GLOBAL ASYMPTOTIC STABILITY OF A DIFFUSIVE SVIR EPIDEMIC MODEL WITH IMMIGRATION OF INDIVIDUALS

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1 Electronic Journal of Differential Equations, Vol. 16 (16), No. 8, pp ISSN: URL: or GLOBAL ASYMPTOTIC STABILITY OF A DIFFUSIVE SVIR EPIDEMIC MODEL WITH IMMIGRATION OF INDIVIDUALS SALEM ABDELMALEK, SAMIR BENDOUKHA Abstract. In this article, e consider a spatially SVIR model of infectious disease epidemics hich allos for continuous immigration of all classes of individuals. We sho that the proposed model has a unique steady state that is asymptotically stable. Using an appropriately constructed Lyapunov functional, e establish its global asymptotic stability. Numerical results obtained through Matlab simulations are presented to confirm the results. 1. Introduction In this article, e are concerned ith reaction-diffusion models of disease epidemics. Of the many models available in the literature, see [3], e ill deal ith one of the suceptible-vaccinated-infectious-recovered (SVIR) type, hich as the name suggests takes into consideration four classes of individuals according to their relation to the disease. Numerous recent publications can be found in the literature regarding the subject. In the folloing is a brief description of the most relevant of these studies. Liu et al. [9] presented to different models to represent the to vaccination strategies: continuous and pulse and shoed that the dynamics of both models depend on the basic reproduction number. The study of Kuniya [8] considered a multi-group SVIR model that allos for the heterogeneity of the population and the effect of immunity induced by the vaccination. Results shoed that the long time behaviour of the model depends on the basic reproductive number. In [5], Duan et al. examined an ODE SVIR model hich allos for the vaccinated individuals to become suceptible again after a certain period of time as the vaccine loses its cover. They studied the dynamics of the model based on LaSalle s invariance principle and appropriately constructed Lyapunov functionals and shoed that the global stability of the equilibriums depend only upon the basic reproductive number. In [7], Hensha and McCluskey studied the local and global asymptotic stability of an ODE SVIR model ith immigration of individuals. The model they proposed is the basis of the ork that ill be presented in this paper. Our aim is to sho that the inclusion of spatial spreading in the model does not affect the asymptotic stability of the equilibrium. The ork carried out here is analogous to that of 1 Mathematics Subject Classification. 35K5, 35K57. Key ords and phrases. Reaction diffusion systems; SVIR; immigration; Lyapunov functional; large time behavior. c 16 Texas State University. Submitted May 1, 16. Published October, 16. 1

2 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 Abdelmalek et al. [], here they studied the asymptotic stability of an SEI model including immigration of all classes of individuals. The remainder of this paper consists of three sections. Section ill present the proposed system model and identify its main characteristics and the conditions on the parameters. Section 3 ill examine the main properties of the steady state solutions. Section ill prove that the unique steady state of the model is globally asymptotically stable using an appropriate Lyapunov functional.. System model In this article, e study the SVIR epidemic model ith immigration of individuals, t u d 1 u = Λ 1 uf() (µ + α)u := f 1 (u, v, ) in R +, t v d v = Λ + αu vg() (µ + β)v := f (u, v, ) in R +, t d 3 = Λ 3 + uf() + vg() (µ + γ + δ) := f 3 (u, v, ) in R +, t R d R = Λ + βv + δ µr := f (v,, R) in R +, (.1) here is an open bounded subset of R n ith pieceise smooth boundary. We assume the initial conditions u (x) = u(x, ), v (x) = v(x, ), (x) = (x, ), R (x) = R(x, ), in, (.) here u (x), v (x), (x), R (x) C () C (), and homogoneous Neumann boundary conditions u ν = v ν = ν = R ν = on R+, (.3) ith ν being the unit outer normal to. We ill also assume that the initial conditions u (x), v (x), (x), R (x) R. Note that this model is similar to that proposed in [7] but ith the inclusion of spatial diffusion. In the proposed model, the positive functions u(x, t), v(x, t), (x, t), R(x, t) represent the population distributions of four classes of people: suceptible, vaccinated, infectious, and recovered, respectively. Hoever, since the recovered class R does not have an impact on the remaining classes, it ill be omitted in the sequel. The parameters Λ i > denote the groth of the different classes of individuals hether through birth or immigration and migration. The parameter α denotes the rate at hich the suceptible population is vaccinated. In this model, death can either be attributed to the infectious disease or to other reasons. The per capita death rate for the former is denoted by γ, hereas the latter is denoted by µ >. Since in reality, it takes a hile for the vaccinated individual to develop full immunity, the parameter β has been introduced here indicating an average duration 1 β. The parameter δ is introduced to allo for some of the infected individuals to recover on their on after a duration 1 δ. We ill assume that α, β, γ, δ. The transfer diagram shon in Figure presents a summary of the proposed model. The model (.1) (.3) includes the spatial spreading of the individuals. The parameters d i represent the diffusivity constants modelling the movement of a certain class as a result of its distribution.

3 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 3 µu (µ + γ) u Λ 1 uf() Λ 3 δ µr R Λ αu vg() βv Λ v µv Figure 1. A transfer diagram of the proposed system model. The functions f() and g() are knon as the incidence functions alloing for a nonlinear relation beteen the first three classes of individuals. We ill assume that the incidence functions satisfy the folloing conditions for all : (H1) f(), g() ith equality if and only if =, (H) f (), g (), (H3) f (), g (), (H) g() f(). In addition, note that for (u, v, ) R 3, e have f 1 (, v, ) = Λ 1, f (u,, ) = Λ + uf(), f 3 (u, v, ) = Λ 3 + βv. Hence, the function (f 1, f, f ) T is essentially nonnegative. Then, the non-negative octant R 3 is an invariant set (see [6, Proposition.1] and [1, page 88]). 3. Steady states and stability 3.1. ODE Case. Before e determine the steady state solutions to our proposed model (.1) (.3) and their asymptotic stability, let us recall the results obtained by Hensha and McCluskey in [7]. We mentioned previously that the fourth equation of the system (.1) (.3) ill be omitted as it has no impact on the remaining three. In the absence of diffusion, the proposed system reduces to First, let us define and for any ɛ, t u = Λ 1 uf() (µ + α)u, t v = Λ + αu vg() (µ + β)v, t = Λ 3 + uf() + vg() (µ + γ + δ). Λ = Λ 1 + Λ + Λ 3, (3.1) D ɛ = { (u, v, ) : u, v, > ɛ and u + v + Λ µ }. (3.)

4 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 System (3.1) as shon in [7] to have the positively invariant non-negative octant R 3 and that there exists a number ɛ > such that D ɛ is non-empty, attracting and positively invariant. Hensha and McCluskey also shoed that the system has a unique equilibrium in the attraction region (u, v, ) D ɛ. This equilibrium is the solution of the system Λ 1 = u f( ) + (µ + α)u Λ = αu + v g( ) + (µ + β)v (3.3) (µ + γ + δ) = Λ 3 + u f( ) + vg( ). To determine the local stability of this unique equilibrium, e need to examine the eigenvalues of the Jacobian. The Jacobian and its second additive compound (see (6.1)) are H µ α F J = α H 1 µ β G H H 1 H and H H 1 α β µ G F J [] = H 1 H H µ α, H α H 1 H µ β respectively, here F = u f ( ), G = v g ( ), H = f( ), H 1 = g( ), H = (µ + γ + δ) u f ( ) v g ( ). (3.) The positivity of the terms F, G, H, H 1 is trivial. The term H, hoever, requires a careful attention. Note that by applying Proposition 6. in the Appendix to f and g and using the third equation of (3.3), e have H (µ + γ + δ) u f( ) v g( ) = (µ + γ + δ) u f( ) v g( ) = Λ 3. For information about the meaning and properties of additive compounds, e refer to [11]. The local stability of the equilibrium can be examined by looking at the determinant of the Jacobian det(j), its trace tr(j), and the determinant of its second additive compound det(j [] ) and ensuring that they are all negative (see Proposition 6.1). We have det J = αf H 1 (H + µ + α)[(h 1 + µ + β)h + G H 1 ] F (H 1 + µ + β)h, (3.5) tr J = (H + H 1 + H + α + β + µ), (3.6) det J [] = F [αh 1 H (H + H + µ + α)] G H 1 (H 1 + H + µ + β) (H + H 1 + α + β + µ)(h + H + µ + α)(h 1 + H + µ + β). (3.7)

5 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 5 It is evident that det J < and tr J <. Hoever, for det J [], the term αh 1 H (H +H +µ+α) needs to be examined. Using condition (H), e have H 1 H, leading to αh 1 H (H + H + µ + α) αh H (H + H + µ + α) = H (H + H + µ). Therefore, e see that det J [] <. Hence, as shon in [7], the unique equilibrium (u, v, ) is in fact locally asymptotically stable. 3.. Properties of the steady states. In this subsection, e shall discuss the basic properties of the non-homogeneous steady states of the proposed epidemic model (.1) (.3). In the presence of diffusion, the steady state solution satisfies d 1 u + Λ 1 u f( ) (µ + α)u =, d v + Λ + αu v g( ) (µ + β)v =, d 3 + Λ 3 + u f( ) + v g( ) (µ + γ + δ) =. (3.8) subject to the homogeneous Neumann boundary condition u ν = for all x. Let = λ < λ 1 λ.... be the sequence of eigenvalues for the elliptic operator ( ) subject to the homogeneous Neumann boundary condition on, here each λ i has multiplicity m i 1. Also let Φ ij, 1 j m i, (recall that Φ = const and λ i at i ) be the normalized eigenfunctions corresponding to λ i. That is, Φ ij and λ i satisfy Φ ij = λ i Φ ij in, ith Φij ν = in, and (x)dx = 1. Φ ij ν = v ν = Theorem 3.1. The constant steady state (u, v, ) is asymptotically stable. Proof. Let us define the linearizing operator d 1 (H + µ + α) F L = α d (H 1 + µ + β) G. H H 1 d 3 H Similar to the ODE case, the asymptotic stability of the steady state solution (u, v, ) can be determined by examining the eigenvalues of the operator L. That is the solution is asymptotically stable if all the eigenvalues of L have negative real parts. In order to achieve that, suppose (φ(x), ψ(x), Υ(x)) is an eigenfunction of L corresponding to an eigenvalue ξ. By definition, e have leading to L(φ(x), ψ(x), Υ(x)) t = ξ(φ(x), ψ(x), Υ(x)) t, φ (L ξi) ψ =. Υ This can be rearranged to the form (A i ξi) i,1 j m i a ij b ij c ij Φ ij =,

6 6 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 here φ = and i,1 j m i a ij Φ ij, ψ = i,1 j m i b ij Φ ij, Υ = i,1 j m i c ij Φ ij, d 1 λ i (H + µ + α) F A i = α d λ i (H 1 + µ + β) G. H H 1 d 3 λ i H The stability of the steady state no reduces to examining the eigenvalues of the matrices A i. The negativity of the real parts of every eigenvalue is ensured if the trace and determinant of A i and the determinant of its second additive compound A [] i are all negative. The trace of A i is given by tr A i = (d 1 + d + d 3 )λ i + tr J, hich is clearly negative for all i since tr J < (see (3.6)). The determinant of A i can be shon to be here det A i = d 1 d d 3 λ 3 i B A λ i C A λ i + det J, (3.9) B A = H d 1 d + (H + µ + α)d d 3 + (H 1 + µ + β)d 1 d 3 >, C A = (G H 1 + (H 1 + µ + β)h )d 1 + ((H + α + µ)h + F H )d + (β + µ + H 1 )(α + µ + H )d 3 >. Clearly, det A i for all i since det J <. The last thing is to examine det A [] i <. The matrix A [] i is the second additive compound of A i given by (d 1 + d )λ i A G F A [] i = H 1 (d 1 + d 3 )λ i B, (3.1) H α (d + d 3 )λ i C here Therefore, ith A = H + H 1 + µ + α + β > B = H + H + µ + α > C = H 1 + H + µ + β >. det A [] i = (d + d 3 )(d 1 + d 3 )(d 1 + d )λ 3 i B A []λ i C A []λ i + det J [], (3.11) B A [] = (B + C)d 1 d + (A + B + C)d d 3 + (A + B + C)d 1 d 3 + Ad 3 + Bd + Cd 1, C A [] = (AC + BC + F H )d 1 + (AB + BC + G H 1 )d + (AB + F H + AC + G H 1 )d 3. We can see that B A [], C A [] >, and since det J [] <, it follos that det A [] i < for all i. Hence, the steady state solution is locally asymptotically stable. This concludes the proof of the Proposition.

7 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 7. Global asymptotic stability In this section, e study the global asymptotic stability of the steady state solutions for the proposed system (.1) (.3). In the ODE case, Hensha and McCluskey [7] established the global asymptotic stability of the unique equilibrium using an appropriate Lyapunov functional. The aim here is to sho that in the presence of diffusion, every solution of the system (.1) (.3) ith a positive initial value that is different from the equilibrium point ill converge to the equilibrium. First, let for x >. L(x) = x 1 ln(x) (.1) Theorem.1. Let V (t) = [u L( u u ) + u L( v v ) + u 3L( )]dx. Then, V (t) is non-negative and is strictly minimized at the unique equilibrium (u, v, ), i.e. it is a valid Lyapunov functional. Hence, (u, v, ) is globally asymptotically stable. Proof. To prove that the steady state solution (u, v, ) is globally asymptotically stable, e need to establish that V (t) is a Lyapunov functional. First, e differentiate V (t) ith respect to time dv dt = [ (1 u u )du dt v + (1 v )dv + (1 dt )d ] dx. dt Substituting the time derivatives ith their values from (.1) yields dv dt = (1 u u )[d 1 u + Λ 1 uf() (µ + α)u]dx + (1 v v )[d v + Λ + αu vg() (µ + β)v]dx + (1 )[d 3 + Λ 3 + uf() + vg() (µ + γ + δ)]dx = I + J. The first part is here I 1 = I = I 3 = I = I 1 + I + I 3, (.) d 1 (1 u ) u dx, u d (1 v ) v dx, v d 3 (1 ) dx.

8 8 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 The second part of the derivative is J = (1 u u )[Λ 1 uf() (µ + α)u]dx + (1 v v )[Λ + αu vg() (µ + β)v]dx + (1 )[Λ 3 + uf() + vg() (µ + γ + δ)]dx, (.3) We start by looking at I. Using Green s formula and assuming the Neumann boundary conditions in (.3), e obtain I 1 = d 1 (1 u u ) udx = d 1 (1 u u ) udx u = d 1 u u dx, I = d (1 v v ) vdx = d v v v dx, and I 3 = d 3 (1 ) dx = d 3 dx. Therefore, by (.), e have [ u v I = d1 u u + d v v + d 3 ] dx <. The second part of the derivative J can be simplified by replacing Λ 1, Λ, and (µ + γ + δ) ith their values from (3.3) and rearranging to the form J = (1 u u )[u f( ) + (µ + α)u uf() (µ + α)u]dx + (1 v v )[v g( ) + (µ + β)v αu + αu vg() (µ + β)v]dx [ + (1 ) Λ 3 + uf() + vg() Λ 3 + u f( ) + v g( ) ] dx [ ( = (1 u u ) u f( ) 1 uf() ) u f( + (µ + α)u (1 u ] ) u ) dx [ ( + (1 v v ) v g( ) 1 vg() ) v g( + (µ + β)v (1 v ) v ) + αu ( u ] u 1) dx [ + (1 ) Λ 3 (1 ( uf() ) + u f( ) u f( ) ) ( vg() + v g( ) v g( ) )] dx. Further simplification yields J = (µ + α)u (1 u u )(1 u u ) + Λ 3(1 )(1 ) (.)

9 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 9 [( uf() + u f( ) u f( ) ) (1 [( + v g( ) 1 vg() v g( ) ( u ) + (1 u ) 1 uf() )] u f( ) )(1 v v ) + ( vg() v g( ) ) (1 ) ] (.5) (.6) + (µ + β)v (1 v v )(1 v v ) + αu ( u v 1)(1 )dx. (.7) u v No, to sho that J is negative, e observe the folloing equalities L( u u ) + L(u u ) = (1 u u )(1 u u ), L( u u ) + L( f() ) L( f( ) ) L( uf() ) u f( ) ( uf() = u f( ) ) (1 u ) + (1 u )( 1 uf() ), u f( ) L ( ) vg() L( v g( ) ) L(v v ) + L( g() ) g( ) ( = 1 vg() ( )(1 v g( v vg() ) v ) + v g( ) ) (1 ), L( u u ) L(uv u v ) + L(v v ) = ( u v 1)(1 u v ). Substituting these in (.7) leads to J = (µ + α)u [ L( u ] u ) + L(u u ) ( ) dx Λ 3 dx u f( ) [L( u f() ) L( u f( ) ) + L( ( uf() ) + L u f( ) v g( ) [L( ( vg() ) ( ) + L v g( + L( v g() ) v ) L g( ) (µ + β) v [ L( v v ) + L(v v )] dx + α u [ L( u u ) L(uv u v ) + L(v v )] dx )] dx )] dx No, using Proposition 6.3 and simplification similar to [7] yields the inequality J (µ + α)u [ L( u ] u ) + L(u u ) ( ) dx Λ 3 dx ( u f( ) [L( u uf() )] u ) + L u f( dx ) [ ( vg() v g( )] ) L v g( dx ) (µ + β) v L( v v )dx α u L( uv u v )dx. It is clear that J, hich leads to dv dt ; dv dt = only at the steady state (u, v, ). Therefore, by Lyapunov s direct method, the steady state solution (u, v, ) is globally asymptotically stable.

10 1 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 5. Numerical examples In this section, e present to numerical examples that illustrate and confirm the findings of this study. The parameters utilized for the examples are stated in Table 1. Table 1. Simulation parameters for the stated numerical examples. Parameters Example 1 Example x x f(x) x+1 x+1 x x g(x) x+ x+ Λ Λ Λ Λ µ.. α.3.3 β.5.5 γ..5 δ.1.1 d d d d 1 5 u 5 sinc[.(x + y )] 5 sinc[.(x + y )] v 15 sinc[.8(x + y )] 15 sinc[.8(x + y )] 1 sinc[.8(x + y )] 1 sinc[.8(x + y )] r.1 sinc[.8(x + y )].1 sinc[.8(x + y )] 5.1. First Example. We use the parameters from the first column of Table 1. Solving the system of equations (3.3) numerically yields the equilibrium solution (u, v, ) = (.796, 1.373, 6.733, 6.876). In the ODE case, the initial data in the ODE case is simply (5, 15, 1) and the equilibrium can be clearly seen to be asymptotically stable as seen in Figure. Figure 3 shos the solutions in the to-dimensional diffusion case and the steady state solution is again asymptotically stable. We see that over-time, the solutions tend to the steady state (u, v,, r ) and become close to uniformly distributed in space. 5.. Second Example. The aim of this example is to sho that high diffusivity constants do not affect the asymptotic stability of the solutions. The system parameters are shon in the second column of Table 1. Figures and 5 sho the solutions in the ODE and to-dimensional cases, respectively. Observe that due to the high diffusivities, the solutions reach the equilibrium (u, v,, r ) = (1.77, , 8.997, ) in a shorter time and that the solutions remain stable in both scenarios.

11 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 11 Population Density u(t) v(t) (t) r(t) Time Figure. Solutions of the proposed system (.1) in the ODE case using parameters from the first column of Table 1. u(x, ). u(x, 1) v(x, ). v(x, 1). (x, ) (x, 1). 6 r(x, ). 5 3 r(x, 1) u(x, 1)..71 v(x, 1). (x, 1). 1 r(x, 1) Figure 3. Solutions of the proposed system (.1) in the todimensional PDE diffusion case using parameters from the first column of Table 1. The snapshots from top to bottom are taken at times t =, t = 1, and t = 1, respectively. 6. Appendix Lemma 6.1 ([1]). Let M be a 3 3 real matrix. If tr(m), det(m), and det(m [] ) are all negative, then all of the eigenvalues of M have negative real parts, here (see [11]) a 11 a 1 a 13 a 11 + a a 3 a 13 M = a 1 a a 3, M [] = a 3 a 11 + a 33 a 1. (6.1) a 31 a 3 a 33 a 31 a 1 a + a 33

12 1 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 Population Density u(t) v(t) (t) r(t) Time Figure. Solutions of the proposed system (.1) in the ODE case using parameters from the second column of Table 1. u(x, ). u(x, 1). 8 6 v(x, ). v(x, 1) (x, ) (x, 1) r(x, )..9.8 r(x, 1) u(x, 1) v(x, 1) (x, 1) r(x, 1) Figure 5. Solutions of the proposed system (.1) in the todimensional PDE diffusion case using parameters from the second column of Table 1. The snapshots from top to bottom are taken at times t =, t = 1, and t = 1, respectively. Proof. Let λ j, j = 1,, 3 be the eigenvalues of M ith R(λ 1 ) R(λ ) R(λ 3 ). It follos from det(m) < that λ 1 λ λ 3 <. Thus, either R(λ j ) < for j = 1,, 3 (hich ould prove the lemma) or R(λ 1 ) < R(λ ) R(λ 3 ). Suppose that the second set of inequalities holds. Since tr(m) <, it follos that λ 1 + λ + λ 3 <, hich implies that R(λ 1 + λ ) < and R(λ 1 + λ 3 ) <. The eigenvalues of M [] are λ i + λ j, 1 i < j 3, and so sgn(det(m [] )) = sgn(r(λ 1 + λ )R(λ 1 + λ 3 )R(λ + λ 3 )) = sgn(r(λ + λ 3 )).

13 EJDE-16/8 GLOBAL STABILITY OF SVIR MODELS 13 It follos from det(m [] ) < that R(λ + λ 3 ) <. Thus, it cannot be that R(λ 1 ) < R(λ ) R(λ 3 ), and therefore R(λ j ) < for j = 1,, 3. Proposition 6. ([13]). f () f() and g () g() for all >. Proof. Let >. Since f() is continuous on [, ] and differentiable on (, ), the mean value theorem implies that there exists c (, ) such that f (c) f() f(). By (H1), e have f (c) = f(). From (H3), f is monotone decreasing. Thus, f () f (c) = f(). The same can be said about g. Proposition 6.3 ([7]). Suppose the incidence functions f and g satisfy the criteria in (H1) (H). It follos that if >, then ( f() ) L f( L ( ), (6.) ) ( g() ) L g( L ( ) ). (6.3) Proof. In this proof, e ill only establish the property (6.). Hoever, the same can be said about (6.3). Let and m() = f(). It follos that m () = f () f() f() f() =. Therefore, e conclude that m is decreasing, hich leads to m() m( ), i.e., and so Since f is increasing, e have f() f( ), f() f( ). 1 f() f( ). Note that L(x) = 1 1 x. Hence, L is increasing for x > 1, and ( f() ) L f( L( ) ). Acknoledgements. The authors ould like to thank Prof. M. Kirane and the anonymous referee for their most insightful suggestions that improved the quality of this article. References [1] S. Abdelmalek, M. Kirane, A. Youkana; A Lyapunov functional for a triangular reaction diffusion system ith nonlinearities of exponential groth, Math. Methods in the Appl. Sci., Vol. 36, 8 85, (13). [] S. Abdelmalek, S. Bendoukha; Global Asymptotic Stability for an SEI Reaction-Diffusion Model of Infectious Diseases ith Immigration, submitted. [3] S. Busenberg, K. Cooke; Vertically Transmitted Diseases, Methods and Dynamics. Biomathematics, Vol. 3. Springer Verlag: Ne York, (1993).

14 1 S. ABDELMALEK, S. BENDOUKHA EJDE-16/8 [] R. G. Casten, C. J. Holland; Stability properties of solutions to systems of reaction-diffusion equations, SIAM J. Appl. Math. 33, (1977). [5] X. Duan, S. Yuan, X. Li; Global stability of an SVIR model ith age of vaccination, Applied Math. and Comp., Vol. 6, 58 5, (1). [6] W. M. Haddad, V. Chellaboina, Q. Hui; Nonnegative and Compartmental Dynamical Systems, Princeton University Press, Princeton, Ne Jersey, (1). [7] S. Hensha, C. C. McCluskey; Global stability of a vaccination model ith immigration, Electron. J. of Diff. Eqs., Vol. 15, No. 9, pp. 1 1, (15). [8] T. Kuniya; Global stability of a multi-group SVIR epidemic model, Nonlinear Anal.: Real World Apps., Vol. 1, , (13). [9] X. Liu, Y. Takeuchib, S. Iami; SVIR epidemic models ith vaccination strategies, J. Theor. Bio., Vol. 53, 1 11, (8). [1] C. C. McCluskey, P. van den Driessche; Global analysis of to tuberculosis models. J. Dynam. Diff. Eqs., Vol. 16(1): , (). [11] J. S. Muldoney; Compound matrices and ordinary differential equations, Rocky Mountain J. Math., Vol. : , (199). [1] P. Quittner, Ph. Souplet; Superlinear Parabolic Problems. Blo-up, Global Existence and Steady States, Birkhauser Verlag AG, Basel. Boston., Berlin, (7). [13] Ram P. Sigdel, C. Connell McCluskey; Global stability for an SEI model of infectious disease ith immigration, Applied Math. and Comp., Vol. 3: , (1). Salem Abdelmalek Department of mathematics, University of Tebessa 1, Algeria address: sallllm@gmail.com Samir Bendoukha Electrical Engineering Department, College of Engineering at Yanbu, Taibah University, Saudi Arabia address: sbendoukha@taibahu.edu.sa