DDNS Discrete Dynamics in Nature and Society 1607-887X 1026-0226 Hindawi Publishing Corporation 10.1155/2015/848623 848623 Research Article Dynamical Analysis of SIR Epidemic Model with Nonlinear Pulse Vaccination and Lifelong Immunity http://orcid.org/0000-0003-4689-695X Zhao Wencai 1, 2 Li Juan 1 http://orcid.org/0000-0002-6553-9686 Meng Xinzhu 1, 2 Niamsup Piyapong 1 College of Mathematics and Systems Science Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2 State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology Shandong University of Science and Technology Qingdao 266590 China sdust.edu.cn 2015 14 5 2015 2015 04 11 2014 10 02 2015 14 5 2015 2015 Copyright © 2015 Wencai Zhao 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.

SIR epidemic model with nonlinear pulse vaccination and lifelong immunity is proposed. Due to the limited medical resources, vaccine immunization rate is considered as a nonlinear saturation function. Firstly, by using stroboscopic map and fixed point theory of difference equations, the existence of disease-free periodic solution is discussed, and the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Moreover, by using the bifurcation theorem, sufficient condition for the existence of positive periodic solution is obtained by choosing impulsive vaccination period as a bifurcation parameter. Lastly, some simulations are given to validate the theoretical results.

1. Introduction and Model Formulation

Infectious disease is one of the greatest enemies of human health. According to the World Health Statistics Report 2013 , nearly 2.5 million persons are infected by HIV each year. Although the number of infectious patients has dropped compared with 20 years ago, the absolute number of people with AIDS is still increasing, due to the fact that there are about 80,000 more infection cases than deaths. At the same time, AIDS has an important impact on adult mortality in high-prevalence countries. For example, the life expectancy in South Africa has fallen from 63 years old (in 1990) to 58 years old (in 2011). In Zimbabwe, the drop is six years during the same period. In recent years, due to the emergence of H7N9 avian influenza and other infectious diseases, the prevention and control situation is extremely grim all over the world. In order to prevent and control infectious diseases, vaccination is widely accepted. Generally, there are two types of strategies: continuous vaccination strategy (CVS) and pulse vaccination strategy (PVS) . For certain kinds of infectious diseases, PVS is more affordable and easier to implement than CVS. Theoretical study about PVS was started by Agur and coworkers in . In Central and South America [3, 4] and UK , PVS has a positive effect on the prevention of measles. With the encouragement of successful applications of PVS, many models are established to study the PVS . Pang and Chen  studied a class of SIRS model with pulse vaccination and saturated contact rate as follows: (1) d S d t = - β S t I t 1 + α S t + ω R ( t ) + μ 1 - S t , d I d t = β S t I t 1 + α S t - μ I ( t ) - λ I ( t ) , d R d t = λ I t - μ R t - ω R t , t k τ , S t + = 1 - p S t , I t + = I ( t ) , R t + = R ( t ) + p S ( t ) , t = k τ . In model (1), S ( t ) , I ( t ) , and R ( t ) represent the number of susceptible, infected, and removed individuals at the time t , respectively. Constant p is the vaccination rate. However, for some emerging infectious diseases, vaccination is often restricted by limited medical resources. The vaccination success rate always has some saturation effect; that is, vaccination rate can be expressed as a saturation function as follows : (2) p ( t ) = p S ( t ) S ( t ) + θ . Here, p is the maximum pulse immunization rate and θ is the half-saturation constant; that is, the number of susceptible when the vaccination rate is half to the largest vaccination rate. Thus we have that (3) S t + = 1 - p t S t . Then, we establish a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity as follows: (4) S ˙ ( t ) = a A - β S ( t ) I ( t ) 1 + α S ( t ) - μ S ( t ) , I ˙ ( t ) = β S ( t ) I ( t ) 1 + α S ( t ) - λ I ( t ) - γ I ( t ) - μ I ( t ) , R ˙ ( t ) = ( 1 - a ) A - μ R ( t ) + γ I ( t ) , t k T , S ( t + ) = ( 1 - p ( t ) ) S ( t ) , I ( t + ) = I ( t ) , R ( t + ) = R ( t ) + p ( t ) S ( t ) , t = k T . Here, constant A represents the total number of input population and a ( 0 < a 1 ) is the proportion of input population without immunity. μ is natural mortality, λ is the death rate due to illness, and γ is the recovery rate. The definitions of other symbols are shown in literature . Note that variable R just appears in the third and the sixth equations of model (4), so we only need to consider the subsystem of (4) as follows: (5) S ˙ ( t ) = a A - β S ( t ) I ( t ) 1 + α S ( t ) - μ S ( t ) , I ˙ ( t ) = β S ( t ) I ( t ) 1 + α S ( t ) - λ I ( t ) - γ I ( t ) - μ I ( t ) , t k T , S ( t + ) = ( 1 - p ( t ) ) S ( t ) , I t + = I t , t = k T .

This paper is organized as follows. In Section 2, we will firstly discuss the existence of the disease-free periodic solution by constructing stroboscopic map and using fixed point theory of difference equations. Then we will discuss the stability of the disease-free periodic solution by using the Floquet multipliers theory and the differential equations comparison theorem. In Section 3, we will discuss the existence of positive periodic solution and bifurcation by using the bifurcation theorem. Finally, we will give some numerical simulations and a brief decision in Section 4.

2. The Existence and Stability of the Disease-Free Periodic Solution

Let the total population number of model (4) be N ( t ) = S ( t ) + I ( t ) + R ( t ) , which satisfies (6) d N d t = A - μ N ( t ) - λ I ( t ) . Clearly, we have limsup t N ( t ) < A / μ , and then system (4) is ultimately bounded. Next, we will discuss the existence of the disease-free periodic solution of model (5). Let I ( t ) = 0 in system (5); then we get the subsystem of system (5) as follows: (7) S ˙ t = a A - μ S t , t k T , S t + = 1 - p t S t , t = k T . We have the following lemma for the property of subsystem (7).

Lemma 1.

System (7) has a unique globally asymptotically stable periodic solution S * ( t ) .

Proof.

Solving the first equation of system (7), we get (8) S t = S k T + - a A μ e - μ t - k T + a A μ , k T < t k + 1 T . From the second equation of system (7), we get the pulse condition (9) S k + 1 T + = 1 - p S 2 k + 1 T + θ S k + 1 T S k + 1 T + θ . Then, we have (10) S k + 1 T + = ( 1 - p ) S k T + - a A μ e - μ T + a A μ 2 + θ S k T + - a A μ e - μ T + a A μ ( 1 - p ) S k T + - a A μ e - μ T + a A μ 2 · S k T + - a A μ e - μ T + a A μ + θ - 1 . Let S ( k T + ) = S k , and construct a stroboscopic map as follows: (11) S k + 1 = ( 1 - p ) S k - a A μ e - μ T + a A μ 2 + θ S k - a A μ e - μ T + a A μ ( 1 - p ) S k - a A μ e - μ T + a A μ 2 · S k - a A μ e - μ T + a A μ + θ - 1 f ( S k ) , then system (11) has a unique fixed point. In fact, assume that S ~ is the fixed point of (11); then S ~ = S ( k T + ) = S ( ( k + 1 ) T + ) , and it satisfies (12) q 1 S ~ 2 + q 2 S ~ + q 3 = 0 , where (13) q 1 = e - μ T - 1 - p e - 2 μ T > 0 , q 2 = e - μ T - 1 a A μ 2 1 - p e - μ T - 1 - θ , q 3 = - 1 - p a A μ 2 e - μ T - 1 2 - a A θ μ 1 - e - μ T < 0 . Since Δ = q 2 2 - 4 q 1 q 3 > 0 , then the equation q 1 S ~ 2 + q 2 S ~ + q 3 = 0 has a unique positive root (14) S ~ = - q 2 + q 2 2 - 4 q 1 q 3 2 q 1 , therefore system (11) has a unique fixed point S ~ .

From system (11), we have (15) f ( S k ) = p θ 2 e - μ T S k - a A / μ e - μ T + a A / μ + θ 2 + 1 - p e - μ T . Clearly, 0 < f S k < 1 ; then 0 < f S ~ < 1 ; thus, S ~ is a stable fixed point of (11). Substituting the expression of S ~ into (8), we have (16) S * ( t ) = a A μ + S ~ - a A μ e - μ ( t - k T ) , k T < t ( k + 1 ) T . Since S ~ is the unique stable fixed point of the difference equations, then S * ( t ) is the unique global asymptotically stable periodic solution of the system (7). The proof is completed.

According to Lemma 1, we have the following theorem.

Theorem 2.

System (5) has a disease-free periodic solution ( S * ( t ) , 0 ) .

Next we will discuss the stability of the periodic solution. Suppose that ( S ( t ) , I ( t ) ) is any positive solution of the system (5); let (17) x t = S t - S * t , y t = I t ; then the linearized system of the system (5) for the disease-free periodic solution ( S * ( t ) , 0 ) is (18) d x ( t ) d t d y ( t ) d t = - μ - β S * t α S * t + 1 0 β S * t α S * t + 1 - λ - γ - μ x ( t ) y ( t ) ; here t k T . Let Φ ( t ) be the fundamental solution matrix of the system; thus (19) d Φ ( t ) d t = - μ - β S * t α S * t + 1 0 β S * t α S * t + 1 - λ - γ - μ Φ t , and Φ ( 0 ) = E , where E is the unit matrix. Then (20) Φ t = e - μ t φ 12 ( t ) 0 φ 22 ( t ) , where d φ 22 ( t ) / d t = β S * t / ( α S * t + 1 ) - λ - γ - μ φ 22 ( t ) . Here, φ 12 ( t ) is not required in the following analysis. Then we get (21) φ 22 t = exp k T k T + t β S * ( u ) α S * ( u ) + 1 - λ - γ - μ d u . For t = k T , the pulse condition is (22) x ( t + ) y ( t + ) = 1 - p + p θ 2 S * t + θ 2 0 0 1 x ( t ) y ( t ) . Let (23) B T = 1 - p + p θ 2 S * T + θ 2 0 0 1 ; then the single-valued matrix of the system (18) is (24) M = B ( T ) Φ ( T ) = 1 - p + p θ 2 S * T + θ 2 e - μ T 1 - p + p θ 2 S * T + θ 2 φ 12 ( T ) 0 φ 22 ( T ) .

The eigenvalues of the matrix M are λ 1 = ( 1 - p + p θ 2 / ( S * ( T ) + θ ) 2 ) e - μ T < 1 and λ 2 = φ 22 ( T ) . By the Floquet multiplier theory , the disease-free periodic solution ( S * ( t ) , 0 ) is locally asymptotically stable if λ 2 < 1 ; that is, (25) k T ( k + 1 ) T β S * ( u ) 1 + α S * ( u ) - λ - γ - μ d u < 0 . Denote (26) R 1 = β α λ + γ + μ · α S ~ - a A / μ + 1 + α a A / μ e μ T α S ~ - a A / μ + 1 + α a A / μ 1 - 1 α a A + μ T · ln α S ~ - a A / μ + 1 + α a A / μ e μ T α S ~ - a A / μ + 1 + α a A / μ ; we can draw a conclusion as follows.

Theorem 3.

The disease-free periodic solution ( S * ( t ) , 0 ) of system (5) is locally asymptotically stable if R 1 < 1 .

Next we will prove the global attractivity of the disease-free periodic solution ( S * ( t ) , 0 ) of system (5).

Theorem 4.

The disease-free periodic solution ( S * ( t ) , 0 ) of system (5) is globally attractive if R 1 < 1 .

Proof.

Let ( S ( t ) , I ( t ) ) be any solution of the system (5). Since R 1 < 1 , one can choose ε > 0 small enough such that (27) σ = 0 T β ( S * ( t ) + ε ) 1 + α ( S * ( t ) + ε ) - λ - γ - μ d t < 0 . From the first and third equations of the system (5), we have (28) S ˙ ( t ) a A - μ S ( t ) , t k T , S ( t + ) = ( 1 - p ) S 2 ( t ) + θ S ( t ) S ( t ) + θ , t = k T . Consider the following impulsive comparison system: (29) d x ( t ) d t = a A - μ x ( t ) , t k T , x t + = 1 - p x 2 t + θ x t x t + θ , t = k T . By the comparison theorem of impulsive differential equation, we have S ( t ) x ( t ) and x ( t ) S * ( t ) as t . Hence there exists ε > 0 such that (30) S ( t ) x ( t ) < S * t + ε , for all t large enough. For simplification we may assume (30) holds for all t 0 . From the second equation of system (5), we have (31) I ˙ ( t ) < β ( S * ( t ) + ε ) 1 + α ( S * ( t ) + ε ) - λ - γ - μ I ( t ) , which leads to (32) I k + 1 T I ( k T + ) exp k T ( k + 1 ) T β ( S * ( t ) + ε ) 1 + α ( S * ( t ) + ε ) - λ - γ - μ d t = I ( k T ) exp k T ( k + 1 ) T β ( S * ( t ) + ε ) 1 + α ( S * ( t ) + ε ) - λ - γ - μ d t = I ( k T ) e σ . Hence I ( k T ) I ( 0 + ) e k σ and I ( k T ) 0 as k . Therefore I ( t ) 0 as t since 0 < I ( t ) I ( k T ) e β T for k T < t ( k + 1 ) T . Without loss of generality, we may assume that 0 < I ( t ) < ε ( ε < α a A / β ) for all t 0 . From the first equation of system (5), we have (33) a A - β ε α - μ S ( t ) S ˙ ( t ) a A - μ S ( t ) . Then, we have z 1 ( t ) S ( t ) z 2 ( t ) and z 1 ( t ) z 1 * ( t ) , z 2 ( t ) S * ( t ) , where z 1 ( t ) and z 2 ( t ) are solutions of (34) d z 1 t d t = a A - β ε α - μ z 1 t , t k T , z 1 t + = 1 - p t z 1 t , t = k T , d z 2 t d t = a A - μ z 2 t , t k T , z 2 ( t + ) = ( 1 - p ( t ) ) z 2 ( t ) , t = k T , respectively. Consider z 1 * ( t ) = ( a A - β ε / α ) / μ + S ~ - ( a A - β ε / α ) / μ e - μ ( t - k T ) , k T < t ( k + 1 ) T . Therefore, for any ε 1 > 0 , there exists a T > 0 such that (35) z 1 * ( t ) - ε 1 < S ( t ) < S * ( t ) + ε 1 , for t > T .

Letting ε 0 , we have (36) S * ( t ) - ε 1 < S ( t ) < S * ( t ) + ε 1 , for t large enough, which implies S ( t ) S * ( t ) as t . So the disease-free periodic solution ( S * ( t ) , 0 ) of system (5) is global attractivity. The proof is completed.

Synthesizing Theorems 3 and 4, we have the following.

Theorem 5.

The disease-free periodic solution ( S * ( t ) , 0 ) of the system (5) is globally asymptotically stable if R 1 < 1 .

3. Existence of Positive Periodic Solution and Bifurcation

In this section, we will discuss the existence of the positive periodic solution and the branch of the system (5) by using the bifurcation theorem .

Obviously, the threshold value R 1 is proportional to the pulse vaccination period T , and R 1 1 for T large enough. In this case, the disease-free periodic solution ( S * ( t ) , 0 ) of the system (5) is unstable. Assume that T = T 0 as R 1 = 1 . We choose the pulse vaccination period T as a bifurcation parameter. Denote x 1 ( t ) = S ( t ) , x 2 ( t ) = I ( t ) , and then the system (5) can be rewritten as (37) x ˙ 1 ( t ) = a A - β x 1 ( t ) x 2 ( t ) 1 + α x 1 ( t ) - μ x 1 ( t ) F 1 ( x 1 , x 2 ) , x ˙ 2 ( t ) = β x 1 ( t ) x 2 ( t ) 1 + α x 1 ( t ) - λ x 2 ( t ) - γ x 2 ( t ) - μ x 2 ( t ) F 2 ( x 1 , x 2 ) , t k T , x 1 ( t + ) = x 1 ( t ) - p x 1 2 ( t ) x 1 ( t ) + θ θ 1 ( x 1 , x 2 ) , x 2 ( t + ) = x 2 ( t ) θ 2 ( x 1 , x 2 ) , t = k T . We assume that X ( t ) = ( x 1 ( t ) , x 2 ( t ) ) T = Φ ( t , X 0 ) = ( Φ 1 ( t , X 0 ) , Φ 2 ( t , X 0 ) ) T is the solution of the system (37) through the initial point X ( 0 ) = X 0 = ( x 10 , x 20 ) T . By Theorem 2, the system (37) has a boundary periodic solution ς ( t ) = ( x 1 * ( t ) , 0 ) = ( S * ( t ) , 0 ) , and ς ( 0 ) = ( x 1 * ( 0 ) , 0 ) ( x 0 , 0 ) . In order to apply the bifurcation theorem (see ), we make the calculations as follows: (38) Φ 1 ( t , X 0 ) x 1 = exp 0 t F 1 ( ς ( τ ) ) x 1 d τ = e - μ t , Φ 1 ( t , X 0 ) x 2 = 0 t exp u t F 1 ς τ x 1 d τ F 1 ( ς ( u ) ) x 2 · exp 0 u F 2 ς τ x 2 d τ d u = - β 0 t e - μ ( t - u ) S * ( u ) 1 + α S * ( u ) · exp 0 u β S * τ 1 + α S * τ - μ - λ - γ d τ d u U ( t ) < 0 , Φ 2 ( t , X 0 ) x 2 = exp 0 t F 2 ( ς ( τ ) ) x 2 d τ = exp 0 t β S * τ 1 + α S * τ - μ - λ - γ d τ , a 0 = 1 - θ 1 x 1 Φ 1 x 1 ( T 0 , X 0 ) = 1 - 1 - p + p θ 2 θ + S * ( T 0 ) 2 e - μ T 0 1 - m e - μ T 0 , b 0 = - θ 1 x 1 Φ 1 x 2 + θ 1 x 2 Φ 2 x 2 ( T 0 , X 0 ) = - 1 - p + p θ 2 θ + S * ( T 0 ) 2 U ( T 0 ) = - m U ( T 0 ) , d 0 = 1 - θ 2 x 2 Φ 2 x 2 ( T 0 , X 0 ) = 1 - exp 0 T 0 ( β S * ( t ) 1 + α S * ( t ) - μ - λ - γ ) d t . Obviously, d 0 = 0 is equivalent to R 1 = 1 . Consider (39) 2 Φ 2 t , X 0 x 1 x 2 = 0 t exp u t F 2 ( ς ( τ ) ) x 2 d τ 2 F 2 ( ς ( u ) ) x 1 x 2 · exp 0 u F 2 ( ς ( τ ) ) x 2 d τ d u = β 0 t 1 1 + α S * ( u ) 2 · exp 0 t ( β S * ( τ ) 1 + α S * ( τ ) - μ - λ - γ ) d τ d u , 2 Φ 2 ( t , X 0 ) x 2 2 = 0 t exp u t F 2 ( ς ( τ ) ) x 2 d τ 2 F 2 ( ς ( u ) ) x 2 2 · exp 0 u F 2 ( ς ( τ ) ) x 2 d τ d u + 0 t exp u t F 2 ( ς ( τ ) ) x 2 d τ 2 F 2 ( ς ( u ) ) x 2 x 1 · 0 u exp p u F 1 ( ς ( τ ) ) x 1 d τ F 1 ( ς ( p ) ) x 2 · exp 0 p F 2 ( ς ( τ ) ) x 2 d τ d p d u = - β 2 0 t exp u t β S * τ 1 + α S * τ - μ - λ - γ d τ · 1 1 + α S * ( u ) 2 · 0 u S * ( p ) 1 + α S * ( p ) · exp β S * τ 1 + α S * τ - μ - λ - γ d τ - μ ( u - p ) + 0 p β S * τ 1 + α S * τ - μ - λ - γ β S * τ 1 + α S * τ d τ d p d u . Then, we have (40) 2 Φ 2 ( T 0 , X 0 ) x 1 x 2 = 0 T 0 β 1 + α S * ( u ) 2 d u · exp 0 T 0 β S * τ 1 + α S * τ - μ - λ - γ d τ , 2 Φ 2 ( T 0 , X 0 ) x 2 2 < 0 , 2 Φ 2 ( t , X 0 ) T ~ x 2 = F 2 ( ς ( t ) ) x 2 exp 0 t F 2 ( ς ( τ ) ) x 2 d τ = β S * ( t ) 1 + α S * ( t ) - μ - λ - γ · exp 0 t β S * τ 1 + α S * τ - μ - λ - γ d τ , Φ 1 ( T 0 , X 0 ) T ~ = d x 1 * ( T 0 ) d t = - μ S ~ - a A μ e - μ T 0 , B = - 2 θ 2 x 1 x 2 Φ 1 ( T 0 , X 0 ) T ~ + Φ 1 ( T 0 , X 0 ) x 1 · 1 a 0 θ 1 x 1 Φ 1 ( T 0 , X 0 ) T ~ Φ 2 ( T 0 , X 0 ) x 2 - θ 2 x 2 2 Φ 2 ( T 0 , X 0 ) T ~ x 2 + 2 Φ 2 ( T 0 , X 0 ) x 1 x 2 1 a 0 θ 1 x 1 Φ 1 ( T 0 , X 0 ) T ~ = exp 0 T 0 β S * ( τ ) 1 + α S * ( τ ) - μ - λ - γ d τ · μ S ~ - a A μ m e - μ T 0 1 - m e - μ T 0 0 T 0 β 1 + α S * ( u ) 2 d u - exp 0 T 0 β S * τ 1 + α S * τ - μ - λ - γ d τ · β S * ( T 0 ) 1 + α S * ( T 0 ) - μ - λ - γ , C = - 2 2 θ 2 x 1 x 2 - b 0 a 0 Φ 1 ( T 0 , X 0 ) x 1 + Φ 1 ( T 0 , X 0 ) x 2 · Φ 2 ( T 0 , X 0 ) x 2 - 2 θ 2 x 2 2 Φ 2 ( T 0 , X 0 ) x 2 2 + 2 θ 2 x 2 · b 0 a 0 · 2 Φ 2 ( T 0 , X 0 ) x 2 x 1 - θ 2 x 2 · 2 Φ 2 ( T 0 , X 0 ) x 2 2 = - 2 m U ( T 0 ) 1 - m e - μ T 0 exp 0 T 0 β S * ( τ ) 1 + α S * ( τ ) - μ - λ - γ d τ · 0 T 0 β 1 + α S * u 2 d u - 2 Φ 2 T 0 , X 0 x 2 2 > 0 . From the bifurcation theorem, the system (37) could produce nontrivial periodic solutions by the boundary solution with the condition B C 0 , and the bifurcation is a supercritical bifurcation if B C < 0 , or a subcritical bifurcation if B C > 0 . Letting (41) H : β S * ( T 0 ) 1 + α S * ( T 0 ) - μ - λ - γ - μ S ~ - a A μ m e - μ T 0 1 - m e - μ T 0 · 0 T 0 β 1 + α S * ( u ) 2 d u > 0 , then the following theorem is gotten.

Theorem 6.

If the condition ( H ) holds, then system (37) has a supercritical branch at the point T 0 ; that is, a nontrivial periodic solution could be produced by the boundary periodic solution ( S * ( t ) , 0 ) . Here, T 0 satisfies R 1 ( T 0 ) = 1 .

4. Discussion and Numerical Simulation

In this paper, we have considered a SIR epidemic model with the nonlinear impulsive vaccination and life-immunity. The whole dynamics of the model is investigated under nonlinear impulsive effect. Firstly, the existence of disease-free periodic solution is discussed by using stroboscopic map and fixed point theory of difference equations, the globally asymptotical stability of disease-free periodic solution is proven by using Floquet multiplier theory and differential impulsive comparison theorem. Then, by choosing impulsive vaccination period as a bifurcation parameter, sufficient condition for the existence of positive periodic solution was obtained by using the bifurcation theorem. We have found that the dynamics of the model (5) depends on the threshold R 1 . If R 1 < 1 , then the disease-free periodic solution ( S * ( t ) , 0 ) of the system (5) is globally asymptotically stable. Otherwise, it is unstable and will show a supercritical branch for R 1 ( T 0 ) = 1 . The threshold R 1 is related to all parameters of the model (5).

Next, we focus on the relations of the R 1 with the parameters θ and T . The model (5) adopts the saturated vaccination rate p ( t ) = p S ( t ) / ( S ( t ) + θ ) . Here, θ represents the degree of restriction about medical resources. The relations of R 1 with θ and T can be seen in Figure 1.

The relations of R 1 with the parameters θ and T .

From Figure 1, if we fix θ , the threshold R 1 is an increasing function of the vaccination period T . And if we fix the vaccination period T , R 1 is an increasing function of the parameter θ . By the meaning of θ and T , if we enrich the medical resources (i.e., decrease θ ) or reduce the vaccination period (i.e., decrease T ), then the disease will be extinction; otherwise, the disease will be permanent.

To show the influence of restriction of medical resources on the model dynamics, we give some numerical simulations. Let p = 0.7 , a = 0.2 , A = 2 , β = 2 , α = 0.7 , μ = 0.4 , λ = 0.2 , γ = 0.2 , and T = 2 . We have an example as follows: (42) S ˙ ( t ) = 0.4 - 2 S ( t ) I ( t ) 1 + 0.7 S ( t ) - 0.4 S ( t ) , I ˙ ( t ) = 2 S ( t ) I ( t ) 1 + 0.7 S ( t ) - 0.8 I ( t ) , t 2 k , S ( t + ) = 1 - 0.7 S ( t ) S ( t ) + θ S ( t ) , I ( t + ) = I ( t ) , t = 2 k .

Case 1.

Let θ = 0 , the initial point is ( 0.8,0.06 ) . By calculation, we obtain R 1 = 0.8324 . Figure 2 shows that the number of susceptible individuals produces periodic oscillation. Figure 3 shows that the disease will eventually be eliminated. Figure 4 shows that system (5) has a disease-free periodic solution ( S * ( t ) , 0 ) , which is globally asymptotically stable.

Time series of S of system (42) while θ = 0 .

Time series of I of system (42) while θ = 0 .

Phase diagram of system (42) while θ = 0 .

Case 2.

Letting θ = 0.9 , the initial point is ( 0.8,0.06 ) . By calculation, we obtain R 1 = 1.1456 . Figure 5 shows that the number of susceptible individuals produces periodic oscillation under the pulsed effect. Figure 6 shows that the disease will be persistent. Figure 7 shows that system (5) has a globally asymptotically stable positive periodic solution.

Time series of S of system (42) while θ = 0.9 .

Time series of I of system (42) while θ = 0.9 .

Phase diagram of system (42) while θ = 0.9 .

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 National Natural Science Foundation of China (no. 11371230), Natural Sciences Fund of Shandong Province of China (no. ZR2012AM012), a Project for Higher Educational Science and Technology Program of Shandong Province of China (no. J13LI05), SDUST Research Fund (2014TDJH102), and Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources, Shandong Province of China.

The World Health Statistics Report 2013, http://www.who.int/research/en/ Agur Z. Cojocaru L. Mazor G. Anderson R. M. Danon Y. L. Pulse mass measles vaccination across age cohorts Proceedings of the National Academy of Sciences of the United States of America 1993 90 24 11698 11702 10.1073/pnas.90.24.11698 2-s2.0-0027136227 de Quadros C. A. Andrus J. K. Olivé J.-M. da Silveira C. M. Eikhof R. M. Carrasco P. Fitzsimmons J. W. Pinheiro F. P. Eradication of poliomyelitis: progress in the Americas The Pediatric Infectious Disease Journal 1991 10 3 222 229 10.1097/00006454-199103000-00011 2-s2.0-0025905391 Sabin A. B. Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control European Journal of Epidemiology 1991 7 1 1 22 2-s2.0-0026035347 Ramsay M. Gay N. Miller E. The epidemiology of measles in England and Wales: rationale for the 1994 national vaccination campaign Communicable Disease Report 1994 4 12 R141 R146 Shulgin B. Stone L. Agur Z. Pulse vaccination strategy in the SIR epidemic model Bulletin of Mathematical Biology 1998 60 6 1123 1148 10.1016/s0092-8240(98)90005-2 2-s2.0-0032213158 Stone L. Shulgin B. Agur Z. Theoretical examination of the pulse vaccination policy in the SIR epidemic model Mathematical and Computer Modelling 2000 31 4 207 215 d'Onofrio A. Stability properties of pulse vaccination strategy in SEIR epidemic model Mathematical Biosciences 2002 179 1 57 72 10.1016/s0025-5564(02)00095-0 MR1908736 2-s2.0-0036291194 d'Onofrio A. On pulse vaccination strategy in the SIR epidemic model with vertical transmission Applied Mathematics Letters 2005 18 7 729 732 10.1016/j.aml.2004.05.012 MR2144719 2-s2.0-18144413585 Jiao J. Chen L. Cai S. An SEIRS epidemic model with two delays and pulse vaccination Journal of Systems Science and Complexity 2008 21 2 217 225 10.1007/s11424-008-9105-y MR2403728 2-s2.0-44449115241 Zhang H. Chen L. Nieto J. J. A delayed epidemic model with stage-structure and pulses for pest management strategy Nonlinear Analysis: Real World Applications 2008 9 4 1714 1726 2-s2.0-43649101953 10.1016/j.nonrwa.2007.05.004 MR2422575 Song X. Jiang Y. Wei H. Analysis of a saturation incidence SVEIRS epidemic model with pulse and two time delays Applied Mathematics and Computation 2009 214 2 381 390 MR2541674 2-s2.0-67649224128 10.1016/j.amc.2009.04.005 Gao S. Chen L. Nieto J. J. Torres A. Analysis of a delayed epidemic model with pulse vaccination and saturation incidence Vaccine 2006 24 35-36 6037 6045 10.1016/j.vaccine.2006.05.018 2-s2.0-33746833233 Liu X. Takeuchi Y. Iwami S. SVIR epidemic models with vaccination strategies Journal of Theoretical Biology 2008 253 1 1 11 10.1016/j.jtbi.2007.10.014 MR2960677 2-s2.0-44849133934 Zhao Z. Chen L. Song X. Impulsive vaccination of SEIR epidemic model with time delay and nonlinear incidence rate Mathematics and Computers in Simulation 2008 79 3 500 510 10.1016/j.matcom.2008.02.007 MR2477543 2-s2.0-55949111153 Zhao W. C. Meng X. Z. An SIR epidemic disease model with vertical transmission and pulse vaccination Mathematica Applicata 2009 22 3 676 682 MR2566859 Meng X. Chen L. Global dynamical behaviors for an SIR epidemic model with time delay and pulse vaccination Taiwanese Journal of Mathematics 2008 12 5 1107 1122 MR2431883 2-s2.0-67349179733 Meng X. Chen L. Wu B. A delay SIR epidemic model with pulse vaccination and incubation times Nonlinear Analysis: Real World Applications 2010 11 1 88 98 2-s2.0-70350731589 MR2570527 10.1016/j.nonrwa.2008.10.041 Meng X. Chen L. The dynamics of a new SIR epidemic model concerning pulse vaccination strategy Applied Mathematics and Computation 2008 197 2 582 597 10.1016/j.amc.2007.07.083 MR2400680 2-s2.0-39449119838 Zhao W. Zhang T. Chang Z. Meng X. Liu Y. Dynamical analysis of SIR epidemic models with distributed delay Journal of Applied Mathematics 2013 2013 15 154387 10.1155/2013/154387 2-s2.0-84881398730 Gao S. Liu Y. Nieto J. J. Andrade H. Seasonality and mixed vaccination strategy in an epidemic model with vertical transmission Mathematics and Computers in Simulation 2011 81 9 1855 1868 10.1016/j.matcom.2010.10.032 MR2799731 2-s2.0-79955561859 Shi R. Jiang X. Chen L. The effect of impulsive vaccination on an SIR epidemic model Applied Mathematics and Computation 2009 212 2 305 311 10.1016/j.amc.2009.02.017 MR2531237 2-s2.0-67349167681 Li J. Yang Y. SIR-SVS epidemic models with continuous and impulsive vaccination strategies Journal of Theoretical Biology 2011 280 1 108 116 10.1016/j.jtbi.2011.03.013 MR2975047 2-s2.0-79955499583 Zhang T. Meng X. Zhang T. Song Y. Global dynamics for a new high-dimensional SIR model with distributed delay Applied Mathematics and Computation 2012 218 24 11806 11819 MR2945184 10.1016/j.amc.2012.04.079 2-s2.0-84863779111 Zhang T. Meng X. Zhang T. SVEIRS: a new epidemic disease model with time delays and impulsive effects Abstract and Applied Analysis 2014 2014 15 542154 10.1155/2014/542154 Zhang J. Sun J. A delayed SEIRS epidemic model with impulsive vaccination and nonlinear incidence rate International Journal of Biomathematics 2014 7 3 1450032 10.1142/s1793524514500326 MR3210481 2-s2.0-84901617594 Pei Y. Liu S. Chen L. Wang C. Two different vaccination strategies in an SIR epidemic model with saturated infectious force International Journal of Biomathematics 2008 1 2 147 160 10.1142/s1793524508000126 MR2444292 2-s2.0-84863115736 Hou J. Teng Z. Continuous and impulsive vaccination of SEIR epidemic models with saturation incidence rates Mathematics and Computers in Simulation 2009 79 10 3038 3054 2-s2.0-67349113472 10.1016/j.matcom.2009.02.001 MR2541316 Pang G. P. Chen L. S. The SIRS epidemic model with saturated contact rate and pulse vaccination Journal of Systems Science and Mathematical Sciences 2007 27 4 563 572 MR2349979 Pang G. Chen L. A delayed SIRS epidemic model with pulse vaccination Chaos, Solitons & Fractals 2007 34 5 1629 1635 10.1016/j.chaos.2006.04.061 MR2335410 2-s2.0-34250220629 Hill A. V. The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves The Journal of Physiology 1910 40 4 4 7 Bainov D. Simeonov P. Impulsive Differential Equations: Periodic Solutions and Applications 1993 Boca Raton, Fla, USA CRC Press Lakmeche A. Arino O. Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment Dynamics of Continuous, Discrete and Impulsive Systems 2000 7 2 265 287 MR1744966