International Journal of Differential Equations

Volume 2018, Issue 1 4168061

Research Article

Open Access

On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

Stanislas Ouaro

orcid.org/0000-0003-4458-6624

LAboratoire de Mathématiques et Informatique (LAMI), Unité de Formation et de Recherche en Sciences Exactes et Appliquées, Département de Mathématiques, Université Ouaga I Pr Joseph KI-ZERBO, BP 7021, Ouagadougou, Burkina Faso univ-ouaga.bf

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Ali Traoré,

Corresponding Author

Ali Traoré

[email protected]

orcid.org/0000-0001-6647-7985

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Stanislas Ouaro,

Stanislas Ouaro

orcid.org/0000-0003-4458-6624

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Ali Traoré,

Corresponding Author

Ali Traoré

[email protected]

orcid.org/0000-0001-6647-7985

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First published: 15 March 2018

https://doi.org/10.1155/2018/4168061

Citations: 3

Academic Editor: Khalid Hattaf

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Abstract

We study a vector-borne disease with age of vaccination. A nonlinear incidence rate including mass action and saturating incidence as special cases is considered. The global dynamics of the equilibria are investigated and we show that if the basic reproduction number is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out, while if the basic reproduction number is larger than 1, then the endemic equilibrium is globally asymptotically stable, which means that the disease persists in the population. Using the basic reproduction number, we derive a vaccination coverage rate that is required for disease control and elimination.

1. Introduction

Many of infections that have the important impact on human health in terms of mortality or morbidity are vector-borne disease. Mosquitoes [1] are perhaps the best known disease vectors, with various species playing a role in the transmission of infections such as malaria, yellow fever, dengue fever, and West Nile virus. One of the effective methods in disease prevention is the vaccination [2–5]. Several studies in the literature have been carried out to investigate the role of treatment and vaccination of the spread of diseases ([6–8] and the references therein). An epidemic model with vaccination for measles is derived by Linda [9]. The effect of vaccination on the spread of periodic diseases, using discrete-time model, was studied by Mickens [10].

The impact of vaccination in two SVIR models with permanent immunity is studied by Liu et al. [11]. Xiao and Tang [12] have shown from an SIV model that complex dynamics are induced by imperfect vaccination. Gumel and Moghadas [13] investigated a disease transmission model by considering the impact of a protective vaccine and found the optimal vaccine coverage threshold required for disease control and elimination. The eradicating of an SEIRS epidemic model by using vaccine was studied by Gao et al. [14]. Yang et al. [8] derived a threshold value for the vaccination coverage of an SIVS epidemic model. Many previous studies have shown that the reemergence of some diseases is caused by the waning of vaccine-induced immunity [15–17]. A consequence of this is that it is important for health authorities to take into account waning of vaccine-induced immunity in the disease control and elimination campaign.

In this paper, we consider a vector-borne disease model such as malaria that incorporates the waning of vaccine-induced immunity. Additionally, we use incidences with a nonlinear response to the number of infectious individuals and infectious vectors. The incidences take the form Sf(I) and Sg(I), respectively, for the human and vector populations. We assume that f and g satisfy the following assumptions:

(H1) For with equality if and only if x = 0, f^′(x) ≥ 0, and f^′′(x) ≤ 0.

(H2) For with equality if and only if x = 0, g^′(x) ≥ 0, and g^′′(x) ≤ 0.

From the above assumptions and the Mean Value Theorem, it follows that

()

Let S_h, I_h, and R_h denote, respectively, the number of susceptible, infectious, and removed host individuals and S_v, I_v the number of susceptible and infectious vectors. The susceptible individuals are vaccinated at the rate θ ≥ 0. v(t, a) denotes the population size of the vaccinated compartment at time t with the vaccine age a. Let ϵ(a) be the rate at which the vaccine-induced immunity wanes. We assume that ϵ(a) and the following assumption:

(H3) ϵ : [0, ∞)→[0, ∞) is bounded, nondecreasing, and piecewise continuous with possibly many finite jumps.

We consider a relatively isolated community where there is no immigration or emigration. Additionally, we assume that all the newly recruited, including the newborns, are susceptibles. Let, at any time t, Λ_h and Λ_v be the recruitment rate of host individuals and vectors, respectively. μ_h and μ_v are, respectively, the natural death rate of host individuals and vectors. Let γ be the natural recovery rate from the infected population and δ the disease induced death rate of host individuals. The number of individuals moving from the vaccinated class into the susceptible class at time t is

. From the above assumptions, we formulate our vector-borne epidemic model in the following way:

()

where

is the set of integrable functions from (0, ∞) into

. Since the removed host individual population does not appear in the remaining equations of system (2), it is sufficient to consider the following system:

()

From [18, 19], we state that system (3) has a unique continuous solution if the initial conditions satisfy the compatibility condition

()

In the remaining part of this paper, we always assume that condition (4) is satisfied. The existence and the nonnegativity of the solution of (3) can be reached in Browne and Pilyugin [20]. We next introduce a semiflow solution of system (3).

Define

()

and consider the linear operator A : dom⁡(A) ⊂ χ → χ defined by

()

with

, where W^1,1 is a Sobolev space. Then,

is not dense in χ. We consider a nonlinear map

which is defined by

()

and let

()

Set

()

Based on the above, we can reformulate system (3) as the following abstract Cauchy problem:

()

By applying the results given in [19, 21], we derive the existence and uniqueness of the semiflow {Φ(t)} _t≥0 on

generated by system (3). By using the theory for dynamical system (see [19]), we can further obtain the following lemma.

Lemma 1. System (3) generates a unique continuous semiflow {Φ(t)} _t≥0 on that is asymptotically smooth and bounded dissipative. Furthermore, the semiflow {Φ(t)} _t≥0 has a compact global attractor .

The total population size of human hosts and vectors is, respectively,

()

Then, from the time derivative of N_h(t) and N_v(t), we get

()

which implies

()

We hence restrict our attention to solutions of (3) with initial conditions in

()

The rest of the paper is structured as follows. In Section 2, we study the existence and local stability of equilibria of system (3). In Section 3, we present the results for the global dynamics of equilibria of system (3). In Section 4, the paper closes with conclusion.

2. Existence and Local Stability of Equilibria

In this part, we state the result about the existence and local stability of equilibria of the model (3). We first start by the existence of equilibria. We define

()

Then,

()

Let

be an equilibrium of (3). This implies

()

From the third and the sixth equations of (18), we deduce that

()

By the first equation of (18), we get

()

From the fourth equation of (18), we have

()

Substituting

and

into the second and the fifth equations of (18) gives

()

From the second equation of (22), we obtain

()

Replacing

in the first equation of (22) yields

()

By (H1) and (H2),

is a solution of the above equation. Thus, system (3) has a disease-free equilibrium

()

Following the same method as [22], the basic reproduction number for model (3) is

()

describes a threshold for endemic persistence/spread of the disease, the rate of increase in the number of cases during an epidemic. Its magnitude allows determining the effort necessary either to prevent an epidemic or to eliminate an infection from a population.

Let

be an endemic equilibrium. Then,

and

, where

()

The function h is continuous with h(0) = 0 and h(Λ_h/μ_h) ≤ 0.

Moreover, for I_h ∈ (0, Λ_h/μ_h),

()

The sufficient condition for h to have a zero in (0, Λ_h/μ_h) is that h is increasing at 0. Thus, there is an endemic equilibrium if

()

which is equivalent to

. Let

be a unique solution in (0, Λ_h/μ_h) of h(I_h) = 0. Then, system (3) admits a unique endemic equilibrium

, where

()

We summarize the above analysis in the following result.

Theorem 2 (consider system (3)). If , then there is a unique equilibrium, which is the disease-free equilibrium .

If , then there are two equilibria, the disease-free equilibrium and the endemic equilibrium .

We now deal with the local stability of the disease-free equilibrium. We show the stability of by linearizing system (3) about . The result is stated as follows.

Theorem 3 (consider system (3)). If , the disease-free equilibrium is locally asymptotically stable.

If , the unique endemic equilibrium is locally asymptotically stable.

Proof. From the linearization of system (3) at , we deduce the following characteristic equation:

()

where

()

From (31), the eigenvalues are −μ_v and solutions of

()

All roots of (33) and (34) have negative real parts; otherwise let λ₀ be a root of (33) with Re⁡(λ₀) ≥ 0. Then, we have

()

This leads to a contradiction.

Now, let λ₀ be a root of (34) with Re⁡(λ₀) ≥ 0. From (26), we have

()

This also leads to a contradiction by using (34) and then proves that

is locally asymptotically stable.

The characteristic equation at is

()

By using

and

, we get

()

We show that the characteristic equation has no eigenvalues with nonnegative real parts. The eigenvalues are −μ_v and solutions of

()

By way of contradiction, assume that there is one eigenvalue λ₁ with Re⁡(λ₁) ≥ 0. Then,

()

From (1), it follows that

()

Since

()

we have

()

This leads to a contradiction.

3. Global Stability Analysis of Equilibria

In this section, we prove the global stability of the equilibria of model (3). We first start by the global stability of the disease-free equilibrium . To attend this, we need the Fluctuation Lemma [23].

Let us introduce the notations

()

The Fluctuation Lemma is stated as follows.

Lemma 4 (See [23]). Let be a bounded and continuously differentiable function. Then, there exist sequences {s_n} and {t_n} such that s_n → ∞, t_n → ∞, ψ(s_n) → ψ_∞, ψ^′(s_n) → 0, ψ(t_n) → ψ^∞, and ψ^′(t_n) → 0 as n → ∞.

We also need the following lemma for establishing the global stability of .

Lemma 5 (See [18]). Suppose that is a bounded function and . Then,

()

We state the stability result of the disease-free equilibrium as follows.

Theorem 6. If , then the disease-free equilibrium is globally asymptotically stable.

Proof. Using Theorem 3, it is sufficient to show that is attractive in Γ.

Let (S_h(t), I_h(t), S_v(t), I_v(t), v(t, a)) be a solution of (3) with (S_h0, I_h0, S_v0, I_v0, v₀(·)) ∈ Γ. We integrate the third equation of (3) with the boundary conditions to obtain

()

Using the Fluctuation Lemma 4, we derive

()

From (1) and (3), we get

()

From (48), we have

()

It is evident that all eigenvalues of the matrix

()

have negative real parts when

. This leads to

()

From Lemma 4, it follows that there exists a sequence {t_n} such that t_n → ∞, S_h(t_n) → S_h,∞, S_v(t_n) → S_v,∞, and

as n → ∞.

Note that

()

Thus,

()

Let n → ∞; then

()

which gives

()

Since

and

, we obtain

()

That is,

()

From (46), it follows that

()

Therefore,

as t → ∞.

We now deal with the global stability of the endemic equilibrium .

A total trajectory of Φ is a function such that Φ(s, X(t)) = X(t + s) for all and all .

We define by ϕ(x) = x − 1 − ln⁡x. ϕ has a strict global minimum at 1 with ϕ(1) = 0 and ϕ(x) > 0, ∀x ≠ 1.

Lemma 7 (see [24].)Define

()

If assumptions (H1) and (H2) are satisfied, then F(X) ≤ 0, ∀X > 0.

The result of the global stability of the endemic equilibrium is stated as follows.

Theorem 8. If , then the endemic equilibrium is globally asymptotically stable in Γ.

Proof. Evaluating both sides of (3) at gives

()

Let

()

Then,

()

Let

()

Define

()

We study the behavior of the Lyapunov functional V(t) given by (68). V(t) is bounded and V(t) ≥ 0 with equality if and only if

For clarity, the derivatives of will be calculated separately and then combined to obtain dV(t)/dt. We first have

()

Using (60) to replace Λ_h in (69) gives

()

Next, we calculate

()

Using (61) to replace

in (71) gives

()

We now calculate the derivative of

()

Using (62) to replace Λ_v in (73) gives

()

Differentiating

with respect to t yields

()

Using (63) to replace

in (75) gives

()

The derivative of V_v(t) is

()

where v_a(t, a) = ∂v(t, a)/∂a.

Using (∂/∂a)ϕ(v(t, a)/v^∗(a)) = (v(t, a)/v^∗(a) − 1)(v_a(t, a)/v(t, a) + μ_h + ϵ(a)), dα(a)/da = −ϵ(a)v^∗(a) and integration by parts, we get

()

From

, we get

()

Combining (70), (72), (74), (76), and (79) and multiplying appropriately by coefficients determined by (68), we obtain

()

, v(t, 0) = θS_h(t), and

, it follows that

()

Thus, from Lemma 7, we deduce that

()

that is, V is nonincreasing. Denote by

the largest invariant subset of {dV(t)/dt = 0}.

Since V is bounded on X(·), the ω-limit set of X(·) must be contained in .

dV(t)/dt = 0 yields , and v(t, a) = v^∗(a).

Thus, dS_h(t)/dt = dS_v(t)/dt = 0 in . This implies that

()

for

, which gives

and

. The monotonicity of f and g stated in (H1) and (H2) implies that

and

. Therefore,

Then, the ω-limit set of X(·) is the endemic equilibrium and hence , . Thus, .

4. Conclusion

We have analysed a vector-borne disease model with nonlinear incidences, in which we have incorporated the waning of vaccine-induced immunity. These nonlinear incidences rates include mass action and saturating incidence as special cases. The basic reproduction number denoted by is derived. The model exhibits two equilibria, namely, the disease-free equilibrium and the endemic equilibrium . We have shown that if is less than 1, then the disease-free equilibrium is globally asymptotically stable; that is, the disease dies out and if is larger than 1, then the endemic equilibrium is globally asymptotically stable; that is, the disease persists in the population.

From these results, a critical vaccine coverage rate is obtained by solving the equation

, which yields

()

Then, if the vaccine coverage rate θ is greater than θ₀, then

and the disease will die out. The critical vaccine coverage rate θ₀ is increasing in the waning of vaccine. Then, neglecting the waning of vaccine (i.e.,

) when applying a vaccination for vector-borne disease will surely not be sufficient to make the disease die out in the population.

Conflicts of Interest

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

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On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

Abstract

1. Introduction

2. Existence and Local Stability of Equilibria

3. Global Stability Analysis of Equilibria

4. Conclusion

Conflicts of Interest

References

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On the Global Dynamics of a Vector-Borne Disease Model with Age of Vaccination

Abstract

1. Introduction

2. Existence and Local Stability of Equilibria

3. Global Stability Analysis of Equilibria

4. Conclusion

Conflicts of Interest

References

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