## 2 Materials and Methods  
### Model Description  
In this paper, we propose a discrete-time SEIRQD-type COVID-19 epidemic model. We take into account that the exposed are asymptomatic, which means that they are also infectious. The model is describe by the following discrete system  
\[\begin{array}{rcl}S_{i+1}&=&\Gamma_{i}+S_{i}+\rho_{1}E_{i}^{q}+\rho_{2}R_{i}+ \rho_{3}I_{i}-\beta_{1}(1-u_{i})\frac{S_{i}I_{i}}{N_{i}}-\beta_{2}(1-v_{i}) \frac{S_{i}E_{i}}{N_{i}}-\alpha\sum_{r=0}^{9}\frac{\beta_{1}S_{i-r}(\gamma_{2} E_{i})}{N_{i-r}}-dS_{i};\\ E_{i+1}&=&E_{i}+\beta_{1}(1-u_{i})\frac{S_{i}I_{i}}{N_{i}}+\beta_{2}(1-v_{i}) \frac{S_{i}E_{i}}{N_{i}}-(\gamma_{1}+\gamma_{2}+d)E_{i};\\ I_{i+1}&=&I_{i}+\gamma_{1}E_{i}-(u_{1}+\rho_{3}+d)I_{i};\\ R_{i+1}&=&R_{i}+TI_{i}^{q}-(\rho_{2}+d)R_{i};\\ E_{i+1}^{q}&=&E_{i}^{q}+\alpha\sum_{r=0}^{9}\frac{\beta_{1}S_{i-r}(\gamma_{2} E_{i})}{N_{i-r}}-(\rho_{1}+\gamma_{3}+d)E_{i}^{q};\\ I_{i+1}^{q}&=&I_{i}^{q}+\gamma_{2}E_{i}+\gamma_{3}E_{i}^{q}-(T+\mu_{2}+d)I_{i} ^{q};\\ D_{i+1}&=&D_{i}+\mu_{2}I_{i}^{q}+\mu_{1}I_{i};\end{array} \tag{1}\]  
subject to non-negative initial conditions. \(S_{i}\), \(E_{i}\), \(I_{i}\), \(R_{i}\), and \(D_{i}\) denote the number of susceptible, exposed, infectious, recovered, and dead persons. \(E_{i}^{q}\) and \(I_{i}^{q}\) denote the number of exposed and infectious persons in quarantine. In such a SEIRQD-model type of the COVID-19, \(N_{i}\) is the total population size, the incubation period is estimated to be 7 days and the infectious period of the virus is estimated as 10 days.  
A susceptible person can infected by contacting either an infected person or an exposed (asymptomatic) person. The number of these contacts is \(\beta_{1}\frac{S_{i}I_{i}}{N_{i}}+\beta_{2}\frac{S_{i}E_{i}}{N_{i}}\).  
The control strategy that was introduced into this system aims to reduce contact between susceptible people and infected or exposed people.  
For a period of 10 days of virus incubation, the symptoms might appear in a portion of \(E_{i}\) individuals. Hence, a fraction of these individuals, \(\gamma_{1}E_{i}\), will choose not to be hospitalized, as their symptoms are mild and can be treated at home. We also assume that infected person become infected again with the rate \(\rho_{3}I_{i}\). There are cases of infected people who died, and their autopsy confirmed COVID-19 as cause of death; therefore, a fraction \(\mu_{1}I_{i}\) will be added to the number of deceased caused by COVID-19. The other fraction of the asymptomatic \(\gamma_{2}E_{i}\) is hospitalized and put under medical care. Once a person is infected and hospitalized, the health service tries to trace the people who contacted the infected person during the 10 days of the virus incubation period. The individuals that are traced will be quarantined and placed under medical care. \(\sum_{r=0}^{9}\frac{S_{i-r}(\gamma_{2}E_{i})}{N_{i-r}}\) is the total number of people who were in contact with infected people before being quarantined. The \(\gamma_{3}E_{i}^{q}\) of cases will have symptoms and will be placed under medical surveillance, or they will have to get medical care if their symptoms are mild. The contacts who tested negative, after an observation period, find themselves newly susceptible with a \(\rho_{1}E_{i}^{q}\) rate. Among these infected cases, a number of \(\mu_{2}I_{i}^{q}\) will have died and a number of \(TI_{i}^{q}\) will be recovered. The recovered people do not have immunity from the COVID-19; therefore, they may be susceptible again. The birth rate \(\Gamma_{i}\) is 16.5% of the total population [12] and the natural mortality rate \(d\) is equal to 5.8%.  
All the above parameters are assumed to be non-negative and defined in Table 1. Schematically, the flow between compartments is represented in Figure 1.  
Figure 1: The flowchart of the model (1).