# PIRB: A Comprehensive Benchmark of Polish Dense and Hybrid Text Retrieval
Methods

###### Abstract

We present Polish Information Retrieval Benchmark (PIRB), a comprehensive
evaluation framework encompassing 41 text information retrieval tasks for
Polish. The benchmark incorporates existing datasets as well as 10 new,
previously unpublished datasets covering diverse topics such as medicine, law,
business, physics, and linguistics. We conduct an extensive evaluation of over
20 dense and sparse retrieval models, including the baseline models trained by
us as well as other available Polish and multilingual methods. Finally, we
introduce a three-step process for training highly effective language-specific
retrievers, consisting of knowledge distillation, supervised fine-tuning, and
building sparse-dense hybrid retrievers using a lightweight rescoring model.
In order to validate our approach, we train new text encoders for Polish and
compare their results with previously evaluated methods. Our dense models
outperform the best solutions available to date, and the use of hybrid methods
further improves their performance.

Keywords: information retrieval, dense retrieval, hybrid retrieval, neural
text encoders

PIRB: A Comprehensive Benchmark of Polish Dense and Hybrid Text Retrieval
Methods

Sławomir Dadas, Michał Perełkiewicz, Rafał Poświata
---
National Information Processing Institute, Warsaw, Poland
{sdadas, mperelkiewicz<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Text information retrieval is a process of retrieving relevant documents from
a large collection of text data in response to a user’s query. It is a
fundamental task in natural language processing and plays a crucial role in
various applications, including search engines, question-answering systems,
and recommendation engines. Despite the fact that research in the field of
information retrieval has a long history, we have recently observed increased
interest in this topic. This is primarily related to the emergence of large
language models (LLMs), whether offered as services like GPT-4 (OpenAI, 2023)
or free and publicly available ones like LLaMA (Touvron et al., 2023a, b) or
Falcon (Penedo et al., 2023). With the popularization of these solutions, more
attention is also being given to retrieval augmented generation systems (Lewis
et al., 2020; Cai et al., 2022), in which a large language model, along with a
user’s query, receives additional context from an external knowledge base.
This context is created based on documents most relevant to the query,
extracted using a retrieval algorithm. Correctly selected documents reduce the
hallucinations of the language model (Shuster et al., 2021) and allow the
injection of additional knowledge that the model may not possess. The quality
of the response of such a system is therefore highly dependent on the
performance of its retrieval component.

## 2\. Related work

In recent years, an active area of research has been the use of neural
networks, particularly Transformer-based language models, to develop novel
text retrieval methods. Many new models have been proposed (Yates et al.,
2021; Zhao et al., 2022; Guo et al., 2022; Fan et al., 2022), leading to
significantly improved results on public benchmarks (Thakur et al., 2021)
compared to classical term-based approaches such as BM25 (Robertson et al.,
2009). However, this progress has been particularly noticeable for high-
resource languages such as English and Chinese, while interest in multilingual
and low-resource language solutions has been considerably less pronounced.
This can be attributed to the limited availability of datasets, making it
challenging to apply the same supervised training methods. This is also
related to the lack of standardized benchmarks that would allow for the
evaluation of model performance across a wide range of retrieval tasks.

The situation for multilingual retrieval has only recently begun to change
with the release of datasets such as mMARCO (Bonifacio et al., 2021), Mr.TyDi
(Zhang et al., 2021) and MIRACL (Zhang et al., 2023), each covering more than
10 languages. In the case of models, multilingual encoders designed for
semantic textual similarity (Reimers and Gurevych, 2020; Yang et al., 2020) or
for bitext mining (Artetxe and Schwenk, 2019; Feng et al., 2022) had been
available for several years, but there was a lack of high-quality models
adapted for text retrieval problems. This improved with the release of the
multilingual versions of the E5 (Wang et al., 2022) models, which were trained
in a weakly supervised manner on a large-scale corpus of text pairs extracted
from various online data sources, and subsequently fine-tuned on several
manually annotated corpora, including retrieval data.

Despite these efforts, there are still languages not covered in multilingual
text retrieval research. One such language is Polish, which has not been
featured in any of the aforementioned datasets. To address this issue, the
Polish research community has taken the initiative to prepare language-
specific corpora. Rybak et al. (2022) introduced PolQA, a manually annotated
dataset for text retrieval containing 7,000 questions matched with passages
from Wikipedia. Additionally, some datasets have been created in an automatic
or semi-automatic way. Wojtasik et al. (2023) translated the original BEIR
benchmark (Thakur et al., 2021) into Polish using Google Translate and
evaluated several baseline retrievers and rerankers. Rybak (2023) made
available the MAUPQA collection, comprising almost 400,000 automatically
generated, translated, or mined question-answer pairs. Since its original
release, the collection has been expanded to include more datasets and
currently has over one million text pairs. It is also worth mentioning the
PolEval-2022 Passage Retrieval challenge (Kobyliński et al., 2023), which
utilized data from the PolQA dataset, and introduced two additional subsets
from new domains: legal questions and e-commerce FAQ.

Although the amount of Polish data for text retrieval is currently
sufficiently large, at least compared to low-resource languages, the vast
majority of available corpora have been generated automatically, primarily
through machine translation. Manually annotated data make up only a small
percentage and are limited to a few thousand queries in a single dataset at
most. Consequently, some of the generated datasets may be noisy and contain
errors resulting from incorrect translations of documents. Therefore, we
believe it would still be beneficial to create new datasets containing real
questions and answers written or labeled by humans.

In contrast to datasets, there have been few Polish text retrieval models
released to date. As part of the work on the MAUPQA collection, two dense
retrievers trained on its question-passage pairs have been published (Rybak,
2023; Rybak and Ogrodniczuk, 2023). Furthermore, the authors of the Polish
BEIR benchmark provided four rerankers trained by
them111https://huggingface.co/clarin-knext.

## 3\. Contributions

The aim of our research is to advance Polish text information retrieval in two
areas. Our first contribution is to propose a unified benchmark covering a
wide range of multi-domain tasks with different characteristics, enabling a
reliable and comprehensive evaluation of existing retrieval methods,
particularly their generalization ability and zero-shot performance. The
second goal is to train and publish new retrieval models for Polish, and then
evaluate their performance on the proposed benchmark. The results of our work
are described in the article in the following order:

  * •

We present Polish Information Retrieval Benchmark (PIRB) covering 41 text
retrieval tasks. The benchmark includes pre-existing dataset collections such
as MaupQA, BEIR-PL, and PolEval-2022 Passage Retrieval. We have also prepared
10 new, previously unpublished tasks specifically for the evaluation. Nine of
these tasks contain sets of actual questions and answers collected from
various Polish websites with diverse topics such as medicine, law, business,
physics, or linguistics. The last dataset was generated semi-automatically
using GPT-3.5 and includes over 8,000 exam-like questions from over 400
university-level courses.

  * •

We perform an evaluation of more than 20 Polish and multilingual text encoders
on the PIRB benchmark. The experiments are performed on already existing dense
retrieval models, as well as on strong baseline solutions trained by us,
including dense retrievers created using scripts from Sentence-Transformers
library and the state-of-the-art sparse retrieval model SPLADE (Formal et al.,
2021b, a).

  * •

In the last part of the publication, we present our recipe for training highly
effective language-specific retrievers. It is a three-step process. First, we
use a multilingual knowledge distillation technique (Reimers and Gurevych,
2020) to transfer knowledge from a high-quality English text encoder to a pre-
trained language model for Polish. In the next step, we perform a supervised
fine-tuning of the created encoder on the annotated retrieval dataset. The
final step is to create a lightweight hybrid retriever, combining the results
of the sparse and dense methods using an additional learning-to-rank model.
Our proposed technique can be an efficient alternative to solutions that
combine the retrieval stage with computationally expensive rerankers. In order
to validate the effectiveness of the proposed solution, we apply it to train
several Polish text encoders and compare their results with previously
evaluated methods on the PIRB benchmark.

Furthermore, we make the developed benchmark publicly
available222huggingface.co/spaces/sdadas/pirb, as well as the source code of
our experiments333github.com/sdadas/pirb, and the checkpoints of all the
models we trained444share.opi.org.pl/s/iS3ziwW9syHdrBj.

## 4\. Overview of the datasets

In this section, we present the datasets comprising the PIRB benchmark. We
begin with a description of the already existing datasets and dataset
collections, and then move on to introduce the corpora we have prepared: web
datasets and GPT-exams.

### 4.1. Pre-existing datasets

Of all the datasets included in the PIRB, 31 were previously available as
separate benchmarks or individual corpora. When building the benchmark, we
sought to collect most of the known publicly available question answering and
text retrieval corpora for Polish. We excluded datasets that overlapped with
others already found in the benchmark, such as PolQA, whose data was mostly
included in queries and passages published for the PolEval-2022 challenge. The
following datasets were included in our evaluation:

  * •

PolEval-2022 Passage Retrieval was a competition that took place from mid-2022
to early 2023, and its results were summarized during a workshop organized as
part of the FedCSIS conference (Kobyliński et al., 2023). During the
competition, successive pieces of data were gradually made available, starting
from the training (train) and validation (dev-0) splits, which included only
data from Wikipedia, to the test sets (test-A and test-B), each containing
three subsets from different domains: Wikipedia, legal questions, and
e-commerce FAQ. As part of PIRB, we added 7 data subsets from PolEval: dev-0,
as well as three parts each for test-A and test-B.

  * •

BEIR-PL (Wojtasik et al., 2023) aimed to replicate the original BEIR (Thakur
et al., 2021) benchmark for Polish. The authors used Google Translate to
translate the datasets and published 11 of them, which we included in PIRB.

  * •

MAUPQA (Rybak, 2023; Rybak and Ogrodniczuk, 2023) is an ongoing project aimed
at assembling a large and diverse corpus of questions and passages that can be
used to train Polish retrievers, rerankers, or generative question answering
models. The data was mostly collected automatically, and the author employed
various techniques to build individual subsets, including machine translation,
generation using large language models, transcription of game show recordings,
and utilizing data available in partially structured sources for question and
answer mining. Some parts of the MAUPQA collection are exact copies of
datasets already present in BEIR-PL, so not all of them were included in PIRB.
We selected 12 datasets that were added to the benchmark.

  * •

MFAQ (De Bruyn et al., 2021) is a multilingual dataset of questions and
answers extracted from FAQ pages found in Common Crawl. While the original
collection covered 21 languages, in our evaluation we used only the Polish
corpus, consisting of more than 60,000 text pairs.

### 4.2. Web datasets

Dataset name |  Domain |  Total queries |  Avg. words per query |  Avg. answers per query |  Total documents |  Avg. words per document
---|---|---|---|---|---|---
abczdrowie | medicine | 224,533 | 82 | 1.22 | 274,687 | 77
e-prawnik | law | 35,994 | 63 | 1.00 | 35,994 | 207
gemini | medicine | 272 | 19 | 1.00 | 272 | 120
odi | bussiness | 969 | 8 | 1.60 | 1,546 | 96
onet | trivia | 10,943 | 10 | 1.00 | 10,943 | 27
pwn | linguistics | 16,197 | 41 | 1.00 | 16,197 | 89
specprawnik | law | 54,783 | 66 | 1.22 | 66,832 | 64
techpedia | trivia | 9,205 | 8 | 1.00 | 9,205 | 43
zapytajfizyka | physics | 1,446 | 50 | 1.00 | 1,446 | 211
Table 1:  Characteristics of the web datasets collected by us. For each
dataset, we report its domain, number of queries and documents, average query
and document length in words, as well as the average number of relevant
documents per query.

The PIRB benchmark includes nine datasets crawled from Polish websites. Our
goal was to enrich the benchmark with real-world data written originally in
Polish. To accomplish this, we identified websites with separate Q&A sections,
some containing questions and answers written directly by the site’s editors,
and others allowing content creation by registered users. We selected a
diverse set of data sources covering various domains such as medicine, law,
business, physics, and linguistics. For each data source, we implemented a
separate parser to enable precise extraction of question and answer content.
We also conducted data cleaning and anonymization, removing personal and
contact information. A summary of statistics regarding the collected datasets
is presented in Table 1.

The largest datasets originate from websites such as abczdrowie and
specprawnik, where content is created by the user community. These are
platforms that connect specialists in a particular field, namely medicine and
law, with users seeking answers to questions in those fields. Anyone can ask a
question, and any registered and verified specialist can provide an answer,
allowing these platforms to build large databases of expert advice. The
content on these platforms is moderated or redacted by site editors to a
minimal extent, resulting in lower data quality compared to other websites we
considered. These datasets also required intensive cleaning and filtering. In
the resulting data, we included only answers longer than 200 characters and
filtered out all unanswered questions. It is also worth noting that in the
case of these two datasets, the average length of questions and answers is
similar, as users often describe their personal situation with details.
Questions have a more individualized character compared to other platforms.

Another group consists of websites that have a dedicated Q&A section managed
by the site staff. Examples of such sites include e-prawnik, gemini, odi, pwn,
and zapytajfizyka. Some of these sites allow users to submit their own
questions or answers, but these contributions go through editors who review
them and decide on their publication. These websites often employ or
collaborate with specialists in their respective fields. Datasets created
based on these sources are of high quality, but their size is smaller compared
to sites relying on user-generated content. Questions are typically shorter
and have a general nature, while answers are comprehensive articles providing
exhaustive descriptions of the given topic.

The last type of websites includes onet and techpedia. The datasets based on
them come from quizzes or collections of trivia questions, the main purpose of
which is not to provide advice in a specific field, but primarily to
entertain. Users of these sites can complete quizzes to test their knowledge
on various topics. Both questions and answers are short. We discarded most
data samples from these sources, as they included answers in the form of a
single phrase or a short sentence. Only those cases that had a descriptive
answer longer than 50 characters were added to the dataset.

### 4.3. GPT-exams

GPT-exams is a dataset created by us in a semi-automatic way utilizing the
gpt-3.5-turbo-0613 model available in the OpenAI API. The dataset contains
8,131 exam-like question-answer pairs covering a wide range of topics. To
build the dataset, we performed the following steps:

  1. 1.

We manually prepared a list of 409 university-level courses from various
fields. For each course, we instructed the model with the prompt: "Wygeneruj
20 przykładowych pytań na egzamin z [nazwa przedmiotu]" (Generate 20 sample
questions for the [course name] exam). We then parsed the outputs of the model
to extract individual questions and performed their deduplication.

  2. 2.

In the next step, we requested the model to generate the answer to each of the
collected questions. We used the following prompt: "Odpowiedz na następujące
pytanie z dziedziny [nazwa przedmiotu]: [treść pytania]" (Answer the following
question from [course name]: [question content]). We sent the following system
message with the prompt: "Jesteś ekspertem w dziedzinie [nazwa przedmiotu].
Udzielasz specjalistycznych i wyczerpujących odpowiedzi na pytania." (You are
an expert in [course name]. You provide knowledgeable and comprehensive
answers to questions).

  3. 3.

In the last step, we manually removed from the dataset the cases in which the
model refused to answer the question. We searched for phrases such as "model
języka" (language model), "nie jestem" (I’m not), or "nie mogę" (I can’t).

## 5\. Evaluation

This section provides a description of the evaluation we conducted on the
Polish Information Retrieval Benchmark (PIRB). The experiments include both
strong baseline models that we trained as a part of this research and other
dense text encoders available for the Polish language. First, we describe the
evaluated methods, and then we proceed to discuss the obtained results.

### 5.1. Baseline methods

Model name |  Average NDCG@10 (41 tasks) |  Avg. without MAUPQA (29 tasks) |  PolEval-2022 (7 tasks) |  Web Datasets (9 tasks) |  BEIR-PL (11 tasks) |  MAUPQA (12 tasks) |  Other (2 tasks)
---|---|---|---|---|---|---|---
Our sparse baselines
BM25 | 41.85 | 43.17 | 45.51 | 47.27 | 33.26 | 38.64 | 71.09
SPLADE++ | 52.93 | 54.06 | 58.92 | 58.60 | 42.47 | 50.22 | 80.39
Our dense baselines
MSE baseline (base) | 45.47 | 47.82 | 51.87 | 55.56 | 33.55 | 39.80 | 77.35
MSE baseline (large) | 49.98 | 52.47 | 57.49 | 61.13 | 37.26 | 43.98 | 79.58
MNR baseline (base) | 46.44 | 48.98 | 51.69 | 55.71 | 36.52 | 40.30 | 77.74
MNR baseline (large) | 48.63 | 51.18 | 54.22 | 57.82 | 38.61 | 42.45 | 79.84
Other Polish and multilingual retrievers
poleval-2022 (base) | 45.57 | 47.62 | 50.45 | 55.96 | 33.84 | 40.63 | 75.98
poleval-2022 (large) | 48.26 | 50.90 | 53.78 | 58.99 | 37.41 | 41.90 | 78.58
silver-retriever-v1 (base) | $*$ | 53.61 | 60.87 | 61.92 | 37.18 | $*$ | 81.18
multilingual-e5 (small) | 50.65 | 52.26 | 57.84 | 53.82 | 42.45 | 46.77 | 79.72
multilingual-e5 (base) | 53.12 | 55.08 | 60.16 | 59.09 | 44.01 | 48.38 | 80.18
multilingual-e5 (large) | 57.29 | 60.09 | 65.86 | 64.35 | 48.99 | 50.53 | 81.81
Table 2:  Evaluation results of our baselines and other available retrieval
models for Polish. We report Normalized Discounted Cumulative Gain at 10
(NDCG@10) for the entire PIRB benchmark consisting of 41 tasks as well as for
separate task groups. The best score in each column is shown in blue, the
second best in red.
$*$ Since silver-retriever-base-v1 (Rybak and Ogrodniczuk, 2023) was trained
on datasets from MAUPQA, it cannot be reliably evaluated on this part of the
benchmark. In order to compare it to the other models, we also provide average
scores on the benchmark with MAUPQA datasets excluded.

In our experiments, we included two baseline methods relying on sparse term-
based vectors, as well as dense retrievers fine-tuned from Transformer
language models. We used the training split of the Polish MS MARCO dataset for
all models that required training. The evaluation includes the following
sparse approaches:

  * •

BM25 (Robertson et al., 2009) is a popular and effective term-based ranking
function used in information retrieval since the 1980s. It is an extension of
TF-IDF weighting scheme, providing a balanced approach to text ranking,
considering both term frequency and document specificity. In our experiments,
we used the implementation of BM25 available in anserini (Yang et al., 2018).

  * •

Sparse Lexical and Expansion (SPLADE) model (Formal et al., 2021b, a) is a
family of modern term-based methods employing Transformer language models. In
this approach, the masked language modeling (MLM) head is optimized to
generate a vocabulary-sized weight vector adapted for text retrieval. SPLADE
is a highly effective sparse retrieval ranking algorithm, achieving better
performance than classic methods such as BM25. We trained the Polish version
following the same procedure as the SPLADE++ variant (Formal et al., 2022),
utilizing hard negatives mined with an ensemble of cross-encoders
(EnsembleDistil). We used Polish DistilRoBERTa (Dadas et al., 2020) as our
base language model.

In addition to the above methods, we also trained Polish text encoders using
scripts provided in the Sentence-Tranformers library. These examples show how
to train high-quality dense retrievers with a set of hard negatives acquired
from a cross-encoder. Authors of the library propose two fine-tuning methods
with different loss functions:

  * •

MSE (Mean Squared Error) baseline is a model fine-tuned using the Margin-MSE
loss (Hofstätter et al., 2020), which aims to reduce the margin difference for
a positive-negative pair between the reference scores obtained from cross-
encoder and scores produced by the optimized model. Specifically, the loss is
calculated using the following formula:

$\mathrm{MSE}\Bigl{(}s(q,d^{+})-s(q,d^{-}),\hat{s}(q,d^{+})-\hat{s}(q,d^{-})\Bigr{)}$
(1)

in which $q$, $d^{+}$ and $d^{-}$ denote query, positive document, and
negative document, respectively, $s$ is the reference similarity between
query-document pair, and $\hat{s}$ is the similarity computed by the trained
model.

  * •

MNR (Multiple Negatives Ranking) baseline uses contrastive loss with in-batch
negatives (Henderson et al., 2017) in addition to hard negatives. Given a
query $q_{i}$ and a mini-batch consisting of $K$ queries, the loss for that
query is calculated with the following formula:

$-\hat{s}(q_{i},d^{+}_{i})+\log\sum_{\begin{subarray}{c}k=1,\\\ k\neq
i\end{subarray}}^{K}e^{\hat{s}(q_{i},d^{+}_{k})}+\log\sum_{k=1}^{K}e^{\hat{s}(q_{i},d^{-}_{k})}$
(2)

in which $d^{+}_{k}$ and $d^{-}_{k}$ denote positive and negative document for
the $k$-th query, and $\hat{s}$ is the computed text pair similarity.

We fine-tuned Polish RoBERTa base and large (Dadas et al., 2020) language
models with both methods. We used the default hyperparameters defined in the
original training scripts, except for the batch size, which we reduced from 64
to 32 for the large models. The models were trained for 10 epochs. We employed
a learning rate scheduler with a warmup phase for the first 1,000 batches, a
peak learning rate of $2e{-5}$, and linear decay for the remainder of the
training. Cosine similarity was used as a similarity metric and mean pooling
as the output pooling method.

### 5.2. Other methods

As part of our experiments, we evaluated over 20 publicly available Polish and
multilingual dense text encoders, but only a few of them proved to be
effective enough for practical use in retrieval tasks. In this publication, we
present results only for the models that achieved an NDCG@10 score higher than
the BM25 baseline. In the full ranking, which we have made available
online555huggingface.co/spaces/sdadas/pirb, we include all of the evaluated
retrievers, along with more detailed results covering individual datasets.
Models offering good performance for Polish text retrieval include:

  * •

Multilingual E5 is a text encoder supporting over 100 languages. It is a
multilingual version of the E5 model (Wang et al., 2022), developed using the
same two-stage training procedure. The first stage involved weakly-supervised
training on a dataset of text pairs extracted from large internet corpora,
such as Common Crawl. In the second stage, the model was fine-tuned in a
supervised manner on several annotated datasets. The multilingual versions,
which were released a few months after the original English models, were made
available in small (118M parameters), base (278M), and large (560M) sizes.

  * •

Silver Retriever (Rybak and Ogrodniczuk, 2023) is a Polish dense retrieval
model trained on MAUPQA datasets with hard negatives mined employing a
combination of heuristic rules and cross-encoders. The model was trained with
constrastive loss for 15,000 steps using a batch size of 1024. Only the base-
sized version of the encoder was released, fine-tuned from the HerBERT
language model (Mroczkowski et al., 2021).

  * •

Although the solutions submitted to the PolEval-2022 Passage Retrieval
competition focused primarily on the reranking phase, a few dense retrievers
were also trained. Kozłowski (2023) provided two text retrieval models, base
and large, fine-tuned on a Polish translation of MS MARCO and training data
from the competition, with a set of negatives generated using BM25.

### 5.3. Results

The results of our evaluation are presented in Table 1. The table compares the
performance of our baseline models and the other models described in the
previous section. The methods are scored according to the Normalized
Discounted Cumulative Gain at 10 (NDCG@10) metric.

We can see that the multilingual E5 models demonstrate high quality, with the
large model significantly outperforming the other evaluated solutions. Among
the smaller models, in addition to E5 base, Silver Retriever and SPLADE++ also
deliver good results. Silver Retriever achieves the second-highest score in
three task groups, although worse performance on the BEIR-PL datasets lowers
its overall rating. SPLADE++ performs well on all task groups, achieving an
average NDCG@10 value close to E5 base, despite being based on a model with
only 82 million parameters, fewer than any of the evaluated dense retrievers.

Models from the PolEval-2022 competition and our dense baselines lag behind
the other methods. All of these retrievers show similar performance, with MSE
large baseline being marginally better than the other models. However, the
difference in NDCG@10 between these models and E5 with a similar number of
parameters is at least a few points in favor of E5.

## 6\. Dense and hybrid retrievers

Figure 1: An overview of our procedure for building effective retrieval
methods. In the first step, we perform knowledge transfer from an English
dense retriever using a bilingual corpus. Next, we fine-tune the obtained
model on an annotated dataset for text retrieval in the target language using
contrastive loss. In the final step, we construct a lightweight hybrid
combining dense and sparse methods, utilizing an additional learning-to-rank
model.

The final stage of our research involved training new dense text encoders for
the Polish language that could compete with the best currently available
models. In this section, we present our recipe for training effective
language-specific retrieval methods. Our proposed process consists of three
steps, as illustrated in Figure 1. We begin this section by providing an
overview of each stage of building our retrieval solution. We then compare the
results achieved by our models on the PIRB benchmark with methods tested
earlier.

### 6.1. Methodology

Below, we present a description of the individual stages of our experiments.
The dense retrieval model resulting from each of the steps is used in the
subsequent stage of the process. The order of these steps is as follows:

  1. 1.

Knowledge distillation \- Our procedure begins with transferring knowledge
from a high-quality English text encoder to a language-specific model,
employing multilingual knowledge distillation technique (Reimers and Gurevych,
2020). In this procedure, the original well-performing encoder is known as the
teacher model, while the new encoder we want to train is the student model. We
initialized the student with the weights of an already trained checkpoint
supporting the target language, in our case Polish. We selected two groups of
models of different sizes as the basis for our experiments: pre-trained Polish
RoBERTa language models (Dadas et al., 2020) and multilingual E5 (Wang et al.,
2022). For teachers, we chose English FlagEmbedding Xiao et al. (2023) models.
The goal of this knowledge distillation method is to fine-tune a student model
using bilingual corpora to approximate text representation generated by the
teacher. During training, the original encoder generates a vector
representation of the English text, while the student produces a
representation for the translation of that text. The difference between the
vectors is then used to compute the mean-squared error (MSE) loss, which aims
to reduce the distance between these representations.

To conduct this stage of training, we collected a Polish-English bilingual
corpus consisting of over 60 million text pairs. The corpus included subsets
representing various types of content: 1) single sentences and short texts
from the OPUS (Tiedemann, 2012) project; 2) longer paragraph-aligned texts
from PELCRA (Przepiórkowski et al., 2010), DGT (Steinberger et al., 2012),
Wikipedia, and bilingual abstracts from scientific publications; 3) a
collection of over 2 million questions extracted from Common Crawl and
question-answering datasets, translated from English to Polish using machine
translation. The collected texts were subsequently filtered by the LaBSE (Feng
et al., 2022) model to exclude noisy and low-quality translations. We
discarded texts for which the similarity between aligned sentences was lower
than 0.7. The models were trained with a batch size of 32. We trained small
and base models for 3 epochs, large models for 2 epochs. A learning rate
scheduler with linear decay and a warmup phase of 10,000 steps was used.

Student model | Teacher model |  Before fine-tuning | Fine-tuned |  Hybrid (BM25) |  Hybrid (SPLADE)
---|---|---|---|---|---
No distillation, only fine-tuning
multilingual-e5 (small) | - | 50.65 | 51.38 (+0.73) | 53.56 (+2.91) | 55.63 (+4.98)
multilingual-e5 (base) | - | 53.12 | 54.13 (+1.01) | 55.59 (+2.47) | 56.79 (+3.67)
multilingual-e5 (large) | - | 57.29 | 57.44 (+0.15) | 58.47 (+1.18) | 58.66 (+1.37)
Distillation + fine-tuning
multilingual-e5 (small) | bge-small-en | 47.64 | 52.25 (+4.61) | 54.25 (+6.61) | 56.04 (+8.40)
multilingual-e5 (base) | bge-base-en | 53.56 | 56.01 (+2.45) | 56.99 (+3.43) | 57.91 (+4.35)
multilingual-e5 (large) | bge-large-en | 56.00 | 58.03 (+2.03) | 58.84 (+2.84) | 59.22 (+3.22)
polish-roberta-v2 (base) | bge-base-en | 53.60 | 56.11 (+2.51) | 57.20 (+3.60) | 58.01 (+4.41)
polish-roberta-v2 (large) | bge-large-en | 55.62 | 58.06 (+2.44) | 58.82 (+3.20) | 59.23 (+3.61)
Table 3:  A comparison of the NDCG@10 metric on the PIRB benchmark for
distilled, fine-tuned and hybrid methods. The best score in each column is
shown in blue, the second best in red.

  2. 2.

Fine-tuning \- The second stage involved further supervised training on an
annotated text retrieval dataset. This step is similar to the previously used
procedure for training baseline MNR models. In this case, we also used the
training split of the MS MARCO dataset, optimizing the models with contrastive
loss utilizing in-batch and hard negatives. However, we applied different
training hyperparameters. Following Wang et al. (2022), we introduced a
temperature hyperparameter and set it to a value of 0.01. We also used larger
batch sizes: 288 for small models, 192 for base models, and 72 for large
models. All models were fine-tuned for 50,000 steps, with a warm-up phase for
the first 1,000 steps to reach a peak learning rate of $2e-6$, which was then
linearly decayed.

  3. 3.

Sparse-dense hybrids \- The final step in our procedure was to build a hybrid
system that combines results from dense and sparse models. Since these
representations can capture different characteristics of the input data,
merging results from two indexes can improve the overall performance of the
text retrieval system (Chen et al., 2022; Luan et al., 2021). Naive
aggregation methods involve computing new weights for query-document pairs as
a weighted average of the scores returned by each index. Unfortunately, due to
the different scales of the scores, it’s difficult to set such weights that
would provide good zero-shot results for different datasets. More complex
systems use two-stage retrieval, in which the first stage retrieves a number
of relevant documents from each index, and then the documents are sorted by
the reranker model. However, such systems are considerably slower than purely
retrieval-based solutions.

In our approach, we propose a lightweight learning-to-rank model that takes
the scores of dense and sparse models as input and returns a new score for
each query-document pair. To train the model, we use the pairwise LambdaMART
(Burges, 2010) algorithm implemented in the XGBoost (Chen and Guestrin, 2016)
library, which optimizes the NDCG metric. For each index that makes up the
hybrid system, four attributes are passed to the learning-to-rank model: the
score for a given query-document pair, the maximum and minimum score returned
for the entire list of relevant documents, and a binary attribute indicating
whether a given document is on the relevant documents list. If the document is
not present in the set of relevant documents, all four attributes for the
index are set to 0. As in the previous stage, we also use a training split of
Polish MS MARCO to optimize the rescoring model. We use the following
hyperparameters for the XGBRanker model: number of gradient boosted trees is
set to 100, max tree depth to 6, subsampling ratio to 0.75, and column
subsampling by tree to 0.9. In our experiments, we evaluate two types of
hybrid indexes: one using the BM25 method as the sparse index and another
utilizing the stronger SPLADE model.

One of the advantages of the proposed rescoring model is its efficiency,
resulting in minimal computational overhead on the entire retrieval system.
For a hybrid system composed of two indexes and a maximum number of relevant
documents set to 200 per query (100 from each index), the throughput of
aggregating results using our lightweight learning-to-rank model was
approximately 1,500 queries per second on a machine with Intel i7-13700 CPU
and Nvidia RTX 3090 GPU. For comparison, using a Transformer-based
reranker666https://huggingface.co/clarin-knext/herbert-base-reranker-msmarco
on the same configuration achieved a throughput ranging from 2 to 5 queries
per second, depending on the dataset and length of the documents in batch.

### 6.2. Results

The results of our experiments are presented in Table 3. We compared the
average NDCG@10 values achieved by distilled, fine-tuned, and hybrid models on
the PIRB benchmark. We also included an evaluation of the original
multilingual E5 models, which exhibited high performance in retrieval tasks.
We fine-tuned them on Polish data and built sparse-dense hybrids, just like
for the distilled models.

We can observe that after the distillation stage, the quality of the obtained
models is lower than that of the available E5 models. An exception is the
base-sized models, which already gain a slight advantage at this stage.
However, distilled models respond significantly better to fine-tuning. Each
model achieves better results after that stage, with distilled models
experiencing an increase in the NDCG@10 metric ranging from 2.5 to over 4.5
points. For the original E5 models, this observed increase is lower, at a
maximum of 1 point.

The use of hybrid solutions allows for further improvement in results. The
observed increases are higher for smaller dense models, for which the
combination with a sparse index yields strong results. For instance, the
hybrid of the small-sized model with the SPLADE model significantly
outperforms standalone base-sized models, even approaching results obtained by
large models by around 1 point. For the other model sizes, an increase is also
noticeable, with base models gaining an additional 2 points and large models
gaining 1 point in NDCG@10 compared to the results from the second stage. As
expected, SPLADE hybrids offer better quality than BM25 hybrids, but the
difference between these methods also decreases as the size of the dense model
increases.

## 7\. Conclusion

In this work, we have introduced PIRB, a text information retrieval benchmark
for the Polish language consisting of 41 tasks, including 10 that incorporate
new, previously unpublished datasets. We conducted a detailed evaluation of
multiple retrievers, including dense and hybrid solutions trained as part of
our research. The methods built using our approach achieved results surpassing
the best text retrieval models for Polish to date. We believe that our work
will help standardize the evaluation of information retrieval systems and
advance research in this area for Polish.

## 8\. Acknowledgements

This research was conducted with the A100 GPU cluster support delivered by the
Gdansk University of Technology within the TASK center initiative.

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##  Appendix A Appendix: Detailed results

In the main part of the publication, we focused only on the subset of models
evaluated by us, which we assessed according to the NDCG@10 metric. Below are
more detailed results of our experiments, demonstrating a wider range of
metrics including NNDCG@10, MRR@10, Recall@100, and Accuracy@1.

Figure 2: NDCG@10 values obtained by the models trained in our study. We
present the results of the original multilingual E5 models, as well as models
distilled from FlagEmbeddings based on E5 and Polish RoBERTa. For each model,
we show its performance before fine-tuning, after fine-tuning on the Polish MS
MARCO dataset, and the result of the sparse-dense hybrid combining the given
model with BM25 and SPLADE methods.

Figure 3: MRR@10 values obtained by the models trained in our study. We
present the results of the original multilingual E5 models, as well as models
distilled from FlagEmbeddings based on E5 and Polish RoBERTa. For each model,
we show its performance before fine-tuning, after fine-tuning on the Polish MS
MARCO dataset, and the result of the sparse-dense hybrid combining the given
model with BM25 and SPLADE methods.

Figure 4: Recall@100 values obtained by the models trained in our study. We
present the results of the original multilingual E5 models, as well as models
distilled from FlagEmbeddings based on E5 and Polish RoBERTa. For each model,
we show its performance before fine-tuning, after fine-tuning on the Polish MS
MARCO dataset, and the result of the sparse-dense hybrid combining the given
model with BM25 and SPLADE methods.

Figure 5: Accuracy@1 values obtained by the models trained in our study. We
present the results of the original multilingual E5 models, as well as models
distilled from FlagEmbeddings based on E5 and Polish RoBERTa. For each model,
we show its performance before fine-tuning, after fine-tuning on the Polish MS
MARCO dataset, and the result of the sparse-dense hybrid combining the given
model with BM25 and SPLADE methods.


# Analysis of an SIR–model with global and local infections

Thomas Götz<EMAIL_ADDRESS>Mathematical Institute, University Koblenz,
D–56070 Koblenz, Germany

###### Abstract.

An epidemic model where disease transmission can occur either through global
contacts or through local, nearest neighbor interactions is considered. The
classical SIR–model describing the global interactions is extended by adding
additional equations for the density of local pairs in different epidemic
states. A locality parameter $p\in[0,1]$ characterizes the probability of
global or local infections. The equilibria of the resulting model are analyzed
in dependence of the locality parameter and the transmission rate of the
pathogen. An explicit expression for the reproduction number in terms of the
locality parameter and the disease parameters is obtained. Transient
simulations confirm these findings. Neighboring pairs of one infected and one
susceptible can be considered as active pairs, since local transmission of the
disease can only occur in that situation. Our analysis shows, that the
fraction of active pairs is minimal for intermediate values of the locality
parameter.

###### Key words and phrases:

SIR–model, Pair Approximation, Epidemiology

##  1\. Introduction

Compartmental models to describe the dynamics of diseases have been analyzed
since decades, starting with the works by Kermack–McKendrick [2]. Classical
SIR models assume a homogeneous mixing of susceptible, infected and recovered
individuals, such that each infected can transmit the disease to any
susceptible with the same rate. However network effects [12, 8] or spatial
inhomogeneity [13, 10] are known to have important effect on the transmission
dynamics. For the current SARS–COV–2 pandemic, household effects have been
identified as a prominent example of transmission networks [9, 12]. These
network effects can be viewed as _local_ transmissions in contrast to the
_global_ transmissions that are included in the classic SIR–model or its
variants with refined compartmental structure.

Spatial PDEs, network–based models or stochastic simulations are typical
approaches to combine local and global transmission mechanisms. Cellular
automata [14, 15, 16] can consider the global, long–range infection of a
susceptible by any arbitrary infected as well as local, short–ranged
infections due to neighboring infected individuals. In the sequel, we will
consider the transmission rate as a mix of the global transmission $p\beta$
and the local transmission $(1-p)\beta$ between the four neighbors on a square
lattice. The probability $p\in[0,1]$ can be viewed as a _locality parameter_
modeling the ratio of global transmissions compared to the overall
transmissions. The case $p=1$, i.e. only global transmissions and no local
transmissions corresponds to the classical spatial homogeneous SIR–model. The
other extremal case $p=0$ describes only local and no global transmissions
where infections occur only between neighboring individuals.

To mitigate the spread of epidemics, non–pharmaceutical interventions (NPI)
are a frequently used measure [18, 17]. Contact restrictions and lock–downs as
they are imposed during the SARS–COV–2 pandemic almost all over the world, can
be viewed as a shift in the locality parameter $p$. The stricter the
lock–down, the less likely global infections should be. However, the local
infections occurring inside households and other closely related groups are
still continuing and rather insensitive to government–imposed measures. Hence,
analyzing the behavior of an extended SIR–model including both, global and
local infections, can provide further insight into the effect of NPI’s when
altering the ratio between global and local transmissions.

Besides the locality parameter $p$, the transmission rate $\beta$ of the
pathogen plays an important role. During the ongoing pandemic, several
variants of the SARS–COV–2 virus appeared like the _alpha_ –, _delta_ – and
_omicron_ –subtype. Concerning their transmissibilty, _omicron_ seems to have
a three–times higher transmission rate than the previously dominant _delta_
–strain [7]. On the other hand, the recovery rate seems to be comparable for
both strains, see e.g. [11]. Therefore we will use the transmission rate
$\beta$ as varying parameter in our analysis to illustrate the sensitivity of
equilibria with respect to varying transmissibility of the pathogen.

The paper is organized as follows: In Section 2 we recall a pair–approximation
model [1, 6, 3] which can be viewed as an extension of the classical SIR–model
[2, 5]. The _density_ of pairs $SS,SI,\dots$ of two susceptible, one
susceptible and one infected individual, etc. are also taken into account.
Depending on the locality parameter $p\in[0,1]$ our model switches between the
SIR–model ($p=1$) and the pure pair approximation ($p=0$). To gain insight
into the behavior of the mixed model $0<p<1$, we investigate its equilibria in
Section 3. We compare the known results for the SIR–model with findings for
the local and mixed model. The next generation matrix approach allows us to
compute the basic reproduction number for the mixed model in Section 4.
Transient simulations of the mixed model are presented in Section 5 to
illustrate the convergence to the respective equilibria for various parameter
regimes.

##  2\. SIR–Model with Global and Local Infections

We consider the global–local SIR model with pair approximation as presented by
Maltz and Fabricius [1]. Introducing the normalized susceptible, infected and
recovered compartments $S,I,R$, pairs of two individuals can be in one of the
following six states: $SS$, $SI$, $SR$, $II$, $IR$ and $RR$. Let $Z_{ij}$
denote the normalized number of pairs in state $ij$. Assuming a constant
normalized population $N=1$, it holds that $S+I+R=1$ as well as
$Z_{SS}+Z_{SI}+Z_{SR}+Z_{II}+Z_{IR}+Z_{RR}=2$, since there are in total $2N$
pairs. Following the derivation presented in [1] we end up with an ODE–system
taking into account the susceptible and infected compartments $S,I$ as well as
the $SS$ and $SI$–pairs denoted by $X=Z_{SS}$ and $Y=Z_{SI}$.

(1a) $\displaystyle S^{\prime}$ $\displaystyle=\mu(1-S)-p\beta
SI-\frac{q\beta}{4}Y$ (1b) $\displaystyle I^{\prime}$ $\displaystyle=p\beta
SI+\frac{q\beta}{4}Y-(\gamma+\mu)I$ (1c) $\displaystyle X^{\prime}$
$\displaystyle=\mu(4S-2X)-2p\beta XI-\frac{3}{8}q\beta\frac{XY}{S}$ (1d)
$\displaystyle Y^{\prime}$
$\displaystyle=\mu(4I-Y)+p\beta(2X-Y)I+\frac{3}{8}q\beta\frac{XY}{S}-\frac{q\beta}{4}Y\left(\frac{3}{4}\frac{Y}{S}+1\right)-(\gamma+\mu)Y$

The remaining compartments $Z_{SR},Z_{II},Z_{IR}$ and $Z_{RR}$ are decoupled
from this four–dimensional system. The transmission rate is denoted by
$\beta$, the recovery rate equals $\gamma$ and $\mu$ denotes the birth and
death rates. By $p$ we denote the probability for a global transmission and
$q=1-p$ describes the probability of a local transmission. In case of $p=1$
and $q=0$, we obtain the standard Kermack–McKendrick SIR–model [2] where only
global infections occur. In the other extremal case $p=0$ and $q=1$ we obtain
the pair approximation model presented by Joo and Lebowitz [3].

For later reference we state the Jacobian of the system (1) depending on the
locality parameter $p$

(2) $\tiny J_{p}=\begin{pmatrix}-\mu-p\beta I&-p\beta S&0&-q\beta/4\\\ p\beta
I&p\beta S-(\gamma+\mu)&0&q\beta/4\\\ 4\mu+\frac{3q\beta XY}{8S^{2}}&-2p\beta
X&-2\mu-2p\beta I-\frac{3q\beta Y}{8S}&-\frac{3q\beta X}{8S}\\\ -\frac{3q\beta
Y}{16S^{2}}(2X-Y)&4\mu+p\beta(2X-Y)&2p\beta I+\frac{3q\beta
Y}{8S}&-(\gamma+2\mu+\tfrac{q\beta}{4})-p\beta
I+\frac{3q\beta(X-Y)}{8S}\end{pmatrix}\;.$

##  3\. Equilibria

To determine equilibria of the above model (1), we consider its stationary
state, i.e. $(S,I,X,Y)^{\prime}=0$. Adding Eqns. (1a) and (1b) we can solve
for $I$ and get

(3) $\displaystyle I$ $\displaystyle=I(S)=\frac{\mu}{\gamma+\mu}(1-S)\;.$ In
case of $q\neq 0$, we use Eqn. (1b) to determine (4) $\displaystyle Y$
$\displaystyle=Y(S)=\frac{4\mu}{q\beta}\left(1-\frac{p\beta}{\gamma+\mu}S\right)\,\left(1-S\right)\;.$

In the local situation $p=0$ and $q=1$, the above equation simplifies to

$Y_{0}=Y_{0}(S)=\frac{4\mu}{\beta}(1-S)\;.$

Next, we use (1c) to determine $X$ as a function of $S$

$\displaystyle X\cdot\left(2\mu+2p\beta
I+\frac{3q\beta}{8}\frac{Y}{S}\right)=4\mu S$ and hence (5) $\displaystyle
X=X(S)=\frac{8S^{2}}{S+3+\frac{p\beta}{\gamma+\mu}S(1-S)}\;.$

In the purely local case $p=0$, we get $X_{0}(S)=8S^{2}/(S+3)$ which is
completely independent of $\beta,\mu$ and $\gamma$. Finally Eqn. (1d) yields
an equation for S

(6)
$0\stackrel{{\scriptstyle!}}{{=}}F_{p}(S):=\mu(4I-2Y)+p\beta(2X-Y)I+\frac{3q\beta}{16}\frac{2X-Y}{S}Y-\left(\frac{q\beta}{4}+\gamma\right)Y\;.$

Since both $I$ and $Y$ contain a factor $\mu(1-S)$, we may write

(7) $F_{p}(S):=\mu(1-S)\cdot G_{p}(S)\;,$

this shows, that $S=1$ is the trivial, disease–free equilibrium
$(S^{0},I^{0},X^{0},Y^{0})=(1,0,2,0)$. Other equilibria must be zeros of the
function $G_{p}$. Before analyzing the function $G_{p}$ in order to determine
the equilibria for the general model, we consider first the two special cases
$p=1$ (classical SIR–model) and $p=0$ (only local infections).

###  3.1. Global Model $p=1$

The case $p=1$ and $q=0$ corresponds to the classical Kermack–McKendrick
equations

$\displaystyle S^{\prime}$ $\displaystyle=\mu(1-S)-\beta SI$ $\displaystyle
I^{\prime}$ $\displaystyle=\beta SI-(\gamma+\mu)I$

In this case, the equation $F_{1}(S)=\mu(1-S)\left[\frac{\beta
S}{\gamma+\mu}-1\right]=0$ determines the equilibria. Following the above
notation in Eqn. (7), the function $G_{1}$ is given by

$G_{1}(S)=\frac{\beta S}{\gamma+\mu}-1\;.$

The endemic equilibrium $S_{1}^{\ast}=\frac{\gamma+\mu}{\beta}$ corresponds to
$I_{1}^{\ast}=\frac{\mu}{\gamma+\mu}-\frac{\mu}{\beta}$. For
$\beta>\tilde{\beta}_{1}=\gamma+\mu$ the endemic equilibrium exists and can be
shown to be asymptotically stable. Two other compartments $X$ and $Y$ read in
the endemic equilibrium as

$X_{1}^{\ast}=2{S_{1}^{\ast}}^{2}=\frac{2(\gamma+\mu)^{2}}{\beta^{2}}\quad\text{and}\quad
Y_{1}^{\ast}=\frac{4\mu+2\beta X_{1}^{\ast}}{\gamma+2\mu+\beta
I_{1}^{\ast}}I_{1}^{\ast}=4S_{1}^{\ast}I_{1}^{\ast}\;.$

###  3.2. Local Model $p=0$

In the local model $p=0$ and $q=1$, we summarize

$I_{0}=\frac{\mu}{\gamma+\mu}(1-S)\;,\quad
Y_{0}=\frac{4\mu}{\beta}(1-S)\;,\quad X_{0}=\frac{8S^{2}}{S+3}$

and

$\displaystyle F_{0}(S)$
$\displaystyle=\mu(4I-2Y)+\left[\frac{3\beta}{16}\frac{2X-Y}{S}-\frac{\beta}{4}-\gamma\right]Y\;.$
Factoring out $\mu(1-S)$ yields $\displaystyle G_{0}(S)$
$\displaystyle=\frac{12S}{S+3}-\frac{3\mu}{\beta S}+C_{0}\;,$

where $C_{0}=\frac{4\mu}{\gamma+\mu}-\frac{4\gamma+5\mu}{\beta}-1$.

Expanding by $S(S+3)$ yields the quadratic polynomial

$P_{0}(S)=S(S+3)\cdot
G_{0}(S)=\left(12+C_{0}\right)S^{2}+\left(3C_{0}-\frac{3\mu}{\beta}\right)S-\frac{9\mu}{\beta}\;.$

Its roots determine the endemic equilibria $S_{0}^{\ast}$ of the local model.
We note, that $P_{0}(0)=-9\mu/\beta<0$ and $P_{0}(1)=12+4C_{0}-12\mu/\beta$.
For $12+C_{0}>0$, i.e.
$\beta>\bar{\beta}=\frac{4\gamma+5\mu}{11\gamma+15\mu}(\gamma+\mu)$ there
exists a unique root $S_{0}^{\ast}>0$ given by

(8)
$S_{0}^{\ast}=\frac{3}{2(12+C_{0})}\left[\frac{\mu}{\beta}-C_{0}+\sqrt{\left(C_{0}-\frac{\mu}{\beta}\right)^{2}+\frac{4\mu}{\beta}(12+C_{0})}\right]\;.$

As can be seen in Fig. 1, for fixed $\mu$ (blue curves) the transmission
parameter $\beta$ needs to be large enough to obtain a root $S_{0}^{\ast}$ in
the relevant interval $[0,1]$. For $\mu=0.5$ and $\beta=2$ (blue dashed
curve), there exists no equilibrium $S_{0}^{\ast}<1$, while for $\beta=3$
(blue solid) the endemic equilibrium $S_{0}^{\ast}\approx 0.8$ exists. The
critical value $\tilde{\beta}_{0}$ such that the endemic equilibrium exists,
is determined by the condition $P_{0}(S=1)=0$ for $\beta=\tilde{\beta}_{0}$,
i.e.

$0=2+\frac{4\mu}{\gamma+\mu}-\frac{4\gamma+8\mu}{\tilde{\beta}_{0}}\quad\iff\quad\tilde{\beta}_{0}=2\frac{(\gamma+2\mu)(\gamma+\mu)}{\gamma+3\mu}\,.$

For $\beta>\tilde{\beta}_{0}$ we obtain $P_{0}(1)>0$ and hence a root
$S_{0}^{\ast}$ in the interval $(0,1)$. For $\gamma=1$ and $\mu=1/2$, this
critical value equals $\tilde{\beta}_{0}=12/5=2.4$.

Figure 1. Plot of $P_{0}(S)$ for $\gamma=1$ and different combinations of
$\beta$ and $\mu$. Note, that for $\mu=0.5$ and $\beta=2<\tilde{\beta}_{0}$
(blue dashed), there exists _no_ equilibrium $P_{0}(S_{0}^{\ast})=0$ in the
interval $0\leq S_{0}^{\ast}\leq 1$.

The equilibrium $S_{0}^{\ast}$, see Eqn. 8, gives rise to the corresponding
equilibrium value $I_{0}^{\ast}=\frac{\mu}{\gamma+\mu}(1-S_{0}^{\ast})$ for
the infected compartment. In Fig. 2 we compare the local equilibrium
$I_{0}^{\ast}$ with its global counterpart
$I_{1}^{\ast}=\frac{\mu}{\gamma+\mu}-\frac{\mu}{\beta}\xrightarrow{\beta\to\infty}\frac{\mu}{\gamma+\mu}$.

Figure 2.  Left: Plot of $I^{\ast}$ vs. $\beta$ for $\mu=1/4$, $1/2$, $3/4$.
The blue curves correspond to the local model $p=0$, the green curves are for
the global model $p=1$.
Right: Plot of $I^{\ast}$ vs. $\mu$ for $\beta=2.5$, $3$, $5$.

To analyze the limiting behavior of $I_{0}^{\ast}$ for $\beta\to\infty$, we
consider the limit of the explicit solution formula (8) for $\beta\to\infty$
in the two cases

$\displaystyle S_{0}^{\ast}$
$\displaystyle\xrightarrow{\beta\to\infty}\begin{cases}3\frac{\gamma-3\mu}{11\gamma+15\mu}&\text{for
}\gamma>3\mu\\\ 0&\text{for }\gamma\leq 3\mu\end{cases}$ and hence
$\displaystyle I_{0}^{\ast}$
$\displaystyle\xrightarrow{\beta\to\infty}\begin{cases}\frac{8\mu(\gamma+3\mu)}{(\gamma+\mu)(11\gamma+15\mu)}&\text{for
}\gamma>3\mu\\\ \frac{\mu}{\gamma+\mu}&\text{for }\gamma\leq
3\mu\end{cases}\;.$

Summing up, there exist two equilibria in the local case:

  1. (1)

the trivial disease–free equilibrium
$(S_{0}^{0},I_{0}^{0},X_{0}^{0},Y_{0}^{0})=(1,0,2,0)$ and

  2. (2)

for $\beta>\tilde{\beta}_{0}$ the endemic equilibrium

$(S_{0}^{\ast},I_{0}^{\ast},X_{0}^{\ast},Y_{0}^{\ast})=\left(S_{0}^{\ast},\frac{\mu}{\gamma+\mu}(1-S_{0}^{\ast}),\frac{8S_{0}^{\ast}{}^{2}}{S_{0}^{\ast}+3},\frac{4\mu}{\beta}(1-S_{0}^{\ast})\right)\;.$

To analyze their stability, consider the Jacobian $J_{p=0}$, see Eqn. (2) at
the disease–free equilibrium

$\displaystyle J_{0}^{0}$ $\displaystyle=\begin{pmatrix}-\mu&0&0&-\beta/4\\\
0&-(\gamma+\mu)&0&\beta/4\\\ 4\mu&0&-2\mu&-3\beta/4\\\
0&4\mu&0&-(\gamma+2\mu)+\beta/2\end{pmatrix}\;.$

The eigenvalues of $J_{0}^{0}$ are given by

$\lambda_{1}=-\mu,\quad\lambda_{2}=-2\mu\quad\text{and}\quad\lambda_{3,4}=\frac{\beta}{4}-\gamma-\frac{3\mu}{2}\pm\frac{1}{4}\sqrt{\beta^{2}+12\beta\mu+4\mu^{2}}\;.$

For $\beta>\tilde{\beta}_{0}=2\frac{(\gamma+2\mu)(\gamma+\mu)}{\gamma+3\mu}$,
the maximal real part gets positive and the disease–free equilibrium gets
unstable. This can be seen in Fig. 3, where we plotted the maximal real part
of the eigenvalues at the disease–free equilibrium (blue curves) and at the
endemic equilibrium (green curves).

Figure 3.  Maximal real part of the eigenvalues of the Jacobian $J_{0}$ vs.
$\beta$ for different values of $\mu$. The blue curves correspond to the
disease–free equilibrium and the green ones are for the endemic equilibrium.
For $\beta>\tilde{\beta}$, the disease–free equilibrium gets unstable and the
endemic equilibrium turns stable.

###  3.3. General Case $0<p<1$

Next we consider the equilibria for the general case $0<p<1$. Recall, that in
this situation we have

$\displaystyle I(S)$ $\displaystyle=\mu(1-S)\cdot\frac{1}{\gamma+\mu}$
$\displaystyle Y(S)$
$\displaystyle=\mu(1-S)\cdot\frac{4}{q\beta}\left(1-\frac{p\beta}{\gamma+\mu}S\right)=\frac{4\mu}{\gamma+\mu}\frac{p}{q}S^{2}+\dots$

Plugging these expressions into (6) and factoring out the common term
$\mu(1-S)$ we obtain

$\displaystyle 0$ $\displaystyle\stackrel{{\scriptstyle!}}{{=}}F_{p}(S)=4\mu
I-2\mu Y+p\beta
I(2X-Y)+\frac{3}{16}q\beta\frac{Y}{S}(2X-Y)-Y\left(\frac{q\beta}{4}+\gamma\right)$
$\displaystyle=\mu(1-S)\left\\{\frac{4\mu}{\gamma+\mu}-\frac{8\mu}{q\beta}\left(1-\frac{p\beta}{\gamma+\mu}S\right)+\frac{p\beta}{\gamma+\mu}(2X-Y)\right.$
$\displaystyle\qquad\qquad\qquad\left.+\frac{3}{4S}\left(1-\frac{p\beta}{\gamma+\mu}S\right)(2X-Y)-\frac{4}{q\beta}\left(1-\frac{p\beta}{\gamma+\mu}S\right)\left(\frac{q\beta}{4}+\gamma\right)\right\\}$
$\displaystyle=:\mu(1-S)\cdot G_{p}(S)$

where

(9) $\displaystyle G_{p}(S)$
$\displaystyle=\frac{4\mu}{\gamma+\mu}-\left(1+\frac{8\mu+4\gamma}{q\beta}\right)\left(1-\frac{p\beta}{\gamma+\mu}S\right)+\left(\frac{1}{4}\frac{p\beta}{\gamma+\mu}+\frac{3}{4S}\right)(2X-Y)\;.$

Multiplying $G_{p}$ with $S\cdot
Q(S):=S\cdot\left(S+3+\frac{p\beta}{\gamma+\mu}S(1-S)\right)>0$ for $S>0$ to
get rid of $S$ in the denominator of $X$ yields the following polynomial in
$S$

$P_{p}(S):=\left[\frac{4\mu}{\gamma+\mu}-\left(1+\frac{8\mu+4\gamma}{q\beta}\right)\left(1-\frac{p\beta}{\gamma+\mu}S\right)\right]\cdot
S\cdot Q(S)\\\
+\left[\frac{p\beta}{4(\gamma+\mu)}S+\frac{3}{4}\right]\cdot\left(16S^{2}-Y\cdot
Q(S)\right)\;.$

Since $Q$ and $Y$ are quadratic in $S$, we see, that
$P_{p}\in\mathcal{P}_{5}(S)$, i.e. a polynomial of degree $5$. Hence we may
expect up to $5$ possible roots. To determine whether some of them are in the
relevant interval $0<S<1$ we note that

$\displaystyle P_{p}(S=0)$ $\displaystyle=-\frac{9\mu}{q\beta}<0$ and
$\displaystyle P_{p}(S=1)$
$\displaystyle=8+\frac{16\mu+8p\beta}{\gamma+\mu}+4\frac{8\mu+4\gamma}{q\beta}\left(\frac{p\beta}{\gamma+\mu}-1\right)\;.$

Recasting the terms, we can write $P_{p}$ as

(10) $P_{p}(S)=4\frac{p\beta}{\gamma+\mu}S^{3}+12S^{2}\\\
+Q(S)\cdot\left[\frac{4\mu}{\gamma+\mu}S-\left(\frac{4\mu}{q\beta}+S+\frac{4(\gamma+\mu)}{q\beta}S-\frac{1}{4}Y\right)\left(1-\frac{p\beta}{\gamma+\mu}S\right)\right]$

The leading coefficient of this polynomial is given by

$a_{5}=\frac{p\beta}{\gamma+\mu}\cdot\frac{1}{4}\cdot\frac{4\mu}{\gamma+\mu}\cdot\frac{p}{q}\cdot\frac{p\beta}{\gamma+\mu}=\frac{p^{3}\beta^{2}}{q(\gamma+\mu)^{3}}>0$

Hence there exists at least one positive root of $P_{p}$, the endemic
equilibrium. Since no analytic solutions are available, we solve the
polynomial $P_{p}$ numerically to identify the equilibira.

In Fig. 4 we have plotted the numerical results. The equilibrium
$I_{p}^{\ast}$ given in Fig. 4(right) for $\beta=3$ corresponds to the
stationary state visible in the transient solution in Fig 7.

Figure 4.  Left: Plot of the polynomial $P_{p}(S)$ for $\beta=3$, $\gamma=1$
and $\mu=1/2$ fixed. The locality parameter $p$ varies between $p=0$ (solid
blue) and $p=0.8$ (dotted magenta). The zeros correspond to the endemic
equilibrium.
Right: Endemic equilibrium $I_{p}^{\ast}$ vs. $\beta$ for $\gamma=1$ and
$\mu=1/2$ fixed and $3/2\leq\beta\leq 5$. Locality parameter $p$ varies
between $p=0$ (solid blue) and $p=1$ (solid green).

To ensure the existence of the endemic equilibrium $I_{p}^{\ast}$, the
following condition $P_{p}(S=1)>0$ is necessary. Solving

$\displaystyle
P_{p}(S=1)=\frac{8}{q\beta(\gamma+\mu)}\left[\beta(3\mu+\gamma+p\beta)+p\beta(\mu+\gamma-p\beta)-(4\mu+2\gamma)(\gamma+\mu)\right]>0$
for $\beta$ while keeping $\gamma,\mu>0$ fixed yields
$\displaystyle\beta^{2}(p-p^{2})+\beta((3+p)\mu+(1+p)\gamma)-2(2\mu+\gamma)(\gamma+\mu)>0\;.$

Since $p-p^{2}>0$, there exists a critical value $\tilde{\beta}_{p}>0$, such
that $P_{p}(S=1)>0$ for $\beta>\tilde{\beta}_{p}$. This critical value is
given by

(11)
$\tilde{\beta}_{p}:=\frac{1}{2(p-p^{2})}\left[\sqrt{((3+p)\mu+(1+p)\gamma)^{2}+8(2\mu+\gamma)(\gamma+\mu)(p-p^{2})}\right.\\\
\left.-((3+p)\mu+(1+p)\gamma)\right]\;.$

In the two extremal cases $p=0$ and $p=1$ we reobtain our previous findings
for the critical value

$\displaystyle\tilde{\beta}_{0}$
$\displaystyle:=2\frac{\gamma+2\mu}{\gamma+3\mu}(\gamma+\mu)\;,$
$\displaystyle\tilde{\beta}_{1}$ $\displaystyle:=\gamma+\mu\;.$

In Figure 5 we have depicted the regions, where the endemic equilibrium
exists. For fixed $\mu$ this region is determined by
$\beta>\tilde{\beta}_{p}$. For smaller values of $\beta$, the only equilibrium
is the disease–free equilibrium.

Figure 5.  Critical value $\tilde{\beta}_{p}$ vs. $\mu$ for different values
of the locality parameter $p$, recovery rate $\gamma=1$ fixed. For
$\beta>\tilde{\beta}_{p}$ (e.g. pale blue area for $p=0$), the endemic
equilibrium exists and is stable. For smaller values of $\beta$, there exists
only the disease–free equilibrium.

Inspecting the Jacobian

$J_{p}^{0}=\begin{pmatrix}-\mu&-p\beta&0&-q\beta/4\\\
0&p\beta-(\gamma+\mu)&0&q\beta/4\\\ 4\mu&-4p\beta&-2\mu&-3q\beta/4\\\
0&4\mu+4p\beta&0&-(\gamma+2\mu)+q\beta/2\end{pmatrix}$

at the disease–free equilibrium yields the following eigenvalues
$\lambda_{1}=-\mu$, $\lambda_{2}=-2\mu$ and

$\lambda_{3,4}=\frac{1}{4}\left[(1+p)\beta-4\gamma-6\mu\pm\sqrt{((1+p)\beta+2\mu)^{2}+8pq\beta^{2}+8q\mu\beta}\right]\;.$

For $\beta>\tilde{\beta}_{p}$, see Eqn. (11), the maximal real part of
$\lambda_{3}$ gets positive. Again, as in the purely local case $p=0$, as soon
as the endemic equilibrium appears, the disease–free equilibrium gets
unstable.

##  4\. Basic Reproduction Number

To compute the basic reproduction number we follow the next generation matrix
approach, see [4, 5]. We subdivide the system (1) into the infected
compartments $\xi=(I,Y)$ and the non–infected ones $\eta=(S,X)$. The equations
for the infected compartments are written as

$\displaystyle\xi^{\prime}$
$\displaystyle=\mathcal{F}(\xi,\eta)-\mathcal{V}(\xi,\eta)$ where
$\displaystyle\mathcal{F}_{p}(\xi,\eta)$
$\displaystyle=\begin{pmatrix}p\beta\xi_{1}\eta_{1}+q\beta\xi_{2}/4\\\
2p\beta\xi_{1}\eta_{2}+\tfrac{3}{8}q\beta\xi_{2}\eta_{2}/\eta_{1}\end{pmatrix}$
describes the new infections and $\displaystyle\mathcal{V}_{p}(\xi,\eta)$
$\displaystyle=\begin{pmatrix}(\gamma+\mu)\xi_{1}\\\
p\beta\xi_{1}\xi_{2}+\frac{q\beta}{4}\xi_{2}\left(\frac{3\xi_{2}}{4\eta_{1}}+1\right)+(\gamma+\mu)\xi_{2}-\mu(4\xi_{1}-\xi_{2})\end{pmatrix}$

describes other transitions. The disease free equilibrium is given by
$\xi^{0}=(0,0)$ and $\eta^{0}=(1,2)$. Introducing the Jacobians
$F_{p}=\frac{\operatorname{\partial}}{\operatorname{\partial}\xi}\mathcal{F}_{p}(0,\eta^{0})$
and
$V_{p}=\frac{\operatorname{\partial}}{\operatorname{\partial}\xi}\mathcal{V}_{p}(0,\eta^{0})$,
the basic reproduction number is given as the spectral radius of the next
generation matrix $M_{p}=F_{p}V_{p}^{-1}$, i.e.

$\mathcal{R}_{0,p}=\rho(M_{p})=\max\left\\{\lvert\lambda\rvert:\
\text{$\lambda$ eigenvalue of $M_{p}$}\right\\}\;.$

In our case

$\displaystyle F_{p}$
$\displaystyle=\begin{pmatrix}p\beta&\frac{1}{4}q\beta\\\\[2.15277pt]
4p\beta&\frac{3}{4}q\beta\end{pmatrix}\;,\quad
V_{p}=\begin{pmatrix}\gamma+\mu&0\\\
-4\mu&\frac{q\beta}{4}+\gamma+2\mu\end{pmatrix}$ and $\displaystyle M_{p}$
$\displaystyle=\begin{pmatrix}p\beta&\tfrac{1}{4}q\beta\\\\[2.15277pt]
4p\beta&\tfrac{3}{4}q\beta\end{pmatrix}\cdot\begin{pmatrix}\frac{1}{\gamma+\mu}&0\\\\[2.15277pt]
\frac{4\mu}{(\gamma+\mu)(\gamma+2\mu+q\beta/4)}&\frac{1}{\gamma+2\mu+q\beta/4}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}pk_{1}+q\mu k_{1}/k_{2}&q\beta/(4k_{2})\\\
4pk_{1}+3q\mu k_{1}/k_{2}&3q\beta/(4k_{2})\end{pmatrix}$

where $k_{1}=\beta/(\gamma+\mu)$ and $k_{2}=\gamma+2\mu+q\beta/4$. Hence

$\mathcal{R}_{0,p}=\frac{1}{8k_{2}}\left[4pk_{1}k_{2}+q(3\beta+4\mu
k_{1})+\sqrt{(4pk_{1}k_{2}+q(3\beta+4\mu k_{1}))^{2}+16pq\beta
k_{1}k_{2}}\right]$

In the global case $p=1$, $q=0$, this reduces to the well–known basic
reproductive number for the classical SIR–system

$\mathcal{R}_{0,p=1}=k_{1}=\frac{\beta}{\gamma+\mu}$

and for the local case $p=0$ and $q=1$ we get

$\mathcal{R}_{0,p=0}=\frac{3\beta+4\mu
k_{1}}{4k_{2}}=\frac{\beta}{\gamma+\mu}\cdot\frac{3\gamma+7\mu}{4\gamma+8\mu+\beta}=\mathcal{R}_{0,1}\cdot\frac{3\gamma+7\mu}{4\gamma+8\mu+\beta}$

It is obvious, that $\mathcal{R}_{0,p=0}<\mathcal{R}_{0,p=1}$. The local
transmission via pair interactions is slower than the global transmission.
This is confirmed by stochastic simulations, see [1, 6].

The critical value $\mathcal{R}_{0,p=0}=1$ for the local model corresponds to

$\displaystyle\beta=\tilde{\beta}_{0}$
$\displaystyle=2\frac{\gamma+2\mu}{\gamma+3\mu}(\gamma+\mu)\;.$

In the general case $p\in(0,1)$ we get

(12)
$\mathcal{R}_{0,p}=\mathcal{R}_{0,1}\cdot\left[\frac{1}{2}p+q\frac{3\gamma+7\mu}{8\gamma+16\mu+2q\beta}\right.\\\
\left.+\sqrt{\left(\frac{1}{2}p+q\frac{3\gamma+7\mu}{8\gamma+16\mu+2q\beta}\right)^{2}+pq\frac{\gamma+\mu}{4\gamma+8\mu+q\beta}}\right]\;.$

In Fig. 6 we show the variation of the basic reproduction number with varying
$\beta$ and $p$.

Figure 6.  Basic reproduction number $\mathcal{R}_{0,p}$ vs. $\beta$ for
$\gamma=1$ and $\mu=1/2$ fixed. The locality parameter $p$ varies between
$p=0$ (blue) and $p=1$ (green).

##  5\. Transient Solutions

We solve the ODE–system (1) numerically using a standard Runge–Kutta–Fehlberg
method of order 5(4). The recovery rate $\gamma=1$ is fixed and the time
interval is chosen as $t\in[0,25]$ to ensure that the transient solutions
reach the possible equilibrium. The initial condition
$(S,I,X,Y)(t=0)=(0.99,0.01,0.0)$ models the scenario of $1\%$ infections in an
otherwise initially naive population. In Figure 7 we show the transient
behavior of the infected compartment $I_{p}$ for different values of the
locality parameter $p\in[0,1]$ and $\beta=3$, $\mu=1/2$. All trajectories
settle at the corresponding endemic equilibrium $I_{p}^{\ast}$ shown in Fig.
4(right).

Figure 7.  Plot of the infected compartment $I_{p}$ vs. time for $\beta=3$,
$\gamma=1$ and $\mu=1/2$. The locality parameter $p$ varies between $p=0$
(blue) and $p=1$ (green).

To visualize the convergence against the respective equilibrium, we consider
in Fig. 8 the situation for $p=0.4$, $\gamma=1$ and $\mu=0.75$ fixed.
According to Eqn. (11) or Fig. 5, for these parameter values, the critical
transmission rate equals $\tilde{\beta}_{p=0.4}=1.978$. For larger values of
$\beta$, the transient solution will converge to the endemic equilibrium
$I_{p}^{\ast}$ and for smaller values of $\beta$ the disease will die out
since the disease–free equilibrium is asymptotically stable. The curves in
Fig. 8 confirm this; the blue trajectory for $\beta=1.75<\tilde{\beta}_{p}$
tends to zero while the two other trajectories for $\beta=2$ (magenta) and
$\beta=2.25$ (green) tend to the respective endemic equilibria.

Figure 8.  Plot of the infected compartment $I_{p}$ vs. time for $p=0.4$,
$\gamma=1$ and $\mu=0.75$. The transmission rate $\beta$ varies between the
subcritical value $1.75<\tilde{\beta}$ (blue) where the disease free
equilibrium is asymptotically stable and the two other cases $\beta=2$ and
$2.25$ (magenta and green), where the transmission rate is above the threshold
$\tilde{\beta}_{0.4}=1.978$ and hence the endemic equilibrium gets the stable
one.

As a final check we investigated the behavior of the transient solutions for
varying initial conditions. In Fig. 9 we plotted the transient solution $I$ in
the mixed model $p=0.4$ for parameters $(\beta,\gamma,\mu)=(3,1,0.5)$ fixed
and various initial conditions. All simulations tend to the same endemic
equilibrium $I^{\ast}_{p=0.4}\simeq 0.146$ as to be expected.

Figure 9.  Plot of the infected compartment $I$ vs. time for varying initial
conditions $(S,I,X,Y)(0)$. The locality parameter $p=0.4$ and the model
parameters $\beta=3$, $\gamma=1$ and $\mu=0.5$ are fixed.

##  6\. Discussion and Conclusion

The pair approximation model considers the infected pairs $SI$, $II$ and $IR$.
The pairs $II$ and $IR$ can be considered as _blocked_ since no transmission
can occur in that situation. The only _active_ pair is the combination $SI$.
Hence we introduce the ratio $\rho_{p}$ as the ratio between active and all
infection pairs. The total number of pairs including infected individuals
equals $Y+2Z_{II}+Z_{IR}=4I$ (the $II$–pair contains two infected) and we
obtain

$\displaystyle\rho_{p}$ $\displaystyle=\frac{Y}{4I}\;.$ It’s equilibrium value
is given by $\displaystyle\rho_{p}^{\ast}$
$\displaystyle=\frac{Y_{p}^{\ast}}{4I_{p}^{\ast}}=\frac{\gamma+\mu}{q\beta}\left(1-\frac{p\beta}{\gamma+\mu}S_{p}^{\ast}\right)$

for $p<1$ and $q>0$. In both extremal cases $p=1$ and $p=0$ we obtain the same
equilibrium value

$\rho_{1}^{\ast}=\rho_{0}^{\ast}=\frac{\gamma+\mu}{\beta}\;.$

Computing $\rho_{p}$ for arbitrary $p\in(0,1)$ seems out of reach, since it
requires the solution an endemic equilibrium $S_{p}^{\ast}$, the root of the
quintic polynomial (10).

In Fig. 10 we show the ratio $\rho_{p}$ vs. the locality parameter $p$ for
different disease parameters $\beta$ and $\mu$ ($\gamma=1$ is fixed). We
observe, that in both extremal cases $p=0$ (local model) and $p=1$ (global
model), the fraction of active pairs is maximal in each situation. For
intermediate values of $p$ more pairs gets blocked and the fraction of active
pairs attains a minimum. Note, that the curves are _not_ symmetric, i.e. the
minimum is not attained for $p=1/2$. Rewriting $\rho_{p}^{\ast}$ as

$\displaystyle\rho_{p}^{\ast}$
$\displaystyle=\frac{1}{1-p}\left(\frac{\gamma+\mu}{\beta}-pS_{p}^{\ast}\right)=\frac{1}{1-p}\left(S_{1}^{\ast}-pS_{p}^{\ast}\right)$
we get $\displaystyle\left.\frac{d}{dp}\rho_{p}^{\ast}\right|_{p=0}$
$\displaystyle=S_{1}^{\ast}-S_{0}^{\ast}\;.$

To show, that $S_{1}^{\ast}<S_{0}^{\ast}$ we consider

$\displaystyle G_{0}(S_{1}^{\ast})$
$\displaystyle=\frac{12S_{1}^{\ast}}{3+S_{1}^{\ast}}-\frac{3\mu}{\beta
S_{1}^{\ast}}+C_{0}=\frac{12(\gamma+\mu)}{3\beta+\gamma+\mu}+\frac{\mu}{\gamma+\mu}-\frac{5\mu+4\gamma}{\beta}-1$
$\displaystyle=-\frac{\beta\gamma^{2}+3\beta\mu^{2}+4\beta\gamma\mu+3\beta^{2}\gamma+(5\mu+4\gamma)(\gamma+\mu)^{2}}{\beta(3\beta+\gamma+\mu)(\gamma+\mu)}$

Hence $G_{0}(S_{1}^{\ast})<0$ and therefore $S_{1}\ast<S_{0}^{\ast}$ (cf. Fig.
1) and finally $\left.\frac{d}{dp}\rho_{p}^{\ast}\right|_{p=0}<0$. To
visualize the emergence of the minimum of $\rho_{p}^{\ast}$ graphically, we
have plotted in Fig. 11 the endemic equilibria $I^{\ast}_{p}$ and
$Y^{\ast}_{p}$ for $p\in[0,1]$. In this graph the model parameters $\beta=3$,
$\gamma=1$ and $\mu=0.5$ are fixed. As we can see, both curves decrease as $p$
decreases from the global model $p=1$ to smaller values. However, the
$SI$–pairs $Y_{p}^{\ast}$ decay faster than the total infected $I^{\ast}_{p}$;
hence the ratio $\rho_{p}=Y_{p}^{\ast}/(4I_{p}^{\ast})$ also decreases as $p$
gets smaller than $1$. In the neighborhood of the local situation $p=0$ we
obtain a similar behavior.

Figure 10.  Equilibrium value of the fraction $\rho_{p}$ of active pairs. The
model parameter $\gamma=1$ is fixed.

Figure 11.  Endemic equilibria $I^{\ast}_{p}$ and $Y^{\ast}_{p}$ vs $p$. The
model parameters $\beta=3$, $\gamma=1$ and $\mu=0.5$ are fixed.

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