Dynamical Analysis of SIR Epidemic Models with Distributed Delay

Abstract

SIR epidemic models with distributed delay are proposed. Firstly, the dynamical behaviors of the model without vaccination are studied. Using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed. The basic reproduction number 𝑅 is got. In order to study the important role of vaccination to prevent diseases, the model with distributed delay under impulsive vaccination is formulated. And the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model are obtained by using Floquet’s theorem, small-amplitude perturbation skills, and comparison theorem. Lastly, numerical simulation is presented to illustrate our main conclusions that vaccination has significant effects on the dynamical behaviors of the model. The results can provide effective tactic basis for the practical infectious disease prevention.

1. Introduction

Infectious diseases have always been the problem that people have to face. Emerging diseases pose a continual threat to public health such as SARS and avian influenza. It is very necessary to establish and study mathematical models which can reflect the spread of the infectious diseases. A famous model in which the population is partitioned into three classes, the susceptible, infectious and recovered, with sizes denoted by 𝑆, 𝐼, and 𝑅, respectively, that could be used to describe an influenza epidemic was developed early in the 20th century by Kermack and McKendrick [1]. This model known as the susceptible-infectious-recovered (SIR) model is as follows: 𝑆^′(𝑡)(𝑡)𝐼(𝑡)(𝑡),𝐼=Λ−𝛽𝑆−𝜇𝑆_′(𝑡)(𝑡)𝐼(𝑡)(𝑡)(𝑡),𝑅=𝛽𝑆−𝑟𝐼−𝜇𝐼_′(𝑡)=𝑟𝐼(𝑡)−𝜇𝑅(𝑡), (1) where 𝑆(𝑡) denotes the number of members of a population susceptible to the disease, 𝐼(𝑡) denotes the total population of infectives with some virus at time 𝑡,𝑅(𝑡) denotes the number of members who have been removed from the possibility of infection through a temporal immunity. In the model, parameters 𝜇,𝛽,𝑟, and Λ are positive constants, where 𝜇 represents the death rates of susceptibles, infectives, and recovered. 𝛽 is the contact rate and 𝑟 is the recovery rate from the infected compartment. Λ is the recruitment rate of the susceptible population. The SIR infectious disease model is a basic but important biologic model and has been studied by many authors [1–17]. Many diseases have the incubation period, such as rabies; the incubation period in rabies ranges from about two weeks to several months, and rarely even to years [18–21]. Diseases with incubation period always lead to time delay in the epidemic models, so delay differential equation is widely used in the epidemic mathematical model, such as [2, 7, 8, 10, 12–16, 22]. Based on model (1), a new epidemic model with distributed delay is as follows: 𝑆^′(𝑡)=Λ−𝛽𝐼(𝑡)^𝑡_−∞𝐹()𝑆(𝜏)(𝑡),𝐼𝑡−𝜏𝑑𝜏−𝜇𝑆_′(𝑡)=𝛽𝐼(𝑡)^𝑡_−∞𝐹()𝑆(𝜏)(𝑡)(𝑡),𝑅𝑡−𝜏𝑑𝜏−𝑟𝐼−𝜇𝐼_′(𝑡)=𝑟𝐼(𝑡)−𝜇𝑅(𝑡), (2) where the function 𝐹(𝑡)=𝑎𝑒^−𝑎𝑡,𝑎>0,∫₀^+∞𝐹(𝜏)𝑑𝜏=1.

As is known to all, one of strategies to control infectious diseases is vaccination. Then a number of epidemic models in ecology can be formulated as dynamical systems of differential equations with vaccination [23–26]. Based on ODE, systems with sudden perturbations lead to impulsive differential equations. The theory of impulsive differential equations has been studied intensively and systematically in [27–34]. Compared to continuous constant vaccination, pulse vaccination seems more reasonable in the real world. Pulse vaccination, the repeated application of vaccine over a defined age range, is gaining prominence as a strategy for the elimination of childhood viral infections such as measles hepatitis, parotitis, smallpox, and phthisis. Under the pulse vaccination strategy (PVS) [9, 11, 17, 28, 35–41], what we are interested is how large a fraction of the population should we keep vaccinated in order to prevent the agent from establishing; that is, it is very important for us to investigate the conditions under which a given agent can invade a partially vaccinated population. Thus, we also need to consider the following epidemic model with distributed delay and pulse vaccination strategy at fixed moments, which is more realistic as 𝑆^′(𝑡)=Λ−𝛽𝐼(𝑡)^𝑡_−∞𝐹()𝑆(𝜏)(𝑡)𝐼𝑡−𝜏𝑑𝜏−𝜇𝑆,𝑡≠𝑛𝑇,_′(𝑡)=𝛽𝐼(𝑡)^𝑡_−∞𝐹()𝑆(𝜏)(𝑡)(𝑡)𝑅𝑡−𝜏𝑑𝜏−𝑟𝐼−𝜇𝐼,𝑡≠𝑛𝑇,_′(𝑡)(𝑡)(𝑡)𝑆𝑡=𝑟𝐼−𝜇𝑅,𝑡≠𝑛𝑇,₊=()𝑆(𝑡)𝐼𝑡1−𝛿,𝑡=𝑛𝑇,₊(𝑡)𝑅𝑡=𝐼,𝑡=𝑛𝑇,₊=𝑅(𝑡)+𝛿𝑆(𝑡),𝑡=𝑛𝑇, (3) where 𝛿 (with 0<𝛿<1) is the proportion of those vaccinated successfully to all of the susceptibles and 𝑇 is a fixed positive constant and denotes the period of the impulsive effect, 𝑛={1,2,…}.

The organization of this paper is as follows. In Section 2, we will show the boundedness of the SIR system and give the transformation of models and some lemmas. In Section 3, we will analyze the local stability of equilibrium of system (7) and give the basic reproduction number. In Section 4.1, we will prove the existence and globally asymptotical stability of the periodic solution of the “infection-free” model. In Section 4.2, we obtain sufficient condition for the permanence of the epidemic model with pulse vaccination. Finally, we give numerical analysis and biological conclusions to show our main results.

2. Transformation of Models and Prerequisites

For system (2), the total population size 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡) satisfies 𝑁^′(𝑡)=Λ−𝜇𝑁(𝑡) and lim_𝑡→∞𝑁(𝑡)=Λ/𝜇. Hence it is sufficient to consider system (2) with respect to 𝐷={(𝑆,𝐼,𝑅)∈𝑅³₊∣𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡)≤(Λ/𝜇)+𝜖}. Note that the variable 𝑅 does not appear in the first and second equations of system (2); hence, we only need to consider the subsystem of (2) as follows: 𝑆^′(𝑡)=Λ−𝛽𝐼(𝑡)^𝑡_−∞𝐹()𝑆(𝜏)(𝑡),𝐼𝑡−𝜏𝑑𝜏−𝜇𝑆_′(𝑡)=𝛽𝐼(𝑡)^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏−𝑟𝐼(𝑡)−𝜇𝐼(𝑡). (4) In order to study the system, we can use the chain transformation 𝑍(𝑡)=∫^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏. Since ^𝑡_−∞𝐹(𝑡−𝜏)𝑑𝜏=lim_{𝐴→−∞}^𝑡_𝐴𝑎𝑒^{−𝑎(𝑡−𝜏)}𝑑𝜏=1 (5) and ∫^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏 is convergent, then Δ𝑍(𝑡)=^𝑡+_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏−^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏=0. (6) Hence, system (4) becomes 𝑆^′(𝑡)(𝑡)𝑍(𝑡)(𝑡),𝐼=Λ−𝛽𝐼−𝜇𝑆_′(𝑡)(𝑡)𝑍(𝑡)(𝑡)(𝑡),𝑍=𝛽𝐼−𝑟𝐼−𝜇𝐼_′(𝑡)=𝑎(𝑆(𝑡)−𝑍(𝑡)). (7) We understand the relationship between the two systems as follows. If (𝑆,𝐼)∶[0,+∞)→𝑅² is the solution of system (4) corresponding to continuous and bounded initial function 𝑆(0)=𝑆₀,𝐼(0)=𝐼₀, then (𝑆,𝐼,𝑍)∶[0,+∞)→𝑅³ is a solution of system (7) with 𝑆(0)=𝑆₀,𝐼(0)=𝐼₀,𝑍(0)=𝑍₀, and 𝑍(0)=∫⁰_−∞𝐹(−𝜏)𝑆(𝜏)𝑑𝜏=∫⁰_−∞𝑎𝑒^𝑎𝜏𝑆(𝜏)𝑑𝜏. Conversely, if (𝑆,𝐼,𝑍) is any solution of system (7) defined on the entire real line and bounded on (−∞,0], then 𝑍 is given by 𝑍(𝑡)=∫^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏, and so (𝑆,𝐼) satisfies system (4).

System (7) will be analyzed with the following initial conditions 𝑆(0)=𝑆₀>0, 𝐼(0)=𝐼₀>0, 𝑍(0)=𝑍₀>0.

By the mean value theorem of integrals, there exists 𝜍∈(−∞,𝑡), such that 𝑍(𝑡)=^𝑡_−∞𝐹(𝑡−𝜏)𝑆(𝜏)𝑑𝜏=𝑆(𝜍)^𝑡_−∞𝐹(𝑡−𝜏)𝑑𝜏. (8) We know that 𝑆(𝑡)<(Λ/𝜇)+𝜀 and ∫^𝑡_−∞𝐹(𝑡−𝜏)𝑑𝜏=1; hence we get 𝑍(𝑡)<(Λ/𝜇)+𝜀.

By the same transformation, from system (3), we have 𝑆^′(𝑡)(𝑡)𝑍(𝑡)(𝑡)𝐼=Λ−𝛽𝐼−𝜇𝑆,𝑡≠𝑛𝑇,_′(𝑡)(𝑡)𝑍(𝑡)(𝑡)(𝑡)𝑍=𝛽𝐼−𝑟𝐼−𝜇𝐼,𝑡≠𝑛𝑇,_′(𝑡)(𝑆(𝑡)(𝑡𝑆𝑡=𝑎−𝑍)),𝑡≠𝑛𝑇,₊=()𝑆(𝑡)𝐼𝑡1−𝛿,𝑡=𝑛𝑇,₊(𝑡)𝑍𝑡=𝐼,𝑡=𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇. (9)

Motivated by the application of systems (9) to population dynamics (refer to [27]), we assume that solutions of systems (9) satisfy the initial conditions 𝜙∈𝐶⁺_ℎ, and 𝜙(0)≥0 lie in 𝐶⁺_ℎ={𝜙=(𝜙₁(𝑠),𝜙₂(𝑠),𝜙₃(𝑠))∈𝐶_ℎ∶𝜙_𝑖(0)≥0(𝑖=1,2,3)}, where 𝜙_𝑖(𝑠) is positive, bounded, and continuous function for 𝑠∈[0,+∞).

We would like to have the following definitions first.

Definition 1. Let 𝑅⁺=[0,∞). The map 𝑉∶𝑅₊×𝑅³₊→𝑅₊ is said to belong to class 𝑉₀ if (i)𝑉is continuous in (𝑘𝑇,(𝑘+1)𝑇]×𝑅³₊,𝑘∈𝑁, and for each 𝑥∈𝑅³₊, lim_{(𝑡,𝑧)→(𝑘𝑇}⁺_,𝑥)𝑉(𝑡,𝑧)=𝑉(𝑘𝑇⁺,𝑥) exist, (ii)𝑉 is locally Lipschitzian in 𝑥.

Also we have the following lemmas.

Lemma 2 (see [31]). Let 𝑉∶𝑅₊×𝑅³₊→𝑅₊, and 𝑉∈𝑉₀. Assume that 𝐷⁺𝑉((𝑡≤(≥)𝑔(((𝑡𝑉𝑡𝑡,𝑧))𝑡,𝑉𝑡,𝑧))),𝑡≠𝑛𝑇,𝑡,𝑧⁺≤(≥)Ψ_𝑛(𝑉(𝑡,𝑧(𝑡))),𝑡=𝑛𝑇, (10) where 𝑔∶𝑅₊×𝑅₊→𝑅 is continuous in (𝑛𝑇,(𝑛+1)𝑇]×𝑅₊ and for each 𝑥∈𝑅₊, 𝑛∈𝑁, lim_{(𝑡,𝑦)→((𝑛𝑇)}⁺_,𝑥)𝑔(𝑡,𝑦)=𝑔((𝑛𝑇)⁺,𝑥) exist; Ψ_𝑛∶𝑅₊→𝑅₊ is nondecreasing. Let 𝑟(𝑡)=𝑟(𝑡,0,𝑢₀) be the maximal (minimal) solution of the scalar impulsive differential equation 𝑢^′()𝑢𝑡=𝑔𝑡,𝑢,𝑡≠𝑛𝑇,₊=Ψ_𝑛(𝑢(𝑡𝑢0)),𝑡=𝑛𝑇,₊=𝑢₀, (11) existing on [0,∞). Then 𝑉(0⁺,𝑧₀)≤(≥)𝑢₀ implies that 𝑉(𝑡,𝑧(𝑡))≤(≥)𝑟(𝑡),𝑡≥0, where 𝑧(𝑡)=𝑧(𝑡,0,𝑧₀) is any solution of (9) existing on [0,∞).

Lemma 3. Consider the following system: 𝑥^′(𝑡)(𝑡)𝑦=Λ−𝜇𝑥,𝑡≠𝑛𝑇,_′(𝑡)𝑥(𝑡)(𝑡)𝑥𝑡=𝑎−𝑦,𝑡≠𝑛𝑇,₊=()𝑥(𝑡)𝑦𝑡1−𝛿,𝑡=𝑛𝑇,₊=𝑦(𝑡),𝑡=𝑛𝑇. (12) Then system (12) has a unique positive 𝑇-periodic solution 𝑥^∗(𝑡),𝑦^∗(𝑡) as 𝑥^∗(𝑡)=Λ𝜇1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇,𝑦^∗(𝑡)=Λ𝜇𝑒1+𝑎𝛿^{−𝑎(𝑡−𝑛𝑇)}1−𝑒^−𝜇𝑇−𝑒^{−𝜇(𝑡−𝑛𝑇)}1−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇,𝑥^∗0⁺=Λ𝜇𝛿1−1−(1−𝛿)𝑒^−𝜇𝑇,𝑦^∗0⁺=Λ𝜇𝑒1−𝑎𝛿^−𝜇𝑇−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇, (13) and for each solution, 𝑥(𝑡)→𝑥^∗(𝑡),𝑦(𝑡)→𝑥^∗(𝑡) as 𝑡→∞.

Proof. Solving the first equation of system (12), we have 𝑥(𝑡)=𝑥𝑛𝑇⁺𝑒^{−𝜇(𝑡−𝑛𝑇)}+Λ𝜇1−𝑒^{−𝜇(𝑡−𝑛𝑇)},𝑛𝑇<𝑡≤(𝑛+1)𝑇. (14) Substituting 𝑥(𝑡) into the second equation of (12), we integrate both sides in interval (𝑛𝑇,(𝑛+1)𝑇], and we get 𝑦(𝑡)=𝑦𝑛𝑇⁺𝑒^{−𝑎(𝑡−𝑛𝑇)}+𝑎𝑥𝑎−𝜇𝑛𝑇⁺−Λ𝜇𝑒^{−𝜇(𝑡−𝑛𝑇)}−𝑒^{−𝑎(𝑡−𝑛𝑇)}+Λ𝜇1−𝑒^{−𝑎(𝑡−𝑛𝑇)}. (15) By using stroboscopic map of difference equation, we have 𝑥(𝑛+1)𝑇⁺=()𝑥)𝑇)=()𝑥1−𝛿((𝑛+11−𝛿𝑛𝑇⁺𝑒^−𝜇𝑇+Λ𝜇(1−𝛿)1−𝑒^−𝜇𝑇,𝑦(𝑛+1)𝑇⁺=𝑦((𝑛+1)𝑇)=𝑦𝑛𝑇⁺𝑒^−𝑎𝑇+𝑎𝑥𝑎−𝜇𝑛𝑇⁺−Λ𝜇×𝑒^−𝜇𝑇−𝑒^−𝑎𝑇+Λ𝜇1−𝑒^−𝑎𝑇. (16) The fixed point of the above mapping is 𝑥^∗=Λ𝜇𝛿1−1−(1−𝛿)𝑒^−𝜇𝑇,𝑦^∗=Λ𝜇𝑒1−𝑎𝛿^−𝜇𝑇−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇. (17) Therefore, we can get the following 𝑇-periodic solution (𝑥^∗(𝑡),𝑦^∗(𝑡)) of system (12): 𝑥^∗(𝑡)=Λ𝜇1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇()𝑦,𝑛𝑇<𝑡≤𝑛+1𝑇,_∗(𝑡)=Λ𝜇1+𝑎𝛿1−𝑒^−𝜇𝑇𝑒^{−𝑎(𝑡−𝑛𝑇)}−1−𝑒^−𝑎𝑇𝑒^{−𝜇(𝑡−𝑛𝑇)}𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇,𝑛𝑇<𝑡≤(𝑛+1)𝑇. (18) Next, we will prove the attractivity of periodic solutions. Let (𝑥(𝑡),𝑦(𝑡)) be an any solution of the system (12); then, for 𝑡∈(𝑛𝑇,(𝑛+1)𝑇], we have 𝑥(𝑡)=𝑥^∗(𝑡)+𝑥𝑛𝑇⁺−𝑥^∗𝑒^{−𝜇(𝑡−𝑛𝑇)},𝑦(𝑡)=𝑦^∗(𝑡)+𝑦𝑛𝑇⁺−𝑦^∗𝑒^{−𝑎(𝑡−𝑛𝑇)}+𝑎𝑥𝑎−𝜇𝑛𝑇⁺−𝑥^∗𝑒^{−𝜇(𝑡−𝑛𝑇)}−𝑒^{−𝑎(𝑡−𝑛𝑇)}. (19) On the one hand, by the recurrence formula, we have 𝑥(𝑛+1)𝑇⁺=(1−𝛿)𝑒^−𝜇𝑇𝑥𝑛𝑇⁺+Λ𝜇(1−𝛿)1−𝑒^−𝜇𝑇. (20) Thus 𝑥𝑛𝑇⁺=(1−𝛿)^𝑛𝑒^{−𝑛𝜇𝑇}𝑥0⁺+1+(1−𝛿)𝑒^−𝜇𝑇+⋯+(1−𝛿)^𝑛−1𝑒^{−(𝑛−1)𝜇𝑇}×Λ𝜇(1−𝛿)1−𝑒^−𝜇𝑇=(1−𝛿)^𝑛𝑒^{−𝑛𝜇𝑇}𝑥0⁺+Λ𝜇(1−𝛿)1−𝑒^−𝜇𝑇1−(1−𝛿)^𝑛𝑒^{−𝑛𝜇𝑇}1−(1−𝛿)𝑒^−𝜇𝑇⟶𝑥^∗(𝑛⟶∞), (21) and then we have 𝑥(𝑡)→𝑥^∗(𝑡)(𝑛→∞).
On the other hand, 𝑦(𝑛+1)𝑇⁺=𝑦((𝑛+1)𝑇)=𝑦𝑛𝑇⁺𝑒^−𝑎𝑇+𝑎𝑥𝑎−𝜇𝑛𝑇⁺−Λ𝜇×𝑒^−𝜇𝑇−𝑒^−𝑎𝑇+Λ𝜇1−𝑒^−𝑎𝑇, (22) and lim_𝑛→∞𝑥(𝑛𝑇⁺)=𝑥^∗, for 𝑛 large enough we have the following approximate recursive formula: 𝑦(𝑛+1)𝑇⁺=𝑦𝑛𝑇⁺𝑒^−𝑎𝑇+𝑎𝑥𝑎−𝜇^∗−Λ𝜇×𝑒^−𝜇𝑇−𝑒^−𝑎𝑇+Λ𝜇1−𝑒^−𝑎𝑇=𝑦𝑛𝑇⁺𝑒^−𝑎𝑇+𝑎𝛿Λ𝑒^−𝑎𝑇−𝑒^−𝜇𝑇𝜇𝑎−𝜇1−(1−𝛿)𝑒^−𝜇𝑇+Λ𝜇1−𝑒^−𝑎𝑇. (23) Thus, for any 𝑚∈𝑁⁺, we get 𝑦(𝑚+𝑁)𝑇⁺=𝑦𝑁𝑇⁺𝑒^{−𝑚𝑎𝑇}+1+𝑒^−𝑎𝑇+⋯+𝑒^{−(𝑚−1)𝑎𝑇}×𝑎𝛿Λ𝑒^−𝑎𝑇−𝑒^−𝜇𝑇𝜇𝑎−𝜇1−(1−𝛿)𝑒^−𝜇𝑇+Λ𝜇1−𝑒^−𝑎𝑇=𝑦𝑁𝑇⁺𝑒^{−𝑚𝑎𝑇}+1−𝑒^{−𝑚𝑎𝑇}1−𝑒^−𝑎𝑇×𝑎𝛿Λ𝑒^−𝑎𝑇−𝑒^−𝜇𝑇𝜇𝑎−𝜇1−(1−𝛿)𝑒^−𝜇𝑇+Λ𝜇1−𝑒^−𝑎𝑇⟶Λ𝜇𝑒1−𝑎𝛿^−𝜇𝑇−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇(𝑚⟶∞). (24) Then we have lim_𝑛→∞𝑦(𝑛𝑇⁺)=𝑦^∗, thus 𝑦(𝑡)→𝑦^∗(𝑡)(𝑡→∞). The proof is completed.

3. The Stability of Equilibrium of System (7) and the Basic Reproduction Number

In this section, we will consider the local stability of equilibrium of system (7) and give the basic reproduction number. Obviously, the system (7) has a disease-free equilibrium 𝐸₀(Λ/𝜇,0,Λ/𝜇) and an endemic equilibrium 𝐸^∗(𝑆^∗,𝐼^∗,𝑍^∗), where 𝑆^∗=(𝜇+𝑟)/𝛽, 𝐼^∗=(𝛽Λ−𝜇(𝜇+𝑟))/𝛽(𝜇+𝑟), 𝑍^∗=(𝜇+𝑟)/𝛽. Let 𝑅=𝛽Λ/𝜇(𝜇+𝑟); we have the following theorem.

Theorem 4. System (7) always has a disease-free equilibrium 𝐸₀. If and only if 𝑅>1, system (7) has an endemic equilibrium 𝐸^∗.

3.1. Local Stability of Disease-Free Equilibrium 𝐸₀

We calculate the Jacobian matrix of system (7) evaluated at 𝐸₀; one gets the following matrix: 𝐽𝐸₀=Λ−𝜇−𝛽𝜇0Λ0𝛽𝜇−𝜇−𝑟0𝑎0−𝑎. (25) Obviously, 𝐸₀ is locally asymptotically stable if 𝛽(Λ/𝜇)−𝜇−𝑟<0 which implies that 𝑅<1, and unstable if 𝑅>1. Then 𝑅=𝛽Λ/𝜇(𝜇+𝑟) can be used as the basic reproductive number. Thus, we obtain the following result.

Theorem 5. If 𝑅<1, then disease-free equilibrium 𝐸₀ of system (7) is locally asymptotically stable and unstable if 𝑅>1.

3.2. Local Stability of Endemic Equilibrium 𝐸^∗

About the local stability of endemic equilibrium 𝐸^∗, we have the following theorem.

Theorem 6. If 𝑅>1 and 𝑎≥𝑟 or 1<𝑅<(𝑟+𝜇)/(𝑟−𝑎), the equilibrium 𝐸^∗ of system (7) is locally asymptotically stable.

Proof. The Jacobian matrix of system (7) evaluated at 𝐸^∗ is 𝐽𝐸^∗=−𝜇−𝛽𝑍^∗−𝛽𝐼^∗00𝛽𝐼^∗𝑎0−𝑎. (26) Let 𝜆_𝑖(𝑖=1,2,3) be its eigenvalues with Re𝜆₁≤Re𝜆₂≤Re𝜆₃. After a simple calculation, it follows that det𝐽𝐸^∗=−𝑎𝛽²𝐼^∗𝑍^∗<0. (27) For det𝐽(𝐸^∗)=𝜆₁𝜆₂𝜆₃<0, there are two cases as follows: (i)Re𝜆_𝑖<0 for 𝑖=1,2,3;(ii)Re𝜆₁<0≤Re𝜆₂≤Re𝜆₃.
Now we prove that the case (ii) is not true. Note that det𝐽(𝐸^∗)=−𝑎𝛽²𝐼^∗𝑍^∗<0; one gets tr𝐽(𝐸^∗)<0, that is, 𝜆₁+𝜆₂+𝜆₃<0. If the case (ii) is true, we have that Re(𝜆₁+𝜆₂)<0 and Re(𝜆₁+𝜆₃)<0. The second additive compound matrix [6] of 𝐽(𝐸^∗) (see the Appendix) is as follows: 𝐽^[2]𝐸^∗=−𝜇𝛽𝐼^∗𝛽𝐼^∗0−𝜇−𝑎−𝛽𝑍^∗,−𝑎0−𝑎det𝐽^[2]𝐸^∗=−𝑎𝛽𝐼^∗+𝜇𝜇+𝑎−𝛽²𝐼^∗𝑍^∗=−𝑎𝜇𝑅(𝑎−𝑟)+𝜇+𝑟, (28) where 𝑆^∗=(𝜇+𝑟)/𝛽,𝐼^∗=(𝛽Λ−𝜇(𝜇+𝑟))/𝛽(𝜇+𝑟), and 𝑍^∗=(𝜇+𝑟)/𝛽 is used. Notice that 𝑅>1, and two cases will happen as follows: (a)if 𝑎≥𝑟, then det𝐽^[2](𝐸^∗)<0;(b)if 𝑎<𝑟, but 1<𝑅<(𝑟+𝜇)/(𝑟−𝑎), then det𝐽^[2](𝐸^∗)<0.
According to the property of the second additive compound matrix [6], the eigenvalues of 𝐽^[2](𝐸^∗) are 𝜆_𝑖+𝜆_𝑗,1≤𝑖<𝑗≤3. Then, we have 𝜆₁+𝜆₂𝜆₁+𝜆₃𝜆₂+𝜆₃<0. (29) Notice that Re(𝜆₁+𝜆₂)<0 and Re(𝜆₁+𝜆₃)<0, then we get Re(𝜆₂+𝜆₃)<0, which contradicts with case (ii). Therefore, Re𝜆_𝑖<0 for 𝑖=1,2,3. So 𝐸^∗ is locally asymptotically stable for 𝑅>1 and 𝑎≥𝑟 or 1<𝑅<(𝑟+𝜇)/(𝑟−𝑎). This completes the proof.

3.3. Analysis at 𝑅=1

In this section, we consider the stability of system (7) under 𝑅=1 using the center manifold theory, as described in [42, Theorem 4.1]. To apply this method, the following simplification and change of variables are made first. Let 𝑆=𝑥₁,𝐼=𝑥₂,𝑍=𝑥₃, and the system (7) becomes 𝑥̇₁=Λ−𝛽𝑥₂𝑥₃−𝜇𝑥₁=𝑓₁,𝑥̇₂=𝛽𝑥₂𝑥₃−𝑟𝑥₂−𝜇𝑥₂=𝑓₂,𝑥̇₃=𝑎𝑥₁−𝑥₃=𝑓₃, (30) with 𝑅=1 corresponding to 𝛽=𝛽^∗=𝜇(𝜇+𝑟)/Λ. The virus-free equilibrium is 𝐸₀(Λ/𝜇,0,Λ/𝜇). The linearization matrix of system (7) around the infection-free equilibrium 𝐸₀ when 𝛽=𝛽^∗ is 𝐽𝑓=𝛽−𝜇−^∗Λ𝜇00𝛽^∗Λ𝜇−0=0.𝜇+𝑟𝑎0−𝑎−𝜇−𝜇+𝑟000𝑎0−𝑎 (31) The matrix 𝐽(𝑓) has eigenvalues (0,−𝜇,−𝑎)^𝑇, which meets the requirement of a simple zero eigenvalue and others having negative real part. A right eigenvector 𝜂 corresponding to the zero eigenvalue is 𝜂=(−(𝜇+𝑟)/𝜇,1,−(𝜇+𝑟)/𝜇)^𝑇 and the left eigenvector satisfying 𝜉𝜂=1 is 𝜉=(0,1,0). For the system (7), we can get 𝑎=³_{𝑖,𝑗,𝑘=1}𝜉_𝑘𝜂_𝑖𝜂_𝑗𝜕²𝑓_𝑘𝜕𝑥_𝑖𝜕𝑥_𝑗Λ𝜇Λ,0,𝜇2=−Λ𝜇+𝑟²<0,𝑏=³_𝑖,𝑘=1𝜉_𝑘𝜂_𝑖𝜕²𝑓_𝑘𝜕𝑥_𝑖Λ𝜕𝛽𝜇Λ,0,𝜇=Λ𝜇>0. (32) Thus, 𝑎<0,𝑏>0, by item (iv) of Theorem 4.1 in [42]; we can give the following result.

Theorem 7. The disease-free equilibrium 𝐸₀ for system (7) is locally asymptotically stable for 𝑅 near 1.

4. Disease Impulsive Control for System (9)

4.1. The Existence and Globally Asymptotical Stability of the “Infection-Free” Periodic Solution of System (9)

We demonstrate the expression of the infectives-free solution of the system (9) firstly, in which the infectives are entirely absent from the population permanently. Consider the infectives-free subsystem of system (9) in the form 𝑆^′(𝑡)(𝑡)𝑍=Λ−𝜇𝑆,𝑡≠𝑛𝑇,_′(𝑡)(𝑆(𝑡)(𝑡𝑆𝑡=𝑎−𝑍)),𝑡≠𝑛𝑇,₊=()𝑆(𝑡)𝑍𝑡1−𝛿,𝑡=𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇. (33) By Lemma 3, system (33) has a unique positive 𝑇-periodic solution (𝑆^∗(𝑡),𝑍^∗(𝑡)) given by 𝑆^∗(𝑡)=Λ𝜇1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇()𝑍,𝑛𝑇<𝑡≤𝑛+1𝑇,_∗(𝑡)=Λ𝜇𝑒1+𝑎𝛿^{−𝑎(𝑡−𝑛𝑇)}1−𝑒^−𝜇𝑇−𝑒^{−𝜇(𝑡−𝑛𝑇)}1−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇,()𝑆𝑛𝑇<𝑡≤𝑛+1𝑇,_∗0⁺=Λ𝜇𝛿1−1−(1−𝛿)𝑒^−𝜇𝑇,𝑍^∗0⁺=Λ𝜇𝑒1−𝑎𝛿^−𝜇𝑇−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇, (34) and for each solution, 𝑆(𝑡)→𝑆^∗(𝑡),𝑍(𝑡)→𝑍^∗(𝑡) as 𝑡→∞. Hence, we have Theorem 8.

Theorem 8. The system (9) has an “infection-free” periodic solution (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) for 𝑡∈(𝑛𝑇,(𝑛+1)𝑇],𝑛∈𝑁.

In next section, we will prove that “infection-free” periodic solution (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) is globally asymptotically stable.

Theorem 9. Let (𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)) be any solution of (9), and then (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) is globally asymptotically stable provided ℜ<1, where ℜ=((𝛽Λ/𝜇)(𝑇−𝛿(1−𝑒^−𝜇𝑇)/𝜇(1−(1−𝛿)𝑒^−𝜇𝑇)))/(𝜇+𝑟)𝑇.

Proof. Firstly, we will prove the local stability. The local stability of periodic solution (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) may be determined by considering the behavior of small amplitude perturbations of the solution. This may be written as 𝑢(𝑡)𝑣(𝑡)𝑤(𝑡)(𝑡)𝑢(0)𝑣(0)𝑤(0)=Φ,0≤𝑡<𝑇, (35) where Φ satisfies 𝑑Φ(𝑡)=𝑑𝑡−𝜇−𝛽𝑍^∗(𝑡)00𝛽𝑍^∗(𝑡)−𝜇−𝑟0Φ(𝑡)𝑎0−𝑎, (36) and Φ(0)=𝐸, the identity matrix. Hence, the fundamental solution matrix is Φ(𝑡)=𝑒^−𝜇𝑡𝐴00𝑒^{∫𝑡0(𝛽𝑍∗(𝑠)−𝜇−𝑟)𝑑𝑠}0𝐶0𝑒^−𝑎𝑡. (37) There is no need to calculate the exact form of 𝐴,𝐶 as it is not required in the analysis that follows. The linearization of the fourth, fifth, and sixth equations of system (9) becomes 𝑢𝑛𝑇⁺𝑣𝑛𝑇⁺𝑤𝑛𝑇⁺=𝑢()𝑣()𝑤()1−𝛿00010001𝑛𝑇𝑛𝑇𝑛𝑇. (38) The stability of the periodic solution (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) is determined by the eigenvalues of 𝑀=Φ(𝑇)1−𝛿00010001, (39) which are 𝜆₁=(1−𝛿)𝑒^−𝜇𝑇<1,𝜆₂=𝑒^−𝑎𝑇𝜆<1,₃=𝑒^{∫𝑇0(𝛽𝑍∗(𝑡)−𝜇−𝑟)𝑑𝑡}. (40) According to Floquet theory (see [43]), (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) is locally stable if |𝜆₃|<1.
Denote ℜ=𝛽Λ/𝜇𝑇−𝛿1−𝑒^−𝜇𝑇/𝜇1−(1−𝛿)𝑒^−𝜇𝑇𝑇,𝜇+𝑟 (41) For ℜ<1, we have ^𝑇₀𝛽𝑍^∗(𝑡)−𝜇−𝑟𝑑𝑡<0, (42) which leads to |𝜆₃|<1. Thus the periodic solution (𝑆^∗(𝑡),0,𝑍^∗(𝑡)) is locally stable.
In the following, we prove the global attractivity. Since ℜ<1 holds, we have 𝛽Λ𝜇𝛿𝑇−1−𝑒^−𝜇𝑇𝜇1−(1−𝛿)𝑒^−𝜇𝑇−𝜇+𝑟𝑇<0. (43) We can choose a 𝜀>0 small enough such that Λ𝜂=𝛽𝜇)𝛿(1+2𝜀𝑇−1−𝑒^−𝜇𝑇𝜇1−(1−𝛿)𝑒^−𝜇𝑇−𝜇+𝑟𝑇<0. (44) Note that 𝑆^′(𝑡)≤Λ−𝜇𝑆(𝑡), by impulsive differential inequalities, we have 𝑆(𝑡)<𝑆^∗(𝑡)+𝜀 (45) for all 𝑡 large enough. For simplification, we may assume that (45) holds for all 𝑡≥0. From the third equation of system (9) and (45), we get 𝑍^′(𝑡)≤𝑎𝑆^∗(𝑡)(𝑡)𝑍𝑡+𝜀−𝑍,𝑡≠𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇. (46) By impulsive differential inequalities, we have 𝑍(𝑡)<𝑃^∗(𝑡)+𝜀, (47) where 𝑃^∗(𝑡) is the periodic solution of the following equation: 𝑃^′(𝑡)=𝑎𝑆^∗(𝑡)(𝑡)𝑃𝑡+𝜀−𝑃,𝑡≠𝑛𝑇,₊(𝑡)𝑃=𝑃,𝑡=𝑛𝑇,_∗(𝑡)=Λ𝜇𝑒1+𝑎𝛿^{−𝑎(𝑡−𝑛𝑇)}1−𝑒^−𝜇𝑇−𝑒^{−𝜇(𝑡−𝑛𝑇)}1−𝑒^−𝑎𝑇𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇+𝜀,𝑛𝑇<𝑡≤(𝑛+1)𝑇. (48)
From the second and fifth equations of the system (9), we have 𝐼^′(𝑡)≤𝛽𝐼(𝑡)𝑃^∗(𝑡)(𝑡)(𝑡)𝐼𝑡+𝜀−𝜇𝐼−𝑟𝐼,𝑡≠𝑛𝑇,₊=𝐼(𝑡),𝑡=𝑛𝑇, (49) which leads to 𝐼((𝑛+1)𝑇)≤𝐼𝑛𝑇⁺exp^(𝑛+1)𝑇_𝑛𝑇𝛽𝑃^∗(𝑡)−()+𝜀𝜇+𝑟𝑑𝑡=𝐼𝑛𝑇exp^(𝑛+1)𝑇_𝑛𝑇𝛽𝑃^∗(𝑡)−()𝑒+𝜀𝜇+𝑟𝑑𝑡=𝐼𝑛𝑇^𝜂. (50) Hence 𝐼(𝑛𝑇)≤𝐼(0)𝑒^𝑛𝜂 and 𝐼(𝑛𝑇)→0 as 𝑛→+∞. Therefore, 𝐼(𝑡)→0 as 𝑡→+∞, since 𝐼(𝑡)≤𝐼(𝑛𝑇) for 𝑛𝑇<𝑡≤(𝑛+1)𝑇.
Next, we prove that 𝑆(𝑡)→𝑆^∗(𝑡),𝑍(𝑡)→𝑍^∗(𝑡) as 𝑡→+∞. For all 𝜀>0, there must exist a 𝑇₁>0 such that 0≤𝐼(𝑡)<𝜀 for 𝑡>𝑇₁. Without loss of generality, we may assume that 0≤𝐼(𝑡)<𝜀 for all 𝑡≥0, and then from system (9), we have Λ−𝛽Λ𝜀𝜇−𝜇𝑆(𝑡)≤𝑆^′(𝑡)≤Λ−𝜇𝑆(𝑡). (51) Then we have 𝑞(𝑡)<𝑆(𝑡)<𝑆^∗(𝑡)+𝜀₁ and 𝑞(𝑡)→𝑞^∗(𝑡) as 𝑡→+∞, where 𝑞(𝑡) is the solution of 𝑞^′(𝑡)=Λ−𝛽Λ𝜀𝜇(𝑡)𝑞𝑡−𝜇𝑞,𝑡≠𝑛𝑇,₊=()𝑞(𝑡)𝑞01−𝛿,𝑡=𝑛𝑇,₊=𝑆0⁺,𝑞^∗(𝑡)=Λ𝜇−𝛽𝜀𝜇²1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇,𝑛𝑇<𝑡≤(𝑛+1)𝑇. (52) Therefore, for any 𝜀₁>0, there exists a 𝑇₂>0 such that 𝑞^∗(𝑡)−𝜀₁<𝑆(𝑡)<𝑆^∗(𝑡)+𝜀₁. Let 𝜀→0; we have 𝑆^∗(𝑡)−𝜀₁≤𝑆(𝑡)≤𝑆^∗(𝑡)+𝜀₁ for 𝑡 large enough, which implies 𝑆(𝑡)→𝑆^∗(𝑡) as 𝑡→+∞.
Similarly, 𝑍(𝑡)→𝑍^∗(𝑡) as 𝑡→+∞ can be analyzed by the same method as the above, so we omit it. This completes the proof.

4.2. Permanence of System (9)

Theorem 10. If ℜ>1, then system (9) is permanent, that is, there exist three positive constants 𝑚₁,  𝑚₂, and 𝑚₃ such that 𝑆(𝑡)≥𝑚₁,𝐼(𝑡)≥𝑚₂, and 𝑍(𝑡)≥𝑚₃ for 𝑡 large enough.

Proof. Suppose that the 𝑋(𝑡)=(𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)) is any positive solution of system (9). From system (9), we can get 𝑆^′(𝑡)≥Λ−𝛽Λ²𝜇²(𝑡)𝑆𝑡−𝜇𝑆,𝑡≠𝑛𝑇,₊=(1−𝛿)𝑆(𝑡),𝑡=𝑛𝑇. (53) Consider the following impulsive differential equation: 𝐺^′(𝑡)=Λ−𝛽Λ²𝜇²(𝑡)𝐺𝑡−𝜇𝐺,𝑡≠𝑛𝑇,₊=()𝐺(𝑡)𝐺01−𝛿,𝑡=𝑛𝑇,₊=𝑆0⁺. (54) Then, we have 𝑆(𝑡)≥𝐺(𝑡)>𝐺^∗(𝑡)−𝜀, where 𝐺^∗(𝑡)=Λ−𝛽Λ²/𝜇²𝜇1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇,()𝐺𝑛𝑇<𝑡≤𝑛+1𝑇,_∗0=Λ−𝛽Λ²/𝜇²(1−𝛿)1−𝑒^−𝜇𝑇𝜇1−(1−𝛿)𝑒^−𝜇𝑇. (55) So we have 𝑆(𝑡)≥𝐺(𝑡)>𝐺^∗(𝑡)−𝜀>𝐺^∗₀−𝜀=𝑚₁>0 (56) for 𝑡 large enough. From (56) and the third and sixth equations of the system (9), we have 𝑍^′(𝑡)>𝑎𝑚₁(𝑡)𝑍𝑡−𝑍,𝑡≠𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇. (57) Consider the following system: 𝐻^′(𝑡)=𝑎𝑚₁(𝑡)𝐻𝑡−𝐻,𝑡≠𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇. (58) Obviously, 𝐻(𝑡)→𝑚₁ as 𝑡→∞, Then there exists 𝜀>0 such that 𝑍(𝑡)>𝑚₁−𝜀=𝑚₃>0. (59)
Next, we prove that there exists a constant 𝑚₂>0 such that 𝐼(𝑡)≥𝑚₂ for 𝑡 large enough. We will do it in the following two steps.
Step (I). Since ℜ>1, we can choose 𝑚₄,𝜀>0 small enough such that 𝜔=𝛽Λ−𝛽𝑚₄𝜇𝛿−2𝜀𝑇−Λ−𝛽𝑚₄1−𝑒^−𝜇𝑇𝜇²1−(1−𝛿)𝑒^−𝜇𝑇−𝜇+𝑟𝑇>0. (60) We will prove that 𝐼(𝑡)<(𝜇/Λ)𝑚₄ cannot hold for all 𝑡≥0. Otherwise, 𝑆^′(𝑡)≥Λ−𝛽𝑚₄(𝑡)𝑆𝑡−𝜇𝑆,𝑡≠𝑛𝑇,₊=(1−𝛿)𝑆(𝑡),𝑡=𝑛𝑇. (61) So we have 𝑆(𝑡)≥𝑚(𝑡)>𝑚^∗(𝑡)−𝜀 for 𝑡 large enough, where 𝑚(𝑡) is the solution of 𝑚^′(𝑡)=Λ−𝛽𝑚₄(𝑡)𝑚𝑡−𝜇𝑚,𝑡≠𝑛𝑇,₊=()𝑚(𝑡)𝑚1−𝛿,𝑡=𝑛𝑇,_∗(𝑡)=Λ−𝛽𝑚₄𝜇1−𝛿𝑒^{−𝜇(𝑡−𝑛𝑇)}1−(1−𝛿)𝑒^−𝜇𝑇,()𝑚𝑛𝑇<𝑡≤𝑛+1𝑇,_∗=Λ−𝛽𝑚₄𝜇𝛿1−1−(1−𝛿)𝑒^−𝜇𝑇. (62)(63) From the third and sixth equations of the system (9), we have 𝑍^′(𝑡)≥𝑎𝑚^∗(𝑡)(𝑡)𝑍𝑡−𝜀−𝑍,𝑡≠𝑛𝑇,₊=𝑍(𝑡),𝑡=𝑛𝑇, (64) and 𝑍(𝑡)>ℎ^∗(𝑡)−𝜀, where ℎ^∗(𝑡)=Λ−𝛽𝑚₄𝜇+Λ−𝛽𝑚₄𝜇×𝑎𝛿1−𝑒^−𝜇𝑇𝑒^{−𝑎(𝑡−𝑛𝑇)}−1−𝑒^−𝑎𝑇𝑒^{−𝜇(𝑡−𝑛𝑇)}𝑎−𝜇1−𝑒^−𝑎𝑇1−(1−𝛿)𝑒^−𝜇𝑇−𝜀. (65) So we have 𝐼^′(𝑡)≥𝛽ℎ^∗(𝑡)𝐼(𝑡)𝐼𝑡−𝜀−𝜇−𝑟,𝑡≠𝑛𝑇,₊=𝐼(𝑡),𝑡=𝑛𝑇. (66) Integrating (66) on (𝑛𝑇,(𝑛+1)𝑇], we have 𝐼(𝑛+1)𝑇≥𝐼𝑛𝑇⁺exp^(𝑛+1)𝑇_𝑛𝑇𝛽ℎ^∗(𝑡)()𝑒−𝜀−𝜇−𝑟𝑑𝑡=𝐼𝑛𝑇^𝜔. (67) Then 𝐼(𝑛𝑇)≥𝐼(0)𝑒^𝑛𝜔→∞ as 𝑛→∞, which is a contradiction to the boundedness of 𝐼(𝑡). Hence, there exists a 𝑡₁>0 such that 𝐼(𝑡₁)≥(𝜇/Λ)𝑚₄. For the sake of simplification, we let (𝜇/Λ)𝑚₄ be 𝑚₄.
Step (II). If 𝐼(𝑡)≥𝑚₄ for all 𝑡≥𝑡₁, and we let 𝑚₂=𝑚₄, then our aim is obtained. Otherwise, let 𝑡^∗=inf_𝑡>𝑡1{𝑡∣𝐼(𝑡)<𝑚₄}, there are two possible case for 𝑡^∗.
Case (I). 𝑡^∗=𝑛₁𝑇,𝑛₁∈𝑍⁺. Then 𝐼(𝑡)≥𝑚₄ for 𝑡∈[𝑡₁,𝑡^∗) and 𝐼(𝑡^∗)=𝑚₄. Choose 𝑛₂,𝑛₃∈𝑍₊, such that 𝑛₂1𝑇>𝜇ln1+𝑚^∗𝜀,𝑒^{−(𝑛2+1)(𝜇+𝑟)𝑇}𝑒^𝑛3𝜔>1. (68) Let 𝑇^′=(𝑛₂+𝑛₃)𝑇, we claim that there exists a 𝑡₂∈(𝑡^∗,𝑡^∗+𝑇^′] such that 𝐼(𝑡₂)>𝑚₄. Otherwise, consider (62) with 𝑚(𝑛₁𝑇⁺)=𝑆(𝑛₁𝑇⁺) for 𝑡∈(𝑛𝑇,(𝑛+1)𝑇] and 𝑛₁≤𝑛≤𝑛₁+𝑛₂+𝑛₃. We have 𝑚(𝑡)−𝑚^∗(𝑡)≤1+𝑚^∗𝑒^{−𝜇(𝑡−𝑡∗)}<𝜀 (69) for (𝑛₁+𝑛₂)𝑇<𝑡≤(𝑛₁+𝑛₂+𝑛₃)𝑇. So as in the above step (I), we have 𝐼𝑡^∗+𝑇^′≥𝐼𝑡^∗+𝑛₂𝑇𝑒^𝑛3𝜔. (70) From system (9), we get 𝐼^′(𝑡)≥𝐼(𝑡)𝐼𝑡−𝜇−𝑟,𝑡≠𝑛𝑇,₊=𝐼(𝑡),𝑡=𝑛𝑇, (71) for 𝑡∈[𝑡^∗,𝑡^∗+𝑛₂𝑇]. Integrating (71) on [𝑡^∗,𝑡^∗+𝑛₂𝑇], we have 𝐼𝑡^∗+𝑛₂𝑇≥𝑚₄𝑒^{(−𝜇−𝑟)𝑛2𝑇}. (72) Thus, we have 𝐼𝑡^∗+𝑇^′≥𝑚₄𝑒^{(−𝜇−𝑟)𝑛2𝑇}𝑒^𝑛3𝜔>𝑚₄, (73) which is a contradiction.
Let 𝑡̃=inf_𝑡>𝑡∗{𝑡∣𝐼(𝑡)>𝑚₄}, and then for 𝑡∈(𝑡^∗,𝑡)̃,𝐼(𝑡)≤𝑚₄ and 𝐼(𝑡)̃=𝑚₄, since 𝐼(𝑡) is continuous and 𝐼(𝑡⁺)=𝐼(𝑡) when 𝑡=𝑛𝑇. For 𝑡∈(𝑡^∗,𝑡)̃, suppose 𝑡̃∈(𝑡^∗+(𝑘−1)𝑇,𝑡^∗+𝑘𝑇],𝑘∈𝑍₊,𝑘≤𝑛₂+𝑛₃; from (71), we have 𝐼(𝑡)≥𝐼𝑡^∗𝑒^{(−𝜇−𝑟)(𝑡−𝑡∗)}≥𝑚₄𝑒^{(𝑛2+𝑛3)(−𝜇−𝑟)𝑇}. (74) Let 𝑚^′₂=𝑚₄𝑒^{(𝑛2+𝑛3)(−𝜇−𝑟)𝑇}, hence; we have 𝐼(𝑡)≥𝑚^′₂ for 𝑡∈(𝑡^∗,𝑡)̃. For 𝑡>𝑡̃, the same arguments can be continued since 𝐼(𝑡)̃≥𝑚₄.
Case (II). 𝑡^∗≠𝑛𝑇,𝑛∈𝑍₊. Then 𝐼(𝑡)≥𝑚₄ for 𝑡∈[𝑡₁,𝑡^∗] and 𝐼(𝑡^∗)=𝑚₄; suppose 𝑡^∗∈(𝑛₄𝑇,(𝑛₄+1)𝑇),𝑛₄∈𝑍₊. There are two possible cases for 𝑡∈(𝑡^∗,(𝑛₄+1)𝑇),𝑛₄∈𝑍₊.
Case (IIa). 𝐼(𝑡)≤𝑚₄ for all 𝑡∈(𝑡^∗,(𝑛₄+1)𝑇). We claim that there must be a 𝑡^′₂∈[(𝑛₄+1)𝑇,(𝑛₄+1)𝑇+𝑇^′] such that 𝐼(𝑡^′₂)>𝑚₄. Otherwise, consider (62) with 𝑚((𝑛₄+1)𝑇⁺)=𝑆((𝑛₄+1)𝑇⁺); we have 𝑚(𝑡)−𝑚^∗(𝑡)≤1+𝑚^∗𝑒^{−𝜇(𝑡−(𝑛4+1)𝑇)} (75) for 𝑡∈(𝑛𝑇,(𝑛+1)𝑇] and 𝑛₄+1≤𝑛≤𝑛₄+1+𝑛₂+𝑛₃. By a similar argument as in step II case (I), we get 𝐼𝑛₄+1+𝑛₂+𝑛₃𝑇≥𝐼𝑛₄+1+𝑛₂𝑇𝑒^𝑛3𝜔. (76) Since 𝐼(𝑡)≤𝑚₄ for 𝑡∈(𝑡^∗,(𝑛₄+1)𝑇), (71) holds on [𝑡^∗,(𝑛₄+1+𝑛₂)𝑇], so we have 𝐼𝑛₄+1+𝑛₂𝑇≥𝑚₄𝑒^{(𝑛2+1)(−𝜇−𝑟)𝑇}. (77) Thus 𝐼𝑛₄+1+𝑛₂+𝑛₃𝑇≥𝑚₄𝑒^{(𝑛2+1)(−𝜇−𝑟)𝑇}𝑒^𝑛3𝜔>𝑚₄, (78) which is a contradiction. Let 𝑡=inf_𝑡>𝑡∗{𝑡∣𝐼(𝑡)>𝑚₄}, and then for ∈(𝑡^∗,𝑡),𝐼(𝑡)≤𝑚₄ and 𝐼(𝑡)=𝑚₄. For 𝑡∈(𝑡^∗,𝑡), suppose 𝑡∈(𝑛₄𝑇+(𝑘^′−1)𝑇,𝑛₄𝑇+𝑘^′𝑇],𝑘^′∈𝑍₊,𝑘^′≤1+𝑛₂+𝑛₃, we have 𝐼(𝑡)≥𝑚₄𝑒^{(1+𝑛2+𝑛3)(−𝜇−𝑟)𝑇}. (79) Let 𝑚₂=𝑚₄𝑒^{(1+𝑛2+𝑛3)(−𝜇−𝑟)𝑇}<𝑚^′₂, so 𝐼(𝑡)≥𝑚₂ for 𝑡∈(𝑡^∗,𝑡). For 𝑡>𝑡, the same arguments can be continued since 𝐼(𝑡)≥𝑚₄.
Case (IIb). There exists a 𝑡∈(𝑡^∗,(𝑛₄+1)𝑇) such that 𝐼(𝑡)>𝑚₄. Let 𝑡^∗∗=inf_𝑡>𝑡∗{𝑡∣𝐼(𝑡)>𝑚₄}, and then for 𝑡∈(𝑡^∗,𝑡^∗∗),𝐼(𝑡)≤𝑚₄ and 𝐼(𝑡^∗∗)=𝑚₄. For 𝑡∈(𝑡^∗,𝑡^∗∗), integrating (71) on (𝑡^∗,𝑡^∗∗), we have 𝐼(𝑡)≥𝐼𝑡^∗𝑒^{(−𝜇−𝑟)(𝑡−𝑡∗)}≥𝑚₄𝑒^{(−𝜇−𝑟)𝑇}>𝑚₂. (80) So, 𝐼(𝑡)≥𝑚₂ for 𝑡∈(𝑡^∗,𝑡^∗∗). Since 𝐼(𝑡^∗∗)≥𝑚₄, for 𝑡>𝑡^∗∗, the same arguments can be continued.
Hence, 𝐼(𝑡)≥𝑚₂ for all 𝑡≥𝑡₁. The proof is completed.

5. Numerical Analysis and Conclusion

To verify the theoretical results obtained in this paper, we will give some numerical simulations. We consider the hypothetical set of parameter values as Λ=0.2, 𝜇=0.2, 𝛽=0.1, 𝑟=0.3, and 𝑎=0.1 with (𝑆₀,𝐼₀,𝑍₀)=(0.7,0.3,0.3). By calculation, we know that 𝑅=0.2<1, and according to Theorem 5, and we know that the disease-free equilibrium 𝐸₀(1,0,1) of system (7) is locally asymptotically stable for this case (see Figure 1). We set the hypothetical set of parameter values as Λ=1.8, 𝜇=0.2, 𝛽=0.08, 𝑟=0.3, 𝑎=0.1 with (𝑆₀,𝐼₀,𝑍₀)=(0.7,0.3,0.3). By calculation, we know that 𝑅=1.44>1, and according to Theorem 6, we know that the endemic equilibrium 𝐸^∗(6.25,1.1,6.25) of system (7) is locally asymptotically stable for this case (see Figure 2).

(a) Time series of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

(b) Phase portrait of 𝑆(𝑡),𝐼(𝑡)

(c) Phase portrait of 𝑆(𝑡),𝑍(𝑡)

(d) Phase portrait of 𝐼(𝑡),𝑍(𝑡)

(e) Phase portrait of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

Figure 1 

Illustration of basic behavior of system (7) on the threshold values 𝑅=0.2<1, where Λ=0.2,𝜇=0.2,𝛽=0.1,𝑟=0.3,𝑎=0.1, the initial value is (𝑆0,𝐼0,𝑍0)=(0.7,0.3,0.3), and 𝐸0=(1,0,1).

(a)   Time series of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

(b) Phase portrait of 𝑆(𝑡),𝐼(𝑡)

(c) Phase portrait of 𝑆(𝑡),𝑍(𝑡)

(d) Phase portrait of 𝐼(𝑡),𝑍(𝑡)

(e) Phase portrait of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

Figure 2 

Illustration of basic behavior of system (7) on the threshold values 𝑅=1.44>1, where Λ=1.8,𝜇=0.2,𝛽=0.08,𝑟=0.3,𝑎=0.1, the initial value is (𝑆0,𝐼0,𝑍0)=(0.7,0.3,0.3), and 𝐸∗=(6.25,1.1,6.25).

On the other hand, we consider the hypothetical set of parameter values as Λ=1,𝜇=0.2,𝛽=0.5,𝑟=0.6,𝑎=3, and 𝑇=1 with (𝑆₀,𝐼₀,𝑍₀)=(0.7,0.3,0.3). If 𝛿=0.8, by calculation, we know that ℜ=0.4155<1. According to Theorem 9, we know that the “infection-free” periodic solution of system (9) is globally asymptotically stable for this case (see Figure 3). We can explain this in the epidemiology that if we take such a strategy by improving vaccination proportion of susceptible persons in practice, as a result, the infectious population vanishes; that is, diseases eliminate. Figures 3(b), 3(c), 3(d), and 3(e) show the “infection-free” periodic solution under ℜ=0.4155<1. In contrast, if we decrease vaccination proportion of susceptible persons to 0.2, the diseases will be permanent (in this case ℜ=1.4831.) (see Figure 4). Comparing Figure 1 with Figure 3 and Figure 2 with Figure 4, we find that the impulse vaccination proportion of susceptible persons has played a very important role in the actual epidemic prevention.

(a)   Time series of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

(b) Phase portrait of 𝑆(𝑡),𝐼(𝑡)

(c) Phase portrait of 𝑆(𝑡),𝑍(𝑡)

(d) Phase portrait of 𝐼(𝑡),𝑍(𝑡)

(e) Phase portrait of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

Figure 3 

Illustration of basic behavior of system (9) on the threshold values ℜ=0.4155<1, where Λ=1,𝜇=0.2,𝛽=0.5,𝑟=0.6,𝑎=3,𝑇=1, and 𝛿=0.8.

(a)   Time series of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

(b) Phase portrait of 𝑆(𝑡),𝐼(𝑡)

(c) Phase portrait of 𝑆(𝑡),𝑍(𝑡)

(d) Phase portrait of 𝐼(𝑡),𝑍(𝑡)

(e) Phase portrait of 𝑆(𝑡),𝐼(𝑡),𝑍(𝑡)

Figure 4 

Illustration of basic behavior of system (9) on the threshold values ℜ=1.4831>1, where Λ=1,𝜇=0.2,𝛽=0.5,𝑟=0.6,𝑎=3,𝑇=1, and 𝛿=0.2.

In this paper, a SIR epidemic model with distributed delay is proposed. The dynamics behavior of the model without vaccination or under impulsive vaccination is studied, respectively. By using the Jacobian matrix, the stability of the equilibrium points of the system without vaccination is analyzed and by using Floquet's theorem, small-amplitude perturbation skills and comparison theorem the sufficient conditions of globally asymptotic stability of “infection-free” periodic solution and the permanence of the model under impulsive vaccination are obtained. Lastly, we give some numerical simulation to illustrate our main conclusions. We think our mathematical results would be helpful in diseases control.

Appendix

A. The Second Additive Compound Matrix

Let 𝐴=(𝑎_𝑖𝑗) be a 3×3 matrix. Then its second additive compound matrix is as follows: 𝐴^[2]=𝑎₁₁+𝑎₂₂𝑎₂₃−𝑎₁₃𝑎₃₂𝑎₁₁+𝑎₃₃𝑎₁₂−𝑎₃₁𝑎₂₁𝑎₂₂+𝑎₃₃. (A.1)

Proposition 5. Let 𝜎𝐴=𝜆_𝑖∶𝑖=1,2,3 be the spectrum of 𝐴. Then the spectrum of 𝐴^[2] is 𝜎𝐴^[2]=𝜆_𝑖+𝜆_𝑗∶1≤𝑖<𝑗≤3.

Acknowledgments

The authors would like to thank the referees and the editor for their careful reading of the paper and many valuable comments and suggestions that greatly improved the presentation of this paper. This work is supported by Shandong Provincial Natural Science Foundation, China (no. ZR2012AM012), a Project of Shandong Province Higher Educational Science and Technology Program, China (no. J13LI05), and the SDUST Research Fund (no. 2011KYTD105).