∎

11institutetext: Dario Ortega Anderez and Eiman Kanjo1School of Science and
Technology, Nottingham Trent University, Clifton Lane, NG11 8NS, UK. 11email:
<EMAIL_ADDRESS>11email<EMAIL_ADDRESS>22institutetext: Shane Johnson 33institutetext: 2University College London
(UCL) Jill Dando Institute, 35 Tavistock Square, London, WC1H 9EZ, UK
44institutetext: Ganna Pogrebna55institutetext: 3The University of Sydney
Business School, Abercrombie Building H70, Darlington NSW 2006, Australia and
the Alan Turing Institute, 96 Euston Road, London, NW1 2DB, UK
66institutetext: John Hunt 77institutetext: 4Medical Technologies Innovation
Facility, Nottingham Trent University, Nottingham, NG11 8NS, U.K. College of
Biomedical Engineering, China Medical University, Taichung 40402, Taiwan

# A Modified Epidemiological Model to Understand the Uneven Impact of COVID-19
on Vulnerable Individuals and the Approaches Required to Help them Emerge from
Lockdown

Dario Ortega Anderez*1 Eiman Kanjo1 Ganna Pogrebna3 Shane Johnson2 John Alan
Hunt4

(Received: date / Accepted: date)

###### Abstract

Background: COVID-19 has shown a relatively low case fatality rate in young
healthy individuals, with the majority of this group being asymptomatic or
having mild symptoms. However, the severity of the disease among the elderly
as well as in individuals with underlying health conditions has caused
significant mortality rates worldwide. Understanding this variance amongst
different sectors of society and modelling this will enable the different
levels of risk to be determined to enable strategies to be applied to
different groups. Long-established compartmental epidemiological models like
SIR and SEIR do not account for the variability encountered in the severity of
the SARS-CoV-2 disease across different population groups.

Objective: The objective of this study is to investigate how a reduction in
the exposure of vulnerable individuals to COVID-19 can minimise the number of
deaths caused by the disease, using the UK as a case study.

Methods: To overcome the limitation of long-established compartmental
epidemiological models, it is proposed that a modified model, namely SEIR-v,
through which the population is separated into two groups regarding their
vulnerability to SARS-CoV-2 is applied. This enables the analysis of the
spread of the epidemic when different contention measures are applied to
different groups in society regarding their vulnerability to the disease. A
Monte Carlo simulation (100,000 runs) along the proposed SEIR-v model is used
to study the number of deaths which could be avoided as a function of the
decrease in the exposure of vulnerable individuals to the disease.

Results: The results indicate a large number of deaths could be avoided by a
slight realistic decrease in the exposure of vulnerable groups to the disease.
The mean values across the simulations indicate 3,681 and 7,460 lives could be
saved when such exposure is reduced by 10% and 20% respectively.

Conclusions: From the encouraging results of the modelling a number of
mechanisms are proposed to limit the exposure of vulnerable individuals to the
disease. One option could be the provision of a wristband to vulnerable people
and those without a smartphone and contact-tracing app, filling the gap
created by systems relying on smartphone apps only. By combining very dense
contact tracing data from smartphone apps and wristband signals with
information about infection status and symptoms, vulnerable people can be
protected and kept safer.

###### Keywords:

COVID-19 Coronavirus Infection Spread Modelling Epidemiological Model

##  1 Introduction

Coronaviruses (CoV) are a large family of enveloped, positive-strand RNA viral
diseases capable of infecting a variety of host species, including humans and
several other vertebrates Channappanavar2017 . CoVs predominantly cause
gastrointestinal and respiratory tract infections, inducing a wide range of
clinical symptotic manifestations Su2016 .

Before the latest strain of coronavirus SARS-CoV-2 in December 2019, six human
CoVs including four endemic (HCoV-OC43, -229E, -NL63, and -HKU1) and two
epidemic (SARS-CoV and MERS-CoV) viruses had been identified.

Despite the recent scientific progress across different domains and the
increasing levels of public hygiene worldwide, the combination of various key
adverse factors has translated the SARS-CoV-2 outbreak into a global epidemic
with a real risk to life. The COVID-19 outbreak originated in Wuhan, China,
which in addition to being the largest city in central China, is home to the
largest deep-water port, airport and train station in that same area
Wilson2020 . The outbreak coincided with increased levels of travel prior to
the Chinese New Year, that resulted in more than five million people
travelling out from Wuhan in the first few days of the outbreak before Wuhan
was put under lockdown Wilder2020 . This facilitated the spread of COVID-19
within Mainland China as well as in countries with high volumes of air traffic
with China, such as South Korea, Japan, Thailand and Singapore Bogoch2020 .
Initial estimates from the World Health Organisation (WHO) for the basic
reproduction number ($R_{0}$) of COVID-19 were in the range of 1.4 to 2.5,
whereas recent studies suggest this estimated basic reproduction number could
have been higher, with statistical studies giving $R_{0}$ values in the range
(2.2-3.58) Liu2020 ; Zhao2020 ; Li2020 , mathematical studies from 1.4 to 6.49
Shen2020 ; Cao2020 ; Tang2020 , and stochastic approaches from 2.2 to 2.68
Wu2020 ; Riou2020 . This implies the reproduction number of COVID-19 may be a
lot higher than originally estimated and therefore significantly greater than
that of MERS ($R_{0}<1$) World2017  and in the range of SARS’ ($R_{0}\approx
3$) World2003 . As the COVID-19 epidemic unfolds, the commonality of mild and
asymptomatic cases has become apparent. This is a key issue, since unlike with
other CoVs like SARS, where peak viral shedding occurred after patients were
severely ill and therefore easily identifiable, preliminary evidence suggests
that COVID-19 can be transmitted at the early phases of the disease where no
symptoms are observed Rothe2020 .

The combination of these factors has led to the rapid international spread of
SARS-CoV-2 and it being declared a Public Health Emergency of International
Concern on 1 February 2020 and thereafter a pandemic on the 11 March 2020. The
dramatic consequences of the pandemic have translated into unprecedented non-
pharmaceutical interventions (NPIs) internationally, including national and
international travel restrictions, quarantine and isolation of large
populations, the closure of universities, schools and social spaces, as well
as a widespread social distancing policy and the banning of public gatherings
and events. Despite these containment measures by different countries, as of 2
June 2020, over 6.4 million cases of COVID-19 have been reported worldwide
causing death to over 378,290 people WorldometersWorld .

While COVID-19 has the potential to infect every individual in the world, the
disease has been extremely dangerous with a greater risk of mortality in
groups older than 70 years of age as well as in people with underlying health
conditions (e.g., high blood pressure; respiratory problems, etc.). Given the
current increasingly aging population in many parts of the world,
particularly, in developed nations such as the UK, where 2018 figures from the
Office for National Statistics Coates2019  reported that 18.3% of people were
65 or over. Put this, with the number of younger individuals who have
underlying health conditions, then there is an approximate 20% of the UK
population having a considerably high risk of death if infected. Motivated by
the disparate mortality rates across different groups in society, this work
proposes a modified compartmental epidemiological model (SEIR-v) to study the
impact of reducing the contact rate of vulnerable individuals on the potential
to reduce the number of fatalities caused by the disease. The motivation
behind the model was that traditional epidemiological models like SIR
Kermack1927  and SEIR Aron1984  assume equal contact rates and death rates for
every individual in the population (i.e., these models, essentially, abstract
from significant individual, behavioral, and spacial heterogeneity observed in
the population of any country). In contrast, SEIR-v provides a means of
studying the progression of the number of fatalities when differing contention
measures are applied across different groups of individuals regarding their
vulnerability to the disease, while also accounting for the characteristic
variability seen in the case fatality rates across these groups. As outlined
in Ferretti2020 , the original mathematical models the British government used
to inform policy in the UK did not account for vulnerable people. In contrast,
SEIR-v provides the opportunity to distinguish between vulnerable groups and
low-risk groups, allowing policy recommendations for each segment, rather than
applying a one-size-fits-all policy. Predictions made with SEIR-v outlined the
importance of minimising the chances of vulnerable people contracting the
disease, with an estimated reduction of 3681 and 7406 further deaths if their
exposure to the virus was decreased by only 10% and 20% respectively. The
widespread policies of physical distancing restrictions, may now have
contributed to another health problem: loneliness. Those who do not have close
family or friends, and rely on the support of voluntary services or social
care, may feel vulnerable, according to recent correspondence published in The
Lancet Armitage2020 . The impact on physical health, arising from mental
health problems due to loneliness should not be underestimated and ignored.

There is therefore a need to help vulnerable people exit the lockdown whilst
addressing their continued protection and provide them with the means to
participate in the contact-tracing process. Here, we consider practical
options for facilitating the exit strategy for vulnerable groups from
lockdown. In line with this, we provide a set of recommendations:

  1. 1.

the use of wearable devices (henceforth, wearables) to enable vulnerable
people to take part in contact tracing,

  2. 2.

the development of effective incentive mechanisms in order to motivate people
to engage in contact tracing,

  3. 3.

the use of digital tools to maintain physical distancing and monitor health
symptoms,

  4. 4.

the use of personal protective equipment,

  5. 5.

the planning for easing the contention measures.

The most important determinants of outcome are:

  1. 1.

reduction of transmission rates post lockdown in the vulnerable populations;

  2. 2.

fewer restrictions on the vulnerable post-lockdown with noticeable improvement
in their well-being (many may already be suffering from loneliness and mental
health problems due to the lockdown);

  3. 3.

maintenance of keeping vulnerable people and the hard-to-reach connected and
closely monitored.

##  2 Methods

###  2.1 Compartmental Epidemiological Models

Generally, the spread of infectious diseases like COVID-19 are studied through
the use of compartmental epidemiological models. Such models provide a
simplified means of describing the transmission of the infectious disease
through the different individuals within a population by dividing the
individuals into different states regarding their current susceptibility to
the disease and their disease transmission capabilities. Within such models,
the SIR Kermack1927  and the SEIR Aron1984  models are widely employed and
published in the literature in the field. A visual representation of the SIR
and SEIR models is depicted in Figure 1.

####  2.1.1 SIR Model

The SIR model divides the entire population into three different compartments,
namely susceptible, infectious and recovered. The transition between one
compartment to its following are controlled by the model transition
parameters:

  1. 1.

Infectious rate ($\beta$): is the rate of spread of the virus given by the
probability of transmitting the disease between an infectious individual and a
susceptible individual. This is subject to the disease transmission
probability and the chance of contact.

  2. 2.

Recovery rate ($\gamma$) = $\frac{1}{T_{lat}}$ is determined by the average
duration of the infectious period of the disease ($T_{lat}$).

  3. 3.

Re-susceptibility rate ($\xi$) is the rate at which recovered individuals
return to the susceptible state due to loss of immunity (normally ignored due
to long-term immunity).

Given the definition of the above parameters, the SIR model can be expressed
by:

$\displaystyle\frac{\partial S}{\partial t}=-\frac{\beta IS}{N},$ (1)
$\displaystyle\frac{\partial I}{\partial t}=\frac{\beta IS}{N}-\gamma I,$
$\displaystyle\frac{\partial R}{\partial t}=\gamma I$

where $S$, $I$ and $R$ are the number of susceptible, infected and recovered
(or removed) individuals respectively, and N is the total population, which
follows $N=S(t)+I(t)+R(t)$.

Provided the set of equations above, the dynamics of the infectious disease
given by its reproduction number ($R_{0}$) is calculated as:

$R_{0}=\frac{\beta}{\gamma}$ (2)

Figure 1: SIR and SEIR epidemiological models.

####  2.1.2 SEIR Model

The SEIR model proposed in Aron1984  is a slight variation of the SIR model,
in that it includes an additional state ‘Exposed’ to the three states used in
the SIR model. The motivation behind this comes from the fact that some
infectious diseases exhibit a considerable post-infection incubation period in
which an infected person (exposed) is not yet infectious, thereby affecting
significantly the dynamics of the transmission of the disease. The SEIR model
can be expressed as follows:

$\displaystyle\frac{\partial S}{\partial t}=-\frac{\beta IS}{N},$ (3)
$\displaystyle\frac{\partial E}{\partial t}=\frac{\beta IS}{N}-\sigma E,$
$\displaystyle\frac{\partial I}{\partial t}=\sigma E-\gamma I,$
$\displaystyle\frac{\partial R}{\partial t}=\gamma I$

###  2.2 The Role of Non Pharmaceutical Interventions and Herd Immunity

Non Pharmaceutical Interventions (NPIs) are those actions or measures employed
with the aim of limiting the spread of a viral disease when pharmaceutical
interventions, such as anti-viral medications and vaccines, are still not
available. A general classification into two main contention strategies,
namely suppression and mitigation, is made based on whether the measures
applied aim at quickly reducing the reproduction number (the average number of
secondary cases generated per typical infectious case), $R$, to values lower
than 1, or to simply slow down the spread of the virus by controlling the
value of $R$, while allowing it to take values in ($R\geq 1$).

In other words, suppression strategies aim to turn the pandemic phase of the
disease ($R>1$) into the endemic phase ($R<1$), where each infectious
individual, on average, spreads the virus to less than one person, thereby
causing a decay in the daily number of new cases. In contrast, mitigation
strategies, unless combined with certain levels of population immunity, are
not aimed at the suppression of the virus per se. Instead, they are employed
to reduce the health impact of the epidemic by controlling the curve through
the contention of $R$, so that even though each individual, on average,
spreads the disease to more than one person, such spread is to some extent
controlled to meet the capacity of the respective health care system, while
building up population immunity along the course of the epidemic phase.
Ultimately, such built up immunity will prevent the disease from spreading any
further, leading thereby to a rapid decline in the number of new infections
and the consequent endemic phase of the disease.

Figure 2: Non-pharmaceutical interventions applied by the UK government.

To date, hybrid strategies through which mitigation and suppression measures
are combined to fight the spread of the epidemic, are being adopted by the
vast majority of countries severely affected by COVID-19. Such strategies have
overall included the implementation of diverse contention measures including
the instruction to self-isolate to confirmed and suspected cases, the
encouragement of social distancing, the banning of non essential travel, mass
gatherings and public events, the closure of schools and universities and the
lockdown of the population. Whilst allowing key workers to carry out their
duties and then broadly the general public in lockdown at home were allowed to
go out for essentials like food and exercise. The analysis of existing data
has already demonstrated that mass testing and the isolation of infected
individuals can on its own have a suppressive impact on the curve, thereby
reducing significantly the size of the peak Aleta2020 . Examples include the
strategies followed by South Korea and Germany. However, it must be noted that
the adoption of suppression strategies is challenging due to the low level of
herd immunity achieved as function of time throughout the progression of the
disease, thereby, to avoid the eventual increase of new infections, the
contention measures have to be maintained until pharmaceutical interventions
are available.

Figure 3: SEIR-v epidemiological model.

Although still no empirical evidence can be found suggesting the dismissal of
the possibility of reinfection from COVID-19, the results reported on studies
investigating the persistence of antibodies in patients exposed to similar
CoVs Mo2006 ; Wu2007 ; Alshukairi2016 ; Payne2016  suggest the antibody
immunity built up by individuals exposed to the SARS-CoV-2 virus may
potentially last until medical interventions are available. For instance, the
study in Mo2006  showed the persistence of antibodies in patients infected by
the severe acute respiratory syndrome coronavirus (SARS-CoV), could last for
at least two years. The results reported in Wu2007 , where 176 patients were
found to maintain SARS-specific antibodies for 2 years were in line with those
of Mo2006 . A significant reduction of immunoglobulin G–positive was the case
in the third year. Thus, SARS patients might be susceptible to reinfection 3
years after the initial exposure to the virus. Regarding the immunity against
MERS-CoV, the work in Alshukairi2016  studied the antibody response in 9
healthcare workers in Jeddah, Saudi Arabia, who had previously suffered from
the disease showing symptoms of severe pneumonia. A further study in Payne2016
, explored the long-term antibody response against MERS-CoV. This research
reported the persistence of antibodies, including neutralising antibodies, in
6 out of the 7 (86%) explored individuals for at least 34 months after the
outbreak.

###  2.3 Proposed Model: SEIR-v

The need to develop the SEIR-v model comes from the consideration of the
characteristic variability encountered in the severity of COVID-19 across
different individuals with respect to their age group and state of health
during the early phase of the infection. Evidence suggests, the mortality rate
in individuals of advanced age and / or with underlying health conditions is
significantly higher than that in younger healthier individuals Ferguson2020 .
Thereby, to reduce the number of fatalities, it is crucial to have detailed
means of looking more closely into the impact of the application of the
different NPIs across the different population groups regarding their
vulnerability to the disease on the overall mortality rate. SEIR-v provides a
means of studying the impact of the different NPIs on the number of deaths,
when these are only applied to people with higher vulnerability to the
disease. The SEIR-v compartmental model can be defined as follows:

$\begin{split}\frac{\partial S}{\partial t}=-(1-p_{v})\frac{\beta
IS}{N}-p_{v}\frac{\beta_{v}IS}{N},\\\ \\\ \frac{\partial E}{\partial
t}=(1-p_{v})\frac{\beta IS}{N}-\sigma E\\\ \\\ \frac{\partial E_{v}}{\partial
t}=p_{v}\frac{\beta_{v}IS}{N}-\sigma E_{v},\\\ \\\ \frac{\partial I}{\partial
t}=\sigma E-\gamma I,\\\ \\\ \frac{\partial I_{v}}{\partial t}=\sigma
E_{v}-\gamma I_{v},\\\ \\\ \frac{\partial R}{\partial t}=\gamma I(1-\mu),\\\
\\\ \frac{\partial R_{v}}{\partial t}=\gamma I(1-\mu_{v}),\\\ \\\
\frac{\partial D}{\partial t}=\gamma I\mu+\gamma I_{v}\mu_{v}\end{split}$ (4)

where $S$ is the number of susceptible individuals. $E$, $I$, $R$ are the
number of non vulnerable exposed, infectious and recovered individuals
respectively. $E_{v}$ $I_{v}$, $R_{v}$ are the number of vulnerable exposed,
infectious and recovered individuals respectively. $D$ is the total number of
deaths caused by the disease. $p_{v}$ is the probability of an individual
being vulnerable to the disease. $\beta$ and $\beta_{v}$ are the contact rates
by non vulnerable and by vulnerable individuals respectively. $\sigma$ is the
rate at which an exposed individual becomes infectious. $\gamma$ is the the
rate at which an infectious individual recovers from the disease. $\mu$ and
$\mu_{v}$ are the case fatality rates for non vulnerable and for vulnerable
individuals respectively. The conceptual diagram of SEIR-v can be seen in
Figure 3.

Table 1: Description of the SEIR-v model parameters

Parameter | Description | Value | Comments
---|---|---|---
N | Population | 67.838.235 WorldometersUK  | Total population in the UK as of 2020
$E_{v_{0}}$ | Vulnerable Exposed | 2 | Vulnerable individuals exposed to the disease at the beginning of the outbreak
$E_{0}$ | Exposed | 4 | Non-vulnerable individuals exposed to the disease at the beginning of the outbreak
$I_{v_{0}}$ | Infected | 0 | Vulnerable infected individuals at the beginning of the outbreak
$I_{0}$ | Infected | 1 | Non-vulnerable infected individuals at the beginning of the outbreak
$T_{inc}$ | Incubation period | 5.6 Linton2020  |  $\sigma=\frac{1}{T_{inc}}$ The time it takes for an exposed individual to become infectious
$T_{lat}$ | Latent period | 7.5 Linton2020  |  $\gamma=\frac{1}{T_{lat}}$ The time it takes for an infectious individual to recover
$\mu_{v}$ |  | Vulnerable
---
Case Fatality Rate
[0.005-0.037, 95% CI]% | Case fatality rate of COVID-19 on vulnerable individuals
$\mu$ |  | Non-vulnerable
---
Case Fatality Rate
[0.000007-0.000011, 95% CI]% | Case fatality rate of COVID-19 on non-vulnerable individuals
$p_{v}$ | Vulnerable probability | 0.2 | Probability of an individual being vulnerable to the disease
$\eta$ | Fear Factor | 0.33 |  | Fear factor caused by the recommendation made by the UK government for
---
vulnerable individuals to stay at home for at least 12 weeks at the beginning
of the outbreak and the widespread severity of the disease within this group
$\beta_{0}$ | Initial Contact Rate | [0.5-2.1, 95% CI] | Contact rate at the beginning of the outbreak
$\beta_{1}$ | Contact Rate 1 | [0.9-0.95, 95% CI]*$\beta_{0}$ | Contact rate after the mandate of case-based self isolation
$\beta_{2}$ | Contact Rate 2 | [0.9-0.95, 95% CI]*$\beta_{1}$ | Contact rate after government encouragement for social distancing
$\beta_{3}$ | Contact Rate 3 | [0.75-0.85, 95% CI]*$\beta_{2}$ | Contact rate after schools closure
$\beta_{4}$ | Contact Rate 4 | [0.40-0.60, 95% CI]*$\beta_{3}$ | Contact rate after lockdown order and banning of public events
$\beta_{5}$ | Contact Rate 5 | [1.1-1.9, 95% CI]*$\beta_{4}$ | Contact rate after recommendation for people to go back to work
$\beta_{v_{i}}$ | Vulnerable Contact Rate | $\eta*\beta_{i}$ | Contact rate of vulnerable individuals.

###  2.4 Model Parameterisation

Definitive values for the different parameters that define the spread of the
COVID-19 and the impact of the contention measures in place are still unknown,
given that the disease is still spreading globally and key information is
still unknown. For instance, the number of asymptomatic cases and consequently
the total number of individuals who are or have been infected by the disease
remains to be determined. Consequently, the case fatality rate of the disease
is also unknown. Motivated by this fact, first efforts were aimed at
estimating the value of the different model parameters through the comparison
of the estimated number of deaths and the real number of deaths as reported in
WorldometersUKDeaths .To do so, the incubation period ($\sigma$) and the
recovery rate ($\gamma$) estimated in previous work were taken into
consideration Linton2020 . With these figures, the best fit parameters were
identified by running a Monte Carlo simulation with 100.000 iterations,
allowing the rest of the parameters to take random values from a Gaussian
distribution within their respective expected intervals. The definitions of
the different model parameters can be found in Table 1.

####  2.4.1 Virus Transmissibility Study

Given the parameters provided in Table 1, the final mortality rate of COVID-19
was studied as a function of the percentage decrease in the contact rate of
vulnerable individuals ($\beta_{v}$). The mean values across the different
Monte Carlo simulations and the reduction in the number of deaths caused by
the disease estimated by the best fit model are provided.

##  3 Results

The results achieved by the analysis of the number of deaths caused by
COVID-19 as a function of the contact rate of vulnerable individuals
($\beta_{v}$) are presented in this section. Two different scenarios were
considered:

  1. 1.

The vulnerable group contact rate, $\beta_{v}$, is decreased from the
beginning of the outbreak. With this scenario, the potential reduction in the
number of deaths if more protective measures for vulnerable groups had been
applied from the beginning of the outbreak was studied.

  2. 2.

The vulnerable group contact rate, $\beta_{v}$, is decreased from now (2 June
2020). With this, the potential reduction in the number of deaths the disease
will cause from today was studied.

It should be noted that the Fear Factor ($\eta$) already takes into
consideration the widespread risk of the disease in vulnerable groups and the
recommendation made by the UK government for vulnerable individuals to stay at
home for at least 12 weeks at the beginning of the outbreak. The decreases in
$B_{v}$ are therefore additional to that caused by $\eta$. Initial simulations
using higher case fatality rates significantly overestimated the number of
deaths caused by the disease in the UK as compared to those reported in
WorldometersUKDeaths . It is thus believed the case fatality rate is
significantly lower than that reported in Ferguson2020 .

Table 2: Relationship between the decrease in $\beta_{v}$ and the resultant number of deaths avoided when this decrease is applied from the beginning of the outbreak expressed as the mean value of the Monte Carlo simulation Decrease in $\beta_{v}$ | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90%
---|---|---|---|---|---|---|---|---|---
Decrease in number of deaths | 7699 | 15512 | 23428 | 31434 | 39519 | 47671 | 55876 | 64122 | 72395

###  3.1 Reduction of the Contact Rate of Vulnerable Individuals from the
Beginning of the Outbreak

Figure 4: Reduction in the number of deaths as a function of the percentage
decrease in $\beta_{v}$, given that this reduction is applied at the beginning
of the outbreak.

The potential reduction in the number of deaths caused by COVID-19 as a
function of the percentage decrease applied to $\beta_{v}$ achieved by the
Monte Carlo simulation is shown in Figure 4. From this data, the reduction of
the exposure of vulnerable groups to the disease at the beginning of the
outbreak greatly decreases the number of deaths, The mean figures for the
prevented deaths for each 10% decrease in $\beta_{v}$ are shown in Table 2.

Similarly, the results obtained by the use of the best fit model across the
Monte Carlo simulation are shown in Figure 5. The performance of this model at
predicting the number of deaths caused by COVID-19 can be seen in Figure 6.
The prediction given by the best fit model for the total number of deaths
caused by COVID-19 in the UK at the end of the outbreak is 39825.

Figure 5: Reduction in the number of deaths as a function of the percentage
decrease in $\beta_{v}$ for the best fit model, given that this reduction is
applied at the beginning of the outbreak. Figure 6: Predicted number of deaths
using the best fit model vs the number of deaths reported (real).

####  3.1.1 Reduction of the Contact Rate of Vulnerable Individuals at Present

While this data from modelling reported the decrease in the number of deaths
if the exposure of vulnerable people to the virus had been reduced from the
beginning of the SARS-CoV-2 outbreak, this section presents the benefits this
reduction would bring if actions were taken from this point in time going
forwards.

The potential reduction in the number of deaths caused by COVID-19 as a
function of the percentage decrease applied to $\beta_{v}$ achieved by the
Monte Carlo simulation is shown in Figure 7. As it can be seen in the figure,
the reduction of the exposure of vulnerable groups to the disease from today
(2 June 2020) greatly decreases the number of further deaths, The mean figures
for the prevented deaths for each 10% decrease in $\beta_{v}$ are shown in
Table 3.

The results obtained by the use of the best fit model are shown in Figure 8.

Figure 7: Reduction in the number of deaths as a function of the percentage decrease in $\beta_{v}$, given that this reduction is applied from 2 June 2020. Table 3: Relationship between the decrease in $\beta_{v}$ and the resultant number of deaths avoided when this decrease is applied from today (2 June 2020) expressed as the mean value of the Monte Carlo simulation. Decrease in $\beta_{v}$ | 10% | 20% | 30% | 40% | 50% | 60% | 70% | 80% | 90%
---|---|---|---|---|---|---|---|---|---
Decrease in number of deaths | 3681 | 7406 | 11172 | 14975 | 18810 | 22673 | 26559 | 30464 | 34383
Figure 8: Reduction in the number of deaths as a function of the percentage
decrease in $\beta_{v}$ for the best fit model, given that this reduction is
applied from 2 June 2020.

##  4 Discussion

From the results of the modelling, it has been evidenced it is crucial to
consider the contrasting vulnerability in different groups of the population
regarding their age and overall health condition when developing plans to
return to normality. Motivated by this, three main points for consideration
are proposed, namely 1) the use of dedicated wristbands for the vulnerable to
enable social distancing and protect their well-being, 2) the use of Personal
Protective Equipment(PPE) and 3) the lockdown easing plan and the benefits
they can bring.

###  4.1 Protecting the Vulnerable People

Contact tracing, which has been used alongside other protection measures
across the world, can keep a record of any new infection cases and anyone who
has been close to them WHOtracing . This could enable uninfected and immune
people to leave their homes, while people who might have been infected
instructed to self-isolate. However, contract tracing is time consuming and
resource intensive and to be strictly accurate and valid requires 100%
adoption. Bluetooth technology has been used previously for messaging and
tracking of nearby devices using proximity detection Viralnet  kanjo12 . Apple
and Google along with many health authorities have proposed software
smartphone hosted apps using Bluetooth Low Energy (BLE) to automate the
contact tracing process Apple . BLE is a form of wireless communication
designed especially for short-range communication suitable for situations
where battery life is preferred over high data transfer speeds. Unfortunately,
vulnerable and older people are more likely to use older smartphones that
don’t come equipped with the BLE feature, e.g. about 9-12% of smartphones in
the UK lack the BLE functionality needed for it to work. In addition, data
from Ofcom shows that while around 80 per cent of all adults owned a smart
mobile phone in 2018, only 47% of 65-74 year old’s, and 26% of over 75s did
Ofcom . This is supported by data published by Statista, which indicates that
only 40% of over 65s used a smart phone to connect to the internet in 2019
statista . Furthermore, many people have privacy concerns about using their
smart phones as a tracer and they might not be willing to download a contact
tracing app. With this background, three possible solutions to support
vulnerable people are proposed 1) contact Tracing wristband or wearable,2)
social distancing alert mechanism and, 3) a wearable to monitor symptoms.

####  4.1.1 Wearables for Contact Tracing

The effectiveness of contact tracing hinges on how many people use it. It is
proposed that governments could provide vulnerable individuals with a BLE
wristband similar to the one in Figure 9), closing a data collection hole
created by systems relying on smartphones and an app only. Aiming to take a
population to 100

The wristband includes a proximity sensor powered by BLE. It also includes a
manual control to self-report and change a wearer’s status, recording states
like self-isolating, symptomatic and, tested negative or tested infected. When
a user updates their state to indicate an infection after testing, that
updates others they have been in close proximity with.

Figure 9: Wristband for Vulnerable People

Figure 10 shows how the wristband concept works. Contacts of an individual
wearing the wristband A are recorded on the wristband. A passer-by (e.g.
postman) with a smartphone and Contact Tracing app comes within read range
(longer range than close proximity) and downloads the records broadcasted by
the wristband A. In turns. The individual wearing the wristband A could be
alerted in real time using LED light (or sound) of close proximity events. We
believe that adopting wearable (e.g. wristbands) solutions is advantageous
over mobile apps for the following reasons:

  1. 1.

Mobile phones might not be always with users. Instead, they might be left at
home, in the car or at work, which means their social encounters don’t always
correspond to actual contact.

  2. 2.

A wristband solution will only have radio technology with a small memory and a
battery with no access to users data which could help to preserve privacy, as
it is low power it can always be on. It does not require setting up or
installation by the wearer.

  3. 3.

Phone apps require to be installed and activated by the users, Bluetooth need
to be switched on. These requirements make the apps unfeasible for vulnerable
people with difficulties in remembering instructions; and lack of digital
literacy, vision or motor control.

  4. 4.

Contact Tracing app can consume more energy as they are often kept active with
battery optimisation features disabled.

  5. 5.

Wearables are more likely to be work in front of the body (e.g. wristbands,
necklace or a keyfob), which could potentially improve the accuracy of the
proximity detection in the case of the face of face contact.

  6. 6.

Smart phones come with different operating systems and settings, which means
each model might require individual calibration and configuration.

Figure 10: A schematic of wearable-based COVID-19 proximity tracing

####  4.1.2 Digital tools to maintain Social distancing

By combining very dense contact tracing data from smartphone apps and
wristbands signals with information about infection status and symptoms,
vulnerable people can be protected and kept safe. Many countries have
introduced wristbands for different purposes. For instance, in South Korea,
people found to be violating lockdown rules can be ordered to wear a tracking
band, which alerts the police if people leave the house bbc_2020 . The
trackers were introduced after people started to leave their phones at home to
avoid detection. The devices also alert the authorities if people try to
remove it. Bulgaria has been testing Comarch LifeWristbands, developed in
Poland bbc_2020 . This system in addition to confirming a person is staying at
home, can monitor the wearer’s heart rate and then be used to call the
emergency services.

Contact Tracing apps and Internet of Things (IOT) such as key fobs, tags or
wrist bands can also be used to alert people (e.g. using vibration) if another
device comes within a specified distance. Figure 11 shows BLE keyfob and
KeepyourDistance app concepts being developed at Nottingham Trent University
(NTU) to maintain social distancing. The KeepyourDistance screenshots show the
signal strengths of nearby devices and it vibrates when other phones with the
same app are in close proximity.

Figure 11: A BLE mobile app and a keyfob concept developed at Nottingham Trent
University (NTU) to alert people when they are within 2m proximity.

####  4.1.3 Wearable to monitor symptoms

Dedicated health wristbands can be provided to vulnerable people with
underling health conditions to track their health including: temperature,
breathing and heart rate, and transmit it to their doctors. In addition to the
above functionalities, most commercially available smart-wristbands and smart-
watches incorporate an inertial measurement unit (IMU) composed of tri-axial
accelerometers and gyroscopes. The work in Kwok2015  reports a face touching
frequency of 23 times per hour with 44% of the face touches involving contact
with a mucous membrane. Wristbands should be programmed to incorporate an
alert mechanism to warn users whenever a potential movement of the hand
towards the face is detected. The accuracy achieved by relevant gesture
recognition work using inertial sensors Anderez2019 ; Anderez2020  suggests
that the development of such alerting features is feasible. These wearables
can then be re-purposed once the COVID-19 epidemic is over. For example they
can be given to older people and individuals with underling heath conditions
to monitor their health and well-being. Also they can be used to keep
vulnerable people connected with local volunteers and community services.

###  4.2 Disease Transmission and the Use of Personal Protective Equipment

As outlined by the WHO World2020transmission , COVID-19 is primarily
transmitted from person to person through small droplets exhaled by the mouth
or nose when an infected individual speaks, sneezes or coughs. Direct
transmission then occurs when such droplets travel onto the mucous membranes
of susceptible recipients, necessitating contact at close range (usually
within 1 meter) Leder2005 . In addition to this, exhaled droplets can also
come to rest on surfaces around the infected individual. According to the WHO,
the COVID-19 virus can survive up to 4 hours on copper, 24 hours on cardboard
and up to 72 hours on plastic and stainless steel surfaces. As a consequence,
these become a potential source of infection when touched by susceptible
individuals prior to touching their nose, mouth or eyes. Although exhaled
droplets are often heavy enough not to travel great distances, sinking quickly
to the ground or surrounding surfaces, viral particles in the form of bio-
aerosols can remain airborne for an extended period of time, particularly when
droplet diameters are either too small for gravitational deposition ($<$2
$\mu$m) or too large for diffusive deposition ($>$200 $\mu$m) Stahlhofen1989 .
Experimental results on SARS during the 2003 epidemic Booth2005  support these
statements, showing viral particles in the form of bio-aerosols were being
emitted by hospitalised patients. Likewise, recent work by Van2020
demonstrates aerosol COVID-19 remains viable and infectious with a half-life
on the order of 1 hour, thus confirming the plausibility of its transmission
via airborne particles.

Given the above, the use of personal protective equipment (PPE) can be a key
aspect to prevent the transmission of COVID-19. In this context, PPE includes,
among other equipment, the use of respiratory protection to safeguard the
mucous membranes and to prevent the transmission of viral droplets such as
masks and respirators, as well as that of physical barriers such as gloves,
goggles and face shields.

####  4.2.1 The Use of Masks and Respirators

According to the European standards EN149:2001 and EN14683, there are four
types of filtering masks, namely Filtering Face Piece 1 (FFP1), FFP2 111The
American and Chinese equivalents for FFP2 are N95 and KN95 respectively, FFP3
and surgical masks, with different models differing primarily in the
filtration efficiency given by their capability to filtrate inwards and
outwards particles. For the scope of this work, the use of surgical and FFP2
masks is discussed. In terms of inwards filtration, surgical masks are
designed to protect against droplets, sprays and any other particle with a
diameter greater than 100$\mu$m Smereka2020 . FFP2 masks, however, retain
$>$94% of the particles smaller than 0.5$\mu$m. As the work in Bake2019
indicated, exhaled particles can range from 0.01 to 1000 $\mu$m, with COVID-19
particles exhibiting a round or elliptic shape with diameters ranging from
0.06 to 0.14 $\mu$m Cascella2020 . As a consequence, surgical masks can
prevent the inhalation of COVID-19 particles when these are expelled in the
form of droplets with diameters greater than 100$\mu$m but not when expelled
in the form of small airborne particles. In contrast, FFP2 masks or
respirators prevent the inhalation of both droplet and airborne viral
particles. Despite the drawbacks encountered on surgical masks when it comes
to the prevention of the filtration of small airborne particles, it must be
noted that they can reduce the emission of viral particles into the
environment Leung2020 . It is worth noting that mask effectiveness decreases
with increasing concentrations of water vapour and carbon dioxide between the
face and the mask/respirator caused by each subsequent exhalation Smereka2020
. Thus, masks should be replaced frequently. I should also be noted face masks
have to fitted correctly and form a seal peripherally to stop air passing
around the mask and not through it.

It should also be stated, the incorrect use of PPE, such as not changing
disposable masks or gloves, can have a counterproductive effect, thus
jeopardising their protective effect and even increasing the risk of infection
Feng2020 . Given this, health organisations and government bodies should be
spreading good clear information covering how to wear and discard the
different recommended protective equipment components properly.

####  4.2.2 Other Personal Protective Equipment

As mentioned above, although COVID-19 is mainly transmitted through direct
contact between an infected and a susceptible individual, indirect
transmission is also plausible. In addition to masks and respirators, the use
of further PPE can help to reduce the risk of the transmission of COVID-19
both directly and indirectly. In this context, face shields and eye protection
equipment such as goggles can play an important role in preventing both direct
and indirect transmission.

A face shield is a PPE component which provides a physical protective barrier
to the facial area and related mucous membranes Roberge2016 . Various
experimental works Lindsley2014 ; Shoham  have shown the potential
effectiveness of face shields against the transmission of viral respiratory
diseases like COVID-19. For instance, the work in Lindsley2014  employed a
cough aerosol simulator filled up with influenza virus (aerosol mean diameter
of 8.5$\mu$m) alongside a breathing simulator to test the effectiveness of a
face shield against the transmission of the virus. The results reported
outline risk reductions of 96% and 92% on the inhalational exposure right
after a cough at distances of 46cm and 183cm respectively. Reducing the
aerosol diameter to 3.4$\mu$m resulted in a reduction of the blocking
effectiveness to 68% at 46 cm right after a cough and to a 23% over a post-
cough period of 1 to 30 minutes. The experimental work in Shoham  used a
fluorescent dye (particle diameter $\approx 5\mu m$, distance $\approx 50cm$)
to evaluate the effectiveness of a face shield at protecting eye contamination
from aerosol particles. The results outlined the use of a full face shield
(FFS) completely prevented eye contamination. Additionally, it was
demonstrated that the combination of the FFS with a N95 respirator offered
full protection of the eyes, nose and mouth from contamination. In contrast,
the use of safety glasses with either a surgical mask or N95 respirator
resulted in some eye contamination.

According to WHO recommendations World2020PPE , gloves should be worn by
medical personnel, laboratory technicians manipulating respiratory samples,
caregivers who provide direct care to confirmed COVID-19 patients and visitors
entering a COVID-19 patient’s room. Therefore, the WHO does not consider the
general use of disposable gloves by the community necessary. Instead, a
thorough hand hygiene is proposed. Besides hand hygiene, a consistent
disinfection of commonly touched surfaces is recommended when an individual is
quarantined after contracting the disease to prevent its spread to susceptible
individuals who may co-handle those surfaces. In line with the WHO, it should
not be recommended that the general public use disposable gloves, as this is
unlikely to reduce the risk of infection and it may lead to a potential
unnecessary shortage for scenarios in which they are indispensable and
absolutely necessary.

####  4.2.3 Final Recommendations on the Use of PPE

Provided the analysis upon the effectiveness of the different PPE components
on the prevention of the spread and on the contraction of COVID-19 carried out
in Sections 4.2.1 4.2.2, the following recommendations are made:

  1. 1.

Masks should be worn by every individual in the population. Non-vulnerable
individuals should use surgical masks. FFP2 masks or respirators should be
worn by vulnerable individuals, especially in closed environment public spaces
such as supermarkets, pharmacies and public transport.

  2. 2.

Face shields are recommended for vulnerable individuals, especially in closed
spaces. Ideally, face shields should be worn in conjunction with an FFP2 mask
or respirator.

  3. 3.

Gloves are not recommended for the general public.

###  4.3 Adoption and Incentive Mechanisms for Behavioral Change

One of the challenges around implementing the proposed solutions is getting
people to adopt and use them. In analogous settings, when young drivers were
encouraged to use tracing technology solutions to improve their driving
habits; young drivers, who participated in the technology trials, only used
behaviour monitoring apps while the incentives lasted and stopped when the
incentives stopped. Moreover, drivers often tried to play and work the system
in order to obtain extra incentive points musicant2016can . While the need to
motivate individual adoption is universal, a segmented reaction to the
proposed solutions and best practise is anticipated, so different strategies
and incentive mechanisms might be required for different groups. Successful
adoption depends on individual motivation, incentives provided through a
variety of strategies. The incentive mechanisms can range from providing
information that is meant to resonate with basic values, such as “using this
technology will save lives” to material and non-material incentives in the
form of (i) paying people for their personal data (e.g., providing tax-
rebates, etc.) or (ii) providing priority access to some services (e.g.,
medical services, etc.). For example, in the past, people successfully adopted
healthy behaviours when they received financial incentives marteau2009using .
From this, it follows that if incentives are offered people (a) adopt a
certain behavior or (b) engage in a behavioral change. Rightly or wrongly
these incentives may be more effective if they appeal to people’s ”present
bias”, which is the tendency to pursue smaller and more immediate rewards
(e.g. getting a small amount of money every day) rather than bigger and more
abstract goals, such ”eradicating COVID-19 in the world”. One potential
incentivization mechanism could be offering a better value proposition to the
user that goes beyond tracing. Considering that the vulnerable population
generally tend to procure care services more often shivayogi2013vulnerable ,
the function of contact tracing is likely to succeed if it is embedded into
the general care applications and services. An example of such general care
services would be framing the contact tracing technology as a “digital nurse”
for the vulnerable groups, that aims to monitor (in real time) the state of
the wearer and potentially offer some desirable care features. Features such
as health monitoring or being able to make automated calls for medical help in
case the wearable detects signs of distress, etc. vegesna2017remote . Having a
clear value proposition, which would go beyond the functional purpose of
tracing would allow the technological solutions to succeed with the elderly
population roupa2010use  more easily, as users will see not only how their
data can benefit society, but also how their data can help them receive
better, more efficient, and higher quality care. In such situations, due to
the behavioural ”privacy paradox” norberg2007privacy , users, who are
concerned about privacy, are also likely to perceive the added value of the
technological solutions as a trade-off between their personal data and
desirable services. If the benefits of using the technology outweighs the cost
(in terms of personal data loss), users will be much more likely to adopt the
new wearables, even if that requires disclosing their location
xu2011personalization . This, in turn, will significantly increase the chances
of technology adoption as well as ensure policy success. Considering that
people tend to attribute greater value and use products that have a cost
associated with them (e.g., people see greater value in goods and services
they need to pay for rather than free goods and services), such value
proposition (i.e., offering medical or care service benefits in exchange for
personal data) is likely to be self-sustainable. An alternative incentive
mechanism can be implemented through material incentives by offering to buy
personal data from the users (i.e., the users will receive financial
remuneration for their personal data) schwartz2003property . This mechanism
will leverage on participants’ willingness to trade privacy in the context of
willingness to accept (WTA) money in exchange for data. The mechanism could
work through offering a certain amount of money or an equivalent in discounts
for users’ personal data in a clear and transparent transaction
bizon2016willingness . The key to success in this case is full transparency
about (i) who is the data buyer (government or third-party organizations);
(ii) how the data buyer will use the data; and (iii) how the data buyer will
protect user data. The third strategy to push for adherence to the use of the
proposed solutions, is through legal means. For example, contact tracing could
be mandated by law and distributed to the eligible population. The use of the
technology can then be monitored and ”fines” for non-compliance could be
introduced. The ”fines” do not necessarily have to be monetary. They can be
implemented as decreased benefits. The success of such this approach greatly
depends on cultural values and social norms (in some societies punishment may
work better than reward in policy implementations) milberg1995values . The
three suggested approaches are summarized on Figure 12.

Figure 12: Adoption Approaches

###  4.4 Lockdown Easing Plan

Once past the peak of the epidemic, that is, the exponential phase of the
curve, it is then time to develop plans for a gradual easing of the imposed
restrictions. Several research works Kermack1927 ; Aron1984 ; Ferretti2020 ,
including the work presented here, have proposed a variety of epidemiological
models to study the spread of the disease and to make the consequent
estimations of the total number COVID-19 cases in different territories.
Having done that, it is possible to have estimations on the herd immunity
gained throughout the progression of the disease. However, there are still
many unknowns of the disease which may potentially compromise the estimations
made by the application of such epidemiological models. Given this, government
bodies should be pressed to carry out prevalence studies to further support
the estimations obtained by the use of epidemiological models, as it is
crucial for the minimisation of further infection cases and consequently for
the reduction of the number of further deaths caused by the disease, to have
accurate numbers for the number of people who have already had and recovered
from the disease and have, thus produced antibodies against it.

###  4.5 Conclusions and Future Work

The key driver for this work has been the large size of the vulnerable
population and their higher risk of severe infection and death. This group is
also prone to suffer from loneliness resulting from the prolonged period of
lockdown. Governments need to balance the need to socially isolate vulnerable
people and shield them while also taking into account their mental health and
well-being which might be severely affected by the isolation measures.

A modified SEIR model, namely SEIR-v, through which the population was
separated into two groups regarding their vulnerability to the disease was
proposed to provide a means of studying the spread and the case fatality rate
of COVID-19 when different contention measures are applied to different groups
regarding that vulnerability. Using SEIR-v, the impact of a reduction in the
exposure of vulnerable individuals to COVID-19 on the number of fatalities
caused by the disease was analysed. The results indicate an average of 3681
deaths can still be saved by only reducing by 10% the exposure of vulnerable
groups to the disease from now. In line with this and also considering the
negative consequences caused by the application of strict isolation measures
on people’s mental health, a set of recommendations including the adoption of
digital tools and protective equipment was proposed. Future work will be
directed towards the analysis of the lockdown easing steps taken by the U.K.
government and their potential impact on further fatalities caused by
COVID-19.

## Conflict of interest

On behalf of all authors, the corresponding author states that there is no
conflict of interest.

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# Multiple measurements on the cosmic curvature using Gaussian process
regression without calibration and a cosmological model

Xiaolong Gong1, Yifei Xu1, Tonghua Liu1111Corresponding author:
<EMAIL_ADDRESS>
Shuo Cao2, 3222Corresponding author<EMAIL_ADDRESS>Jianyong Jiang2,
Yalong Nan2, 3, Ruobin Ding2, 3 1\. School of Physics and Optoelectronic
Engineering, Yangtze University, Jingzhou 434023, China;
2\. School of Physics and Astronomy, Beijing Normal University, Beijing
100875, China;
3\. Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal
University, Beijing 102206, China.

###### Abstract

In this letter, we propose an improved cosmological model-independent method
to measure cosmic curvature, combining the recent measurements of transverse
and line-of-sight directions in the baryon acoustic oscillations (BAO) with
cosmic chronometers (CC) datasets. Considering that the CC dataset is discrete
and includes only 32 $H(z)$ measurements, we apply Gaussian process (GP)
regression to fit the CC dataset and reconstruct them. Our methodology, which
does not need the calibration or selection of any cosmological model, provide
multiple measurements of spatial curvature ($\Omega_{K}$) at different
redshifts (depending on the redshift coverage of BAO dataset). For combination
of all BAO data, we find that the constraint result on cosmic curvature is
$\Omega_{K}=-0.096^{+0.190}_{-0.195}$ with $1\sigma$ observational
uncertainty. Although the measured $\Omega_{K}$ is in good agreement with zero
cosmic curvature within 1$\sigma$ confidence level, our result revels the fact
of a closed universe. More importantly, our results show that the obtained
$\Omega_{K}$ measurements are almost unaffected by different priors of the
Hubble constant. This could help solve the issue of the Hubble tension that
may be caused by inconsistencies in the spatial curvature between the early
and late universes.

##  I Introduction

The cosmic curvature, $\Omega_{K}$, is arguably the most fundamental and
crucial parameter in our universe which affects the evolution of our universe
and the dark energy properties. Any deviation from a flat universe would bring
far-reaching thoughts and new understandings in the inflation theory and
fundamental physics. Recently, some works suggested that spatial curvature of
the universe is still an open issue (Handley, 2021; Di Valentino et al.,
2021). More specifically, the combination of Planck lensing and low redshift
baryon acoustic oscillations (BAO) datasets revealed a flat universe with
$\Omega_{K}=0.0007\pm 0.0019$ Planck Collaboration et al. (2020). However, the
work (Di Valentino et al., 2020) claimed that a close universe
($\Omega_{K}=-0.044^{+0.018}_{-0.015}$) was supported by the pure Planck
Cosmic Microwave Background Radiation (CMBR) temperature and polarization
power spectra data. It should be emphasized that both methods depend, to some
extent, on a particular cosmological model assumed, i.e. the non-flat
$\Lambda$ cold dark matter ($\Lambda$CDM) model. This tension, like Hubble
constant (more discussions on Hubble tension please see the references Poulin
et al. (2019); Feeney et al. (2019); Dainotti et al. (2021); Liu et al.
(2022); Liao et al. (2020) ), their results are based on $\Lambda$CDM model.
Thus, cosmological model-independent curvature measurements can also test the
$\Lambda$CDM model and are worth developing. Similarly, any curvature tension
related to the CMB could be another evidence of $\Lambda$CDM model violation.

All these crises may be caused by measurement inconsistencies between the
early universe and the late universe, and many works (Qi et al., 2023;
Vagnozzi et al., 2021; Zuckerman & Anchordoqui, 2022) point to the fact that
relying on $\Lambda$CDM model will lead to such discrepancies. To solve the
crisis posed by the measurement of the curvature of the universe, a necessary
approach is to confirm the value of $\Omega_{K}$ at late cosmic times using a
method that is independence of the cosmological model. A non-exhaustive of
references in this direction can be found in (Collett et al., 2019; Wei &
Melia, 2020; Qi et al., 2021; Liu et al., 2022). A typical curvature
measurement idea is to obtain constraints on the geometrical quantity
$\Omega_{K}$ using recent observational data provided by the high precision
cosmological probes, namely, the type Ia supernovae (SNe Ia) distance modulus
data, the cosmic chronometer and the radial BAO measurements of the Hubble
parameter (Mukherjee & Banerjee, 2022). However, it should be noted that the
BAO data are not completely cosmological model-independent, because one need
to assume a prior on the radius of the sound horizon $r_{s}$, which is
inferred from the CMB observations, i.e., the prior of $r_{s}$ obtained from
the CMB analysis is used to calibrate the BAO distance. Secondly, previous
work has generally used non-parametric Gaussian Processes (GP) method to
reconstruct the expansion history of the universe. However, as mentioned in
many works Mertens et al. (2018); Mukherjee et al. (2022), one of the most
recently debated topics for non-parametric reconstruction in cosmology with GP
is that this technique is exposed to several foundational issues such as
overfitting and kernel consistency problems Colgáin & Sheikh-Jabbari (2021).

Inspired by the above, we propose an improved cosmological model-independent
method to measure cosmic curvature combining the observations of BAO in the
line-of-sight and transverse directions with CC datasets by using GP
regression. The main advantage of this work is that it does not require the
use of a priori values for the radius of the sound horizon $r_{s}$. We can
perform measurements of cosmological model-independent curvature at multiple
redshifts (depend on the BAO dataset). Moreover, the GP regression is powerful
tool for reconstruction of function since the regression occurs in an
infinite-dimensional function space without over-fitting problem (Joudaki et
al., 2018). This paper is organized as follows: in Section 2 we briefly
introduce the methodology and observational data used in our work. In Section
3, we show our results and discussion in Section 4.

##  II Methodology and observational data

###  II.1 Methodology for measuring on cosmic curvature

The foundation of modern cosmology is based on the basic principles of
cosmology, i.e., our universe is homogeneous and isotropic at large scales.
The space-time of our universe can be described by Friedmann-Lematre-
Robertson-Walker (FLRW) metric

$ds^{2}=dt^{2}-\frac{a(t)^{2}}{1-Kr^{2}}dr^{2}-a(t)^{2}r^{2}d\Omega^{2},$ (1)

where $a(t)$ is the scale factor, and $K$ is dimensionless curvature taking
one of three values $\\{-1,0,1\\}$ corresponding to a close, flat and open
universe, respectively. The cosmic curvature parameter $\rm\Omega_{K}$ is
related to $K$ and the Hubble constant $H_{0}$, as $\rm{\Omega_{K}}$
$=-c^{2}K/a^{2}_{0}H_{0}^{2}$. The comoving distance $d_{C}$ is related to the
evolution of the scale factor

$D_{C}=c\int^{z}_{0}\frac{dz^{\prime}}{H(z^{\prime})},$ (2)

where the redshift $z$ replaces the scale factor via relation $a=1/(1+z)$, the
$c$ represents the speed of light, and the $H(z^{\prime})$ denotes the Hubble
parameter at redshift $z^{\prime}$. However, we cannot actually obtain the
comoving distance from observations. In fact, the cosmological distances we
observe are luminosity or angular diameter distances. Since the luminosity
distance is not used in this work, its relation to the comoving distance is
not given. The comoving distance $D_{C}$ and the angular diameter distance
$D_{A}$ satisfy the following relation Weinberg (1972)

${D_{A}}(z)=\left\\{\begin{array}[]{lll}\frac{c}{(1+z)H_{0}\sqrt{|\rm\Omega_{\rm
K}|}}\sinh\left[\sqrt{|\rm\Omega_{\rm K}|}{H_{0}D_{C}/c}\right]~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}>0,\\\
D_{C}/(1+z)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}=0,\\\ \frac{c}{(1+z)H_{0}\sqrt{|\rm\Omega_{\rm
K}|}}\sin\left[\sqrt{|\rm\Omega_{\rm K}|}H_{0}D_{C}/c\right]~{}~{}~{}~{}{\rm
for}~{}~{}\rm\Omega_{K}<0.\\\ \end{array}\right.$ (3)

As can be seen from the Eq. (3), if we want to achieve the measurement of
curvature in a cosmological model independent way, one need to obtain the
comoving distance and the angular diameter distance at the same redshift. We
will seek for cosmic chronometer (CC) and BAO observations to obtain the
comoving distances and the angular diameter distances, respectively.

###  II.2 Reconstructed comoving distance from cosmic chronometer observations

Cosmic chronometers (CC) as the standard clocks provide the measurements of
Hubble parameter, which derive from the combination of differential age
estimates of systems with passively evolving star populations (e.g. globular
clusters) and their corresponding spectral redshifts Jimenez & Loeb (2002),

$H(z)=-\frac{1}{1+z}\frac{\Delta z}{\Delta t}.$ (4)

The CC data were obtained by measuring red envelope galaxies of different
ages, i.e. different age method. The aging of stars can be seen as an
indicator of the aging of the universe. The spectra of stars can be converted
into information about their age, because the evolution of stars is well
known. Since stars cannot be observed individually on a cosmic scale, it is
common to use relatively homogeneous galaxy spectra of stellar populations.
This approach relies on the detailed shape of the galaxy spectra rather than
the luminosity of the galaxies. Therefore, CC data do not need to be
calibrated and do not depend on underlying cosmological models Avgoustidis et
al. (2010); Liao et al. (2015). The full information for total number of 32 CC
is listed in Table 1 of the literature Jimenez & Loeb (2002). The 32 CC data
covers the redshift range of $0.07\leq z\leq 1.965$ Zhang et al. (2014); Stern
et al. (2010); Moresco et al. (2012, 2016); Ratsimbazafy et al. (2017);
Moresco (2015). The scatter diagram for 32 CC data is shown in the Fig. 1. At
each redshift, the providing us with a continuous set of probability
distribution functions (PDFs) at any given redshift.

To work with the redshift points corresponding to the BAO dataset, and we
apply the Gaussian process regression to fit to CC dataset in order to
reconstruct them at the chosen BAO redshifts. For measuring curvature, we
generate samples of the $H(z)$ from the posterior of the CC dataset. In order
to perform the posterior sampling in a cosmological model-independent way, the
GP regression (Holsclaw et al., 2010, 2010b; Joudaki et al., 2018; Shafieloo
et al., 2012, 2013; Keeley et al., 2019) is considered here by using the code
GPHist333https://github.com/dkirkby/gphist (Kirkby & Keeley, 2017). The GP
regression works by generating a large sample of functions $\gamma(z)$
determined by the covariance function. The covariance between these functions
can be described by a kernel function. We adopt a squared-exponential kernel
to parameterize the covariance

$\langle\gamma(z_{1})\gamma(z_{2})\rangle=\sigma_{f}^{2}\,\exp\\{-[s(z_{1})-s(z_{2})]^{2}/(2\ell^{2})\\},$
(5)

with the hyperparameters $\sigma_{f}$ and $\ell$ that are marginalized over.
The $\gamma(z)$ is a random function inferred from the distribution defined by
the covariance, and we adopt $\gamma(z)=\ln([H^{\rm
fid}(z)/H_{0}]/[H(z)/H_{0}])$ to generate expansion histories $[H(z)]^{\rm
CC}/H_{0}$ by using CC dataset. Here $H^{\rm fid}(z)/H_{0}$ is chosen to be
the best fit $\Lambda$CDM model for the CC dataset and serves the role of the
mean function for GP regression. Since the final reconstruction result is not
completely independent of the mean function, it has some influence on the
final reconstruction result because the value of the hyperparameter helps to
track the deviation from the mean function. The true model should be very
close to flat $\rm{\Lambda}CDM$, hence, our choice for the mean function is
very reasonable (Shafieloo et al., 2012, 2013; Aghamousa et al., 2017). More
detailed description of the hyperparameter, kernel and and mean functions and
the GP regression method can be found in the work Liao et al. (2019); Liu &
Liao (2023); Li et al. (2023, 2021). It should be noted that we adopt a prior
value on $H_{0}=74.03$ $km/s/Mpc$ from the Supernova $H_{0}$ for the Equation
of State (SH0ES) collaboration Riess et al. (2019). In fact, the a priori
value of $H_{0}$ does not have much influence on the measurement of the
curvature of the universe. We will discuss this point further later. The final
reconstructed 1000 curves (or realizations) $[H(z)]^{\rm CC}$ from the CC
dataset are shown in the left panel of the Fig. 1. By using a trapezoidal
integral method, we obtain the 1000 realizations for the comoving distances
$[D_{C}(z)]^{\rm CC}$, and show in the right panel of the Fig. 1.

Figure 1: Left panel: The reconstructed 1000 realizations $H(z)$ from the CC
dataset . Right panel: Based on reconstructed 1000 realizations of
$[H(z)]^{\rm CC}$, the 1000 realizations for comoving distances
$[D_{C}(z)]^{\rm CC}$ obtained with the trapezoidal integral method.

###  II.3 BAO observation

For angular diameter distance $D_{A}$, we seek for seven BAO data points from
the last data release of the BOSS and eBOSS collaboration Alam et al. (2017);
Bautista et al. (2021); Gil-Marín et al. (2020); Tamone et al. (2020); de
Mattia et al. (2021); Neveux et al. (2020); Hou et al. (2021); de Sainte
Agathe et al. (2019); Blomqvist et al. (2019). The observations of BAO provide
line-of-sight and transverse measurements, but these entangle with the radius
of the sound horizon $r_{s}$, i.e., $D_{M}/r_{s}$ and $D_{H}/r_{s}$, where the
$D_{M}=(1+z)D_{A}$ and $D_{H}=cH(z)^{-1}$ is transverse and line-of-sight
directions measurement in BAO observations. To obtain the angular diameter
distance, it is common procedure to use the prior value of radius of the sound
horizon $r_{s}$ obtained from Planck CMB observations. However, it is to some
extent cosmological model-dependent, i.e., partly dependent on the assumptions
of cosmological models. Therefore, we use the BAO observations in the
transverse and line-of-sight directions, combined with observations from the
$[H(z)]^{\rm CC}$ provided by the CC dataset, to obtain angular diameter
distances independent of the cosmological model.

$[D_{A}(z)]^{\rm BAO+CC}=\frac{c}{(1+z)\,[H(z)]^{\rm
CC}}\cdot\frac{D_{M}}{r_{s}}\cdot\frac{r_{s}}{D_{H}}.$ (6)

For the BAO data, we do not perform any fit and use their seven redshift
points as reference points. Further, by combining the observed data from BAO
given in Table 1 with the GP of reconstructing 1000 $[H(z)]^{\rm CC}$ from the
CC dataset, we obtain the five angular diameter distances at the chosen
redshifts. It should be noted that the redshift of CC dataset not full covers
range of all BAO redshifts, considering the uncertainty of redshift
extrapolation, we therefore rejected two BAO data point, i.e., $z=2.34$ and
$z=2.35$. For the observational uncertainties of the BAO data, we assuming
they follow gaussian PDFs (based on the symmetry of their 68% confidence
intervals), to obtain PDFs of $[D_{A}(z)]^{\rm BAO+CC}$ at the five redshifts.

Data source  |  $z$ |  $D_{M}/r_{s}$ |  $D_{H}/r_{s}$ | Reference
---|---|---|---|---
BOSS galaxy–galaxy  |  $0.38$ |  $10.27\pm 0.15$ |  $24.89\pm 0.58$ | Alam et al. (2017)
eBOSS galaxy–galaxy  |  $0.51$ |  $13.38\pm 0.18$ |  $22.43\pm 0.48$ | Alam et al. (2017)
|  $0.70$ |  $17.65\pm 0.30$ |  $19.78\pm 0.46$ | Bautista et al. (2021); Gil-Marín et al. (2020)
|  $0.85$ |  $19.50\pm 1.00$ |  $19.60\pm 2.10$ | Tamone et al. (2020); de Mattia et al. (2021)
|  $1.48$ |  $30.21\pm 0.79$ |  $13.23\pm 0.47$ | Neveux et al. (2020); Hou et al. (2021)
eBOSS Ly-$\alpha$–Ly-$\alpha$ |  $2.34$ |  $37.41\pm 1.86$ |  $8.86\pm 0.29$ | de Sainte Agathe et al. (2019)
eBOSS Ly-$\alpha$–quasar  |  $2.35$ |  $36.30\pm 1.80$ |  $8.20\pm 0.36$ | Blomqvist et al. (2019)
Table 1: The BAO data used in this work given in terms of $D_{M}=(1+z)D_{A}$
and $D_{H}=c\,H(z)^{-1}$.

##  III Results and discussions

The steps for constraining $\Omega_{K}$ are summarized as follows:

  1. 1.

We draw 1000 reconstructed realizations $[H(z)]^{\rm CC}$ from CC dataset, and
convert to comoving distances $[D_{C}(z)]^{\rm CC}$ by using the trapezoidal
integral method at the five BAO redshifts based on Eq. (2), respectively;

  2. 2.

Based on Eq. (6), we calculate 1000 values of $[D_{A}(z)]^{\rm BAO+CC}$ at the
five BAO redshifts by combing the 1000 $[H(z)]^{\rm CC}$ realizations and the
five BAO observations, respectively;

  3. 3.

We combine the 1000 realizations of the $[D_{C}(z)]^{\rm CC}$ with the
realizations of $[D_{A}(z)]^{\rm BAO+CC}$ to get the estimations of the PDFs
of $\Omega_{K}$ at the five BAO redshifts based on the Eq. (3), respectively;

  4. 4.

With the above procedure, the uncertainty on $[D_{C}(z)]^{\rm CC}$ and
$[D_{A}(z)]^{\rm BAO+CC}$ , as well as their mutual correlation are
intrinsically included in the respective PDFs. Then, we use these samples to
reconstruct the multi-variate distribution of the five $\Omega_{K}$
measurements.

  5. 5.

Finally, we extract the marginal distribution of the individual $\Omega_{K}$
thus leaving us with five uncorrelated PDFs.

Figure 2: Left panel: The individual measurement on $\Omega_{K}$ using the
$H_{0}=74.03$ $km/s/Mpc$ prior reported by the SH0ES collaboration. Right
panel: The individual measurement on $\Omega_{K}$ using the $H_{0}=67.4$
$km/s/Mpc$ prior reported by the Planck CMB observations.

The final step is similar to the process used in Markov Chain Monte Carlo
(MCMC) to derive constraints on the likelihood parameter and to allow for a
direct conversion of the correlation and uncertainty of the PDFs of the
distances into the PDFs of $\Omega_{K}$. The whole method is similar to a
combination of MCMC-like parameter estimation and Gaussian process
reconstruction, which is called as Gaussian Process Monte Carlo (GPMC). The
GPMC method was first proposed by the work Renzi & Silvestri (2023).

In order to perform the above procedure, we have to take a prior value for the
Hubble constant. We first adopt a prior value on $H_{0}=74.03$ $km/s/Mpc$
reported by the SH0ES collaboration. The individual $\Omega_{K}$ PDFs and the
final results obtained combining them are reported in the left panel of the
Fig. 2. We note that our results for $\Omega_{K}$ are individual measurements,
which means that we achieve measurements of $\Omega_{K}$ at different
redshifts. Although the final result obtained combining five individual
$\Omega_{K}$ PDFs reports that our universe has a flat spatial curvature with
the $\Omega_{K}=-0.096^{+0.190}_{-0.195}$ at $1\sigma$ confidence level, the
precision of our constrained results on $\Omega_{K}$ is not particularly high.
It is reasonable to get such precision that we used only five points of BAO
data. Furthermore, we obtain a negative value for the centre of curvature,
which is consistent with recently reported work, i.e., combining the Planck
temperature and polarization power spectra data, the work showed that a closed
universe ($\rm\Omega_{K}=-0.044^{+0.018}_{-0.015}$) was supported. Our work
also reinforces that the universe is likely to be a closed universe.

Some recent works suggested that the $H_{0}$ tension could possibly be caused
by the inconsistency of spatial curvature between the early-universe and late-
universe Di Valentino et al. (2020); Handley (2021); Di Valentino et al.
(2021). An effective way to study this problem is to simultaneously constrain
the curvature of the universe and the Hubble constant, and to use a
cosmological model-independent method (Collett et al., 2019; Wei & Melia,
2020; Qi et al., 2021; Liu et al., 2022). This is due to the strong degeneracy
between the cosmic curvature and the Hubble constant. Inspired by these, we
further adopt a prior value on $H_{0}=67.4$ $km/s/Mpc$ reported by the Planck
CMB observations. The final results of the PDFs of $\Omega_{K}$ are shown in
the right panel of the Fig. 2. The result is
$\Omega_{K}=-0.101^{+0.207}_{-0.212}$ at $1\sigma$ confidence level by
combining five individual $\Omega_{K}$ PDFs. Compared with the results using
the SH0ES prior, our results show that the obtained values of curvature are
almost unaffected with the different prior of Hubble constant by using our
method. This conclusion can also be seen from the Eq. 2, the combination of
$[D_{A}(z)]^{BAO+CC}$ requires the $[H(z)]^{CC}$ component. If we write $H(z)$
in the form of the $H_{0}*E(z)$, where the $E(z)$ is dimensionless Hubble
parameter, we can see that the Hubble constant on the left and right sides of
the equation is cancelled out. Irrespective of the prior of $H_{0}$ used, our
results demonstrate the validity and robustness of the method of measuring
$\Omega_{K}$ in this work.

As a final remark, we also look forward to a large amount of future data, not
only from the observations of BAO, but also from the cosmic chronometer,
allowing us to further improve the precision of $\Omega_{K}$ measurements.
Future surveys, such as Dark Energy Spectroscopic Instrument (DESI), will
measure $H(z)r_{s}$ at several redshifts and be able to bring down those
errors, thus to improve the constrain on the cosmic curvature in a wider range
and with much higher precision and accuracy without cosmological model
assumptions by using our method. Meanwhile, considering the existence of many
data reconstruction methods, such as machine learning methods, the rapid
development and technical improvement of these methods makes us optimistic
about measuring curvature with higher precision and accuracy in the future.

##  IV Acknowledgments

Liu. T.-H was supported by National Natural Science Foundation of China under
Grant No. 12203009; Chutian Scholars Program in Hubei Province (X2023007). Cao
S. was supported by the National Natural Science Foundation of China under
Grants No. 12021003, 11633001, and 11920101003; the Strategic Priority
Research Program of the Chinese Academy of Sciences, Grant No. XDB23000000;
and the Interdiscipline Research Funds of Beijing Normal University.

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