$

and also

$v_{i}=\begin{bmatrix}\beta L+\mu L+\eta(1+k_{1})L\\\ \mu I+\gamma
I+\alpha(1+k_{2})I-\beta L-\theta S(1-k_{2})\\\ \mu T+\omega
T-\eta(1+k_{1})L-\alpha(1+k_{2})I\end{bmatrix}$

so that

$\frac{\partial v_{i}}{\partial
x_{j}}E_{0}=V=\begin{pmatrix}(\beta+\mu+\eta(1+k_{1}))&0&0\\\
-\beta&(\mu+\gamma+\alpha(1+k_{2}))&0\\\
-\eta(1+k_{1})&-\alpha(1+k_{2})&(\mu+\omega)\end{pmatrix}.$

The co-factors matrix is then

$\small{\begin{pmatrix}(\mu+\gamma+\alpha(1+k_{2}))(\mu+\omega)&-\beta(\mu+\omega)&\alpha\beta(1+k_{2})+\eta(1+k_{1})(\mu+\gamma+\alpha(1+k_{2})\\\
0&(\beta+\mu+\eta(1+k_{1}))(\mu+\omega)&-\alpha(1+k_{2})(\beta+\mu+\eta(1+k_{1}))\\\
0&0&(\beta+\mu+\eta(1+k_{1})(\mu+\gamma+\alpha(1+k_{2}))\end{pmatrix}}.$

So, following the same procedure for the computation of reproduction number,
we get the eigenvalues:

$\lambda^{2}=0~{}~{}\text{or}~{}~{}\lambda=\frac{(\beta\theta
S)}{(\beta+\mu+\eta(1+k_{1}))(\mu+\gamma+\alpha(1+k_{2}))}.$

Therefore, the effective reproduction number

$R_{e}=\frac{(\beta\theta
S)}{(\beta+\mu+\eta(1+k_{1}))(\mu+\gamma+\alpha(1+k_{2}))}.$ (3.2)

It can be observed here that the effective reproduction number is far less
than the basic reproduction number i.e. $R_{e}<<R_{0}$ since

$\frac{(\beta\theta
S)}{(\beta+\mu+\eta(1+k_{1}))(\mu+\gamma+\alpha(1+k_{2}))}<\frac{(\beta\theta
S)}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}.$

Using the data in Table 1, we see that

$\displaystyle R_{e}$ $\displaystyle=$
$\displaystyle\frac{\sigma\beta\theta((\mu+\upsilon)-\mu
f)}{\mu(\mu+\alpha+\gamma)(\mu+\beta+\eta)(\mu+\upsilon)}<\frac{(1-k_{2})(\alpha+\gamma+\mu)(\beta+\mu+\eta)}{(\alpha(1+k_{2})+\gamma+\mu)(\eta(1+k_{1})+\beta+\mu))}$
$\displaystyle=$ $\displaystyle 0.09700176367<0.1566070960\ll 1.$

This gives the result. ∎

One can as well use the Routh-Hurwitz criteria, see [23], to assess the local
stability of the model. In this technique, an equilibrium point is called
asymptotically stable if the trace of the Jacobian of the next generation
matrix of the model is less than zero. This means that all the roots of the
characteristic polynomial have negative real parts. We have the next result.

######  Theorem 3.5.

The extended model, (2.14), is asymptotically stable.

###### Proof.

From the model system (2.7) we set

$\displaystyle f_{1}$ $\displaystyle=$ $\displaystyle f\sigma-(\upsilon+\mu)Q$
$\displaystyle f_{2}$ $\displaystyle=$ $\displaystyle\upsilon
Q+(1-f)\sigma-\theta(1-k_{2})S+\delta R-\mu S-\theta IS$ $\displaystyle f_{3}$
$\displaystyle=$ $\displaystyle\theta SI-(\beta+\mu+\eta(1+k_{1}))L$
$\displaystyle f_{4}$ $\displaystyle=$ $\displaystyle\beta L+\theta
S(1-k_{2})-(\mu+\gamma+\alpha(1+k_{2}))I$ $\displaystyle f_{5}$
$\displaystyle=$ $\displaystyle\omega T-(\mu+\delta)R$ $\displaystyle f_{6}$
$\displaystyle=$ $\displaystyle\eta(1+k_{1})L+\alpha(1+k_{2})I-\mu T-\omega
T.$

At the disease-free equilibrium, the Jacobian matrix, $J|_{(DFE)},$ of the
model and the associated characteristics determinant $|J-\lambda I|=0$ implies

$\displaystyle(-(\mu+\upsilon)-\lambda)(-(\mu+\theta(1-k_{2}))-\lambda)(-(\beta+\mu+\eta(1+k_{1})-\lambda)(-\alpha(1+k_{2})$
$\displaystyle+\gamma+\mu)-\lambda)(-(\mu+\delta)-\lambda)(-(\mu+\omega)-\lambda)=0.$

Therefore,

$\displaystyle\lambda_{1}$ $\displaystyle=$ $\displaystyle-(\mu+\upsilon),$
$\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle-(\mu+\theta(1-k_{2})),$ $\displaystyle\lambda_{3}$
$\displaystyle=$ $\displaystyle-(\beta+\eta(1+k_{1})+\mu),$
$\displaystyle\lambda_{4}$ $\displaystyle=$
$\displaystyle-(\gamma+\alpha(1+k_{2})+\mu),$ $\displaystyle\lambda_{5}$
$\displaystyle=$ $\displaystyle-(\mu+\delta),$ $\displaystyle\lambda_{6}$
$\displaystyle=$ $\displaystyle-(\mu+\omega);$

where $\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5}$ and
$\lambda_{6}$ corresponding to $\lambda_{j},j=1,2,\cdots 6$ are the
eigenvalues. Since all the eigenvalues are negative, we conclude that the
model is asymptotically stable. ∎

We observe that this result indicates that the control interventions in the
model such as the use of condom, education enlightenment programme and
treatment are effective enough to almost wipe out the disease in a limited
time.

One can also discuss the global stability of the Model. Here, we use Lyapunov
direct method, (see e.g. [23]), to derive sufficient condition for the global
stability of the system.

######  Theorem 3.6.

[23]. Let $D$ be an open subset of $\mathbb{R}^{6}$ and $\dot{x}=f(x)$ a
system of differential equations for $x\in D.$ Let
$V:D\rightarrow\mathbb{R}^{6}$ be a smooth function. Suppose $V$ is positive
definite around a critical point $x^{\ast}\in D,$ i.e., $V(x^{\ast})=0$ and
$V(x)>0,$ for all $x\in D$ with $x\neq x^{\ast}.$ Then, the critical solution
$x=x^{\ast}$ is asymptotically stable in the sense of Lyapunov if the time
derivative $\dot{V}<0.$

For Proof, see e.g. [23]. We can now present one of the main results of this
study.

######  Theorem 3.7.

The equilibrium point solution is globally asymptotically stable.

###### Proof.

In $\displaystyle{R_{0}=\frac{\beta\theta
S}{(\alpha+\gamma+\mu)(\beta+\eta+\mu)}},$ let
$\displaystyle{\mu_{0}=\alpha+\gamma+\mu}$ and
$\displaystyle{\mu_{1}=\beta+\eta+\mu}.$

Then $\displaystyle{R_{0}=\frac{\beta\theta S}{\mu_{0}\mu_{1}}}$ so that
$\displaystyle{\frac{\mu_{0}\mu_{1}R_{0}}{\beta}=\theta S}.$ Since
$\displaystyle{R_{0}<1},$ then $\displaystyle{R_{0}-1<0}$ and so
$\displaystyle{\frac{\mu_{0}\mu_{1}}{\beta}(R_{0}-1)<0}.$

If we choose $\displaystyle{\varepsilon>0}$ sufficiently small such that
$\sigma=\frac{\mu_{1}}{\beta}-\varepsilon$ where
$\frac{\beta+\eta+\mu}{\beta}$ is the mean number of susceptible people that
are infected by one infectious individual during the infectious time and
$\sigma$ is the recruitment rate direct to susceptible class.

Now, define a Lyapunov function $V(L,I)$ by

$V=L+\sigma I.$ (3.3)

Clearly, $V$ is a positive definite function and so we can show that its time
derivative $\dot{V}$ is negative definite. That is,

$\displaystyle\frac{dV}{dt}$ $\displaystyle=$
$\displaystyle\frac{dL}{dt}+\frac{dI}{dt}\sigma$ $\displaystyle=$
$\displaystyle\theta IS-(\beta+\eta+\mu)L+\sigma(\beta
L-(\alpha+\mu+\gamma)I)$ $\displaystyle=$ $\displaystyle\theta
IS-\mu_{1}L+\sigma\beta L-\sigma\mu_{0}I<0.$

Take co-efficients of $L$ and $I$ to have that

$\displaystyle L_{c}$ $\displaystyle=$ $\displaystyle\sigma\beta-\mu_{1}$
$\displaystyle I_{c}$ $\displaystyle=$ $\displaystyle\theta S-\mu_{0}\sigma.$

From $L_{c}$ we have that

$\displaystyle(\frac{\mu_{1}}{\beta}-\varepsilon)\beta-\mu_{1}=-\varepsilon\beta<0\Rightarrow\theta
S-\mu_{0}(\frac{\mu_{1}}{\beta}-\varepsilon)$ $\displaystyle=$
$\displaystyle\theta S-\frac{\mu_{0}\mu_{1}}{\beta}+\varepsilon\mu_{0}$
$\displaystyle\frac{\mu_{0}\mu_{1}R_{0}}{\beta}-\frac{\mu_{0}\mu_{1}}{\beta}+\varepsilon\mu_{0}$
$\displaystyle=$
$\displaystyle\frac{\mu_{0}\mu_{1}}{\beta}(R_{0}-1)+\varepsilon\mu_{0}<0.$

This implies,

$\frac{dV}{dt}=\frac{dL}{dt}+\sigma\frac{dI}{dt}<0.$

Hence, $\dot{V}<0$ as required. ∎

Finally, we conduct the sensitivity analysis of the model on the effective
reproduction number.

######  Theorem 3.8.

The dynamics of the model is variable and dependent on the waning rate, the
control parameters and the interaction between the latent and infected
classes.

###### Proof.

Following [5], we prove Theorem (3.8) using the sensitivity index function of
the effective reproduction number given by

$\gamma_{R_{e}}(\cdot)=\frac{\partial
R_{e}}{\partial(\cdot)}\times\frac{(\cdot)}{R_{e}}.$ (3.4)

So, the variation of the treatment rate of the infected class is

$\gamma_{R_{e}}(\alpha)=\frac{\alpha}{R_{e}}\times\frac{\partial}{\partial\alpha}R_{e}=-(\frac{\alpha((\beta+\mu+\eta(1+c))(1+k))}{(\beta+\mu+\eta(1+c))(\mu+\Omega+\alpha(1+k))}).$

Now taking $\Omega=0.01,\alpha=0.2,\beta=0.01,c=0.5,\eta=0.1,k=0.8,\mu=0.2],$
we have

$\gamma_{R_{e}}(\alpha)=-0.6315789474.$

Similarly, the variation of the rate at which the the recovered class becomes
susceptible again is

$\gamma_{R_{e}}(\delta)=\frac{\delta}{R_{[{e}]}}\times\frac{\partial}{\partial\delta}R_{[{e}]}=\lambda((\frac{\alpha((\beta+\mu+\upsilon(1+f))(1+k))}{(\beta+\mu+\upsilon(1+f))(\mu+\delta+\theta(1+k))})).$

Take $\delta=0.4,\theta=0.5,\beta=0.01,f=0.91,\upsilon=0.4,k=0.8,\mu=0.2$ to
have

$\gamma_{R_{e}}(\delta)=+1.000000000.$

Continuing this way, we obtain the sensitivity index table of the model as
Table 2:

Parameters | Sensitivity indices $\gamma_{R_{e}}(\cdot)$
---|---
$\theta$ | $+1$
$\sigma$ | $+1$
$\beta$ | $+0.97222222218$
$\upsilon$ | $+0.2902711325$
$\gamma$ | $-0.01754385965$
$k_{1}$ | $-0.1388888889$
$\mu$ | $-2.176703880$
$\eta$ | $-0.4166666667$
$\alpha$ | $-0.6315789474$
$f$ | $-0.4354066982$
$k_{2}$ | $-0.4280701754$
Table 2: Sensitive parameters and values.

Graphically, we have the effect of increasing waning rate on the susceptible
and immune classes as Figure 3 while Figure 4 shows the effect of a higher
waning rate.

Figure 3: Waning rate $\upsilon=0.2$.  Figure 4: Waning rate $\upsilon=0.6.$

The graphs indicate that when the waning rate $\upsilon$ is low (i.e.,
$\upsilon=0.2$), the passive immune population decreases exponentially with
time, while when the waning rate is high, (i.e., $\upsilon=0.6$), the passive
immune population decreases faster and varnishes with time. The continuous
decay in the population of the immune class (Q) with time is due to the fact
that the immunity conferred on the individuals in this class is temporal and
hence, expires with time. However, the susceptible population increases slower
to the turning point at about one year and three months as the waning rate
$\upsilon$ is low and increases faster as the waning rate $\upsilon$ is high
as shown in Figure 3 and Figure 4 respectively. In both cases, the susceptible
class later decreases with time due to the interaction among the latent,
infected and the susceptible classes coupled with the natural mortality rate
$\mu.$

Next figures show the effect when the control measures are removed and when
they are introduced. Figure 5 shows when $k_{1}=k_{2}=0$ that the susceptible
individuals first increased after $45$ days due to recruitment into it, and
the trajectory decreases with time as more people are getting infected with
the disease.

Figure 5: When $k_{1}=0=k_{2}.$

In the same way the latent and the infected population show exponential
increase because more people are getting infected without control measure.
However, the treated and the recovered population show drastic exponential
decay with time as a result of no control measure.

Figure (6) suggests that susceptible population increases exponentially with
time since more people get treated, recovered and join the susceptible class
again because of increase in control measure. Also the recovered class
increases with time for about $7$ months as the control measure is high (i.e.,
$k_{1}=0.7$ and $k_{2}=0.9$), and started decreasing again asymptotically with
time because recovery from gonorrhoea is with temporal immunity. However, the
trajectory of the latent and the infected classes decreased to zero with time
due to increase in control measure.

Figure 6: Effect when $k_{1}=0.7$ and $k_{2}=0.9$.

Figure (7) indicates that when the interaction rate is low (i.e.,
$\theta=0.3$), the latent and the infected classes decrease exponentially with
time, and even varnishes in the long run since there will be almost nobody to
contact and suffer the disease. It is also shown that when the interaction
rate $\theta=0$, the basic reproduction number of the disease becomes zero.

Figure 7: Effect when $\lambda=\theta I,~{}~{}\theta=0,~{}~{}\text{and
when}~{}~{}\theta=0.3.$

We conclude that

$\lim_{\theta\rightarrow 0}R_{0}=\lim_{\theta\rightarrow 0}\frac{(\beta\theta
S)}{(\beta+\mu+\eta)(\mu+\gamma+\alpha)}=0.$

At this point, the contact rate $\lambda$ becomes zero and hence, nobody
suffers the disease. ∎

##  4  Conclusion

Based on the analysis and results of this work, we observed that the solution
set of the state variables of the extended model are all positive for $t\geq
0.$ This clearly showed that the population we are studying is not a zero
population. We obtained an important threshold parameter called the effective
reproduction number $\mathbb{R}_{e}$ using the next generation matrix method.
The result of the effective reproduction number is less than one. The local
and global stability were investigated using Routh-Hurtz criteria and Lyapunov
method respectively and both were asymptotically stable for $R_{e}<1.$

From the graphical illustrations, we concluded that immune population
continues to decay exponentially due to temporal immunity conferred on the
individuals in the immune class. We also concluded that reproduction number of
the infection grows when there is no control measure in the model and decays
when control measure is applied in the model. Also from the analysis, it can
be seen that the disease can be totally eliminated from the community, because
the sensitivity index shows that the lower the waning rate $\nu,$ the more the
exponential decrease in the passive immunity but the susceptible population
increases to the equilibrium and wanes asymptotically due to the presence of
the control parameters and restricted interaction of the latent and infected
classes.

The sensitivity analysis of the effective reproduction number further shows
that the rate of expiration of immunity $\upsilon,$ coupled with the
interaction rate $\theta,$ recruitment rate $\sigma$ and the infectious rate
$\beta$ are the most sensitive parameters to be targeted by way of
intervention strategy.

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