The spreading frontiers of avian-human influenza described by the free boundary

In this paper, a reaction-diffusion system is proposed to investigate avian-human influenza. Two free boundaries are introduced to describe the spreading frontiers of the avian influenza. The basic reproduction numbers r 0F (t) and R 0F(t) are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem, respectively. Properties of these two time-dependent basic reproduction numbers are obtained. Sufficient conditions both for spreading and for vanishing of the avian influenza are given. It is shown that if r 0F (0) < 1 and the initial number of the infected birds is small, the avian influenza vanishes in the bird world. Furthermore, if r 0F (0) < 1 and R 0F(0) < 1, the avian influenza vanishes in the bird and human worlds. In the case that r 0F (0) < 1 and R 0F(0) > 1, spreading of the mutant avian influenza in the human world is possible. It is also shown that if r 0F (t 0) ⩾ 1 for any t 0 ⩾ 0, the avian influenza spreads in the bird world.


Introduction
In history, epidemic diseases swept through countries and made people suffer from big disasters. The plague flourished four times in Europe, and it killed more than 15 percent population in the affected areas each time [2,6]. In 6th century, the first outbreak caused about half of European people dead; the second outbreak occurred in 1346-1350 and the rate of the mortality was 33.33%, the third outbreak killed 1 6 population of London in 1665-1666. The last happened in 1720-1722, the death rate was very high, which resulted in half population dead in Marseilles [2]. The prevalent of the disease usually leads the population declined sharply. Beside the Black Death, there are many contagious diseases, such as smallpox, malaria, yellow fever and so on.
At present, about half of the world's population is threatened with different infectious diseases [5]. Today the WHO (World Health Organization) claims that the infectious diseases are still hostile in the human world. From now on, people should be cautious about the diseases and scientists should contribute themselves to eradicate epidemic diseases. Recently, great attention has been paid to study the contagious diseases. In early 20th century, mathematical models have been constructed to study the dynamics of H(t) = β 2 SH + εB − (μ + α + γ)H, R(t) = γH − μR, (1.1) where the first two equations form the SI model, which describes the bird system, no recovered bird exists since the highly pathogenic viruses have high death rates. X and Y are the population of the susceptible birds and the infective birds, respectively. The last four equations are SIR model for humans, S, B, H and R denote, respectively, the population of the susceptible humans, infected with avian influenza, infected with mutant avian influenza and recovered humans from mutant avian influenza. The reproductive number r 0 = cω b(b+m) in the bird system and the reproductive number R 0 = β2λ μ(μ+α+γ) in the human system are defined. The authors showed that when r 0 1 and R 0 > 1, the human-endemic equilibrium (i.e., the boundary equilibrium) E h := (X 0 , 0, S, 0, H, R) is global asymptotically stable, while if r 0 > 1, the full-endemic equilibrium (i.e., the interior equilibrium) is global asymptotically stable. In other words, mutant avian influenza spreads in the human world in the above two cases.
Considering spatial spreading and time delay, Kim et al. [12] introduced diffusion terms and extended the system (1.1) to the following reaction diffusion system: K(x, y, t − s)H(s, y)dsdy + εB − (μ + α + γ)H, for t > 0, x ∈ Ω. The system (1.2) with a no-flow boundary condition is discussed, and the basic reproductive numbers r 0 and R 0 are defined the same as those in [8]. They concluded that the diseasefree equilibrium E 0 := (c/b, 0, λ/μ, 0, 0, 0) is locally asymptotically stable if r 0 < 1 and R 0 < 1; and human-endemic equilibrium E h is locally asymptotically stable if r 0 < 1 and R 0 > 1; while r 0 > 1 the full-endemic equilibrium E + is locally asymptotically stable. Furthermore, they proved the disease-free equilibrium E 0 is globally asymptotically stable when r 0 < 1 and R 0 < 1. It is known that the solution to System (1.2) with Dirichlet or Neumann boundary condition in a bounded domain is always positive for any time t > 0 no matter what the nonnegative nontrivial initial date is. It means that the avian influenza spreads to the whole area immediately even when the infectious is confined to a small part of the area at the beginning. It does not match the observed fact that the avian influenza always spreads gradually. To describe the gradual progress of avian influenza spreading, we are attempting to consider an avian-human influenza model with free boundaries, which describe the spreading frontiers of the virus. In the next section, a mathematical model of the free boundary in both the bird world and the human world is constructed first.
In Section 3, we give the local existence and uniqueness of the solution to Problem (2.4), then prove that the right free boundary x = h(t) is increasing and the left boundary x = g(t) is decreasing. Finally, we conclude that the solution to (2.4) is global and unique. Section 4 is devoted to the basic reproduction numbers r F 0 (t) for avian influenza in the bird system with free boundary and R F 0 (t) for the mutant avian influenza in the human system with free boundary. It is shown that r F 0 (t) and R F 0 (t) are strictly increasing with respect to t. Moreover, if lim t→∞ (h(t)−g(t)) = ∞, then lim t→∞ r F 0 (t) = r 0 and lim t→∞ R F is the basic reproduction number for avian influenza in the bird system (4.1), and R 0 (= βm d h +γ h ) is the basic reproduction number for the mutant avian influenza in the human system (4.1).
Sections 5-7 deal with the asymptotic behaviors of the free boundaries and the solution. In Section 5, we first prove that the free boundaries x = h(t) and x = g(t) are either finite or infinite at the same time. Then we show that if ,h(t)]) = 0, i.e., the avian influenza vanishes in the bird world. Finally by constructing suitable upper solution to problem (2.4), we give the sufficient conditions for the avian influenza to vanish in the bird and human worlds, i.e., if r F 0 (0) < 1, R F 0 (0) < 1 and the initial infected data are small enough, then h ∞ − g ∞ < ∞ and the solution decays gradually toward zero.
In Section 6, we investigate the case that r F 0 (0) < 1 and .
In this situation, the mutant avian influenza transmits in the human. In Section 7, we show that if r F 0 (t 0 ) 1 for any t 0 0, the avian influenza spreads in the bird world. Furthermore, we demonstrate that if r F 0 (0) < 1 < r 0 and B i0 (x) is big enough, then In other words, the avian influenza can spread in the whole area even that r F 0 (0) < 1. A short discussion is given in Section 8.

Model formulation
We consider a general avian-human model and classify the birds and humans as the following categories: • Susceptible birds B s (t), infected birds B i (t) with avian influenza; • Susceptible humans H s (t), infected humans H a (t) with avian influenza, infected humans H m (t) with mutant avian influenza and recovered ones H r (t) from mutant avian influenza.
Here, we assume that all birds infected with avian influenza are dead or remain infected and can be never recovered. Then the bird system can be described by SI model. But the human infected with mutant avian influenza can be cured, then the person has immunity against avian influenza virus in all his/her life. So we can use the SIR model to represent the human world. If we do not consider the spatial spreading of the virus, an ODE model reads where G(B s , B i ) is the per capita reproduction rate of the adult birds which can be taken as r b (1− Bs+Bi K b ), or simply a constant r b ; d b and d h are the death rates with the bird and the human; β a , β b and β m denote the contact rates of the virus from infected bird with avian influenza to human, infected bird to bird and infected human with mutant avian influenza to human, respectively; r h is the birth rate of human; γ h denotes the recover rate of human from the mutant avian influenza virus; ε h is the mutant rate with the avian influenza virus. N b and N h denote the total numbers of the bird and the human, respectively.
In general, the individual disperses randomly in the habitat. Therefore, we consider not only the individual's activity in temporal dimension, but also the distribution of the individual in the spatial and the dynamic characteristic of the avian influenza. To describe the diffusion of the disease, we introduce the spatial diffusion terms. We start with one-dimensional case: −∞ < x < ∞, thus an extended version of the avian-human model can be described by for −∞ < x < ∞ and t > 0, where D b and D h are positive diffusion coefficients for the birds and humans, respectively. For simplicity, we take G(B s , B i ) = r b , and let In other words, we assume the total number of birds and that of humans remain constant.
This research is devoted to the transmission of avian influenza. Despite the infected is limited in a small district, the avian influenza can spread in the whole habitat immediately. Obviously, it cannot be used to describe the real spreading of the avian influenza virus. We now use the free boundary to describe the spreading frontier of the disease. Assume that the birds and humans migrate in the whole habitat (−∞, ∞), and some birds are infected in g(t) < x < h(t), there is only the susceptible birds or humans on the rest part. The right spreading frontier is represented by the free boundary x = h(t). Assume that h(t) grows at a rate that is proportional to the population gradient at the frontier [7,11,17,18,28]. Then the conditions on the right frontier (free boundary) is Similarly, the conditions on the left frontier (free boundary) is In such a case, we have the problem for where x = g(t) and x = h(t) are the moving left and right boundaries to be determined, h 0 and μ are positive constants, and the initial functions B i0 , H a0 , H m0 and H r0 are nonnegative and satisfy Noting that the first equation of the system (2.4) is independent of the last three equations, we can consider the bird system as the following: In the rest part, we will consider (2.4) and (2.6), then give the properties of the solution and the free boundaries.

Existence and uniqueness
In this section, a contraction mapping argument is used to show the local existence and uniqueness of the solution to (2.4). Then we use standard methods (such as Hopf Lemma, the maximum principle) to derive the estimates of the solution and present the global existence and uniqueness of the solution. First, the following local existence result can be proved by the contraction mapping argument as in [4].
According to (2.5) and the equation of By using maximum principle, it is easy to see that 0 To explain that the local solution to (2.4) can be extended to all t > 0, we need the following lemma to show the free boundaries for problem (2.4) are strictly monotonous.
Using the Hopf lemma to the equation of B i at x = g(t) and x = h(t) yields that Hence, g (t) < 0 and h (t) > 0 for t ∈ (0, T 0 ) by using the free boundary conditions in (2.4). It remains to show that −g (t), h (t) C 2 for t ∈ (0, T 0 ) and some C 2 independent of T 0 . Inspired by [7], define For any t ∈ (0, ∞), the solution to (2.4) exists and is unique.

Proof.
It follows from the uniqueness of the solution that there is a number T max such that [0, T max ) is the maximal time interval in which the solution exists. Now, we show that T max = ∞ by the contradiction argument.
Assume that T max < ∞. By Lemma 3.2, there exists C 2 independent of T max such that We now fix δ 0 ∈ (0, T max ) and M > T max . By standard parabolic regularity, we can find

Basic reproduction numbers
In this section, we denote the basic reproduction numbers for different systems and present their properties which will be used in the sequel. Now we study the full system (2.3). The corresponding ODE system is governed by Similarly to that in [8], the basic reproduction number r 0 of the bird and the basic reproduction number R 0 for mutant avian influenza in the human world are represented by According to [8,Theorem 3.2], if r 0 < 1 and R 0 < 1, then DFE is locally asymptotically stable; the HEE is locally asymptotically stable if r 0 < 1 and R 0 > 1, and if r 0 > 1, the FEE is locally asymptotically stable.
If the environment Ω is heterogeneous and the boundary of region Ω is hostile to the population for living (see [23]), then the corresponding system is given by Let us introduce the basic reproduction numbers for the bird r D 0 (Ω) and the mutant human R D 0 (Ω), respectively, as follows: and λ R0 be the first eigenvalue of Furthermore, its corresponding eigenfunctions ϕ λr 0 and φ λR 0 can be chosen to be positive on Ω. Obviously, λ r0 and λ R0 can be given by variational characterization,

Lemma 4.2.
The following properties hold: One can directly verify this statement (see [ where λ(Ω) is the principle eigenvalue of the following eigenvalue problem: It is well known that λ(Ω) is a strictly decreasing continuous function and that From the direct calculations, the following lemma holds.
(iv) Let B ρ be a ball with radius ρ. Then r D 0 (B ρ ) and R D 0 (B ρ ) are strictly monotonically increasing function of ρ. That is to say, By the free boundary problem (2.4), we know that the domain (g(t), h(t)) is changing with respect to t. Thus the basic reproduction number is a function with t. Now, we define the basic reproduction numbers r F 0 (t) and R F 0 (t) for the free boundary problem (2.4) by .
This conclusion follows directly from Lemmas 3.2 and 4.3.

Vanishing of the avian influenza
In this section, we will consider the vanishing of the avian influenza in the bird and in the human world. First, we show that the double free boundary frontiers x = g(t) and x = h(t) are either finite or infinite at the same time.  Suppose that (B i , H a , H m , H r ; h, g) is a solution to (2.4) defined for x ∈ [g(t), h(t)] and t ∈ [0, ∞). Then we have

Proof.
For small t > 0, we have g(t) + h(t) > −2h 0 by the continuity. Define Now, we show that T = ∞ by the contradiction argument. Assume that there exists T with 0 < T < ∞ such that Furthermore, we get Direct calculation shows that Noting that C(x, t) ∈ L ∞ and then by using the comparison principle, we obtain Applying the strong maximum principle gives that However, By the Hopf lemma, we have B x (g(T ), T ) < 0. Moreover, Then we have which is a contradiction to (5.1). Hence we have shown and g(t) + h(t) < 2h 0 for all t 0 can be proven by the same argument.
It follows from the regularity of the parabolic equation that {B n } has a subsequence {B ni } such that Moreover, B(x 0 , 0) σ 2 , and thus we can get Applying the Hopf lemma yields that there exists > 0 such that B x (h ∞ , 0) − . It follows from Theorem 3.1 and Lemma 3.2 that −g(t) and h(t) are monotonically increasing and bounded. By the standard L p theory and the Sobolev imbedding theorem (see [13]), for any 0 < α < 1, there exists a constant C which depends on α, h 0 , B i0 C 1+α ([−h0,h0]) , g ∞ and h ∞ such that On the other hand, since h(t) is bounded, we have h (t) → 0 as t → +∞. So ∂Bi ∂x (h(t n ), t n ) → 0 as n → +∞ by the free boundary condition. Noting This is a contradiction to that B x (h ∞ , 0) − . Hence, lim t→+∞ B i (·, t) C([g(t),h(t)]) = 0.
Lemma 5.2 implies the following result.
To give the sufficient conditions for the avian influenza to vanish, we first give the definition of the upper and the lower solutions to the bird system, then give the comparison principle.
is an upper solution to the bird system (2.6).

h(t) h(t), g g(t) g(t), t ∈ (0, T ],
Now, we give the sufficient condition for the avian influenza to vanish in the bird world. h0,h0]) is sufficiently small.
The following comparison principle for the full system can be obtained similarly to [7,Lemma 3.5].

Discussion
Most prior work related to the disease transmitting has focused on the understanding of the disease dynamic in the fixed domain. There is almost no work which has investigated the characteristics of disease spreading in a moving area.
In this paper, we introduce two moving boundaries, which are called free boundaries, to describe the avian influenza virus transmitting in the habitat. The dynamical behaviors of the solution have been discussed. Two basic reproduction numbers r F 0 (t) and R F 0 (t) are given for the free boundary problem (2.4) in Section 4. The infected domain (g(t), h(t)) changes with t, so the basic reproduction numbers depend on time t. It has been shown that the two reproduction numbers r F 0 (t) and R F 0 (t) are monotonically increasing with respect to t, and r F 0 (t) and R F 0 (t) approach to r 0 and R 0 as t → ∞, respectively, where r 0 and R 0 are the responding basic reproduction numbers of avian influenza in the homogeneous bird world and mutant avian influenza in the homogeneous world, respectively.
Recently, much work with the basic reproduction number has been done to study the diseases dynamic properties. The basic reproduction numbers are usually defined by constants. But our definition of the basic reproduction numbers are time-dependent, and thus r F 0 (t) and R F 0 (t) can be utilized to predict spreading or vanishing of the avian influenza virus.
The sufficient conditions are given to determine the transmission of the avian influenza virus. Our results show that spreading or vanishing of the avian influenza is not only related to the basic reproduction numbers r F 0 (t) and R F 0 (t), but the initial infection. If r F 0 (0) < 1, R F 0 (0) < 1 and the initial infection value is small, the avian influenza disease vanishes, i.e., h ∞ − g ∞ < ∞, lim t→+∞ ( B i (·, t) C , H a (·, t) C , H m (·, t) C , H r (·, t) C ) = (0, 0, 0, 0) uniformly for x ∈ [g(t), h(t)] (see Theorem 5.9). It is shown that the mutant avian influenza spreads in the human world if r F 0 (0) < 1 and (see Theorem 6.1). If r F 0 (t 0 ) 1 for any t 0 0, then the avian influenza transmits in the bird world (see Corollary 7.2).
Nowadays, epidemic models have attracted much attention [14], especially, avian influenza model. The study of our work tries to describe the spreading process of the disease, and encourages people to take good strategies to prevent the avian influenza virus from transmitting to human world. There are some work to do, for example, the spreading speed when spreading happens.