# A mathematical model of Breast cancer (ER+) with excess estrogen: Mixed
treatments using Ketogenic diet, endocrine therapy and Immunotherapy

Hassnaa AKIL  Laboratory LIPIM
ENSA Khouribga
University of sultan moulay slimane
Khouribga, Morocco
<EMAIL_ADDRESS>Nadia IDRISSI FATMI  Laboratory LIPIM
ENSA Khouribga
University of sultan moulay slimane
Khouribga, Morocco
<EMAIL_ADDRESS>
###### Abstract

Breast Cancer is a major public health problem and the most common diagnosed
malignancy in woman. There have been significant developments in clinical
approaches and theoretical experimental to understand the interactions of
cancer cells dynamics with the immune system, also developments on analytical
and computational models to help provide insights into clinical observations
for a better understanding of cancer cells, but more are needed, especially at
the genetic and molecular levels mathematically. Treatments such as
immunotherapy, chemotherapy, hormone therapy, radiotherapy, and gene therapy
are the main strategies in the fight against breast cancer. The present study
aims at investigating the effects of estrogen derived from recent models, but
this time combined with immunotherapy as a way to treat or inhibit the cancer
growth by a mathematical model of breast cancer in situ, governed by a
simplified model of nonlinear-coupled ordinary differential equations, that
combines important interactions between natural cells, tumor cells, immune
cells, ketogenic diet in the presence of an anticancer drug. Another
contribution was to introduce the inhibition effect $\epsilon$ for new results
and conclusions, A qualitative study was performed and biological
interpretations were included to understand the conditions of stability in a
realistic way.

###### Index Terms:

Mathematical modelling, ODE’s, Dynamical systems, Breast cancer, ER-Positive,
Ketogenic diet, Estrogen, Immunotherapy.

##  I Introduction

Female breast cancer was the most common obstructive diseases in women
worldwide in 2020 according to the Global Cancer Statistics. In 2020, as
reported in CA: A Cancer Journal for Clinicians, an estimated 19.3 million new
cancer cases will be diagnosed, with about 10 million people dying from
cancer-related causes. Overall, the data revealed that one out of every five
men and women in the world will develop cancer over their lifetime. In the
future, the analysis estimates that 28.4 million additional cancer cases will
be diagnosed globally in 2040, up 47% from 2020[1]. The most commonly
diagnosed cancer kind in women is breast cancer, with a projected 1.8 million
deaths. mainly, rates are high in North America, Australia, New Zealand, and
northern and western Europe; lowest in much of Africa and Asia (Torre et
al.[2]). It is also occurring in North Africa, although its incidence is lower
than in Western countries (about 1.5 to 2 times lower in women under 50 and 3
to 4 times lower in women over 50) change.

According to The World Health Organization (WHO), in 2020, 2.3 million women
worldwide were diagnosed with breast cancer and 685,000 died. By the end of
2020, 7.8 million women had been diagnosed with breast cancer in the past 5
years, making it the most common cancer in the world and the second-largest
cancer after lung cancer. Globally, women lose more disability-adjusted life
years to breast cancer than any other cancer .

In Morocco there are approximately 50,000 new cancer cases each year. Cancer
also accounts for 13.4% of deaths in the country. Accounting for 20% of cases.
Breast cancer was the most common disease among Moroccan women in 2016,
according to a research issued by the Greater Casablanca Cancer Registry of
Morocco, accounting for 35.8% of all new malignancies among women [4].

Breast cancer is a disease that affects breast cells and causes unhindered
division of mitosis, which can be malignant in breast tissue, it occurs when
the cells in the lobules or the ducts become abnormal and divide
uncontrollably. These abnormal cells begin to invade the surrounding breast
tissue and may eventually spread (Metastasis) via blood vessels and lymphatic
channels to the lymph nodes, lungs, bones, brain and liver [5]. There are many
different types of breast cancer, about 70% of them are sensitive to the
female sex hormone called estrogen. Cells from these cancers have receptor
sites that bind to estrogen, thereby promoting their growth and spread [6].
These cancers are called estrogen receptor-positive cancers (or ER-positive
cancers) and it’s the type we are discussing in this paper. Cells from tumors
are tested to see if they have these receptors. Hormonal (or endocrine)
therapy can be used as a treatment in this case such as Tamoxifen [7].

However, information on the etiology of this disease remains scarce. For many
years Now, there are challenges in researching new ways to understand and
fight breast cancer, the causes of breast cancer are not fully understood and
not well known yet, so it’s hard to say why one woman might develop it and
another doesn’t. In either way, some risk factors are known to affect the
likelihood of developing it. Some of them you can’t change, but some you can,
after all the three main risk factors for breast cancer namely are, hormonal
imbalances, genetical and environmental. Some of the therapies to suppress
tumor growth Or inhibit the dynamic of cancer cell are surgery, chemotherapy,
radiation therapy, endocrine therapy, targeted therapy and immunotherapy,
despite the fact that breast cancer was formerly thought to be difficult to
treat with the latter therapy, because to its immunological ”coldness,”
clinical research and novel medications have proven that immunotherapy
treatment can enhance breast cancer patient outcomes [8]. Yet, every treatment
has side effects such as hair loss, mood swings, nausea, vomiting, and
fatigue, etc… As with most cancers, the earlier breast cancer is detected and
diagnosed, the better the chances of successful treatment.

Over the years, breast cancer modeling has become an invaluable tool for
understanding dynamics Behavior of tumor growth during treatment. Some studies
show that mathematical modeling helps solve epidemiological problems. Oke et
al. [9] and Bozkurt Yousef et al.[10] Improved Mudufza’s model[11], they have
built-in control parameters (ketogenic diet, immune booster and anticancer
drugs) hypothetically have interactions between normal and malignant cells.
Anyhow, these Studies do not include the concept of a nutritious diet combined
with Tamoxifen and immunotherapy in their mathematical models, and even in
traditional current treatments used by oncologists and doctors. According to
the breast oncologist Dr. Hung Khong, MD [12], the majority of research
looking at these types of combinations have focused on an other type of
cancer: triple-negative breast cancer, however, there is no indication that
ER-positive illness is immune to immunotherapy. In fact, there is evidence
that ER-positive breast cancer responds to immunotherapy, and it’s our main
goal using mathematical tools.

This paper is inspired from mathematical models of [11, 13, 25], but here we
introduced the saturation effect of cancer cells and added the immunotherapy
drug (ICB Immune checkpoint blockade’s type [14]) interaction with the immune
cells to conclude new results. This work is organized as follow: first
section, we construct the mathematical Model of breast dynamics with the
presence of cancer treatment, excess estrogen, and a ketogenic diet as a
therapeutic tool to help shrink the tumor size [16] (many researches are still
needed to see the effect of this diet on cancer patients). The five
compartements are modeled by forming a system of differential equations. A
qualitative study consists of the existence of equilibria and the study of
their local stability, basic reproduction numbers are discussed.

##  II Model description

In Oke et al.[9] and de Pillis et al. [25] models, and also previous
mathematical models published lately, never used the combination of endocrine
therapy with a diet and immunotherapy, so we propose a mathematical model to
study the dynamics of breast cancer with excess estrogen by considering the
saturation effect with the presence of treatments. We present a system of
differential equations describing the interactions between normal cells N,
tumor cells T , immune cells I, estrogen E and immunotherapy M, then we study
the dynamic behavior described by the system bellow:

$\left\\{\begin{array}[]{l}\begin{aligned}
\dfrac{dN}{dt}={}&N(t)(a_{1}-b_{1}N(t))-\\\ &\dfrac{d_{1}T(t)N(t)}{1+\epsilon
T(t)}-l_{1}N(t)E(t)(1-k),\end{aligned}\\\ \\\ \begin{aligned}
\dfrac{dT}{dt}={}&T(t)(a_{2}d-b_{2}T(t))-g_{1}I(t)T(t)-m_{d}T(t)+\\\
&l_{1}N(t)E(t)(1-k),\end{aligned}\\\ \\\ \begin{aligned}
\dfrac{dI}{dt}={}&s+\dfrac{rI(t)T(t)}{o+T(t)}-g_{2}I(t)T(t)-mI(t)-\\\
&\dfrac{l_{3}I(t)E(t)}{g+E(t)}(1-k)+\dfrac{p_{M}I(t)M(t)}{j_{M}+M(t)},\end{aligned}\\\
\\\ \dfrac{dE}{dt}=p(1-k)-\theta E(t),\\\ \\\
\dfrac{dM}{dt}=v_{M}(t)-n_{M}M(t)+\dfrac{\chi
M(t)I(t)}{\xi+I(t)}.\end{array}\right.$ (1)

We now explain the model parameters and describe the terms biologically for a
better understanding of the interactions presented in each equation.

###  II-A Modeling normal cells

In system (1), normal cells are represented by the first equation, $a_{1}$ is
the logistic growth rate of the normal cells, $b_{1}$ is the natural death,
therefore; it is very reasonable to model the infection rate by a saturated
incidence of the form $\dfrac{d_{1}TN}{1+\epsilon T}$, where $d_{1}$ and
$\epsilon$ are positive constants which, respectively describe the inhibition
rate of normal cells due to the DNA damage, and the saturation effect. Excess
estrogen leads to DNA mutation and thus normal cells population will be
reduced and transformed into tumor cells by $l_{1}NE$ term.

###  II-B Modeling tumor cells

In the second equation, the first term is the limited growth term for tumor
cells that depends on the ketogenic diet doze $d$, $m_{d}$ is the death rate
of tumor cells as a result of starvation of nutrients, glucose, note that keto
diet [18] put the body in ketosis that is a metabolism that turns fat into
ketones, which helps protect against some cancers. Tumor cells will be removed
due to the immune response by $g_{1}$.

###  II-C Modeling Lymphocytes(Immune cells)

Lymphocytes, including T-cells, T-regulatory cells, and natural killer cells
(NK), and their cytokine release patterns are implicated in both primary
prevention and recurrence of breast cancer [19]. In the third equation $s$ is
the constant source rate of immune response, the presence of tumor cells
stimulates and activate the immune response; resulting in growth of immune
cells, this is represented by a positive nonlinear growth term also called the
Michaelis–Menten interaction term for immune cells $\dfrac{rIT}{o+T}$, where
$r$ is the immune response rate, and $o$ is the immune threshold rate, $g_{2}$
represents the inactivation of immune cells due tumour cells as the
interaction coefficient, $m$ is the natural death rate of immune cell, the
next term is the limited rate suppression of $I$ due to excess estrogen, where
$l_{3}$ is the suppression rate, and $g$ is the estrogen threshold rate, the
last term represente a Michaelis-Menten interaction, it’s the activation of
immune cells by immunotherapy.

###  II-D Modeling estrogen

The last equation represents the estrogen dynamics, this hormone plays various
roles in the reproductive system of a female, many Studies show higher blood
levels of the estrogen called estradiol increase the risk of breast cancer in
postmenopausal women [17], here $p$ is the source rate of estrogen, and
$\theta$ is the decay rate after being washed out from the body.

###  II-E Immunotherapy

We differ our work from previous models such as Oke et al. by adding this
equation gonverning the amount of the drug injected per day per litre of body
volume, its turnover represented by $n_{M}M$, and finally a Michaelis–Menten
term in the drug used $\frac{\chi M(t)I(t)}{\xi+I(t)}$, representing the
production of immunotherapy from activated immune cells ($CD8^{+}T$).

##  III Model analysis

###  III-A Positiveness and existence of equilibria:

######  Theorem 1.

System (1) has one unique solution $(N,T,I,E,M)$ in $\mathbb{R}_{+}^{5}$.

###### Proof.

See [22]. ∎

######  Remark 1.

The aim of our model is to investigate cellular populations, all variables and
parameters are non-negative. Based on the biological findings, the system of
Equation (1) will be evaluated in $\Omega$ discused in the next theorem, that
guarantees that the system is well-posed in such a way that solutions with
non-negative initial conditions persist non-negative for all $t>0$, making the
variables biologically meaningful. As a result, we get the theorem below.

######  Theorem 2.

The following set,

$\Omega=\left\\{(N,T,I,E,M)\in\mathbb{R}_{+}^{5}\right\\},$

is a positively invariant set of model (1).

###### Proof.

Let $P(t)=(N(t),T(t),I(t),E(t),M(t))$ be any positive solution of the model,
with initial condition $P(0)=(N(0),T(0),I(0),E(0),M(0))\in\mathbb{R}_{+}^{5},$
from the model (1) we get:

$\displaystyle
N(t)=N(0)e^{\int_{0}^{t}a_{1}-b_{1}N(\tau)-\frac{d_{1}T(\tau)}{1+\epsilon
T(\tau)}-l_{1}E(\tau)(1-k)\mathrm{d}\tau},$ $\displaystyle
T(t)=T(0)e^{\int_{0}^{t}a_{2}d-b_{2}T(\tau)-g_{1}I(\tau)-m_{d}+\frac{l_{1}N(\tau)E(\tau)}{T(\tau)(1-k)}\mathrm{d}\tau},$
$\displaystyle
I(t)=I(0)e^{\int_{0}^{t}\frac{s}{I(\tau)}+\frac{rT(\tau)}{o+T(\tau)}-g_{2}T(\tau)-m-\frac{l_{3}E(\tau)}{g+E(\tau)}(1-k)+\frac{p_{M}M(\tau)}{j_{M}+M(\tau)}\mathrm{d}\tau},$
$\displaystyle
E(t)=E(0)e^{\int_{0}^{t}\frac{p(1-k)}{E(\tau)}-\theta\mathrm{d}\tau},$
$\displaystyle M(t)=M(0)e^{\int_{0}^{t}\frac{v_{M}}{M(\tau)}-n_{M}+\frac{\chi
I(\tau)}{\xi+I(\tau)}\mathrm{d}\tau}.$

By starting from a positive initial condition each solution of the model
remain positive for all $t\geq 0$, thus $\Omega$ is a positively invariant set
of our model.

From the first and second equation of $N(t)$ and $T(t)$, and by using the
comparison theorem we obtain:

$\frac{\mathrm{d}N(t)}{\mathrm{d}t}\leq N(t)(a_{1}-b_{1}N(t)),$

Thus

$\limsup_{t\rightarrow+\infty}N(t)\leq\frac{a_{1}}{b_{1}}.$

and,

$\limsup_{t\rightarrow+\infty}T(t)\leq\frac{a_{2}}{b_{2}}.$

Consequently, it can be shown that $I,E,M$ are also bounded, which completes
the proof. ∎

###  III-B Equilibria

In this section we study the existence and stability of the equilibria, which
also represents the critical points of system (1). The model system admits
seven steady states in which there are one tumor-free equilibrium point, five
dead equilibria, and finally the coexisting equilibria discussed in a brief
way.

Let $\bar{P}=(\bar{N},\bar{T},\bar{I},\bar{E},\bar{M})$ be an equilibrium
point, We have $\bar{N}$, $\bar{T}$, $\bar{I}$, $\bar{E}$ and $\bar{M}$ are
positive variables since cell populations are non-negative and real.
Therefore, all parameters considered in this model are positive.

####  III-B1 Tumor free equilibrium point

$P_{0}=\left(\bar{N}_{0},0,\bar{I}_{0},\bar{E}_{0},\bar{M}_{0}\right),$

where,

  * •

$\bar{N}_{0}=\dfrac{a_{1}-l_{1}\bar{E}_{0}(1-k)}{b_{1}},$

  * •

$\bar{I}_{0}=\dfrac{s}{m_{+}\dfrac{l_{3}\bar{E}_{0}}{g+\bar{E}_{0}}(1-k)-\dfrac{p_{M}\bar{M}_{0}}{j+\bar{M}_{0}}},$

  * •

$\bar{E}_{0}=\dfrac{(1-k)p}{\theta},$

  * •

$\bar{M}_{0}=\dfrac{v_{M}}{n_{M}-\dfrac{\chi\overline{I_{0}}}{\xi_{+}\overline{I_{0}}}}.$

######  Remark 2.

$CD8^{+}T$ cells also called cytotoxic T lymphocytes, are a type of immune
cells that have the capacity to react to pathogens like infections, virus and
cancer [26].
If we consider $CD8^{+}T$ as the only type of immune cells for our model, then
during the tumor-free state, $I(t)$ will be zero for all $t$, since the
activation of these effector cells depends on tumor.

######  Lemma 1.

$P_{0}$ exists if and only if :

  1. i)

$\bar{E}_{0}\leq\dfrac{a_{1}}{l_{1}(1-k)},with\quad k<1,$

  2. ii)

$\bar{M}_{0}\leq\dfrac{m}{p_{M}},$

  3. iii)

$\bar{I}_{0}\leq\dfrac{n_{M}\xi}{\chi-n_{M}},$

  4. iv)

$l_{1}\leq\dfrac{\theta a}{p(1-k)^{2}}.$

Biologically, it means that the growth rate of normal cells must be more than
the estrogen amount, also we notice that the $\bar{I}_{0}$ in this case
depends on the suppression of estrogen and the amount of immunotherapy dose,
unlike the the estrogen-free model studied by T. Sundaresan et al. [13], where
it depends only on the nature of the dynamics.

####  III-B2 Death equilibrium points type 1

Dead equilibria are states when both normal cells and tumor cells have died
off, maybe by a mastectomy surgery [23], or death, and it’s given by:

$P_{di}=\left(0,0,\bar{I}_{i},\frac{(1-k)p}{\theta},\bar{M}_{i}\right),$

such that,

$\bar{M_{i}}=\dfrac{v_{M}}{n_{M}-\dfrac{\chi\bar{I_{i}}}{\xi_{i}+\bar{I_{i}}}}\text{
for }i\in\\{1,2\\},$

and $\bar{I}_{i}$ verifies this equality:

$\bar{I}^{2}\left(A\chi+p_{M}v_{M}\right)-\bar{I}\left(An_{M}-p_{M}v_{M}\xi+s\chi\right)+sn_{M}=0,$
(2)

such that,

$A=m+\frac{l_{3}\bar{E}}{g+\bar{E}}(1-k),$

therefore, the discriminant of (2) is:

$\displaystyle\Delta=v_{M}(t)^{2}\chi^{2}p_{M}^{2}-2Av_{M}(t)\chi
n_{M}p_{M}-2v_{M}(t)\chi s\xi p_{M}\left(An_{M}\right)^{2}$ (3)
$\displaystyle-2As\xi n_{M}-4v_{M}(t)sn_{M}p_{M}+s^{2}\xi^{2},$

we set $B=A\chi+p_{M}v_{M},$
$B\geq 0\Rightarrow\Delta\geq 0$, which means that equation (2) has two
distinct real roots:

$\bar{I}_{1,2}=\dfrac{-p_{M}v_{M}(t)\chi+An_{M}+s\xi_{\mp}\sqrt{\Delta}}{2\left(A\xi+p_{M}v_{M}(t)\right)}.$
(4)

We summerize then the existence of the two dead equilibrium points in the
following lemma,

######  Lemma 2.

$P_{di}$ for $i=1,2.$ exists if and only if :

  1. i)

$\dfrac{p_{M}v_{M}\xi}{s\chi An_{M}}<1,$

  2. ii)

$k<1,$

  3. iii)

$n_{M}\geq\dfrac{\chi\bar{I}_{i}}{\xi+\bar{I}_{i}}.$

Biologically, it means that the existence of these equilibria depends on the
natural elimination or excretion of IL-2, which must be more than its
production from the activated $CD8^{+}T$.

####  III-B3 Death equilibrium points type 2

$P_{di}=\left(0,\bar{T}_{i},\frac{a_{2}d-b_{2}\bar{T}_{i}-m_{d}}{g_{1}},\frac{(1-k)p}{\theta},\frac{v_{M}}{n_{M}-\frac{\chi\bar{I}_{i}}{\xi+\bar{I}_{i}}}\right),$

for $i={3,4,5}.$

To find the expressions of $\bar{T_{i}}$, we solve the above quadratic
equation:

$\displaystyle\nu(\bar{T})=$
$\displaystyle\bar{T}^{3}\left(b_{2}g_{2}\right.)+\bar{T}^{2}(b_{2}m+b_{2}g_{2}o-b_{2}r+b_{2}C$
(5) $\displaystyle-a_{2}dg_{2}-a_{2}dC+m_{d}g_{2})+\bar{T}(b_{2}mo+b_{2}Co$
$\displaystyle-a_{2}dm-a_{2}dg_{2}o+ra_{2}d-a_{2}doC+g_{1}s$
$\displaystyle+m_{d}m+m_{d}g_{2}\theta-rm_{d}+m_{d}C)$
$\displaystyle+m_{d}Co+m_{d}mo+g_{1}so-a_{2}dmo,$

where,

$C=\frac{l_{3}\bar{E}}{g+\bar{E}}(1-k)-\frac{p_{M}\bar{M}}{j+\bar{M}},$

thus,

$\bar{T}_{i}=Roots(\nu(\bar{T}))\quad\text{for}\quad i=3,4,5.$

We finally can summarize the expression of the death equilibrium points of
type 2 as follows:

$P_{d3,d4,d5}=\left(0,\bar{T}_{3,4,5},\bar{I}_{3,4,5},\frac{(1-k)p}{\theta},\bar{M}_{3,4,5}\right).$

######  Lemma 3.

The dead equilibria of type 2 exist if:

  1. i)

$k<1,$

  2. ii)

$\dfrac{a_{2}d-m_{d}}{b_{2}}-\dfrac{g_{1}^{2}n_{M}}{b_{2}\left(\chi-
g_{1}n_{M}\right)}\leqslant\bar{T}_{3,4,5}\leqslant\dfrac{a_{2}d-m_{d}}{b_{2}}.$

####  III-B4 Co-existing point

In this case all cell populations survive the competition and they coexist, it
is given by:

$P_{e}=\left(N_{e},T_{e},I_{e},E_{e},M_{e}\right),$

such that,

  * •

$N_{e}=\dfrac{1}{b_{1}}\left(a_{1}-\dfrac{d_{1}T_{e}}{1+\epsilon
T_{e}}-l_{1}E_{e}(1-k)\right)=\psi_{1}(I_{e}),$

  * •

$T_{e}=Roots(\mu(T_{e}))=\psi_{2}(I_{e}),$

  * •

$I_{e}=\dfrac{s}{g_{2}T_{e}-\dfrac{rT_{e}}{o+T_{e}}+m+\dfrac{l_{3}E_{e}}{g+E_{e}}(1-k)-\dfrac{p_{M}M_{e}}{j+M_{e}}},$

  * •

$E_{e}=\dfrac{(1-k)p}{\theta},$

  * •

$M_{e}=\dfrac{v_{M}}{n_{M}-\dfrac{\chi I_{e}}{\xi+I_{e}}}=\psi_{3}(I_{e}),$

where,

$\mu(T_{e})=b_{2}T_{e}^{2}+(g_{1}I_{e}+m_{d}-a_{2}d)T_{e}-l_{1}N_{e}E_{e}(1-k),$
(6)

we let,

$\displaystyle a$ $\displaystyle=b_{2},$ $\displaystyle b$
$\displaystyle=g_{1}I_{e}+m_{d}-a_{2}d,$ $\displaystyle c$
$\displaystyle=l_{1}N_{e}E_{e}(1-k),$

the existence of the coexisting point depends on the sign of (6) roots, we
discuss three cases using its coefficients, when $b$ and $c$ are positive we
get two negative roots, means there exist no realistic equilibria in this
case, therefore, for $c<0$ we get one equilibrium point, and for $b$ either
positive, or negative we obtain two roots with opposite signs, which means
there exist at least one coexisting point, finally the case where $b<0$ and
$c>0$ will allow us to get two coexisting points besides the other equilibria
we found in our previous parts.

We summarize the existence of the coexisting point in the following lemma,

######  Lemma 4.

$P_{e}$ exists if and only if:

$I_{e}\in\left]0,\dfrac{a_{2}d-m_{d}}{g_{1}}\right[\text{and}\quad k<1.$

##  IV Stability analysis

In this section we analyze the equilibria in terms of their local stability by
means of eigenvalues to identify conditions that can help eliminate tumor
cells, We apply the Hartman Grobman theorem [21]. Let
$\bar{P}=(\bar{N},\bar{T},\bar{I},\bar{E},\bar{M})$ be an arbitrary
equilibrium of model (1).

Hence, the associated characteristic equation of the jacobian matrix of our
system at $P$is given by,

$\left|\begin{array}[]{ccccc}{a_{1}-2b_{1}\bar{N}-\dfrac{d_{1}\bar{T}}{1+e\bar{T}}-l_{1}\bar{E}(1-k)-\lambda}&{\dfrac{d_{1}\bar{N}}{(1+e\bar{T})^{2}}}&{0}&{-l_{1}(1-k)\bar{N}}&{0}\\\
\\\
{l_{1}(1-k)\bar{E}}&{a_{2}d-2b_{2}\bar{T}-g_{1}\bar{I}-m_{d}-\lambda}&{-g_{1}\bar{T}}&{l_{1}\bar{N}(1-k)}&{0}\\\
\\\
{0}&{{\dfrac{ro\bar{I}}{(o+\bar{T})^{2}}-g_{2}\bar{I}}}&{J_{33}-\lambda}&{-\dfrac{l_{3}\bar{I}g}{(g+\bar{E})^{2}}(1-k)}&{\dfrac{P_{\bar{M}}\bar{I}j_{\bar{M}}}{(j_{\bar{M}}+\bar{M})^{2}}}\\\
\\\ {0}&{0}&{0}&{-\theta-\lambda}&{0}\\\ \\\
{0}&{0}&{\dfrac{\chi\bar{M}\xi}{(\xi+\bar{I})^{2}}}&{0}&{-n_{\bar{M}}+\dfrac{\chi\bar{I}}{\xi+\bar{I}}-\lambda}\end{array}\right|=0.$
(7)

where,

$J_{33}=\dfrac{r\bar{T}}{o+\bar{T}}-g_{2}\bar{T}-m-\dfrac{l_{3}\bar{E}}{g+\bar{E}}(1-k)+\dfrac{P_{\bar{M}}\bar{M}}{j_{\bar{M}}+\bar{M}}.$

Now, we introduce reproductive numbers and conditions for the stability of
tumor-free state, denote:

$\mathscr{R}_{0}=\dfrac{A_{6}A_{9}}{A_{10}A_{5}},\quad\text{and}\quad\mathscr{R}_{1}=\dfrac{A_{1}A_{2}}{A_{0}A_{3}},$

where,

$\displaystyle A_{0}$ $\displaystyle=a_{1}-2b_{1}N-l_{1}E(1-k),$
$\displaystyle A_{1}$ $\displaystyle=l_{1}E(1-k),$ $\displaystyle A_{2}$
$\displaystyle=d_{1}N,$ $\displaystyle A_{3}$
$\displaystyle=a_{2}d-g_{1}I-m_{d},$ $\displaystyle A_{4}$
$\displaystyle=\dfrac{rI}{o}-g_{2}I,$ $\displaystyle A_{5}$
$\displaystyle=-m-\dfrac{l_{3}E(1-k)}{g+E}+\dfrac{P_{M}M}{j_{M}+M},$
$\displaystyle A_{6}$ $\displaystyle=\dfrac{\chi M\xi}{(\xi+I)^{2}},$
$\displaystyle A_{7}$ $\displaystyle=l_{1}N(1-k),$ $\displaystyle A_{8}$
$\displaystyle=\dfrac{l_{3}Ig}{g+E}(1-k),$ $\displaystyle A_{9}$
$\displaystyle=\dfrac{P_{M}Ij_{M}}{(j_{M}+M)^{2}},$ $\displaystyle A_{10}$
$\displaystyle=-n_{M}+\dfrac{\chi I}{\xi+I}.$

######  Theorem 3.

The tumor-free equilibrium point $P_{0}$ of system (1) is locally
asymptotically stable if $\mathscr{R}_{0}<1$ and $\mathscr{R}_{1}<1$,
otherwise it is unstable.

###### Proof.

Let $J_{p_{0}}$ be the jacobian matrix around $P_{0}$:

$J_{p_{0}}=\left(\begin{array}[]{ccccc}A_{0}&A_{2}&0&-A_{7}&0\\\
A_{1}&A_{3}&0&A_{7}&0\\\ 0&A_{4}&-A_{5}&-A_{8}&A_{9}\\\ 0&0&0&-\theta&0\\\
0&0&A_{6}&0&A_{10}\end{array}\right)$

and the characteristic equation at $P_{0}$ is given by:

$\mathbb{P}(\lambda)=(\lambda+\theta)\left(\lambda^{2}-A_{10}\lambda+A_{5}\lambda-
A_{10}A_{5}-A_{6}A_{9}\right)\\\ \left(\lambda^{2}-A_{0}\lambda-A_{3}\lambda-
A_{1}A_{2}+A_{0}A_{3}\right),$

we can clearly see that this equation has five eigenvalues

$\displaystyle\lambda_{1}=-\theta,$ (8)
$\displaystyle\lambda^{2}-\left(A_{10}-A_{5}\right)\lambda-
A_{10}A_{5}\left(1-\dfrac{A_{6}A_{9}}{A_{10}A_{5}}\right)=0,$ (9)
$\displaystyle\lambda^{2}-\left(A_{0}+A_{3}\right)\lambda+A_{0}A_{3}\left(1-\dfrac{A_{1}A_{2}}{A_{0}A_{3}}\right)=0,$
(10)

equation (9) admits two eigenvalues: $\lambda_{2}$ and $\lambda_{3}$, to study
their nature we apply the Routh hurwitz criteria, we let :

$H_{1}=\left(\begin{array}[]{ll}a_{1}&a_{0}\\\ a_{3}&a_{2}\end{array}\right),$

where,

$\displaystyle a_{0}$ $\displaystyle=1,$ $\displaystyle a_{1}$
$\displaystyle=-(A_{10}-A_{5}),$ $\displaystyle a_{2}$
$\displaystyle=-A_{10}A_{5}(1-\mathscr{R}_{0}),$ $\displaystyle a_{3}$
$\displaystyle=0.$

The roots of (9) has negative real parts only if all the principle diagonal
minors of the Hurwitz matrix are positive, provided that:

$a_{0}>0,\quad\Delta_{1}=a_{1}>0,\quad\Delta_{2}=\left|\begin{array}[]{ll}a_{1}&a_{0}\\\
a_{3}&a_{2}\end{array}\right|>0.$

Therefore, the eigenvalues of (9) are negative only if $\mathscr{R}_{0}<1$.

We repeat the same process for (10), we get two eigenvalues: $\lambda_{4}$,
$\lambda_{5}$ and the other reproductive number $\mathscr{R}_{1}$. We conclude
that our system is stable at the tumor free equilibrium point if and only if :

$\mathscr{R}_{0}<1\quad\text{and}\quad\mathscr{R}_{1}<1.$

∎

######  Remark 3.

$\lambda_{3}$ and $\lambda_{5}$ are negative if and only if :

$\dfrac{sM}{V_{M}}<{I}<\dfrac{a_{2}(1+d)-2b_{1}N-l_{1}E(1-k)-m_{d}}{g_{1}}.$

Biologically its means that the immune response must be greater than the
immunotherapy dose taken during the treatment.

Now we study the stability of the death equilibrium points of type 1
$P_{d1,d2}$, and we let $\mathscr{R}_{IM}=\dfrac{B_{5}B_{7}}{B_{4}B_{8}}$ be
the reproduction number for the immune system response such that,

$\displaystyle B_{0}$ $\displaystyle=a_{1}-l_{1}E(1-k),$ $\displaystyle B_{1}$
$\displaystyle=l_{1}E(1-k),$ $\displaystyle B_{2}$
$\displaystyle=a_{2}d-g_{1}I-m_{d},$ $\displaystyle B_{3}$
$\displaystyle=\dfrac{rI}{o}-g_{2}I,$ $\displaystyle B_{4}$
$\displaystyle=-m-\dfrac{l_{3}E(1-k)}{g+E}+\dfrac{P_{M}M}{j_{M}+M},$
$\displaystyle B_{5}$ $\displaystyle=\dfrac{\chi M\xi}{(\xi+I)^{2}},$
$\displaystyle B_{6}$ $\displaystyle=-\dfrac{l_{3}Ig}{(g+E)^{2}}(1-k),$
$\displaystyle B_{7}$ $\displaystyle=\dfrac{P_{M}Ij_{M}}{(j_{M}+M)^{2}},$
$\displaystyle B_{8}$ $\displaystyle=-n_{M}+\dfrac{\chi I}{\xi+I}.$

Therefore equation (7) becomes:

$(\lambda+\theta)(B_{4}B_{8}-B_{4}\lambda-
B_{5}B_{7}-B_{8}\lambda+\lambda^{2})(B_{2}-\lambda)(B_{0}-\lambda)=0$ (11)

######  Theorem 4.

Death equilibria of type 1 are stable if and only if:

  * •

$\mathscr{R}_{IM}<1,$

  * •

$B_{0},B_{2},B_{4},B_{8}<0,$

###### Proof.

The characteristic equation (11) has five eigenvalues:

$\displaystyle\lambda_{1}=-\theta,$
$\displaystyle\lambda_{2,3}=Roots(F(\lambda)=B_{4}B_{8}-B_{4}\lambda-
B_{5}B_{7}-B_{8}\lambda+\lambda^{2}),$ $\displaystyle\lambda_{4}=B_{2},$
$\displaystyle\lambda_{5}=B_{0}.$

For the system to be stable we must have all the eigenvalues with positive
real parts. First let’s work on the $\lambda_{2,3}$ sign:

$\displaystyle F(\lambda)$ $\displaystyle=B_{4}B_{8}-B_{4}\lambda-
B_{8}B_{+}-B_{8}\lambda_{+}\lambda^{2}$
$\displaystyle=\lambda^{2}-\lambda(B_{4}+B_{8})+B_{4}B_{8}(1-\dfrac{B_{5}B_{7}}{B_{4}B_{8}}),$

using Routh-Hurwitz criteria, we summarize that the death equilibria of type 1
are stable only if:

$\left\\{\begin{aligned} &B_{4}<0\quad\text{and}\quad B_{8}<0,\\\
&B_{4}+B_{8}<0,\\\ &\mathscr{R}_{IM}<1.\end{aligned}\right.$ (12)

In a more detailed way, we will determine necessary conditions and also their
biological interpretations for each part as follow:

$B_{4}<0\iff\dfrac{P_{M}M}{J_{M}+M}<m+\dfrac{l_{3}E(1-k)}{g+E},$

biologically means that the activation of the $CD8^{+}T$ by the immunotherapy
must be smaller than the deactivation of the immune response by estrogen.

$B_{8}<0\iff\dfrac{\chi I}{\xi_{1}+I}<n_{M},$

here we should see that the production of $IL_{2}$ from activated immune cells
must be less than the excretion of the $IL_{2}$.

$B_{0}<0\iff a_{1}<l_{1}E(1-k),$

this condition shows that the damaged natural cells due to excess estrogen
should be more than it’s natural growth rate. and finally,

$B_{2}<0\iff I>\dfrac{a_{2}d-m_{d}}{g_{1}},$

in this case the immune response needs to be very strong so it affect and
blocks the tumor cells growth. ∎

Next we study the behavior of the death equilibria of type 2, in this case (7)
becomes,

$\begin{multlined}\mathbb{P}(\lambda)=(\lambda+\theta)(C_{0}-\lambda)[\lambda^{2}\left(C_{2}+C_{5}+C_{9}\right)-\lambda^{3}+\\\
\lambda\left(C_{6}C_{8}-C_{5}C_{9}-C_{3}C_{4}-C_{2}C_{9}-C_{2}C_{5}\right)+\\\
C_{2}C_{5}C_{9}-C_{2}C_{6}C_{8}+C_{3}C_{4}C_{9}]=0,\end{multlined}\mathbb{P}(\lambda)=(\lambda+\theta)(C_{0}-\lambda)[\lambda^{2}\left(C_{2}+C_{5}+C_{9}\right)-\lambda^{3}+\\\
\lambda\left(C_{6}C_{8}-C_{5}C_{9}-C_{3}C_{4}-C_{2}C_{9}-C_{2}C_{5}\right)+\\\
C_{2}C_{5}C_{9}-C_{2}C_{6}C_{8}+C_{3}C_{4}C_{9}]=0,$ (13)

where,

$\displaystyle C_{0}=a_{1}-\frac{d_{1}T_{1}}{1+eT}-l_{1}E,$ $\displaystyle
C_{1}=l_{1}E(1-k),$ $\displaystyle C_{2}=a_{2}d-2b_{2}T-g_{1}I-m_{d},$
$\displaystyle C_{3}=\frac{rIo}{(o+T)^{2}}-g_{2}I,$ $\displaystyle
C_{4}=g_{1}T,$ $\displaystyle
C_{5}=\frac{rT}{o+T}-g_{2}T-m-\dfrac{l_{3}E(1-k)}{g+E}+\dfrac{P_{M}M}{j_{M}+M},$
$\displaystyle C_{6}=\dfrac{\chi M\xi}{(\xi+I)^{2}},$ $\displaystyle
C_{7}=-\dfrac{l_{3}Ig}{(g+E)^{2}}(1-k),$ $\displaystyle
C_{8}=\dfrac{P_{M}Ij_{M}}{(j_{M}+M)^{2}},$ $\displaystyle
C_{9}=-n_{M}+\dfrac{\chi I}{\xi+I}.$

######  Theorem 5.

The death equilibria of type 2 are stable only if:

  1. i)

$a_{1}<\dfrac{d_{1}T_{1}}{1+eT}-l_{1}E,$

  2. ii)

$C_{6}C_{8}-C_{5}C_{9}-C_{3}C_{4}-C_{2}C_{9}-C_{2}C_{5}>0,$

  3. iii)

$C_{2}C_{5}C_{9}-C_{2}C_{6}C_{8}+C_{3}C_{4}C_{9}>0.$

###### Proof.

The characteristic equation (7) has five eigenvalues:

$\displaystyle\lambda_{1}=-\theta,$ $\displaystyle\lambda_{2}=C_{0},$
$\displaystyle\lambda_{3,4,5}=Roots(G(\lambda)),$

such that,

$\begin{multlined}G(\lambda)=-\lambda^{3}+\lambda^{2}(C_{2}+C_{5}+C_{9})+\\\
\lambda(C_{6}C_{8}-C_{5}C_{9}-C_{3}C_{4}-C_{2}C_{9}-C_{2}C_{5})\\\
+C_{2}C_{5}C_{9}-C_{2}C_{6}C_{8}+C_{3}C_{4}C_{9},\end{multlined}G(\lambda)=-\lambda^{3}+\lambda^{2}(C_{2}+C_{5}+C_{9})+\\\
\lambda(C_{6}C_{8}-C_{5}C_{9}-C_{3}C_{4}-C_{2}C_{9}-C_{2}C_{5})\\\
+C_{2}C_{5}C_{9}-C_{2}C_{6}C_{8}+C_{3}C_{4}C_{9},$
in a short form we let:

$f(\lambda)=-a_{0}\lambda^{3}+a_{1}\lambda^{2}+a_{2}\lambda+a_{3},$

$\mathfrak{Re}(\lambda_{i})<0$ for $i={3,4,5}$ if and only if $a_{i}>0$ and
the principal minors of the Hurwitz matrix of $f$ are positive.

$\lambda_{1}<0$, $\lambda_{2}$ has positive real part if:

$a_{1}<\frac{d_{1}T_{1}}{1+eT}-l_{1}E,$

biologically it means that the growth rate of normal cells $a_{1}$, must be
smaller than it’s damage rate by excess estrogen and tumor cells effect; to
reach one of the conditions of stability where $C_{0}<0$. ∎

Finally we study the stability of our system at $P_{e}$, let

$P_{e}=\left(\phi_{1}\left(I_{e}\right),\phi_{2}\left(I_{e}\right),I_{e},\frac{p(1-k)}{\theta},\phi_{3}\left(I_{e}\right)\right),$

such that $\phi_{i}$ for $i={1,2,3}$ are functions of $I_{e}$.

Repeating the same process as before, we develop our calculus, and from the
jacobian matrix at $P_{e}$ we get the following characteristic equation:

$\displaystyle\mathbb{P}(\lambda)=(D_{10}-\lambda)[(D_{0}-\lambda)(D_{3}-\lambda)(D_{6}-\lambda)(D_{12}-\lambda)$
(14) $\displaystyle-D_{11}D_{7}-D_{5}D_{4}(D_{12}-\lambda)(D_{0}-\lambda)$
$\displaystyle-D_{2}D_{1}(D_{6}-\lambda)(D_{12}-\lambda)+D_{2}D_{11}D_{7}]=0,$

such that the $D_{i}$ are the jacobian matrix elements at $P_{e}$,

the equation (14) admets five eignevalues, using mathematical calculus tools
such as Maple, we can define each eignevalues, therefore we can conclude the
stability of this system at this point in a brief way in the theorem bellow:

######  Theorem 6.

The system is stable at $P_{e}$ if and only if: $\mathfrak{Re}(\lambda_{i})<0$
for $i={2,3,4,5}$, while $\lambda_{1}=-\theta<0.$

######  Remark 4.

To study the stability of the system at $P_{e}$, we can also develop its
characteristic equation (14), and study the sign of its coefficients using the
Routh-Hurwitz criteria using the same previous calculus.

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# Privacy-Preserving Social Distancing Bracelet
††thanks: The authors would like to express their sincere gratitude to the
Deanship of Scientific Research at KFUPM for the immense support provided. Our
gratitude is extended to the Technology Advancement and Prototyping Centre
(TAPC) at the Dhahran Techno Valley for the technical support and facilities.

Asaad AlGhamdi  Information & Computer Science Department
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia
<EMAIL_ADDRESS>Louai Al-Awami  Computer Engineering Department
King Fahd University of Petroleum and Minerals
Dhahran, Saudi Arabia
<EMAIL_ADDRESS>
###### Abstract

This demo presents a functional Proof-of-Concept prototype of a smart bracelet
that utilizes IoT and ML to help in the effort to contain pandemics such as
COVID-19. The designed smart bracelet aids people to navigate life safely by
monitoring health signs; and detecting and alerting people when they violate
social distancing regulations. In addition, the bracelet communicates with
similar bracelets to keep track of recent contacts. Using RFID technology, the
bracelet helps in automating access control to premises such as workplaces.
All this is achieved while preserving the privacy of the users.

###### Index Terms:

IoT, contact tracing, e-Health, COVID-19.

##  I Introduction

Technology can play a crucial rule at times of crises. This is particularly
true in the current intricate case of the COVID-19 pandemic where the
challenge is of perplexing nature. Technologies such as the Internet-of-Things
(IoT), Artificial Intelligence (AI), Big Data, and Robotics, to mention a few,
can provide valuable insights and reduce dependence on human actors leading to
a safer and faster containment.

Contact tracing aims to identify potentially infected individuals in a timely
fashion to properly treat and isolate the spread of infections. While contact
tracing has long been in practice, it has - until recently - been manually
performed. Many automated contact tracing protocols have been recently
proposed in the literature such Pan-European Privacy-Preserving Proximity
Tracing (PEPP-PT) [1], Coalition [2], and the Decentralized Privacy-Preserving
Proximity Tracing (DP-3T) [3]. Compared to other protocols, (DP-3T) has been
more successful where privacy is a key concern. In DP-3T, devices regularly
broadcast unique and continuously-changing tags. Each device stores both the
set of tags it broadcasted and the tags it heard. When a user is tested
positive, he/she can voluntarily permit the device to send the tags it
broadcasted to a central database that stores all “infected” tags. Other
devices regularly check and compare the tags they heard against the database.
If a match is found, the device alerts the user to take proper measures. The
main strength of DP-3T is that no information that can be used to identify the
user is collected, hence, providing strict privacy. However, the protocol
lacks a mechanism to track duration of exposure when tags change. In this
work, we use the DP-3T protocol due to its strict privacy measures. Our
implementation in contrast adds an extension to enable tracing the duration of
exposure even when the tags change without breaking the privacy promise. Our
system also uploads contacts automatically when infection symptoms are
detected which improves the reaction time of the system.

In this context, we have successfully developed an IoT-based solution to
enable automatic tracing of potential COVID-19 cases. The solution is
comprised of a smart bracelets equipped with sensors to monitor the vital
signs of its wearer and provide an early warning when abnormal symptoms arise.
In addition, the bracelet can detect when another bracelet is physically too
close and reminds the user to respect social distancing regulations. The
bracelet also listens and stores messages form nearby bracelets which are then
used to warn those who had contacted a confirmed COVID-19 patient; in order to
take necessary measures such as executing a quarantine. As privacy is a key
concern in such applications, we have adopted a fully privacy-preserving
approach. Yet, if necessary, the bracelet can also be used by authorities and
employers to control access to certain places. Using Radio Frequency
IDentification (RFID), the bracelet can be scanned using an electronic reader
to determine the users’ risk level, and consequently grant or deny him/her
access.

##  II Research Components and Issues

The presented application combines a number of interesting research issues.
The first is the privacy preservation while maintaing the ability to track the
identity of the subjects and the duration of exposure. The second issue is the
proximity detection and computation of separation distance using RSSI. As RSSI
is typically unreliable, machine learning was used to build a model to perform
this task. The third is chrachterising couhging and health condition through
fusion of sensor data. In addition to presenting exciting research issues, the
application demonistrates a well-rounded example of integration between many
interrelated technologies, namely IoT, ML, Cloud Computing, RFID, and BLE;
working hand-in-hand to deliver a timely and crucial application. The demo
includes a functional prototype demonistrating all the described
functionalites working toegther to achieve the claimed application.

##  III Design & Implementation

###  III-A System Model

Figure 1: Smart Bracelet System Architecture.

As shown in Figure 1, the system is comprised of a number of smart bracelets
each is equipped with a Bluetooth Low-Energy (BLE) and a sensing modules. The
BLE module provides a beaconing service and enables pairing the bracelet to a
smart phone. On the other hand, the sensing module encompasses a number of
sensors used to monitor the health status of the wearer. The system also
relies on a cloud service that is reachable over the Internet to test tags.

###  III-B Health Monitoring

Figure 2: Health Monitoring Module.

The role of the health monitor is to track readings from various sensors to
infer the health status of the user. The module also detects when the bracelet
wearer coughs, or when another person is too close to the bracelet. The module
can be seen in Figure 2. User risk level can be either: 1) No-risk: a person
who has no symptoms, no excessive violations, and no infected contacts 2) Low-
risk: a person who had either been in contact with an infected person or have
excessive social distancing violations 3) High-risk: a person with abnormal
health symptoms, or who has been in contact with an infected case and has
excessive social distancing violations.

We define abnormal health symptoms as follows: 1) Detection of high
temperature (Fever greater than 38.0 °C) 2) Excessive coughing 3) Blood Oxygen
less than 90%. Once any of the aforementioned symptoms is detected, the user
is classified as a high-risk individual. Temperature and Blood Oxygen readings
are sampled periodically using medical grade external sensors. Social
distancing violations and coughing are detected through the combination of
general-purpose sensors.

###  III-C Contact Tracing

Figure 3: Contact Tracing Module.

The contact tracing module frequently sends BLE beacons while listening to
beacons coming from other bracelets. The module alerts the user when violating
social distancing by inferring the distance form the RSSI of the received
beacon. We developed a supervised machine learning model to estimate the
distance based on the transmitted power and the RSSI. When the risk level of
the wearer changes, the module - if permitted by the user - uploads the user’s
generated tags to the cloud and stores the new risk level on the RFID tag.
This module has been implemented using Raspberry Pi as seen in Figure 3. The
software of the module includes multiple threads for various tasks. The
software maintains two tables: contacts and tags. The contacts table contains
all the heard tags, the time each tag was first heard, and the total exposure
time; while the tags table stores the tags sent by the node itself.

###  III-D Privacy Preservation

To preserve privacy, each device uses a random BLE address when sending
beacons. In addition, devices generate random tags generated using a secure
hash function to avoid collisions. Both BLE addresses and tags change every
$15$ minutes. When devices hear beacons from a nearby device, the random tags
heard are stored locally in addition to the duration of exposure. When a user
is deemed infected, the tags he used are uploaded to the cloud. Other devices
constantly contact the cloud to retreive the list of infected tags and compare
them to the locally stored ones. This way, exposure detection is identified by
the end devices rather than the cloud which improves privacy.

###  III-E Cloud Service

The cloud service includes a database holding all the infected tags and two
main server-side scripts: one to receive and store tags in the database and
the other to process and compare overheard tags against infected tags. When a
bracelet wants to test its contacts, it uploads the tags to the cloud. The
service then checks if any of the contacts is in the infected tags database.
Infected tags coming from the same node are linked together. This allows
computing the exposure duration across different tags coming from the same
bracelet. If an exposure with an infected contact for 15 minutes or more is
found, the scripts returns a positive response implying that the user may be
infected and needs to take proper measures.

###  III-F Access Control

The purpose of the access control module is to enable scanning the health
status of the user using RFID technology. This module includes a physical RFID
tag that can be programmed by software to reflect the value of the risk level
of the user. The module can be used in conjunction with an RFID reader to
automate access control at work, or public places if required.

##  IV Conclusion

This paper demonstrates an important application of e-Health using IoT. Such
an application can play a crucial role in controlling the spread of vicious
diseases such as COVID-19. While a standard application of contact tracing can
easily be implemented using off-the-shelf technologies, this specific target
application combines a number of challenging aspects: 1) Privacy preservation
2) Exposure duration determination 3) Physical distance estimation 4) Health
status determination. The prototype demonstrates how an interesting mix of
technologies such as IoT, RFID, ML, BLE, and localization can all be combined
to achieve effective applications.

## References

  * [1] “HOME — PEPP-PT.” [Online]. Available: https://www.pepp-pt.org/
  * [2] L. Loiseau, V. Bellet _et al._ , “Whisper Tracing - An Open and Privacy First Protocol for Contact Tracing,” pp. 1–5, 2020. [Online]. Available: https://docsend.com/view/nis3dac
  * [3] C. Troncoso, M. Payer _et al._ , “Decentralized Privacy-Preserving Proximity Tracing,” no. May, pp. 1–46, 2020. [Online]. Available: http://arxiv.org/abs/2005.12273