The Effect of Impulsive Vaccination on Delayed SEIRS Epidemic Model Incorporating Saturation Recovery
Abstract
A delay SEIRS model with pulse vaccination incorporating a general media coverage function and saturation recovery is investigated. Using the discrete dynamical system determined by the stroboscopic map, we obtain the existence of the disease-free periodic solution and its exact expression. Further, using the comparison theorem, we establish the sufficient conditions of global attractivity of the disease-free periodic solution. Moreover, we show that the disease is uniformly persistent if vaccination rate is less than θ*. Finally, we discuss the effect of media coverage on controlling disease.
1. Introduction
In recent years, controlling infectious disease is a very important issue; vaccination is a commonly used method for controlling disease; the study of vaccines against infectious disease has been a boon to mankind. There are now vaccines that are effective in preventing such viral infections as rabies, yellow fever, poliovirus, hepatitis B, parotitis, and encephalitis B. Eventually, vaccines will probably prevent malaria, some forms of heart disease, and cancer. Vaccines have been very important to people. Theoretical results show that pulse vaccination strategy can be distinguished from the conventional strategies leading to disease eradication at relatively low values of vaccination [1]. Theories of impulsive differential equations are found in the books [2, 3]. In recent years, their applications can be found in the domain of applied sciences [4–7]. In this paper, we consider impulsive vaccination to susceptible individuals.
In the classical endemic models, the incidence rate is assumed to be mass action incidence with bilinear interactions given by βSI, where β is the probability of transmission per contact, and S and I represent the susceptible and infected populations, respectively. If the population is saturated with infective individuals, there are three kinds of incidence forms that are used in epidemiological model: the proportionate mixing incidence βSI/N [8], nonlinear incidence βIpSq [9], and saturation incidence βIS/(1 + αS) [10] or βIlS/(1 + αIh) [11]. However, some factors such as media coverage, manner of life, and density of population may affect the incidence rate directly or indirectly; nonlinear incidence rate can be approximated by a variety of forms, such as β(1 − cI)IS, (c > 0), (β1 − β2(I/(m + I)))SI, (β1 > β2 > 0, m > 0) which were discussed by [12–14].
In this paper, we suggest a general nonlinear incidence rate (β1 − β2(Ih/(c + Ih)))SI, (β1 > β2 > 0, c > 0, h ≥ 1) which reflects some characters of media coverage, where β1 = pc1, β2 = pc2, p is the transmission probability under contacts in unit time, c1 is the usual contact rate, c2 is the maximum reduced contact rate through actual media coverage, and c is the rate of the reflection on the disease. Again, media coverage can not totally interrupt disease transmission, so we have β1 > β2. We use β2(Ih/(c + Ih)) to reflect the amount of contact rate reduced through media coverage. When infective individuals appear in a region, people reduce their contact with others to avoid being infected, and the more infective individuals being reported, the less contact with others; hence, we take the above form. Few studies have appeared on this aspect.
In the classical disease transmission models, the recovery from infected class per unit of time is assumed to be proportional to the number of infective individuals (denoted by I); say γI, where γ > 0 is the removal rate. This is a reasonable approximation to the truth when the number of the infectious individuals is not too large and below the capacity of health care settings. If the number of illness exceeds a fixed large size, then the number of recovered is independent of further changes in infectious size. We adopt the Verhulst-type function g(I) = γI/(d + I) to model the recovered part which increases for small infectives and approaches a maximum for large infectives. Here, γ gives the maximum recovery per unit of time, and d, the infected size at which is 50% saturation (g(b) = c/2), measures how soon saturation occurs. Cui et al. studied this removal rate [15].
The main purpose of this paper is to establish sufficient conditions that the disease dies out and show that the disease is uniformly persistent under some conditions.
2. Notations and Definitions
We introduce some notations and definitions and state some results which will be useful in subsequent sections.
The solution of system (3) is a piecewise continuous function is continuous on (kT, (k + 1)T], k ∈ Z+, and exists. Obviously, the smooth properties of f guarantee the global existence and uniqueness of solution of system (3) (see [3], for details on fundamental properties of impulsive systems). Since , and whenever I(t) = 0, for t ≠ kT, k ∈ Z+. Moreover, S(kT+) = (1 − θ)S(kT−) and I(kT+) = I(kT−) for k ∈ Z+. Therefore, we have the following lemma.
Lemma 1. Suppose Z(t) is a solution of system (3) with initial conditions (4); then, Z(t) ≥ 0 for all t ≥ 0.
Denote that Ω = {(S, I) ∈ R2∣S ≥ 0, I ≥ 0, S + I ≤ 1}. Using the fact that S(t) + E(t) + I(t) + R(t) = 1, it is easy to show that Ω is positively invariant with respect to (3).
Lemma 2 (see [10].)Consider the following impulsive system:
Lemma 3 (see [17].)Consider the following delayed differential equation:
- (i)
if a1 < a2, then limt→∞x(t) = 0;
- (ii)
if a1 > a2, then limt→∞x(t) = +∞.
Definition 4. System (3) is said to be uniformly persistent if there is an η > 0 (independent of the initial conditions) such that every solution (S(t), I(t)) with initial conditions (4) of system (3) satisfies
3. Global Attractivity of Infection-Free Periodic Solution
About the global attractivity of infection-free periodic solution of system (3), we have the following theorem.
Theorem 6. The infection-free periodic solution of system (3) is globally attractivity provided that R* < 1, where R* = β1e−bw(b + γe−bτ/(d + 1))(1 − e−bT)/(b(b − β2/(c + 1))(1 − (1 − θ)e−bT)).
Proof. Since R* < 1, we can choose ε0 > 0 sufficiently small such that
Let (S(t), I(t)) be the solution of system (3) with initial values (4) and let S(0+) = S0 > 0, x(t) be the solution of system (13) with initial values x(0+) = S0. By the comparison theorem in impulsive differential equation [3], there exists an integer k1 > 0 such that for t > k1T; thus,
Consider the following comparison system:
Let (S(t), I(t)) be the solution of system (3) with initial values (4) and let I(ζ) = φ(ζ) > 0 (ζ ∈ [−w, 0]), y(t) be the solution of system (17) with initial values y(ζ) = φ(ζ) > 0 (ζ ∈ [−w, 0]). By the comparison theorem, we have limt→∞supI(t) ≤ limt→∞supy(t) = 0. Incorporating into the positivity of I(t), we know that limt→∞I(t) = 0. Therefore, there exists an integer k2 > k1 (where k2T > k1T + w) such that I(t) < ε0 for all t > k2T.
From the first equation of system (3), we have
Let (S(t), I(t)) be the solution of system (3) with initial values (4) and S(0+) = S0 > 0, z1(t) and let z2(t) be the solutions of system (19) and (20) with initial values z1(0+) = z2(0+) = S0, respectively. By the comparison theorem in impulsive differential equation, there exists an integer k3 > k2 such that k3T > k2T + τ and
Denote that θ* = 1 − ebT + (β1e−bw(1 + (γ/b)e−bτ)(ebT − 1))/b(b − β2/(c + 1)), , and c* = β2b(1 − (1 − θ)e−bT)/(b2(1 − (1 − θ)e−bT) − β1e−bw(b + γe−bτ/(d + 1))(1 − e−bT)) − 1.
According to Theorem 6, we can obtain the following result.
4. Permanence
In this section, we say the disease is endemic if the infectious population persists above a certain positive level for sufficiently large time.
Denote that R* = (β1 − β2/(c + 1))e−bw(1 − θ)(1 − e−bT)/(b + γ/d)(1 − (1 − θ)e−bT) and I* = (b/β)[(β1 − β2/(c + 1))e−bw(1 − θ)(1 − e−bT)/(b + γ/d)(1 − (1 − θ)e−bT) − 1].
Theorem 8. If R* > 1, then there is a positive constant q such that each positive solution (S(t), I(t)) of system (3) satisfies I(t) > q for t large enough.
Proof. From R* > 1, we easily know that I* > 0, and there exists sufficiently small ε > 0 such that
Consider the following comparison impulsive system for t ≥ t0:
Let (S(t), I(t)) be the solution of system (3) with initial values (4) and let S(0+) = S0 > 0, u(t) be the solution of system (27) with initial values u(0+) = S0. By the comparison theorem for impulsive differential equation, there exists an integer k1(>t0 + w) such that for t ≥ k1T; thus,
The second equation of system (3) can be rewritten as
Let ; in the following, we will show that I(t) ≥ Il for t ≥ t1. Suppose the contrary; then, there is a T0 ≥ 0 such that I(t) ≥ Il for t1 ≤ t ≤ t1 + w + T0, I(t1 + w + T0) = Il and I(t1 + w + T0) ≤ 0. However, the second equation of system (3) and (4) imply that
- (i)
I(t) ≥ I* for t large enough;
- (ii)
I(t) oscillates about I* for t large enough.
Evidently, we only need to consider the case (ii). Let t* > 0 and ξ > 0 satisfy I(t*) = I(t* + ξ) = I*, and let I(t) < I* for t* < t < t* + ξ, where t* is sufficiently large such that S(t) > δ1 for t* < t < t* + ξ. Since I(t) is continuous and ultimately bounded and is not effected by impulses, we conclude that I(t) is uniformly continuous. Hence, there exists a constant λ (with 0 < λ < w and λ is independent of the choice of t*) such that I(t) > I*/2 for t* < t < t* + λ. If ξ ≤ λ, our aim is obtained. If λ < ξ ≤ w, from the second equation of (3), we have that and I(t*) = I*; then, we have I(t) ≥ q for t ∈ [t*, t* + w], where q = min{I*/2, q*}, q* = I*e−(b+γ)w). The same arguments can be continued, and we can obtain I(t) ≥ q for t ∈ [t* + w, t* + ξ]. Since the interval is chosen in an arbitrary way, we get that I(t) ≥ q for t large enough. In view of our arguments above, the choice of q is independent of the positive solution of (3) which satisfies that I(t) ≥ q for sufficiently large t. This completes the proof.
Denote that θ* = [(β1 − β2/(c + 1))e−bw − (b + γ/d)](1 − e−bT)/((β1 − β2/(c + 1))e−bw(1 − e−bT) + (b + γ/d)e−bT), β2* = (c + 1)(β1 + (b + γ/d)(1 − (1 − θ)e−bT)/e−bw(1 − θ)(1 − e−bT)), and c* = β2e−bw(1 − θ)(1 − e−bT)/(β1e−bw(1 − θ)(1 − e−bT) + (b + γ/d)(1 − (1 − θ)e−bT)) − 1.
It follows from Theorem 8 that the disease is uniformly persistent provided that θ < θ* or β2 < β2* or c > c*.
Theorem 9. If R* > 1, then system (3) is permanent.
Proof. Suppose that (S(t), I(t)) is any positive solution of system (3) with initial conditions (4). From the first equation of system (3), we have . By similar arguments as those in the proof of Theorem 6, we have that
Set Ω0 = {(S, I) ∈ R2∣p ≤ S, q ≤ I, S + I ≤ 1}. From Theorem 8 and (36), we know that the set Ω0 is a global attractor in Ω, and, of course, every solution of system (3) with initial conditions (4) will eventually enter and remain in region Ω0. Therefore, system (3) is permanent. The proof is complete.
From Theorem 9, we can obtain the following result.
Corollary 10. Assume that θ < θ* or β2 < β2* or c > c*; then, system (3) is permanent.
5. Conclusion
In this paper, we introduce media coverage and saturation recovery in the delayed SEIRS epidemic model with pulse vaccination and analyze detailedly in theory that media coverage and pulse vaccination bring effects on infection-free and the permanence of epidemic disease. We suggest the probability of transmission per contact β(I) = β1 − β2(Ih/(c + Ih)), which reflects some characters of media coverage. We can get that ∂β(I)/∂β2 = −(Ih/(c + Ih)) < 0, so β(I) is a monotone decreasing function on β2; that is, if β2 (the reduced valid contact rate through actual media coverage) is larger, then infection rate of disease is smaller. Again, , so β(I) is a monotone increasing function on c; that is, if c is smaller (refection on the disease is quickly), then infection rate of disease is smaller. When infective individuals appear in a region, people reduce their contact with others to avoid being infected, and the more infective individuals being reported, the less contact with others. From above analysis, we know that media coverage is very important on controlling disease, and media coverage should be considered in incidence rate.
By Theorem 6, the infection-free periodic solution of system (3) is globally attractivity provided that R* < 1. By Theorem 9, system (3) is permanent if R* > 1.
From Corollaries 7 and 10, we can choose the proportion of those vaccinated successfully to all of newborns such that θ > θ* in order to prevent the epidemic disease from generating endemic, and the epidemic is permanent if θ < θ*. But for θ ∈ [θ*, θ*], the dynamical behavior of model (3) has not been studied, and the threshold parameter for the vaccination rate between the extinction of the disease and the uniform persistence of the disease has not been obtained. These issues will be considered in our future work.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the NNSF of China (11371048, 11201433), NSF of Henan Province (112300410156, 122300410117), NSF of the Education Department Henan Province (2011A110022), GGJS of Henan Province (2013GGJS-110), and XGGJS of Zhengzhou University of Light Industry (2012XGGJS003).
