Permanence for a Delayed Nonautonomous SIR Epidemic Model with Density-Dependent Birth Rate

Abstract

Based on some well-known SIR models, a revised nonautonomous SIR epidemic model with distributed delay and density-dependent birth rate was considered. Applying some classical analysis techniques for ordinary differential equations and the method proposed by Wang (2002), the threshold value for the permanence and extinction of the model was obtained.

1. Introduction

For understanding the spread of infectious diseases in population, mathematical models that use the theories of ordinary differential equations in epidemiology have been developed rapidly. Epidemic models with delay, including either the autonomous continuous systems or the discrete ones, were discussed by many authors [1–13]. The important subjects for this models are looking for the threshold value that determines whether the infectious disease will be permanent or extinct. Hence, permanence of disease plays an important role in epidemiology. Furthermore, it is well known that models with distributed delay are more appropriate than the discrete ones because it is considered more realistic to assume the infectivity to be a function of the duration since infection and up to some maximum duration.

Recently, Song and Ma in [4] and Song et al. in [5] discussed the permanence of disease in a generalized autonomous SIR epidemic model with density dependent birth rate. On the other hand, for the nonautonomous systems [1] (and the reference therein), the literature is still very inadequate. Motivated by the works in [1, 2, 4, 5, 9, 10], in this paper, we will consider the following nonautonomous delayed systems with density-dependent birth rate: 𝑆̇(𝑡)=−𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₁(𝑡)𝑆(𝑡)+𝑏(𝑡)1−𝛽₁(𝑡)𝑁(𝑡)(𝑡),𝐼̇(𝑡)(𝑡)𝑆(𝑡)1+𝑁=𝛽^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₂(𝑡)𝐼(𝑡)(𝑡)𝐼(𝑡),𝑅̇(𝑡)(𝑡)𝐼(𝑡)−𝜆=𝜆−𝜇₃(𝑡)𝑅(𝑡), (1.1) where 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡) denote the total number of a population at time 𝑡, 𝑆 is the susceptible population, 𝐼 is the infective population and 𝑅 is the removed population. It is assumed that all newborns are susceptible. Functions 𝜇₁(𝑡),𝜇₂(𝑡) and 𝜇₃(𝑡) are instantaneous per capita mortality rates of susceptible, infective, and recovered population, respectively; functions 𝑏(𝑡) and 𝜆(𝑡) represent the birth rate of the population and the recovery rate of infectives, respectively; function 𝛽₁(𝑡)(0≤𝛽₁(𝑡)<1) reflects the relation between the birth rate and the density of population. The nonnegative constant ℎ is the time delay. The function 𝜂(𝑠)∶[0,ℎ]→[0,+∞) is nondecreasing and has hounded variation such that 𝑑𝜂(^ℎ₀𝑠)=𝜂(ℎ)−𝜂(0)=1. (1.2)

2. Preliminaries

Firstly, we give some notations for convenience: if 𝑓(𝑡) is a continuous bounded function defined on [0,+∞), then we set 𝑓^𝐿=inf_𝑡≥0𝑓(𝑡),𝑓^𝑀=sup_𝑡≥0𝑓(𝑡). (2.1) Secondly, for system (1.1), we introduce the following assumption: functions 𝑏(𝑡),𝛽(𝑡),𝛽₁(𝑡),𝜆(𝑡),𝜇₁(𝑡),𝜇₂(𝑡),𝜇₃(𝑡) are nonnegative continuous bounded functions and have positive lower bounds. It is biologically natural to assume that 𝜇₂(𝑡),𝜇₃(𝑡)≥𝜇₁(𝑡) (i.e., epidemics will increase the death rates of the infective and removed).

The initial condition of (1.1) is given as 𝑆(𝜃)=𝜑₁(𝜃),𝐼(𝜃)=𝜑₂(𝜃),𝑅(𝜃)=𝜑₃(𝜃),−∞<𝜃≤0, (2.2) where 𝜙=(𝜑₁,𝜑₂,𝜑₃)^𝑇∈𝐶 such that 𝜑_𝑖(𝜃)≥0(𝑖=1,2,3) for all −∞<𝜃≤0, and 𝐶 denotes the Banach space 𝐶((−∞,0],ℝ) of bounded continuous functions mapping the interval (−∞,0] into ℝ with the toplogy of uniform convergence, that is, for 𝜑∈𝐶, we designate the norm of an element 𝜙∈𝐶 by ‖𝜙‖=sup_{−∞<𝜃≤0}{|𝜑₁(𝜃)|,|𝜑₂(𝜃)|,|𝜑₃(𝜃)|}. By a biological meaning, we further assume that 𝜑_𝑖(𝜃)>0 for 𝑖=1,2,3.

Now, a very impotent lemma for investigating the dynamics of the above system will be given.

Lemma 2.1. Any solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) of system (1.1) with (2.2) is positive for all 𝑡≥0.

Proof. Because the right-hand side of system (1.1) is completely continuous and locally Lipschitzian on 𝐶, the solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) of system (1.1) with initial condition (2.2) exists and is unique on [0,𝛼), where 0<𝛼≤+∞. Firstly, we show that 𝑆(𝑡)>0 for all 𝑡∈[0,𝛼). Otherwise, there is a 𝑡₁∈(0,𝛼) such that 𝑆(𝑡₁)=0, 𝑆(̇𝑡₁)≤0, and 𝑆(𝑡)>0 for all 𝑡∈[0,𝑡₁). Hence, we have 𝐼(𝑡)>0 for all 𝑡∈[0,𝑡₁). If the statement is not true, then there is a 𝑡₂∈(0,𝑡₁) such that 𝐼(𝑡₂)=0,𝐼(̇𝑡₂)≤0, and 𝐼(𝑡)>0 on 𝑡∈[0,𝑡₂). Integrating the second equation of (1.1) from 0 to 𝑡₂, we get 𝐼𝑡₂=𝐼(0)𝑒^{−∫𝑡20[𝜇2(𝑠)+𝛾(𝑠)]𝑑𝑠}+^𝑡2₀^ℎ₀𝛽(𝑢)𝑆(𝑢)𝐼𝑢−𝜌𝑒^{∫𝑢𝑡2[𝜇2(𝑠)+𝛾(𝑠)]𝑑𝑠}𝑑𝜌𝑑𝑢>0, (2.3) which is a contradiction. This shows that 𝐼(𝑡)>0 for all 𝑡∈[0,𝑡₁). From the third equation of (1.1), we also get that 𝑅(𝑡)≥𝑅(0)exp−^𝑡2₀𝜇₃(𝑠)𝑑𝑠>0 (2.4) on [0,𝑡₁). Therefore, from the first equation of system (1.1) 𝑆̇𝑡₁=𝑏𝑡₁1−𝛽₁𝑡₁+𝑏𝑡₁𝛽₁𝑡₁1+𝐼𝑡₁+𝑅𝑡₁>0, (2.5) which also is a contradiction. This shows that 𝑆(𝑡)>0 on [0,𝛼). By the above calculation, it is obvious that 𝐼(𝑡)>0,𝑅(𝑡)>0 for all 𝑡∈[0,𝛼).

3. Permanence and Extinction

In this section, we investigate the permanence of system (1.1) and demonstrate how the disease will be extinct under some conditions.

Theorem 3.1. System (1.1) is permanent if 𝑆⁰=𝑏^𝑀1−𝛽₁^𝑀−𝜇^𝐿₁+𝑏^𝑀1−𝛽₁^𝑀−𝜇^𝐿₁²+4𝜇^𝐿₁𝑏^𝑀2𝜇^𝐿₁>𝜇₂+𝜆𝛽^𝑀. (3.1)
Proof. We will give the following several propositions to complete the proof of this theorem.
Proposition 3.2. Assume that (3.1) holds. Thenlimsup_𝑡→+∞𝑁(𝑡)≤𝑆⁰. (3.2)Proof. From system (1.1), we have that 𝑁̇(𝑡)=−𝜇₁(𝑡)𝑆(𝑡)−𝜇₂(𝑡)𝐼(𝑡)−𝜇₃(𝑡)𝑅(𝑡)+𝑏(𝑡)1−𝛽₁(𝑡+𝑏(𝑡)𝛽)¹(𝑡)(𝑡)≤11+𝑁1+𝑁(𝑡)−𝜇₁(𝑡)𝑁(𝑡)[1+𝑁(𝑡)]+𝑏(𝑡)1−𝛽₁(𝑡)[1+𝑁(𝑡)]+𝑏(𝑡)𝛽₁(𝑡≤1)1+𝑁(𝑡)−𝜇^𝐿₁𝑁²(𝑡)−𝜇^𝐿₁𝑁(𝑡)+𝑏^𝑀1−𝛽₁^𝑀𝑁(𝑡)+𝑏^𝑀1=−(𝑡)𝑁(𝑡)1+𝑁−𝑆⁰𝑁(𝑡)−𝑆⁰_∗, (3.3) where 𝑆⁰_∗=𝑏^𝑀1−𝛽₁^𝑀−𝜇^𝐿₁−𝑏^𝑀1−𝛽₁^𝑀−𝜇^𝐿₁²+4𝜇^𝐿₁𝑏^𝑀2𝜇^𝐿₁<0. (3.4) Now, we consider only the following three cases.(i)𝑁(𝑡)≤𝑆⁰ for all sufficiently large 𝑡. Obviously, for the case (3.2) is true. (ii)𝑁(𝑡)≥𝑆⁰ for all sufficiently large 𝑡. In this case, 𝑁̇(𝑡)≤0 for all sufficiently large 𝑡 from (3.2). Hence, there is a constant 𝑁 such that lim_𝑡→+∞𝑁(𝑡)=𝑁≥𝑆⁰. (3.5) In fact, we can obtain that 𝑁=𝑆⁰. If 𝑁>𝑆⁰, for all sufficiently small 𝜀∶0<𝜀<𝑁−𝑆⁰, there is a sufficiently large 𝑡^∗ such that for any 𝑡>𝑡^∗, 0<𝑁−𝜀≤𝑁(𝑡)≤𝑁+𝜀. (3.6) From (3.3), we have that for all 𝑡≥𝑡^∗, 𝑁̇(𝑡)≤−11+𝑁+𝜀𝑁−𝑆⁰−𝜀𝑁−𝑆⁰_∗−𝜀<0, (3.7) which implies that lim_𝑡→+∞𝑁(𝑡)=−∞. This is a contradiction to lim_𝑡→+∞𝑁(𝑡)=𝑁. Hence, 𝑁=𝑆⁰.
(iii)There is a time sequence {𝑡_𝑛}∶𝑡_𝑛→+∞ as 𝑛→+∞ such that 𝑁(𝑡_𝑛)>𝑆⁰ and 𝑁̇(𝑡_𝑛)=0. Then, for 𝑡=𝑡_𝑛,𝑁̇(𝑡)=−1(𝑡)𝑁(𝑡)1+𝑁−𝑆⁰𝑁(𝑡)−𝑆⁰_∗<0, (3.8) which is a contradiction to 𝑁̇(𝑡_𝑛)=0. This completes the proof.Proposition 3.3. Assume that (3.1) holds. Then any positive solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) of system (1.1) with (2.2) satisfiesliminf_𝑡→+∞𝑆(𝑡)≥𝑏1−𝛽₁𝛽𝑆⁰+𝜇₁^𝐿≡𝜈₁. (3.9)Proof. For any sufficiently small 𝜀>0, from Proposition 3.2, there is a large 𝑡₁>0 such that for all 𝑡>𝑡₁>0,𝐼(𝑡)≤𝑆⁰+𝜀. Hence, for 𝑡>𝑡₁+ℎ,𝑆̇(𝑡)≥−𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₁(𝑡)𝑆(𝑡)+𝑏(𝑡)1−𝛽₁(𝑡(𝑡)≥𝑏)1−𝛽₁(𝑡)−𝜇₁(𝑡)+𝛽(𝑡)𝑆⁰+𝜀𝑆(𝑡), (3.10) which clearly implies that liminf_𝑡→+∞𝑆(𝑡)≥𝑏1−𝛽₁𝛽𝑆⁰+𝜀+𝜇₁^𝐿≡𝜈₁. (3.11) Noting that 𝜀 can be arbitrarily small, the conclusion is valid. This completes the proof of Proposition 3.3.Proposition 3.4. Assume that (3.1) holds. Then for any positive solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) of system (1.1) with (2.2), one has liminf_𝑡→+∞𝐼(𝑡)≥𝜚𝑒^{−(𝜇1+𝜆)𝑀𝑑}≡𝜈₂, (3.12) where 𝜚>0, 𝜀₀>0 and 𝑑>0 satisfy1𝑝=𝜇₁+𝜚𝛽^𝑀𝑀𝑏1−𝛽₁+𝑏𝛽₁1+𝑆⁰+𝜀₀^𝐿>𝜇₂+𝜆𝛽^𝑀,𝑆^Δ=𝑝1−exp−𝜇^𝐿₁+𝜚𝛽^𝑀𝑑>𝜇₂+𝜆𝛽^𝑀. (3.13)Proof. By (3.3) and (3.1) it is obvious that 𝑆⁰=1𝜇^𝐿₁𝑏1−𝛽₁^𝑀+𝑏𝛽₁^𝑀1+𝑆⁰>𝜇₂+𝜆𝛽^𝑀. (3.14) Then there exist two positive constants 𝜚>0, 𝜀₀>0 and 𝑑>0 such that 𝑝>((𝜇₂+𝜆)/𝛽)^𝑀 and 𝑆^Δ>((𝜇₂+𝜆)/𝛽)^𝑀.
Let us consider the following differential function 𝑉(𝑡):𝑉(𝑡)=𝐼(𝑡)+^ℎ₀^𝑡_𝑡−𝑠𝛽(𝑢+𝑠)𝑆(𝑢+𝑠)𝐼(𝑢)𝑑𝑢𝑑𝜂(𝑠). (3.15) Then, the derivative of 𝑉(𝑡) along with the solution of (1.1) with (2.2) is 𝑉̇(𝑡)=𝐼̇(𝑡)+^ℎ₀𝛽(𝑡+𝑠)𝑆(𝑡+𝑠)𝐼(𝑡)𝑑𝜂(𝑠)−^ℎ₀𝛽(𝑡)𝑆(𝑡)𝐼()(𝑠)=𝑡−𝑠𝑑𝜂_ℎ0𝛽(𝑡+𝑠)𝑆(𝑡+𝑠)𝑑𝜂(𝑠)−𝜇₂(𝑡)(𝑡𝐼(𝑡)≥𝛽+𝜆)_𝐿ℎ0𝑆(𝑡+𝑠)𝑑𝜂(𝑠)−𝜇₂+𝜆^𝑀𝐼(𝑡). (3.16) We claim that it is impossible that 𝐼(𝑡)≤𝜚 for all 𝑡≥𝑡₁ (𝑡₁ is any nonnegative constant). In fact, supposing the contrary, it follows from the first equation of (1.1) and (3.1) that for any 𝑡≥𝑡₁+ℎ, 𝑆̇(𝑡)=−𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₁(𝑡)𝑆(𝑡)+𝑏(𝑡)1−𝛽₁(𝑡)+𝑏(𝑡)𝛽₁(𝑡)1+𝑁(𝑡)≥𝑏₁(𝑡)1−𝛽₁(𝑡+𝑏(𝑡)𝛽)¹(𝑡)1+𝑆⁰+𝜀₀−𝜇₁(𝑡)+𝜚𝛽^𝑀𝑆(𝑡). (3.17) Integrating the above inequality from 𝑡₁+ℎ to 𝑡, we obtain 𝑆(𝑡)≥𝑆𝑡₁+ℎexp^𝑡1_𝑡^+ℎ𝜇₁(𝑠)+𝜚𝛽^𝑀+𝑑𝑠_{𝑡𝑡1+ℎ}𝑏(𝑡)1−𝛽₁(𝑡+𝑏(𝑡)𝛽)¹(𝑡)1+𝑆⁰+𝜀₀exp^𝑠_𝑡𝜇₁(𝜏)+𝜚𝛽^𝑀≥∫𝑑𝜏𝑑𝑠_{𝑡𝑡1+ℎ}𝑏(𝑡)1−𝛽₁(𝑡)+𝑏(𝑡)𝛽₁(𝑡)/1+𝑆⁰+𝜀₀𝜇₁(𝜏)+𝜚𝛽^𝑀exp∫^𝑠₀𝜇₁(𝜏)+𝜚𝛽^𝑀𝑑𝜏𝑑𝑠𝜇₁(𝜏)+𝜚𝛽^𝑀exp∫^𝑡₀𝜇₁(𝜏)+𝜚𝛽^𝑀≥1𝑑𝜏𝜇₁+𝜚𝛽^𝑀𝑀𝑏1−𝛽₁+𝑏𝛽₁1+𝑆⁰+𝜀₀^𝐿1−exp−𝜇^𝐿₁+𝜚𝛽^𝑀𝑡−𝑡₁−ℎ. (3.18) Thus, for any 𝑡≥𝑡₁+ℎ+𝑑, 𝑆(𝑡)≥1𝜇₁+𝜚𝛽^𝑀𝑀𝑏1−𝛽₁+𝑏𝛽₁1+𝑆⁰+𝜀₀^𝐿1−exp−𝜇^𝐿₁+𝜚𝛽^𝑀𝑑=𝑆^Δ. (3.19) Then, by the above inequality, from (3.16) and (3.19), we get, for any 𝑡≥𝑡₁+ℎ+𝑑, 𝑉̇(𝑡)≥𝛽^𝐿𝑆^Δ−𝜇₂+𝜆^𝑀𝐼(𝑡). (3.20) Set 𝑖=min_{[]𝜃∈−ℎ,0}𝐼𝑡₁+𝑑+2ℎ+𝜃. (3.21) Next, we will prove that 𝐼(𝑡)≥𝑖 for all 𝑡≥𝑡₁+𝑑+ℎ. Suppose that this is not true. Then there is a 𝑇≥0 such that 𝐼(𝑡)≥𝑖 for all 𝑡∈[𝑡₁+𝑑+ℎ,𝑡₁+𝑑+2ℎ+𝑇], 𝐼(𝑡₁+𝑑+2ℎ+𝑇)=𝑖 and 𝐼(̇𝑡₁+𝑑+2ℎ+𝑇)≤0. On the other hand, by the second equation of (1.1) and (3.16), for 𝑡=𝑡₁+𝑑+2ℎ+𝑇, 𝐼̇(𝑡)=𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₂(𝑡)𝐼(𝑡)(𝑡)𝐼(𝑡)≥𝛽−𝜆_𝐿𝑆^Δ−𝜇₂+𝜆^𝑀𝑖>0. (3.22) This is a contradiction to 𝐼(̇𝑡₁+𝑑+2ℎ+𝑇)≤0. Hence, 𝐼(𝑡)≥𝑖 for all 𝑡≥𝑡₁+𝑑+ℎ.
Consequently, for all 𝑡≥𝑡₁+𝑑+ℎ, we obtain that 𝑉̇(𝑡)≥𝛽^𝐿𝑆^Δ−𝜇₂+𝜆^𝑀𝑖>0, (3.23) which implies that 𝑉(𝑡)→+∞ as 𝑡→+∞. In fact, from Proposition 3.2, V(𝑡) is bounded, which is a contradiction. Consequently, the claim is proved. From the claim, we will discuss the following possible cases:
(i)𝐼(𝑡)≥𝜚 for all large 𝑡;(ii)𝐼(𝑡) oscillates about 𝜚 for all large 𝑡.We show that 𝐼(𝑡)≥𝜚e^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} for all large 𝑡. Evidently, we only need to consider case (ii). Let 𝑡₁ and 𝑡₂ be sufficiently large such that𝐼𝑡₁=𝐼𝑡₂=𝜚, (3.24)𝐼(𝑡)<𝜚𝑎𝑠𝑡∈𝑡₁,𝑡₂. (3.25)If 𝑡₂−𝑡₁≤𝑑+ℎ and from the second equation of (1.1) we have that 𝐼(̇𝑡)≥−(𝜇₂+𝜆)^𝑀𝐼(𝑡), it is obvious that 𝐼(𝑡)≥𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} for 𝑡∈(𝑡₁,𝑡₂); if 𝑡₂−𝑡₁>𝑑+ℎ, and then it is clear that 𝐼(𝑡)≥𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} for 𝑡∈[𝑡₁+𝑑+ℎ,𝑡₂]; in fact, if not, there exists a 𝑇^∗≥0, such that 𝐼(𝑡)≥𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} for 𝑡∈[𝑡₁,𝑡₁+𝑑+ℎ+𝑇^∗]⊆[𝑡₁,𝑡₂], 𝐼(𝑡₁+𝑑+ℎ+𝑇^∗)=𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} and 𝐼(̇𝑡₁+𝑑+ℎ+𝑇^∗)≤0. Using the second equation of system (1.1), as 𝑡=𝑡₁+𝑑+ℎ+𝑇^∗, 𝐼̇(𝑡)=𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₂(𝑡)𝐼(𝑡)(𝑡)𝐼(𝑡)≥𝛽−𝜆_𝐿𝑆^Δ−𝜇₂+𝜆^𝑀𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)}>0. (3.26) This is a contradiction to 𝐼(̇𝑡₁+𝑑+ℎ+𝑇^∗)≤0. So, 𝐼(𝑡)≥𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)} is valid for all 𝑡∈[𝑡₁,𝑡₂]. Since this kind of interval [𝑡₁,𝑡₂] is chosen as an arbitrary way, liminf_𝑡→+∞𝐼(𝑡)≥𝜚𝑒^{−(𝜇2+𝜆)𝑀(𝑑+ℎ)}=𝜈₂>0. (3.27) The proof of Proposition 3.4 is completed.
Form the third equation of system (1.1) and Proposition 3.4, we easily obtain liminf_𝑡→+∞𝑅(𝑡)≥𝜈₂𝜆𝜇₂^𝐿=𝜈₃>0. (3.28) Thus, the system (1.1) is permanent by Propositions 3.2–3.4. The proof of Theorem 3.1 is completed.

Next, we give a useful lemma discussing the extinction of the disease.

Lemma 3.5 (see [12]). Consider an autonomous system of delayed differential equation 𝑥̇(𝑡)=𝑎₁^ℎ₀𝑥(𝑡−ℎ)𝑑𝜂(𝑠)−𝑎₂𝑥(𝑡), (3.29) where 𝑎₁,𝑎₂ are two constants. If 0≤𝑎₁<𝑎₂, then for any solutions 𝑥(𝑡) with initial condition 𝜙(𝜃)≥0, 𝜃∈[−ℎ,0], one has lim_𝑡→+∞𝑥(𝑡)=0. (3.30)

Theorem 3.6. If 𝑆⁰<𝜇₂+𝜆𝛽^𝐿, (3.31) then lim_𝑡→+∞𝑥(𝑡)=0; that is, the disease in system (1.1) will be extinct.

Proof. From Lemma 3.5, for any sufficiently small 𝜀>0, there exists a 𝑇>0, for 𝑡>𝑇, such that 𝑆(𝑡)<𝑆⁰+𝜀. (3.32) Therefore, from the second equation of (1.1), we have 𝐼̇(𝑡)=𝛽(𝑡)𝑆(𝑡)^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₂(𝑡)+𝜆(𝑡)𝐼(𝑡)≤𝛽^𝑀𝑆⁰+𝜀^ℎ₀𝐼(𝑡−𝑠)𝑑𝜂(𝑠)−𝜇₂+𝜆^𝐿𝐼(𝑡). (3.33) Using the comparison theorem of functional differential equations and Lemma 3.5, we can easily get lim_𝑡→+∞𝐼(𝑡)=0. The proof is completed.

4. Discussion

In this paper, we discussed a revised nonautonomous SIR epidemic model with distributed time delay and density-dependent birth rate. Theorems 3.1 and 3.6 show that the disease will be permanent when 𝑆⁰>((𝜇₂+𝜆)/𝛽)^𝑀, and the disease will be extinct when 𝑆⁰<((𝜇₂+𝜆)/𝛽)^𝐿. From the point of view of biology, Theorems 3.1 and 3.6 suggest that the inequality 𝑆⁰>((𝜇₂+𝜆)/𝛽)^𝑀 may be the threshold value for an epidemic disease to occur. The technique for obtaining the sufficient condition on the permanence of the disease in this paper is the same as that in [4, 5] if all coefficients in system (1.1) are constants and the delay is discrete.

Acknowledgment

The authors are grateful to the anonymous referee for his/her many valuable comments and suggestions.