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Research Article | Open Access
Stability and Persistence of an Avian Influenza Epidemic Model with Impacts of Climate Change
The growing number of reported avian influenza cases has prompted awareness of the importance of research methods to control the spread of the disease. Seasonal variation is one of the important factors that affect the spread of avian influenza. This paper presents a “nonautonomous” model to analyze the transmission dynamics of avian influenza with the effects of climate change. We obtain and discuss the global stability conditions of the disease-free equilibrium; the threshold conditions for persistence, permanence, and extinction of the disease; and the parameters with periodicity for controlling and eliminating the avian influenza.
The avian influenza (avian flu or bird flu) refers to the influenza that is caused by viruses adapted to birds. The influenza A virus, only species of influenza virus, can cause influenza in birds and some mammals. The strains of all subtypes of influenza A virus have been isolated from wild birds and at the current understanding some isolates of influenza A virus can cause severe disease both in domestic poultry and in humans though rarely  since there is always a possibility of the transmission of viruses from wild aquatic birds to domestic poultry and through them it can cause an outbreak or give rise to human influenza pandemics.
A human was infected with avian influenza from birds in Hong Kong in 1997 and earlier in Hong Kong flu of 1968. Since then the infection to human from the avian influenza has occurred successively. The WHO has warned of a substantial risk of pandemic avian influenza in the near future. The outbreak of avian influenza is related to the changes in climate and usually it happens in spring and winter. The influenza A viruses are highly sensitive to temperature and with the increase of ambient temperature the survival time of the virus is shortened since in summer and autumn the sunlight is stronger and the sun’s ultraviolet rays are able to kill the virus. However, the same is not true in winter and rainy weather and these weather conditions offer the virus an opportunity to thrive. In this regard, it is important to analyze the impact of the weather seasons on spread of the avian influenza. There exist many diseases that show seasonal behavior, for example, allergic rhinitis, pityriasis rosea, coronary heart disease, psoriasis, and avian influenza. These infections are also periodic and the transmission rates and other parameters can vary because of the force of season [2–10].
Mathematical modelling of avian influenza is not new. In recent years, Iwami et al. [11–14] formulated and analyzed several mathematical models for avian flu for better understanding of transmission and control of this disease. In 2007, Iwami et al.  studied a simple mathematical model for avian influenza by considering both human and bird populations. Their findings suggested that culling of infected bird will not be enough to control the avian influenza when mutant avian influenza has already occurred. The human infected with mutant avian influenza must be quarantined to control the further occurrence of this disease. In 2008, Iwami et al.  further extended their previous model and focused on the effectivity of the two prevention policies, namely, the elimination policy and the quarantine policy, and it is found that the effectivity of the policies depends on the rate of transmission, rate of mutation, and the virulence evolution. Later, in 2009 , they investigated the relations between evolution of the virulence and effectiveness of pandemic control measures after the emergence of mutant avian influenza. Also, they presented a deterministic path-structured model in heterogeneous areas in the same year . Kwon et al.  discussed the immune responses and pathogenesis in immune compromised chickens in response to the infection with the H9N2 low pathogenic avian influenza virus. Wang et al.  studied the person to person transmission of H5N1 in China. Some additional results have also been obtained in [17–19]. But in all of the above discussed models impact of climate is not taken into account and models were based on system of autonomous differential equations. In the present work, we have framed a mathematical model incorporating seasonal variations in the parameters which makes this system nonautonomous but it is more realistic. It can be noted here that some of our results are comparable with the results of  because our approach is similar to them. However, our contribution is that our presented model is more complicated as compared to  and we consider avian influenza model involving both human world and bird world. Also, in the human world we have class of humans infected with mutant avian influenza in addition to usual infectious class. We have analyzed the system for the permanence and extinction of the infectives in both populations.
The rest of this paper is organized as follows: Section 2 presents the model; Section 3 discusses the global stability conditions of the disease-free equilibrium in the absence of human infected with the mutant avian influenza; Section 4 presents the global stability of disease-free equilibrium in the presence of human infected with the mutant avian influenza; Section 5 deals with the persistence, permanence, and extinction analysis of the bird world and human world; and finally Section 6 concludes the paper and summarizes our major results and contributions.
2. The Model
In this section, we are presenting a model to interpret the spread of avian influenza between the bird world and the human world. The epidemic model considered here is as follows: System (1) is divided into the bird world and the human world. In the bird world, avian influenza-free birds and the infectious birds at time are denoted by and , respectively. Here is the recruitment rate of the birds; and represent the natural death rate and the disease-related death rate, respectively. The incidence of newly infected birds is given by . The human world can be divided into four classes: susceptible , infectious , human infected with mutant avian influenza , and recovered human who is recovered from mutant avian influenza, respectively. Here is the rate of recruitment into the human world and it is assumed that all newly recruited humans are susceptible. is the natural death rate; and are the disease-related death rates in and classes, respectively; is the rate of recovery in class; is the mutation rate from to class at time ; and represent the force of infection at time by humans infected with avian influenza and mutant avian influenza, respectively.
In this paper, we consider the infection rate to be time dependent; for example, infection rates and other parameters vary with the time, and the periodicities are the same due to the effect of social and other factors. We assume that are the periodic function of the same period and are given in the form of a sinusoidal function as follows: In fact this kind of periodic function is considered by several other researchers in different context. Li and Qi  formulated the peripheral blood in the following form: . Also in , authors define the periodic function with and is the constant vector population , but with replaced by the average vector population.
First we analyze the model system (1) by assuming Hence the model system (1) now reduces to the following system: By straightforward calculation, it is easy to see that system (4) has the disease-free equilibrium . Let where denotes the average value of . We denotewhere
3. The Global Asymptotic Stability and the Persistence of System (4) with ()
Proof. We note that where and
We consider the bird world first. Since is a constant solution of system (4), by the uniqueness and continuity of the solutions with respect to the initial conditions, we have for all
Next, we will prove that is nonnegative. Suppose that is not always positive. Let be the first time such that . By the first equation of system (4), we have , which means for , where is an arbitrarily small positive constant. It leads to a contradiction. Thus is always positive. Let . Since all solutions of (4) are positive, we have . Therefore, for . Since , , and are ultimately bounded. There exists a positive constant such that and .
Similarly, in the human world, is a constant solution of system (4). By the uniqueness and continuity of the solutions with respect to the initial conditions, we get for any Similarly, suppose that is not always positive. Let be the first time such that . Then from the third equation of system (4), which means for ; is an arbitrarily small positive constant. It leads to a contradiction. Thus is always positive. In the same way, we can show that and are always positive. Let . Then we have . Therefore, for . Thus the human system is ultimately bounded for some positive constant Let . The bird-human system (4) is bounded by for This completes the proof of Theorem 1.
Theorem 2. Let Then ,
Proof. By the first equality of system (4), we have We integrate the inequality over : Thus, we have We choose , which is arbitrarily small, and is large enough, such that Hence for Thus, for , . Let , and thus we have , and hence . We can proof by using the similar method given in the proof of This completes the proof of Theorem 2.
Theorem 3. The disease-free equilibria and are globally asymptotically stable if
Proof. By the second equality of system (4), we haveIntegrating the above differential equation over , we haveBy Theorem 2, we know that given there exists such that for all . HenceWe have if and ; supposing is small enough, we assume that From (16), we obtain Hence as . We suppose that is arbitrarily small. There exists such that for all .
Since as , there exists such that for all , where is arbitrarily small. For , by the first equality of system (13), we haveSimilarly, we have for , where . Hence by Theorem 2, as and is arbitrarily small. Since as , we assume that there exists , such that as
From the third equality of system (13) we haveBy comparison theorem, we obtain as . By Theorem 2, we have as Similarly, We assume as Hence Thus as is arbitrarily small. Hence When , . This completes the proof of Theorem 3.
Theorem 4. If , system (13) is uniformly persistent; thus there exists such that
Proof. By the variation of the constants method for nonhomogeneous linear ordinary differential equations, the solution to the first equality of system (13) has the form Thus,as There exists , such that for all .
Choosing which is arbitrarily small and in Theorem 1, inequality (27) can be written asNow, we consider the following differential equation:with , when that is,It is easy to see thatBy standard comparison theorem, we haveHence, we haveas is large enough. Thus By the third equality of system (13), we haveBy the comparison theorem we haveSimilarly,Hence Let Thus, we have
Theorem 5. The equilibrium point is unstable if .
Proof. It is easy to be seen from the proof of Theorem 3.
4. Analysis of System (4) When and (Single Mutation Case)
Under this condition, system (4) reduces to Let where and
Theorem 6. All solutions of system (42) with initial values , , , , , and are positive for all and they are all ultimately bounded.
Theorem 7. The disease-free equilibrium of system (42) is globally asymptotically stable if .
Proof. Here we proceed in the same manner as given in the proof of Theorem 3. From Theorem 3 we have for all . and are as stated in Theorem 3 and as .
From the fifth equation of system (42), we haveIntegrating this inequality from to , we get Since , and Let From (45), we have Hence as . So there exists , such that for all . We supposed that as . ConsiderBy comparison theorem, we have as By Theorem 2, we have as .
Similarly, we have as .
Hence . Thus as . Thus as . We assume that as . Hence . Consider as is small enough. Thus Consider as . This completes the proof of Theorem 7.
5. The Permanence and Extinction of the Bird World and Human World
In this section we prove the permanence and extinction results of bird world and human world by extending earlier results of  where the authors dealt with the simplified system.
We consider the following bird and human systems which are the subsystems of system (1): Let be the solutions for model (51) and the solutions for model (52). We consider in system (51) and in system (52) for the persistence, permanence, and extinction of the infectives. We have the following definitions.
If then we say that the infectives are strongly persistent.
If there are constants such that we say that the infectives are permanent, if as for any solution of model (51), of model (52) with initial values , , , , and ; for some , we say that infectives go extinct.
For model (51), we introduce the following assumption. Functions are nonnegative, continuous, and bounded on . There exist positive constants , such that
It is easy to prove that assumption is equivalent to In particular, when system (51) degenerates into -periodic system, that is, and and are all nonnegative continuous periodic functions with period , then is equivalent to the following cases: and (, , , , and ). Let us denote the average value of by for any continuous periodic function ; that is,
Then we consider the following nonautonomous linear equations:We have the following results.
Theorem 8. Suppose that assumptions and hold; then consider the following.
(a) Each fixed solution of (58) (see (59)) with initial value is bounded and globally uniformly attractive on .
(b) Let be a solution of (58) (see (59)) and a solution obtained in (58) (see (59)); when is replaced by another continuous function and , then there is a constant depending only on such that .
(c) If there is a constant such that then there are constants such that for any solution of (58) (see (59)) with initial value .
(d) When (58) (see (59)) is -periodic, then (58) (see (59)) has a unique nonnegative almost periodic solution which is globally uniformly attractive.
(e) When (58) (see (59)) is almost periodic, then (58) (see (59)) has a unique nonnegative almost periodic solution which is globally uniformly attractive.
(f) For bird system, if then for any solution of (58) with the initial value where For human system, if for and , then for any solution of (59) with initial value , we have where
5.1. Bird World
Using the variation of constants method, comparison theorem, and the method of Lyapunov function, we can prove this lemma very easily; here, we omit it.
Theorem 9. Suppose that and hold. Then the following statements are equivalent for model (51):(a)Infective y is permanent.(b)Infective y is strongly persistent.(c)There is a constant such that
Proof. Let with the initial value . Firstly, we prove that the number is independent of the choice of . In fact, by (a) of Theorem 8, for arbitrary small and any solution of (58) with initial value , there exists such that for all . Hence, for all . Since we have From the arbitrariness of , we finally obtain This shows that is independent of the choice of . Let be any solution of system (51) with and for some . We next prove that and are nonnegative and is ultimately bounded on . In fact, from the second equation of system (51), we have for all Hence, is nonnegative on .
Similarly We have for all since for all Using the comparison theorem and conclusion (a) of Theorem 8, we can obtain that there is a constant , such that for any solution of system (51) there is such that for all . Since , we further obtain and for . Therefore, all solutions of system (51) with initial values and for some are ultimately bounded. Now, we prove . If (c) is true, then, by and , we can choose small enough positive constants and a large enough constant , such thatfor all .
For any positive constant , we consider the following equation:for any and . Let be the solution of (58) with the initial value and the solution of (58) with initial value . By conclusion (b) of Theorem 9 there is a constant , depending only on , such that for all . We choose sufficiently small and is independent of any and , such that for all . By conclusion (a) of Theorem 9 we know that is globally uniformly attractive on . Thus, for every constant , there is , such that for all and for some . Let be any solution of system (51) with initial values and for some . Then, we prove Suppose that (81) is not true; there exists such that for all . From the first equation of system (51), we have for all . Let be the solution of (58) with initial value . By comparison theorem we have for all . From (79) for , we can obtain for all . By (76), we have for all