Mathematical analysis of a MERS-Cov coronavirus model

Mahmoud H. DarAssi; Taqi A. M. Shatnawi; Mohammad A. Safi

doi:10.1515/dema-2022-0022

Open Access Published by De Gruyter Open Access July 13, 2022

Mathematical analysis of a MERS-Cov coronavirus model

Mahmoud H. DarAssi , Taqi A. M. Shatnawi and Mohammad A. Safi

From the journal Demonstratio Mathematica

https://doi.org/10.1515/dema-2022-0022

Abstract

In this study, we have proposed a mathematical model to describe the dynamics of the spread of Middle East Respiratory Syndrome disease. The model consists of six-coupled ordinary differential equations. The existence of the corona-free equilibrium and endemic equilibrium points has been proved. The threshold condition for which the disease will die out or becomes permanent has been computed. That is the corona-free equilibrium point is locally asymptotically stable whenever the reproduction number is less than unity, and it is globally asymptotically stable (GAS) whenever the reproduction number is greater than unity. Moreover, we have proved that the endemic equilibrium point is GAS whenever the reproduction number is greater than unity. The results of the model analysis have been illustrated by numerical simulations.

Keywords: MERS-Cov; coronaviruses; reproduction number; equilibria; stability

MSC 2010: 34D23; 92B05; 37N25; 93D05

1 Introduction

In recent years, the threat of emerging and re-emerging viruses on public health and economics has increased drastically. In the past few decades, human beings have suffered from several coronaviruses such as severe acute respiratory syndrome (SARS), Middle East Respiratory Syndrome (MERS-Cov), and recently the coronavirus disease (COVID-19) pandemic. Coronaviruses are single RNA viruses that cause respiratory tract infections. Those viruses were discovered in the 1960s [1,2].

The MERS-Cov is a respiratory disease caused by a coronavirus that attacks the respiratory tract. The main symptoms of MERS-Cov are fever, shortness of breath, and cough from mild to severe with an incubation period of 2–14 days. In some cases, it may cause diarrhea, nausea, vomiting, and myalgia [2].

The first case was recorded in Saudi Arabia in 2012 as it was reported by the World Health Organization (WHO). Moreover, the number of confirmed cases since 2012 is 2,464 with a death rate of 34% in 27 countries. 2073 confirmed cases were recorded in Saudi Arabia with a death rate of 37% [3].

In 2015, South Korea has suffered from the MERS-Cov outbreak. The reported number of confirmed cases was 168 cases. The latter was the largest in the region other than Middle East [4].

Mathematical models have been used to explore the dynamical spread of disease, such as hepatitis, cholera, HIV, SARS, Ebola, yellow fever, smallpox, and MERS [5,6, 7,8,9, 10,11,12, 13,14,15, 16,17,18, 19,20,21, 22,23,24, 25,26]. Tang et al. [27] have studied the spread of MERS-Cov and the expression of dipeptidyl peptidase 4 (DPP4). They have discussed the global stability of the equilibrium points of the model, and they have shown that the spread of MERS-Cov can be controlled by decreasing the rate of DPP4. Berrai et al. [28] have used the semi-group theory and optimal control to show the existence of a solution for the proposed model and to prove the existence of optimal control. Manaqib et al. [29] have studied the effect of the medical mask usage and vaccination on the spread of MERS-Cov. They have found that if the reproduction number is less than one, then the disease will die out. On the other hand, if the reproduction number is more than unity, the disease will be permanent. DarAssi et al. [9] have discussed a discrete-time MERS-Cov model. They have studied the global stability of the equilibrium points, and they have shown that the disease will die out if the reproduction number is less than unity.

In this paper, we have proposed a mathematical model to describe the spread of MERS-Cov. coronavirus. This model consists of six compartments: susceptible, exposed, infected with symptoms, infected without symptoms, hospitalized, and recovered or removed individuals. This paper is organized as follows: In Section 2, we formulate the model. In Section 3, the local and global stability of corona-free equilibrium is investigated. In Section 4, we have studied the existence of the endemic equilibrium point, and we have discussed the local stability at this point. The numerical simulation and the conclusion have been discussed in Section 5.

2 Mathematical model and basic properties

The SEAIHR epidemic model for MERS-Cov has been proposed. We have divided the total population P ( t ) into six compartments as follows: susceptible individuals ( S ( t ) ) , exposed individuals ( E ( t ) ) , the asymptomatic infected individuals ( A ( t ) ) , the symptomatic infected individuals ( I ( t ) ) , the hospitalized individuals ( H ( t ) ) , and the recovered or removed individuals ( R ( t ) ) . Therefore, the total population P ( t ) = S ( t ) + E ( t ) + A ( t ) + I ( t ) + H ( t ) + R ( t ) .

The susceptible individuals ( S ( t ) ) will increase by the recruitment rate of ( Π ) , and it will decrease by an infection rate of ( α ) and by a death rate of ( μ ) . We assumed that all compartments in this population have the same death rate of ( μ ). The exposed individuals ( E ( t ) ) will increase by the infected rate of ( α ) and will decrease to asymptomatic ( A ( t ) ) at a rate of ( σ ), and it will decrease to symptomatic ( I ( t ) ) at a rate of ( r ) . The asymptomatic people ( A ( t ) ) will increase to the exposed individuals ( E ( t ) ) at the rate of ( σ ) , and it will be reduced by progression to the exposed at the rate of ( r ) and to the recovered individuals ( R ( t ) ) at a rate of ( γ 1 ) . The symptomatic people ( I ( t ) ) will increase by the exposed individuals at a rate of ( r σ ), and will decrease to the hospitalized individuals (isolated people) ( H ( t ) ) at a rate of ( ϕ ) , to the recovered people at a rate of ( γ 2 ) , and disease-induced death at a rate of ( δ 1 ). The hospitalized (isolated) people will increase by the symptomatic individuals ( I ( t ) ) at a rate of ( ϕ ) and will decrease to recovery at a rate of ( γ 3 ) and disease-induced death at a rate of ( δ 2 ). Therefore, the SEAIHR epidemic model is governed by the following system of ordinary differential equations:

(1) d S d t = Π − α ( t ) S ( t ) − μ S ( t ) , d E d t = α ( t ) S ( t ) − ( σ + μ ) E ( t ) , d A d t = ( 1 − r ) σ E ( t ) − ( γ 1 + μ ) A ( t ) , d I d t = r σ E ( t ) − ( ϕ + γ 2 + μ + δ 1 ) I ( t ) , d H d t = ϕ I ( t ) − ( γ 3 + μ + δ 2 ) H ( t ) , d R d t = γ 1 A ( t ) + γ 2 I ( t ) + γ 3 H ( t ) − μ R ( t ) ,

where

α ( t ) = β [ A ( t ) + η 1 I ( t ) + η 2 H ( t ) ] P ( t ) .

The aforementioned model is subject to the following initial conditions:

(2) S ( 0 ) > 0 , E ( 0 ) > 0 , A ( 0 ) > 0 , I ( 0 ) > 0 , H ( 0 ) > 0 , R ( 0 ) > 0 .

Let k 1 = σ + μ , k 2 = μ + γ 1 , k 3 = μ + ϕ + δ 1 + γ 2 , and k 4 = μ + δ 2 + γ 3 , then model (1) becomes

(3) d S d t = Π − α ( t ) S ( t ) − μ S ( t ) , d E d t = α ( t ) S ( t ) − k 1 E ( t ) , d A d t = ( 1 − r ) σ E ( t ) − k 2 A ( t ) , d I d t = r σ E ( t ) − k 3 I ( t ) , d H d t = ϕ I ( t ) − k 4 H ( t ) , d R d t = γ 1 A ( t ) + γ 2 I ( t ) + γ 3 H ( t ) − μ R ( t ) .

Figure 1 illustrates the flow diagram of the model (1).

Figure 1

The schematic diagram for the model (1).

2.1 Basic properties

All the parameters of the model (1) are non negative because it is a human population model. Moreover, the following result holds.

Lemma 1

The variables of the model (1) are non negative for all time t.

Proof

Let y = sup { x > 0 : S > 0 , E > 0 , A > 0 , I > 0 , H > 0 , R > 0 ∈ [ 0 , x ] } . Upon considering the first equation of the system (1), we have

d d x S ( x ) exp μ x + ∫ 0 x α ( τ ) d τ = Π exp μ x + ∫ 0 x α ( τ ) d τ .

Thus,

S ( y ) exp μ y + ∫ 0 y α ( τ ) d τ − S ( 0 ) = ∫ 0 y Π exp μ z + ∫ 0 z α ( τ ) d τ d z ,

so that,

S ( y ) = S ( 0 ) exp − μ y − ∫ 0 y α ( τ ) d τ + exp − μ y − ∫ 0 y α ( τ ) d τ ∫ 0 y Π exp μ z + ∫ 0 z α ( τ ) d τ d z > 0 .

In a similar manner, we can prove that E > 0 , A > 0 , I > 0 , H > 0 , and R > 0 for all time t > 0 .□

Moreover, the method in Appendix A of [30] can be used to obtain the same results mentioned earlier. The following result can be claimed.

Lemma 2

The closed set

D = ( S , E , A , I , H , R ) ∈ R + 6 : S + E + A + I + H + R ≤ Π μ

is positively invariant.

Proof

By adding all the equations in (1), we have,

(4) d P d t = Π − μ P − ( δ 1 I + δ 2 H ) .

If P ≥ Π / μ , then d P / d t ≤ 0 (Since d P / d t ≤ Π − μ P ). By the use of the standard comparison theorem [31], we can conclude that P ≤ P ( 0 ) e − μ t + Π μ ( 1 − e − μ t ) . Therefore, if P ( 0 ) ≤ Π / μ , then P ( t ) ≤ Π / μ . This means that the region D is positively invariant. Furthermore, if P ( 0 ) > Π / μ , then either the solution enters D in finite time or P ( t ) approaches Π / μ asymptotically. Thus, the region D attracts all solutions in R + 6 .□

It is sufficient to study the dynamics of the flow generated by the model (1) in D , since the region D is positively invariant. Moreover, the existence and uniqueness of the equilibrium points for the proposed model can be investigated as in [32].

3 Stability of corona-free equilibrium

3.1 Local stability of disease free equilibrium (DFE)

The model (1) has DFE given by

(5) E 0 = ( S ∗ , E ∗ , A ∗ , I ∗ , H ∗ , R ∗ ) = ( Π / μ , 0 , 0 , 0 , 0 , 0 ) .

The next-generation method [33,34] will be used to investigate the local stability of the corona-free equilibrium point (CFE). Upon using the same notations in [34], the matrix of the new infection terms, F , and the matrix of the transition terms, V , associated with the model (1) are given as follows:

F = 0 β η 1 β η 2 β 0 0 0 0 0 0 0 0 0 0 0 0

and

V = σ + μ 0 0 0 − ( 1 − r ) σ γ 1 + μ 0 0 − r σ 0 ϕ + γ 2 + μ + δ 1 0 0 0 − ϕ γ 3 + μ + δ 2 .

It is known that the basic reproduction number [35,36] is denoted by ℛ 0 = ρ ( F V − 1 ) , where ρ is the spectral radius. For the proposed model (1), the basic reproduction number is given by the following expression:

ℛ 0 = β σ [ r ϕ η 2 k 2 + r η 1 k 2 k 4 + ( 1 − r ) k 3 k 4 ] k 1 k 2 k 3 k 4 .

Using Theorem 2 in [34], the following result is established.

Lemma 3

The CFE of the model (1) is locally asymptotically stable if ℛ 0 < 1 and unstable if ℛ 0 > 1 .

The value of ℛ 0 is a dimensionless rate that measures the average number of secondary Corona cases generated when a Coronavirus-infected individual is introduced into a susceptible population during his or her lifetime of infections. From Lemma 3, it is clear that if the initial sizes of the subpopulations of the model are in the basin of attraction of the CFE ( E 0 ), then the Coronavirus disease vanishes out from the community.

3.2 Global stability of CFE

Theorem 1

The CFE of the model (1), given by (5), is globally asymptotically stable (GAS) in D whenever ℛ 0 ≤ 1 .

Proof

The proof of the aforementioned theorem is established using a suitable Lyapunov function [39] and LaSalle’s invariance principle [39,40] by defining the following Lyapunov function:

ℱ = α 1 E ( n ) + α 2 A ( n ) + α 3 I ( n ) + α 4 H ( n ) ,

where

α 1 = [ ( 1 − r ) σ k 3 k 4 + r σ η 1 k 2 k 4 + r σ ϕ η 2 k 2 ] k 1 k 2 k 3 , α 2 = k 4 k 2 , α 3 = η 1 k 4 + η 2 ϕ k 3 and α 4 = η 2

and the derivative of ℱ is given by

ℱ ˙ = [ ( 1 − r ) σ k 3 k 4 + r σ η 1 k 2 k 4 + r σ ϕ η 2 k 2 ] k 1 k 2 k 3 E ˙ + k 4 k 2 A ˙ + η 1 k 4 + η 2 ϕ k 3 I ˙ + η 2 H ˙ , = [ ( 1 − r ) σ k 3 k 4 + r σ η 1 k 2 k 4 + r σ ϕ η 2 k 2 ] k 1 k 2 k 3 [ α S − k 1 E ] + k 4 k 2 [ ( 1 − r ) σ E − k 2 A ] + η 1 k 4 + η 2 ϕ k 3 [ r σ E − k 3 I ] + η 2 [ ϕ I − k 4 H ] ,

= k 4 ℛ 0 ( A + η 1 I + η 2 H ) S N − k 4 ( A + η 1 I + η 2 H ) , since S ≤ N in D ℱ ˙ ≤ k 4 ℛ 0 ( A + η 1 I + η 2 H ) − k 4 ( A + η 1 I + η 2 H ) , = k 4 ( A + η 1 I + η 2 H ) ( ℛ 0 − 1 ) ≤ 0 , whenever ℛ 0 ≤ 1 . □

4 Existence and stability of endemic equilibria

In this section, the possibility of the existence and stability of endemic (positive) equilibria of the model (1) will be investigated (i.e., equilibria where at least one of the infected components of the model is nonzero).

4.1 Existence of endemic equilibrium point

The endemic equilibrium point (EEP) of model (1) is denoted by ℰ 1 = ( S ∗ , E ∗ , A ∗ , I ∗ , H ∗ , R ∗ ) , so that N ∗ = S ∗ + E ∗ + A ∗ + I ∗ + H ∗ + R ∗ and α ∗ = A ∗ + η 1 I ∗ + η 2 H ∗ N ∗ . Thus, the following lemma is concluded

Lemma 4

The model (1) has a unique endemic equilibrium point ℰ 1 ∈ R + 6 , whenever ℛ 0 ≥ 1 .

Proof

Consider the following steady-state equations of model (1)

(6) Π − α ∗ S ∗ − μ S ∗ = 0 , α ∗ S ∗ − k 1 E ∗ = 0 , ( 1 − r ) σ E ∗ − k 2 A ∗ = 0 , r σ E ∗ − k 3 I ∗ = 0 , ϕ I ∗ − k 4 H ∗ = 0 , γ 1 A ∗ + γ 2 I ∗ + γ 3 H ∗ − μ R ∗ = 0 .

On solving the equations (6), we obtain

(7) S ∗ = Π α ∗ + μ , E ∗ = α ∗ S ∗ k 1 , A 1 = ( 1 − r ) σ α ∗ S ∗ k 1 k 2 , I ∗ = r σ α ∗ S ∗ k 1 k 3 , H ∗ = r ϕ σ α ∗ S ∗ k 1 k 3 k 4 , R ∗ = r σ α ∗ S ∗ μ k 1 k 2 k 3 k 4 [ ( 1 − r ) γ 1 k 3 k 4 + r γ 2 k 2 k 4 + r γ 3 ϕ k 2 ] .

Substituting equations (7) in the expression of α ∗ , we obtain

(8) ω ( α ∗ ) 2 + ( 1 − ℛ 0 ) α ∗ = 0 ,

where

ω = μ k 2 k 3 k 4 + ( 1 − r ) σ ( μ + γ 1 ) k 3 k 4 + r σ ( μ + γ 2 ) k 2 k 4 + r ϕ σ ( μ + γ 3 ) k 2 μ k 1 k 2 k 3 k 4 .

Equation (8) has two solutions, namely, α 1 ∗ = 0 , which is corresponding to the case when ℛ 0 = 1 and α 2 ∗ = ℛ 0 − 1 ω . Since ω > 0 , then α 2 ∗ = ℛ 0 − 1 ω > 0 if and only if ℛ 0 > 1 , which implies that S ∗ > 0 , E ∗ > 0 , A ∗ > 0 , I ∗ > 0 , H ∗ > 0 , and R ∗ > 0 if and only if ℛ 0 > 1 . Therefore, the endemic equilibrium point exists and is unique whenever ℛ 0 ≥ 1 , which completes the proof.□

4.2 Local stability of endemic equilibrium point

Theorem 2

The associated unique endemic equilibrium of the model (1) is GAS whenever ℛ 0 > 1 .

To show the global stability of the endemic equilibrium ℰ 1 , we will apply the algebraic approach in [37,38]. Thus, we define the Lyapunov function as follows:

(9) L ( S , E , A , I , H ) = S − S ∗ − S ∗ ln S S ∗ + a 1 E − E ∗ − E ∗ ln E E ∗ + a 2 A − A ∗ − A ∗ ln A A ∗ + a 3 I − I ∗ − I ∗ ln I I ∗ + a 4 H − H ∗ − H ∗ ln H H ∗ ,

where a i > 0 ( i = 1 , 2 , 3 , 4 ) are unknown to be determined.

The derivative L along the solutions of system (1) with respect to t is given by:

d L d t = Π + ( α + μ ) S ∗ + a 1 ( σ + μ ) E ∗ + a 2 ( γ 1 + μ ) A ∗ + a 3 ( ϕ + γ 2 + μ + δ 1 ) I ∗ + a 4 ( γ 3 + μ + δ 2 ) H ∗ − ( α + μ + α a 1 ) S + ( a 3 r σ + a 2 ( 1 − r ) σ − a 1 ( σ + μ ) ) E − a 2 ( γ 1 + μ ) A + [ a 4 ϕ − a 3 ( ϕ + γ 2 + μ + δ 1 ) ] I − a 4 ( γ 3 + μ + δ 2 ) H − Π S ∗ S − a 1 α S E ∗ E − a 2 ( 1 − r ) σ E A ∗ A − a 3 r σ E I ∗ I − a 4 ϕ I H ∗ H ≕ G ( S , E , A , I , H ) .

Now, we assume that

(10) y 1 = S S ∗ , y 2 = E E ∗ , y 3 = A A ∗ , y 4 = I I ∗ , y 5 = H H ∗ .

Thus, the function G ( S , E , A , I , H ) can be written as follows:

G ( y 1 , y 2 , y 3 , y 4 , y 5 ) = Π + ( α + μ ) S ∗ + a 1 ( σ + μ ) E ∗ + a 2 ( γ 1 + μ ) A ∗ + a 3 ( ϕ + γ 2 + μ + δ 1 ) I ∗ + a 4 ( γ 3 + μ + δ 2 ) H ∗ − ( α + μ + α a 1 ) y 1 S ∗ + ( a 3 r σ + a 2 ( 1 − r ) σ − a 1 ( σ + μ ) ) y 2 E ∗ − a 2 ( γ 1 + μ ) y 3 A ∗ + [ a 4 ϕ − a 3 ( ϕ + γ 2 + μ + δ 1 ) ] y 4 I ∗ − a 4 ( γ 3 + μ + δ 2 ) y 5 H ∗ − Π 1 y 1 − a 1 α S ∗ y 1 y 2 − a 2 ( 1 − r ) σ E ∗ y 2 y 3 − a 3 r σ E ∗ y 2 y 4 − a 4 ϕ I ∗ y 4 y 5 .

Next, construct the function set ξ

(11) ξ = y 1 , y 2 , y 3 , y 4 , y 5 , 1 y 1 , y 1 y 2 , y 2 y 3 , y 2 y 4 , y 4 y 5 .

There are five groups associated with ξ , and each one has a product of all functions equals to unity. These groups are given by

y 1 , 1 y 1 , y 2 , y 1 y 2 , 1 y 1 , y 3 , y 2 y 3 , y 1 y 2 , 1 y 1 , y 4 , y 2 y 4 , y 1 y 2 , 1 y 1 , y 5 , y 4 y 5 , y 2 y 4 , y 1 y 2 , 1 y 1 .

Define the function:

(12) H ( y 1 , … , y j ) = ∑ i = 1 j b k ( n i − ( h i , 1 + h i , 2 + ⋯ + h i , n i ) ) ,

where b i ≥ 0 ( i = 1 , 2 , … , j ), h i , k is an expression only including multiplication and division of y j satisfying ∏ k = 1 n i h i , k = 1 , and n i is the number of terms h i , 1 , h i , 2 , … , h i , n i .

According to the aforementioned group, we define the function H as follows:

H ( y 1 , y 2 , y 3 , y 4 , y 5 ) = b 1 2 − y 1 − 1 y 1 + b 2 3 − y 2 − y 1 y 2 − 1 y 1 + b 3 4 − y 3 − y 2 y 3 − y 1 y 2 − 1 y 1 + b 4 4 − y 4 − y 2 y 4 − y 1 y 2 − 1 y 1 + b 5 5 − y 5 − y 4 y 5 − y 2 y 4 − y 1 y 2 − 1 y 1 .

To find appropriate parameters a i > 0 and b k ≥ 0 such that G = H . The following equations are obtained by equating the coefficients of the similar terms on both sides.

(13) b 1 = ( α + μ + α a 1 ) S ∗ , b 2 = − ( a 3 r σ + a 2 ( 1 − r ) σ − a 1 ( σ + μ ) ) E ∗ , b 3 = a 2 ( γ 1 + μ ) A ∗ , b 4 = − ( a 4 ϕ − a 3 ( ϕ + γ 2 + μ + δ 1 ) ) I ∗ , b 5 = a 4 ( γ 3 + μ + δ 2 ) H ∗ , b 1 + b 2 + b 3 + b 4 + b 5 = Π , b 2 + b 3 + b 4 + b 5 = a 1 α S ∗ , b 3 = a 2 ( 1 − r ) σ E ∗ , b 4 + b 5 = a 3 r σ E ∗ , b 5 = a 4 ϕ I ∗ , 2 b 1 + 3 b 2 + 4 ( b 3 + b 4 ) + 5 b 5 = K .

This system has the following solution set:

(14) a 1 = a 2 = a 3 = 1 and a 4 = r σ E ∗ − ( ϕ + γ 2 + μ + δ 1 ) I ∗ ( γ 3 + μ + δ 2 ) H ∗ − ϕ I ∗ .

It is clear that a 1 , a 2 , a 3 , and are positive, and it is also straightforward to show that a 4 is positive. This shows that L is a positive definite function.

Upon substituting the values of a i in (14) into the function G ( y 1 , y 2 , y 3 , y 4 , y 5 ) and by using the endemic equilibria in (5), we obtain

G ( y 1 , y 2 , y 3 , y 4 , y 5 ) = Π + ( α + μ ) S ∗ + α ∗ S ∗ + σ E ∗ − ( 2 α + μ ) y 1 S ∗ + ( σ E ∗ − α ∗ S ∗ ) y 2 − ( 1 − r ) σ E ∗ y 3 − r σ E ∗ y 4 − Π 1 y 1 − α S ∗ y 1 y 2 − ( 1 − r ) σ E ∗ y 2 y 3 − r σ E ∗ y 2 y 4 .

The equating of the coefficients for the same terms between G = H yields

(15) b 1 = ( 2 α + μ ) S ∗ , b 2 = α ∗ S ∗ − σ E ∗ , b 3 = ( 1 − r ) σ E ∗ , b 4 = r σ E ∗ , b 5 = 0 , b 1 + b 2 + b 3 + b 4 = Π , b 2 + b 3 + b 4 = − α S ∗ , b 3 = ( 1 − r ) σ E ∗ , b 4 + b 5 = r σ E ∗ .

The completion of the preceding steps indicates that the function G may be rewritten in the form of the function H . Therefore, the values of b k are determined by (15), and then,

H ( y 1 , y 2 , y 3 , y 4 , y 5 ) = ( 2 α + μ ) S ∗ 2 − y 1 − 1 y 1 + ( α ∗ S ∗ − σ E ∗ ) 3 − y 2 − y 1 y 2 − 1 y 1 + ( 1 − r ) σ E ∗ 4 − y 3 − y 2 y 3 − y 1 y 2 − 1 y 1 + r σ E ∗ 4 − y 4 − y 2 y 4 − y 1 y 2 − 1 y 1 .

According to the property that the arithmetic mean is greater than or equal to the geometric mean, we have G = H ≤ 0 and the equality holds for y 1 = y 2 = 1 y 3 = y 2 = 1 , y 2 = y 4 = 1 that is the set { ( y 1 , y 2 , y 3 , y 4 , y 5 ) ∈ D : H ( y 1 , y 2 , y 3 , y 4 , y 5 ) = 0 } is equivalent to { ( y 1 , y 2 , y 3 , y 4 , y 5 ) : y 1 = y 2 = 1 , y 3 = y 2 = 1 , y 2 = y 4 = 1 } . That means it corresponds to the set D ′ = { ( S , E , A , I , H ) : S = S ∗ , E = E ∗ , A = A ∗ , H H ∗ = I I ∗ } ⊂ D .

Since the singleton { ℰ 1 } is the maximum invariant set on the set D ′ , then the endemic equilibrium ℰ 1 is globally stable in D by LaSalle’s invariable principle [40].

5 Discussion and conclusion

In this paper, we have proposed a mathematical model to study the dynamics of the MERS-Cov coronavirus. The population has been divided into six compartments which are susceptible, exposed, asymptomatic, symptomatic, hospitalized, and recovered. Figure 1 shows the proposed model. The latter has been analyzed and two equilibrium points were found. The existence of the corona-free equilibrium point (CFP) and the endemic equilibrium point (EEP) have been discussed. Both locally and globally stability analyses of the CFP and EEP have been conducted. Figure 2 shows the behavior of the population compartments when ℛ 0 < 1 (left) and ℛ 0 > 1 (right). The parameters’ values are obtained from Table 1.

$Figure 2 (Left) The plot of the population compartments as functions of time when ℛ 0 = 0.7203 {{\mathcal{ {\mathcal R} }}}_{0}=0.7203 . (Right) The plot of the population compartments as functions of time when ℛ 0 = 1.4406 {{\mathcal{ {\mathcal R} }}}_{0}=1.4406 .$

Figure 2

(Left) The plot of the population compartments as functions of time when ℛ 0 = 0.7203 . (Right) The plot of the population compartments as functions of time when ℛ 0 = 1.4406 .

Table 1

Estimated values for the parameters of the model (1)

Parameters	Values (per day)	Sources
β	[0.05, 0.1]	[41]
μ	0.0000351	[42]
γ 1	0.03521	[43]
γ 2	0.042553	[43]
γ 3	0.0535	[44]
δ 1	0.04227	[45]
δ 2	0.027855	[43]
ϕ	0.20619	[43]
Π	136	[41]
σ	0.1	[41]
r	0.5	Assumed
η 1 and η 2	(0, 1]	Variable

Figure 3 depicts the effect of the hospitalization (isolation) on the virus spread curve in both cases when ℛ 0 < 1 and ℛ 0 > 1 . The figure on the right shows that when the reproduction number ℛ 0 < 1 , the cumulative number of cases of infections is decreased whenever they are isolated or hospitalized. On the contrary, when the reproduction number ℛ 0 > 1 , the cumulative number of cases of infections is increased as far as they are not isolated or hospitalized, which is depicted in the figure on the left.

$Figure 3 (Left) The plot of the cumulative infected compartments as functions of time when ℛ 0 = 0.7203 {{\mathcal{ {\mathcal R} }}}_{0}=0.7203 . (Right) The plot of the cumulative infected compartments as functions of time when ℛ 0 = 1.4406 {{\mathcal{ {\mathcal R} }}}_{0}=1.4406 .$

Figure 3

(Left) The plot of the cumulative infected compartments as functions of time when ℛ 0 = 0.7203 . (Right) The plot of the cumulative infected compartments as functions of time when ℛ 0 = 1.4406 .

Figure 4 illustrates the contour plots of the reproduction number ℛ 0 with the symptomatic to hospitalization rate ϕ and the hospitalization reduction factor in transmission rate η 2 . Both plots show that as the symptomatic to hospitalization rate ϕ increases, the reproduction number ℛ 0 decreases. On the other hand, when the hospitalization reduction factor in transmission rate η 2 increases, the reproduction number ℛ 0 increases, which is consistent with the provided analysis.

$Figure 4 (Left) The contour plots of the reproduction number ℛ 0 < 1 {{\mathcal{ {\mathcal R} }}}_{0}\lt 1 with the symptomatic to hospitalization rate ϕ \phi and the hospitalization reduction factor in transmission rate η 2 {\eta }_{2} . (Right) The contour plots of the reproduction number ℛ 0 > 1 {{\mathcal{ {\mathcal R} }}}_{0}\gt 1 with the symptomatic to hospitalization rate ϕ \phi and the hospitalization reduction factor in transmission rate η 2 {\eta }_{2} .$

Figure 4

(Left) The contour plots of the reproduction number ℛ 0 < 1 with the symptomatic to hospitalization rate ϕ and the hospitalization reduction factor in transmission rate η 2 . (Right) The contour plots of the reproduction number ℛ 0 > 1 with the symptomatic to hospitalization rate ϕ and the hospitalization reduction factor in transmission rate η 2 .

Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no coflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2022-04-25

Revised: 2022-05-22

Accepted: 2022-05-24

Published Online: 2022-07-13

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mathematical analysis of a MERS-Cov coronavirus model

Abstract

1 Introduction

2 Mathematical model and basic properties

2.1 Basic properties

Lemma 1

Proof

Lemma 2

Proof

3 Stability of corona-free equilibrium

3.1 Local stability of disease free equilibrium (DFE)

Lemma 3

3.2 Global stability of CFE

Theorem 1

Proof

4 Existence and stability of endemic equilibria

4.1 Existence of endemic equilibrium point

Lemma 4

Proof

4.2 Local stability of endemic equilibrium point

Theorem 2

5 Discussion and conclusion

References

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