STABILITY ANALYSIS OF SLIV R COVID-19 EPIDEMIC MODEL WITH QUARANTINE POLICY

. In this paper, we present a mathematical model illustrating the dynamics of the COVID-19 disease with vaccination and quarantine strategies. The presented model contains ﬁve equations that describe the interaction between individuals who are susceptible, exposed, infected, vaccinated, and recovered. We start the study by verifying the positivity and boundedness of solutions. The existence and the stability of both disease-free equilibrium and endemic equilibrium are proved. Finally, numerical simulations are performed to demonstrate the behavior of the infection over time and to say the inﬂuence of quarantine and vaccination on both the COVID-19 dynamics and the basic reproduction number mathcalR 0 for controlling the disease’s spread.


INTRODUCTION
SARS-CoV-2, a coronavirus discovered in 2019, has produced a respiratory disease pandemic known as COVID-19.The virus spreads between people through direct contact or via contaminated surfaces.This disease is currently spreading rapidly in many countries, and the global number of COVID-19 cases is rapidly increasing.
Mathematical modelling in epidemiology is a source of knowledge for understanding the spread of an disease and an effective tool for controlling and predicting the dynamics of diseases such as Cancer [1], VIH [2], influenza [3], Tuberculosis [4], HBV [5], COVID-19 [11,12,13] and the co-infection of COVID-19 and HBV [6], other researchers are interested in modeling multistrain diseases [7,8,10,9].Many active studies are currently being conducted to investigate various epidemic models applied to the spread of COVID-19 by scientists all over the world.
Ian et al. [14] developed a susceptible-infected-removed (SIR) model that provides a theoretical framework to investigate the time evolution of different populations and monitor diverse significant parameters for the spread of the disease COVID-19 in various communities.In addition, the SARS-COV-2 virus has a long incubation period, which refers to the time between being exposed to the virus and developing symptoms.The average incubation period is 6 days, with recorded variations ranging from 2 to 27 days [15].As a result, many earlier studies considered a new compartment to the classic SIR in model to account for the exposed population.An SLIR model was implimented by Shaobo et al. [19], to analyse the spread of COVID-19 in Hubei province.Additionally many mathematical models were constructed to study the outbreak of COVID-19 in many countries [16,17,18].
Countries worldwide have implemented strict and adequate precautions to prevent and control the spread of COVID-19, including early detection approaches and social distancing [20] to limit contact between individuals as much as possible, as well as medical treatment, to reduce the number of infected citizens.Vaccination is a crucial strategy in fighting against many previously infected diseases.Recently, authors in [21] constructed a SLIR model by considering vaccination and isolation factors as model parameters.Moreover, Amouch et al. [22] proposed a new epidemiological mathematical model for the spread of the COVID-19 disease with a special focus on the transmissibility of individuals with severe symptoms, mild symptoms, and asymptomatic symptoms and take into consideration the vaccination of a portion of susceptible individuals.More recently, a COVID-19 vaccine epidemic model has been tackled [23].In this work, we continue the investigation of the effect of vaccination by taking into account the effect of quarantine measures on SLIVR COVID-19 epidemic model presented in [23].Therefore, the SLIVR COVID-19 epidemic model is formulated as follows (1) This model includes six variables: susceptible individuals (S ), the population that can make contact with the infection, the exposed individuals (L ), the population exposed to the virus but without developing clinical symptoms.The infectious individuals (I ), the population with fully developed corona-virus symptoms.The vaccinated individuals (V ), and finally removed individuals (R).All model parameters are assumed to be positive and are described as follows: Λ denotes the population recruitment rate and d is natural mortality in all compartments.β represent the effective contact rate, α represents relative transmissibility rate.r s represents rate of infection development with symptoms, d 0 is death rate due to infection and r c is rate of recovery from infection.r v denote vaccination rate, w v is vaccine waning rate and l m is loss of disease acquired immunity.The parameter ρ represent the efficiency of quarantine in reducing the latent and infected individuals.The flowchart of our model is illustrated in Fig. 1.
The current work is divided into different sections.In the following section, we will demonstrate the existence, the positivity and boundedness results.In Section 3, we will establish both the local and global stability of both equilibrium.In Section 4, we will present some numerical simulations to verify the theoretical results.The conclusion is stated in Section 5.
The diagram of the COVID-19 model.

THE PROBLEM WELLPOSEDNESS AND STEADY STATES
In this section, we will prove that the system is biologically meaningful and mathematically well posed.To do this, it is required to prove that the solutions of the system of ordinary differential equation ( 1) are positive and bounded for all time.
Proposition 2.1.For all non-negative initial condition, the solutions of the system (1) exist, non-negative and remain bounded for all t > 0.Moreover, Proof.In order to prove the positivity result, we will show that any solution starting from nonnegative orthant R 5 + remains there forever.
Let us now show that T = +∞.Suppose the contrary.By continuity of solutions, we have -If S (T ) = 0 before the other variables L , I , V , R, becomes zero.Hence, Using the first equation of system (1), we obtain -If L (T ) = 0 before the other variables S , I , V , R, becomes zero.Hence, Using the first equation of system (1), we obtain Therefore, T could not be finite, which implies that S , L , I , V and R are all positive for all positive time.This proof the positivity of solutions.
To prove the subsequent part of Proposition (2.1), let the total population By adding equations involved in the system (1), we have By simple manipulation, we have Moreover, the closed region define by Ω ⊂ R 5 + is positive imvariant for the model ( 1) with nonnegative initial conditions in R 5 + .
Proof.As we know that but the solution of ( 5) is Therefore, Thus, the region Ω is positive invariant, and all the solutions trajectories are attracted in R 5 +

ANALYSIS OF THE MODEL
In this section, we will first calculate the basic reproduction number R 0 for The COVID-19 model (1).Next, we will present the steady states and finally we will demonstrate the local and global stability of all steady states.
3.1.The Basic Reproduction Number.Biologically, the basic reproduction number denoted R 0 represents the average number of new infections generated by each infected person in a population where all individuals are susceptible to infection.We will use the next generation matrix to calculate the basic reproduction number R 0 [24].The necessary matrices denoted by F and Y are given by given by Therefore, The basic reproduction number R 0 is obtained as the spectral radius of FY −1 .Hence, we get the following expression of R 0 where , 0 .

Local Stability.
Theorem 3.2.The disease-free equilibrium, P 0 is locally asymptotically stable if R 0 < 1 and else unstable.
Proof.The Jacobian of model ( 1) at disease-free equilibrium, , 0 is, , and the corresponding characteristic polynomial is where a 2 = 1, When R 0 < 1, the coefficients a 0 and a 1 are positive.By Routh-Hurwitz stability criteria, the disease-free equilibrium is locally asymptomatically stable in Ω.
Moreover, P * is otherwise unstable.
Proof.The Jacobian of system (1) at where The corresponding characteristics equation is given by It is obvious to show that the necessary conditions of Routh-Hurwitz stability criteria for degree 0 holds.Therefore, the P * is locally asymptotically stable in Ω.

Global stability.
The following Theorem investigates the global dynamics of diseasefree equilibrium, P 0 , of the COVID-19 epidemic model described in (1).

Proof. Consider the Lyapunov function given by
Hence, the Lyapunov derivative is Since, and Therefore, by LaSalle's Invariance Principle, it follows that the disease-free equilibrium point is globally asymptotically stable in Ω.
The following Theorem investigates the global dynamics of The endemic equilibrium, P * , of the COVID-19 epidemic model described in (1).
Proof.The appropriate Lyapunov function is defined as where Therefore From (6), we have by using the system (1) and the fact that S (t), L (t), I (t), V (t), R(t) are all non-negative for t > 0, we get From ( 8) and ( 9), we have If R 0 > 1, we have dF * dt < 0. Therefore, according to LaSalle's Invariance Principle, the endemic equilibrium point P * is globally asymptotically stable in Ω.

NUMERICAL SIMULATIONS
In this section, we will perform some numerical simulations in order to check the impact of quratine and vaccination measures in controlling the spread of the COVID-19.Fig. 2 show the evolution of the exposed and infected individuals for the following parameters: Λ = 8939, β = 0.4114, α = 0.3131, d = 1/(67.7 * 365), r s = 0.0164, d 0 = 0.022, w v = 0.0057, r c = 0.1, l m = 0.1762.In the first case, we ignore the effect of quarantine and take the vaccination baseline, ρ = 0, r v = 0.0380, the disease persist and the exposed and infected cases reach a very high level (the blue curve).In the second case, we increase the vaccination rate r v by 50 percent from the baseline value, ρ = 0, r v = 0.057, we can see that the disease persists with a significant reduction in both exposed and infected individuals (the read curve).Furthermore, in the third and the fourth cases, when the quarantine is maximally implemented simultaneously with population vaccination, the disease dies out and the infected, as well as exposed population, decreases very quickly (the yellow and purple curves).

CONCLUSION
In this paper, we have studied the mathematical model illustrating the dynamics of the COVID-19 disease with both vaccination and quarantine strategies.This model includes five equations describing the interaction between susceptible, exposed, infected, vaccinated, and recovered individuals.This study is oriented primarily toward to verify the positiveness and the boundedness of solutions that are established to have the well-posedness of the formulation.
Furthermore, we have studied the existence and the stability of both disease-free equilibrium and endemic equilibrium, the Disease-free equilibrium always exists, and it is stable when R 0 < 1 but for the endemic equilibrium point exists and is stable when R 0 > 1.Finally, the numerical simulations are carried out in order to show the behavior of the infection over time and to proclaim the effect of the quarantine and vaccination on both the COVID-19 dynamics and the basic reproduction number R 0 for controlling the spread of the disease.We have concluded that the combination of quarantine and vaccination policies is the key aspect of infection control related to the spread of the COVID-19 outbreak.

3. 2 .
Steady states.In the next Theorem, the steady states of the the COVID-19 epidemic model (1) are given.Theorem 3.1.-The COVID-19 epidemic model (1) has a disease free equilibrium defined by

FIGURE 2 . 1 FIGURE 3 .
FIGURE 2. Impact of quarantine and vaccination strategies on exposed and infected population.