00footnotetext: A.D. and L.T.H. are supported by the grant
FRGS/1/2019/STG06/UM/02/6.

# A classification of conformally flat generalized Ricci recurrent pseudo-
Riemannian manifolds

Tee-How Loo and Avik De  T. H. Loo
Institute of Mathematical Sciences
Universiti Malaya
50603 Kuala Lumpur
Malaysia<EMAIL_ADDRESS>A. De
Department of Mathematical and Actuarial Sciences
Universiti Tunku Abdul Rahman
Jalan Sungai Long
43000 Cheras
Malaysia<EMAIL_ADDRESS>
###### Abstract.

Conformally flat pseudo-Riemannian manifolds with generalized Ricci recurrent,
$(GR)_{n}$ structure are completely classified in this short report. A
conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is
shown to be either a de Sitter space or an anti-de Sitter space. In
particular, a conformally flat generalized Ricci recurrent spacetime must be
either a de Sitter spacetime or an anti-de Sitter spacetime.

##  1\. Introduction

The importance of the Generalized Ricci recurrent structure and its
interaction with several gravity theories, like the standard theory of gravity
[5], the modified $f(R)$-theory [1], the modified Gauss-Bonnet $f(R,G)$-theory
[2] are well established. And of course, mathematically the structure raised
curiosity among researchers in several occasions, details can be seen in [5]
and the references therein.

De et al. [3] introduced the notion of $(GR)_{n}$ as an $n$-dimensional non-
flat pseudo-Riemannian manifold whose Ricci tensor $R_{ij}$ satisfies the
following:

$\nabla_{i}R_{jl}=A_{i}R_{jl}+B_{i}g_{jl},$ (1)

where $A_{i}$ and $B_{i}$ are two non-zero 1-forms. The structure is
considered to be a generalization of Patterson introduced Ricci recurrent
manifolds $R_{n}$ [7] in which the Ricci tensor satisfies
$\nabla_{i}R_{jl}=A_{i}R_{jl}$, with a non-zero 1-form $A_{i}$. Obviously, if
the one-form $B_{i}$ vanishes, it reduces to a $R_{n}$.

A $(GR)_{4}$ spacetime is a generalized Robertson Walker spacetime with
Einstein fibre for a Codazzi type $R_{ij}$ [4]. In [5], a conformally flat
$(GR)_{4}$ was shown to be a perfect fluid. Its interaction with general
relativistic cosmology was discussed thoroughly in the same paper. Continuing
to this study, in [1], the authors studied a conformally flat $(GR)_{4}$ with
constant $R$ as a solution of $f(R)$-gravity theory. The presently accepted
homogeneous and isotropic model of our universe, the Robertson-Walker
spacetime is $(GR)_{4}$ if and only if it is Ricci symmetric. The equation of
state (EoS) was shown to have $\omega=-1$. Several energy conditions were also
analyzed and validated by current observational dataset. Very recently, the
impact of $(GR)_{4}$ structure is investigated in modified Gauss-Bonnet,
$f(R,G)$ theory of gravity [2]. The obtained results were examined for two
particular $f(R,G)$-models and for them both, the weak, null and dominant
energy conditions were validated while the strong energy condition was
violated, which is a good agreement with the recent observational studies
revealing that the current universe is in accelerating phase. Despite having
such recognised physical significance, a complete classification of
conformally flat Lorentzian manifolds with generalized Ricci recurrent
structure is still due. This motivated us to do a careful scrutiny through all
the possible cases. We shall give a complete classification of this structure
in a general setting as below:

######  Theorem 1.1.

A conformally flat generalized Ricci recurrent pseudo-Riemannian manifold is
an Einstein manifold, in particular, it is either a de Sitter space or an
anti-de Sitter space.

We can then deduce a classification of conformally flat generalized Ricci
recurrent spacetime.

######  Corollary 1.1.

A conformally flat generalized Ricci recurrent spacetime is either a de Sitter
spacetime or an anti-de Sitter spacetime.

##  2\. The Proof

Let $M$ be an $n$-dimensional pseudo-Riemannian manifold, $n\geq 3$, with
index $q$ ($0\leq q\leq n$). Suppose $M$ is a conformally flat $(GR)_{n}$,
that is, the Ricci tensor satisfies (1) and the Riemannian curvature tensor
satisfies [6, pp.302]

$\displaystyle
R^{l}{}_{ijk}=\frac{R_{ki}\delta^{l}_{j}-R_{ji}\delta^{l}_{k}+g_{ki}R^{l}_{j}-g_{ji}R^{l}_{k}}{n-2}-\frac{g_{ki}\delta^{l}_{j}-g_{ji}\delta^{l}_{k}}{n-1}\frac{R}{n-2}.$
(2)

The divergence of the above equation gives

$\displaystyle\nabla_{j}R_{ik}-\nabla_{k}R_{ij}=\frac{1}{2}\frac{\nabla_{j}Rg_{ki}-\nabla_{k}Rg_{ji}}{n-1}.$
(3)

Contracting $j$ and $l$ in (1) we obtain

$\nabla_{i}R=RA_{i}+nB_{i}.$ (4)

Contracting $i$ and $l$ in (1) we obtain

$\frac{1}{2}\nabla_{j}R=A^{l}R_{lj}+B_{j}.$ (5)

It follows from (4)–(5) that

$2A^{l}R_{lj}=RA_{j}+(n-2)B_{j}.$ (6)

By using (1), (3)–(4) we calculate

$2(n-1)\\{A_{i}R_{kl}-A_{k}R_{il}\\}=RA_{i}g_{kl}-RA_{k}g_{il}-(n-2)B_{i}g_{kl}+(n-2)B_{k}g_{il}.$
(7)

Next we split into two cases: $A^{i}$ is not a lightlike vector and $A^{i}$ is
a lightlike vector.

(I) $A^{i}$ is either a timelike or spacelike vector. Write

$A_{i}=\varepsilon\alpha U_{i};\quad U_{l}U^{l}=\varepsilon=\pm
1;\quad\alpha\neq 0.$

Then (6)–(7) become

$\displaystyle 2\varepsilon\alpha U^{l}R_{ij}=$
$\displaystyle\varepsilon\alpha RU_{j}+(n-2)B_{j},$ (8) $\displaystyle
2(n-1)\varepsilon\alpha\\{U_{i}R_{kl}-U_{k}R_{il}\\}=$
$\displaystyle\varepsilon\alpha\\{RU_{i}g_{kl}-RU_{k}g_{il}\\}-(n-2)B_{i}g_{kl}+(n-2)B_{k}g_{il}.$
(9)

Transvecting (9) with $U^{i}$ we obtain

$\displaystyle 2(n-1)\alpha R_{kl}=\\{\alpha
R-(n-2)\beta\\}g_{kl}+(n-2)\\{\varepsilon\alpha
RU_{k}U_{l}+U_{l}B_{k}+(n-1)U_{k}B_{l}\\},$ (10)

where $\beta=B_{l}U^{l}$. The skew-symmetry part gives

$U_{l}B_{k}+(n-1)U_{k}B_{l}=U_{k}B_{l}+(n-1)U_{l}B_{k},$

which implies that

$B_{i}={\varepsilon}\beta U_{i},\quad(\beta=U^{l}B_{l}).$ (11)

Armed with this significant result, using (4), (8) and (10) we can conclude
that the Ricci curvature tensor and the Ricci scalar satisfy

$\displaystyle 2(n-1)R_{kl}=$
$\displaystyle\\{R-(n-2)\lambda\\}g_{kl}+(n-2)\\{R+n\lambda\\}\varepsilon
U_{k}U_{l},$ (12) $\displaystyle\nabla_{i}R=$
$\displaystyle\\{R+n\lambda\\}\alpha\varepsilon U_{i},$ (13) $\displaystyle
2U^{l}R_{lj}=$ $\displaystyle\\{R+(n-2)\lambda\\}U_{j},$ (14)

where

$\lambda=\frac{\beta}{\alpha}.$

Covariantly differentiating (12) with respect to $i$ we obtain

$\displaystyle 2(n-1)\nabla_{i}R_{kl}=$
$\displaystyle\\{\nabla_{i}R-(n-2)\nabla_{i}\lambda\\}g_{kl}+(n-2)(\nabla_{i}R+n\nabla_{i}\lambda)\varepsilon
U_{k}U_{l}$
$\displaystyle+(n-2)(R+n\lambda){\varepsilon}\nabla_{i}(U_{k}U_{l})$
$\displaystyle=$ $\displaystyle(n-2)\\{-\nabla_{i}\lambda
g_{kl}+n\nabla_{i}\lambda\varepsilon
U_{k}U_{l}+(R+n\lambda){\varepsilon}\nabla_{i}(U_{k}U_{l})\\}$
$\displaystyle+\nabla_{i}R\\{g_{kl}+(n-2)\varepsilon U_{k}U_{l}\\}.$

On the other hand, (1), (11)–(12) give

$\displaystyle 2(n-1)\nabla_{i}R_{kl}=$ $\displaystyle 2(n-1)\alpha\varepsilon
U_{i}R_{kl}+2(n-1)\lambda\alpha\varepsilon U_{i}g_{kl}$ $\displaystyle=$
$\displaystyle\\{R-(n-2)\lambda\\}\alpha\varepsilon
U_{i}g_{kl}+(n-2)\\{R+n\lambda\\}\alpha\varepsilon U_{i}\varepsilon
U_{k}U_{l}$ $\displaystyle+2(n-1)\lambda\alpha\varepsilon U_{i}g_{kl}$
$\displaystyle=$ $\displaystyle(R+n\lambda)\alpha\varepsilon
U_{i}\\{g_{kl}+(n-2)\varepsilon U_{k}U_{l}\\}$ $\displaystyle=$
$\displaystyle\nabla_{i}R\\{g_{kl}+(n-2)\varepsilon U_{k}U_{l}\\}.$

These two equations give

$\displaystyle-\nabla_{i}\lambda g_{kl}+n\nabla_{i}\lambda\varepsilon
U_{k}U_{l}+(R+n\lambda){\varepsilon}\nabla_{i}(U_{k}U_{l})=0.$

Transvecting with $U^{k}$ and $U^{l}$ we obtain

$\displaystyle\nabla_{i}\lambda=0.$ (15)

It follows from these two equations that

$\displaystyle(R+n\lambda)\nabla_{i}(U_{k}U_{l})=0.$

We consider two subcases: $R+n\lambda\neq 0$ and $R+n\lambda=0$.

a:

If $R+n\lambda\neq 0$, we have $\nabla_{i}U_{l}=0$ and so $U_{l}R_{i}^{l}=0$.
It follows from (14) that

$R+(n-2)\lambda=0.$

It follows from this equation and (15) that

$\nabla_{i}R=-(n-2)\nabla_{i}\lambda=0.$

After applying this to (13) gives $R+n\lambda=0$. Hence we have $R=\lambda=0$
and so $R_{lk}=0$; violating the hypothesis of being $(GR)_{n}$. Hence this
case is impossible.

b:

Suppose that

$R+n\lambda=0.$

Then, using (12) we obtain

$R_{kl}=-\lambda g_{kl},$

meaning that it is an Einstein space.

(II) $A^{i}$ is a lightlike vector. Take another lightlike vector $K^{i}$ such
that

$A_{l}A^{l}=K_{l}K^{l}=0;\quad A_{l}K^{l}=-1.$

Transvecting (7) with $A^{i}$, with the help of (6), we obtain

$-A^{j}B_{j}g_{kl}+RA_{k}A_{l}+A_{l}B_{k}+(n-1)A_{k}B_{l}=0.$ (16)

The skew-symmetry part gives

$A_{l}B_{k}=A_{k}B_{l},$

which implies that $A^{k}B_{k}=0$ and

$B_{i}=-\lambda A_{i},\quad(\lambda=K^{l}B_{l}).$ (17)

Substituting these into (4) and (16) respectively give

$\displaystyle\nabla_{i}R=R-n\lambda=0,$ (18)

and so

$\nabla_{i}\lambda=0.$ (19)

By using (17)–(18), we can simply (7) as

$A_{i}R_{kl}-A_{k}R_{il}=\lambda A_{i}g_{kl}-\lambda A_{k}g_{il}.$ (20)

Transvecting with $K^{i}$ and $K^{l}$ gives

$K^{l}R_{lk}=-\tau A_{k}+\lambda K_{k};\quad(\tau=K^{i}K^{j}R_{ij}).$

Transvecting (20) with $K^{i}$ and applying the above equation gives

$R_{lk}=\lambda g_{lk}+\tau A_{l}A_{k}.$ (21)

Suppose

$\tau\neq 0.$ (22)

Applying (18) and (21), we can simplfy (2) as

$\displaystyle
R^{l}{}_{ijk}=\frac{\lambda}{n-1}\\{g_{ki}\delta^{l}_{j}-g_{ji}\delta^{l}_{k}\\}+\frac{\tau}{n-2}\\{A_{k}A_{i}\delta^{l}_{j}-A_{j}A_{i}\delta^{l}_{k}+g_{ki}A^{l}A_{j}-g_{ji}A^{l}A_{k}\\}.$
(23)

On the other hand, by substituting (17) and (21) into (1) give

$\nabla_{i}R_{jk}=\tau A_{i}A_{j}A_{k}.$ (24)

It follows from (2), (18)–(19) and (21) that

$\nabla_{i}R_{hj}=\nabla_{h}R_{ij}=\nabla_{h}(\tau A_{i}A_{j}).$

Covariantly differentiating (24) with respect to $h$ gives

$\nabla_{h}\nabla_{i}R_{jk}=\nabla_{h}(\tau A_{i}A_{j})A_{k}+\tau
A_{i}A_{j}\nabla_{h}A_{k}=\nabla_{h}R_{ij}A_{k}+\tau
A_{i}A_{j}\nabla_{h}A_{k},$

and so

$\displaystyle\tau A_{j}A_{i}\nabla_{h}A_{k}-\tau A_{j}A_{h}\nabla_{i}A_{k}=$
$\displaystyle(\nabla_{h}\nabla_{i}-\nabla_{i}\nabla_{h})R_{jk}=-R^{l}{}_{jhi}R_{lk}-R^{l}{}_{khi}R_{lj}$

Next, by applying (21), the properties $R_{kjhi}+R_{jkhi}=0$ and (22) we have

$\displaystyle A_{j}A_{i}\nabla_{h}A_{k}-A_{j}A_{h}\nabla_{i}A_{k}=$
$\displaystyle-R^{l}{}_{jhi}A_{l}A_{k}+R^{l}{}_{khi}A_{l}A_{j}.$

Furthermore, by using (23) we obtain

$\displaystyle
A_{j}A_{i}\Big{\\{}\nabla_{h}A_{k}-\frac{\lambda}{n-1}g_{hk}\Big{\\}}-A_{j}A_{h}\Big{\\{}\nabla_{i}A_{k}-$
$\displaystyle\frac{\lambda}{n-1}g_{ik}\Big{\\}}$ $\displaystyle=$
$\displaystyle\frac{\lambda}{n-1}\\{-g_{ij}A_{h}+g_{hj}A_{i}\\}A_{k}.$ (25)

Transvecting with $K^{i}$ and $K^{j}$ we obtain

$\displaystyle\nabla_{h}A_{k}-\frac{\lambda}{n-1}g_{hk}=-A_{h}\Omega_{k}-\frac{\lambda}{n-1}K_{h}A_{k};\quad\left(\Omega_{k}=K^{i}\nabla_{i}A_{k}-\frac{\lambda}{n-1}K_{k}\right).$

Substituting this into (2) gives

$\frac{\lambda}{n-1}\\{-A_{j}A_{i}K_{h}+A_{j}A_{h}K_{i}\\}A_{k}=\frac{\lambda}{n-1}\\{-g_{ij}A_{h}+g_{hj}A_{i}\\}A_{k}.$

Since $\lambda\neq 0$, it is descended to

$-A_{j}A_{i}K_{h}+A_{j}A_{h}K_{i}=-g_{ij}A_{h}+g_{hj}A_{i}.$

Transvecting with $K^{h}$ and $g^{ij}$ we obtain $1=n-1$. This is impossible.
Hence (22) is false or we must have $\tau=0$. It follows from (21) that it is
also an Einstein space in this case.

Hence we conclude that $M$ is an Einstein space in both cases. As a conformaly
flat Einstein space is of constant sectional curvature and the manifold is
non-flat, We conclude that a conformally flat generalized Ricci recurrent
spacetime is either a de Sitter space or an anti-de Sitter space.

## References

  * [1] A. De, T. H. Loo, R. Solanki and P. K. Sahoo, A conformally flat generalized Ricci recurrent spacetime in $f(R)$ gravity, Phys. Scripta, 96, 085001 (2021).
  * [2] A. De, T. H. Loo, R. Solanki, P.K. Sahoo, How a conformally flat (GR)4 impacts Gauss-Bonnet gravity? arXiv:2107.02889.
  * [3] U. C. De, N. Guha and D. Kamilya, On generalized Ricci-recurrent manifolds, Tensor (N.S.), 56, 312–317 (1995).
  * [4] C. Dey, On Generalized Ricci Recurrent Spacetimes, Kyungpook Math. J., 60, 571–584 (2020).
  * [5] S. Mallick, A. De and U. C. De, On Generalized Ricci Recurrent Manifolds with Applications To Relativity, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 83, 143–152 (2013).
  * [6] B. O’Neill, The Geometry of Kerr Black holes, A K Peters., Wellesly, Massasuchetts, (1995).
  * [7] E. M. Patterson, Some theorems on Ricci-recurrent spaces, J. London Math. Soc., 27, 287–295 (1952).


# AI Use in Manuscript Preparation
for Academic Journals††thanks: We thank many members of the UCSB Economics
Department, the Decision Theory Forum, and the Economic Science Association,
especially Monica Capra, for completing our survey and for very helpful
suggestions. We also thank Ambreen Chaudhri at Kellogg Research Support for
her help in building our manuscript database, and to Ruixi (Raye) Zhu, Marisa
Weng, and Bonny Wang for research assistance.

Nir Chemaya and Daniel Martin University of California, Santa Barbara,
<EMAIL_ADDRESS>of California, Santa Barbara and Kellogg School of
Management, Northwestern University<EMAIL_ADDRESS>
(November 18, 2023)

###### Abstract

The emergent abilities of Large Language Models (LLMs), which power tools like
ChatGPT and Bard, have produced both excitement and worry about how AI will
impact academic writing. In response to rising concerns about AI use, authors
of academic publications may decide to voluntarily disclose any AI tools they
use to revise their manuscripts, and journals and conferences could begin
mandating disclosure and/or turn to using detection services, as many teachers
have done with student writing in class settings. Given these looming
possibilities, we investigate whether academics view it as necessary to report
AI use in manuscript preparation and how detectors react to the use of AI in
academic writing.

##  1 Introduction

There is both excitement and concern about the impact that artificial
intelligence (AI) could have on our world. In academia, this has inspired an
explosion of research around AI. Some researchers are leveraging AI to help in
answering research questions; for example, as a tool for performing
statistical or textual analyses,111For example, [32], [4], [21], [18], [35],
[6], [10], [20], [38]. or for the design or implementation of
experiments.222For example, [5], [11], [25]. Others have focused more on the
interaction of humans with AI systems333For example, [16], [15], [23], [42],
[43], [44], [47], [48]. and the larger societal ramifications of AI.444For
example, [1], [30], [33], [49], [12], [37], [50], [34], [36], [41].

In terms of societal ramifications, one topic of interest is how AI will
impact academia itself, especially given the emergent abilities of Large
Language Models (LLM), which power tools like ChatGPT and Bard. Most of this
research has centered on student use of AI to complete course or degree
requirements.555 For example, [13], [14], [22], [26], [28], [31], [39], [40].
However, there has been less focus on the impact that AI is having and might
have on manuscript preparation for academic journals, even though the impact
this could have on science, and the propagation of scientific results, is
potentially large.

An exception is [29], who documents several use cases for LLMs for
researchers, including academic writing.666[46] consider a related issue:
using AI to perform literature reviews. He notes, “LLMs can edit text for
grammatical or spelling mistakes, style, clarity, or simplicity.” As a result,
this class of tools “allows researchers to concentrate their energy on the
ideas in their text as opposed to the writing process itself.” In particular,
“this set of capabilities are perhaps most useful for non-native speakers who
want to improve their writing.” The potential impact of these tools is
validated by the emergence of companies that help academics leverage LLM in
their writing.777For example: https://speakai.co/ChatGPT-for-academic-papers/.
Some of those initiatives are even run by professors from academia, such as
online workshops that teach researchers how to use ChatGPT for academic
publishing.888Some workshops are privately-run
(https://instats.org/structuredcourse/chatgpt-for-academic-paper-writing2515),
others by institutions (https://ctl.columbia.edu/announcements/faculty-weekly-
april18-2023), and others are run by assocations (https://www.acs.org/acs-
webinars/library/ai-in-education.html).

However, an emerging set of papers (e.g., [3], [45], [40] and [24]) detail
some practical and ethical concerns with the use of these tools in preparing
manuscripts for academic journals. For example, AI can generate a text with
mistakes, including incorrect math, reasoning, logic, factual information, and
citations (even producing references to scientific papers that do not exist).
There are many well-documented examples where LLM’s “hallucinate” and provide
completely fictitious information. On top of this, these tools may also
produce text that is biased against particular groups. These issues are
exacerbated by the “black box” nature of LLM suggestions, meaning that we lack
an understanding of how they work (e.g., [8]). Given these issues, liabilities
arise when authors submit papers without fully vetting the text generated by
LLMs. In addition, given that they are trained on a corpus of other writing
and not the author’s own writing, using the output of tools like ChatGPT and
Bard without proper attribution could be considered plagiarism.

Because of these concerns, authors may voluntarily choose to disclose whether
they have used LLMs in preparing their manuscripts, as suggested by [7].
However, given well-documented failures of disclosure in the field and lab
(e.g., [17], [27]), journals, conferences, and associations may respond by
mandating disclosure through reporting requirements, as is already done for
conflict-of-interest issues. For example, Elsevier limits the use of AI by
authors “only to improve the language and readability of their paper” and
requires “the appropriate disclosure… at the bottom of the paper in a separate
section before the list of references.”999See Table 6 in Appendix B for a list
of major publishers who have voluntary and mandatory disclosure policies for
AI use. Further, to enforce these reporting requirements, journals,
conferences, and associations might turn to using detection services, as many
teachers have done with student writing in class settings. The use of LLMs in
writing is potentially detectable because as [29] notes, “Some observe that
[LLM writing] is, naturally, a bit sterile and lacks the idiosyncrasies and
elements of surprise that characterize human writing – a feature that
detectors of LLM-written text zero in on.”

Given these looming possibilities, we investigate whether academics view
reporting AI use in manuscript preparation as necessary and how detectors
react to the use of AI in manuscript preparation. We first conducted a survey
of academics in which we elicited perceptions about reporting assistance in
manuscript preparation.101010This sample was primarily composed of academic
economists specializing in lab and field experiments. We then used GPT-3.5 to
revise abstracts from the past 10 years of Management Science by asking the
software program to either fix grammar or rewrite text. Finally, we ran the
original abstracts and revised abstracts through a leading paid AI detection
service.

We have three main findings. First, as shown partially in Figure 1, the
academics we surveyed were less likely to think that using AI to fix the
grammar in manuscripts should be reported than using AI to rewrite
manuscripts, but detection software did not always draw this distinction, as
abstracts for which GPT-3.5 was used to fix grammar were often flagged as
having a high chance of being written by AI. Second, we found little
difference in preferences for reporting ChatGPT and research assistant (RA)
help, but significant differences in reporting preferences between these
sources of assistance and paid proofreading and other AI assistant tools
(Grammarly and Word).111111Recently, the new version of Word offers an auto-
completion feature for sentences. Grammarly uses AI models as described in
[19]. Third, we found disagreements among the academics we surveyed on whether
using ChatGPT to rewrite text needs should be reported, and differences were
related to perceptions of ethics, academic role, and English language
background.

(a) Fraction of survey respondents indicating that ChatGPT use in fixing
grammar or rewriting text should be reported, with 95% confidence intervals.

(b) 75th percentile of AI score for original abstracts and the versions that
GPT-3.5 was used to either fix grammar or rewrite text.

Figure 1: Reporting views (a) vs. detection results (b).

The rest of the paper is structured as follows. Section 2 introduces our
survey design and our method for testing ChatGPT use with AI detection
algorithms. Section 3 shows our main results. Finally, section 4 discusses
some limitations of our work, open questions arising from our results, and
suggestions for future research.

##  2 Design

In order to answer our main research questions, we split our design into two
parts. The first part was a survey of the academic community that captured
academics’ views on reporting AI use in manuscript preparation. The second
part was to test an AI detector’s reaction to AI use in manuscript
preparation. Our primary focus was on using AI to fix grammar and rewrite
text, which we consider to be important intermediate cases between not using
AI at all and using AI to do all of the writing based on limited inputs (e.g.,
only a title and/or a set of results). We asked academics for their
perceptions about these two kinds of AI usage in manuscript preparation and
investigated how the detector responded to these different kinds of usage. All
of this will be explained in more detail in the following subsections.

###  2.1 Survey Design

In August of 2023, we conducted a brief unpaid survey about perceptions of AI
use in manuscript preparation. We sent this survey to a convenience sample of
academics; specifically, we sent it to three listservs: the UCSB Economic
Department listserv,121212This connects graduate students, professors, and
postdocs at the University of California, Santa Barbara. the Economics Science
Association (ESA) announcement listserv,131313ESA is the leading organization
for experimental economics. and the Decision Theory (DT) Forum
listserv.141414The DT Forum is a listserv of academics working on decision
theory that is run by Itzhak Gilboa. We sent the survey with a few days delay
between each group, which allows for a rough measure of the respondents from
each source. We had a total of 271 respondents complete the survey: 38 from
UCSB, 199 from UCSB and the ESA community, and 34 from UCSB, the ESA
community, and the DT Forum.151515This number does not include 20 individuals
who did not specify an academic role. The median time to answer the survey was
1.21 minutes.

In this survey, we focused on two main aspects related to perceptions of AI
use in manuscript preparation. The first aspect is whether authors should
acknowledge using ChatGPT to fix grammar or to rewrite text.161616Our use of
the term “rewrite” is intentionally vague in order to span a wide range of
possibilities. However, this might lead to differences between respondents if
they can imagine different uses of ChatGPT. We asked this over two separate
questions, as shown in Figure 10 in Appendix A. These two uses of ChatGPT
reflect to many of the use cases of LLMs suggested in articles, websites, and
online courses. The second aspect is whether using ChatGPT to modify an
academic manuscript is unethical. Again, we split this into two questions –
one for grammar and one for rewrite – to allow researchers to address whether
ethics also depends on usage. Because we suspected that there might be
differences in these perceptions and an academic’s role and language
background, we asked respondents if they were a native speaker of English and
their current role (postdoc, student, untenured professor, tenured professor,
and/or other). Of the 271 respondents who completed our survey, 83 reported
being a native English speaker, and we categorized 67 respondents as students,
32 as postdocs, 59 as untenured professors, and 113 as tenured
professors.171717If someone reported multiple roles, we took the “highest”
report, ranked in this order.

In addition to these questions, we added follow-up questions on a second page
of the survey about whether authors should acknowledge other writing services
and tools such as Word, Grammarly, proofreading, and the work of research
assistants (see Figure 11 in Appendix A). We asked this question only after
the ChatGPT questions and on a new page because our main focus was the
perception of AI (e.g., ChatGPT) in academic writing, and because we wanted to
avoid these follow-up questions influencing the answers given about ChatGPT.
In addition, we did not randomize the order between ChatGPT and other services
because the convenience sample already knew this was a survey about ChatGPT,
so that would have been in their mind already when completing the survey.
However, by asking questions about ChatGPT first, we might have reduced
differences in reported perceptions with other writing services, as people
might feel the need to report consistently for a given use.

To make the questions about other sources comparable the questions about
ChatGPT, we used the same question format and split the comparison into two
groups: fixing grammar and rewriting text. The tools we included for fixing
grammar included Word, Grammarly, and RA help. Importantly, we wrote
explicitly in the question that these tools were to be used for fixing
grammar. For rewriting text, we used proofreading and RA help. Once more, we
explicitly asked whether authors should acknowledge using these tools and
services for rewriting text for an academic journal.

###  2.2 Detection Design

Next, we tested how an AI detector would react to using ChatGPT to fix grammar
in academic text and to rewrite academic text. We first collected titles and
abstracts from 2,716 papers published in the journal Management Science from
January 2013 to September 2023.181818We excluded articles with the following
words in their titles, as they did not appear to be original articles:
“Erratum,” “Comment on,” “Management Science,” “Reviewers and Guest Associate
Editors,” and “Reviewers and Guest Editors.” We intentionally included papers
that were published before the launch of ChatGPT in November 30, 2022 in order
to have source text that was plausibly unimpacted by LLM use.191919Date
source: https://en.wikipedia.org/wiki/ChatGPT.

We revised these abstracts using a variety of prompts, as seen in Table 1
below. Rather than selecting the prompts ourselves, we wanted to use an
external source. We decided to use prompts from the online GitHub page ChatGPT
Prompts for Academic Writing, which advises researchers on how to use
ChatGPT.202020See: https://github.com/ahmetbersoz/ChatGPT-prompts-for-
academic-writing. We chose this source because it was the first link returned
from a Google search of “ChatGPT Prompts for Academic Writing.”

|  Prompt
---|---
Grammar 1 |  Correct the grammar. Give a version in one paragraph based on this paragraph: “[PARAGRAPH]”
Rewrite 1 |  Rewrite this paragraph in an academic language. Give a version in one paragraph based on this paragraph: “[PARAGRAPH]”
Grammar 2 |  Act as a language expert, proofread my paper on “[TITLE]” while putting a focus on grammar and punctuation. Give a version in one paragraph based on this paragraph: “[PARAGRAPH]”
Rewrite 2 |  Improve the clarity and coherence of my writing. Give a version in one paragraph based on this paragraph: “[PARAGRAPH]”
Grammar 1b |  Correct the grammar: “[PARAGRAPH]”
Rewrite 1b |  Rewrite this paragraph in an academic language: “[PARAGRAPH]”
Table 1: Prompts used to revise abstracts with GPT-3.5.

Our baselines prompts were “Grammar 1” and “Rewrite 1,” and for robustness we
also considered two functionally related prompts, “Grammar 2” and “Rewrite 2.”
We added a restriction at the end of these prompts to “Give a version in one
paragraph based on this paragraph” in order to meet the requirement that the
abstract be only one paragraph in length. To consider the robustness of our
results to this additional language, we also ran versions of our baseline
prompts that did not include it: “Grammar 1b” and “Rewrite 1b.”

It has been shown that there are ways to fool certain detectors, such as by
adding strange characters to the text, and this is a constantly evolving game
of cat-and-mouse as detectors evolve. Although this could be of interest for
researchers, we do not study this particular phenomenon. Instead, it is our
goal to see whether an academic who uses ChatGPT for widely-proposed purposes
– without further edits or detection avoidance strategies – would have their
writing be flagged as AI generated.

To revise these abstracts at scale, we leveraged the GPT API with settings
that would produce output that closely mirrors the output of a researcher
using ChatGPT to revise academic writing. The specific model we used was
gpt-3.5-turbo-0613 (GPT-3.5 Turbo released in June 13 2023), and we kept the
default settings for temperature, top p, frequency penalty, and presence
penalty.212121Source: https://github.com/langchain-
ai/langchain/blob/master/libs/langchain/langchain/llms/openai.py. We also used
the default system prompt, “You are a helpful assistant,” which is a
background prompt that can be changed in the API but not in ChatGPT
itself.222222For more details, see https://zapier.com/blog/openai-playground/.
We did a fresh call each time we used a prompt to avoid learning and history
effects.

Finally, we used a leading paid service (Originality.ai) to see how AI
detection algorithms might react to this use of LLMs.232323There are other
popular and free services, including GPTzero (https://gptzero.me/) We first
evaluated the original Management Science abstracts, and then the abstracts
revised by GPT-3.5 based on all of the prompts.242424[48] also provides a
comparison of human and AI editors, but in the context of human delegation
decisions. Originality.ai provides an “AI score,” which is a value between 0%
and 100% that is interpreted as the likelihood that AI wrote the text being
evaluated. A high score means a high likelihood that AI generated the text.
Specifically, the company states: “If an article has an AI score of 5%… there
is a 95% chance that the article was human-generated (NOT that 5% of the
article is AI generated).”252525Source: https://originality.ai/blog/ai-
content-detection-score-google. [2] studied a number of popular detection
tools and found that Originality.ai had the highest accuracy rate (97%).

##  3 Results

###  3.1 Reporting Views and Detection Evaluations

Figure 1 presents the fraction of survey respondents who indicated that using
ChatGPT to fix grammar or rewrite text should be acknowledged. We find
substantial differences in reporting views between using ChatGPT for fixing
grammar and using it to rewrite text, with 22% of the respondents indicating
that grammar correction should be reported relative to 52% for text rewriting
(a two-sided test of proportions gives p<0.0001). These perceptions were
cleanly nested, as 95% of respondents who thought grammar should be reported
also thought rewriting should be reported (only 3 respondents thought that
fixing grammar should be reported but not rewriting.)

On aggregate, survey respondents viewed these types of AI use differently, but
did the AI detector we study treat them differently? Figure 2 shows that the
distribution of AI scores is skewed more to the right for abstracts revised
using the Rewrite 1 prompt than for those revised using the Grammar 1 prompt.
However, both Grammar 1 and Rewrite 1 produce a large number of high AI
scores, and if we look again in Figure 1, the 75th percentile values of the AI
scores for the abstracts produced by these two prompts are both near the
maximal value. It is worth noting that while the detector gave abstracts
revised using both prompts high scores, gave much lower scores to the original
abstracts. Thus, it had a high degree of accuracy in separating manuscripts
that were not revised by ChatGPT from those that were revised in some way.

(a) Original abstracts.

(b) Grammar 1 abstracts.

(c) Rewrite 1 abstracts.

Figure 2: The distribution of AI scores for the original abstracts (a),
abstracts revised using the Grammar 1 prompt (b), and abstracts revised using
the Rewrite 1 prompt (c).

However, this analysis does not indicate how Grammar 1 and Rewrite 1 compare
for a given abstract. It could be that abstracts always had higher AI scores
when the Rewrite 1 prompt was used. Looking at the abstract level, 2.2% of
abstracts had the same AI score for both types of prompts and 24.2% of
abstracts had an higher AI score when the Grammar 1 prompt was used than when
the Rewrite 1 prompt was used.

These results combine to produce our first main finding:

###### Main Finding 3.1

The academics we surveyed were less likely to think that using AI to fix the
grammar in manuscripts should be reported than using AI to rewrite
manuscripts, but detection software did not always draw this distinction, as
abstracts for which GPT-3.5 was used to fix grammar were often flagged as
having a high chance of being written by AI.

###  3.2 Heterogeneity of Perceptions

Figure 3: Fraction of survey respondents indicating that ChatGPT, RA,
Grammarly, and Word use in fixing grammar or ChatGPT, RA, or Proofreading use
in rewriting text should be reported, with 95% confidence intervals.

When we compare our survey results for different sources of assistance, we
find very similar reporting preferences between ChatGPT and the help of a
research assistant (RA) for grammar correction or for rewriting text, which is
illustrated in Figure 2.262626More detailed summary statistics are provided in
Appendix B. Comparing the difference between RA and ChatGPT for fixing grammar
and rewriting text, we find no significant difference (p=0.5418 for fixing
grammar and p=0.6062 for rewriting for two-sided tests of proportions).

However, respondents indicated that other tools used for fixing grammar, such
as Grammarly and Word, should be acknowledged at even lower rates than
ChatGPT, even though these tools might provide users with similar grammar
corrections as ChatGPT. Only 14% (5%) of those completing the survey responded
that researchers should report using Grammarly (Word) to fix grammar for
academic text. A two-sided test of proportions between reporting ChatGPT and
Grammarly (Word) for fixing grammar gives p=0.0100 (p<0.0001). This might
indicate that researchers are unfamiliar with the differences between these
tools, especially given that Grammarly uses some AI models ([19]).
Alternatively, respondents may be so familiar with those tools that they are
less worried about their influence on academic writing compared to ChatGPT,
which is relatively new.

Finally, only 27% of those completing the survey responded that proofreading
services should be acknowledged for rewriting academic text, which is much
lower than for ChatGPT (52%) and for RA help (49.6%). A two-sided test of
proportions comparing ChatGPT to proofreading for rewriting text gives
p<0.0001. One explanation for this result could be that proofreading might be
considered to be closer to fixing grammar than rewriting text (even that we
explicitly stated that proofreading would be used for rewriting text). Another
potential explanation could stem from an academic norm related to
acknowledging this service or the fact that this service is paid.

This leads to our second main finding:

###### Main Finding 3.2

We found little difference in preferences for reporting ChatGPT and RA help,
but significant differences in reporting preferences between these sources of
assistance and paid proofreading and other AI assistant tools (Grammarly and
Word).

Next, we investigate the potential reasons why some academics we surveyed
thought that using ChatGPT to rewrite text should be reported and why others
did not. Figure 4 shows how reporting preferences differ by English language
background (native speaker or not), academic role (professor or not), and
perceptions of ethics (whether using ChatGPT to rewrite text is unethical or
not).272727To increase statistical power, we collapse role into professor or
not and pool together those who answered either “yes” or “maybe” to the survey
question on the ethics of using ChatGPT to rewrite text. When we compare
native English speakers to non-native ones, we find that on average native
speakers are more inclined towards reporting the use of ChatGPT for rewriting
text. When we compare professors (both tenured and untenured) to students and
postdocs we find that, on average, postdocs and students are more likely to
believe that authors should acknowledge the use of ChatGPT when rewriting
text. One possible explanation is that early career researchers are more
conservative because they are unfamiliar with the norms in the profession and
prefer to take the safer option and report. On the other hand, they may have a
better sense of the power of these tools, and hence might feel that reporting
is more necessary. Finally, we find large differences in reporting preferences
based on perceptions of ethics. Survey respondents who believe that it is
unethical to use ChatGPT to rewrite text are almost three times more likely to
believe that it should be reported.282828A more detailed analysis of
perceptions of ethics is provided in Appendix B.

Figure 4: Fraction of survey respondents indicating that ChatGPT use in fixing
grammar or rewriting text should be reported, with 95% confidence intervals,
by English background, academic role, and perceptions of ethics.

When we run an OLS regression analysis with only English language background
(dummy variable Native=1) and academic role (dummy variable Professor=1) as
the explanatory variables, as presented in first column of Table 2, we find
that the coefficients are significant at the 5% and 10% level for Native and
Professor respectively. However, when we add perceptions of ethics to the
regression, the effect for Native becomes weaker and less significant. This
suggests that some of this effect was due to differences in perceptions of
ethics between native and non-native speakers of English.

| (1) | (2)
---|---|---
| Report GPT Rewrite | Report GPT Rewrite
Native | 0.142∗∗ | 0.0851
| (0.065) | (0.055)
Professor | -0.111∗ | -0.102∗
| (0.063) | (0.052)
Unethical |  | 0.557∗∗∗
|  | (0.052)
Constant | 0.547∗∗∗ | 0.343∗∗∗
| (0.053) | (0.048)
$N$ | 271 | 271
Note: Standard errors in parentheses. ∗ $p<0.10$, ∗∗ $p<0.05$, ∗∗∗ $p<0.01$.
Table 2: Regressions of whether using GPT to rewrite should be reported (dummy
variable) onto being a native English speaker or not (dummy variable), being a
professor or not (dummy variable), and in the second specification, whether
using ChatGPT to rewrite text is unethical or not (dummy variable).

###### Main Finding 3.3

We found disagreements among the academics we surveyed about whether using
Chat-GPT to rewrite text needs to be reported, and differences were related to
perceptions of ethics, academic role, and English language background.

###  3.3 Detection Robustness

First, we compare the 75th percentile of AI scores across all of the prompts
provided in Table 1, and Figure 5 shows that we find no perceptible
differences for Grammar 2 and Rewrite 2. However, we find that dropping the
requirement that the GPT-3.5 output be only one paragraph dramatically reduces
the 75th percentile value when revising grammar (Grammar 1b). However, the
result is more robust if we consider the 90th percentile instead, as shown in
Figure 6.

Figure 5: 75th percentile of AI score for original abstracts and the versions
that were revised by GPT-3.5 for all of the prompts. Figure 6: 90th percentile
of AI score for original abstracts and the versions that were revised by
GPT-3.5 for all of the prompts.

Second, we checked if the 75th percentile values of the AI scores were
different in the years before and after the launch of ChatGPT. Looking at the
75th percentile of AI scores in Figure 7, we find a very slight decrease in
the 75th percentile AI scores for original abstracts that appeared in 2023.
Given this small change for original abstracts, it might not be surprisingly
that we do not see much of a difference for Grammar 1 or Rewrite 1.

Figure 7: 75th percentile of AI score for original abstracts and the versions
that GPT-3.5 was used to either fix grammar or rewrite text for abstracts
appearing in print before and after 2023.

Third, we ran 1,016 of our abstracts through ChatGPT twice to test for the
variability in AI scores due to any stochasticity in ChatGPT. Figure 8 shows
that there is very little difference in the 75th percentile of scores.

Figure 8: 75th percentile of AI score for original abstracts and the versions
that GPT-3.5 was used to either fix grammar or rewrite text for abstracts in
our main run and in our secondary run.

##  4 Discussion

Our findings are a starting point for future research and suggest that several
issues need to be carefully considered by the academic community. For
instance, the community needs to decide which forms of assistance should be
reported, whether it be ChatGPT or some other source, such as RA help or help
from another AI tool. Additionally, it needs to decide which types of
assistance to be reported, be it fixing grammar or something more extensive.
We found more consensus that using ChatGPT to fix grammar does not need to be
reported than when it comes to rewriting text. One avenue could be to disclose
the actual prompts that are used to revise the paper.292929Grammarly has a new
feature that allows users to acknowledge the usage of AI and the actual
prompts that the users used. For more info: https://www.grammarly.com/blog/ai-
for-homework/.

In addition, our research raises the question of how journals, conferences,
and associations can enforce differences in what should be reported. While the
detection tool that we employed was able to detect relatively accurately
whether AI was used at all, abstracts that were rewritten by GPT-3.5 were
sometimes given a lower chance of being written by AI than the grammar-fixed
abstracts. This also opens the question of whether using ChatGPT to fix
grammar might inadvertently make more substantial changes than desired. How
can researchers be sure that they use AI in the desired way? One solution
could be to use ChatGPT to point out grammar errors, but to fix them manually,
so that ChatGPT does not actually revise the text.

These tools and our perceptions of them will surely evolve, but the aim of
this paper is to determine how they are perceived and detected in this moment
in time, as it appears to be an inflection point in AI ability and in its use
to revise text.

###  4.1 Limitations and Future Directions

In terms of our survey design, the use of a convenience sample may have
introduced selection issues that complicate our comparisons by role and
English language background. In addition, our convenience sample was largely
composed of economists, and because views might differ across fields, it would
be valuable to also consider what perceptions look like more generally and to
illuminate differences across fields. An alternative approach that allows for
an assessment of perceptions across fields is offered by [9], who runs a
sentiment analysis on papers written about AI use in manuscript preparation
and finds that the sentiment in those papers is generally positive. Also, the
use of a convenience sample did not lend itself to randomizing on the form of
assistance (ChatGPT, RA assistance, etc.).

In addition, another limitation of our survey is that we do not dig deeply
into the nature of ethical perceptions. Since these perceptions were such an
important predictor of reporting preferences, it might be insightful to know
why academics feel that using AI tools for manuscript preparation is
unethical. For example, is someone harmed by their use – such as other
academics, science in general, or the authors whose material is used in
training the AI – or is there a deeper moral question at play?303030We thank
Christina Botros for raising this issue. One way to tease apart these
subtleties would be to have academics evaluate a number of detailed
vignettes.313131We thank Monica Capra for this excellent suggestion. It might
also be interesting to determine the role of payment in ethical
considerations, as ChatGPT, RAs, and proofreaders sometimes require payment
and sometimes do not.

In terms of our detection design, we just considered published papers, at a
top journal, and for the field of management. To understand whether these
results hold more generally, it would be necessary to look at papers published
in other fields, perhaps using a service like Scopus, and to look at working
papers, perhaps sourced from SSRN or arXiv. In addition, it might be valuable
to consider full papers instead of just abstracts, or as a middle case between
abstracts and full papers, introductions might also be useful to examine.

Additionally, our results are limited to one AI detection service, so we do
not know if they extend to other services, such as GPTZero. Also, it might be
of interest to see if other AI-based revision services, such as Grammarly, are
flagged by AI detectors too. Along these lines, it might be insightful to
consider other forms of writing assistance that might be taken besides fixing
grammar and rewriting text, or specific forms of rewriting text. One important
dimension could be whether use of chat-based and completion-based AI tools
lead to different detection rates.

Finally, and related to the last point, it would be valuable to consider a
range of different prompts, especially given the sensitivity that we found to
relatively small changes in prompt language. One systematic way to choose the
prompts could be to hold focus groups of researchers or ChatGPT users or by
having a way for researchers to vote on the prompts that are tested. Another
dimension of investigation could be to see if our results are sensitive to
other important features of the GPT API inputs, such as increasing the model
to GPT-4, increasing the temperature to increase the hallucination rate, or by
using other system prompts. Given the black box nature of LLMs, a robust
empirical analysis is needed to inform policymakers at associations, journals,
and conferences of the link between how AI is used and how it is flagged by
detection software.

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Appendix for AI Use in Manuscript Preparation
for Academic Journals

##  Appendix A Survey Screenshots

Figure 9: Consent form Figure 10: Survey questions (page 1) Figure 11: Survey
questions (part 2)

##  Appendix B Additional Tables

Table 3 provides summary statistics for whether authors should acknowledge
different forms of assistance. The top panel provides the average responses
broken out by being a native English speaker or not, and the bottom panel by
being a professor or not.

| Grammar | Rewrite | Respondents
---|---|---|---
| ChatGPT | RA | Word | Grammarly | ChatGPT | RA | Proofreading |
| (mean) | (mean) | (mean) | (mean) | (mean) | (mean) | (mean) | (N)
Native |  |  |  |  |  |  |  |
Yes | 0.27 | 0.27 | 0.05 | 0.17 | 0.61 | 0.53 | 0.28 | 83
No | 0.2 | 0.23 | 0.05 | 0.12 | 0.48 | 0.48 | 0.27 | 188
Total | 0.22 | 0.24 | 0.05 | 0.14 | 0.52 | 0.5 | 0.27 | 271
Role |  |  |  |  |  |  |  |
Professor | 0.2 | 0.21 | 0.04 | 0.1 | 0.48 | 0.47 | 0.2 | 172
Student |  |  |  |  |  |  |  |
or Postdoc | 0.25 | 0.3 | 0.07 | 0.2 | 0.59 | 0.56 | 0.38 | 99
Total | 0.22 | 0.24 | 0.05 | 0.14 | 0.52 | 0.5 | 0.27 | 271
Table 3: The survey averages responses to whether authors should acknowledge
using ChatGPT, RA, Grammarly, and Word for grammar correction, as well as
ChatGPT, RA, and proofreading for text rewriting based on the respondent’s
role and whether the respondent is a native English speaker.

Table 4 shows the results of asking if it is unethical to use ChatGPT to
correct the grammar or rewrite the text in a manuscript for an academic
journal. For fixing grammar, there is a consensus that it is ethical to use
ChatGPT, with 94% of the researchers replying that it is not unethical.
However, for rewriting, this becomes more open questions with only 61%
respondents thinking that is not unethical.323232We allow respondents to
answer “maybe” to these questions to identify which researchers are uncertain
about this question. However, in our analysis, we categorize respondents into
those who believe it is ethical and those who either answered that it is
unethical or indicate that it is “maybe” unethical.

| Unethical Grammar | Unethical Rewrite | Respondents
---|---|---|---
| (mean) | (mean) | (N)
Native |  |  |
Yes | 0.07 | 0.46 | 83
No | 0.05 | 0.36 | 188
Total | 0.06 | 0.39 | 271
Role |  |  |
Professor | 0.06 | 0.38 | 172
Student or Postdoc | 0.05 | 0.39 | 99
Total | 0.06 | 0.39 | 271
Table 4: The survey averages responses to the questions: “Do you think it is
unethical to ask ChatGPT to fix grammar (rewrite text) in a manuscript for an
academic journal?” based on one’s role and whether they are a native English
speaker.

When we compare native English speakers to non-native ones, we find that there
is not a significant difference in perceptions of ethics around fixing
grammar: 7% compared to 5% (two-sample test of proportions yields p=0.211).
However, when it comes to rewriting text, 46% of native English speakers think
that it is unethical to use ChatGPT, compared to 36% of non-native speakers. A
two-sample test of proportions yields p = 0.121 on a two-tailed test or p =
0.060 on the one-tailed. When we compare professors to students and postdocs,
we do not find any significant difference for either fixing grammar and
rewriting text (for grammar p=0.76 and for rewrite p=0.977).

Finally, Table 5 shows the perceptions of respondents regarding whether one
should report the use of the ChatGPT for assistance in manuscript preparation.
This includes those who consider it ethical to use it and those who believe it
to be unethical or are unsure (answered “maybe”). A significant difference
exists among these groups. When researchers perceive it as ethical to use
ChatGPT, only 21% (30%) believe that reporting is necessary when using it for
grammar correction (rewriting). In contrast, researchers who view it as
unethical or are uncertain about its ethics have a higher percentage, with 47%
(87%) believing that reporting is necessary. This difference among the groups
is significant. A two-sample test of proportions yields p=0.017 (0.0001).

| Acknowledge | Respondents
---|---|---
| (mean) | N
Grammar |  |
Unethical (Yes or Maybe) | 0.47 | 15
Unethical (No) | 0.21 | 256
Total | 0.22 | 271
Rewrite |  |
Unethical (Yes or Maybe) | 0.87 | 105
Unethical (No) | 0.3 | 166
Total | 0.52 | 271
Table 5: The average perception of researchers regarding whether authors should report using the ChatGPT for assistance in manuscript preparation is grouped by whether the respondent thinks it is unethical to use it to fix grammar or rewrite the academic text. Publisher | Disclosure | Link
---|---|---
Science | Mandatory | https://www.science.org/content/page/science-journals-editorial-policies
Elsevier | Mandatory | https://www.elsevier.com/about/policies-and-standards/publishing-ethics
Springer Nature | Voluntary | https://www.nature.com/nature/editorial-policies
Taylor & Francis | Mandatory | https://newsroom.taylorandfrancisgroup.com/taylor-francis-clarifies-the-responsible-use-of-ai-tools-in-academic-content-creation/
JAMA Network | Voluntary | https://jamanetwork.com/journals/jama/fullarticle/2801170
American Chemistry Society (ChemArxiv) | Mandatory | https://axial.acs.org/publishing/new-chemrxiv-policy-on-the-use-of-ai-tools
International Committee of Medical Journal | Mandatory | https://www.icmje.org/icmje-recommendations.pdf
World Association of Medical Editors | Mandatory | https://wame.org/page3.php?id=106
International Conference on Machine Learning | Mandatory | https://www.acm.org/publications/policies/new-acm-policy-on-authorship
Table 6: AI disclosure policy for several major publishers as of November 18,
2023.


Human society under the COVID-19 as a complex system. Mean first-passage time
methodology

Anatolii V. Mokshin1, Yana A. Shadrina1 and Vladimir G. Sherputovskiy2

1Kazan Federal University, 420008 Kazan, Russia
2Republican Medical Information and Analytical Centre of the Republic of
Tatarstan, 420073 Kazan, Russia
E-mail<EMAIL_ADDRESS>
> Human society under the COVID-19 pandemic can be viewed as a complex system,
> the evolution of which is characterized by such the parameters as the number
> of newly diseased, the number of seriously ill patients, the number of those
> who were identified as diseased, etc. To analyze such a complex system, we
> propose to apply the methodology known in statistical physics as the
> methodology of the _mean first-passage times_. As will be shown, this
> methodology makes it possible to determine properly the so-called threshold,
> at which a spreading viral disease goes into an epidemic regime with
> accelerated progression. The efficiency of the methodology is demonstrated
> by the example of data analysis on the spread of the COVID-19 in the Russian
> Federation as well as separately in one of the regions of the Russian
> Federation. The methodology can serve as an additional useful tool for
> solving optimization problems.
>
> Key words: complex system, mean first-passage times, infectious diseases,
> critical values, disease patterns
>
> PACS numbers:

##  1 Indroduction

The definition of “ _complex system_ ” is very subjective in its common usage:
each individual (person) can imply and put something of his own into this
definition. At the same time, the concept of “ _complexity_ ” can be
associated with different aspects of a considered system: its internal
structure, form, reaction to external disturbances, unpredictability of its
behaviour (evolution), impossibility or complexity of its control. Currently,
in natural scientific researches, the definition of “ _complex system_ ” is
becoming one of the basic definitions in the context of characterizing systems
of a certain type [1]. The nature of these systems can be very diverse: these
can be physical, biological, economic, social, physiological, etc. Despite the
fact that the definition of “ _complex system_ ” is widely used in the
scientific vocabulary and there is a substantive field of science dedicated to
the study of complex systems, there is still no universally accepted
definition. A discussion of this issue can be found, for example, in the
papers [2, 3, 4]. Thus, nine completely different definitions are given in
Ref. [2]. Complex systems are assumed to have the following features. First,
any aspect or set of aspects of a complex system is inherently _nonlinear_.
Mathematically, it means that the structure, form, dynamics of a considered
system, interaction between its constituent elements can be described, in
particular, by nonlinear differential equations. Second, a complex system is,
generally, _non-Hamiltonian_. In other words, it is difficult or impossible to
determine unambiguously the total energy of such a system. Third, such a
system is characterized by _non-integrability_ of features of its elements.
This means that the sum of the properties of individual components does not
necessarily determine the properties of system as a whole. In addition, a
typical property of such a system is its _ability to self-organize_ , that is
expressed in the fact that the system can maintain a certain resistance to
external perturbations [5, 6]. The aforementioned features are probably the
basic features of complex systems of any nature.

In this study, we assume that a complex system is a system that can be
_accurately described_ only by an infinitely large number of degrees of
freedom (factors, parameters) [7, 8]. On the one hand, such a definition does
not contradict the features given above. It also provides a basis for applying
the methodology of statistical physics to the analysis and/or description of a
complex system. Finally, this definition best reflects the fact that the
appropriate tools for studying a complex system are methods of metaheuristics
and machine learning (artificial neural networks, genetic algorithms, etc.)
that operate with sets of various factors and establish links between these
factors.

According to the definition we have given, human society, evolving on its own
or under any conditions, can be seen as a complex system. For example, a human
society in the face of the spread of an infectious disease (ID) will be
determined by an infinitely large number of factors related to demography,
migration, climate/weather, economy, level of development of its health care
system, and the restrictive regulations in the region concerned. Thus, the
character of the spreading ID depends on all these factors to some degree.
Some of these factors are the most important, while the impact of other
factors may be insignificant. On the other hand, it is possible to focus on
studying only one _key_ factor that characterizes directly the spread of an
ID. In this particular case, it is appropriate to use the number of confirmed
cases currently identified as a factor. By analyzing the dynamics of this
factor, one can try to determine a trend of ID development, its peculiarities,
as well as to make a probabilistic forecast of further ID development, to
estimate the probability of ID transition to epidemic and/or pandemic regime
[9, 10]. Currently, the powerful mathematical methods as well as the machine
learning tools are widely used to analyze evolution of ID’s [12, 11, 13, 14,
15, 16, 17, 18].

There is the specific feature of such a complex system as human society under
conditions of spread of ID. Namely, when ID is absent or the degree of its
development has not reached some threshold at which an epidemic/pandemic
regime emerges, it can be assumed that the system is in some stable state.
Then, the emergence of an epidemic regime will correspond to escape of the
system from this stable state. Here, there is an analogy with the so-called
activation-type processes that are well known in physics, chemistry,
engineering and biology [19, 20, 21]. In this study, it will be shown that the
analysis of the spreading ID, which can go into epidemic or pandemic regimes
in human society, can be carried out using the so called _mean first-passage
time_ (MFPT) method well known in modern statistical physics [22, 23].
Quantitative results are demonstrated by the example of the COVID-19 spread in
the Russian Federation (RF) and in a specific region of the RF.

##  2 Methodology

Suppose there is a complex system whose evolution is tracked by means of the
time-dependent parameter $n(t)$. The system is assumed to have two stationary
(quasi-stationary) states where it can exist and in which the parameter $n$
takes the values $n^{(1)}$ and $n^{(2)}$, respectively. Therefore, we can say
that its evolution takes place on a hypothetical free energy landscape $U(n)$,
where the values $n^{(1)}$ and $n^{(2)}$ define the positions of the minima.
Then, the maximum value of the quantity $U(n)$ enclosed between these minima
is determined by the conditions of transition from one state to another, as
shown in Fig. 1b.

FIG. 1: Schematic drawings showing the transition of the system from a local
minimum, where the order parameter $n$ takes the value $n^{(1)}$, to another
local minimum with the value $n^{(2)}$. (a) Three independent trajectories of
the order parameter $n(t)$. The horizontal dotted lines correspond to the
values of the order parameters $n^{(1)}$, $n^{(c)}$, $n^{(j)}$, $n^{(j+1)}$,
$n^{(2)}$. (b) Free energy landscape $U(n)$, where the quantity $n$
corresponds the abscissa axis, whereas the quantity $U(n)$ is given on the
ordinate axis.

Here, the quantity $n(t)$ takes the meaning of the reaction coordinate or
order parameter characterizing the transition between states. It should be
noted that the question of the nature of the free energy $U(n)$ is not
fundamental in this case. In physics, this situation corresponds to a two-
level system whose evolution happens in a double-well potential [19]. The
solution corresponding to this situation is used, for example, in the well-
known physical problem of a particle overcoming an energy barrier during the
transition from one local energy minimum at $n=n^{(1)}$ to another one at
$n=n^{(2)}$.

Stay of the system in each of the minima $U(n^{(1)})$ and $U(n^{(2)})$ is
determined by the time scales $\tau^{(1)}$ and $\tau^{(2)}$, and the mean
(most probable) time to reach the maximum $U(n^{(c)})$ is given by the
quantity $\tau^{(c)}$. Thus, $\tau^{(c)}$ determines the most probable time at
which the system reaches the critical value $n^{(c)}$, and the inverse
quantity $1/\tau^{(c)}$ characterizes the transition rate. The process of
transition from one minimum to another is stochastic and is driven by some
forces, the nature of which is not crucial in the context of our topic.
However, if the transition process has the character of an activation-type,
the energy brought by these forces should be much smaller than $U(n^{(c)})$.
For example, from the point of view of classical thermodynamics, the
transition between different aggregate states is due to the thermal energy
$k_{B}T$, where $k_{B}$ is the Boltzmann’s constant and $T$ is the
temperature.

The characterization and probabilistic description of such a system can be
performed through statistical consideration of various partial realizations,
from which we obtain a set of $M$ independent trajectories:

$\left\\{n_{1}(t),\ n_{2}(t),\ \ldots,n_{i}(t),\ \ldots,\ n_{M}(t)\right\\}.$

The lower index $i$ is the label of the $i$th realization. This set can be
analyzed using the procedure of _inverted averaging_ [24, 25, 26, 27], which
is based on the MFPT method [22].

The main idea of the MFPT method is to determine the sequence of the first
passage times for all possible values of the parameter $n$ for each $i$th
trajectory, i.e.:

$n_{i}^{(1)}(t),\ n_{i}^{(2)}(t),\ \ldots,n_{i}^{(j)}(t),\ldots$

The upper index $(j)$ defines the $j$th value of parameter $n$. In other
words, for each $i$th trajectory, $t(n_{i}^{(j)})$ is defined that has the
form of a non-decreasing step function (for details, see Fig. 1). Thus, we
find the distribution of the average first occurrence time by inverted
averaging over the results for $M$ realisations:

$\tau(n)=\frac{1}{M}\sum_{i=1}^{M}t(n_{i}^{(j)}).$ (1)

For activation processes, the MFPT curve has a sigmoidal shape. The larger the
ratio of the time scale $\tau^{(c)}$ to the time scale $\tau^{(1)}$, i.e.
$\tau^{(c)}/\tau^{(1)}\gg 1$, the steeper the bend will be characterized by
this curve. The more pronounced and higher is the energy barrier, the more
smooth dependence characterizes the curve $tau(n)$ in the plateau regime (see
Fig. 2a).

FIG. 2: (a) MFPT curve obtained by averaging the different trajectories. The
position of the red point on the curve corresponds to the critical value
$n^{(c)}$ and the time scale $\tau^{(c)}$. (b) Derivative of the MFPT curve
$\partial\tau(n)/\partial n$ whose maximum position corresponds to the
critical value $n^{(c)}$.

Finally, the position of the first inflection point of the MFPT curve
corresponds to the critical value $n^{(c)}$ and the transition time
$\tau^{(c)}$ (see Fig. 2). Both of these quantities can be accurately
determined by the position of the maximum on the graph of the derivative
$\partial\tau(n)/\partial n$ (see Fig. 2b). Thus, the MFPT methodology allows
us to accurately determine the critical value of the order parameter $n$
corresponding to a transition from one state to another and the time scale
$\tau$ that characterizes this transition. The applicability of this
methodology is not limited to any particular nature of the system. The
analysis by means of this technique is based only on a set of independent
trajectories for the order parameter $n$, and the number of these trajectories
should be enough to obtain a reliable probabilistic estimate.

##  3 Description of data

This paper performs an analysis of daily data describing the spread of the
COVID-19 for the period from March $12$, $2020$ to November $16$, $2022$. Two
situations are considered: (1) the RF under the COVID-19 and (2) the Republic
of Tatarstan (RT), as a separate entity of the RF, also under the COVID-19.
Six separate waves of ID were captured during this time period. Statistical
analysis has been performed separately for each wave. The parameter $n$ is a
quantity that characterizes the number of new confirmed cases of the disease
per population of the territorial unit. Thus, quantity $n$ denotes the
fraction of diseased persons from the total population for the considered
territorial unit.

(1) _The Russian Federation_. – The analysis is based on the data for $72$
regions of the RF (see Fig. 3a).

FIG. 3: Dynamics of the COVID-19 incidence in the RF and the RT for the period
from March $12$, $2020$ to November $16$, $2022$. Individual trajectories
correspond to different regions in the case of the RF and different health
facilities in the case of the RT (for details, see the text).

The data for the RF is taken from the official portal “stopcoronavirus.rf”
111https://xn–80aesfpebagmfblc0a.xn–p1ai/, where information on disease
incidence is updated daily. Various regions were considered, including the
largest, such as Moscow, the Moscow Region and St. Petersburg, the average
populations of which are $12\,678\,079$, $7\,690\,863$ and $5\,398\,064$
people, respectively. The average population of such small regions as the
Nenets Autonomous Area, the Chukotka Autonomous Area and the Jewish Autonomous
Region for the period under review was $44\,111$, $50\,288$ and $158\,305$
people, respectively. The population density of the regions of the RF
considered in the given study ranges from $0.1$ to $4950.4$ people/km2. The
regions with the lowest population density are the Chukotka Autonomous Area
($0.1$ people/km2), the Nenets Autonomous Area ($0.2$ people/km2) and the
Republic of Sakha (Yakutia) ($0.3$ people/km2). The highest densities are in
such the regions as Moscow ($4\,950.4$ people/km2), St. Petersburg ($3\,858.5$
people/km2) and the Moscow Region ($173.5$ people/km2). The data source is the
Federal State Statistics Service for 2020
222https://rosstat.gov.ru/folder/210/document/13205.

(2) _The Republic of Tatarstan_. – The study considers data for $97$ health
facilities of the RT (see Fig. 3b), where confirmed cases of the disease were
registered. The values of the number of diseased persons were recalculated
taking into account the population attached to the respective health facility
333Population of municipalities of the Republic of Tatarstan at the beginning
of 2020. Statistical Bulletin.,444https://msu.tatarstan.ru/mregions.htm. The
data were relevant to both large settlements with several health facilities
represented and those with small numbers of people living in them.

The period taken into consideration for both the systems was $980$ days. As
shown in Fig. 3, the dynamics of daily detected cases both nationally (Fig.
3a) and regionally (Fig. 3b) for a given time period reveals six spikes/waves
where there is a sharp increase in prevalence. It should be noted that it is
practically impossible to determine the boundaries between individual waves
with sufficient accuracy from the original data. In this case, the third and
fourth waves practically merge.

##  4 Results for the Russian Federation

###  4.1 The start of the next wave of the epidemic

The key point of the analysis is to determine the time moment that can be
taken as the beginning of the next wave. The first wave was excluded from our
analysis because the information on morbidity at the initial stages of the
COVID-19 was inconsistent and incomplete.

An accurate statistical estimate of the time at which the next wave begins can
be made on the basis of the distribution of times when the number of cases in
the regions was minimal. The position of the maximum on this distribution can
be taken as the time of the beginning of the next wave of the COVID-19. Thus,
Fig. 4 shows such a distribution for the stage corresponding to the beginning
of the second wave of ID progression.

FIG. 4: Distribution of times of minimum number of cases in different regions
of the RF before the second wave of the COVID-19.

The time step of the distribution (i.e. bin width) is $5$ days. This quantity
determines the size of the confidence interval around the resulting estimates
for the beginning date of the corresponding wave. In addition, this time step
appears to be optimal for reasons of the statistics used to construct the
distribution based on data for $100$ independent objects, as well as for
reasons of the resulting ratio between disease wave size and time step. From
this distribution, we find that the date corresponding to $162.5$ ($\pm 2.5$)
days from the beginning of the COVID-19 leakage registration can be chosen as
the initial moment of the second wave. Similar probabilistic estimates were
performed for all other waves of the COVID-19. Thus, for the third wave we
find that its onset is 427.5 days, for the fourth wave – $547.5$ days, for the
fifth and sixth waves – $667.5$ and $837.5$ days, respectively.

Based on the results obtained, the full time interval of $980$ days was
divided into fragments corresponding to individual disease waves. Subsequent
analysis was performed separately for each wave of the COVID-19 (except for
the first wave).

###  4.2 Critical values of the transition to the epidemic regime for
different waves

The time distributions of $\tau(n)$ for each wave were obtained using the MFPT
analysis. The obtained results are shown in Fig. 5.

FIG. 5: MFPT $\tau(n)$ curves and derivatives $\partial\tau(n)/\partial n$
determined for different disease waves. Here, $\tau_{1}^{(c)}$ and
$\tau_{2}^{(c)}$ define the critical time range, $n_{1}^{(c)}$ and
$n_{2}^{(c)}$ define the boundaries of the parameter $n$ with the critical
values corresponding to this range. Insert in panel (d): enlarged fragment of
the main graph.

In the case when the variable $n$ represents a characteristic of the spread of
ID - the daily number of newly infected people, the quantity of $\tau$ will
make sense of the time moment when the number of infected people $n$ takes
some specific value. As can be seen from the shapes of the obtained
distributions, the evolution of the next wave follows the scenario typical for
activation-type processes. This is indicated by the fact that all the
distributions have a pronounced sigmoidal shape, similar to one in Fig. 2a.
Thus, the smooth form of each obtained distribution of $\tau(n)$ for the
initial stage of the disease wave, is replaced by a sharp increase, which, in
turn, passes into a saturation regime (plateau).

It turns out to be very interesting that the inflection point in each
distribution $\tau(n)$, defined by the critical values of $\tau(n)$ and
$n^{(c)}$, is not characterized by single values of these quantities, but by
ranges of values $\tau^{(c)}\in[\tau_{1}^{(c)};\tau_{2}^{(c)}]$ and
$n^{(c)}\in[n_{1}^{(c)};n_{2}^{(c)}]$. The determined critical values are
given in Tab. 1.

Table 1: Parameters characterizing the course of the different the COVID-19 waves in the RF: start date of the $i$th wave, ranges of values of the critical time $\tau^{(c)}$ (in days) and the parameter $n^{(c)}$ for the $i$th wave, average peak characteristics of the $i$th wave: $\bar{\tau}_{i}^{(max)}$ and $\bar{n}_{i}^{(max)}$. $i$th wave | start date | $\tau_{i}^{(c)}$ | $n_{i}^{(c)},\cdot 10^{-5}$ | $\bar{\tau}_{i}^{(max)}$ | $\bar{n}_{i}^{(max)},\cdot 10^{-4}$
---|---|---|---|---|---
2 | 162.5$\pm$2.5 (20.08.2020) | [14; 30] | [2.76; 4.54] | 117 | 2.29
3 | 427.5$\pm$2.5 (13.05.2021) | [18; 36] | [3.97; 6.41] | 81 | 1.98
4 | 547.5$\pm$2.5 (10.09.2021) | [11; 16] | [11.6; 13.5] | 63 | 2.95
5 | 667.5$\pm$2.5 (08.01.2022) | [7; 9] | [11.8; 14.2] | 37 | 16.7
6 | 837.5$\pm$2.5 (27.06.2022) | [9; 23] | [1.96; 3.29] | 82 | 4.71

This result seems rather unexpected, although understandable. It indicates
that the onset of an irreversible increase in the number of new cases falls
within a time range of a few days to three weeks, depending on the particular
wave. Thus, this range is the largest in the case of the third wave of the
disease. It also turns out that the critical proportion of all confirmed cases
of the disease, the excess of which correlates with the explosive spread of
the disease, is about $1\cdot 10^{-5}$ of the total population in the
corresponding region. For example, in the case of the second wave for Moscow,
we find that the absolute critical values of the confirmed cases of the
disease are in the range [$350$; $576$] people, and for the Nenets Autonomous
Area we get the range [$2$; $3$] people. These absolute critical values are
quite commensurable if we take into account the total population living in the
respective regions. The obtained values clearly reflect the fact that the
COVID-19 belongs to extremely highly ID’s.

###  4.3 Results for the Republic of Tatarstan as a region of the Russian
Federation

In the case of the data for the RT, the time moments of the start of each
disease wave were estimated using the same scheme as for the RF (see
description 4.1). The values of the time moments were obtained which were
found to correlate with the results for the RF (see Tab. 2) were obtained.

Table 2: Parameters characterizing the course of different waves of the COVID-19 in the RT (by analogy with Tab. 1). $i$th wave | start date | $\tau_{i}^{(c)}$ | $n_{i}^{(c)},\cdot 10^{-5}$ | $\bar{\tau}_{i}^{(max)}$ | $\bar{n}_{i}^{(max)},\cdot 10^{-4}$
---|---|---|---|---|---
2 | 176$\pm$2 (03.09.2020) | [18; 26] | [0.49; 0.85] | 117 | 5.24
3 | 426$\pm$2 (11.05.2021) | [27; 35] | [1.59; 2.07] | 104 | 6.73
4 | 550$\pm$2 (12.09.2021) | [8; 13] | [4.77; 6.99] | 36 | 13.4
5 | 656$\pm$2 (27.12.2021) | [6; 11] | [0.58; 1.27] | 36 | 34
6 | 856$\pm$2 (15.07.2022) | [7; 17] | [0.55; 1.47] | 58 | 7.23

The obtained MFPT-curves also indicate the activation character of the
progression of each wave of the COVID-19 in the RT. Analysis of these curves
reveals that there are the ranges of critical values of $\tau^{(c)}$ and
$n^{(c)}$ that generally correlate with the results obtained for the country
as a whole.

###  4.4 Epidemic patterns

A common feature of most ID’s is the occurrence of separate waves. At the same
time, the course of diseases is largely determined by the way these waves
evolve, i.e. how the disease spreads at the stage of separate waves. In this
regard, it is useful to consider the so-called disease patterns, which
represent the correspondence of different characteristics for a sequence of
waves. Thus, Fig. 6 shows the patterns for the critical wave characteristics –
the critical value $n_{i}^{(c)}$ and the critical time $\tau_{i}^{(c)}$.

FIG. 6:  Patterns of the critical values of $\tau^{(c)}$ and $n^{(c)}$ of
individual waves determined for the second ($i=2$), third ($i=3$), $\ldots$
and sixth ($i=6$) waves of the COVID-19 in the RF (solid black line) and in
the RT (dashed red line). The value of $i$ indicates the order number of the
considered wave. Insert: The main directions of the trajectory in the pattern
according to the development of the epidemic (see Sec. 4.4).

In the figure, the abscissa axis represents the time characteristics and the
ordinate axis displays the parameters characterizing the fractions of infected
individuals. As the ID progresses and moves from one wave to the next, some
directional trajectories will emerge in these patterns.

For such the patterns, the following main scenarios can be highlighted. The
upward direction of the trajectory (labelled “direction $3$”) implies an
increase in the critical value of the subsequent wave. This scenario is
possible if the system adapts to an ID and a virus mutates accordingly. In
this case, the transition of the system into the epidemic regime at the stage
of the next wave is possible only at a larger critical value of $n^{(c)}$
compared to the critical value of the previous wave. Here, the adaptation of
the system may be largely due to the actions implemented, as the system is
“trained” to cope with the challenges caused by the ID. On the other hand, the
downward direction of the trajectory (“direction $5$”) may indicate mutations
in the virus that make it more infectious, as well as a weakening of the
system’s “immunity”. The direction of the trajectory to the right side of the
pattern (“direction $1$”) indicates that the value of the critical time $\tau$
of the next wave exceeds the critical value of the previous wave. This is
evidence that the spread of the ID is slowing down, which may be due to both
mutations in the virus as well as the system’s response to it. On the other
hand, the direction of the trajectory to the left (“direction $7$”) reflects
the fact that the ID is becoming more contagious and the virus is spreading
faster in the system. Explanations are provided in insert to Fig. 6.

The diagonal directions of the pattern will be interpreted as follows:

  1. $\bullet$

The right and up (“direction $2$”). This trajectory direction corresponds to a
situation where a virus is present and active, but from wave to wave the
system is trained to deal with it more successfully.

  2. $\bullet$

The left and up (“direction $4$”). This direction of the trajectory represents
the case when the infectivity of the virus becomes higher as it progresses to
the next wave, but the system, as a whole, copes with the virus.

  3. $\bullet$

The left and down (“direction $6$”). This is possible when the virus is
“successful”. This is the worst-case scenario for the spread of an ID in the
system and can lead to a fatal ending.

  4. $\bullet$

The right and down (“direction $8$”). In this case, the virus remains highly
contagious. However, specific conditions are required for this contagion to be
realized. This situation is possible when the system successfully fights a
virus and defeats it.

On the basis of this, it becomes clear that for a young, active virus, the
initial stages of its progression and spread will be associated with movement
along the directions $2\rightarrow 3\rightarrow 4$. On the other hand, the
implementation of “direction $8$” may indicate the finality of virus activity
and may be due to virus degeneration.

It can be seen in Fig. 6 that the initial stages of the trajectories related
to the $2$nd, $3$rd and $4$th waves for the RF and the RT proceed in a similar
way. Moreover, the character of the direction of these trajectories is typical
for a young ID such as the COVID-19. Further, the transition from the $4$th
wave to the $5$th wave occurs in the directions (“direction $7$” for the RF
and “direction $5$” for the RT), which are typical for the cases when an ID
becomes highly contagious. This situation was largely due to the spread of the
“Omicron” strain. However, the high infectiousness of the “Omicron” strain was
compensated for by its milder course. As a result, the further spread of the
ID, corresponding to the transition from the $5$th wave to the $6$th wave,
began to evolve according to “direction $8$” for the RF and according to
“direction $1$” for the RT, which are associated with a significant decrease
in virus activity and a slowing of the spread of the ID.

The obtained results do indicate that with the emergence of the “Omicron”
strain, the character of the spread of the COVID-19 is changing dramatically.
At the same time, the results for the RT indicate that there is a certain
preparatory stage for these changes, which takes place during the transition
from the $4$th to the $5$th wave. The same stage is clearly manifested in the
direction of the trajectory, which characterizes the attainment of peak values
of $\bar{\tau}^{(max)}$ and $\bar{n}^{(max)}$ by separate waves (see Fig. 7).

FIG. 7: Patterns of peak values of $\bar{\tau}^{(max)}$ and $\bar{n}^{(max)}$
of individual waves determined for the second, third, $\ldots$ and sixth waves
of the COVID-19 in the RF (solid black line) and in the RT (dashed red line).

Here $\bar{n}^{(max)}$ is the fraction of the total population that is ill
when the wave attains its peak; and $\bar{\tau}^{(max)}$ is the time at which
an individual wave peaks (averaged values of both the parameters are
considered). In the case of the RF, we have a trajectory typical for a
situation, when the activity of the virus increases from second wave to fifth
wave; and only the last stage displays a decrease in its activity. For the RT,
a separate stage is distinguished at the transition from the $4$th to the
$5$th wave, where the trajectory is significantly different from what it was
before. The trajectories at the last stage have a completely similar
character.

##  5 Conclusion

We conclude with the following main results.

  1. (i)

It is shown that the MFPT method can be used to analyze the evolution of a
specific complex system such as human society under conditions of an ID
spread.

  2. (ii)

The COVID-19 propagation in the RF as well as in the RT is an activation-type
process.

  3. (iii)

It is found that the critical values of $\tau^{(c)}$ and $n^{(c)}$ associated
with the transition of the ID to the epidemic regime are given by certain
ranges of values that are accurately determined by the MFPT method.

  4. (iv)

The results obtained for a single region (RT) were found to be fully
correlated with the results for the country (RF) as a whole.

  5. (v)

We propose a methodology for analyzing the development of ID based on a
detailed examination of the patterns that represent the correspondence between
the critical characteristics for a series of ID waves.

  6. (vi)

Based on the character of the obtained patterns, it was directly found that
the COVID-19 spreads in both the RF and the RT according to a scenario typical
for a young ID.

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