# A new fractional model in Caputo sense for studying the dynamics of COVID-19
spread in France

Mahmoud H. A. Saleh<EMAIL_ADDRESS>Tarek M. Abed-Elhameed
<EMAIL_ADDRESS>Department of Physics, Faculty of Science, Assiut
University, Assiut 71516, Egypt.  Department of Mathematics, Faculty of
Science, Assiut University, Assiut 71516, Egypt.

(Received: date / Accepted: date)

###### Abstract

The COVID-19 pandemic has rapidly spread around the world and burdened public
health in almost all countries involving France. After the spread of SARS-
CoV-2, France harvested many deaths in total. In this paper, we develop models
with integer and fractional orders to investigate the dynamics of COVID-19
transmission in French hospitals and intensive care units (ICUs). Moreover,
this paper aims to explore the impact of precautionary measures on the total
infected cases in hospitals and ICUs of COVID-19 for the entire France by
using available actual data.

Keywords: _Mathematical modeling; SIHUR model; COVID-19; Intensive Care Unit;
Social distancing; Numerical results._

##  1 Introduction

The novel coronavirus has been commonly known as a severe acute respiratory
syndrome novel coronavirus 2 (SARS-CoV-2) or coronavirus disease 2019
(COVID-19). The pandemic onset began on 31 December 2019 when the first case
was detected in Wuhan city, Hubei Province of China [1]. From January 2020,
the number of COVID-19 cases has increased rapidly. The SARS-CoV-2 has
affected 213 countries and territories around the world besides two
international conveyances. By 27 July 2020, the total cumulative COVID-19
cases reached 16,638,667, and the total number of deaths became 656,924 [2].

On 24 January 2020, the virus was confirmed to have reached France where the
first three cases of COVID-19 were identified. Two cases of them were
diagnosed in Paris and one in Bordeaux [3, 4]. But it appeared to us,
subsequently, that there were already some cases exhibiting in December 2019
[5]. Also, the basic reproduction number value in France was estimated to
$R_{0}\approx 2.96$ as in [6] and $R_{0}\approx 3.2$ in [7] before lockdown on
17 March 2020. After lockdown in France, we found that the effective in
reducing the spread of COVID-19 appeared and the effective reproduction
number, called $R_{t}$, divided by a factor 5–7 at the country scale by 11 May
[8, 6].

In general, intensive-care units (ICUs) treated highly miscellaneous patients
at high peril of death-rate [9]. Previous studies [10, 11, 12, 13] of many
critically ill patients who survived from intensive care suffered from
subtending impaired outcomes in long-lasting physical, cognitive and/or mental
health. In addition, the study [14] recommended the rehabilitation of
critically ill COVID-19 survivors. The study [5] showed that the survivors
from ICU hospitalization showed a delay in returning to work, resulting in
substantial economic sequels. Another study [15] showed that the mortality
rate of COVID-19 patients in ICUs, across the international studies, was found
to be $41.6(34.0-49.7)\%$ ($95\%$CI).

Mathematical modeling has a significant effect on providing simulating for the
epidemic diseases and other several complex phenomena that vividly appeared in
the literature such as [16, 17]. Many papers, such as [18, 7, 19] [20, 21],
were published to analyze the spread of SARS-CoV-2 using mathematical models.
Also, some papers dealt with mathematical modeling of COVID-19 involving
hospitalization class as in [22, 23] and including critically-infected cases
such as [24].

In recent years, with the continued development of fractional calculus theory
and fractional-order system modeling approaches, one can see how important it
is in various engineering and other fields [25, 26]. Besides that, many
scholars have researched COVID-19 in the context of fractional mathematical
modeling [27, 28, 29]. Fractional derivatives have more accuracy in describing
the behavior of biological systems. And that returns to it has the information
history of the whole time interval or long memory than the short one of
integer derivatives.

This manuscript is organized in the following manner:

  * 1.

Sec. (2), we present the equations of the integer-order SIHUR model and its
initial conditions.

  * 2.

In Sec. (3), we study the essential analytical procedures such as positivity
and boundedness, in subsection (3.1), in addition to the invariant region set
in (3.1.1).

  * 3.

Sec. (4), we propose the fractional-order SIHUR model in Caputo sense together
with the initial conditions (4.1).

  * 4.

With Sec. (4.2), we determine the disease-free equilibrium point (DFEP) to get
the basic reproduction number $R_{0}$ and study its stability as in
subsections (5) and (5.1).

  * 5.

In Sec. (5.1.1), sensitivity analysis and estimation of the SIHUR parameters
are determined.

  * 6.

The last two remaining sections (5.1.2) and (6), and we discuss the numerical
simulations and conclude the core of this study, respectively.

##  2 Model formulation

We designed our SIHUR model of SARS-CoV-2 spread, which corresponds to the
data collection, as shown in the following equations. 1.

Figure 1: SIHUR model.

$\displaystyle\frac{dS}{dt}$ $\displaystyle=\Lambda-\varphi S-\mu S,$
$\displaystyle\frac{dI}{dt}$ $\displaystyle=\varphi S-k_{1}I,$
$\displaystyle\frac{dH}{dt}$ $\displaystyle=\gamma_{1}I-k_{2}H,$ (1)
$\displaystyle\frac{dU}{dt}$ $\displaystyle=\gamma_{2}I+\tau H-k_{3}U,$
$\displaystyle\frac{dR}{dt}$
$\displaystyle=\theta_{1}I+\theta_{2}H+\theta_{3}U-\mu R.$

Subject to the initial conditions to the model (1):
$S(0)\geq 0,~{}I(0)\geq 0,~{}H(0)\geq 0,~{}U(0)\geq 0,~{}R(0)\geq 0$.
Where:
$\varphi=\frac{\beta(I+qH+mU)}{N}$,
$k_{1}=(\gamma_{1}+\gamma_{2}+\theta_{1}+\delta_{1}+\mu)$,
$k_{2}=(\tau+\theta_{2}+\delta_{2}+\mu)$ and
$k_{3}=(\theta_{3}+\delta_{3}+\mu)$.

  1. 1.

The description of the divided classes of the whole human population$N(t)$,
where
$N(t)=S(t)+I(t)+U(t)+H(t)+R(t)$, is as below:

     * (a)

The susceptible individuals density class is S(t).

     * (b)

The infected class with the COVID-19 symptoms is I(t).

     * (c)

The hospitalized patients class is H(t).

     * (d)

The patients in intensive care units (ICUs) or (critically infected cases)
class is U(t).

     * (e)

The recovered$/$removed individuals class is R(t).

  2. 2.

The description of the parameters:

     * (a)

$\Lambda$ is the birth rate.

     * (b)

$\mu$ is the natural death rate in all classes.

     * (c)

$\theta_{1}$, $\theta_{2}$ and $\theta_{3}$ represent the recovery rates of
all infected cases in the class $I$, hospitalized (in-hospital) patients in
the class $H$ and in-ICUs patients in the class $U$ respectively.

     * (d)

$\delta_{1}$, $\delta_{2}$ and $\delta_{3}$ are the mortality rates due to the
COVID-19 in the $I$, $H$ and $U$, respectively.

     * (e)

$\beta$ shows the transmission rate of the infectious disease, COVID-19.

     * (f)

$\gamma_{1}$ denotes the transmissibility rate of the infected people with
COVID-19 who need hospitalization.

     * (g)

$\gamma_{2}$ denotes the transmissibility rate of the infected people with
COVID-19 who need the ICUs.

     * (h)

$\tau$ denotes the transmissibility rate of the hospitalized people with
COVID-19 to the ICUs.

     * (i)

$q$ is the infection rate due to health care workers (HCWs) interacting with
hospitalized COVID-19 patients in hospitals.

     * (j)

$m$ hints at the infection rate because of contact between the HCWs and the
critical care patients in ICUs.

###### Remark 1

We assume $q$ and $m$ represent the infection rate parameters between HCWs and
infected patients in hospitals and ICUs, respectively. That interaction
between them generates new COVID-19 ones among the HCWs. Then, this possibly
generates further new cases out of hospitals and ICUs.

##  3 Some basic analytical results

Herein, we present some necessary analytical results of the SIHUR model (1) of
COVID-19, such as positivity and boundedness for solutions of our model, and
stability of the disease-free equilibrium point (DFEP) after presenting the
fractional model. Moreover, we complete the basics by establishing the
theoretical formula of the crucial biological parameter called the basic
reproduction number, $R_{0}$, of the fractional SIHUR model.

###  3.1 Positivity and boundedness of the solution

For the first part, we examine the solution positivity of our model.

###### Lemma 1

Let the initial data be $P(0)\geq 0$ and $P(t)=(S,I,H,U,R)$ are the variables
of the model. Then, all the solutions of the model (1) will be non-negative
for all $t>0$. Further,

$\lim_{t\to\infty}\sup N(t)\leq\frac{\Lambda}{\mu}.$

Proof: Let $t_{*}=\sup\\{t>0:P(t)>0\in[0,t]\\}$. Then, it follows the first Eq
of our model written as shown:

$\frac{dS}{dt}=\Lambda-(\varphi+\mu)S.$ (2)

Multiply both sides of the above Eq. (2) by the integrating factor,
$\mathrm{exp}\left(\mu t+\int_{0}^{t}\varphi(\zeta)d\zeta\right)$. Then, it
can be further written as below,

$\frac{d}{dt}\left\\{S(t)\times\mathrm{exp}\left(\mu
t+\int_{0}^{t}\varphi(\zeta)d\zeta\right)\right\\}=\Lambda\times\mathrm{exp}\left(\mu
t+\int_{0}^{t}\varphi(\zeta)d\zeta\right).$

In consequence,

$S(t_{*})\times\mathrm{exp}\left(\mu
t_{*}+\int_{0}^{t_{*}}\varphi(\zeta)d\zeta\right)-S(0)=\Lambda\int_{0}^{t_{*}}\mathrm{exp}\left(\mu
t+\int_{0}^{t}\varphi(\nu)d\nu\right)dt.$ (3)

Then, the solution of Eq. (3) is

$\displaystyle S(t_{*})=S(0)$
$\displaystyle\times\mathrm{exp}\left\\{-\left(\mu
t_{*}+\int_{0}^{t_{*}}\varphi(\zeta)d\zeta\right)\right\\}+\mathrm{exp}\left\\{-\left(\mu
t_{*}+\int_{0}^{t_{*}}\varphi(\zeta)d\zeta\right)\right\\}$ (4)
$\displaystyle\times\Lambda\int_{0}^{t_{*}}\mathrm{exp}\left(\mu
t+\int_{0}^{t}\varphi(\nu)d\nu\right)dt>0.$

Doing similar procedure for the rest of the Eqs. of the model (1), it can be
proved that $P(t)>0$, where $\forall t>0$.

For examining the second part, the boundedness property, we have $0<P(0)\leq
N(t)$. Adding all the equations (1) of the system , we get

$\displaystyle\frac{dN(t)}{dt}$ $\displaystyle=\Lambda-\mu
N(t)-\delta_{1}I-\delta_{2}H-\delta_{3}U$ $\displaystyle\leq\Lambda-\mu N(t).$

Hence,

$\lim_{t\to\infty}\sup{N(t)}\leq\frac{\Lambda}{\mu}.$

$\hfill\Box$

####  3.1.1 Invariant region

Over here, we will study the dynamics of the COVID-19 mathematical model in
the following closed biologically feasible region.

$\Upsilon\subset\mathbb{R}_{+}^{5},$

where,
$\Upsilon=\left\\{(S,I,H,U,R)\in\mathbb{R}_{+}^{5}:S+I+H+U+R\leq\frac{\Lambda}{\mu}\right\\}$.

###### Lemma 2

The region given by $\Upsilon$, where $\Upsilon\subset\mathbb{R}_{+}^{5}$, is
positively invariant set for the model (1) with non-negative initial
conditions in $\mathbb{R}_{+}^{5}$

Proof: As in Lemma (1), it follows from the adding of all equation of the
human population of the model (1)

$\frac{dN(t)}{dt}\leq\Lambda-\mu N(t).$ (5)

So, it is vividly that

$\frac{dN(t)}{dt}\leq 0,~{}if~{}N(0)\geq\frac{\Lambda}{\mu}.$

And the solution of (5) is given by the following inequality,

$N(t)\leq N(0)e^{-\mu t}+\frac{\Lambda}{\mu}(1-e^{-\mu t}).$

Then, $N(t)\leq\frac{\Lambda}{\mu}$, if $N(0)\leq\frac{\Lambda}{\mu}$. Thus,
the region $\Upsilon$ is positively invariant set and also attract all the
possible solution trajectories in $\mathbb{R}_{+}^{5}$. $\hfill\Box$

##  4 Mathematical formulation of the fractional SIHUR model of COVID-19

In this section, we discuss the basic literature related to the fractional
operator and its applications to the SIHUR model of COVID-19.

###  4.1 Preliminaries of fractional calculus

Some useful definitions and lemmas are shown as follows

###### Definition 1

[30] The definition of Caputo fractional-order derivative is given by

$\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}f(t)=\begin{cases}\text{$\frac{1}{\Gamma(\nu-\alpha)}\int_{0}^{t}\frac{f^{(\nu)}(\xi)}{(t-\xi)^{1+\alpha-\nu}}d\xi$,}&\quad\text{$\nu-1<\alpha<\nu$,}\\\
\text{$\frac{d^{\nu}}{dt^{\nu}}f(t)$,}&\quad\text{$\alpha=\nu$},\\\
\end{cases}$ (6)

where $\nu$ presents the smallest positive integer not less than $\alpha$.

###### Definition 2

[30, 31] The Mittag–Leffler function $E_{\alpha}(Z)$ is given by

$E_{\alpha}(Z)=\sum_{\varkappa=0}^{\infty}\frac{Z^{\varkappa}}{\Gamma(\varkappa\alpha+1)},$

where $\nu-1\leq\alpha\leq\nu$

###  4.2 SIHUR model

We present the dynamics of the COVID-19 model (1) in Caputo sense (6) and we
have the following model:

$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}S(t)$
$\displaystyle=\Lambda-\varphi S-\mu S,$ (7)
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}I(t)$
$\displaystyle=\varphi S-k_{1}I,$ (8)
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}H(t)$
$\displaystyle=\gamma_{1}I-k_{2}H,$ (9)
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}U(t)$
$\displaystyle=\gamma_{2}I+\tau H-k_{3}U,$ (10)
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}R(t)$
$\displaystyle=\theta_{1}I+\theta_{2}H+\theta_{3}U-\mu R.$ (11)

The initial conditions for the model of Eqs. (7-11) are:
$S(0)\geq 0,~{}I(0)\geq 0,~{}H(0)\geq 0,~{}U(0)\geq 0,~{}R(0)\geq 0$,
where:
$\alpha$ represents the fractional-order, $\varphi=\frac{\beta(I+qH+mU)}{N}$,
$k_{1}=(\gamma_{1}+\gamma_{2}+\theta_{1}+\delta_{1}+\mu)$,
$k_{2}=(\tau+\theta_{2}+\delta_{2}+\mu)$ and
$k_{3}=(\theta_{3}+\delta_{3}+\mu)$.

##  5 The basic reproduction number

The basic reproduction number, which is known as $R_{0}$, is defined as the
predictable number of secondary cases generated by an ideal infected
individual in a totally susceptible population [32]. We calculate $R_{0}$ via
next-generation matrix approach, see [33, 34]. In order to calculate $R_{0}$,
we first need to determine the disease-free equilibrium point (DFEP) point by
setting the right-hand side of the Eqs. (7-11) equal to zero and obtain

$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}S(t)$
$\displaystyle=\Lambda-\varphi S-\mu S=0,$
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}I(t)$
$\displaystyle=\varphi S-k_{1}I=0,$
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}H(t)$
$\displaystyle=\gamma_{1}I-k_{2}H=0,$
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}U(t)$
$\displaystyle=\gamma_{2}I+\tau H-k_{3}U=0,$
$\displaystyle\mathrm{}_{0}^{C}\mathrm{D}_{t}^{\alpha}R(t)$
$\displaystyle=\theta_{1}I+\theta_{2}H+\theta_{3}U-\mu R=0.$

Then, the DFEP is the point in which there does not exist disease among all
individuals in the studied population, so the disease-free equilibrium point
is given by
$D_{0}=(S,I,H,U,R)=(\frac{\Lambda}{\mu},0,0,0,0)$.
The basic reproduction number is very crucial for the qualitative analysis of
our model. We compute $R_{0}$ by the following steps:

First, we divide our system into infected class Eqs. ($\ref{E101}-\ref{E103}$)
and non-infected class Eqs. (7) and (11). Next,

$\frac{d}{dt}\begin{bmatrix}I\\\ H\\\ U\end{bmatrix}=\psi-\zeta,$ (12)

where

$\psi=\begin{bmatrix}\frac{\beta S(I+qH+mU)}{N}\\\ 0\\\ 0\end{bmatrix}$ and
$\zeta=\begin{bmatrix}k_{1}I\\\ -\gamma_{1}I+k_{2}U\\\ -\gamma_{2}I-\tau
H+k_{3}U\end{bmatrix}$.
Then,
$F=J_{D_{0}}(\psi)=\begin{bmatrix}\beta&\beta q&\beta m\\\ 0&0&0\\\
0&0&0\end{bmatrix},$ $V=J_{e_{0}}(\zeta)=\begin{bmatrix}k_{2}k_{3}&0&0\\\
\gamma_{1}k_{3}&k_{1}k_{3}&0\\\
(\gamma_{1}\tau+\gamma_{2}k_{2})&k_{1}\tau&k_{1}k_{2}\end{bmatrix},$
and $V^{-1}=(J_{D_{0}}(\zeta))^{-1}=\begin{bmatrix}\frac{1}{k_{1}}&0&0\\\
\frac{\gamma_{1}}{k_{1}k_{2}}&\frac{1}{k_{2}}&0\\\
\frac{(\gamma_{1}\tau+\gamma_{2}k_{2})}{k_{1}k_{2}k_{3}}&\frac{1}{k_{2}k_{3}}&\frac{1}{k_{3}}\end{bmatrix}.$

Thereafter, the basic reproduction number is determined from the formula
$R_{0}\coloneqq\rho[FV^{-1}]$. Where $\rho(.)$ is the spectral of the matrix
$F=J_{D_{0}}(\psi)$ and $V^{-1}=(J_{D_{0}}(\zeta))^{-1}$. $J_{D_{0}}$
represents the Jacobian matrix of the matrices $\psi$ and $\zeta$ at DFEP,
$D_{0}$. $FV^{-1}$ is the next generation matrix. Accordingly, $R_{0}$ can be
written afterwards:

$R_{0}=\frac{\beta}{k_{1}}\bigg{(}1+q\frac{\gamma_{1}}{k_{2}}+m\frac{(\gamma_{1}\tau+\gamma_{2}k_{2})}{k_{2}k_{3}}\bigg{)}.$
(13)

Further, we can write $R_{0}$ in the following form:

$R_{0}=R_{1}+R_{2}+R_{3}+R_{4},$

where,

$R_{1}=\frac{\beta}{k_{1}},~{}~{}~{}~{}~{}R_{2}=\beta
q\frac{\gamma_{1}}{k_{1}k_{2}},~{}~{}~{}~{}~{}R_{3}=\beta
m\frac{\gamma_{1}\tau}{k_{1}k_{2}k_{3}},~{}~{}~{}~{}~{}R_{4}=\beta
m\frac{\gamma_{2}}{k_{1}k_{3}}.$

###  5.1 Stability of disease-free equilibrium point

In this section, we calculate both local and global stability of our system
(7-11), see section 2, around the disease-free equilibrium point (DFEP). We
then prove the local and global stability of our proposed model around the
DFEP. The epidemiological effect of the stability of the output of the DFEP
state is when a small influx of COVID-19 infections cases does not cause a
COVID-19 outbreak in the case $R_{0}<1$.

####  5.1.1 Local stability of disease-free equilibrium, $D_{0}$

###### Theorem 1

If $R_{0}<1$, then the DFEP, $D_{0}$ of the model (7-11) is locally
asymptotically stable if the condition $\mid
arg(\lambda)\mid\textgreater\alpha\frac{\pi}{2}$ is satisfied and unstable
otherwise.

Proof: The Jacobian matrix $J_{D_{0}}$ obtained at the DFE $D_{0}$ for our
system (7-11) as:

$J_{D_{0}}=\begin{bmatrix}-\mu&-\beta&-\beta q&-\beta m&0\\\ 0&\beta-
k_{1}&\beta q&\beta m&0\\\ 0&\gamma_{1}&-k_{2}&0&0\\\
0&\gamma_{2}&\tau&-k_{3}&0\\\ 0&\theta_{1}&\theta_{2}&\theta_{3}&-\mu\\\
\end{bmatrix},$ (14)

Vividly, from the matrix (14) the eigenvalues, $-\mu$ and$-\mu$ are negative
values. The remaining eigenvalues obtained from the below equation:

$\lambda^{3}+C_{1}\lambda^{2}+C_{2}\lambda+C_{3}=0.$ (15)

The coefficients involved in Eq. (14) are as follow:

$\displaystyle C_{1}$ $\displaystyle=k_{1}+k_{2}+k_{3}-k_{1}R_{1},$
$\displaystyle C_{2}$
$\displaystyle=k_{2}k_{3}+k_{1}k_{2}(1-R_{2})+k_{1}k_{3}(1-R_{4})-k_{1}(k_{2}+k_{3})R_{1},$
$\displaystyle C_{3}$ $\displaystyle=k_{1}k_{2}k_{3}(1-R_{0}).$

Vividly, $\forall~{}C_{i}$, where $i=1,2,3$ are positive if $R_{0}<1$. Then,
the eigenvalues from Eq. (15) are negative or have a negative real part. Since
all the eigenvalues $\lambda_{i}$, ($i=1,2,...,6$), of 14 satisfy the
condition $\mid arg(\lambda)\mid\textgreater\alpha\frac{\pi}{2}$, the DFEP
($D_{0}$) is locally asymptotically stable. $\hfill\Box$

####  5.1.2 Global stability of disease-free equilibrium, $D_{0}$

The global stability of the system at the DFEP, $D_{0}$, of the COVID-19
transmission model is investigated in the following consequence:

###### Theorem 2

The non-linear dynamical system (7-11) at the DFEP ($D_{0}$) is globally
asymptotically stable in $\Upsilon$ when $R_{0}<1$, and unstable in the case
$R_{0}>1$.

Proof: Let we suppose the following Lyapunov function to prove the desired
consequence:

$L(t)=\Psi_{1}I+\Psi_{2}H+\Psi_{3}U,$

where $\Psi_{j}$, for $j=1,2,3$, are used for unknown positive constants. By
differentiating the function $L(t)$ with respect to time $t$ and using the
Eqs. (8-9), we get:

$\displaystyle\frac{dL(t)}{dt}$ $\displaystyle=\Psi_{1}\bigg{\\{}\frac{\beta
S(I+qH+mU)}{N}-(\gamma_{1}+\gamma_{2}+\theta_{1}+\delta_{1}+\mu)I\bigg{\\}}+\Psi_{2}\bigg{\\{}\gamma_{1}I-(\tau+\theta_{2}+\delta_{2}+\mu)H\bigg{\\}}$
$\displaystyle+\Psi_{3}\bigg{\\{}\gamma_{2}I+\tau
H-(\theta_{3}+\delta_{3}+\mu)U\bigg{\\}}.$

As $\frac{S}{N}<1$. Then,

$\displaystyle\frac{dL(t)}{dt}$
$\displaystyle\leq\Psi_{1}\bigg{\\{}\beta(I+qH+mU)-(\gamma_{1}+\gamma_{2}+\theta_{1}+\delta_{1}+\mu)I\bigg{\\}}+\Psi_{2}\bigg{\\{}\gamma_{1}I-(\tau+\theta_{2}+\delta_{2}+\mu)H\bigg{\\}}$
$\displaystyle+\Psi_{3}\bigg{\\{}\gamma_{2}I+\tau
H-(\theta_{3}+\delta_{3}+\mu)U\bigg{\\}}=(\Psi_{1}\beta-\Psi_{1}\gamma_{1}-\Psi_{1}\gamma_{2}-\Psi_{1}\theta_{1}-\Psi_{1}\delta_{1}-\Psi_{1}\mu+\Psi_{2}\gamma_{1}+\Psi_{3}\gamma_{2})I$
$\displaystyle+(\Psi_{1}\beta
q-\Psi_{2}\tau-\Psi_{2}\theta_{2}-\Psi_{2}\delta_{2}-\Psi_{2}\mu+\Psi_{3}\tau)H+(\Psi_{1}\beta
m-\Psi_{3}\theta_{3}-\Psi_{3}\delta_{3}-\Psi_{3}\mu)U.$

Now, let $\Psi_{1}$, $\Psi_{2}$ and $\Psi_{3}$ values as below,
$\Psi_{1}=\frac{1}{k_{1}}$, $\Psi_{2}=\beta m\frac{\tau}{k_{1}k_{2}k_{3}}$ and
$\Psi_{3}=\beta m\frac{1}{k_{1}k_{3}}$.
Then, we get,

$\frac{dL(t)}{dt}\leq(R_{1}+R_{3}+R_{4}-1)I+q(R_{2})H.$

Obviously, $\frac{dL(t)}{dt}<0$ when $R_{0}<1$. Thus, the the largest compact
invariant set in $\Upsilon$ is the singleton set $D_{0}$ according to the
LaSalle’s invariant principle [35]. Thus, $D_{0}$ is entirely globally
asymptotically stable in $\Upsilon$. $\hfill\Box$

##  6 Sensitivity analysis and estimation of SIHUR parameters

In the previous Section 3, we derived the basic reproduction number for the
system (7-11) proposed in Section 2. The sensitivity analysis for the
threshold of pandemic ($R_{0}$) is significant. It is usually used to
determine the robustness of the model predictions to parameter values, since
there are common errors or uncertainty in data collection and assumed
parameter values [36, 37, 38]. The importance of sensitivity indices arises in
measuring the relative change in a variable when a parameter value changes.
Therefore, here we use the normalized forward sensitivity index of a variable
with respect to a given parameter. This is defined as the ratio of the
relative change in some variable to the relative change in the given
parameter. When the variable is a differentiable function of the parameter,
the sensitivity index maybe, on the other hand, be defined as the partial
derivative of the variable with respect to the given parameter. We can write
the definition as follows.

###### Definition 6.1

[37]: The normalized forward sensitivity index of a variable, $R_{0}$, that
depends differentiably on a given parameter, $\chi$, which is defined as:

$\Omega_{\chi}^{R_{0}}=\frac{\partial
R_{0}}{\partial\chi}\times\frac{\chi}{R_{0}}$ (16)

The sensitivity indices, sources, parameters values and initial values are
arranged in tables 1-2.

France
---
Symbol | Initial value | Method/ Source
$S(0)$ | 19948051 | $S(0)=N(0)-(I(0)+H(0)+U(0)+R(0))$
$I(0)$ | 29440000 |  [39]
$H(0)$ | 16785 |  [39]
$U(0)$ | 1074 |  [39]
$R(0)$ | 15867601 |  [40]
Table 1: Initial values of the SIHUR model for France. France
---
Parameter | Value | Sensitivity index | Method/ Source
$\mu$ | $\frac{1}{83.13\times 365}$ | $-6.5048\times 10^{-4}$ |  [41]
$\Lambda$ | $0.011$ | $-$ |  [42]
$\beta$ | $0.204$ | 1 |  [43]
$\delta_{1}$ | $5.027\times 10^{-3}$ | $-0.0992$ | [39]
$\delta_{2}$ | $0.2$ | $-2.33\times 10^{-5}$ |  [44]
$\delta_{3}$ | $0.416$ | $-1.4257\times 10^{-5}$ |  [15]
$\theta_{1}$ | $0.045$ | $-0.8882$ | Assumption
$\theta_{2}$ | $0.8$ | $-9.3242\times 10^{-5}$ |  [45]
$\theta_{3}$ | $0.6$ | $-2.0564\times 10^{-5}$ |  [45]
$\gamma_{1}$ | $5.701\times 10^{-4}$ | $-0.0111$ |  [39]
$\gamma_{2}$ | $-7.3793\times 10^{-4}$ | $-7.0205\times 10^{-4}$ |  [39]
$\tau$ | $0.064$ | $9.4130\times 10^{-6}$ |  [39]
$q$ | $0.2$ | $1.0714\times 10^{-4}$ | Assumption
$m$ | $0.5$ | $3.4822\times 10^{-5}$ | Assumption
Table 2: Sensitivity indices, data and values of parameters in the SIHUR model
for France.

###### Remark 2

The natural mortality rate is defined as
$\mu=\frac{1}{\emph{Life}~{}\emph{expectancy}\times 365}$.

##  7 Discussion and numerical results

This section was devoted to discussing the numerical simulation results of our
integer order system of Eqs. (7-11) and fractional-order one 6. We performed
the simulation results by using ode45 and fde12 packages in MATLAB, subjected
to the parameters and initial values listed in Tables 2 and 1, respectively.
The number of population was from [41] for France ($N(0)=65,273,511$). We
studied the corresponding dynamics of the coronavirus spread in France for
total COVID-19 hospitalized cases and critical care patients in ICUs for the
period from $24^{th}$ May 2022 to $13^{th}$ July 2022 (50 days). And we also
investigated the impact of precautionary measures, e. g., social distancing
and wearing masks, on hospitals and ICUs in France for the same period.

We predicted the COVID-19 development trends as shown in Figs. 2 with integer
and non-integer order. In Fig. 2(a), we see that the total number of
hospitalized cases decreases from $16785$ to nearly $16000$ in the first $4$
days, and then the number of hospitalizations is approximately constant until
the $7$ day. After that, we observe a notable exponential decay until the day
$50$. We also notice an acceptable match between the two carves for integer
and non-integer one. We show the two curves are identical to nearly the day
$10$. Next, from day $10$ to day $50$, a suitable match is indeed observed.
For Fig. 2(b), a vast increase in the total number of critical care cases in
ICUs is revealed, exceeding $2000$ cases, until the day $5$. Following this is
a major decrease in cases of ICUs until the last day of our study. And, again,
a satisfactory match between the two carves is noticed.

Precautionary measures are essential measures that several governments
imposed. Naturally, these should be directly taken when an epidemic is at the
onset. We, here, only study the impact of precautionary measures on the
cumulative number of hospitalized and critical care COVID-19 cases, see Figs.
3 and 4, by using our models proposed in Sec. 2 and 4. We assume that the
mitigation in precautionary measures, the transmission rate ($\beta$),
increases by $0.05$. So, we have analyzed the impact of basic ($\beta=0.204$),
below moderate ($\beta=0.254$), moderate ($\beta=0.304$), upper moderate
($\beta=0.354$) and high ($\beta=0.404$) precautionary measures.

In Figs. 3(a) and 4(a), it is observed that mitigation in precautionary has a
major effect on increasing the total number of COVID-19 cases in the hospitals
for all values of $\beta$. This major effect is for the first $10$ days and
the first $15$ days by the integer and fractional order models, respectively.
After these periods, a minor effect of the mitigation is shown. We see a rapid
decrease proceeds from days $10$ to $50$ in Fig. 3(a) and from days $15$ to
$50$ in Fig. 4(a). In Fig. 3(a), we find that for different values of $\beta$,
all curves intercept in the period from days $24$ to $28$, where the number of
cases is nearly $0.8\times 10^{4}$. The same intersection is shown in Fig.
4(a), but for a period from days $28$ to $32$, where cases are also ranging in
$0.8\times 10^{4}$. In addition, for both Figs. 3(a) and 4(a), after the
intersection region, we see all curves very slightly separate from each other.

In Figs. 3(b) and 4(b), we vividly see that the cumulative number of COVID-19
cases in French ICUs accelerates significantly on account of the decrease in
precautionary measures, i. e., increment in the value of $\beta$. That is,
where the number of cases surpasses $2000$. This increment is still until the
first $5$ and $6$ days for Fig. 3(b) and 4(b), respectively. And this is
followed by a major decrease, continuing to the end of the period in both
graphs, 3(b) and 4(b). The curves of both mentioned two graphs, for different
values of $\beta$, are nearly identical for period ($24$-$28$) days in 3(b)
and ($28$-$33$) days in 4(b). From days $28$ (Fig. 3(b)) and $33$ (Fig. 4(b)),
for different values of $\beta$, curves are very close together from one
another.

(a) Graphical description of hospitalized cases versus time.

(b) Graphical description of critical care cases versus time.

Figure 2: Simulation results of integer-order and fractional-order SIHUR model
for France.

(a) Graphical description of precautionary measures mitigation ($\beta$) on
hospitalized cases of COVID-19.

(b) Graphical description of precautionary measures mitigation ($\beta$) on
critical care cases of COVID-19.

Figure 3: Impact of precautionary measures mitigation ($\beta$) on
hospitalized and ICUs’ cases of COVID-19 for France by integer order SIHUR
model.

(a) Graphical description of precautionary measures mitigation ($\beta$) on
hospitalized cases of COVID-19.

(b) Graphical description of precautionary measures mitigation ($\beta$) on
critical care cases of COVID-19.

Figure 4: Impact of precautionary measures mitigation ($\beta$) on
hospitalized and ICUs’ cases of COVID-19 for France by fractional-order SIHUR
model.

##  8 Conclusion

In this work, we investigate the problem of COVID-19 breaking out in the
entire French hospitals and ICUs. The integer order and fractional-order
models are proposed to predict the COVID-19 spread in France from $24^{th}$
May 2022 to $13^{th}$ July 2022. In this paper, with the SIHUR model, we also
study the impact of the precautionary measures. We find that the increase in
the total number of hospitalized and critical care cases is in the first days
of our study. And this is followed by a significant decrease in the cumulative
number of COVID-19 cases. Clearly, we show that implementing the precautionary
measures has an apparent minor effect on the total number of patients in
hospitals and ICUs.

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# Using Multiple RDF Knowledge Graphs for Enriching ChatGPT Responses

Michalis Mountantonakis and Yannis Tzitzikas

Institute of Computer Science - FORTH-ICS, Greece
and Computer Science Department - University of Crete, Greece
{mountant<EMAIL_ADDRESS>
(April, 2023)

###### Abstract

There is a recent trend for using the novel Artificial Intelligence ChatGPT
chatbox, which provides detailed responses and articulate answers across many
domains of knowledge. However, in many cases it returns plausible-sounding but
incorrect or inaccurate responses, whereas it does not provide evidence.
Therefore, any user has to further search for checking the accuracy of the
answer or/and for finding more information about the entities of the response.
At the same time there is a high proliferation of RDF Knowledge Graphs (KGs)
over any real domain, that offer high quality structured data. For enabling
the combination of ChatGPT and RDF KGs, we present a research prototype,
called $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$, which is able to enrich any
ChatGPT response with more information from hundreds of RDF KGs. In
particular, it identifies and annotates each entity of the response with
statistics and hyperlinks to LODsyndesis KG (which contains integrated data
from 400 RDF KGs and over 412 million entities). In this way, it is feasible
to enrich the content of entities and to perform fact checking and validation
for the facts of the response at real time.

Demo Video: https://youtu.be/H30bSv9NfUw.

Demo URL: https://demos.isl.ics.forth.gr/GPToLODS.

##  1 Introduction

ChatGPT is a novel Artificial Intelligence (AI) chatbox
(https://chat.openai.com/), which is built on GPT-3.5 and GPT-4 families of
large language models (LLMs) [2], and provides detailed responses and human-
like answers across many domains of knowledge. However, it has not been
designed to store or retrieve facts, e.g., like a relational database or a
Knowledge Graph (KG). For this reason, in many cases it returns plausible-
sounding but incorrect or inaccurate responses [3]. Thereby, it is hard for
the user to check the validity of the answers returned by ChatGPT. Sometimes
the responses contain entities that do not exist, URLs that are wrong, facts
that cannot be verified, outdated data, and many others. On the contrary,
there are available numerous RDF KGs [4] (e.g., DBpedia [5], Wikidata [17],
YAGO [15], etc), that provides high quality structured data (that are updated
at least periodically), by using Linked Data techniques [1].

Concerning some key issues of ChatGPT (that can be improved by using KGs),
Fig. 1 shows a real conversation with ChatGPT (in March 27, 2023). First, we
asked about the birth place of Aristotle (issue A), and the output was a plain
text, without evidence or annotations. Afterwards, we desired to find
available RDF links for Aristotle (issue B), and only one of them was correct,
whereas the URI of Wikidata refers to a completely different entity. Then, we
asked about sources verifying that Stagira is located in Chalkidiki, and all
the returned URIs were invalid (issue C). In the last case, we asked a
question from a different domain, i.e., “Who scored the goal in UEFA Euro 2004
Final?”, and we retrieved erroneous facts (issue D); its response was “Angelos
Basinas from the penalty”, however the correct answer is “Angelos Charisteas
with a header”.

The objective of this paper is to aid the above issues, however, there are
several challenges that should be tackled. Indeed, it is quite difficult to
check the validity of entities and URIs, since it requires access to numerous
KGs, sources and resources in general. For tacking these challenges, we
demonstrate the research prototype $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$,
that enables the user to make a question, and instead of getting the raw
answer from ChatGPT, it retrieves the ChatGPT response annotated with the
identified entities, and with relevant data about these entities (URIs, facts
and KGs), For making this feasible at real time, we exploit LODsyndesis suite
of services [9, 11] including an Information Extraction (IE) service [12].
These services rely on LODsyndesis KG; a large KG, equipped with special
indexes and algorithms, that has integrated 2 billion facts for 412 million
entities, from 400 real RDF KGs. In this way, the user is able to speed up the
validation of the ChatGPT response and to retrieve more information.
Concerning the novelty, to the best of our knowledge it is the first system
offering annotation and linking of the ChatGPT responses to hundreds of KGs.

Figure 1: ChatGPT issues through real conversations (March 27, 2023)

The rest of this demo paper is organized as follows; 2 describes the related
work, 3 presents the process and the use cases and 4 concludes the paper.

##  2 Related Work

First, concerning ChatGPT and KGs, [13] provides a comparison for the Question
Answering task and they concluded that ChatGPT can have high precision in
general knowledge domains but very low scores in unseen domains, compared to a
KG-based approach. Regarding IE tools for Entity Recognition, there are
available approaches from several areas, i.e., from Natural Language
Processing (NLP)[6], from KGs [7, 14, 8], and from Neural Networks [16].
However, these tools link the entities to a single KG, and for this reason
$\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ uses the machinery of LODsyndesisIE
[12], which combines tools from NLP and KG, and links the entities to 400 RDF
KGs. Regarding the novelty, to the best of our knowledge there is no other
related system that annotates the response of ChatGPT and provides links and
services using hundreds of KGs at real time.

Figure 2: Screenshots from $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ research
prototype

##  3 The Process of $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ and Use Cases

First, the name of the prototype $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ comes
from the mathematical notation for function composition, i.e. ($GPT$ $\bullet$
$LODS$)($x$) =$LODS(GPT(x))$, where $LODS$ comes from LODSyndesis [11].
Concerning the process (see Fig. 2), the user submits through the web
application a question (in english), $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$
sends the question to the ChatGPT API and retrieves the response. Afterwards,
it applies the machinery of LODsyndesisIE (described in [12]) for recognizing
the entities of the response, i.e., it combines widely used Entity Recognition
tools (i.e., DBpedia Spotlight [7], WAT [14] and Stanford CoreNLP[6]) for
recognizing the entities. The next step is to further process the response for
creating the annotations and for adding statistics and links to LODsyndesis KG
for each recognized entity, and finally it returns the annotated response to
the user. Fig. 2 shows that by clicking on each entity, one can see its name,
image, type and statistics, such as the number of its RDF KGs, URIs and facts
that occur in LODSyndesis. Moreover, by clicking on the links the user can
browse (or download) all this data (e.g., the list of all the URIs of each
entity). Finally, several other services are offered, including a fact
checking service (see Fig. 2) that shows all the relations between any pair of
entities of the response (for fact validation).
Scenario & Use Cases. Below, we see a scenario which is planned to be also
demonstrated in the conference. The target audience can be any researcher of
AI area, since $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$
(https://demos.isl.ics.forth.gr/GPToLODS) is a research prototype that
combines tools and techniques of several AI components, including LLMs, NLP,
and Knowledge Representation and Reasoning (Linked Data and KGs). The scenario
is about the user questions (and needs) that are shown in Fig. 1, i.e.,
starting with the question “Which is the birth Place of Aristotle”. For
tackling the needs of this scenario, the results include: the annotated
entities, related information (identifiers, facts and datasets) about these
entities and validation of facts. The scenario can be accessed in an online
video (https://youtu.be/H30bSv9NfUw), which presents the issues of Fig. 1 and
how they can be solved through the following use cases.
Use Case 1. Annotation, Evidence and Linking. This refers to the issues A and
B of Figure 1, i.e., by having the annotation of the entities of the response,
we are able to find more information (links, datasets and facts) for each of
the entities of the response. Moreover, we can have access to the correct RDF
links for each entity, e.g., see in the left part of Fig. 2, that we retrieved
the correct Wikidata and VIAF link for Aristotle (e.g.,
https://www.wikidata.org/wiki/Q868), and in total 31 URIs.
Use Case 2. Fact Validation and Correct Answer. In Fig. 1 we can see that a
ChatGPT response can either provide wrong links for validation or even
erroneous facts (issues C,D). In the first case,
$\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ can be used for fact validation, e.g.,
in Fig. 2 it verified 2 facts of the response from popular KGs, like DBpedia
and Wikidata. Regarding issue D, $\mathtt{GPT}$$\bullet$$\mathtt{LODS}$ will
not find the erroneous fact in LODsyndesis KG. For finding the correct answer,
the user can further browse all the facts of an entity (by clicking on the
link).
Use Case 3. Dataset Discovery and Enrichment. The user can discover all the
datasets of each entity (e.g., see the lower right part of Fig. 2), and all
(or a part of) the facts of that entity in the KGs that are included in
LODsyndesis. This can be useful for enriching the available content of each
entity, e.g., for creating an application, a data warehouse [10], for
performing an analysis, etc.

##  4 Concluding Remarks

In this paper, we presented the research prototype
$\mathtt{GPT}$$\bullet$$\mathtt{LODS}$, which enables the real time annotation
and linking of a ChatGPT response to hundreds of RDF KGs, the enrichment of
its entities and the validation of its facts. As a future work, we plan to
improve the GUI, and the fact checking service by performing relation
extraction, to offer a REST API and to support multilinguality.

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