Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 7871251
http://dx.doi.org/10.1155/2016/7871251

Research Article

Stability and Persistence of an Avian Influenza Epidemic Model with Impacts of Climate Change

Xiao-Yan Zhao,¹ Shu-Min Guo,² Mini Ghosh,³ and Xue-Zhi Li⁴

¹Department of Basic Courses, Henan Quality Engineering Vocational College, Pingdingshan 467001, China
²College of Mathematics and Statistics, Shaoguan University, Shaoguan 512005, China
³School of Advanced Sciences, VIT University, Chennai Campus, Chennai 600127, India
⁴College of Mathematics and Physics, Anyang Institute of Technology, Anyang 455000, China

Received 14 December 2015; Accepted 11 April 2016

Academic Editor: Ciprian G. Gal

Copyright © 2016 Xiao-Yan Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. Introduction

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 [1] 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. [11] 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. [12] 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 [13], 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 [14]. Kwon et al. [15] 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. [16] 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 [3] because our approach is similar to them. However, our contribution is that our presented model is more complicated as compared to [3] 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 [20] formulated the peripheral blood in the following form: . Also in [3], 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 denote

where

3. The Global Asymptotic Stability and the Persistence of System (4) with ()

Theorem 1. All solutions of system (4) are positive and ultimately bounded; that is, there exists a positive constant , such that all solutions of system (4) satisfy , , , , , and for all large .

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.

We will prove the global stability results of model (4) for . When , system (4) reduces to a new system:

We have the disease-free equilibrium (bird world) and (the bird-human world).

Theorem 3. The disease-free equilibria and are globally asymptotically stable if

Proof. By the second equality of system (4), we have

Integrating the above differential equation over , we have

By Theorem 2, we know that given there exists such that for all . Hence

We 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 have

Similarly, 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 have

By 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 as

Now, we consider the following differential equation:

with , when

that is,

It is easy to see that

By standard comparison theorem, we have

Hence, we have

as is large enough. Thus

By the third equality of system (13), we have

By the comparison theorem we have

Similarly,

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.

Proof. The theorem can be easily proved by using the similar method given in the proof of Theorem 1. So we omit the proof of Theorem 6.

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 have

Integrating 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 . Consider

By 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 [3] 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.

Let , and let be some fixed solution of (58) with initial value . Firstly, on the persistence and permanence of infective in model (51), we have the following theorem.

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 that

for 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 , where . Integrating the second equation of system (51) from to and using (75) and (83), we have

for all . By (75) we have as . This is contradictory with for all . So we conclude that . Then we prove that there exists such that

In fact, from (75) and (76) we can obtain that there is a positive constant such that

for all and , where the constant ; we assume that there is a sequence of initial value with and such that

From , for every there are two time sequences and satisfying

and , such that

for all . By the ultimate boundedness of solution of system (42), for each we can choose positive integer such that , , for all and . Let , and then, for any , we have

where .
Integrating the inequality from to , we have

By (79), we have

Hence,

For each , from the first equation of system (51), it follows that

for all and . Let be the solution of (77) with initial value , and by the comparison theorem we have for all and .
From (79), for and , we have

for all , where is the solution of (58) with the initial value . Since is globally uniformly attractive on for (58) there is a constant and is independent of any and , such that

for all . From (93), we can choose a large enough integer such that

for all and . Hence, by (95) and (96), when and , for any , we have

Integrating the second equation of system (51) from to and having , we finally have

This leads to a contradiction. Thus, we finally prove that inequality (85) is true. At last, we prove that . In fact, if (c) is not true, then we have

for any constant . Let be the solution of system (51) with the initial values and . Since , there is a constant such that for all . Letting , then . Since , there are constants and , such that for we have . We choose a constant , such that . For this constant , we further choose an integer such that

Thus, for any and , since there is an integer , such that , we obtain

From this inequality and by (100), for any constant , we have

Hence, there is , such that

Since for all , by and comparison theorem,

Now using (104), we obtain , which leads to a contradiction. This shows that if (b) holds, then (c) also holds, since is obvious. This completes the proof of Theorem 9.

Theorem 10. Suppose that assumptions and hold and infective is permanent in model (51). Then susceptible in model (51) is also permanent.

This theorem can be proved easily by using the similar method given in the proof of Theorem 9.

Theorem 11. Suppose that assumptions and hold. If there is a constant , such that

then infective in model (51) is extinct.

Proof. If (106) holds, then by assumptions and , for any constant , we can choose constants , , and positive constant which is large enough such that

for all . If (107) holds, then by assumptions and we can directly choose , , and , where , are small enough and is large enough, such that

for all .
For any solution of system (51) with initial values and for some , since for all , we have for all , where is the solution of (58) with the initial value . Since is uniformly globally attractive on for (58), we obtain that, for given above, there exists , such that

for all . Hence, we have for all .
Since

for all , by integrating we further have

for all . Suppose holds. If for all , then from (112) we obtain

By (109), it follows that as . This leads to a contradiction. Hence, there exists , such that . Letting

then is bounded for all . We will prove

for all .
If (115) is not true, then there is , such that . Hence, there exists , such that and for all . Let be a nonnegative integer such that

Then from (110) and (113) we have

This leads to a contradiction. Hence (115) holds. Furthermore, as is arbitrary, we conclude that as . Supposing that (107) holds, then from (113) we get

for all . From (109), we have as .

5.2. Human World ()

In this section, we will give conditions under which the solutions exist on and are positive. The main result is as follows.

Theorem 12. Suppose that assumptions and hold. The solution with initial conditions , , , and is nonnegative and uniformly bounded on .
We can easily prove this theorem using Theorem 1. We define

In this section, we will discuss the permanence of the disease for system (52) and will demonstrate how the disease for system (58) will be permanent under certain conditions. Let the function

And let be some fixed solution of (59) with initial value . We have the following theorem.

Theorem 13. Suppose that assumptions and hold and there is a constant , such that

and then the infective is permanent.

Proof. Since is the solution of system (45), we know is a solution of (59). System (52) is equivalent to the following system:

Firstly, we prove that the number is independent of the choice of . In fact, Theorem 8 implies that, for any sufficiently small and any solution of (59) with initial value , there exists such that . Consider

Hence

For , we obtain

We obtain

We finally obtain

So is independent of . Therefore,

By assumptions , and (128), we choose , , , , which are small enough positive constants; then there exist and satisfying

for all , where . Firstly, we will prove

For any solution of (58), suppose that (133) is not true; then there exists a solution of system (58) and such that for all . If for all , then from the first equation of system (52) we have

for all . Then , as by (129). This is a contradiction. Hence there is such that . Next, we will prove

for all . Otherwise, there is , such that . Hence, there must be such that . Choose an integer such that . Integrating the second equation of system (122) from to , we obtain

This is a contradiction. Hence, (135) is valid.
If for all , then from the fourth equation of system (58) we have

for all . By (130), it follows that as . This is a contradiction. Hence, there is such that . In the following, we prove

for all . If it is not true, then there is satisfying . Hence, there must be such that and for all . Choose an integer such that

Integrating the fourth equation of system (122) from to , we obtain

This is a contradiction. Hence (138) is valid. From this we conclude that there exists such that (135) and (138) are both true for all . For , we define a differentiable function

Integrating the above inequality from to , we have

By (131), we obtain . This contradicts the boundedness of and . From this, we have . Secondly, we will prove that there is a constant such that

From (129)–(132) and , we have that there exist , , and such that

for all and . Choose an integer such that

where . By any , we claim that it is impossible that for all . From this claim, we will discuss the following two possibilities.

(i) for all large .(ii) oscillates about for all large .

Finally, we will show that as is large sufficiently. Let and be sufficiently large time satisfying

for all .
If , then . And , and we have for all . If , then it is clear that for all . If for all , then

This is a contradiction. Hence, there is such that .
From (135), we can obtain

for all Similarly, there is , such that and

for all . Obviously, as ,

Therefore, from the second equation of system (122), (132), (151), and (152), we have

For all , integrating the above inequality from to and using (147) we have

We claim that for all . If it is not true, then there is such that and on . Let , and , and the derivative of along solution of (122) satisfies

for all . Integrating the above inequality from to , we further have

This contradicts (151). Hence is valid for any and we have

Theorem 14. Suppose that assumptions and hold. If there is a constant , such that

then infective H in system (122) is extinct.

Proof. From assumption we choose small enough and big enough, satisfying

for all . For any constant , we set

If (159) holds, then there exists such that

for all . Choose an integer satisfying . Set , and then

Then . Set function

and differentiate along a solution of (122) obtaining

If for all , then from (166) we obtain

By (164) and (167), it follows that as . This is a contradiction with . Hence, there must be such that . Let

be bounded for each . Finally, we will prove

for all . If it is not true, then there exists , such that

Hence, there exists such that and for all . Let be a nonnegative integer such that . From (167), we have

This leads to a contradiction. Hence, inequality (169) holds. Furthermore, since can be arbitrarily small and and , we conclude that and as . Suppose that (160) holds. There exist and such that

for all . From (167) we directly obtain

for all . By (172), as . Therefore, we finally also have and as . This completes the proof of Theorem 14.

6. Concluding Remarks

Birds and poultry are the source of food and livelihoods in many parts of the world where avian influenza is endemic. The avian influenza virus is mainly transmitted directly from birds or from avian virus-contaminated environments to humans. However, at least one instance of the human-to-human spread is thought to have occurred in Thailand. We assume that avian influenza can be transmitted between people. Based on the reported data and experiments, we construct the nonautonomous avian influenza model which incorporates the effect of climate change as parameters become time dependent. These models are more reasonable and closely match with the realistic situation. When there is no infected individual with mutant avian influenza (i.e., ), system (13) is ultimately bounded and the disease-free equilibrium is globally stable if If , system (13) is uniformly persistent. When mutant avian influenza occurs in human, that is, when , is globally stable when . In human-poultry system, we have given the conditions for the permanence of the system; that is, if and hold, is permanent and strongly persistent. Thus we have . If or , the infective is extinct. In human system, if and , the infective is permanent under the condition of and . If there is a constant such that or , the infective is extinct.

Competing Interests

The authors declare that there is no conflict of interests.

Acknowledgments

This research is supported by the NNSF of China (no. 11271314), Plan for Scientific Innovation Talent of Henan Province (144200510021), and Plan for Research Projects of Shaoguan University (no. S201501014).

References

World Health Organization, Avian Influenza Fact Sheet, World Health Organization, Geneva, Switzerland, 2014, http://www.who.int/mediacentre/factsheets/avian_influenza/en/.

Z. Teng, Y. Liu, and L. Zhang, “Persistence and extinction of disease in non-autonomous SIRS epidemic models with disease-induced mortality,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2599–2614, 2008.

T. Zhang and Z. Teng, “On a nonautonomous SEIRS model in epidemiology,” Bulletin of Mathematical Biology, vol. 69, no. 8, pp. 2537–2559, 2007.

N. Bacaer and S. Guernaoui, “The epidemic threshold of vector-borne diseases with seasonality. The case of cutaneous leishmaniasis in Chichaoua, Morocco,” Journal of Mathematical Biology, vol. 53, no. 3, pp. 421–436, 2006.

J. Ma and Z. Ma, “Epidemic threshold conditions for seasonally forced SEIR models,” Mathematical Biosciences and Engineering, vol. 3, no. 1, pp. 161–172, 2006.

F. A. Coutinho, M. N. Burattini, L. F. Lopez, and E. Massad, “Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue,” Bulletin of Mathematical Biology, vol. 68, no. 8, pp. 2263–2282, 2006.

H. R. Thieme, “Uniform persistence and permanence for non-autonomous semiflows in population biology,” Mathematical Biosciences, vol. 166, no. 2, pp. 173–201, 2000.

T. Zhang, J. Liu, and Z. Teng, “Differential susceptibility time-dependent SIR epidemic model,” International Journal of Biomathematics, vol. 1, no. 1, pp. 45–64, 2008.

X. Abdurahman, “On the persistence and extinction for a non-autonomous SIRS epidemic model,” Journal of Biomathematics, vol. 21, no. 2, pp. 167–176, 2006.

K. Wang and A. Fan, “Uniform persistence and periodic solution of chemostat-type model with antibiotic,” Discrete and Continuous Dynamical Systems, vol. 4, no. 3, pp. 789–795, 2004.

S. Iwami, Y. Takeuchi, and X. Liu, “Avian–human influenza epidemic model,” Mathematical Biosciences, vol. 207, no. 1, pp. 1–25, 2007.

S. Iwami, Y. Takeuchi, A. Korobeinikov, and X. Liu, “Prevention of avian influenza epidemic: what policy should we choose?” Journal of Theoretical Biology, vol. 252, no. 4, pp. 732–741, 2008.

S. Iwami, Y. Takeuchi, and X. Liu, “Avian flu pandemic: can we prevent it?” Journal of Theoretical Biology, vol. 257, no. 1, pp. 181–190, 2009.

S. Iwami, Y. Takeuchi, X. Liu, and S. Nakaoka, “A geographical spread of vaccine-resistance in avian influenza epidemics,” Journal of Theoretical Biology, vol. 259, no. 2, pp. 219–228, 2009.

J.-S. Kwon, H.-J. Lee, D.-H. Lee et al., “Immune responses and pathogenesis in immunocompromised chickens in response to infection with the H9N2 low pathogenic avian influenza virus,” Virus Research, vol. 133, no. 2, pp. 187–194, 2008.

H. Wang, Z. Feng, Y. Shu et al., “Probable limited person-to-person transmission of highly pathogenic avian influenza A(H5N1) virus in China,” The Lancet, vol. 371, no. 9622, pp. 1427–1434, 2008.

L. Bourouiba, A. Teslya, and J. Wu, “Highly pathogenic avian influenza outbreak mitigated by seasonal low pathogenic strains: insights from dynamic modeling,” Journal of Theoretical Biology, vol. 271, pp. 181–201, 2011.

T. M. Uyeki and N. J. Cox, “Global concerns regarding novel influenza A (H7N9) virus infections,” The New England Journal of Medicine, vol. 368, no. 20, pp. 1862–1864, 2013.

Y. Chen and Y. Wen, “Global dynamic analysis of a H7N9 avian-human influenza model in an outbreak region,” Journal of Theoretical Biology, vol. 367, pp. 180–188, 2015.

F. Li and A. Qi, “Mathematical model of the effects of plasma cortisol on human T lymphocytes recirculation,” Acta Biochimica et Biophysica Sinica, vol. 16, pp. 586–594, 2000.