www.biomathforum.org/biomath/index.php/biomath
ORIGINAL ARTICLE
Modeling, analysis and simulations of MERS
outbreak in Saudi Arabia
Nofe Al-Asuoad, Meir Shillor
Department of Mathematics and Statistics
Oakland University
Rochester, MI, USA
nalasuoa@oakland.edu, shillor@oakland.edu
Received: 5 October 2017, accepted: 27 February 2018, published: 9 March 2018
Abstract—This work describes a continuous dif-
ferential equations model for the dynamics of Mid-
dle Eastern Respiratory Syndrome (MERS) and
provides its computer simulations. It is a con-
tinuation of our previous paper Al-Asuoad et al.
(Biomath 5, 2016) and it extends the simulations
results provided there, which were restricted to
the city of Riyadh, to the whole of Saudi Arabia.
In addition, it updates the results for the city of
Riyadh itself. Using an optimization procedure, the
system coefficients were obtained from published
data, and the model allows for the prediction of
possible scenarios for the transmission and spread
of the disease in the near future. This, in turn,
allows for the application of possible disease control
measures. The model is found to be very sensitive
to the daily effective contact parameter, and the
presented simulations indicate that the system is
very close to the bifurcation of the stability of the
Disease Free Equilibrium (DFE) and appearance of
the Endemic Equilibrium (EE), which indicates that
the disease will not decay substantially in the near
future. Finally, we establish the stability of the DFE
using only the stability number Rc, which simplifies
and improves one of the main theoretical results in
the previous paper.
Keywords-MERS model; stability of DFE and EE;
simulations; sensitivity analysis;
I. INTRODUCTION
This work uses the mathematical model con-
structed in [2] to study the dynamics of the Middle
East Respiratory Syndrome (MERS) in Saudi Ara-
bia. It also expands the study that was performed
there of the disease in the city of Riyadh, since
new data became available since the publishing
of the paper. The aim of this work is to provide
the health care community and related authorities
with a predictive tool that allows to assess various
MERS scenarios and the effectiveness of various
intervention practices.
MERS is a new respiratory disease caused by
the newly discovered Middle East Respiratory
Syndrome Coronavirus (MERS-CoV). The first
case of the disease was reported in Saudi Arabia
in June 2012, when a 60-year-old man died of
progressive respiratory and renal failure 11 days
Copyright: c© 2017 Al-Asuoad et al. This article is distributed under the terms of the Creative Commons Attribution License
(CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and
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Citation: Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in
Saudi Arabia, Biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
after hospital admission. He had a 7-day history of
fever, cough, expectoration and shortness of breath
[26], [30].
In September 2012 a case of a 49-year old man
from Qatar with pneumonia and kidney failure,
who was treated in an intensive care unit at a
London hospital, was reported. He had a history
of travel to Saudi Arabia. Further laboratory tests
revealed a positive MERS-CoV infection [6]. Ret-
rospectively, the infection was found in stored
respiratory and serum samples on two deceased
patients from Jordan, where in April 2012 an
outbreak of acute respiratory illness occurred in
a public hospital [17].
In 2003, a previously unknown coronavirus, the
Severe Acute Respiratory Syndrome Coronavirus
(SARS-CoV), caused a global outbreak of pneu-
monia that resulted in approximately 800 deaths
[22]. MERS-CoV that has effects similar to those
of SARS-CoV, is classified as a coronavirus, which
is a family of single-stranded RNA viruses. This
family includes viruses that cause mild illness such
as common cold as well as severe illness such as
SARS in humans. MERS-CoV is a beta coron-
avirus which has not been identified in humans be-
fore 2012 and is different from any coronaviruses
(including SARS-CoV) that have been found in
humans or animals [10], [31].
Within a year from its discovery, a total of 130
MERS-CoV cases were identified, 58 of which
died, which means that the case fatality rate is
45%, much higher than SARS-CoV, which has a
case fatality of 15% [24], [27]. Up to date (August
31, 2017) 2067 confirmed cases of MERS-CoV
have been reported worldwide, out of these 1679
were reported from Saudi Arabia where the case
fatality rate has been 40.6% [28]. The infection has
been a global threat due to continuous outbreaks in
the Arabian Peninsula and international spread to
26 countries including Qatar, Jordan, United Arab
Emirates, United Kingdom, the Philippines, United
States and other countries, by infected travelers
[14]. In 2015, a large outbreak happened in South
Korea, which was the first outbreak outside the
Arabian Peninsula [19], [23], and 186 people were
infected, 38 of whom died.
After intensive search, camels were found to
have a high rate of anti-MERS-CoV antibodies,
which indicates that they were infected with the
virus. Then, definite evidence of camel-to-human
transmission of the virus has been reported re-
cently [5], [29]. Moreover, there is clear evidence
that the infection is transmitted from person to
person upon close contact, including from patients
to healthcare workers [4], [12].
The incubation period from exposure to the
development of clinical disease is from five to
14 days. MERS-CoV is typically characterized
by cough, fever, sore throat, chills, myalgia and
shortness of breath [11], [13]. One-third of the pa-
tients had also gastrointestinal symptoms such as
vomiting and diarrhea. The common complications
of the MERS-CoV infection include pneumonia,
acute respiratory distress syndrome and respiratory
failure. Although it is known that asymptomatic
infection occurs, the percentage of patients who
have it is unknown, yet, [3], [7].
No specific treatment is available for the MERS-
CoV infection. Currently, the management of the
disease is done by supportive therapy that mini-
mizes the symptoms. Some patients require me-
chanical ventilation or extra-corporal membrane
oxygenation. Since no vaccine exists for MERS
[9], [18], once a case is identified, the individual
and those connected to them are being isolated to
minimize the spread of the disease.
Part of the content of this article can be found
in the recent Doctoral Dissertation [1] where ad-
ditional information can also be found.
To help assessing the threat of the spread of
MERS, we constructed a mathematical model as a
tool to predict possible future scenarios and the ef-
fectiveness of various intervention procedures. The
model is in the form of a coupled system of five
nonlinear ordinary differential equations (ODEs)
for the susceptible, asymptomatic, clinically symp-
tomatic, isolated and recovered populations. It is of
a rather standard MSEIR type (see, e.g., [8], [15],
[16], [21] and the many references therein). The
novelty in this work lies in the theoretical proof
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
of the global stability of the Endemic Equilibrium
(when it exists), and simulations for the whole of
Saudi Arabia.
First, we mathematically analyze the model and
provide a new proof of the global stability of
disease free equilibrium (DFE) and of the endemic
equilibrium (EE), when it exists (and then the DFE
becomes unstable). Moreover, the new proof of
the global stability of the EE is simpler and more
elegant than the one in Proposition 4 in [2], since
it uses only the effective reproduction number or
the stability control parameter Rc.
Second, the model is used for simulations of
the overall outbreak in Saudi Arabia, and it also
extends the study of MERS done in [2] of the
city of Riyadh, as more data has been collected
since its publication. Indeed, currently there exists
data that spans 1550 days, [25]. For the whole of
Saudi Arabia, the model parameters were fitted to
the data using the first 865 days, then runs for
1690 days, until Nov. 4, 2020, were performed,
allowing for the prediction of the disease spread
in the next three years. In the previous paper, we
fitted our model to the daily reported cumulative
cases of MERS data for Riyadh for the period
from Nov. 4, 2013 to March 17, 2016 (865 days).
Here, we fitted the model to the data from Nov.
4, 2013 to July 11, 2017 (1346 days). The model
was found to be very sensitive to the scaled contact
parameter that is directly related to the number of
individuals a person is in contact with each day.
Nevertheless, the simulations provide a very good
fit with what has been observed and are similar in
their predictions of the near future, say the next
two years.
The rest of the paper is structured as follows.
The model is described in Section II, where its
compartmental structure and flow chart are also
provided. The stability analysis of the DFE and EE
is done in Section III, where the local and global
stability of the DFE are studied. Our new results
on the global stability of the EE are summarized
in Proposition 4 and proved using the Lyapunov
method and LaSalle’s principle. The description
of the simulation results can be found in Section
IV. First the optimized parameters for the baseline
for Saudi Arabia are presented and the simula-
tion results depicted. Then, the extended study of
Riyadh is described. The sensitivity of the model
to the contact parameter β is done in Section V,
which is one of the main characteristics of the
model. In Section VI we depict graphically the
errors, i.e., the difference between the data and
model predictions. The paper concludes in Section
VII where the results are summarized and some
unresolved issues indicated.
II. THE MODEL
We use the basic model of MERS dynamics de-
veloped in [2], where full details of the model and
its underlying assumptions can be found so, the
description of the model here follows very closely
the one in [2]. The model represents the disease
dynamics of five populations of individuals: sus-
ceptible S(t), asymptomatic E(t), symptomatic
I(t), isolated J(t), and recovered R(t), where the
time t is measured in days. Also, N(t) denotes the
total population at time t and is given by
N(t) = S(t) + E(t) + I(t) + J(t) + R(t). (1)
The model flow diagram is depicted in Fig. 1.
P
- S
µS
-
?
E
µE
-
?
I
kE
µI
��
γI
@@R
-
?
R
σ1I
σ2J
d1I
d2J
µR
HHj
J
µJ
?
6
@@R
Fig. 1: Compartmental structure and flow chart for
the model
The mathematical model for the MERS disease
(the basic model in [2]), which consists of five rate
equations for the dynamics of S,E,I,J and R, is
as follows.
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
Model 1: Find five functions (S,E,I,J,R),
defined on [0,T] and values in R+ ∪ {0}, such
that for 0 < t ≤ T , the following hold:
dS
dt
= P −
S(βI + �EβE + �JβJ)
N
−µS, (2)
dE
dt
=
S(βI + �EβE + �JβJ)
N
− (k + µ)E, (3)
dI
dt
= kE − (γ + σ1 + d1 + µ)I, (4)
dJ
dt
= γI − (σ2 + d2 + µ)J, (5)
dR
dt
= σ1I + σ2J −µR, (6)
together with the initial conditions
S(0) = S0, E(0) = E0, I(0) = I0,
(7)
J(0) = J0, R(0) = R0.
Here, we denote by S0 > 0 the initial population
before the disease outbreak, and E0,I0,J0 and
R0 are nonnegative populations that satisfy (1) at
t = 0. It is appropriate, within the context of
Saudi Arabia to assume that that initially S0 =
N(0) > 0, and the others vanish, meaning that at
first there are only susceptibles in the population.
However, for the sake of generality, we assume
that the initial populations are nonnegative.
The rate of change of the susceptible popu-
lation S(t) is given in equation (2), where P
represents the recruitment rate and is assumed
to be a constant. We denote by β (1/day) the
effective contact rate. The dimensionless param-
eters �E and �J are the transmission coefficients
from asymptomatic and symptomatic individuals,
respectively. Thus, the second term on the right-
hand side of equation (2) describes the rate at
which the susceptibles become infected with the
virus as a result of contact with asymptomatic,
infected, and isolated individuals. The population’s
natural death rate is µ (1/day). Equation (3)
describes the rate of change of the asymptomatic
population E(t). These individuals carry the virus
but have not yet developed clinical symptoms of
MERS, which means that they can infect suscep-
tibles unintentionally. Following [2], k (1/day)
is the rate of development of clinical symptoms
in asymptomatic population. The last term on the
right-hand side of (3) describes both the mortality
rate due to development of clinical symptoms at
rate k and the natural mortality rate. We turn
now to equations (4) and (5). It is assumed that
infectives are isolated at rate γ (1/day). The
parameters σ1,σ2 (1/day) denote the recovery
rate of symptomatic and isolated populations,
while d1,d2 (1/day) are the disease-induced death
for symptomatic and isolated populations, respec-
tively. The first term on the right-hand side of the
equation (4) represents the asymptomatic individu-
als who developed clinical symptoms and become
infected, while the first term on the right-hand
side of the equation (5) represents the isolated-
infected individuals. Finally, the rate of change of
the recovered population R(t) is given in equation
(6) where the first and second terms on the right-
hand side represent the recover-infected and the
recover-isolated individuals, respectively. We note
that since there is no data, yet, about possible re-
infection of the Recovered, we assume that they
are permanently immune.
We recall that µ denotes the natural death rate.
Thus, if a person has a a life expectancy of 80
years, then the natural death rate µ is 0.000034
per day. It was assumed that in the absence of
disease, the total Saudi population was N = P
µ
=
32 million for P = 1088 people and µ = 0.000034
per day and the total population of Riyadh was
N = P
µ
= 5 million for P = 170 people and
µ = 0.000034 per day.
A full description of the variables, parameters,
and the parameters’ values considered in the model
can be found in Table (I).
Finally, the cumulative cases of MERS up to
time t, were obtained from the expression
CT(t) =
∫ t
0
(kE(τ)) dτ, (8)
with the initial value CT(0) = 0, while the
cumulative recovered from the disease up to time
t, were obtained from
CR(t) =
∫ t
0
(σ1I(τ) + σ2J(τ))) dτ, (9)
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
with the initial value CR(0) = 0. Similarly, the
cumulative deaths induced by the disease up to
time t, were obtained from
D(t) =
∫ t
0
(d1I(τ) + d2J(τ))) dτ, (10)
with the initial value D(0) = 0.
We note, for the sake of completeness, that the
following results, in addition to those mentioned
and improved in Section III, were established in
[2]: the existence and uniqueness, positivity and
boundedness of the solutions. Indeed, it was found
that all the trajectories of the system lie in the set
Ω = {(S,E,I,J,R) ∈ R5+ :
0 ≤ S + E + I + J + R = N ≤
P
µ
+N0},
which is invariant and compact.
III. STABILITY ANALYSIS
We first analyze the stability of the Disease-free
Equilibrium (DFE) and of the Endemic Equilib-
rium (EE), both locally and then globally. How-
ever, we note that the local stability of the EE has
already been done in [2]. These results improve
considerably the results there, and also simplify
them as they show that the effective reproduction
or control number Rc controls the stability, and
this closes a gap described there. Thus, there is no
need for the basic stability number R0 that was
introduced there. We have,
Rc = λ1 =
�Eβ
D1
+
βk
D1D2
+
�Jβkγ
D1D2D3
. (11)
Here, λ1 is the largest eigenvalue of the Jacobian
matrix J(P0) given shortly, and
D1 = k + µ,
D2 = γ + d1 + σ1 + µ, (12)
D3 = σ2 + d2 + µ.
A. Local stability of the DFE
We begin with the local stability. It is straight-
forward to see that the DFE is given by
P0 = (S0, 0, 0, 0, 0),
and S0 = P/µ.
In our previous paper ( [2]), the local stability
of the P0 was proved in term of R0. Here, we
used the Routh-Hurwitz criterion (see e.g., [21])
to prove the following stability result using Rc.
Our local result is the following.
Proposition 2: The disease-free equilibrium of
the model is locally asymptotically stable when
Rc < 1 and is unstable when Rc > 1.
Proof: The Jacobian matrix of the system at
the disease-free equilibrium P0 = (Pµ , 0, 0, 0, 0),
when γ > 0, is given by
J(P0) =
−µ −�Eβ −β −�Jβ 0
0 −D1 + �Eβ β �Jβ 0
0 k −D2 0 0
0 0 γ −D3 0
0 0 σ1 σ2 −µ
.
The characteristic equation is
(λ + µ)2(λ3 + Aλ2 + Bλ + C) = 0,
where
A = D1 + D2 + D3 − �Eβ
=
1
D3D2
(
D3(D3D2 +D
2
2 +kβ)
+�Jβkγ+D1D2D3(1−Rc)) ,
B = D1D2 + D1D3 + D2D3
−((D2 + D3)�Eβ + kβ)
= D3D2 +
D3
D2
kβ + �Jβkγ
(
1
D2
+
1
D3
)
+(D1(D2 + D3))(1 −Rc),
C = D1D2D3 (1 −Rc) .
Now, since (λ + µ)2 = 0, there are two equal and
negative eigenvalues, λ1,2 = −µ. The remaining
three eigenvalues are determined from the cubic
equation
λ3 + Aλ2 + Bλ + C = 0.
It follows from the Routh-Hurwitz criterion ( [21])
that the solutions of this equation have negative
real parts when A > 0,B > 0,C > 0, and AB >
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
C. Clearly, A,B,C > 0 when Rc < 1. It remains
to show that AB > C. We have,
AB = 3D1D2D3 (1 −Rc)
+(1−Rc)(D21(1−Rc)(D2 +D3)+D1(D
2
2 +D
2
3))
+�Jβkγ
(
3+
(
1
D2
+
1
D3
)
+
�Jβkγ
D3D2
+2D1(1−Rc)
)
+D3D2(D2 + D3)
+kβ
(
D3
(
kβ
D22
+
D3
D2
+2
)
+D1(1−Rc)
(
2D3
D2
+1
))
+
k2β2�Jγ
D2
(
1
D3
+
2
D2
)
.
It is seen that AB > C since
3D1D2D3 (1 −Rc) > D1D2D3 (1 −Rc).
Thus, AB > C, when Rc < 1, and it follow
from the Routh-Hurwitz criterion that all the
eigenvalues have negative real parts in this case.
We conclude that P0 is locally asymptotically
stable.
B. Global stability of the disease-free equilibrium
The global stability of the disease-free equilib-
rium is shown in the following proposition, based
on the construction of an appropriate Lyapunov
function and the use of LaSalle’s invariance prin-
ciple.
Proposition 3: The disease-free equilibrium of
the model is globally asymptotically stable in R5+
when Rc ≤ 1.
Proof: To show the global stability of the
disease-free equilibrium P0, we construct the fol-
lowing Lyapunov function
L(E,I,J) = ω1E + ω2I + ω3J,
in which we only considered the variables rep-
resenting the infected components of the model,
where,
ω1 = �ED2D3 + kD3 + �Jkγ,
ω2 = D1(D3 + �Jγ), ω3 = �JD1D2.
Next, we let
Λ =
βI + �EβE + �JβJ
N
, (13)
where N = S + E + I + J + R.
Calculating the derivative of L along the solu-
tion (E(t),I(t),J(t)) of the system (3)–(5), we
obtain
dL
dt
= ω1
dE
dt
+ ω2
dI
dt
+ ω3
dJ
dt
= ω1(SΛ−D1E)+ω2(kE−D2I)+ω3(γI−D3J)
= ω1SΛ−ω1D1E+ω2kE−ω2D2I+ω3γI−ω3D3J
= ω1ΛS −D1D2D3(I + �EE + �JJ)
= ω1ΛS −D1D2D3
N(βI + β�EE + β�JJ)
Nβ
= ω1ΛS −
ΛND1D2D3
β
=
ΛND1D2D3
β
(
ω1ΛSβ
ΛND1D2D3
− 1
)
=
ΛND1D2D3
β
(
Sβ(�ED2D3 +kD3 +�Jkγ)
ND1D2D3
− 1
)
≤
ΛND1D2D3
β
(
β(�ED2D3 +kD3 +�Jkγ)
D1D2D3
−1
)
=
ΛND1D2D3
β
(
�Eβ
D1
+
βk
D1D2
+
�Jβkγ
D1D2D3
−1
)
=
ΛND1D2D3
β
(Rc − 1) ≤ 0.
Therefore, since all the parameters are non-
negative,
dL
dt
≤ 0
when Rc ≤ 1. We note that dL/dt = 0 if and
only if E = I = J = 0 i.e., it vanishes only at the
disease-free equilibrium. Therefore, if we let
Γ = {(S,E,I,J,R) ∈ R5+ :
dL
dt
≤ 0},
then the largest compact and invariant set in Γ
is the singleton {P0}. By LaSalle’s invariance
principle ( [20]), every solution of the equations
(2)-(6), with initial conditions in ΩN , approaches
P0 as t → ∞, whenever Rc ≤ 1. This completes
the proof.
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
C. Global stability of the endemic equilibrium
The global asymptotic stability of the endemic
equilibrium when it exists, is also proved by
constructing an appropriate Lyapunov function and
using LaSalle’s invariance principle. We note that
it follows from the local stability analysis (see
[2][Proposition 5]) that the EE exists and is unique
only when Rc > 1. Therefore, we deal with the
case Rc > 1.
Theorem 4: The the endemic equilibrium P∗ =
(S∗,E∗,I∗,J∗,R∗) exists and is globally asymp-
totically stable when Rc > 1, and does not exist
when Rc < 1.
Proof: To show the global stability of the
endemic equilibrium P∗, we consider change of
variables and construct the following Lyapunov
function L,
L(W1,W2,W3,W4,W5)
= W∗21 + W
∗2
2 + W
∗2
3 + W
∗2
4 + W
∗2
5 ,
where,
W∗1 = S −
P
Λ + µ
,
W∗2 = E −
ΛS
D1
,
W∗3 = I −
kE
D2
,
W∗4 = J −
γI
D3
,
W∗5 = R−
σ1I + σ2J
µ
.
We note that L(0, 0, 0, 0, 0) = 0 and
L(W1,W2,W3,W4,W5) is positive. Calculating
the derivative of L about the system (2)–(6), we
obtain
dL
dt
=2
(
S −
P
Λ + µ
)
dS
dt
+ 2
(
E −
ΛS
D1
)
dE
dt
+ 2
(
I −
kE
D2
)
dI
dt
+ 2
(
J −
γI
D3
)
dI
dt
+ 2
(
R−
σ1I + σ2J
µ
)
dR
dt
.
Then, using the equations, we obtain(
S−
P
Λ+µ
)
dS
dt
=PS−(Λ+µ)S2−
P2
Λ+µ
+(Λ+µ)S
P
Λ+µ
,(
E−
ΛS
D1
)
dE
dt
= ΛSE−D1E2−
(ΛS)2
D1
+D1E
ΛS
D1
,(
I−
kE
D2
)
dI
dt
=kEI−D2I2−
(kE)2
D2
+D2I
kE
D2
,
(
J−
γI
D3
)
dJ
dt
=γIJ−(D3)J2−
(γI)2
D3
+D3J
γI
D3
,(
R−
σ1I+σ2J
µ
)
dR
dt
= (σ1I+σ2J)R
−µR2−
(σ1I+σ2J)
2
µ
+µR
(σ1I+σ2J)
µ
.
It follows that
dL
dt
=− 2(Λ+µ)
(
S−
P
Λ+µ
)2
−2D1
(
E−
ΛS
D1
)2
− 2D2
(
I −
kE
D2
)2
− 2D3
(
J −
γI
D3
)2
− 2µ
(
R−
σ1I + σ2J
µ
)2
.
Hence, each term of dL/dt < 0, and thus the
largest invariant set at which dL/dt = 0 is the
equilibrium point P∗ and it follows from LaSalle’s
invariant principle [20] that P∗ is globally asymp-
totically stable.
IV. NUMERICAL SIMULATIONS
We turn to describe the numerical simulations of
the model. We used the same numerical algorithm
that was developed and implemented in MAPLE in
[2]. Then, we run extensive numerical simulations,
using the values of the parameters given in Table I
for the baseline simulations. A number of other
sets of parameters were also used, as explained be-
low. The simulations were run for both the city of
Riyadh and the whole of Saudi Arabia. Those for
Riyadh were an extension of the simulation in [2],
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
since new cases were found since the last day of
simulation reported there, and the additional data
has been incorporated into the simulations below.
The simulations of Saudi Arabia were new and
motivated by the fact that the model predictions
in [2] were very close to what subsequently has
been observed in the field.
The data currently available for the outbreaks of
MERS in Riyadh and in Saudi Arabia was from
November 4, 2013 till February 1, 2018, a total
of 1550 days (∼ 51 months). The parameters P
and µ, which were not associated with the disease,
were readily available for both places. The other
model parameters were obtained by fitting the
numerical solutions to the data of the first 865
days of the disease breakout using the optimization
routine lsqcurvefit in MATLAB. This generated the
baseline case parameters that are provided in Table
I.
Then, we run the simulations for a longer time
and observed the model predictions in the follow-
ing 540 days for which data were available but
not used in the parameters fitting. This provided
an insight into how well the model predicted the
disease dynamics. We would like to point out here,
as was noted in [2], that the additional data was
found to fit very well into the model, without any
need to change the previously fitted parameters,
and we describe these results in detail below.
We first present the baseline simulations for the
whole country, and this is completely a novel ad-
dition to the literature. Then, we study the disease
spread in the city of Riyadh, where additional
information was provided. Finally, we perform a
reduced sensitivity analysis for the model with
respect to the scaled effective contact number
β, which shows that the simulation results are
extremely sensitive to its value. We discuss it in
Section V.
A note on the optimization for the model
parameters. The optimization program found
a number of local minima that provided, for the
cases of Saudi Arabia and Riyadh, results in which
Rc had a value close to one, both below and above
one. We chose the baseline simulations in both
cases to be those with Rc < 1, since these lead
to a slightly better fit. However, below we depict
simulation results for Saudi Arabia with either
Rc = 0.99704 for the baseline case, in which
the DFE is stable and attracting, or Rc = 1.004,
in which the EE is asymptotically stable and the
DFE unstable. Similarly, for the city of Riyadh,
we used the baseline case with Rc = 0.9928,
which is related to a stable and attracting DFE, or
Rc = 1.0045 that has an unstable DFE and stable
and attracting EE. These are directly related to the
sensitivity of the model to β and as we explain
below, it was found that a change of 0.3% leads
to the change in the stability, hence in the disease
dynamics. We note in the cases when Rc < 1,
when there in EE, the decay to the disease free
equilibrium to the DFE is slow, over a period of
more than a hundred years, and since our interest
in this work is only the next few years, we do not
depict the long time results.
A. Baseline simulations – Saudi Arabia
In this subsection, we describe the baseline
simulations for the whole of Saudi Arabia with
Rc = 0.99704. Then, for the sake of completeness,
below we describe another set of simulations with
Rc > 1. Both agree well with the data in the short
term (1550 days ≈51 months), but differ in long
term behavior, as was to be expected, since the
baseline is associated with Rc < 1 in which there
is no EE, while the second set with Rc > 1 is
associated with stable and attracting EE. However,
both values are very close to 1.
The daily reported new cases of MERS were ob-
tained from the Saudi Arabian Ministry of Health
website [25]. More specifically, we considered
the period of 1550 days from Nov. 4, 2013 to
February 1, 2018. A nonlinear least square fit,
using lsqcurvefit - a Matlab function contained in
the optimization toolbox- was performed to obtain
the model parameter values. As was noted above,
we fitted the basic model parameters of (2)–(6)
to the data from Nov. 4, 2013 until Mar. 17,
2016 (a period 865 days) using reasonable initial
guesses for the parameter values and obtained
better estimates of the same parameters from the
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
TABLE I: Model baseline parameters for Saudi Arabia and Riyadh
Parameter Description Saudi Arabia Riyadh
S susceptible population S(0) = 31,999,990 S(0) = 4,999,990
E asymptomatic population E(0) = 0 E(0) = 0
I symptomatic population I(0) = 10 I(0) = 10
J isolated population J(0) = 0 J(0) = 0
R recoverd population R(0) = 0 R(0) = 0
P recruitment rate of susceptible individuals 1088 170
β effective contact rate 0.1818 0.1222
�E reduction factor in transmission rate by exposed per day 0.3688 0.2956
�J reduction factor in transmission rate by exposed per day 0.1 0.0901
k rate of development of clinical 0.6937 0.1529
symptoms in asymptomatic population
µ natural death rate 0.000034 0.000034
d1 disease-induced death for symptomatic population 0.0191 0.0110
d2 disease-induced death for isolated population 0.1260 0.0516
σ1 recovery rate in symptomatic population 0.0336 0.02913
σ2 recovery rate in Isolated population 0.2472 0.1098
γ isolation rate 0.1577 0.1335
optimization fit, which are given in Table I. The
results of model fitting, which is the baseline, are
depicted in Fig. 2.
0 200 400 600 800
0
200
400
600
800
1000
1200
Time in Days
C
u
m
u
la
ti
v
e
c
a
s
e
s
o
f
M
E
R
S
Fig. 2: MERS model parameters fit to daily re-
ported cumulative new cases data - red points -
obtained from Saudi Arabian Ministry of Health
website during the first 865 days of the disease
outbreak. The solid blue line represents the base-
line model prediction. The estimated parameters
are provided in Table I.
We next describe the simulations of the MERS
model, equations (2)–(6) with the initial condi-
tions S(0) = 31, 999, 990, E(0) = 0,I(0) =
10,J(0) = 0,R(0) = 0. This choice of these
initial conditions was made based on the data or
the lack of it on Nov. 4, 2013 when it became
available and when the simulations start. The
results of the numerical simulation, depicted in
Fig. 3, show a very good agreement between the
model predictions- smooth colored curves- and the
observed data -red dots (red curves on this scale).
We emphasize again that the curve fitting was done
on the first 865 days and the next 683 days are
the model predictions, and they agree with the
field date very well, indeed. Then, in the figure we
depict the model prediction for another three years,
or 1006 days, until Nov. 4, 2020, which means
cumulative results for 2555 days of simulations.
In the simulations, Fig. 3, the cumulative num-
ber of: infected cases is depicted in the top (T),
the recovered in the middle (M), and the deaths
on the bottom (B). The model predicts, that if the
epidemic continues its current trajectory, by Nov.
4, 2020 (another 33 months), there will be about
2200 new cases (M), the cumulative recovered will
be about 1449 (M), and the cumulative deaths will
be around 760, (B).
We note that there is an under-reporting issue
with the cumulative number of recovered (M) for
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
TABLE II: Model parameter for Saudi Arabia, the
case Rc = 1.004.
Parameter Parameters value Units
P 1088 individual/day
β 0.1284 1/day
�E 0.0490 -
�J 0.1077 -
k 0.2915 1/day
µ 0.000034 1/day
d1 0.00490 1/day
d2 0.1072 1/day
σ1 0.0490 1/day
σ2 0.1077 1/day
γ 0.0542 1/day
the first 183 days, since the data was not available,
so the number was set as zero and this explains
why the whole red graph is below the blue curve.
However, by raising the red dots curve to agree
with the blue curve on day 183 led to a very good
fit on the cumulative recovered, too.
Although in this case the DFE is stable and
attracting, and the disease will die out in about
20 years, we did not show the long term behavior
since at this stage it seems not to be very relevant.
It is seen that if MERS continues in the current
trajectory, in the next three years one can expect
another 548 cases or so in the whole country.
Although the number is not large relatively to the
size of the population of the whole country, the
possible epidemic-like spread of the disease must
be taken into account by the authorities. We return
to this point in the conclusions section.
Next, as was noted above, running the opti-
mization program with different initial conditions
yielded a number of sets of values, related to local
minima of the optimization function. So for the
sake of completeness, we present simulation re-
sults with somewhat different parameters, provided
in Table II in which case Rc = 1.004. Thus, the
EE is stable and attracting, and the disease cannot
be eradicated.
We note that in the case when the DFE is
asymptotically stable we had β = 0.1818 and here
Fig. 3: Saudi baseline simulations of cumulative cases
of MERS (T) - green curve; cumulative number of
recovered (M) - blue curve; and cumulative number of
death (B) - brown curve. The red dots are the field data.
The run was for 2555 days (∼ 84 months).
we have β = 0.1284, which is very interesting and
this point is discussed further in Section 5, when
we study the sensitivity with respect to β.
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
Since in this case Rc = 1.004, the endemic
equilibrium exists and is asymptotically stable.
Indeed, we found that the EE values P∗ =
(S∗,E∗,I∗,J∗,R∗) were
P∗ = (31, 772, 111; 27; 51; 13; 113, 994).
Note that the population of the country was taken
as |P∗| = 31, 886, 195.
What these numbers mean is that when the
disease is close to the EE, one would have on each
day about 27 asymptomatics, 51 people with the
disease symptoms, 13 isolated, and 113, 994 had
just recovered. Clearly these numbers, if MERS
would takes such a turn, pose significant chal-
lenges to the authorities and the whole society.
The eigenvalues corresponding to the Jacobian
matrix J(P∗) were found to be
− 0.000034, −0.399388, −0.225192
− 0.000017 ± 0.0001127i,
indicating that the EE is locally stable and attract-
ing, as was claimed above.
We solved the system with the same initial
conditions as above. The simulations results are
depicted in Fig. 4, where the run was for 2555
days (∼ 84 months). We note that the endemic
equilibrium is approach in about 80 years.
The cumulative infected cases of MERS are
depicted in the upper left (T), the cumulative
number recovered in the upper right (M), and the
cumulative number of deaths on the bottom (B). If
the epidemic switches to such a trajectory, by the
year 2020 (another 33 months), the model predicts
a bit over 4000 new cases, the cumulative death
will be around 2000, and there will be about 2000
recovered.
It seems, comparing the two simulations, that
the fit for Rc = 0.99704 is better than the one with
Rc = 1.004, however, visually it is also related to
the vertical scales in the figures. The difference
in the fit is actually quite small, although the
difference in the predictions is larges.
The simulations in Saudi Arabia show that even-
tually the disease will either disappear, or stabilize.
At this stage it is impossible do decide which is
Fig. 4: Saudi simulations with Rc = 1.004. Cumulative
cases of MERS (T), cumulative number of recovered
(M), and cumulative number of death (B). The red dots
are the field data. The run was for 2555 days (∼ 84
months).
the case based on the model predictions. However,
in either case, in the next few years MERS will be
spreading, possibly between 620 and 1240 deaths
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
per year. This we would like the authorities to be
aware of it.
B. Baseline simulations - Riyadh
We turn to the simulations of Riyadh, and as was
noted above, these include more data that became
available since our paper [2]. Moreover, the pa-
rameters are somewhat different than those in the
article. Indeed, as was the case with Saudi Arabia,
we present simulations with two sets of parameters
obtained by the optimization subroutine. The first,
which seems to be a better fit with the field data
has Rc = 0.9928, while the other one, whose fit
is slightly worse has Rc = 1.0045. Therefore, the
first simulations are in the case when MERS will
eventually disappear, while in the second case the
endemic equilibrium exists and the disease will
persist.
In the first case, the values of the parameters
obtained from the fit for Rc = 0.9928 are given in
Table I. We solved the MERS model (2)–(6) with
the initial conditions S(0) = 4, 999, 990, E(0) =
0,I(0) = 10,J(0) = 0,R(0) = 0. The results of
numerical simulation, depicted in Fig. 5, seem to
agree well with the observed data. Again, we note
that the fit was found using the field data reported
in [25] for the first 865 days, and the very good
agreement in the following 683 days just supports
the model predictions.
The cumulative infected cases of MERS are
depicted on the upper left (T) of Fig. 5, the cu-
mulative number of recovered in the upper right
(M), and the cumulative number of deaths on the
bottom (B). If the epidemic continues at its current
trajectory, by 2020 (another 33 months), the model
predicts about 840 new cases, the cumulative death
will be around 260, and there will be about 580
recovered.
As was done above, we now present another
simulation in the case when Rc = 1.0045. The
values of the parameters obtained from the opti-
mization fit are given in Table III.
We solved the MERS model with the same ini-
tial conditions and simulation results are depicted
Fig. 5: Riyadh baseline simulations of cumulative cases
of MERS (T) - green curve; cumulative number of
recovered (M) - blue curve; and cumulative number of
death (B) - brown curve. The red dots are the field data.
The run was for 2555 days (∼ 84 months).
in Fig. 6. The model predicts that if the epidemic
would proceed on this trajectory, by the year 2020
(another 33 months), there would be about 1440
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
TABLE III: Model parameter for Riyadh, the case
Rc = 1.0045.
Parameter Parameters value Units
P 170 individual/day
β 0.04 1/day
�E 0.7105 -
�J 0.7107 -
k 0.3165 1/day
µ 0.000034 1/day
d1 0.0077 1/day
d2 0.0426 1/day
σ1 0.0089 1/day
σ2 0.0449 1/day
γ 0.0316 1/day
new cases, the cumulative death will be around
740, and there will be about 680 recovered.
When comparing with the predictions of the
case with Rc < 1 above, it is found that there
would be 480 more deaths in the next three years.
We conclude that wether the DFE is stable and
attracting and eventually the disease will disappear
or the EE is stable and attracting and the disease
will be active for along time, in the near future,
say the next three years, one can expect at least a
hundred to two hundred and fifty deaths per year
in the city of Riyadh. However, these scenarios
depend crucially on the effective contact number
β. So we turn to discuss this dependence next.
V. SENSITIVITY TO β
This section deals with the considerable sensi-
tivity of the model to the scaled contact number
β. First, we describe the mathematical aspects,
then we remark on the possible disease control
implications of this model sensitivity. We note that
we already performed a similar study in [2] for the
city of Riyadh, and it was found that the system
was very sensitive with respect to β. Here, we
perform it for the whole country of Saudi Arabia
and very similar results are obtained. For the sake
of completeness, we do it for the city of Riyadh
too. This sensitivity may have considerable policy
implications.
We selected three typical examples that predict
very different scenarios, with very close values of
Fig. 6: Riyadh simulations with Rc = 1.0045. Cu-
mulative cases of MERS (T), cumulative number of
recovered (M), and cumulative number of death (B).
The red dots are the field data. The run was for 2555
days (∼ 84 months).
the contact rate β, and we run the simulations
for over seven year, actually 2600 days (since the
beginning of MERS in Saudi Arabia). We used the
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
Fig. 7: Saudi Arabia. Simulation results of the
first case, for the cumulative reported cases of
MERS (T) and the cumulative number of deaths
(B). Parameter values used are β = 0.1818 – solid
curves; β = 0.1824 –dashed; and β = 0.1830 –
dash-dot curves.
baseline value β = 0.1818 (Table I), and the two
additional values β = 0.1824 and β = 0.1830.
The choice was such that the stability numbers
were Rc < 1, Rc = 1 and Rc > 1, respectively,
but the first and last with values very close to 1.
The simulations are depicted in Fig. 7, where the
case with β = 0.1824 is depicted in solid curves,
β = 0.1824 in dashed curves, and β = 0.1830 in
dash-dot curves. The predicted cumulative cases of
MERS at the end of the seven years (T) were found
to be about 2200, 4200 and 9580, respectively. The
cumulative deaths were found to be about 760,
1450 and 3280, respectively. A noticeable differ-
ence among the three cases was found, indeed,
the numbers more than doubled from the first to
the second and from the second to the third case,
while the difference between consecutive values of
β was just 0.3%.
This clearly indicates that the model is very
sensitive to the value of β. As was pointed out in
[2], it is very unlikely that this just a mathemat-
ical model. This belief is also supported by the
description in the literature on the virulent spread
of MERS in confined places.
We conducted a similar study of seven years
for the city of Riyadh, with results depicted in
Fig. 8. We used three scenarios with the values
β = 0.1222 (solid lines), β = 0.1231 (dashed
lines), and β = 0.1240 (dash-doted lines). The
choice was based on the same considerations as
for the whole country. The results were about
850, 1740 and 4442 cumulative cases of MERS,
respectively, shown in Fig. 8 (T); and about 265,
540, and 1365 cumulative deaths, respectively,
shown in Fig. 8 (B).
It is seen that changes of 0.7% lead to the
doubling of the cumulative cases of MERS and
of the cumulative numbers of deaths. Again, this
reinforces the issue of the extreme sensitivity of
the model to the scaled contact number.
VI. THE MODEL ERRORS
In this short section we provide a graphic repre-
sentations of the errors, which are the differences
between the data points and the model solution
results. These are depicted in Figs. 9 to 11. In Fig.
9 we show the difference between the cumulative
reported cases of MERS in Saudi Arabia, for the
865 days used to find the system coefficients using
the optimization routine in MATLAB. It represents
the errors in the results depicted in Fig. 2 above.
Next, Fig. 10 presents the errors for Saudi Arabia
in the cumulative numbers of reported cases and
the deaths. These are the details presented in Fig.
3 above. Finally, the errors between the data and
model simulations for the city of Riyadh in the
cumulative numbers of reported cases and the
deaths are depicted in Fig. 11. These are taken
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Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
Fig. 8: Riyadh: Simulation results for the cu-
mulative reported cases of MERS (left) and the
cumulative number of deaths (right). Parameter
values used are β = 0.1222 – dashed curves,
β = 0.1231 – solid curves, and β = 0.124 –
dashed dot curves.
from the results in Fig. 5. It seems that the errors
do not have any noticeable pattern, with mean
about zero, which reinforces our confidence in the
model predictions.
VII. CONCLUSIONS
This work deals with the possible trajectories
of the MERS disease in Saudi Arabia and in the
city of Riyadh. It is a continuation of the study
in [2] where the basic model was constructed and
simulations of the outbreak of MERS in Riyadh
were conducted.
Our aim in this work was two-fold. First, we
established the local stability of the DFE and the
0 100 200 300 400 500 600 700 800 900
Time in Days
-80
-60
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-20
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20
40
60
80
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u
m
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v
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r
e
p
o
rt
e
d
c
a
se
s
o
f
M
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R
S
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Fig. 9: The difference between the data and the
model solution in Fig.2
0 200 400 600 800 1000 1200 1400 1600
Time in Days
-150
-100
-50
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50
100
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200
C
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Fig. 10: Saudi Arabia: The difference between the
data and the solution.
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Time in Days
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Fig. 11: Riyadh: The difference between the data
and the solution.
global stability of the both the DFE and the EE
by using only the effective reproduction number or
stability control number Rc, and this improves the
theoretical results in [2], Section 3.3. The analysis
of the stability of the model’s equilibrium points
can be found in Section 3.
The second and more important aim was to
simulate and predict the disease outbreak in Saudi
Arabia in the very near future, actually, the next
two years. We fit the model parameters to a part
of the available data, from the first 865 days since
the disease was identified, to see how does the
model compare with the data for the next 683
days for which data is available, and then used the
model to predict the disease outcomes for the next
three years, assuming that its trajectory remains
the same. It was seen that for both Saudi Arabia
and the city of Riyadh, the model predictions for
the last 683 days were excellent. Nevertheless,
considering the future predictions of the model
some caution is in order. Indeed, the baseline
simulations for Saudi Arabia, which agreed very
well with the data for the 1550 days since the dis-
ease was identified, were with the control number
Rc = 0.99704. However, another parameter fit,
which was almost as good, was with Rc = 1.004.
In the first case there was no endemic equilibrium
(EE), while in the second case the EE was found
to be asymptotically stable, and these explain why
the predictions for the next three years somewhat
diverge. In the first case the model predictions for
the next three years, until Nov. 4, 2020, there will
be about 2200 new cases, the cumulative recovered
will be about 1449, and the cumulative deaths
will be around 760, Fig. 3. In the second case the
model predictions were over 4000 new cases, the
cumulative death will be around 2000, and there
will be about 2000 recovered. The difference in the
predictions is noticeable, although in a country of
32 million population these do not seem to be too
divergent.
However, at this stage of the research it is not
clear which scenario will play itself in the long
run, the one with Rc = 0.99704 in which the
disease dies (although in 20 years or so) or the
one with Rc = 1.004, in which the disease is
endemic and lingers for a long time. Nevertheless,
both predictions seem very reasonable and only
time will tell which would be closer to field ob-
servations. We stress again that these observations
depend crucially on the assumption that the disease
continues its current trend.
Similar observations were found for Riyadh,
provided in Section 5.
One of the main mathematical features of the
model, already pointed out in [2], is its consider-
able sensitivity to the value of the scaled contact
number β. The number measures the probability
that one contact between a susceptible individual
and a sick one results in infection, and therefore
it includes the rate at which people meet each
other. Indeed, as was seen in Section 5, Figs. 7
Biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 Page 16 of 18
http://dx.doi.org/10.11145/j.biomath.2018.02.277
Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
and 8, for Saudi Arabia and for Riyadh, changes of
0.3% and 0.7%, respectively, led to quite different
predictions, increases of more than 100%. This
sensitivity may have considerable policy implica-
tions. In settings where many people congregate
and contact is high, the value of may be β higher
with possibly severe outbreaks of MERS.
Acknowledgement. The authors would like to
thank the reviewers for their comments that im-
proved the presentation of the work, and made it
easier to read.
REFERENCES
[1] N Al-Asuoad, “Mathematical Models, Analysis and
Simulations in Epidemiology, Population Dynamics, and
Heart Tissue,” Ph.D. thesis, Oakland University, USA,
2017.
[2] N Al-Asuoad, L Rong, S Alaswad and M Shillor,
“Mathematical model and simulations of MERS out-
break: Predictions and implications for control mea-
sures,” Biomath (2016) 5,
http://dx.doi.org/10.11145/j.biomath.2016.12.141
[3] YM Arabi, AA Arifi, HH Balkhy, et al., “Clinical course
and outcomes of critically ill patients with Middle East
respiratory syndrome coronavirus infection,” Ann Intern
Med., 60(6) (2014), 389–397.
http://dx.doi.org/10.7326/M13-2486
[4] A Assiri, A McGeer, TM Perl, et al., “Hospital outbreak
of Middle East Respiratory Syndrome Coronavirus”, N.
Engl. J. Med., 369 (2013), 407–416.
http://dx.doi.org/10.1056/NEJMoa1306742
[5] EI Azhar, SA El-Kafrawy, SA Farraj, et al., ”Evi-
dence for camel-to-human transmission of MERS coro-
navirus,” N. Engl. J. Med., 370 (2014), 2499–2505.
http://dx.doi.org/10.1056/NEJMoa1401505
[6] A Bermingham, MA Chand, CS Brown, et al., “Severe
respiratory illness caused by a novel coronavirus, in
a patient transferred to the United Kingdom from the
Middle East, September 2012”, Euro Surveill., 17(40)
(2012).
[7] SR Bialek, D Allen, F Alvarado-Ramy, et al.,; Cen-
ters for Disease Control and Prevention (CDC), “First
confirmed cases of Middle East Respiratory Syndrome
Coronavirus (MERS-CoV) infection in the United
States,” Updated Information on the Epidemiology of
MERS-CoV Infection, and Guidance for the Public,
Clinicians, and Public Health Authorities-May 2014”,
MMWR, 63(19) (2014), 431–436.
[8] F Brauer, P van den Driessche, J Wu, (Eds.) Mathemat-
ical Epidemiology, Lecture Notes in Mathematics 1945,
Springer, Berlin, Heidelberg, 2008.
http://dx.doi.org/10.1007/978-3-540-78911-6
[9] C Brown, G Carson, M Chand, et al., “Treatment
of MERS-CoV: information for clinicians. Clinical
decision-making support for treatment of MERS-CoV
patients,” Public Health England, 3 (2014).
[10] VM Corman, I Eckerle, T Bleicker, et al., “Detection
of a novel human coronavirus by real-time reverse-
transcription polymerase chain reaction,” Euro Surveill.,
17 (39) (2012). pii=20285.
[11] M Gańczak, “Etiological, epidemiological and clinical
aspects of coronavirus infection MERS-CoV,” in Pol
Merkur Lekarski, 38(223) (2015), 46–50.
[12] P Gautret, GC Gray, RN Charrel, et al., “Emerging viral
respiratory tract infectionsenvironmental risk factors and
transmission,” The Lancet, 14 (2014), 1113–1122.
http://dx.doi.org/10.1016/S1473-3099(14)70831-X
[13] H Geng and W Tan, “A novel human coronavirus:
Middle East respiratory syndrome human coronavirus,”
Sci China Life Sci., 56(8) (2013), 683–687.
http://dx.doi.org/10.1007/s11427-013-4519-8
[14] LO Gostin and D Lucey, “Middle East respiratory
syndrome: a global health challenge”, JAMA, 314(8)
(2015), 771–772.
http://dx.doi.org/10.1001/jama.2015.7646
[15] HW Hethcote, “The mathematics of Infectious Dis-
eases,” SIAM Rev., 42(4), 599–653.
https://doi.org/10.1137/S0036144500371907
[16] HW Hethcote, “The basic epidemiology models: mod-
els, expressions for R0, parameter estimation, and ap-
plications,” a chapter in Mathematical Understanding of
Infectious Disease Dynamics, S Ma and Y Xia (Eds.)
Lecture Notes Series 16, World Scientific, Singapore,
2009.
[17] B Hijawi, M Abdallat, A Sayaydeh, et al., “Novel
coronavirus infections in Jordan, April 2012: epidemi-
ological findings from a retrospective investigation,”
Eastern Mediterranean Health J., 19 (2013), 12–18.
[18] I Khalid, BM Alraddadi, Y Dairi Y, et al., “Acute man-
agement and long-term survival among subjects with
severe Middle East respiratory syndrome coronavirus
pneumonia and ARDS,” Respir Care., 61(3) (2016),
1340–348. http://dx.doi.org/10.4187/respcare.04325
[19] A Khan, A Farooqui, Y Guan and DJ Kelvin, “Lessons
to learn from MERS-CoV outbreak in South Korea,” J
Infect Dev Ctries, 9(6) (2015), 543–546.
http://dx.doi.org/10.3855/jidc.7278
[20] JP LaSalle, “The Stability of Dynamical Systems,”
Regional Conference Series in Applied Mathematics,
SIAM, 1976; https://doi.org/10.1137/1.9781611970432
[21] JD Murray, Mathematical Biology, Springer 1989,
http://dx.doi.org/10.1007/b98868
[22] L Nelson, H David, A Wu,et al., “A major outbreak of
severe acute respiratory syndrome in Hong Kong,” N.
Engl. J. Med., 348 (2003), 1986–1994.
http://dx.doi.org/10.1056/NEJMoa030685
[23] HY Park, EJ Lee, YW Ryu, et al., “Epidemiological
investigation of MERS-CoV spread in a single hospital
in South Korea,” Euro Surveill,20 (2015), 1–6.
Biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 Page 17 of 18
http://dx.doi.org/10.11145/j.biomath.2016.12.141
http://dx.doi.org/10.7326/M13-2486
http://dx.doi.org/10.1056/NEJMoa1306742
http://dx.doi.org/10.1056/NEJMoa1401505
http://dx.doi.org/10.1007/978-3-540-78911-6
http://dx.doi.org/10.1016/S1473-3099(14)70831-X
http://dx.doi.org/10.1007/s11427-013-4519-8
http://dx.doi.org/10.1001/jama.2015.7646
https://doi.org/10.1137/S0036144500371907
http://dx.doi.org/10.4187/respcare.04325
http://dx.doi.org/10.3855/jidc.7278
https://doi.org/10.1137/1.9781611970432
http://dx.doi.org/10.1007/b98868
http://dx.doi.org/10.1056/NEJMoa030685
http://dx.doi.org/10.11145/j.biomath.2018.02.277
Nofe Al-Asuoad, Meir Shillor, Modeling, analysis and simulations of MERS outbreak in Saudi Arabia
[24] JS Peiris, Y Guan and KY Yuen, “Severe acute respira-
tory syndrome,” Nat Med., 10 (2004), 88–97.
http://dx.doi.org/10.1038/nm1143
[25] Saudi Arabian Ministry of Health.
http://www.moh.gov.sa/en/CCC/PressReleases/Pages/
default.aspx?PageIndex=1
[26] L Van der Hoek, K Pyrc, MF Jebbink, et al., “Identifi-
cation of a new human coronavirus,” Nat. Med., 10(4)
(2004), 368–373. http://dx.doi.org/10.1038/nm1024
[27] World Health Organization, “Middle East respiratory
syndrome coronavirus (MERS-CoV) summary
and literature update-as of 20 September 2013”.
http://www.who.int/csr/disease/coronavirus infections/
update 20130920/en/
[28] World Health Organization, “Middle East respiratory
syndrome coronavirus (MERS-CoV)-update”.
http://www.who.int/csr/don/2014 06 13 mers/en/
[29] World Health Organization, “Update on MERS-CoV
transmission from animals to humans, and interim
recommendations for at-risk groups.” WHO June 2014.
http://www.who.int/csr/disease/coronavirus infections/
MERS CoV RA 20140613.pdf
[30] AM Zaki, S van Boheemen, TM Bestebroer, ADME
Osterhaus, and RAM Fouchier, “Isolation of a novel
coronavirus from a man with pneumonia in Saudi
Arabia,” N. Engl. J. Med., 367 (2012), 1814–1820.
http://dx.doi.org/10.1056/NEJMoa1211721
[31] A Zumla, DS Hui and S Perlman, “Middle East res-
piratory syndrome,” The Lancet, 386(9997) (2015),
995–1007. http://dx.doi.org/10.1016/S0140-6736(15)
60454-8
Biomath 7 (2017), 1802277, http://dx.doi.org/10.11145/j.biomath.2018.02.277 Page 18 of 18
http://dx.doi.org/10.1038/nm1143
http://www.moh.gov.sa/en/CCC/PressReleases/Pages/default.aspx?PageIndex=1
http://www.moh.gov.sa/en/CCC/PressReleases/Pages/default.aspx?PageIndex=1
http://dx.doi.org/10.1038/nm1024
http://www.who.int/csr/disease/coronavirus_infections/update_20130920/en/
http://www.who.int/csr/disease/coronavirus_infections/update_20130920/en/
http://www.who.int/csr/don/2014_06_13_mers/en/
http://www.who.int/csr/disease/coronavirus_infections/MERS_CoV_RA_20140613.pdf
http://www.who.int/csr/disease/coronavirus_infections/MERS_CoV_RA_20140613.pdf
http://dx.doi.org/10.1056/NEJMoa1211721
http://dx.doi.org/10.1016/S0140-6736(15)60454-8
http://dx.doi.org/10.1016/S0140-6736(15)60454-8
http://dx.doi.org/10.11145/j.biomath.2018.02.277
Introduction
The Model
Stability analysis
Local stability of the DFE
Global stability of the disease-free equilibrium
Global stability of the endemic equilibrium
Numerical Simulations
Baseline simulations – Saudi Arabia
Baseline simulations - Riyadh
Sensitivity to
The model errors
Conclusions
References