Abstract and Applied Analysis
Volume 2018 (2018), Article ID 8479638, 21 pages
https://doi.org/10.1155/2018/8479638

Research Article

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

Rodrigue Yves M’pika Massoukou,¹ Suares Clovis Oukouomi Noutchie,¹ and Richard Guiem1,2

¹School of Mathematical and Statistical Sciences, North-West University, Mafikeng campus, Private Bag X2046, Mmabatho 2735, South Africa
²National 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

Academic Editor: Jaume Giné

Copyright © 2018 Rodrigue Yves M’pika Massoukou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Vaccine-induced protection is substantial to control, prevent, and reduce the spread of infectious diseases and to get rid of infectious diseases. In this paper, we propose an SVEIR epidemic model with age-dependent vaccination, latency, and infection. The model also considers that the waning vaccine-induced immunity depends on vaccination age and the vaccinated individuals fall back to the susceptible class after losing immunity. The model is a coupled system of (hyperbolic) partial differential equations with ordinary differential equations. The global dynamics of the model is established through construction of appropriate Lyapunov functionals and application of Lasalle’s invariance principle. As a result, the global stability of the infection-free equilibrium and endemic equilibrium is obtained and is fully determined by the basic reproduction number .

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-age-dependent 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 . Similarly, in [20] the latency, relapse, and vaccination ages are denoted by the same variable . 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 , 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

where and are defined below, instead of the classical incidence rate of the form

where denotes the coefficient of disease transmission from infective individual, with infection age , 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 and be the number of individuals in the susceptible and removed compartments at time , respectively. Let , , and be the density of vaccinated, (latently) infected, and (actively) infected individuals with vaccination, latency, and infection age at time , respectively. It follows that , , and defined by

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

where , , and , with boundary conditions

And initial conditions

where and are initial size of susceptible and removed individuals, respectively, and , , and are initial age-density of vaccinated, latent, and infective individuals, respectively. Moreover, , , and 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:

 : constant recruitment rate of susceptible individuals : rate of vaccination of susceptible individuals : natural mortality rate of individuals : age-specific rate of waning vaccine-induced immunity : age-specific natural mortality rate : age-specific removal rate : age-specific rate moving from latent to infective : age-specific infection rate of susceptible individuals by infective individuals (of the same age–intracohort contagion) : probability that an infective individual of age will successfully infect a susceptible individual of age , after contact

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

A1

(i) with .(ii) and with essential upper bounds and , respectively.(iii) are Lipschitz continuous on with coefficients , and , respectively.(iv)There exists such that .

A2. is a decreasing function of for any constant removal rate such that .

A3. , for every .

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 (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

endowed with the norm

where , for any . Let us denote by the positive cone of the Banach space such that

For any initial value satisfying the conditions

system (4) is well-posed, under assumption , due to [21]. Thus, a continuous semiflow is obtained and it is defined by (4) such that

Now, we introduce the functions

By integrating the second, third, and fourth equations of (4) along the characteristic line , respectively, we get

where

Taking the norm of and using the positiveness of the components of , we get

Differentiating (15) with respect to leads to

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

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

Likewise, we also have

and

Therefore, we get

Using (iv) of , (21) yields

that is,

Thus, we obtain

where .

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

we get

for any , whenever . Moreover

for any .

Then, we state the following result.

Lemma 1. The set is positively invariant for ; that is,

Moreover, is point dissipative and the set attracts all points in .

As we aim to make use of the Lasalle’s Invariance Principle, we are required to establish the relative compactness of the orbit in due to the infinite dimensional Banach space . For this, we consider two operators and , (), such that

where

Thus, , for any ; and from the proof of [22, Proposition 3.13] and Lemma 1, we get to the following result.

Theorem 2. For , the orbit has a compact closure in if the following conditions are satisfied:

(i)There exists a function such that, for any , , and if with , then for any .(ii)For any , maps any bounded sets of into a set with compact closure in .

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

Lemma 3. For , let . Then . Then (i) of Theorem 2 holds.

Proof. Clearly, we see that . Making use of some equations in (13), we get

Taking the initial condition such that and , we have

Lemma 4. For , maps any bounded set of into a set with a compact closure in .

Proof. Since and remain in the compact set by Lemma 1, it is sufficient to show that , , and remain in a precompact subset of , which does not depend on the initial data . To achieve this, the following conditions (see [23, Theorem B.2]) must be satisfied for , , and :

(i)The supremum of with respect to is finite;(ii) uniformly with respect to ;(iii) uniformly with respect to ;(iv) uniformly with respect to ,

where . It follows from (13) and (30) that

and hence, using Lemma 1, we get

and hence, (i), (ii), and (iv) follow. To establish (iii), we take a sufficiently small such that and we show that

Indeed, is a nondecreasing function of such that satisfying

On the other hand, the Lipschitz continuity of is obtained from the first equation of (4), using the boundedness of the solution of (4) (see Lemma 1); that is, there exists such that for any .
Since does not depend on the initial data and as , 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 has a global attractor in , which attracts any bounded subset of .

3. Equilibria and Basic Reproduction Number

System (4) has a unique disease-free equilibrium , where

Apart from , system (4) could also have an endemic equilibrium. We suppose that there exists an endemic equilibrium for system (4) denoted by . Therefore,

are satisfied. In addition, also satisfies (5), i.e.,

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

It follows from the third equation of (38) that

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

We introduce parameters , , , and such that

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

And hence,

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

Therefore, it easily follows that

A threshold condition is derived from the existence condition for the endemic equilibrium such that . Thus, the parameter , given by the first equation of (43), can be called the basic reproduction number of system (4). Moreover, can also be expressed as

where

and 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 , then system (4) has only a disease-free equilibrium , while if , then system (4) also has an endemic equilibrium in addition to the disease-free equilibrium .

4. Uniform Persistence

This section is devoted to the uniform persistence of system (4) under the condition . For this, we introduce a function defined by

where . Furthermore, we consider the set defined by

such that as whenever .

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

for .

Theorem 8. If , then system (4) is uniformly weakly -persistent.

Proof. We assume that, for any , we can find such that

Since , then we can find a small enough such that

In particular, we can find such that

We can assume that, for any , . It follows from the first equation of (4) that

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

It follows that

And hence,

This yields . We can assume that, for any ,

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

We apply the Laplace Transform to the above inequality so that

Dividing the above inequality by yields

First, taking the limit inferior as on both sides of the above inequality, we obtain

Next, take the limit as on both sides of the above inequality, we get

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) -persistence as follows.

Theorem 9. If , then the semiflow is uniformly (strongly) -persistent.

Definition 10. A total trajectory of a continuous semiflow , defined by (11), is a function such that for any .

Note that a global attractor will only contain points with total trajectories through them as it needs to be invariant. So, the -limit point of a total trajectory , passing through , is given by

A total trajectory satisfies

Corollary 11. Let and be, respectively, a global attractor of in and a total trajectory of in . If , then there exists such that

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

From the first equation of (4), we have

that is,

This yields

Taking the limit inferior as in (72) leads to

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

And hence,

It follows from Theorem 9 and Definition 7 that .
Further, considering the boundary condition given by the third equation of (5), we obtain

Finally, the fifth equation of (4) leads to

And hence,

Taking the limit inferior as in (78), we obtain

And therefore . By choosing such that , for , we get

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 can be investigated through the following result.

Theorem 12. If then is locally asymptotically stable; if then is unstable.

Proof. To investigate the stability of the disease-free equilibrium , we denote by , , , , and the perturbations of , , , , and , respectively, such that

The perturbations satisfy the following:

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

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

where , , , , and (real or complex number) satisfy the following system:

with boundary conditions

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

respectively, where , , and 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

where

such that . It is not difficult to see that . Thus, is a decreasing continuous function of which approaches as and as . Hence, the characteristic equation (88) admits a real solution such that whenever and whenever .
On the other hand, by assuming a complex solution of the characteristic equation , we can notice that is always true. Thus, we clearly get . It follows from the characteristic equation that and . Therefore, we obtain , i.e., . Hence, , since is a decreasing function.
It results from the above statements that all eigenvalues of the characteristic equation have negative real part whenever , i.e., . Thus, the disease-free equilibrium is locally asymptotically stable if . Otherwise, , i.e., the unique real solution of the characteristic equation is positive, and hence the disease-free equilibrium is unstable.

Theorem 13. If then is locally asymptotically stable.

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

Substituting , , , and , into (4) and neglecting the terms of second order and above, the perturbations satisfy the linear system

with boundary conditions

after substituting , , , , and into (5) and neglecting the terms of second order and above.
Now, we consider the exponential solutions of systems (91)-(92) of the form

where , , , , and (real or complex number) satisfy the system

with boundary conditions

Similarly to the process leading to the characteristic equation (88), the characteristic equation at the disease-endemic equilibrium is given by

where

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

where and . It follows from (96) that

where

and hence

where

and

with

Since is always true and ( is a decreasing function of ), thus .
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

We state the following results.

Theorem 14. The disease-free equilibrium of (4) is globally asymptotically stable if .

Proof. A Lyapunov function of the form

is considered, where

The functions and are nonnegative functions to be chosen suitably and carefully. , , denote the derivatives of with respect to along the solution to (4) and are given by

Therefore,

i.e.,

after choosing and such that

Note that

after using . Thus, (110) becomes

Using assumption , we show that

where

By assumption , obtain

Next, we choose functions and such that

and

It follows from (117) and (118) that and

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

and

respectively. Therefore, (113) is reduced to

We denote

Since

it is easy to check that has two negative (real) roots. Moreover, for every . Therefore, the sign of will be determined by the sign of

Thus, three cases occur:
Case 1 (). It is easy to see that and therefore if .
Case 2 (). For this value of , we have and therefore if .
Case 3 (). For such values of we have . Since , it follows that for any . Therefore if .
It results from the above that the derivative of along the solutions of (4) is . If , , and are simultaneously satisfied with , then holds. Moreover, it can be verified that . Therefore, it results from Lasalle’s Invariance Theorem [26, p. 200] that is globally asymptotically stable, if .

Theorem 15. The endemic equilibrium of (4) is globally asymptotically stable on the set, if .

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

where

with , for any .
Thus, we have

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

We add and subtract, carefully, some terms to the above expression, and we also identify group of terms in the form given by . We obtain

By differentiating with respect to , we obtain

Since

thus (131) yields

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

Moreover,

yields

Similarly to , from and we get, respectively,

and

while using the fifth equation of (38) leads to

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

This yields

Since

thus

Using assumption , it is easy to see that . Hence, for ; i.e., for . Moreover,

(i) = ⇔ or ;(ii) = ⇔ or ;(iii)⇔ or ;(iv)⇔ or ;(v) = ⇔ or .

Therefore, we obtain

in the following cases:

(i), , , , and ;(ii), , , , and ;(iii), , , , and ;(iv), , , , and .

From (144), we say that the derivative of along the solutions of (4) is . If , , , , and are simultaneously satisfied, then we obtain from (76). Moreover, it can be verified that . Therefore, it results from Lasalle’s Invariance Theorem [26, p. 200] that is globally asymptotically stable, if .

The results from Theorems 14 and 15 show that there exists a threshold parameter, called the basic reproduction number and denoted by , 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

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.

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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