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Global Dynamics of an SVEIR Model with Age-Dependent Vaccination, Infection, and Latency.

1. Introduction

Protection induced by vaccines plays a significant role in preventing and reducing the transmission of infectious diseases. One of the greatest successful events of vaccination is illustrated through the eradication of small-pox. It is reported in [1] that the case of small-pox was last recorded in 1977. Immunity conveyed by vaccination depends on different vaccines and vaccination policies--lifelong immunity occurs for certain vaccines while immunization period is induced by some vaccines. However, waning vaccine-induced immunity takes place (naturally) after immunization process. It is reported in [2] that a significant decay in the proportion of chicken pox took place in US in 1995 after conducting a universal vaccination campaign. But, surprisingly new cases of chicken pox appeared mainly in highly vaccinated school communities in US. Some studies were conducted and revealed waning vaccine-induced immunity in children under protection induced by vaccine. Moreover, this was also investigated in [3-5] and it was proved that such waning of immunity is attached to the time since vaccination and the age at vaccination. In this regard, it was published in [6, 7] that the time of vaccine-induced immunity depends on individuals features and vaccine age.

From the abovementioned statements and citations, we see that it is necessary to associate waning vaccine-induced immunity with vaccination in infectious disease models and it is interesting to investigate the impact of waning vaccine-induced immunity on the dynamics of infectious diseases. Many mathematical models on vaccination have been already investigated and, to cite a few of them, see [2, 8-16]. Some of the above-cited model considered either age-dependent vaccination, while some did not.

Despite vaccination age-structure being the main and appropriate feature required in the dynamics of infectious diseases with waning induced-vaccine immunity, most of epidemiological models with vaccination including waning induced-vaccine immunity were studied after assuming a constant rate of immunity loss (to name a few, see [9, 10, 15]). Age-dependent vaccination was considered in some epidemiological models studied recently in [1, 2, 17-20]. However, some of these works considered either waning vaccine-induced immunity or not, either vaccine-agedependent waning vaccine-induced immunity or not, either age-dependent latency or not, either age-dependent relapse or not, and either age-dependent infection or not. In [17] an SVIR epidemic model with continuous age-dependent vaccination was formulated to establish the global stability of equilibria. In [18] an SVIJS epidemic model with age-dependent vaccination was considered to study the asymptotical behavior of the equilibria, after assuming that age-dependent vaccine-induced immunity decays with time after vaccination. In [19] an SVIS epidemic model with age-dependent vaccination and vaccine-age-dependent waning vaccine-induced immunity and treatment was formulated to investigate backward bifurcations. In [2] an SVIR epidemic model with vaccination age was considered to establish global stability of equilibria, after assuming that vaccine-induced immunity decays with time after vaccination. In [20] an SVEIR epidemic model with age-dependence vaccination, latency, and relapse was formulated to establish the global stability of the equilibria. In [1] a multigroup SVEIR epidemic model with latent class and vaccination age was formulated to study global stability of equilibria, after assuming that vaccine-induced immunity decays with time after vaccination and likewise in [17-19].

Recently, in [14] an SVEIR epidemic model with continuous age-structure in the latent and infectious classes and without continuous age-structure in the vaccinated class was formulated to prove the global stability of equilibria, while in [20] an SVEIR epidemic model with continuous age-structure in the latent, infectious, and recovered classes and with vaccine-age-dependent waning vaccine-induced immunity was formulated. Moreover, in [14] the latency and infection ages are denoted by the same variable a. Similarly, in [20] the latency, relapse, and vaccination ages are denoted by the same variable a. In spite of this, to the best of our knowledge, the global dynamics of an SVEIR epidemic model with continuous age-structure in latency, infection, vaccination, and vaccine-age-dependence waning vaccine-induced immunity has not yet been neither considered nor investigated using the approach of Lyapunov functionals. The aim of this work is to fill this gap by investigating the global dynamics of an SVEIR epidemic model as above defined. Motivated by [14, 20], we propose a new SVEIR epidemic model originated from an existing SVEIR formulated in [1], by considering continuous age-structure in latency and infection in addition to age-dependence vaccination and vaccine-age-dependence waning vaccine-induced immunity (which the authors took into account in [1]). However, the latency, infection, and vaccination ages are all denoted by a, as in [14,20]. Moreover, in this paper, we also consider a more significant incidence rate (taking into account transmission by both age-mates infective individuals and infective individuals of any age) of the form

[mathematical expression not reproducible], (1)

where [K.sub.0](a) and K(a, a') are defined below, instead of the classical incidence rate of the form

S(t) [[integral].sup.[infinity].sub.0] [beta] (a) i (a, t) da, (2)

where [beta](a) denotes the coefficient of disease transmission from infective individual, with infection age a, to susceptible individual. The latter is considered in the references therein where continuous age-structure in infection is taken into account.

The model splits the total population into five epidemiological compartments, namely, the susceptible compartment, the vaccinated compartment, the latent compartment, infected compartment, and the removed compartment. Let S(t) and R(t) be the number of individuals in the susceptible and removed compartments at time t, respectively. Let v(a, t), e(a, t), and i(a, t) be the density of vaccinated, (latently) infected, and (actively) infected individuals with vaccination, latency, and infection age a at time t, respectively. It follows that V(t), E(t), and I(t) defined by

[mathematical expression not reproducible], (3)

are the number of individuals in the vaccinated, latent, and infected compartments, respectively.

The model to be investigated consists of a hybrid system of nonlinear coupled ordinary differential equations and partial differential equations of the form

[mathematical expression not reproducible] (4)

where [eta](a) = [alpha](a) + [mu](a), [??](a) = [epsilon](a) + p(a), and a(a) = [gamma](a) + [mu](a), with boundary conditions

[mathematical expression not reproducible]. (5)

And initial conditions

[mathematical expression not reproducible], (6)

where [S.sub.0] and [R.sub.0] are initial size of susceptible and removed individuals, respectively, and [v.sub.0](a), [e.sub.0](a), and [i.sub.0](a) are initial age-density of vaccinated, latent, and infective individuals, respectively. Moreover, [v.sub.0], [e.sub.0], and [i.sub.0] are Lebesgue integrable functions, and it is assumed that the recruitment of newly vaccinated individuals in the vaccinated compartment is done at age zero.

The meaning of parameters in (4)-(5) is given below:

[LAMBDA]: constant recruitment rate of susceptible individuals

v: rate of vaccination of susceptible individuals

[[mu].sup.0]: natural mortality rate of individuals

[alpha](a): age-specific rate of waning vaccine-induced immunity

[mu](a): age-specific natural mortality rate

[gamma](a): age-specific removal rate

[epsilon](a): age-specific rate moving from latent to infective

[K.sub.0](a): age-specific infection rate of susceptible individuals by infective individuals (of the same age-intracohort contagion)

K(a, a'): probability that an infective individual of age a' will successfully infect a susceptible individual of age a, after contact

In the sequel, we make following assumptions on parameters in (4)-(5):

A1

(i) [LAMBDA], v, [[mu].sup.0] > 0 with v < [[mu].sup.0].

(ii) [mathematical expression not reproducible] with essential upper bounds [mathematical expression not reproducible], respectively.

(iii) [mathematical expression not reproducible], respectively.

(iv) There exists [mathematical expression not reproducible].

A2. R(*) is a decreasing function of t for any constant removal rate [[gamma].sup.0] such that [[gamma].sup.0] [greater than or equal to] [bar.[gamma]].

A3. [mathematical expression not reproducible], for every a.

This paper is composed sections, in addition to the introduction, which are structured as follows. In Section 2, we present some preliminary result on compactness property of the semiflow generated by (4)-(5) and we discuss its asymptotic smoothness property. The uniform persistence property of (4)-(5) is addressed in Section 4. Section 3 deals with the existence of equilibria and the formulation of the threshold parameter [R.sub.0] (the basic reproduction number). Local stability of equilibria for (4) is established in Section 5, while global stability of equilibria for (4) is examined in Section 6.

2. Preliminaries

We consider the Banach space

X = R [lambda] [L.sup.1] (0, [infinity]) x [L.sup.1] (0, [infinity]) x [L.sup.1] (0, [infinity]) x R (7)

endowed with the norm

[mathematical expression not reproducible], (8)

where [mathematical expression not reproducible] the positive cone of the Banach space [lambda] such that

[mathematical expression not reproducible]. (9)

For any initial value [mathematical expression not reproducible] satisfying the conditions

[mathematical expression not reproducible] (10)

system (4) is well-posed, under assumption A1, due to [21]. Thus, a continuous semiflow O: [R.sub.+] x [X.sub.+] [right arrow] [X.sub.+] is obtained and it is defined by (4) such that

[mathematical expression not reproducible]. (11)

Now, we introduce the functions

[mathematical expression not reproducible]. (12)

By integrating the second, third, and fourth equations of (4) along the characteristic line t-a = constant, respectively, we get

[mathematical expression not reproducible], (13)

where

[mathematical expression not reproducible]. (14)

Taking the norm of [[PHI].sub.t]([X.sub.0]) and using the positiveness of the components of [[PHI].sub.t]([X.sub.0]), we get

[mathematical expression not reproducible]. (15)

Differentiating (15) with respect to t leads to

[mathematical expression not reproducible]. (16)

Next, we seek for the estimates of each time-derivative on the right hand side (16). First, we have

[mathematical expression not reproducible]. (17)

Applying the Leibniz Integral Rule to the first integral in (17) yields

[mathematical expression not reproducible]. (18)

Likewise, we also have

[mathematical expression not reproducible] (19)

and

[mathematical expression not reproducible]. (20)

Therefore, we get

[mathematical expression not reproducible]. (21)

Using (iv) of A1, (21) yields

[mathematical expression not reproducible], (22)

that is,

[mathematical expression not reproducible]. (23)

Thus, we obtain

[mathematical expression not reproducible]. (24)

where [mathematical expression not reproducible].

If we consider the state space r of (4), defined by

[mathematical expression not reproducible], (25)

we get

[mathematical expression not reproducible] (26)

for any t [greater than or equal to] 0, whenever [X.sub.0] [member of] [GAMMA]. Moreover

[mathematical expression not reproducible], (27)

for any [X.sub.0] [member of] [X.sub.+].

Then, we state the following result.

Lemma 1. The set r is positively invariant for [PHI]; that is,

[mathematical expression not reproducible]. (28)

Moreover, [[PHI].sub.t]([X.sub.0]) is point dissipative and the set [GAMMA] attracts all points in [X.sub.+].

As we aim to make use of the Lasalle's Invariance Principle, we are required to establish the relative compactness of the orbit {[[PHI].sub.t]([X.sub.0]) : t [greater than or equal to] 0} in [X.sub.+] due to the infinite dimensional Banach space x. For this, we consider two operators [THETA] and [PSI] ([THETA], [PSI] : [R.sub.+] x [X.sub.+] [right arrow] [X.sub.+]), such that

[mathematical expression not reproducible], (29)

where

[mathematical expression not reproducible]. (30)

Thus, [[PHI].sub.t]([X.sub.0]) = [[THETA].sub.t]([X.sub.0]) + [[PSI].sub.t]([X.sub.0]), for any t [greater than or equal to] 0; and from the proof of [22, Proposition 3.13] and Lemma 1, we get to the following result.

Theorem 2. For [X.sub.0] [member of] r, the orbit {[[PHI].sub.t]([X.sub.0]) : t [greater than or equal to] 0} has a compact closure in [X.sub.+] if the following conditions are satisfied:

(i) There exists a function [mathematical expression not reproducible].

(ii) For any t [greater than or equal to] 0, [[PSI].sub.t](x) maps any bounded sets of [GAMMA] into a set with compact closure in [X.sub.+].

For verifying conditions (i) and (ii) of Theorem 2, we need lemmas.

Lemma 3. For r > 0, let [DELTA](t, r) = [e.sup.-[??]t]r. Then [lim.sub.t[right arrow][infinity]] [DELTA](t, r) = 0. Then (i) of Theorem 2 holds.

Proof. Clearly, we see that [lim.sub.t[right arrow][infinity]][DELTA](t, r) = 0. Making use of some equations in (13), we get

[mathematical expression not reproducible]. (31)

Taking the initial condition [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (32)

Lemma 4. For t [greater than or equal to] 0, [[PSI].sub.t](x) maps any bounded set of [GAMMA] into a set with a compact closure in [X.sub.+].

Proof. Since S(t) and R(t) remain in the compact set [0, [LAMBDA]/[??]] by Lemma 1, it is sufficient to show that [mathematical expression not reproducible] remain in a precompact subset of [L.sup.1.sub.+](0, [infinity]), which does not depend on the initial data [X.sub.0] [member of] [GAMMA]. To achieve this, the following conditions (see [23, Theorem B.2]) mustbe satisfied for [mathematical expression not reproducible]:

(i) The supremum of [parallel][??](x, t)[parallel][sub.1] with respect to [X.sub.0] [member of] [GAMMA] is finite;

(ii) [lim.sub.h[right arrow][infinity]] [[integral].sup.[infinity].sub.h] [??](a, t)da = 0 uniformly with respect to [X.sub.0] [member of] [GAMMA];

(iii) [mathematical expression not reproducible] uniformly with respect to [X.sub.0] [member of] r;

(iv) [mathematical expression not reproducible] uniformly with respect to [X.sub.0] [member of] [GAMMA],

where [mathematical expression not reproducible]. It follows from (13) and (30) that

[mathematical expression not reproducible], (33)

and hence, using Lemma 1, we get

[mathematical expression not reproducible], (34)

and hence, (i), (ii), and (iv) follow. To establish (iii), we take a sufficiently small h such that h [member of] (0, t) and we show that

[mathematical expression not reproducible]. (35)

Indeed, [chi](a) is a nondecreasing function of a such that 0 < [chi](a) [less than or equal to] 1 satisfying

[mathematical expression not reproducible]. (36)

On the other hand, the Lipschitz continuity of S(x) is obtained from the first equation of (4), using the boundedness of the solution of (4) (see Lemma 1); that is, there exists [l.sub.s] > 0 such that [mathematical expression not reproducible].

Since v(2[LAMBDA] + [l.sub.s])h/[??] does not depend on the initial data [mathematical expression not reproducible], it follows from (35) that (iii) is satisfied.

Therefore, From Lemma 1 and Theorem 2, the existence result of global attractors (see [24]) follows.

Theorem 5. The semiflow {[[PHI].sub.t]([X.sub.0]) : t [greater than or equal to] 0} has a global attractor in [X.sub.+], which attracts any bounded subset of [X.sub.+].

3. Equilibria and Basic Reproduction Number

System (4) has a unique disease-free equilibrium [E.sup.0] = ([S.sup.0], [v.sup.0](a),[e.sup.0](a),[i.sup.0],[R.sup.0]), where

[mathematical expression not reproducible]. (37)

Apart from [E.sup.0], system (4) could also have an endemic equilibrium. We suppose that there exists an endemic equilibrium for system (4) denoted by [E.sup.*] = ([S.sup.*], [v.sup.*], [e.sup.*], [i.sup.*], [R.sup.*]). Therefore,

[mathematical expression not reproducible]. (38)

are satisfied. In addition, [E.sup.*] also satisfies (5), i.e.,

[mathematical expression not reproducible]. (39)

The second equation of (38) and the first equation of (39) give

[mathematical expression not reproducible]. (40)

It follows from the third equation of (38) that

[mathematical expression not reproducible]. (41)

Equations (5) and (41) together with the fourth equation of (38) yield

[mathematical expression not reproducible]. (42)

We introduce parameters [R.sub.0], L, J, and P such that

[mathematical expression not reproducible]. (43)

By substituting (42) into the second equation of (39), we have

[S.sup.*] = [S.sub.0]/[R.sub.0] (44)

And hence,

[v.sup.*] (a) = [v.sup.0](a)/[R.sub.0]. (45)

Then, substituting (37), (40), (42), and (44) into the first equation of (38) yields

[e.sup.*] (0) = [LAMBDA](1 - 1/[R.sub.0]). (46)

Therefore, it easily follows that

[mathematical expression not reproducible]. (47)

A threshold condition is derived from the existence condition for the endemic equilibrium [E.sup.*] such that [R.sub.0] > 1. Thus, the parameter [R.sub.0], given by the first equation of (43), can be called the basic reproduction number of system (4). Moreover, [R.sub.0] can also be expressed as

[R.sub.0] = [R.sub.intra] + [R.sub.inter], (48)

where

[mathematical expression not reproducible]. (49)

[R.sub.intra] and [R.sub.inter] can be understood, respectively, as the basic reproduction numbers for the corresponding model with purely intracohort infection mechanism (i.e., situation in which individuals can only be infected by their age-mates) and for the corresponding model with purely intercohort infection mechanism (i.e., situation in which individuals can be infected by those of any age).

Therefore, we state the following.

Theorem 6. If [R.sub.0] [less than or equal to] 1, then system (4) has only a disease-free equilibrium [E.sup.0], while if [R.sub.0] > 1, then system (4) also has an endemic equilibrium [E.sup.*] in addition to the disease-free equilibrium [E.sup.0].

4. Uniform Persistence

This section is devoted to the uniform persistence of system (4) under the condition [R.sub.0] > 1. For this, we introduce a function [rho] : [X.sub.+] [right arrow] [R.sub.+] defined by

[mathematical expression not reproducible], (50)

where [mathematical expression not reproducible]. Furthermore, we consider the set [x.sub.0] defined by

[mathematical expression not reproducible] (51)

such that [[PHI].sub.t]([X.sub.0]) [right arrow] as t-whenever [X.sub.0] [member of] [X.sub.+] \ [x.sub.0].

Definition 7 (see [23, p. 61]). System (4) is uniformly weakly [rho]-persistent (respectively, uniformly strongly [rho]-persistent) if there exists a positive [[epsilon].sup.*], independent of initial conditions, such that

[mathematical expression not reproducible] (52)

for [X.sub.0] [member of] [X.sub.+].

Theorem 8. If [R.sub.0] > 1, then system (4) is uniformly weakly [rho]-persistent.

Proof. We assume that, for any [[epsilon].sup.*] > 0, we can find [X.sub.0] [member of] [X.sub.+] such that

[mathematical expression not reproducible]. (53)

Since [R.sub.0] > 1, then we can find a small enough [[epsilon].sup.*.sub.0] >0 such that

[mathematical expression not reproducible]. (54)

In particular, we can find [mathematical expression not reproducible] such that

[mathematical expression not reproducible]. (55)

We can assume that, for any [mathematical expression not reproducible]. It follows from the first equation of (4) that

[mathematical expression not reproducible]. (56)

We apply the Laplace transform L to the above inequality and we obtain

[mathematical expression not reproducible]. (57)

It follows that

[mathematical expression not reproducible] (58)

And hence,

[mathematical expression not reproducible]. (59)

This yields [mathematical expression not reproducible]. We can assume that, for any t [greater than or equal to] 0,

[mathematical expression not reproducible]. (60)

Now, we consider the boundary condition defined by the second equation of (5) and we get

[mathematical expression not reproducible]. (61)

We apply the Laplace Transform L to the above inequality so that

[mathematical expression not reproducible]. (62)

Dividing the above inequality by L{e(0, t)} yields

[mathematical expression not reproducible]. (63)

First, taking the limit inferior as t [right arrow] [infinity] on both sides of the above inequality, we obtain

[mathematical expression not reproducible]. (64)

Next, take the limit as [lambda] [right arrow] 0 on both sides of the above inequality, we get

[mathematical expression not reproducible], (65)

which contradicts the inequality given in (54).

Combining the results from Theorems 5 and 8 with [25, Theorem 3.2] leads to the uniform (strong) [rho]-persistence as follows.

Theorem 9. If [R.sub.0] > 1, then the semiflow [PHI] is uniformly (strongly) [rho]-persistent.

Definition 10. A total trajectory of a continuous semiflow [PHI], defined by (11), is a function [lambda] : R [right arrow] [X.sub.+] such that [[PHI].sub.t](X(r)) = X(t + r) for any (t, r) [member of] [R.sub.+] x R.

Note that a global attractor will only contain points with total trajectories through them as it needs to be invariant. So, the [alpha]-limit point of a total trajectory X, passing through X(0) = [X.sub.0], is given by

[mathematical expression not reproducible]. (66)

A total trajectory X(t) = (S(t), v(x, t), e(x, t), i(x, t), R(t)) satisfies

[mathematical expression not reproducible]. (67)

Corollary 11. Let A and X(t) be, respectively, a global attractor of [PHI] in [X.sub.+] and a total trajectory of [PHI] in A [intersection] [X.sub.+]. If [R.sub.0] > 1, then there exists [epsilon] > 0 such that

[mathematical expression not reproducible]. (68)

Proof. We consider the boundary condition given by the second equation of (5). Using (25) and (ii) of A1, we get

[mathematical expression not reproducible]. (69)

From the first equation of (4), we have

[mathematical expression not reproducible]; (70)

that is,

[mathematical expression not reproducible]. (71)

This yields

[mathematical expression not reproducible]. (72)

Taking the limit inferior as t [right arrow] [infinity] in (72) leads to

[mathematical expression not reproducible]. (73)

Therefore, S(t) [greater than or equal to] [[epsilon].sub.1]. It follows that v(0, t) [greater than or equal to] [[epsilon].sub.1] v := [[epsilon].sub.2].

Now, we consider again the boundary condition given by the second equation of (5). It is easy to see that

e(0,t) = [rho]([[PHI].sub.t] ([X.sub.0])) = [rho](X (t)) (74)

And hence,

[mathematical expression not reproducible]. (75)

It follows from Theorem 9 and Definition 7 that e(0, t) [greater than or equal to] [[epsilon].sup.*] = [[epsilon].sub.3].

Further, considering the boundary condition given by the third equation of (5), we obtain

[mathematical expression not reproducible]. (76)

Finally, the fifth equation of (4) leads to

[mathematical expression not reproducible] (77)

And hence,

[mathematical expression not reproducible]. (78)

Taking the limit inferior as t [right arrow] [infinity] in (78), we obtain

[mathematical expression not reproducible]. (79)

And therefore R(t) [greater than or equal to] [[epsilon].sub.5]. By choosing [epsilon] such that [epsilon] = [min.sub.i] {[[epsilon].sub.i]}, for i [member of] {1,2, 3,4, 5}, we get

S(t), v (0,t), e (0,t), i(0,t),R [greater than or equal to] [epsilon]. (80)

5. Local Stability of Equilibria

The conditions of stability for each equilibrium will be derived through linearization technique around the equilibrium.

The conditions of stability for the disease-free equilibrium [E.sup.0] can be investigated through the following result.

Theorem 12. If [R.sub.0] < 1 then [E.sup.0] is locally asymptotically stable; if [R.sub.0] > 1 then [E.sup.0] is unstable.

Proof. To investigate the stability of the disease-free equilibrium [E.sup.0], we denote by [mathematical expression not reproducible], and [??](t) the perturbations of S(t), v(a, t), e(a, t), i(a, t), and R(t), respectively, such that

[mathematical expression not reproducible]. (81)

The perturbations satisfy the following:

[mathematical expression not reproducible], (82)

after substituting (81) into (4) and neglecting the terms of order higher or equal to two, with boundary conditions

[mathematical expression not reproducible], (83)

after substituting (81) into (5) and neglecting the terms of order higher or equal to two.

Now, we consider the exponential solutions of systems (82)-(83) of the form

[mathematical expression not reproducible], (84)

where [bar.S], [bar.v](a), [bar.e](a), [bar.i](a), and [lambda] (real or complex number) satisfy the following system:

[mathematical expression not reproducible], (85)

with boundary conditions

[mathematical expression not reproducible]. (86)

From the second, third, and fourth equations of (85), we

[mathematical expression not reproducible], (87)

respectively, where [bar.v](0), [bar.e](0), and [bar.i](0) are given by (86).

Substituting the last equation of (87) into the boundary condition given by the second equation of (86) yields the characteristic equation

[C.sup.0] ([lambda]) = 1, (88)

where

[mathematical expression not reproducible], (89)

such that [C.sup.0](0) = [R.sub.0]. It is not difficult to see that (d/d[lambda])[C.sup.0]([lambda]) = -[C.sup.0]([lambda]) < 0. Thus, [C.sup.0]([lambda]) is a decreasing continuous function of [lambda] which approaches [infinity] as [lambda] [right arrow] -[infinity] and 0 as [lambda] [right arrow] [infinity]. Hence, the characteristic equation (88) admits a real solution [[lambda].sup.*] such that [[lambda].sup.*] <0 whenever [C.sup.0](0) < 1 and [[lambda].sup.*] >0 whenever [C.sup.0](0) > 1.

On the other hand, by assuming a complex solution [lambda] = [alpha] + i[beta] of the characteristic equation C[degrees]([lambda]) = 1, we can notice that r([e.sup.[lambda]]) [less than or equal to] [e.sup.R([lambda])] is always true. Thus, we clearly get R[C.sup.0]([lambda]) < [C.sup.0](R[lambda]). It follows from the characteristic equation [C.sup.0]([lambda]) = 1 that r[C.sup.0]([lambda]) = 1 and I[C.sup.0]([lambda]) = 0. Therefore, we obtain 1 [less than or equal to] [C.sup.0](R[lambda]), i.e., [C.sup.0]([[lambda].sup.*]) [less than or equal to] [C.sup.0](R[lambda]). Hence, R[lambda] [less than or equal to] [[lambda].sup.*], since [C.sup.0]([lambda]) is a decreasing function.

It results from the above statements that all eigenvalues of the characteristic equation [C.sup.0] ([lambda]) = 1 have negative real part whenever [C.sup.0](0) < 1, i.e., [R.sub.0] < 1. Thus, the disease-free equilibrium [E.sup.0] is locally asymptotically stable if [R.sub.0] < 1. Otherwise, [C.sup.0](0) [greater than or equal to] 1, i.e., the unique real solution of the characteristic equation [C.sup.0]([lambda]) = 1 is positive, and hence the disease-free equilibrium [E.sup.0] is unstable.

Theorem 13. If [R.sub.0] > 1 then [E.sup.*] is locally asymptotically stable.

Proof. Likewise for the disease-free equilibrium, we perturb the disease-endemic equilibrium by letting

[mathematical expression not reproducible]. (90)

Substituting [mathematical expression not reproducible] into (4) and neglecting the terms of second order and above, the perturbations satisfy the linear system

[mathematical expression not reproducible], (91)

with boundary conditions

[mathematical expression not reproducible], (92)

after substituting [mathematical expression not reproducible] and neglecting the terms of second order and above.

Now, we consider the exponential solutions of systems (91)-(92) of the form

[mathematical expression not reproducible], (93)

where [bar.S], [bar.v](a), [bar.e](a), [bar.i](a), and [lambda] (real or complex number) satisfy the system

[mathematical expression not reproducible], (94)

with boundary conditions

[mathematical expression not reproducible]. (95)

Similarly to the process leading to the characteristic equation (88), the characteristic equation at the disease-endemic equilibrium [E.sup.*] is given by

[mathematical expression not reproducible], (96)

where

[mathematical expression not reproducible]. (97)

It is sufficient to prove that (96) has no root with nonnegative real part. So, we suppose that (96) has a complex root with nonnegative real part denoted by

[lambda] = [alpha] + i[beta], (98)

where [alpha] [greater than or equal to] 0 and [beta] [not equal to] 0. It follows from (96) that

[mathematical expression not reproducible], (99)

where

[mathematical expression not reproducible], (100)

and hence

[mathematical expression not reproducible], (101)

where

[mathematical expression not reproducible] (102)

and

[mathematical expression not reproducible], (103)

with

[mathematical expression not reproducible]. (104)

Since [mathematical expression not reproducible] is a decreasing function of A), thus R[C.sup.*]([alpha] + i[beta]) [less than or equal to] 1.

Therefore, the latter statement contradicts (102).

6. Global Stability of Equilibria

To investigate the global asymptotic stability of equilibria of system (4) we shall use suitable Volterra-type Lyapunov functions of the form

G (X) = X - 1 - ln X, X > 0. (105)

We state the following results.

Theorem 14. The disease-free equilibrium [E.sup.0] of (4) is globally asymptotically stable if [R.sub.0] [less than or equal to] 1.

Proof. A Lyapunov function [L.sup.0] of the form

[mathematical expression not reproducible] (106)

is considered, where

[mathematical expression not reproducible]. (107)

The functions [phi] and [omega] are nonnegative functions to be chosen suitably and carefully. [[??].sup.0.sub.i], i = 1,..., 5, denote the derivatives of [L.sup.0.sub.i] with respect to t along the solution to (4) and are given by

[mathematical expression not reproducible]. (108)

Therefore,

[mathematical expression not reproducible], (109)

i.e.,

[mathematical expression not reproducible], (110)

after choosing a and a' such that

i(a',f)/[zeta](a') = I (a,t)/[zeta](a). (111)

Note that

[mathematical expression not reproducible], (112)

after using [mathematical expression not reproducible]. Thus, (110) becomes

[mathematical expression not reproducible]. (113)

Using assumption A2, we show that

[mathematical expression not reproducible], (114)

where

I(t) = [[integral].sup.[infinity].sub.0] i (a, x) da. (115)

By assumption A3, obtain

[mathematical expression not reproducible]. (116)

Next, we choose functions [omega] and [phi] such that

[mathematical expression not reproducible] (117)

and

[mathematical expression not reproducible]. (118)

It follows from (117) and (118) that [phi](0) = [R.sub.0] and

[mathematical expression not reproducible]. (119)

Moreover, by differentiation of (117) and (118) with respect to age a, we obtain

[mathematical expression not reproducible] (120)

and

[phi]' (a) - [??](a) [phi] (a) + [omega] (0) [epsilon] (a), (121)

respectively. Therefore, (113) is reduced to

[mathematical expression not reproducible]. (122)

We denote

[mathematical expression not reproducible]. (123)

Since

[LAMBDA]/[[mu].sup.0] + v < [S.sup.0] < A/[[mu].sup.0] - v, (124)

it is easy to check that F(S) has two negative (real) roots. Moreover, F(S) > 0 for every S [greater than or equal to] 0. Therefore, the sign of [L.sup.0](x) will be determined by the sign of

[mathematical expression not reproducible]. (125)

Thus, three cases occur:

Case 1 [mathematical expression not reproducible].

Case 2 [mathematical expression not reproducible].

Case 3 [mathematical expression not reproducible].

It results from the above that the derivative of [L.sup.0](i) along the solutions of (4) is [mathematical expression not reproducible] holds. Moreover, it can be verified that {(S, v, e, i, R) : [[??].sup.0](t) = 0} = {[E.sup.0]}. Therefore, it results from Lasalle's Invariance Theorem [26, p. 200] that [E.sup.0] is globally asymptotically stable, if [R.sub.0] [less than or equal to] 1.

Theorem 15. The endemic equilibrium [E.sup.*] of (4) is globally asymptotically stable on the set, if [R.sub.0] > 1.

Proof. To prove the above result we consider a Lyapunov function of the form given below

[mathematical expression not reproducible], (126)

where

[mathematical expression not reproducible], (127)

with G(X) = X - 1 - ln X, for any X > 0.

Thus, we have

[mathematical expression not reproducible] (128)

Using (37), (44), and (45) together with the first equation of (38), we obtain

[mathematical expression not reproducible]. (129)

We add and subtract, carefully, some terms to the above expression, and we also identify group of terms in the form given by G(X) = X - 1 - ln X. We obtain

[mathematical expression not reproducible]. (130)

By differentiating [L.sup.*.sub.2] with respect to t, we obtain

[mathematical expression not reproducible]. (131)

Since

[mathematical expression not reproducible]. (132)

thus (131) yields

[mathematical expression not reproducible]. (133)

Using the second equation of (38) after integrating by part, we get

[mathematical expression not reproducible]. (134)

Moreover,

[mathematical expression not reproducible], (135)

yields

[mathematical expression not reproducible]. (136)

Similarly to [L.sup.*.sub.2], from [L.sup.*.sub.3] and [L.sup.*.sub.4] we get, respectively,

[mathematical expression not reproducible] (137)

and

[mathematical expression not reproducible], (138)

while using the fifth equation of (38) [L.sup.*.sub.5] leads to

[mathematical expression not reproducible]. (139)

By combining (130), (136), (137), (138), and (139), we obtain

[mathematical expression not reproducible]. (140)

This yields

[mathematical expression not reproducible]. (141)

Since

[mathematical expression not reproducible], (142)

thus

[mathematical expression not reproducible]. (143)

Using assumption A3, it is easy to see that [mathematical expression not reproducible]. Moreover,

(i) [mathematical expression not reproducible];

(ii) [mathematical expression not reproducible];

(iii) [mathematical expression not reproducible];

(iv) [mathematical expression not reproducible];

(v) [mathematical expression not reproducible].

Therefore, we obtain

[[??].sup.*] (t) < 0, (144)

in the following cases:

(i) [mathematical expression not reproducible];

(ii) [mathematical expression not reproducible];

(iii) [mathematical expression not reproducible];

(iv) [mathematical expression not reproducible].

From (144), we say that the derivative of [L.sup.*](t) along the solutions of (4) is [mathematical expression not reproducible] are simultaneously satisfied, then we obtain [[??].sup.*] (t) = 0 from (76). Moreover, it can be verified that {(S, v, e, i, P) : [[??].sup.*] (t) = 0} = {[E.sup.*]}. Therefore, it results from Lasalle's Invariance Theorem [26, p. 200] that [E.sup.*] is globally asymptotically stable, if [R.sub.0] > 1.

The results from Theorems 14 and 15 show that there exists a threshold parameter, called the basic reproduction number and denoted by [R.sub.0], that is essential in the stability analysis of the global dynamics of the model defined by system (4). Moreover, such a parameter can play a crucial role in the implementation of human vaccination policies. As we can see in

[mathematical expression not reproducible], (145)

the rise in rate of vaccination of (infant) susceptible individuals against an SEIR infection can reduce the spread of the infection and help to elaborate control measures to prevent and reduce the spread of the infection.

https://doi.org/10.1155/2018/8479638

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

The work done in this paper was funded by a research grant from Material Science Innovation and Modelling (MaSIM), North-West University, South Africa.

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Rodrigue Yves M'pika Massoukou (iD), (1) Suares Clovis Oukouomi Noutchie (iD), (1) and Richard Guiem (1,2)

(1) School of Mathematical and Statistical Sciences, North-West University, Mafikeng campus, Private BagX2046, Mmabatho 2735, South Africa

(2) National Advanced School of Engineering of Maroua, University of Maroua, P.O. Box 46, Maroua, Cameroon

Correspondence should be addressed to Rodrigue Yves M'pika Massoukou; rodrigue@aims.ac.za

Received 9 April 2018; Revised 26 June 2018; Accepted 18 July 2018; Published 14 August 2018

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