Research Article | Open Access
Permanence for a Delayed Nonautonomous SIR Epidemic Model with Density-Dependent Birth Rate
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.
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  and Song et al. in  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  (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: where denote the total number of a population at time , is the susceptible population, is the infective population and is the removed population. It is assumed that all newborns are susceptible. Functions and are instantaneous per capita mortality rates of susceptible, infective, and recovered population, respectively; functions and represent the birth rate of the population and the recovery rate of infectives, respectively; function reflects the relation between the birth rate and the density of population. The nonnegative constant is the time delay. The function is nondecreasing and has hounded variation such that
Firstly, we give some notations for convenience: if is a continuous bounded function defined on , then we set Secondly, for system (1.1), we introduce the following assumption: functions are nonnegative continuous bounded functions and have positive lower bounds. It is biologically natural to assume that (i.e., epidemics will increase the death rates of the infective and removed).
The initial condition of (1.1) is given as where such that for all , and denotes the Banach space of bounded continuous functions mapping the interval into with the toplogy of uniform convergence, that is, for , we designate the norm of an element by . By a biological meaning, we further assume that for .
Now, a very impotent lemma for investigating the dynamics of the above system will be given.
Proof. Because the right-hand side of system (1.1) is completely continuous and locally Lipschitzian on , the solution of system (1.1) with initial condition (2.2) exists and is unique on , where . Firstly, we show that for all . Otherwise, there is a such that , , and for all . Hence, we have for all . If the statement is not true, then there is a such that , and on . Integrating the second equation of (1.1) from 0 to , we get which is a contradiction. This shows that for all . From the third equation of (1.1), we also get that on . Therefore, from the first equation of system (1.1) which also is a contradiction. This shows that on . By the above calculation, it is obvious that for all .
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
Proof. We will give the following several propositions to complete the proof of this theorem.
Proposition 3.2. Assume that (3.1) holds. ThenProof. From system (1.1), we have that where Now, we consider only the following three cases.(i) for all sufficiently large . Obviously, for the case (3.2) is true. (ii) for all sufficiently large . In this case, for all sufficiently large from (3.2). Hence, there is a constant such that In fact, we can obtain that . If , for all sufficiently small , there is a sufficiently large such that for any , From (3.3), we have that for all , which implies that . This is a contradiction to . Hence, .
(iii)There is a time sequence as such that and . Then, for , which is a contradiction to . This completes the proof.Proposition 3.3. Assume that (3.1) holds. Then any positive solution of system (1.1) with (2.2) satisfiesProof. For any sufficiently small , from Proposition 3.2, there is a large such that for all . Hence, for , which clearly implies that Noting that 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 of system (1.1) with (2.2), one has where , and satisfyProof. By (3.3) and (3.1) it is obvious that Then there exist two positive constants , and such that and .
Let us consider the following differential function : Then, the derivative of along with the solution of (1.1) with (2.2) is We claim that it is impossible that for all ( is any nonnegative constant). In fact, supposing the contrary, it follows from the first equation of (1.1) and (3.1) that for any , Integrating the above inequality from to , we obtain Thus, for any , Then, by the above inequality, from (3.16) and (3.19), we get, for any , Set Next, we will prove that for all . Suppose that this is not true. Then there is a such that for all , and . On the other hand, by the second equation of (1.1) and (3.16), for , This is a contradiction to . Hence, for all .
Consequently, for all , we obtain that which implies that as . In fact, from Proposition 3.2, V() is bounded, which is a contradiction. Consequently, the claim is proved. From the claim, we will discuss the following possible cases:
(i) for all large ;(ii) oscillates about for all large .We show that for all large . Evidently, we only need to consider case (ii). Let and be sufficiently large such thatIf and from the second equation of (1.1) we have that , it is obvious that for ; if , and then it is clear that for ; in fact, if not, there exists a , such that for , and . Using the second equation of system (1.1), as , This is a contradiction to . So, is valid for all . Since this kind of interval is chosen as an arbitrary way, The proof of Proposition 3.4 is completed.
Form the third equation of system (1.1) and Proposition 3.4, we easily obtain 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 ). Consider an autonomous system of delayed differential equation where are two constants. If , then for any solutions with initial condition , , one has
Theorem 3.6. If then ; that is, the disease in system (1.1) will be extinct.
Proof. From Lemma 3.5, for any sufficiently small , there exists a , for , such that Therefore, from the second equation of (1.1), we have Using the comparison theorem of functional differential equations and Lemma 3.5, we can easily get . The proof is completed.
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 , and the disease will be extinct when . From the point of view of biology, Theorems 3.1 and 3.6 suggest that the inequality 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.
The authors are grateful to the anonymous referee for his/her many valuable comments and suggestions.
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