A Model for the Transmission of Covid-19 in a Mass Gathering

Recently the pandemic disease Covid-19 has spread rapidly around the world, the disease has some similar characteristics with a previous endemic disease, the Middle East Respiratory Syndrome Coronavirus (MERS-CoV), which has received recently the attention of many researchers. Here, a two patch SEIR model is introduced in order to explain the importance of the impact of the mass gathering of susceptible individuals when the infected individuals and the exposed individuals are taking separately its role in the dynamics of those diseases that involve the two groups. One or more endemic equilibrium points could appear when the reproduction number exceeds the unity, and the model undergoes a forward bifurcation. A re-formulation of the model as an optimal control problem will permit to evaluate the impact of adequate control strategies for the disease.


Introduction
This article proposes a model for virus transmission such as those that have been reported in the current Covid-19 pandemic or in the MERS-CoV epidemic. Two populations are considered and a SEIR model is proposed in each population. It is assumed that at the time of the meeting of the two groups there are no people in quarantine nor is it well known that the infection is in process. The disease evolves in such a way that the viruses are efficiently transmitted by mean of close person-to-person contact, it is presumed that this occurs through droplets of respiratory secretions, or by contact with contaminated surfaces. A previous model given in [1] analyzes the lives of millions of pilgrims going to Mecca for Hajj and Umrah (due to the associated mass meetings) are threatened by MERS-CoV. lowed to travel in either direction between the two groups.
The total population of Group 1 at time t, denoted by ( ) 1 N t , is sub-divided into four mutually exclusive compartments of susceptible ( ) 1 S t , exposed (i.e., Thus, the total population at time t, ( ) N t , is given by where ( ) ( ) ( ) ( ) ( ) .

N t S t E t I t R t
It should be emphasized that, in this paper, individuals in the exposed i E , symptomatic i I classes 1, 2 i = are infected with the disease, and can transmit it to susceptible individuals. Furthermore, it should be mentioned that the model does not explicitly account for screening. Table 1 describes the populations and parameters by mean of which the model will be described. The population of susceptible individuals in Group 1,1 S , is increased by the recruitment of individuals at a rate 1 π , by migration from The population of exposed individuals in Group 1, 1 E is generated by the Journal of Applied Mathematics and Physics The population of symptomatic individuals in Group 1, 1 I is generated by the exposed population at the rate 1 α and decreased, natural recovery at a rate 1 κ . Hence, are given by (4). The model proposed here is in the same line of the models that appear in the literature such as [3] [4] [5], and characterized by: 1) Includes two groups instead of a single group/patch as is considered in [3] [4].
2) Allows for disease transmission by latently-infected individuals i E which is is not considered in [5].
3) Including heterogeneity in the transmission based on groups. That is, susceptible individuals can acquire infection from infected individuals in Group 1 at rate 1 λ or Group 2 at rate 2 λ , defined in (4), so the infection rate is given as , not considered in [3] [4] [5].
4) The population of the infected asymptomatic individuals is considered here as included in the exposed group.
The flow diagram of the model is represented in Figure 1.

Basic Properties
The parameters of Equation (5) are supposed to be non-negative and it is well known that the model is well behaved, i.e., solutions with positive initial conditions stay positive for all 0 t > and can be stated following the methods of [6]. . min , For the region defined with the set  the following lemma holds. Lemma 2. The region described by the set  is positively invariant for the system given by Equation (5). Journal of Applied Mathematics and Physics Proof. By mean of the notations given in (1), (2), (3) and afterwards of adding Equation (5), the following equality can be obtained ≤ . Hence by standard comparison arguments [7] [8] the following inequality holds Therefore the region  is positively invariant which means that solutions that enter in  remain there for all subsequent time t. QED It is well known that systems that have the property as in the previous lemma are said to be mathematically and epidemiologically well-posed [9].

Existence and Asymptotic Stability of Equilibria
In this section the equilibria of the system (5) will be studied.

Local Asymptotic Stability of Disease-Free Equilibrium
For the system given by Equation (5)  The equilibrium solution for and ( ) is the disease-free equilibrium for the system. In order to establish the local-asymptotic stability of the DFE equilibrium (6), the next operator theory, whose methods and notations are given in [10], will be used here. In the case of the model investigated here, the non-negative,  , and the M-matrix,  , are given respectively by Hence the matrix 1 −  is given by As is established in [10], the reproduction number v  of the system is given by the spectral radius of the next generation matrix Clearly, the spectral radius of the matrix 1 −  is therefore given by ( ) It can be seen that the following relations hold: Lemma 3. 1) The following equality holds, vw ru = .
2) The reproduction number, v  , is given by v r u = +  . 3) As a consequence of the previous relation, Then according with the theory of the next generation operator, see [10], the following theorem holds Theorem 4. The DFE equilibrium (6) of the model given by the system (5) is locally-asymptotically stable for , and unstable for From a epidemiological point of view the previous theorem says that a small influx of infected individuals does not produce a large Coronavirus outbreak in a mass gatherings for values of 1 v <  and therefore, the disease is subject to control whenever the sizes of the initial sub-populations of the model belong to the attraction basin of the equilibrium DFE (6).  , , n c a n c a n c a n c a d d n n n n κ κ κ κ = + = + , , n c a n c a n c a n c a f f n n n n κ κ κ κ = + = + , .

Existence of the Endemic Equilibrium Point
, the condition given in (12) and any of the Cases 1, 2, 3, 4 of Table 2 hold; 2) has one or multiple EEP if the conditions given in (12) and any of the Cases 5, 6, 7, 8 of Table 2 hold; 3) may have more than one EEP if the conditions given in (12) and any of the Cases 2 to 8 of Table 3 hold.
From the epidemiological point of view this result implies that the viruses disease will establish itself in the population, when 1 v >  whenever the initial states of the sub-populations of the system (5) belong to the basin of attraction of the unique endemic equilibrium 1  . In the case when 1 v >  and the endemic equilibrium of the system (5) is stable.

Stability Analysis
, the equilibrium 0  is asymptotically stable and for values of the populations in the attraction basin of this point, the control and elimination of the disease is guaranteed. For 1 v >  the equilibrium 0  lost its stability and    [11], and the Theorem 4.1 of [12].

Changes of sign
Theorem 7. The model described for mean of the system (5) exhibits a forward bifurcation at in the sense previously described. Proof. It is convenient to make the following change of variables: , and suppose that 1 β is chosen as a bifurcation parameter. The transformed system (14), with * 1 1 β β = , has a hyperbolic equilibrium point (i.e., the corresponding associated linear system has a simple eigenvalue with zero real part, and all other eigenvalues have negative real part), so that the center manifold theory [12] can be used to analyze the dynamics of system (14) Similarly, has a left eigenvector, v , given by ( ) The computation of the required bifurcation coefficients a and b gives the following results according with Theorem 4.1 of [12].
The sign of the parameters 0, 0 a b < > . When the bifurcation parameter changes from negative to positive, the equilibrium point 0  changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium be-comes positive and locally asymptotically stable. QED

Optimal Control
In this section an optimal control based in the exposed strategies considered in the system (5) will be carried out. The system will be reformulated as an optimal control problem in the following way,  x ∈  represents the initial conditions of the state variable x. The choice of the cost functional is very common in the subject, see [14] [15] [16].

Existence of an Optimal Control
The following theorem guarantees that the system (16) is controllable this open the possibility to take a strategy to prevent the spread of the disease.

Conclusions
A two-group deterministic model is proposed and analyzed for the transmission dynamics of Covid-19 or MERS-CoV (coronaviruses), taking into account fundamentally that the epidemic is in a beginning stage and the groups converge in a mass gathering. The main results are the following: 1) The model has one disease-free equilibrium which is locally asymptotically stable when the reproduction number 1 v <  . Epidemiologically it implies that a small influx of infected individuals does not produce a large Coronavirus outbreak in mass gatherings, therefore, the disease is subject to control whenever the sizes of the initial sub-populations of the model belong to the attraction basin of the equilibrium DFE (6). When the reproduction number . According to the proof of the Theorem 7, the model exhibits exclusively the phenomenon of forward bifurcation and no backward bifurcation are possible.
2) According to the Pontryagin's Maximum Principle, the system (5) has a pair of optimal controls for the susceptible populations in both groups. This result suggests that with other actions such as quarantine and vaccination the disease could prevent the disease from spreading.
3) The results obtained in this investigation are purely theoretical. In a research in progress related to the model proposed here, a comparison with experimental results will be established, as well as consideration of delays in the system.

Conflicts of Interest
The author declares no conflicts of interest regarding the publication of this paper.