STABILITY AND HOPF BIFURCATION OF AN EPIDEMIOLOGICAL MODEL WITH EFFECT OF DELAY THE AWARENESS PROGRAMS AND VACCINATION: ANALYSIS AND SIMULATION

. The present research is concerned with studying the media coverage delay impact and vaccine impact to control outbreaks the infectious disease in the population, we suggested that our mathematical modeling divide the population into three classes: susceptible persons S(t) , vaccinated person V(t) and infected persons I(t) . The susceptible individuals are also divided due to media programs into awareness and unawareness of disease danger and its mode of transmission. We studied the inﬂuence of time delay in response to the media program on awareness of the epidemic. We ﬁrst discussed the existence of the equilibrium points of the system and obtain sufﬁcient conditions for the local asymptotic stability of all equilibrium points. Further, the existence of Hopf bifurcation is shown near endemic equilibrium. It is interesting to note that there exists at least one limit cycle around the unstable endemic equilibrium. In particular, sufﬁcient conditions for a unique stable limit


INTRODUCTION
The study of epidemic systems using mathematical models becomes one of the most important research topics among applied mathematicians, biologists since the pioneering work of Kermark and Mckendrick [1]. It is a well-known fact, there are many viral diseases that infect humans and animals for example dengue is caused by a virus of the Flaviviridae family and there are four distinct. Also, influenza is caused by four serotypes of viruses. As for the COVID-19 virus being caused by the SARS-CoV-2 virus, many strains have now been detected.
The presence of multiple strains of the pathogen complicates our ability to combat these diseases. Because different genetic strains may have different characteristics. For example, it may spread more easily, may cause more severe disease, or it may evade the host's immune response [2].
Recently, several epidemic models have been proposed with the aim to understand, describe and control the spread of the SARS-CoV-2 disease. These models analyze the evolution of the disease over time by dividing the communities into some compartments, which mainly include the susceptible class (S), the carrier (C), the exposed class (E), the infected class (I) and the removed class (R).
After the rapid spread of the epidemic, the World Health Organization (WHO) accelerated the first measures, which is awareness through media programs such as wearing masks, social distancing to 1.5 m, washing hands for 30 seconds, and others. On the other hand, vaccination is an important control measure to reduce the spread of infectious diseases. Thus, studying the effect of vaccines on the disease dynamics is crucial. Many studies on modeling have focused on the influence of media programs and vaccination such as Mohsen et al. [3], presented a mathematical model to study two different measures to control the COVID-19. The spread of SARS-CoV-2 in china by Ivorra et al. [4]. Zeb et al. [5] described a nonlinear SEIQR COVID-19epidemic model. Hattaf et al. [6] studied the transmission dynamics of COVID- 19 propagation due to environmental contamination and carrier effect. In [7], Mohsen et al., formulated a SARS-CoV-2 model to consider the effect of curfew strategy. Dehingia et al. [8] developed a mathematical model for within-host SARS-CoV-2. Bai et al. [9] have proposed and analyzed an epidemic-economic model for SARS-CoV-2 by mathematical model. Also, the following authors have formulated and explored some COVID-19 model see [10]- [20]. The remainder of this work is organized as follows. In Section 2, we introduce the proposed SARS-CoV-2 model with vaccination strategies and awareness measures. In section 3, we present the existence analysis of the equilibrium solutions. In Section 4, we study the dynamical behaviors and stability analysis of the proposed model. In Section 5, we discuss the properties of the Hopf bifurcation. In section 6, we describe the numerical simulations and sensitive analysis to validate the theoretical results. Finally, conclusions and discussions are presented in Section 7.

THE MATHEMATICAL MODEL INVOLVING AWARENESS PROGRAMS
In this work, we investigate the influence of both vaccination and awareness by media on the infectious diseases. So, it is formulated by the following nonlinear system of ordinary differential equations, where the 1st part susceptible unaware of the diseases and denoted by S u (t) with time t, the 2nd part is vaccinated individuals referred by V (t) at time t, the 3rd part is susceptible of the diseases but has awareness about it and referred by S a (t) at time t, the 4th part is referred by I(t) at time t represented by infected persons and the last part M(t) is the level of awareness that population need. In addition to the above that the total population is N(t) at time t and the natural birth of susceptible is Λ with fraction 0 < p < 1 (i.e that is mean p the rate of vaccination from S u (t)). µ the death rate of each population. We assume that the disease spread by contact between the susceptible population (unaware and aware) and infected population are denoted by β i for i=1,3 respectively with it is noted that β 3 < β 1 . As well as the unaware susceptible become aware due to of the awareness programs at rate β 2 , with the constant q such that 0 ≤ q < 1, is the media-induced vaccination coverage. Moreover, v and d are the recovery rate and mortality due to infection respectively. Also, we have some people convert from awareness level to unaware level due to confuse from media programs at rate ρ.
The influence rate of media campaigns is denoted by γ and the reduction rate is represented by θ . finally, in some time when the transfer the unaware susceptible to aware susceptible due to any loss or random in awareness occurs of delay in time, we denoted for this delay by t − τ (for τ > 0). Now, from the above facts we can write the model by the following system of nonlinear (DDE s ): and all the variables and parameter are presumed to be positive, since (1) is a description of human population.

PRELIMINARIES
3.1. Positive Invariance. We study the positive invariance in the following theorem.
Proof: We begin the proof by contradiction(i.e suppose that the theorem is incorrect), then there exists t * > 0 be the first time verifying S u (t * ) = 0,V (t * ) = 0, S a (t * ) = 0, I(t * ) = 0 and M(t * ) = 0. Now, by the 1st eq. of (1) we havė Hence, we should have S u (t) < 0 for all t ∈ (t * − ε,t * ) with ε > 0. This is contradicts that S u (t) > 0 for al t ∈ [0,t * ) we have S u (t) > 0 for all t > 0. Also, from 2nd eq. we get: Clearly, by same way above and hence we obtain that V (t) > 0 for all t > 0. Now,from 3rd eq. of (1) we get:Ṡ . STABILITY AND HOPF BIFURCATION OF AN EPIDEMIOLOGICAL MODEL 5 Hence we obtain that S a (t) > 0. Therefore, we have I(t) > 0 as shown in the following: Finally, from the last eq. in (1) and using the same way above we havė Consequently, M(t) > 0 for all t > 0. (2): All the solutions of (1) with initial conditions are bounded.

Boundedness. Theorem
Proof: Let(S u (t),V (t), S a (t), I(t), M(t)) be any solution of (1) with initial conditions, hence, by adding the equations is (1) to each other we get: It easy see that, by fact the total population is denoted by N(t) we get: Then solving the above inequality with result of Gronwall lemma we get N ≤ A/µ + ε.

3.3.
The existence the all possible equilibrium points of System (1). In this subsection, we see that system (1) has only two equilibria: (1) The first point is called the disease-free equilibrium point (DFE) and denoted by E • = (S u• ,V • , 0, 0, 0).
(2) The second point is called the endemic equilibrium point (EE) and denoted by Now, discussing the existence conditions of each equilibrium point of (1), and hence the system (1) has the (DFE) only in case I = 0, then we can write it by E • = ((1 − p)A/µ, pA/µ, 0, 0, 0), and it is exists when R 0 < 1, where The threshold parameter R 0 is called the basic reproduction number. On the other hand, if I = 0, we need to check the existence conditions of the (EE). Note that, S * u ,V * , S * a , I * and M * represented the positive solution of the following set of equations.
After solving the above system we get value of last equation in above system in following Now, substituting M * in the 1st, 2nd and 3rd equations in above system and solving its we get: Now, we can simplify the 4th equation in the same system in following: Clearly, by help the Descartes rule, (6) has unique positive real root I * if the following condition holds (7) R 0 > 1.
Hence, we can find the values of S * u ,V * , S * a and M * if we know the value of I * . Remark. System(1) has more than one endemic equilibrium point (EE) if the (7) not be satisfied.

STABILITY ANALYSIS
In this section, to discuss the stability of the equilibrium points, we need the calculating the Jacobian matrix at each points (DFE) and (EE), thus, we state the following theorems. It is well known that, the general form Jacobian matrix for system (1) at any equilibrium point in the following: (3)The (DFE) is locally asymptotically stable if:

Stability of (DFE). Theorem
Proof: The equation (8) at (DFE), we can be reduces to: Then the characteristic equation of (10): Consequently, if (9) holds we get the each eigenvalues are always negative and given the (DFE) is asymptotic stable.

Stability and Hopf
Bifurcation of (EE). Now, we discuss the existence of Hopf bifurcation of the (EE) point of system (1), we can rewrite the matrix (8) with the characteristic equation around the (EE) in this following form. Here: Hence, the characteristic equation of (12) can be written by form: Here: It is easy to verify that condition b 44 < 0 guarantees that B 1 is positive and the condition is always negative. Then to discuss the stability of (EE) without delay (i.e τ = 0). Clearlly, we can rewrite the equation (13) in the following format: Then, according to Routh-Hurwtiz criterion, we see that all roots of equation (14), have We obtain that the (EE) without delay is locally asymptotically stable. Now, to discuss the stability of (EE) when τ > 0 depends on the roots of equation (13). We assume that λ = iω (ω > 0) in equation (13) we have: Now, by isolation the real and imaginary parts, which gives: By squaring the two equation in (15) and adding to each other we get: put h = ω 2 , equation(16) becomes as follow: Now, it easy by according to Descarte rule of sign the equation (16) and (17) has a positive root denoted by h = ω • provided that the following condition holds: Clearly, we get the characteristic equation (13) has at least a pair of pure imaginary roots ±iω • , corresponding to the time delay τ. Hence, we can rewritten the system (15) but substituting ω • instead of ω and solving the resulting system for τ, we have the following result.
where j=0,1,2,...... Now, since equation (13) has least two pure imaginary roots if the condition (18)  Provided that Proof: By deriving equation (13) with respect to τ, the following is obtain the result: Thus, Since, λ = iω • at τ = τ 0 , with using equation (13), we can rewrite (23) as following: We have: We get Hence,the transversality condition is satisfied if the condition (22) holds. Now, from the above result on the local stability and Hopf bifurcation at (EE), we have the following remark: Remark. Let the condition (7) with conditions of Routh-Hurwtiz criterion and (18) are satisfied then: i The (EE) point is locally asymptotically stable for τ < τ 0 .
In addition of above, we using the adjoint property ϕ, Aφ = A * ϕ, φ it easy to show that Hence, we obtain K * ,K = 0. The proof completed. Next, we will compute the coordinates to describe the center manifold at α = 0. Assuming Y t be a solution of (28) when α = 0, we define in following (38) Now, From (28), with α = 0 and equation (38), we get On the center manifold, we get Where, z andz are local coordinates for center manifold in the direction of K * andK * . From (38), clearly we know that W is real if Y t is real. If we consider only the real solutions, we can rewritten the (40) becomes with (42) g(z,z) = g 20 z 2 2 + g 11 zz + g 02z 2 2 + g 21 z 2z 2 + .....

Numerical simulation.
Here, the analytical results we have obtained will be illustrated by numerical simulations by taking hypothetical values of the parameters in system (1), with different initial conditions in order to determine the effect of each of them on the dynamic behavior of the system (1). In the following data with initial point (0.7,0.9,0.6,0.5,0.5), we obtain that the trajectory of system (1) convergent towards stability of (DFE) point when any τ ≥ 0, as shown that in figure 1.  6.2. Sensitivity Analysis. The goal of discussing sensitivity analysis is to reveal a parametric effect of disease prevalence in epidemiological modeling. Our goal is to find out the parameters that control R 0 and through which the dynamics behavior of the epidemiological model can be predicted. And all of this can be studied through the global sensitivity index of R 0 and according to the parameter φ and it is denoted by Π φ R 0 and is defined by From the Eq (54), we note that as there is an increase or decrease in parameter rate φ by a certain percentage say m then the reproduction number R 0 also increases or decreases by the same percentage m. Now, an increase of the value of any one of these parameters β 1 and A will increase the basic reproduction number. While, an increase of the value of other parameters p, d, ν and µ will decrease R 0 . Thus, we confirm that results and illustrate in the following figures (5) and (6)

CONFLICT OF INTERESTS
The authors declare that there is no conflict of interests.