SVEIRS: A New Epidemic Disease Model with Time Delays and Impulsive Effects

Abstract

We first propose a new epidemic disease model governed by system of impulsive delay differential equations. Then, based on theories for impulsive delay differential equations, we skillfully solve the difficulty in analyzing the global dynamical behavior of the model with pulse vaccination and impulsive population input effects at two different periodic moments. We prove the existence and global attractivity of the “infection-free” periodic solution and also the permanence of the model. We then carry out numerical simulations to illustrate our theoretical results, showing us that time delay, pulse vaccination, and pulse population input can exert a significant influence on the dynamics of the system which confirms the availability of pulse vaccination strategy for the practical epidemic prevention. Moreover, it is worth pointing out that we obtained an epidemic control strategy for controlling the number of population input.

1. Introduction

In epidemic modeling, susceptible-infectious-recovered type of models is well known [1–18] although such models very often ignore the incubation period in the development of mathematical models for some diseases. However, recent research shows for certain diseases, such as smallpox, rabies, BSE, and some skin diseases, the incubation period has significant effect on the epidemic dynamics so that it is nonnegligible. The incubation period varies greatly from a couple of days (e.g., H1N1 outbreaking worldwide has generally an incubation period of one to seven days) to several years (e.g., AIDS virus sometimes can be several years). When taking the incubation period into account in the development of models, we reach SEIR model, which is short for susceptible, exposed, infectious, and recovered [19–30]. And some researchers used time delay to describe the incubation period; for example, Cooke [31], Beretta and Takeuchi [4], Takeuchi et al. [32], and Ma et al. [5] studied a SIR model with time delay and nonlinear incidence rate 𝛽𝑆(𝑡)𝐼(𝑡−𝜏). Liu et al. [33, 34] used a nonlinear incidence rate 𝛽𝑆^𝑝(𝑡)𝐼^𝑞(𝑡), and Meng et al. [35] and Jiang et al. [30], respectively, studied an impulsively vaccinating SIR model with nonlinear incidences 𝛽𝑆^𝑞(𝑡)𝐼(𝑡−𝜏) and 𝛽𝑆^𝑞(𝑡−𝜏)𝐼(𝑡−𝜏), which are better to describe the spread process of diseases than linear one.

In order to prevent infectious diseases, [36, 37] suggested that vaccination to the susceptible population is an important strategy. The traditional vaccinations are applied to each individual, while impulsive ones are to periodically vaccinate people within certain age groups [7–10, 38]. Some diseases may have a vaccination period after being cured but may cause losing immunity gradually. In this case, people might be infected again. So it is of great significance to investigate epidemic models with time delay and impulsive effects due to the incubation period and vaccination period [26–29]. For some certain regional systems, the immigrations can be periodic impulsive population input because the immigratory population might be susceptible. Certainly two different impulsive effects for periodic vaccination and population input do not usually happen simultaneously. Therefore, motivated by Jiang et al. [30] and Song et al. [19], we built a new mathematical model: susceptible, vaccinated, exposed, infectious, recovered, and susceptible epidemic model with two time delays and two nonlinear incidences with pulse vaccination and a constant periodic population input at two different moments as follows: 𝑑𝑆(𝑡)(𝑡)𝑑𝑡=−𝑏𝑆−𝛽𝑆^𝑝(𝑡)𝐼(𝑡)+𝛾𝐼(𝑡−𝜔)𝑒^−𝑏𝜔,𝑑𝑉(𝑡)𝑑𝑡=−𝛿𝛽𝑉^𝑞(𝑡)𝐼(𝑡)−𝛾₁𝑉(𝑡)(𝑡),(𝑡)−𝑏𝑉𝑑𝐸𝑑𝑡=−𝑏𝐸+𝛽𝑆^𝑝(𝑡)𝐼(𝑡)+𝛿𝛽𝑉^𝑞(𝑡)𝐼(𝑡)−𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)𝐼(𝑡−𝜏)−𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)𝐼(),(𝑡)𝑡−𝜏𝑑𝐼𝑑𝑡=𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)𝐼(𝑡−𝜏)+𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)𝐼()−𝐼(𝑡),(𝑡)𝑡−𝜏𝛾+𝑏+𝛼𝑑𝑅𝑑𝑡=𝛾₁𝑉(𝑡)+𝛾𝐼(𝑡)−𝑏𝑅(𝑡)−𝛾𝐼(𝑡−𝜔)𝑒^−𝑏𝜔,()(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆,Δ𝐸=0,Δ𝐼=0,Δ𝑅=0,𝑡=𝑛+𝑙−1𝑇,Δ𝑆=𝜇,Δ𝑉=0,Δ𝐸=0,Δ𝐼=0,Δ𝑅=0,𝑡=𝑛𝑇. (1) Here all parameters of system (1) are nonnegative constants. For the significance of parameters in (1), please see literatures Jiang et al. [30] and Song et al. [19]. Terms 𝛽𝑆^𝑝𝐼 and 𝑉^𝑞𝐼 are the nonlinear incidence rates, and in our paper we only discuss the case 1≤𝑞≤𝑝. (2)

2. Preliminaries

Let 𝑁(𝑡)=𝑆(𝑡)+𝑉(𝑡)+𝐸(𝑡)+𝐼(𝑡)+𝑅(𝑡), and then it is easy to see that 𝑁(𝑡) satisfies the following: 𝑁^′(𝑡)≤𝑏(1−𝑁(𝑡)),lim_𝑡→∞sup𝑁(𝑡)≤1. (3) Hence, for time 𝑡 which is large, we obtain 0≤𝑆(𝑡)+𝑉(𝑡)+𝐼(𝑡)≤1. Let 𝜛=max{𝜏,𝜔} and 𝐶⁺={𝜑=(𝜑₁(𝑠),…,𝜑₅(𝑠))∈𝐶∶𝜑_𝑖(0)>0}; here 𝜑_𝑖(𝑠)>0 is bounded function on interval [−𝜛,0]. Since variable 𝑅(𝑡) only appears in the fifth equation, system (1) can be further reduced as 𝑑𝑆(𝑡)(𝑡)𝑑𝑡=−𝑏𝑆−𝛽𝑆^𝑝(𝑡)𝐼(𝑡)+𝛾𝐼(𝑡−𝜔)𝑒^−𝑏𝜔,𝑑𝑉(𝑡)𝑑𝑡=−𝛿𝛽𝑉^𝑞(𝑡)𝐼(𝑡)−𝛾₁𝑉(𝑡)(𝑡),(𝑡)−𝑏𝑉𝑑𝐼𝑑𝑡=𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)𝐼(𝑡−𝜏)+𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)𝐼()−𝐼(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)(𝑡)𝑡−𝜏𝛾+𝑏+𝛼𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆,Δ𝐼=0,𝑡=𝑛+𝑙−1𝑇,Δ𝑆=𝜇,Δ𝑉=0,Δ𝐼=0,𝑡=𝑛𝑇, (4) with the initial conditions 𝜑₁(𝑠),𝜑₂(𝑠),𝜑₄(𝑠)∈𝐶⁺,𝜑_𝑖(0)>0,𝑖=1,2,4. (5)

Lemma 1 (see [39, 40]). For the following impulse differential inequalities 𝑠^′(𝑡)≤(≥)𝑞(𝑡)𝑠(𝑡)+𝑟(𝑡),𝑡≠𝑡_𝑘,𝑠𝑡⁺_𝑘≤(≥)𝑏_𝑘𝑠𝑡_𝑘+𝑝_𝑘,𝑡=𝑡_𝑘,𝑘∈𝑁, (6) where 𝑞(𝑡), 𝑟(𝑡)∈𝐶(𝑅₊,𝑅), 𝑏_𝑘≥0, and 𝑝_𝑘 are constants.
Assume the following:(𝐴₀)the sequence {𝑡_𝑘} satisfies 0≤𝑡₀<𝑡₁<𝑡₂<⋯, with lim_𝑡→∞𝑡_𝑘=∞;(𝐴₁)𝑤∈𝑃𝐶^′(𝑅₊,𝑅) and 𝑠(𝑡) is left-continuous at 𝑡_𝑘,𝑘∈𝑁.Then 𝑠(𝑡)≤(≥)𝑠𝑡_{0𝑡0<𝑡𝑘<𝑡}𝑏_𝑘exp^𝑡_𝑡0𝑞(𝑢)𝑑𝑢+_{𝑡0<𝑡𝑘<𝑡𝑡𝑘<𝑡𝑗<𝑡}𝑏_𝑗exp^𝑡_𝑡𝑘𝑞(𝑢)𝑑𝑢𝑝_𝑘+^𝑡_{𝑡0𝑢<𝑡𝑘<𝑡}𝑏_𝑘exp^𝑡_𝑢𝑞(𝜃)𝑑𝜃𝑟(𝑢)𝑑𝑢,𝑡≥𝑡₀. (7)

Lemma 2 (see [41]). For the following delay differential equation 𝑑𝑧(𝑡)()(𝑡),𝑑𝑡=𝑎𝑧𝑡−𝜃−𝑏𝑧 (8) where 𝑎, 𝑏, and 𝜃 are all positive constants and 𝑧(𝑡)>0 for 𝑡∈[−𝜃,0], then we have lim_𝑡→∞𝑧(𝑡)=0,𝑖𝑓𝑎<𝑏;+∞,𝑖𝑓𝑎>𝑏. (9)

Lemma 3 (see [42]). The following system, 𝑑𝑥(𝑡)(𝑡),(𝑡)𝑑𝑡=−𝑏𝑥𝑑𝑦()𝑦(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡)()(𝑡)(𝑡)𝑑𝑡=−𝑎+𝑏𝑡≠𝑛𝑇,𝑡≠𝑛+𝑙−1𝑇,Δ𝑥=−𝜃𝑥,Δ𝑦=𝜃𝑥,𝑡=𝑛+𝑙−1𝑇,Δ𝑥=𝜇,Δ𝑦=0,𝑡=𝑛𝑇, (10) has a unique positive 𝑇-periodic solution: 𝑥^∗(𝑡)=𝜇exp(−𝑏(𝑡−(𝑛−1)𝑇))()(),)()𝑇],𝜇()((()𝑇1−1−𝜃exp−𝑏𝑇𝑡∈((𝑛−1𝑇,𝑛+𝑙−11−𝜃exp−𝑏𝑡−𝑛−1))()(),)],𝑦1−1−𝜃exp−𝑏𝑇𝑡∈((𝑛+𝑙−1𝑇,≤𝑛𝑇_∗(𝑡)=𝜇𝜃exp(−𝑏𝑙𝑇)exp(−(𝑎+𝑏)(𝑡−(𝑛+𝑙−1)𝑇))(−()𝑇()(,)()𝑇],1−exp𝑎+𝑏)1−1−𝜃exp−𝑏𝑇)𝑡∈((𝑛+𝑙−1𝑇,𝑛+𝑙 (11) and we further have 𝑥(𝑡)→𝑥^∗(𝑡) and 𝑦(𝑡)→𝑦^∗(𝑡) as 𝑡→+∞.

3. The Existence and Global Attractivity of “Infection-Free” Periodic Solution

3.1. Existence

In this section, we are committed to investigate the existence of “infection-free” periodic solution. In this case, we have 𝐼(𝑡)=0,𝑡≥0. (12) From systems (4) and (12), we obtain 𝑑𝑆(𝑡)(𝑡),(𝑡)𝑑𝑡=−𝑏𝑆𝑑𝑉𝛾𝑑𝑡=−₁𝑉(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)+𝑏𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝜇,Δ𝑉=0,𝑡=𝑛𝑇,𝑛∈𝑁. (13) By Lemma 3, system (13) has a unique positive 𝑇-periodic solution: 𝑆^∗(𝑡)=𝜇exp(−𝑏(𝑡−(𝑛−1)𝑇))()(),)()𝑇],𝜇()((()𝑇1−1−𝜃exp−𝑏𝑇𝑡∈((𝑛−1𝑇,𝑛+𝑙−11−𝜃exp−𝑏𝑡−𝑛−1))()(),)],𝑉1−1−𝜃exp−𝑏𝑇𝑡∈((𝑛+𝑙−1𝑇,≤𝑛𝑇_∗(𝑡)=𝜇𝜃exp(−𝑏𝑙𝑇)exp(−(𝑎+𝑏)(𝑡−(𝑛+𝑙−1)𝑇))(−()𝑇()(,)()𝑇].1−exp𝑎+𝑏)1−1−𝜃exp−𝑏𝑇)𝑡∈((𝑛+𝑙−1𝑇,𝑛+𝑙 (14) Furthermore, we can prove that it is the unique globally asymptotically stable positive periodic solution of system (4). We summarize this conclusion in the following lemma.

Lemma 4. The system (4) has an “infection-free” periodic solution (𝑆^∗(𝑡),𝑉^∗(𝑡),0), for 𝑡∈((𝑛+𝑙−1)𝑇,(𝑛+𝑙)𝑇] and 𝑛∈𝑁; for any solution (𝑆(𝑡),𝑉(𝑡),𝐼(𝑡)) of it, the following holds true: 𝑆(𝑡)⟶𝑆^∗(𝑡),𝑉(𝑡)⟶𝑉^∗(𝑡) (15) as 𝑡→∞.

This lemma indicates that in between the vaccination the susceptible and vaccinated populations oscillate with period 𝑇 in synchronization with the periodic pulse vaccination. Next we prove the global attractivity of such solution.

3.2. Global Attractivity

In this section, we will prove our main result on the global attractivity of the infection-free solution. It is stated in the following theorem.

Theorem 5. The system (4) has a unique infection-free periodic solution (𝑆^∗(𝑡),𝑉^∗(𝑡),0), and when it exists, it is globally attractive if ℛ₁<1, (16) where ℛ₁=𝛽𝑒^−𝑏𝜏𝐴^𝑝₁+𝛿𝐴^𝑞₂,𝛾+𝑏+𝛼 (17) with 𝐴₁=𝛾𝑒^−𝑏𝜔𝑏+𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇,𝐴−1₂=𝜃𝑒^{−𝑏𝑙𝑇}(1−𝜃)1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇}. (18)

Proof. Let (𝑆(𝑡),𝑉(𝑡),𝐼(𝑡)) be a solution of (4) satisfied initial condition (5). Since ℛ₁<1, one can choose an 𝜀>0 small enough such that 𝛽𝑒^−𝑏𝜏Δ₁^𝑝+𝛿Δ₂^𝑞−𝛾+𝑏+𝛼<0, (19) where Δ₁=𝛾𝑒^−𝑏𝜔𝑏+𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇Δ−1+𝜀,₂=𝜃𝑒^{−𝑏𝑙𝑇}(1−𝜃)1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇}+𝜀. (20) For 𝑛>𝑛₁, we have 𝑑𝑆(𝑡)(𝑡)𝑑𝑡≤𝑏−𝑏𝑆+𝛾𝑒^−𝑏𝜔,()(𝑡)(𝑡)()(𝑡)𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝜇,𝑡=𝑛𝑇,𝑛∈𝑁. (21) By impulsive differential inequality Lemma 1, we have 𝑆(𝑡)≤𝑆𝑛₁𝑇⁺_{𝑛1𝑇+<𝑛𝑇<𝑡}exp^𝑡_𝑛1𝑇(−𝑏)𝑑𝑠+_{𝑛1𝑇<𝑛𝑇<𝑡𝑛𝑇<𝑡𝑗<𝑡}exp^𝑡_𝑛𝑇()𝜇+−𝑏𝑑𝑠_{𝑡𝑛1𝑇𝑠<𝑛𝑇<𝑡}exp^𝑡_𝑠(−𝑏)𝑑𝜃𝛾𝑒^−𝑏𝜔𝑑𝑠=𝑆₁+𝑆₂+𝑆₃, (22) where 𝑆₁=𝑆𝑛₁𝑇⁺_{𝑛1𝑇+<𝑛𝑇<𝑡}exp_𝑛^𝑡/𝑇₁()𝑛−𝑏𝑑𝑇𝜉=𝑆₁𝑇⁺𝑒^{−𝑏(𝑡−𝑛1𝑇)},𝑆₂=_{𝑛1𝑇<𝑛𝑇<𝑡𝑛𝑇<𝑡𝑗<𝑡}exp^𝑡_𝑛𝑇()𝜇=−𝑏𝑑𝑠_{𝑛1𝑇<𝑛𝑇<𝑡}𝜇𝑒^{−𝑏(𝑡−𝑛𝑇)}𝑒=𝜇^{−𝑏(𝑡−𝑛1)𝑇}−𝑒^{−𝑏(𝑡−(𝑛+𝑛1))𝑇}1−𝑒^𝑏𝑇,𝑆₃=^𝑡_{𝑛1𝑇𝑠<𝑛𝑇<𝑡}exp^𝑡_𝑠(−𝑏)𝑑𝜃𝛾𝑒^−𝑏𝜔𝑑𝑠=𝛾𝑒^{−𝑏𝜔𝑡}_{𝑛1𝑇𝑠<𝑛𝑇<𝑡}exp^𝑡_𝑠()=−𝑏𝑑𝜃𝑑𝑠𝛾𝑒^−𝑏𝜔𝑒^−𝑏𝑡𝑏^𝑡_{𝑛1𝑇𝑠<𝑛𝑇<𝑡}𝑒^𝑏𝑠𝑑()=𝑏𝑠𝛾𝑒^−𝑏𝜔𝑒^−𝑏𝑡𝑏𝑒^𝑏𝑡−𝑒^{𝑏𝑛1𝑇}. (23) Thus 𝑆(𝑡)≤𝑆₁+𝑆₂+𝑆₃=𝑆𝑛₁𝑇⁺𝑒^{−𝑏(𝑡−𝑛1𝑇)}𝑒+𝜇^{−𝑏(𝑡−𝑛1)𝑇}−𝑒^{−𝑏(𝑡−(𝑛+𝑛1))𝑇}1−𝑒^𝑏𝑇+𝛾𝑒^−𝑏𝜔𝑒^−𝑏𝑡𝑏𝑒^𝑏𝑡−𝑒^{𝑏𝑛1𝑇}≤𝑒^−𝑏𝑡𝑆𝑛₁𝑇⁺𝑒^{𝑛1𝑏𝑇}+𝜇𝑒^{−𝑏(𝑡−𝑛1𝑇)}1−𝑒^𝑏𝑇+𝛾𝑒^−𝑏𝜔𝑏𝑒^{−𝑏(𝑡−𝑛1𝑇)}+𝛾𝑒^−𝑏𝜔𝑏+𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇,−1 (24) and then we have lim_𝑡→∞sup𝑆(𝑡)<𝛾𝑒^−𝑏𝜔𝑏+𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇.−1 (25) Thus there exists a positive integer 𝑛₂>𝑛₁ and constant 𝜀>0 small enough such that, for all 𝑡>𝑛₂𝑇, 𝑆(𝑡)≤𝛾𝑒^−𝑏𝜔𝑏+𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇−1+𝜀=Δ₁. (26)
For 𝑛>𝑛₁, system (4) yields 𝑑𝑉(𝑡)𝛾𝑑𝑡≤−₁𝑉(𝑡)()(𝑡)(𝑡)()+𝑏,𝑡≠𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑉=𝜃𝑆,𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁. (27) We obtain the following comparison impulsive differential system: 𝑑𝑥(𝑡)𝛾𝑑𝑡=−₁𝑥(𝑡)()(𝑡)(𝑡)()+𝑏,𝑡≠𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑥=𝜃𝑆,𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁. (28) By Lemma 3, the system has a periodic solution given by 𝑥^∗(𝑡)=𝜃𝑒^{−𝑏𝑙𝑇}𝑒^{−(𝛾1+𝑏)(𝑡−(𝑛+𝑙−1)𝑇)}(1−𝜃)1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇},)()𝑇],𝑥0𝑡∈((𝑛+𝑙−1𝑇,𝑛+𝑙₊=𝜃𝑒^{−𝑏𝑙𝑇}(1−𝜃)1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇}, (29) which is globally asymptotically stable.
Now, assume that 𝑥(𝑡) is the solution of system (28) with initial value 𝑥(0⁺)=𝑉₀. Then by Lemma 1, we know there exists a positive integer 𝑛 such that 𝑉(𝑡)<𝑥(𝑡)<𝑥^∗(𝑡)+𝜀,𝑡∈(𝑛𝑇,(𝑛+1)𝑇]. (30) Hence, 𝑉(𝑡)<𝑥(𝑡)<𝑥^∗(𝑡)<+𝜀𝜃𝑒^{−𝑏𝑙𝑇}(1−𝜃)1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇}+𝜀₀=Δ₂. (31) From (27), (31), and the third equation in (4), for 𝑡>𝑛₂𝑇+𝜏 we have 𝑑𝐼(𝑡)𝑑𝑡≤𝛽𝑒^−𝑏𝜏Δ^𝑝₁+𝛿Δ^𝑞₂𝐼(𝑡−𝜏)−𝛾+𝑏+𝛼𝐼(𝑡). (32) Consider the comparison equation: 𝑑𝑦(𝑡)𝑑𝑡≤𝛽𝑒^−𝑏𝜏Δ^𝑝₁+𝛿Δ^𝑞₂𝑦(𝑡−𝜏)−𝛾+𝑏+𝛼𝑦(𝑡). (33) From (19), we have 𝛽𝑒^−𝑏𝜏Δ^𝑝₁+𝛿Δ^𝑞₂−𝛾+𝑏+𝛼<0. (34) According to Lemma 2, we then obtain lim_𝑡→∞𝑦(𝑡)=0. (35) Notice the fact that 𝐼(𝑠)=𝑦(𝑠)=𝜙₃(𝑠)>0 for all 𝑠∈[−𝜏,0] and 𝐼(𝑡)≥0, and the comparison theorem implies 𝐼(𝑡)→0 as 𝑡→∞. Without loss of generality, we may assume that 0<𝐼(𝑡)<𝜀₁ for all 𝑡≥0. By using the first and second equations in (4), we reach 𝑑𝑆(𝑡)(𝑡)𝑑𝑡≥−𝑏𝑆−𝛽𝜀₁𝑆^𝑝(𝑡),𝑑𝑉(𝑡)𝑑𝑡≥−𝛿𝛽𝜀₁𝑉^𝑞(𝑡)−𝛾₁𝑉(𝑡)(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)−𝑏𝑉𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝜇,Δ𝑉=0,𝑡=𝑛𝑇,𝑛∈𝑁. (3.2) For 1<𝑞<𝑝, we have 𝑑𝑆(𝑡)(𝑡)𝑑𝑡≥−𝑏𝑆−𝛽𝜀₁𝑆(𝑡),𝑑𝑉(𝑡)𝑑𝑡≥−𝛿𝛽𝜀₁𝑉(𝑡)−𝛾₁𝑉(𝑡)(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)−𝑏𝑉𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝜇,Δ𝑉=0,𝑡=𝑛𝑇,𝑛∈𝑁 (37) considering the following system: 𝑑𝑓(𝑡)(𝑡)(𝑡)𝑑𝑡=−𝑏𝑓−𝛽𝜀₁𝑓(𝑡),𝑑𝑔(𝑡)𝑑𝑡=−𝛿𝛽𝜀₁𝑔(𝑡)−𝛾₁𝑔(𝑡)(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)−𝑏𝑔𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑓=−𝜃𝑓,Δ𝑔=𝜃𝑓𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑓=𝜇,Δ𝑔=0,𝑡=𝑛𝑇,𝑛∈𝑁. (38) We obtain 𝑓(𝑡)=𝑏+𝛽𝜀₁𝜇𝑏+𝛽𝜀₁1−(1−𝜃)𝑒^{−(𝑏+𝛽𝜀1)𝑇}×𝑒^{−(𝑏+𝛽𝜀1)(𝑡−(𝑛−1)𝑇)},𝑡∈((𝑛−1)𝑇,(𝑛+𝑙−1)𝑇],𝑏+𝛽𝜀₁𝜇(1−𝜃)𝑒^{−(𝑏+𝛽𝜀1)𝑙𝑇}𝑏+𝛽𝜀₁1−(1−𝜃)𝑒^{−(𝑏+𝛽𝜀1)𝑇}×𝑒^{−(𝑏+𝛽𝜀1)(𝑡−(𝑛+𝑙−1)𝑇)},)],𝑔(𝑡)=𝜃𝑡∈((𝑛+𝑙−1𝑇,𝑛𝑇𝑏+𝛽𝜀₁𝜇𝑒^{−(𝑏+𝛽𝜀1)𝑙𝑇}−𝛽𝜀₁𝜃1−𝑒^{−(𝑏+𝛽𝜀1)𝑇}×𝑒^{−(𝛾1+𝑏+𝛿𝛽𝜀1)(𝑡−(𝑛+𝑙−1)𝑇)}×𝑏+𝛽𝜀₁1−(1−𝜃)𝑒^{−(𝑏+𝛽𝜀1)𝑇}×1−𝑒^{−(𝛾1+𝑏)𝑇−1},𝑡∈((𝑛+𝑙−1)𝑇,(𝑛+𝑙)𝑇]. (39)
Now by using comparison theorem of impulsive equations, for any 𝜀₂>0 there exists a 𝑇₁>0 such that 𝑆(𝑡)>𝑓(𝑡)−𝜀₂,𝑉(𝑡)>𝑔(𝑡)−𝜀₂, (40) for 𝑡>𝑇₁. On the other side, from the first and second equations of (4), we have 𝑑𝑆(𝑡)(𝑡)𝑑𝑡≤−𝑏𝑆+𝛾𝜀₁𝑒^−𝑏𝜔,𝑑𝑉(𝑡)𝑑𝑡≤−𝛾₁𝑉(𝑡)(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)−𝑏𝑉𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝑏,Δ𝑉=0,𝑡=𝑛𝑇,𝑛∈𝑁. (41) Then we have 𝑆(𝑡)≤ℎ(𝑡), 𝑉(𝑡)≤𝑔(𝑡) and ℎ(𝑡)→𝑆^∗(𝑡), 𝑔(𝑡)→𝑉^∗(𝑡), as 𝜀₁→0, where (ℎ(𝑡),𝑔(𝑡)) is a unique positive periodic solution of 𝑑ℎ(𝑡)(𝑡)𝑑𝑡=−𝑏ℎ+𝛾𝜀₁𝑒^−𝑏𝜔,𝑑𝑔(𝑡)𝑑𝑡=−𝛾₁𝑔(𝑡)(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)−𝑏𝑔𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δℎ=−𝜃ℎ,Δ𝑔=𝜃ℎ𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δℎ=𝜇,Δ𝑔=0,𝑡=𝑛𝑇,𝑛∈𝑁, (42) from which we have that, for 𝑛𝑇<𝑡≤(𝑛+1)𝑇, ℎ(𝑡)=𝛾𝜀₁𝑒^−𝑏𝜔𝜃𝑒^{−𝑏(1−𝑙)𝑇}+𝑏𝜇𝑏1−(1−𝜃)𝑒^−𝑏𝑇×𝑒^{−𝑏(𝑡−(𝑛−1)𝑇)}−𝛾𝜀₁𝑒^−𝑏𝜔𝑏,()()()𝑒𝑛−1𝑇<𝑡≤𝑛+𝑙−1𝑇,𝑏𝜇1−𝜃^{−𝑏𝑙𝑇}+𝛾𝜀₁𝑒^−𝑏𝜔𝜃𝑏1−(1−𝜃)𝑒^−𝑏𝑇×𝑒^{−𝑏(𝑡−(𝑛+𝑙−1)𝑇)}−𝛾𝜀₁𝑒^−𝑏𝜔𝑏,()𝑔(𝑡)=𝑛+𝑙−1𝑇<𝑡≤𝑛𝑇,𝜃𝑏𝜇𝑒^{−𝑏𝑙𝑇}−𝛾𝜀₁𝑒^−𝑏𝜔𝜃1−𝑒^−𝑏𝑇𝑒^{−(𝛾1+𝑏)(𝑡−(𝑛+𝑙−1)𝑇)}𝑏1−(1−𝜃)𝑒^−𝑏𝑇1−𝑒^{−(𝛾1+𝑏)𝑇},(𝑛+𝑙−1)𝑇<𝑡≤(𝑛+𝑙)𝑇. (43) Applying the comparison theorem again, for any 𝜀₂>0, there exists a 𝑇₂>0 such that 𝑆(𝑡)<ℎ(𝑡)−𝜀₂,𝑉(𝑡)<𝑔(𝑡)−𝜀₂, (44) for 𝑡>𝑇₂. Let 𝜀₁→0, and then from (40) and (44) we have 𝑆^∗(𝑡)−𝜀₂<𝑆(𝑡)<𝑆^∗(𝑡)−𝜀₂,𝑉^∗(𝑡)−𝜀₂<𝑉(𝑡)<𝑉^∗(𝑡)−𝜀₂, (45) for 𝑡 large enough, which implies 𝑆(𝑡)→𝑆^∗(𝑡), 𝑉(𝑡)→𝑉^∗(𝑡) as 𝑡→∞. This completes the proof.

Corollary 6. If 𝜏>𝜏^∗ or 𝜇<𝜇^∗, then the infection-free periodic solution (𝑆^∗(𝑡),𝑉^∗(𝑡),0) is globally attractive, where the critical values are given below: 𝜏^∗=1𝑏𝛽𝐴ln^𝑝1+𝛿𝐴^𝑞₂,𝜇𝛾+𝑏+𝛼_∗=1−𝑒^−𝑏𝑇×^𝑝𝛾+𝑏+𝛼𝛽𝑒^𝑏𝜏−𝛿𝐴^𝑞₂−𝛾𝑒^−𝑏𝜔𝑏−𝜇𝑒^𝑏𝑇𝑒^𝑏𝑇.−1 (46)

4. Permanence

In this section, we discuss the permanence of the infectious population. First, we introduce the following definition.

Definition 7. System (4) is said to be permanent if there exist positive constants 𝑚_𝑖, 𝑀_𝑖, 𝑖=1,2,3 (independent of initial value), and a finite time 𝑇₀, which may depend on the initial condition, such that every positive solution (𝑆(𝑡),𝑉(𝑡),𝐼(𝑡)) with initial condition (5) satisfies 𝑚₁≤𝑆(𝑡)≤𝑀₁, 𝑚₂≤𝐼(𝑡)≤𝑀₂, 𝑚₃≤𝑉(𝑡)≤𝑀₃ for all 𝑡>𝑇₀.

Let 𝑆^∗=^𝑝𝛾+𝑏+𝛼𝛽𝑒^−𝑏𝜏,𝑉^∗=^𝑞𝛾₁+𝑏𝛿𝛽𝑒^−𝑏𝜏,𝑚^∗=(1/𝑇)lnℜ₁𝑒^𝑏𝑇−1+𝜃+1−𝜃+𝑒^𝑏𝑇/2−𝑏𝛽,ℜ₁=^𝑝𝛽𝑒^−𝑏𝜏𝜇()𝑒𝛾+𝑏+𝛼1−𝜃^−𝑏𝑇1−(1−𝜃)𝑒^−𝑏𝑇,ℜ₂=^𝑞𝛽𝑒^−𝑏𝜏𝛾₁+𝑏𝜇𝜃𝑒^{−(𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^−𝑏𝑇,ℛ₂=minℜ₁,ℜ₂. (47) Then we have our main result of this section.

Theorem 8. Let 1≤𝑞≤𝑝, if ℛ₂>1, and then there exists a positive constant 𝜂 small enough such that 𝐼(𝑡)≥min𝜂𝑚^∗2,𝜂𝑚^∗𝑒^{−(𝛾+𝑏+𝛼)𝜛}=𝑚₁ (48) with 𝑡 large enough.

Proof. As before, we suppose that 𝑋(𝑡)=(𝑆(𝑡),𝑉(𝑡),𝐼(𝑡)) is a positive solution of system (4) with initial condition (5). Then for 𝑡≥0, we construct a function as follows: 𝑈(𝑡)=𝐼(𝑡)+𝑉(𝑡)+𝛽𝑒^−𝑏𝜏𝑆^∗𝑝×^𝑡_𝑡−𝜏𝐼𝜚𝑑𝜚+𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞𝑡_𝑡−𝜏𝑉𝜚𝑑𝜚. (49) And then differentiating 𝑈(𝑡) along the trajectory of (4) yields 𝑈̇(𝑡)=𝐼̇(𝑡)+𝑉̇(𝑡)+𝛽𝑒^−𝑏𝜏𝑆^∗𝑝𝐼(𝑡)−𝛽𝑒^−𝑏𝜏𝑆^∗𝑝𝐼(𝑡−𝜏)+𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞𝐼(𝑡)−𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞𝐼(𝑡−𝜏)=𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)−𝑆^∗𝑝𝐼(𝑡−𝜏)+𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)−𝑉^∗𝑞𝐼()+𝑡−𝜏𝛽𝑒^−𝑏𝜏𝑆^∗𝑝−𝐼(𝑡)+𝛾+𝑏+𝛼𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞−𝛾₁+𝑏𝐼(𝑡)=𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)−𝑆^∗𝑝𝐼(𝑡−𝜏)+𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)−𝑉^∗𝑞𝐼(𝑡−𝜏)=𝛽𝑒^−𝑏𝜏𝑆^𝑝−1(𝑡)+𝑆^𝑝−2(𝑡)𝑆^∗(𝑡)𝑆+⋯+𝑆^∗𝑝−2+𝑆^∗𝑝−1×𝑆(𝑡)−𝑆^∗𝐼(𝑡−𝜏)+𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞−1(𝑡)+𝑉^𝑞−2(𝑡)𝑉^∗(𝑡)𝑉+⋯+𝑉^∗𝑞−2+𝑉^∗𝑞−1×𝑉(𝑡)−𝑉^∗𝐼(𝑡−𝜏) (50) for 𝑡≥0. Let 𝑚^∗=(1/𝑇)lnℜ₁𝑒^𝑏𝑇−1+𝜃+1−𝜃+𝑒^𝑏𝑇/2−𝑏𝛽,𝑆^∗=^𝑝𝛾+𝑏+𝛼𝛽𝑒^−𝑏𝜏. (51) Since ℛ₂>1, we get ℜ₁>1, ℜ₂>1. Then we have 𝑚^∗>0. And from ℜ₁>1, we can get _𝑝𝛽𝑒^−𝑏𝜏𝜇()𝑒𝛾+𝑏+𝛼1−𝜃^−𝑏𝑇1−(1−𝜃)𝑒^−𝑏𝑇>1. (52) Thus, we have 𝜇(1−𝜃)𝑒^−𝑏𝑇1−(1−𝜃)𝑒^−𝑏𝑇>^𝑝𝛾+𝑏+𝛼𝛽𝑒^−𝑏𝜏=𝑆^∗. (53) Form ℜ₂>1, we have _𝑞𝛽𝑒^−𝑏𝜏𝛾₁+𝑏𝜇𝜃𝑒^{−(𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^−𝑏𝑇>1; (54) that is, 𝜇𝜃𝑒^{−(𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^−𝑏𝑇>^𝑞𝛾₁+𝑏𝛽𝑒^−𝑏𝜏=𝑉^∗. (55) We can take 𝜂 small enough such that 𝜇(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}>𝑆^∗,𝜇𝜃𝑒^{−(𝛽(𝛿+𝑙)𝜂𝑚∗+𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛿𝛽𝜂𝑚∗+𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}>𝑉^∗. (56) Thus we can choose 𝜀₁,𝜀₂>0 to be small enough such that 𝑆^∗<𝜇(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}−𝜀₁≡𝑆_Δ,𝑉^∗<𝜇𝜃𝑒^{−(𝛽(𝛿+𝑙)𝜂𝑚∗+𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛿𝛽𝜂𝑚∗+𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}−𝜀₂≡𝑉_Δ. (57)
Then we claim that there exists an 𝑚₂>0 such that 𝐼(𝑡)>𝑚₂ for 𝑡 is large enough. We next prove this claim in two steps.
Step I. For any positive constant 𝑡₀, that 𝐼(𝑡)≤𝜂𝑚^∗ for all 𝑡≥𝑡₀ is not true.
Otherwise, there is a positive constant 𝑡₀, such that 𝐼(𝑡)≤𝜂𝑚^∗ for all 𝑡≥𝑡₀. First, if 𝐼(𝑡)<𝜂𝑚^∗ for all 𝑡≥𝑡₀, it follows from the first, fourth, and fifth equations of (4) that, for 𝑡≥𝑡₀, 𝑑𝑆(𝑡)𝑑𝑡≥−𝛽𝜂𝑚^∗𝑆(𝑡)()(𝑡)(𝑡)()(𝑡)+𝑏,𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,Δ𝑆=−𝜃𝑆,𝑡=𝑛+𝑙−1𝑇,Δ𝑆=𝜇,𝑡=𝑛𝑇. (58) By Lemma 1, there exists 𝑇₁>𝑡₀+𝜏 so that for 𝑡>𝑇₁𝑆(𝑡)>𝜇(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}−𝜀≡𝑆_Δ. (59) Similarly, from the second and the fourth equations of (4), we have 𝑑𝑉(𝑡)𝑑𝑡≥−𝛿𝛽𝜂𝑚^∗+𝛾₁𝑉(𝑡)()(𝑡)(𝑡)()+𝑏,𝑡≠𝑛+𝑙−1𝑇,Δ𝑉=𝜃𝑆,𝑡=𝑛+𝑙−1𝑇, (60) and for 𝑡>𝑇₁, 𝑉(𝑡)≥𝜇𝜃𝑒^{−(𝛽(𝛿+𝑙)𝜂𝑚∗+𝛾1+𝑏+𝑏𝑙)𝑇}1−𝑒^{−(𝛿𝛽𝜂𝑚∗+𝛾1+𝑏)𝑇}1−(1−𝜃)𝑒^{−(𝛽𝜂𝑚∗+𝑏)𝑇}−𝜀≡𝑉_Δ. (61) Then, by (50), for 𝑡≥𝑇₁, 𝑈̇(𝑡)=𝛽𝑒^−𝑏𝜏𝑆^𝑝−1(𝑡)+𝑆^𝑝−2(𝑡)𝑆^∗(𝑡)𝑆+⋯+𝑆^∗𝑝−2+𝑆^∗𝑝−1×𝑆(𝑡)−𝑆^∗𝐼(𝑡−𝜏)+𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞−1(𝑡)+𝑉^𝑞−2(𝑡)𝑉^∗(𝑡𝑉+⋯+𝑉)^∗𝑞−2+𝑉^∗𝑞−1×𝑉(𝑡)−𝑉^∗𝐼(𝑡−𝜏)>𝑝𝛽𝑒^−𝑏𝜏𝑆^∗𝑝−1𝑆_Δ−𝑆^∗𝐼(𝑡−𝜏)+𝑞𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞−1𝑉_Δ−𝑉^∗𝐼(𝑡−𝜏). (62) Let 𝐼_𝐿=min_{𝑡∈[𝑇1,𝑇1+𝜏]}𝐼(𝑡). (63) We can prove that 𝐼(𝑡)≥𝐼_𝐿 for all 𝑡≥𝑇₁. Otherwise, there exists a nonnegative constant 𝑇₂ such that 𝐼(𝑡)≥𝐼_𝐿 for 𝑡∈[𝑇₁,𝑇₁+𝜏+𝑇₂],𝐼(𝑇₁+𝜏+𝑇₂)=𝐼_𝐿, and 𝐼(̇𝑇₁+𝜏+𝑇₂)≤0. Then from the second equation of (4) and (37), we easily see that 𝐼̇𝑇₁+𝜏+𝑇₂≥𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)+𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)−𝛾+𝑏+𝛼𝐼_𝐿=𝛾+𝑏+𝛼𝛽𝑒^−𝑏𝜏𝑆^𝑝(𝑡)+𝛾+𝑏+𝛼𝛿𝛽𝑒^−𝑏𝜏𝑉^𝑞(𝑡)𝐼𝛾+𝑏+𝛼−1_𝐿>𝑆𝛾+𝑏+𝛼^Δ𝑆^∗𝑝+𝛾₁+𝑏𝑉𝛾+𝑏+𝛼𝑉^Δ∗𝑞−1𝐼_𝐿>0, (64) which is a contradiction. Hence 𝐼(𝑡)≥𝐼_𝐿>0 for all 𝑡≥𝑇₁. Equation (62) implies 𝑑𝑈(𝑡)𝑑𝑡>𝑝𝛽𝑒^−𝑏𝜏𝑆^∗𝑝−1𝑆_Δ−𝑆^∗𝐼(𝑡−𝜏)+𝑞𝛿𝛽𝑒^−𝑏𝜏𝑉^∗𝑞−1𝑉_Δ−𝑉^∗𝐼(𝑡−𝜏)>0. (65) It then follows that 𝑈(𝑡)→+∞ as 𝑡→+∞. This is a contradiction to 𝑈(𝑡)≤(𝛼+𝛾+𝛾₁+2𝑏)𝜏+2. Therefore, for any positive constant 𝑡₀, the inequality 𝐼(𝑡)<𝜂𝑚^∗ cannot hold for all 𝑡≥𝑡₀.
Step II. From Step I, we only need to consider the followng: (i) 𝐼(𝑡)>𝜂𝑚^∗ for all 𝑡 large enough and (ii) 𝐼(𝑡) oscillates about 𝜂𝑚^∗ for all large 𝑡. However, Case (i) is obvious in the result of this theorem, so we only need to consider Case (ii), in which we will show that 𝐼(𝑡)≥𝑚₁ for all large 𝑡 where 𝑚₁=min𝜂𝑚^∗2,𝜂𝑚^∗𝑒^{−(𝛾+𝑏+𝛼)𝜛}. (66) First, we notice there exist two positive constants 𝑡, 𝜑 such that 𝐼𝑡=𝐼𝑡+𝜑=𝐼^∗,𝐼(𝑡)<𝜂𝑚^∗,for𝑡<𝑡<𝑡+𝜑. (67) Second, because 𝐼(𝑡) is bounded continuous function and 𝐼(𝑡) has no pulse, we can get that 𝐼(𝑡) is uniformly continuous. Therefore there exists a constant 𝑇₃ (with 0<𝑇₃<𝜛 and 𝑇₃ is independent of the choice of 𝑡) such that 𝐼(𝑡)>𝜂𝑚^∗/2 for all 𝑡≤𝑡≤𝑡+𝑇₃.
If 𝜑≤𝑇₃, our aim is obtained.
If 𝑇₃<𝜑≤𝜛, from the second equation of (4) we have that 𝐼(̇𝑡)≥−(𝛾+𝑏+𝛼)𝐼(𝑡) for 𝑡<𝑡≤𝑡+𝜑. Then we have 𝐼(𝑡)≥𝜂𝑚^∗𝑒^{−(𝛾+𝑏+𝛼)𝜛} for 𝑡<𝑡≤𝑡+𝜑≤𝑡+𝜛 since 𝐼(𝑡)=𝜂𝑚^∗. It is clear that 𝐼(𝑡)≥𝑚₁ for 𝑡<𝑡≤𝑡+𝜑.
If 𝜑≥𝜛, then we have 𝐼(𝑡)≥𝑚₂ for 𝑡<𝑡≤𝑡+𝜛. We then can easily prove 𝐼(𝑡)≥𝑚₁ for 𝑡+𝜛≤𝑡≤𝑡+𝜑. Since the interval [𝑡,𝑡+𝜑] is arbitrarily chosen, we know that 𝐼(𝑡)≥𝑚₁ holds for 𝑡 large enough. Finally, noticing the choice of 𝑚₁ is independent of the positive solution of (4), we completed our proof.

Theorem 9. Let 1≤𝑞≤𝑝, if ℛ₂>1, and then system (4) is permanent.

Proof. Suppose that 𝑋(𝑡)=(𝑆(𝑡),𝑉(𝑡),𝐼(𝑡)) is a positive solution of system (4) with initial conditions (5). Then from system (4), we have 𝑑𝑆(𝑡)𝑆(𝑡),(𝑡)𝑑𝑡≥−𝑏+𝛽𝑑𝑉𝑑𝑡≥−𝛿𝛽+𝛾₁𝑉(𝑡),()(𝑡)(𝑡)(𝑡)(𝑡),()(𝑡)(𝑡)+𝑏𝑡≠𝑛+𝑙−1𝑇,𝑡≠𝑛𝑇,𝑛∈𝑁,Δ𝑆=−𝜃𝑆,Δ𝑉=𝜃𝑆𝑡=𝑛+𝑙−1𝑇,𝑛∈𝑁,Δ𝑆=𝜇,Δ𝑉=0,𝑡=𝑛𝑇,𝑛∈𝑁. (68) As what we did in the proof of Theorem 5, we can prove that there exist 𝑡 large enough and 𝜀>0 small enough such that 𝑆(𝑡)≥𝜇(1−𝜃)𝑒^{−(𝑏+𝛽)𝑇}1−(1−𝜃)𝑒^{−(𝑏+𝛽)𝑇}−𝜀=𝑚₃,𝑉(𝑡)≥𝜇𝜃𝑒^{−(𝑏+𝛾1+𝛿𝛽+(𝑏+𝛽)𝑙)𝑇}1−(1−𝜃)𝑒^{−(𝑏+𝛽)𝑇}1−𝑒^{−(𝑏+𝛾1+𝛿𝛽)𝑇}−𝜀₂=𝑚₄. (69) Then for 𝒟={(𝑆,𝑉,𝐼)∈𝑅³₊∣𝑆(𝑡)+𝑉(𝑡)+𝐼(𝑡)≤1}, by Theorem 8, we have 𝑚₃≤𝑆(𝑡)≤1,𝑚₁≤𝐼(𝑡)≤1,𝑚₄≤𝑉(𝑡)≤1 (70) for 𝑡 large enough. Thus the system (4) is uniformly permanent.

5. Numerical Simulations and Discussions

Next, we carry out numerical simulations to illustrate the theoretical results obtained in the previous sections. We first set the parameters as follows: 𝑏=0.2, 𝛽=0.5, 𝛼=0.05, 𝛾=0.04, 𝛿=0.02, 𝛾₁=0.06, 𝑝=1.5, 𝑞=1.25, 𝑇=1.5, 𝜔=1, 𝜏=1, 𝜃=0.4, and 𝜇=1.4. Straightforward calculation shows ℛ₂=2.6155>1. Then by Theorem 8, the disease will be permanent (please see Figures 1(a), 1(b), 1(c), and 1(d)). In order to show the effect of 𝜏, we decrease 𝜏 to 4, and other parameters are the same with those in Figure 1, and the infection-free periodic solution of system (4) is globally attractive. This phenomenon is also seen from our theoretical analysis as in this case ℛ₁=0.0339<1 and then according to Theorem 5, the disease will be eradicated; please see Figure 2(a).

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease will be permanent

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 1 

The results of numerical simulation on the threshold values ℛ2=2.6155>1, where 𝑝=1.5, 𝑞=1.25.

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease dies out

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 2 

The results of numerical simulation on the threshold values ℛ1=0.0339<1, where 𝑝=1.5, 𝑞=1.25.

If we keep 𝜏=𝜔=1 and 𝜇=1, as the same with those in Figure 1, but increase vaccination proportion of susceptible persons 𝜃 to 0.9, then the disease will be eradicated; see Figure 3(a). If we keep 𝜏=𝜔=1 and 𝜃=0.4 and decrease 𝜇 to 0.2, then the disease also will be eradicated; see Figure 4(a).

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease dies out

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 3 

The results of numerical simulation on the threshold values ℛ1=0.0449<1, where 𝑝=1.5, 𝑞=1.25.

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease dies out

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 4 

The results of numerical simulation on the threshold values ℛ1=0.0352<1, where 𝑝=1.5, 𝑞=1.25.

And if we keep 𝜏=𝜔=4, 𝜇=1 but decrease 𝜃 to 0.1, then the disease will be permanent; see Figure 5. If we keep 𝜏=𝜔=4 and 𝜃=0.4 and increase 𝜇 to 2, then the disease also will be permanent; see Figure 6. For details please see Table 1.

Table 1 

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease will be permanent

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 5 

The results of numerical simulation on the threshold values ℛ2=1.1036>1, where 𝑝=1.5, 𝑞=1.25.

(a) Time series of 𝑆, 𝑉, and 𝐼. The disease will be permanent

(b) Phase portrait of 𝑆(𝑡), 𝑉(𝑡) of the system  (4)

(c) Phase portrait of 𝑆(𝑡), 𝐼(𝑡) of the system  (4)

(d) Phase portrait of 𝑆(𝑡), 𝑉(𝑡), and 𝐼(𝑡) of the system  (4)

Figure 6 

The results of numerical simulation on the threshold values ℛ2=2.3121>1, where 𝑝=1.5, 𝑞=1.25.

Lastly, we conclude our paper as follows. In this paper, we proposed an SVEIRS model, which is a new epidemic model with periodic pulse vaccination and pulse population input at two different fixed moments. Our primary result is to investigate the effect of impulsive vaccination, pulse population input, and time delays to the dynamics of population model. With the help of comparison theorems, we proved the existence of the “infection-free” periodic solution and obtained the conditions for global attractivity of the “infection-free” periodic solution and the conditions for the permanence of the system. All the theoretical results show that we believe it might be helpful in disease control: people can select appropriate vaccination rate and population input rate according to our theoretical results to control diseases.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. T. Q. Zhang and X. Z. Meng are financially supported by the National Natural Science Foundation of China (no. 11371230), the Shandong Provincial Natural Science Foundation of China (no. ZR2012AM012), and the Project of Shandong Province Higher Educational Science and Technology Program of China (no. J13LI05).