The role of media coverage on the dynamical behavior of smoking model with and without spatial diffusion

The spread of epidemic diseases still a major threat to the life of communities. Therefore, with the great development of the technology, the spread of diseases can be reduced by using media coverage awareness. In this paper a smoking model incorporating media coverage for warranting the population is proposed and studied. The dynamics of the model is investigated in two different cases: nonexistence and existence of diffusion. The existence, positivity and bounded-ness of solutions are investigated. The local and global stability by the help of Lyapunov function of all possible equilibrium points are investigated. Moreover, numerical simulations are carried out to validate the analytical results and specify the effect of varying the parameters.


Introduction
The smoke from the Cigarette is a very complex chemical mixture that is dangerous to human health and all the elements of the environment. It contains more than 3,800 toxic chemicals,

Construction of the model
The mathematical model offer us more understand about spread the infection disease, we know that the disease is transmitted by direct contact between healthy individuals with infected individuals. In fact, outbreak the smoking is very similar to the spread of epidemic and hence some populations start smoking due to contact with smokers. Consider a population of size at time . It is assumed that, the population divided into four classes: the 1 st class consisting of individuals who do not smoke tobacco and maybe become smokers in future (potential smokers) and the size of individuals at time for this class denoted by ; 2 nd class involving the smoker individuals and denoted their size at time by ; represents the size of individuals at time in the 3 rd class that contains individuals who temporarily quit smoking; stands for the size of individuals at time in the 4 th class, which contains the recovery from smoking. On the other hand, the efficiency of awareness by media coverage to reducing the number of smokers (or smoking prevention) at time will be denoted by . Accordingly, the dynamics of smoking model with the effect of awareness by media coverage to outbreak the smoking can be describe by the following system of nonlinear ODEs.
As the fourth equation is a linear differential equation with respect to variable , which is not appear in the other equations of system (1), hence system (1) can be reduced to the following system: with initial condition , + , + and . Therefore, by solving system (2) and substituting the solution, say % % % % , of it in the fourth equation of system (1) and solving the obtained linear differential equation we get for & that: Moreover, all the parameters are assumed to be nonnegative with, 0 ψ represents the recruitment of potential smokers population, 0 μ represents the natural death rate of the human populations. The parameter 0 β is the contact rate between potential smokers and smokers. On other hand, the awareness level through media coverage that reached to the individuals is denoted by 0 γ , however portion of individuals who received awareness transfers to smoker class and temporarily quit smoking class with rates ) 1 0 ( σ and ) 1 0 ( e respectively. The parameter 0 δ represents the rate of losing the temporary quitters smoking individuals, in fact fraction of them with rate ) 1 0 ( ε transfers to smoker's class while the rest of individuals will transfer to recovery from smoking class. The parameter 0 α represents media campaigns rate performed by both smokers and nonsmokers, however the rate of disappearance of media coverage represented by 0 θ .

The existence of equilibrium points of system (2)
In this section, the existence conditions of all possible equilibrium points are determine. It is easy to shows that system (2) has three equilibrium points. The points and their existence conditions can be described as following: In the absence of smokers, that is . Then, system (2) has a unique positive equilibrium point in the interior of positive quadrant of $plane, namely smoking free equilibrium point (SFEP), which denoted by where (4a) provided that the following condition holds (4b) In the absence of temporarily quit smokers ( ). Hence, system (2) has an equilibrium point in the interior of positive octant of $space, namely free temporarily quit smoking equilibrium point (FTQSEP), which denoted by ) , 0 , , ( where: (5a) while is a positive root to the following two isoclines: (5c) Clearly, as , the two isoclines reduced to: $ $ (5d) (5e) Obviously, Eq. (5d) has a unique intersection positive point with $axis that given by (6) while, Eq. (5e) has zero root on the $axis. Therefore, straightforward computation shows that the two isoclines (5b) and (5c) have a unique intersection positive point provided that: $ $ (7a) Consequently, in addition to condition (7a), the following condition guarantees the existence of FTQSEP.
(7b) The coexistence equilibrium point or endemic equilibrium point (EEP), which denoted by ) , , , while represents a positive intersection point of the two isoclines , which is given by Eq. (5b), while the other isocline is given by Clearly, as , the last two isocline reduced to the same polynomial equation given in Eq. (5d) and (5e). Hence they have the same nonnegative roots fall on the $axis. Accordingly, exists uniquely in the interior of positive quadrant $plane provided that Hence the EEP exists uniquely in the provided that in addition to condition (9b) the following condition holds (9c) Note that, the EEP and FTQSEP are coinciding in the interior of positive octant of $space under the condition (7b).

Stability analysis of system (2)
In this section, the stability analysis of all equilibrium points of system (2) is studied. The Jacobian matrix of system (2) at can be written in the following form.
, , $ Consequently, the local stability of SFEP is investigated in the following theorem. Theorem 1: The SFEP of system (2) is locally asymptotically stable (LAS) if the following sufficient conditions hold The Jacobian matrix of system (2) at can be written: Hence, the characteristic equation can be written as (13) Such that $ $ $ $ $ $ $ $ while by using some algebraic computation we obtain that can be written as: ! $ $ $ $ $ $ $ Note that, according to the Routh-Hurwitz criterion, all the eigenvalues of have negative real parts and then the SFEP of system (2) is locally asymptotically stable provided that for ; $ and ! . It is easy to verify that condition (11a) guarantees that the element is negative and condition (11b) guarantees that the term $ . Hence due to the sign of matrix elements and the sufficient conditions (11a) and (11b) all the Routh-Hurwitz conditions are satisfied. Therefore, the proof is complete.
The Jacobian matrix of system (2) at can be written: Hence, the characteristic equation can be written as $ (16) where the eigenvalue in the $ direction is given by Note that, according to the Routh-Hurwitz criterion, all the eigenvalues of have negative real parts and then the FTQSEP of system (2) is locally asymptotically stable provided that for and $ . It is easy to verify that condition (14a) guarantees that the element is negative and condition (14b) guarantees that the element is negative, while condition (14c) guarantees that the term $ . On the other hand condition (14d) ensure that $ . Hence due to the sign of matrix elements and the sufficient conditions (14a)-(14d) all the Routh-Hurwitz conditions are satisfied. Therefore, the proof is complete.
The Jacobian matrix of system (2) at is written as . Hence, the characteristic equation can be written as can be written as: $ $ $ Note that, according to the Routh-Hurwitz criterion, all the eigenvalues of have negative real parts and then the EEP of system (2) is locally asymptotically stable provided that for ; $ and ! . It is easy to verify that condition (17a) and (17b) guarantees that the elements and are negative respectively and condition (17c) guarantees that the term $ . While, the term $ if the condition (17d) holds. Hence due to the sign of matrix elements and the sufficient conditions (17a) and (17d) all the Routh-Hurwitz conditions are satisfied. Therefore, the proof is complete.
It is well known that, for each equilibrium point there is a specific basin of attraction and the point will be a globally asymptotically stable if and only if their basin of attraction is the total domain. Therefore, in the following theorems, the basin of attraction or the global stability conditions of each point is determined.

Theorem 4:
Assume that the SFEP is LAS. Then it has a basin of attraction that satisfies the following conditions (20b) Proof: Consider the following positive definite Lyapunov function, which is defined for all and in the domain of system (2).
Clearly, by differentiating with respect to along the solution curve of system (2), it's obtaining that: Therefore by using the above conditions, it's observed that Obviously, at , moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to SFEP. Hence the proof is complete.
Consider the following positive definite Lyapunov function, which is defined for all and in the domain of system (2).
Clearly, by differentiating with respect to along the solution curve of system (2), it's obtaining that: Therefore by using the above conditions, it's observed that Obviously, at , moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to FTQSEP. Hence the proof is complete.
Furthermore, in the following theorem the conditions that specify the basin of attraction of EEP are established. Theorem 6: Assume that the EEP is LAS. Then it has a basin of attraction that satisfies the following conditions M P γ μ β 2 (22a) V with respect to along the solution curve of system (2), we get Therefore by using the above conditions, it's observed that . Obviously, at moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to EEP. Hence the proof is complete.

Smoking model with diffusion
Obviously, system (1) does not consider the structure of smokers spreading and hence it is not suitable to understand the transmission of smoking in case of moving the individuals. Therefore, it is important to consider the diffusion terms in the model structure in order to investigate whether and how spatial heterogeneity can affect the smoking transmission dynamics. Consequently, the smoking model with diffusion is considered in this section, which is extended to the smoking model given in Eq. (1). Let is a bounded domain in with smooth boundary and is the outward unit normal vector on the boundary, then the smoking model with diffusion can be written as: with homogeneous Neumaun boundary condition and initial conditions , denoted the numbers of potential smokers, smokers, temporary quit smoking, recovery and media at location x and time t. All parameters in system (23) have same meaning as those in system (1). However, the parameters 5 , 4 , are the diffusion coefficients of population respectively; while, is Laplacian operator. Similarly as in system (1), we can reduce system (23), by removing Eq. (23d) (recovery equation) from it, since the other equations in this system are independent of the recovery equation and hence system (23) becomes So that can be determined from (27) As the initial values are positive and the growth functions in the interaction functions of system (26) are assumed to be sufficiently smooth in then standard partial differential equations theory shows that the solution of (26) is unique and continuous for all the positive time in . Furthermore, we recall the positivity lemma in order to using it to proof the positivity and the uniformly bounded of the solution of (26).

Lemma 7 [17]:
Hence, according to lemma (7), we have the following theorem. T τ , we get from 1 st equation of system (26) that: As well, by the same way we have shown that the media equation is bounded by Thus the proof is complete.

Stability analysis of system (26)
In this section, the local and global stabilities of the equilibrium points of diffusion system (26) are discussed. It is easy to verify that the equilibrium points of diffusion system (26) and those of system (2) are the same. Then the stability analysis for each of them can be study as in the following theorems where are given by Eq. (12). Then the characteristic equation can be written as $ Note that, all the Routh-Hurwitz conditions that guarantee the LAS of the SFEP of system (26) are satisfied provided that the conditions (29a)-(29b) hold.  13 Again by using Routh-Hurwitz criterion, we get that the EEP is LAS if the sufficient conditions (33a)-(33c) with (17b) hold.
Note that, according to the above theorems it's clear that, the equilibrium points of diffusion system (26) are always LAS if they are stable in system (2), that is mean without diffusion, but the converse is not necessarily true.
Next, in following theorems the globally asymptotically stability (GAS) of diffusion system (26) at SFEP, FTQSEP and EEP is carried out using the method described in [19]. Theorem 12: Assume that the SFEP of the diffusion system (26) is LAS, then it is GAS if the conditions (20a)-(20b) hold Proof: Consider the following candidate Lyapunov function with is a solution of diffusion system (26 where is a continuously differentiable function defined on some . Then the time derivative of 1 W along the positive solution of system (26) is written as is the vector field that given in right hand side of system (26) without diffusion, while is the diffusion term with and + . Therefore, we obtain that Assume that, the integrand of the first term in Eq. (36) is already calculated as that for the system (2) given by theorem (4). However, the second term is simplified by using Green's formula, and we obtain Accordingly, by using Eq. (38) in Eq. (36), it's obtain that Therefore, in order to construct the function we should have Now by using the function ) , that given in theorem (4 where . Therefore by using the conditions (20a)-(20b), it's observed that Obviously, at , moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to SFEP. Hence the proof is complete.
Theorem 13: Assume that the FTQSEP of the diffusion system (26) is LAS, then it is GAS if the conditions (21a)-(21d) hold Proof: Similarly as in proof of theorem (12), we consider the following candidate Lyapunov function with be a solution of diffusion system (26).
with the function that given in theorem (5). Therefore, direct computation gives that where . Therefore by using the conditions (21a)-(21d), it's observed that where , and are given theorem (5). Obviously, at , moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to FTQSEP. Hence the proof is complete.
with the function that given in theorem (6). Therefore, direct computation gives that where . Therefore by using the conditions (22a)-(22f), it's observed that here ; ; ; and are given in theorem (6). Obviously, at , moreover otherwise. Hence is negative definite and then the solution starting from any initial point satisfy the above conditions will approaches asymptotically to EEP. Hence the proof is complete.

Numerical simulation of systems (1)
In a bid to check our computation, some numerical simulations are carried out. The objective is to understand the global dynamics if system (1) and then study the effects of varying the parameters values. For the following set of hypothetical values of the parameters with different initial conditions the dynamical behavior of system (1) is investigated using the following sets of initial conditions (0.7,0.9,0.6,0.5,0.5), (1,2,3,1,4) and (3,0.5,5,3,1) respectively. The obtained trajectories are drawn in Fig . (1)  .

(a) Trajectory of P(t), (b) Trajectory of S(t), (c) Trajectory of Q(t), (d) Trajectory of R(t), (e) Trajectory of M(t).
Clearly, as shown in Fig. (1), system (1) has a globally asymptotically stable SFEP for the data (43). Now, for the following set of hypothetical parameters values with the same initial sets of values used in Fig. (1), the trajectories of system (1) are drawn in Fig. (2)