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Though medicine and living conditions have been constantly improving, infectious diseases are still a global concern. Mathematical modeling can not only enhance our understanding of the transmission mechanisms underlying them but also help us assess the efficacy of control strategies. Among the deterministic models described by ordinary differential equations are compartmental models. One of the basic models is the Kermack-McKendric model,
{dSdt=−βSI,dIdt=βSI−γI,dRdt=γI, |
where S, I, and R are the densities (or numbers) of susceptible, infectious, and recovered individuals, respectively; β is the transmission rate while γ is the recovery rate. The incidence rate is the bilinear one, βSI. To better reflect the actual biology of a given disease, the above model has been significantly modified.
In this paper, we consider the factor of relapse. For certain diseases such as herpes, tuberculosis, simplex virus type 2 (a human disease transmitted by close physical or sexual contacts), recovered individuals may experience relapse, which means that they can revert to the infectious class with the reactivation of a latent infection. For example, this feature of recurrence for tuberculosis is often due to incomplete treatment. Tudor [1] was the first to study relapse, who built the so-called SIRI model. In this model, the bilinear incidence rate is used. Tudor investigated the existence and local stability of equilibria. Later on, Moreira and Wang [2] modified this model with an incidence rate depending on the size of the susceptible population. By means of an elementary analysis of Liénard's equation and Lyapunov's direct method, they established sufficient conditions on the global asymptotic stability of the disease-free and endemic equilibria.
In the above mentioned studies on relapse, the population size is constant. In particular, there are no disease-induced deaths. Thus, in 2013, Vargas-De-León [3] proposed two epidemiological models with relapse and disease-induced deaths. One of them is the following one with the bilinear incidence rate,
{dSdt=Λ−βSI−μS,dIdt=βSI−(α+γ+μ)I+ηR,dRdt=γI−(μ+η)R, | (1.1) |
where Λ represents the recruitment rate, β is the transmission rate, μ is the natural death rate, α is the disease-induced death rate, γ is the recovery rate, and η is the relapse rate. All the parameters are positive. They constructed suitable Lyapunov functions to obtain threshold dynamics determined by the basic reproduction number R0. If R0<1, the disease-free equilibrium is globally asymptotically stable and hence the disease dies out. On the other hand, if R0>1, the endemic equilibrium is globally asymptotically stable and hence the disease remains endemic. For more works on SIRI models described by ordinary differential equations, we refer to [4,5,6] and references therein.
Note that, due to mobility, the distribution of individuals in an area is not even. Modeling this phenomenon often results in reaction-diffusion equations. Consequently, inspired by [3], we have formulated a diffusive epidemic model with relapse and bilinear incidence as follows,
{St(x,t)=dΔS(x,t)+Λ−βS(x,t)I(x,t)−μS(x,t),t>0,x∈Ω,It(x,t)=dΔI(x,t)+βS(x,t)I(x,t)−(α+γ+μ)I(x,t)+ηR(x,t),t>0,x∈Ω,Rt(x,t)=dΔR(x,t)+γI(x,t)−(μ+η)R(x,t),t>0,x∈Ω,S(x,0)=S0(x)≥0,I(x,0)=I0(x)≥,≢0,R(x,0)=R0(x)≥0,x∈¯Ω;∂S∂n(x,t)=∂I∂n(x,t)=∂R∂n(x,t)=0,t>0,x∈∂Ω. | (1.2) |
Here S(x,t), I(x,t), and R(x,t) are the densities of susceptible, infective, and recovered individuals at time t and position x∈Ω, respectively; Ω is a bounded domain in Rn with a smooth boundary ∂Ω; Δ is the usual Laplacian operator; ∂∂n is the outward normal derivative to ∂Ω; d is the diffusion rate which represents the ability of random mobility of individuals; and the meanings of the other parameters are the same as those in (1.1). Note that the Neumann boundary conditions imply that individuals cannot move across the boundary ∂Ω.
It should be pointed out that any solution of (1.2) is always positive for any time t>0 no matter what the nonnegative nontrivial initial condition is. Thus the disease spreads to the whole area immediately, even though the infectious are confined to a quite small part of the habitat at the beginning. This does not agree with the observed fact that diseases always spread gradually. To compensate for the gradual disease spreading progress, a better modeling technique is to introduce free boundary.
The equation governing the free boundary, h′(t)=−μIx(h(t),t), is a special case of the well-known Stefan condition, which has been established in [7] for diffusive populations and used in the modeling of a number of applied problems. For example, it was used to describe the melting of ice in contact with water [8] and to model oxygen in muscles [9] as well as wound healing [10]. There is a vast literature on Stefan problems. Some important recent theoretical advances can be found in [11]. As a typical case, in 2013, Kim et al. [12] studied a diffusive SIR epidemic model in a radially symmetric domain with free boundary. They provided sufficient conditions on disease vanishing and spreading.
Motivated by the above discussion, in this paper, we investigate the behavior of nonnegative solutions (S(x,t),I(x,t),R(x,t);h(t)) of the following reaction-diffusion SIRI epidemic with free boundary,
{St(x,t)=dSxx(x,t)+Λ−βS(x,t)I(x,t)−δS(x,t),x>0,t>0,It(x,t)=dIxx(x,t)+βS(x,t)I(x,t)−(α+γ+δ)I(x,t)+ηR(x,t),0<x<h(t),t>0,Rt(x,t)=dRxx(x,t)+γI(x,t)−(δ+η)R(x,t),0<x<h(t),t>0,Sx(0,t)=Ix(0,t)=Rx(0,t)=0,t>0,I(x,t)=R(x,t)=0,x≥h(t),t>0,h′(t)=−μIx(h(t),t),t>0,h(0)=h0,S(x,0)=S0(x)≥0,I(x,0)=I0(x)≥0,R(x,0)=R0(x)≥0,x≥0, | (1.3) |
where x=h(t) is the moving boundary to be determined, μ represents the moving rate of the free boundary, δ is the natural death rate, and the meanings of the rest parameters are the same as those in model (1.2). All parameters are assumed to be positive. The nonnegative initial functions S0, I0 and R0 satisfy
{S0∈C2([0,+∞)),I0,R0∈C2([0,h0]),I0(x)=R0(x)=0 for x∈[h0,+∞) and I0(x)>0 for x∈[0,h0). | (1.4) |
In reality, I0(x)=0 for x∈[h0,+∞) and I0≢0 on [0,h0). Since for t>0, the solution though the initial condition (S0,I0,R0;h0) with such an I0 satisfy I(x,t)>0 on [0,h(t)) and I(x,t)=0 for x∈[h(t),+∞). Thus, without loss of generality, we make the assumption (1.4). Biologically, model (1.3) means that beyond the free boundary x=h(t), there are only susceptible individuals. We will also consider the case without relapse, that is, η=0. In this case, (1.3) reduces to
{St(x,t)=dSxx(x,t)+Λ−βS(x,t)I(x,t)−δS(x,t),x>0,t>0,It(x,t)=dIxx(x,t)+βS(x,t)I(x,t)−(α+γ+δ)I(x,t),0<x<h(t),t>0,Rt(x,t)=dRxx(x,t)+γI(x,t)−δR(x,t),0<x<h(t),t>0,Sx(0,t)=Ix(0,t)=Rx(0,t)=0,t>0,I(x,t)=R(x,t)=0,x≥h(t),t>0,h′(t)=−μIx(t,h(t)),t>0,h(0)=h0,S(x,0)=S0(x)≥0,I(x,0)=I0(x)≥0,R(x,0)=R0(x)≥0,x≥0. | (1.5) |
The remainder of this paper is organized as follows. In Section 2, we prove some general results on the existence and uniqueness of solutions to (1.3)–(1.4). In particular, solutions are global. Then, in Section 3, we provide some sufficient conditions on disease spreading and vanishing. For (1.5), the disease will die out either if the basic reproduction number R0<1 or if R0>1 and the initial infected area, boundary moving rate, and initial value of infected individuals are sufficiently small; while the disease will spread to the whole area if R0>1 and either the initial infected area is suitably large or the diffusion rate is suitably small. For (1.3), when the basic reproduction number ˜R0≤1, the disease will disappear, whereas when ˜R0>R0>1 and the initial infected area is suitably large, the disease will successfully spread. The paper ends with a brief conclusion and discussion.
First, we state the result on the local existence of solutions to (1.3)–(1.4), which can be proved with some modifications of the arguments in [10] and [13]. Hence we omit the proof to avoid repetition.
Theorem 2.1. For any given (S0,I0,R0) satisfying (1.4) and any r∈(0,1), there is a T>0 such that problem (1.3) admits a unique bounded solution
(S,I,R;h)∈C1+r,(1+r)2(D∞T)×[C1+r,(1+r)2(DT)]2×C1+r2([0,T]); |
moreover,
‖S‖C1+r,(1+r)2(D∞T)+‖I‖C1+r,(1+r)2(DT)+‖R‖C1+r,(1+r)2(DT)+‖h‖C1+r2([0,T])≤C, |
where D∞T={(x,t)∈R2:x∈[0,+∞),t∈[0,T]} and DT={(x,t)∈R2:x∈[0,h(t)],t∈[0,T]}. Here C and T only depend on h0, r, ‖S0‖C2([0,+∞)), ‖I0‖C2([0,h0]), and ‖R0‖C2([0,h0]).
Next we make some preparations to show the global existence of solutions.
Lemma 2.1. Problem (1.3)–(1.4) admits a unique and uniformly bounded solution (S,I,R;h) on (0,T0) for some T0∈(0,+∞], that is, there exists a constant M independent of T0 such that
0<S(x,t)≤Mfor0≤x<+∞,t∈(0,T0).0<I(x,t),R(x,t)≤Mfor0≤x<h(t),t∈(0,T0). |
Proof. As long as the solution exists, it is easy to see that S≥0, I≥0, and R≥0 on [0,+∞)×[0,T0]. By applying the strong maximum principle to the equations on {(x,t):x∈[0,h(t)],t∈[0,T0]}, we immediately obtain
S(x,t)>0 for 0≤x<+∞, 0<t<T0, I(x,t),R(x,t)>0 for 0≤x<h(t), 0<t<T0. |
It remains to prove the uniform boundedness of the solution (S(x,t),I(x,t),R(x,t);h(t)). For this purpose, define
U(x,t)=S(x,t)+I(x,t)+R(x,t),0≤x<+∞,t∈(0,T0). |
A direct calculation gives
dUdt=dSxx+dIxx+dRxx+Λ−δS−(δ+α)I−δR=dUxx+Λ−δ(S+I+R)−αI≤dUxx+Λ−δU, |
which gives U(x,t)≤max{‖U0‖∞,Λδ}≜M, where
‖U0‖∞=‖S(x,0)+I(x,0)+R(x,0)‖∞. |
Now the required result follows immediately.
Finally, we show that the free boundary of (1.3)–(1.4) is strictly monotonically increasing.
Lemma 2.2. Let (S,I,R;h) be a solution to problem (1.3)–(1.4) defined for t∈(0,T0) for some T0∈(0,+∞]. Then there exists a constant C1 independent of T0 such that
0<h′(t)≤C1 fort∈(0,T0). |
Proof. Using the strong maximum principle and Hopf boundary lemma to the equation of I, we can obtain Ix(h(t),t)<0 for t∈(0,T0). This, combined with the Stefan condition h′(t)=−μIx(h(t),t), gives h′(t)>0 for t∈(0,T0).
In order to get a bound for h′(t), we denote
ΩN:={(x,t):h(t)−N−1<x<h(t),0<t<T0}, |
and construct an auxiliary function
ωN(x,t):=M[2N(h(t)−x)−N2(h(t)−x)2]. |
We will choose N so that ωN(x,t)≥I(x,t) holds over ΩN.
Clearly, for (x,t)∈ΩN,
(ωN)t=2MNh′(t)[1−N(h(t)−x)]≥0,−(ωN)xx=2MN2,βSI−(α+γ+δ)I+ηR≤βM2+ηM. |
Therefore, if N2≥βM+η2d then
(ωN)t−d(ωN)xx≥2dMN2≥βM2+ηM. |
On the other hand, we have the boundary condition
ωN(h(t)−N−1,t)=M≥I(h(t)−N−1,t),ωN(h(t),t)=0=I(h(t),t). |
To employ the maximum principle to (ωN−I) over ΩN to deduce that I(x,t)≤ωN(x,t), we only have to find some N independent of T0 such that I0(x)≤ωN(x,0) for x∈[h0−N−1,h0]. It would then follow that
Ix(h(t),t)≥(ωN)x(h(t),t)=−2NM,h′(t)=−μIx(h(t),t)≤2μNM. |
Note that
I0(x)=I0(x)−I0(h0)=−∫h(t)xI′0(s)ds≤(h0−x)‖I′0‖C[0,h0] |
and
ωN(x,0):=M[2N(h0−x)−N2(h0−x)2]≥MN(h0−x),x∈[h0−N−1,h0]. |
It suffices to have
(h0−x)‖I′0‖C[0,h0]≤MN(h0−x). |
Thus choosing
N:=max{√βM+η2d,‖I′0‖C([0,h0])M} |
completes the proof.
By a similar argument as the one in [12,13], we can have the following result.
Theorem 2.2. The solution of problem (1.3)–(1.4) exists and is unique for all t∈(0,+∞).
This section is devoted to the spreading-vanishing dichotomy. We distinguish two cases, η=0 and η>0. We start with a sufficient condition on disease vanishing, which will be used in the coming discussion.
It follows from Lemma 2.2 that if x=h(t) is monotonically increasing, then h∞:=limt→∞h(t)∈(h0,+∞] is well defined.
Lemma 3.1. If h∞<+∞, then limt→+∞‖I(⋅,t)‖C([0,h(t)])=0. Moreover, limt→+∞‖R(⋅,t)‖C([0,h(t)])=0 and limt→+∞S(x,t)=Λδ uniformly in any bounded subset of [0,+∞).
Proof. Define
s=h0xh(t),u(s,t)=S(x,t),v(s,t)=I(x,t),w(s,t)=R(x,t). |
Then it is easy to see that
It=vt−h′(t)h(t)svs,Ix=h0h(t)vs,Ixx=h20h2(t)vss. |
It follows that v(s,t) satisfies
{vt−h′(t)h(t)svs−dh20h2(t)vss=v[βu−(α+δ+γ)]+ηw,0<s<h0,t>0,vs(0,t)=v(h0,t)=0,t>0,v(s,0)=I0(s)≥0,0≤s≤h0. |
This means that the transformation changes the free boundary x=h(t) into the fixed line s=h0 and hence we have an initial boundary value problem over a fixed area s<h0.
Since h0≤h(t)<h∞<+∞, the differential operator is uniformly parabolic. With the bounds in Lemma 2.1 and Lemma 2.2, there exist positive constants M1 and M2 such that
‖v(βu−(α+μ+γ))+ηw‖L∞≤M1and‖h′(t)h(t)s‖L∞≤M2. |
Applying the standard Lp theory and the Sobolev embedding theorem [14], we obtain that
‖v‖C1+α,1+α2([0,h0]×[0,+∞))≤M3 |
for some constant M3 depending on α, h0, M1, M2, and ‖I0‖C2[0,h0]. It follows that there exists a constant ˜C depending on α, h0, (S0,I0,R0), and h∞ such that
‖h‖C1+α2([0,+∞))≤˜C. | (3.1) |
Assume lim supt→+∞‖I(⋅,t)‖C([0,h(t)])=ϖ>0 by contradiction. Then there exists a sequence {(xk,tk)} in [0,h∞)×(0,+∞) such that I(xk,tk)≥ϖ2 for all k∈N and tk→+∞ as k→+∞. Since I(h(t),t)=0 and since (3.1) indicates that ∣Ix(h(t),t)∣ is uniformly bounded for t∈[0,+∞), there exists σ>0 such that xk≤h(tk)−σ for all k≥1. Then there is a subsequence of {xk} which converges to x0∈[0,h∞−σ]. Without loss of generality, we assume xk→x0 as k→+∞. Correspondingly,
sk:=h0xkh(tk)→s0:=h0x0h∞<h0. |
Define Sk(x,t)=S(x,tk+t), Ik(x,t)=I(x,tk+t), and Rk(x,t)=R(x,tk+t) for (x,t)∈(0,h(tk+t))×(−tk,+∞). It follows from the parabolic regularity that {(Sk,Ik,Rk)} has a subsequence {(Ski,Iki,Rki)} such that (Ski,Iki,Rki)→(˜S,˜I,˜R) as i→+∞. Since ‖h‖C1+α2([0,+∞))≤˜C, h′(t)>0, and h(t)≤h∞<+∞, it is necessary that h′(t)→0 as t→+∞. Hence (˜S,˜I,˜R) satisfies
{˜St−d1˜Sxx=Λ−β˜S˜I−δ˜S,0<x<h∞,t∈(−∞,+∞),˜It−d2˜Ixx=β˜S˜I−(α+γ+δ)˜I+η˜R,0<x<h∞,t∈(−∞,+∞),˜Rt−d3˜Rxx=γ˜I−(δ+η)˜R,0<x<h∞,t∈(−∞,+∞). |
Since ˜I(x0,0)≥ϖ2, the maximum principle implies that ˜I>0 on [0,h∞)×(−∞,+∞). Thus we can apply the Hopf lemma to conclude that σ0:=∂˜I∂s(h0,0)<0. It follows that
vx(h(tki),tki)=∂Iki(h0,0)∂sh0h(tki)≤σ02h0h∞<0 |
for all large i. Hence h′(tki)≥−μσ02h0h∞>0 for all large i, which contradicts with h′(t)→0 as t→+∞. This proves limt→+∞‖I(⋅,t)‖C([0,h(t)])=0.
Using a simple comparison argument, we can deduce that limt→+∞‖R(⋅,t)‖C([0,h(t)])=0 and limt→+∞S(x,t)=Λδ uniformly in any bounded subset of [0,+∞). In fact, for any ε>0, there exists a T0≥0 such that I(x,t)≤ε for t≥T0. Then, for t≥T0, we have
St≥dSxx+Λ−(βε+δ)S(x,t) |
and
Rt≤dRxx+γε−(δ+η)R(x,t). |
It follows that
lim inft→+∞S(x,t)≥Λβε+δuniformlyinanyboundedsubsetof[0,+∞) |
and
lim supt→+∞‖R(,⋅,t)‖C([0,h(t)])≤γεδ+η. |
As ε is arbitrarily, letting ε→0+ gives us
lim inft→+∞S(x,t)≥Λδuniformlyinanyboundedsubsetof[0,+∞) |
and
lim supt→+∞‖R(,⋅,t)‖C([0,h(t)])≤0. |
This immediately gives limt→+∞‖R(,⋅,t)‖C([0,h(t)])=0. Moreover, for t≥0, we have
St≤dSxx+Λ−δS(x,t). |
Then S(x,t)≤ˉS(t) for x∈(0,+∞) and t∈(0,+∞), where
ˉS(t):=Λδ+(ˉS(0)−Λδ)e−δt |
is the solution of the problem
dˉS(t)dt=Λ−δˉS(t),t>0;ˉS(0)=max{Λδ,‖S0‖∞}. |
Since limt→+∞ˉS(t)=Λδ, we deduce that
lim supt→+∞S(x,t)≤limt→+∞ˉS(t)=Λδuniformly forx∈[0,+∞). |
Therefore, we have limt→+∞S(x,t)=Λδ uniformly in any bounded subset of [0,+∞).
Consider the following eigenvalue problem,
{dϕxx+βΛδϕ−(α+γ+δ)ϕ+λϕ=0,x∈(0,h0),ϕx(0)=0,ϕ(h0)=0. | (3.2) |
It admits a principal eigenvalue λ1, where
λ1=dπ24h20−βΛδ+(α+γ+δ). |
The basic reproduction number of (1.5) denoted by R0 is given by
R0=βΛδ(γ+α+δ). |
With the assistance of the expression of R0, we can rewrite the expression of λ1 as
λ1=dπ24h20−βΛδ+(α+γ+δ)=dπ24h20−(1−1R0)βΛδ. |
It follows that λ1>0 either if R0≤1 or if R0>1 and h0<√dδπ24βΛ(1−1R0).
We first give some sufficient conditions on disease vanishing.
Theorem 3.2. If R0<1, then limt→+∞‖I(⋅,t)‖C([0,h(t)])=0 and limt→+∞‖R(⋅,t)‖C([0,h(t)])=0. Moreover, limt→+∞S(x,t)=Λδ uniformly in any bounded subset of [0,+∞).
Proof. From the proof of Lemma 3.1, we have obtained that
lim supt→+∞S(x,t)≤Λδuniformlyforx∈[0,+∞). |
Since R0<1, there exists T0 such that S(x,t)≤Λδ1+R02R0 on [0,+∞)×(T0,+∞). Then I(x,t) satisfies
{It(x,t)≤dIxx+[βΛδ1+R02R0−(α+γ+δ)]I(x,t),0<x<h(t),t>T0,Ix(0,t)=0,I(h(t),t)=0,t>T0,I(x,T0)>0,0≤x≤h(T0). |
We know that I(x,t)≤ˉI(x,t) for (x,t)∈{(x,t):x∈[0,h(t)],t∈(T0,+∞)}, where ˉI(x,t) satisfies
{ˉIt(x,t)=dˉIxx+[βΛδ1+R02R0−(α+γ+δ)]ˉI(x,t),0<x<h(t),t>T0,ˉIx(0,t)=ˉI(h(t),t)=0,t>T0,ˉI(x,T0)≥‖I(⋅,T0)‖∞>0,0≤x≤h(T0). |
Since βΛδ1+R02R0−(α+γ+δ)=(α+γ+δ)(R0−1)2<0, we have limt→+∞‖ˉI(⋅,t)‖C[0,h(t)]=0. Then it follows from I(x,t)≤ˉI(x,t) that ‖I(⋅,t)‖C[0,h(t)]→0 as t→+∞. The remaining part follows from Lemma 3.1.
Theorem 3.2. Suppose R0>1. Then h∞<+∞ for given initial condition (S0,I0,R0;h0) satisfying h0≤min{√d16k0,√d16γ} and μ≤d8K, where k0=βM−α−γ−δ>0, M=max{‖S0‖∞,Λδ}, and K=43max{‖I0‖∞,‖R0‖∞}.
Proof. Since R0>1, one can easily see that k0>0. Inspired by [13], we define
ˉS(x,t)=M,ˉI(x,t)={Ke−θtV(xˉh(t)),0≤x≤ˉh(t),0,x>ˉh(t),ˉR(x,t)={Ke−θtV(xˉh(t)),0≤x≤ˉh(t),0,x>ˉh(t),V(y)=1−y2,0≤y≤1,ˉh(t)=2h0(2−e−θt),t≥0, |
where θ is a constant to be determined later. In the following, we show that (ˉS,ˉI,ˉR;ˉh) is an upper solution to (1.5).
For 0<x<ˉh(t) and t>0, direct computations yield
ˉSt−dˉSxx=0≥Λ−δˉS,ˉIt−dˉIxx−(βˉS−α−γ−δ)ˉI=ˉIt−dˉIxx−k0ˉI=Ke−θt[−θV−xˉh′ˉh−2V′−dˉh−2V″−k0V]≥Ke−θt[d8h20−θ−k0],ˉRt−dˉRxx−(γˉI−δˉR)≥Ke−θt[d8h20−θ−γ],ˉh′(t)=2h0θe−θt,−μˉIx(ˉh(t),t)=2Kμˉh−1(t)e−θt. |
Moreover,
ˉS(x,0)≥S0(x),ˉI(x,0)=K(1−x24h20)≥34Kfor x∈[0,h0], ˉR(x,0)=K(1−x24h20)≥34Kfor x∈[0,h0]. |
Choose \theta = \frac{d}{16h_0^2} . Noting \bar h(t)\leq4h_0 , we have
\begin{equation*} \begin{cases} \bar{S}_t-d \bar{S}_{xx} \geq \Lambda-\delta \bar{S}, &x \gt 0, t \gt 0 , \\ \bar{I}_t-d\bar{I}_{xx} \geq \beta \bar{S}\bar{I}-(\alpha+\gamma+\delta)\bar{I}, &0 \lt x \lt \bar{h}(t), t \gt 0, \\ \bar{R}_t-d\bar{R}_{xx} \geq \alpha \bar{I}-\delta\bar{R}, &0 \lt x \lt \bar{h}(t), t \gt 0, \\ \bar{S}_x(0, t) \geq 0, \; \bar{I}_x(0, t) \geq 0, \; \bar{R}_x(0, t)\geq 0, &t \gt 0, \\ \bar{I}(x, t) = \bar{R}(x, t) = 0, &x \geq \bar{h}(t), 0 \lt t \leq T, \\ \bar{h}'(t) \geq -\mu \bar{I}_x(\bar{h}(t), t), \; \bar{h}(0) = 2h_0\geq h_0, &t \gt 0 , \\ \bar{S}(x, 0) \geq S_0(x), \; \bar{I}(x, 0) \geq I_0(x), \; \bar{R}(x, 0) \geq R_0(x), \quad &0 \leq x \leq h_0. \end{cases} \end{equation*} |
This verifies that (\bar{S}, \bar{I}, \bar{R}; \bar{h}) is an upper solution to (1.5). Then we can apply a result similar as [12,Lemma 4.1] (which can be proved in the same manner as [13,Lemma 5.6]) to conclude that h(t)\leq\bar h(t) for t > 0 . This implies that h_\infty\leq \lim\limits_{t\rightarrow +\infty}\bar h(t) = 4h_0 < +\infty .
Theorem 3.3. Assume that \mathcal{R}_0 > 1 . For given initial condition (S_0, I_0, R_0;h_0) , we have h_\infty < +\infty provided that h_0 < h_*: = \min\Big\{\sqrt{\frac{d\pi^2}{4[\beta N-(\alpha+\gamma+\delta)]}}, \frac{\sqrt{d\gamma}}{4\gamma}\Big\} and both \|I_0\|_{\infty} and \|R_0\|_{\infty} are sufficiently small (which is specified in the proof), where N = \max\{\frac{\Lambda}{\delta}, \|S_0\|_{\infty}\} .
Proof. Note that h_* is well defined since \mathcal{R}_0 > 1 . As in the proof of Theorem 3.2, we will construct a suitable upper solution to (1.5). Since h_0 < h_* , there exists \varepsilon_1 > 0 such that h_0 < \sqrt{\frac{d\pi^2}{4[\beta(N+\varepsilon_1)-(\alpha+\gamma+\delta)]}} . Then the principal eigenvalue of the eigenvalue problem
\begin{equation*} \begin{cases} d\phi_{xx}+\beta (N+\varepsilon_1)\phi-(\alpha+\gamma+\delta)\phi+\lambda \phi = 0, \quad & 0 \lt x \lt h_0 \\ \phi_{x}(0) = \phi (h_0) = 0. \end{cases} \end{equation*} |
is
\widetilde{\lambda}_1 = \frac {d\pi^2}{4h_0^2}-\beta (N+\varepsilon_1)+\alpha+\delta+\gamma \gt 0 |
and it is has a normalized positive eigenfunction \tilde{\phi} on (0, h_0) . Moreover, \tilde{\phi}_x < 0 on (0, h_0] . Choose \varepsilon_2\in (0, \gamma) such that
\widetilde{\lambda}_1 \gt [\beta(N+\varepsilon_1)+\varepsilon_2](1+\varepsilon_2)^2-\beta(N+\varepsilon_1) \gt 0. |
Recall that \limsup\limits_{t\to+\infty}S(t, x)\le \frac{\Lambda}{\delta} uniformly for x\in [0, +\infty) . Thus there exists a T_0 > 0 such that 0 < S(x, t)\leq (N+\varepsilon_1) in [0, +\infty)\times[T_0, +\infty) . As in [15], we define
\begin{eqnarray*} \vartheta(t)& = & h_0\left(1+\varepsilon_2-\frac{\varepsilon_2}{2}e^{-\varepsilon_2 t}\right), \\ \bar{S}(x, t) & = & (N+\varepsilon_1), \qquad t \geq T_0, \\ \bar{I}(x, t) & = & \begin{cases} \iota e^{-\varepsilon_2 t}\tilde{\phi} (\frac{xh_0}{\vartheta(t)}), \qquad & 0 \leq x \leq \vartheta(t), t \geq T_0, \\ 0, & x \gt \vartheta (t), t \geq T_0, \end{cases} \\ \bar{R}(x, t) & = & \begin{cases} \iota e^{-\varepsilon_2 t}V (\frac{x}{\vartheta (t)}), \qquad & 0 \leq x \leq \vartheta(t), t \geq T_0, \\ 0, & x \gt \vartheta (t), t \geq T_0. \end{cases}\\ V(y) & = & 1-y^2, \qquad 0\leq y \leq 1, \end{eqnarray*} |
where \iota is a positive number to be determined later. As \tilde{\phi}(h_0) = 0 , it follows that \bar{I}(\vartheta (t), t) = 0 for t\ge T_0 , which implies that the function \bar{I}(x, t) is continuous on [0, +\infty)\times [0, +\infty) . Similarly, as V(1) = 0 , we know that \bar{R} is also continuous on [0, +\infty)\times [0, +\infty) . Detailed calculations yield \bar{S}_t-d\bar{S}_{xx} = 0\ge \Lambda-\delta\bar{S} and, for 0\leq x\leq\vartheta(t) ,
\begin{eqnarray*} & & \bar{I}_{t}-d \bar{I}_{xx}-\beta \bar{S} \bar{I}+(\alpha+\gamma+\delta) \bar{I} \\ & = & \iota e^{-\varepsilon_2 t}\left[-\varepsilon_2 \tilde{\phi} -\frac{ x h_0 \vartheta'(t)}{\vartheta^2(t)}\tilde{\phi}_x-\frac{dh^{2}_{0}}{\vartheta^2(t)}\tilde{\phi}_{xx}-\beta (N+\varepsilon_1)\tilde{\phi}+(\alpha+\gamma+\delta)\tilde{\phi}\right] \\ & = & \iota e^{-\varepsilon_2 t}\left\{-\varepsilon_2\tilde{\phi} -\frac{ x h_0 \vartheta'(t)}{\vartheta^2(t)}\tilde{\phi}_x-\frac{h^{2}_{0}}{\vartheta^2(t)}\left[-\beta (N+\varepsilon_1)\tilde{\phi}+(\alpha+\gamma+\delta)\tilde{\phi}-\widetilde{\lambda}_1 \tilde{\phi}\right]\right. \\ && \left.-\beta (N+\varepsilon_1)\tilde{\phi}+(\alpha+\gamma+\delta)\tilde{\phi}\right\} \\ & = & \iota e^{-\varepsilon_2 t}\left[-\varepsilon_2\tilde{\phi} -\frac{ x h_0 \vartheta'(t)}{\vartheta^2(t)}\tilde{\phi}_x+\left(\frac{h^{2}_0}{\vartheta^2(t)}-1\right)\beta (N+\varepsilon_1)\tilde{\phi} +\left(1-\frac{h^{2}_0}{\vartheta^2(t)}\right)(\alpha+\gamma+\delta)\tilde{\phi} +\frac{h^{2}_0}{\vartheta^2(t)}\widetilde{\lambda}_1\tilde{\phi}\right] \\ & \geq & \tilde{\phi}\iota e^{-\varepsilon_2 t} \left\{-\varepsilon_2+\frac{h^{2}_0}{\vartheta^2(t)}\left[\beta (N+\varepsilon_1)+\widetilde{\lambda}_1\right] -\beta (N+\varepsilon_1)\right\} \\ &\geq & \tilde{\phi}\iota e^{-\varepsilon_2 t}\left\{-\varepsilon_2+\frac{h^{2}_0}{h_0^2(1+\varepsilon_2)^2}\left [\beta (N+\varepsilon_1)+\widetilde{\lambda}_1\right] - \beta (N+\varepsilon_1)\right\} \\ &\geq & \tilde{\phi}\iota e^{-\varepsilon_2 t}\left\{-\varepsilon_2 +\frac{1}{(1+\varepsilon_2)^2}\left [\beta (N+\varepsilon_1)+\widetilde{\lambda}_1\right]-\beta (N+\varepsilon_1)\right\}. \end{eqnarray*} |
Here we have used the fact that \tilde{\phi}_x < 0 for x\in (0, h_0] . It follows that \bar{I}_{t}-d \bar{I}_{xx}-\beta \bar{S} \bar{I}+(\alpha+\gamma+\delta) \bar{I}\ge 0 . On the other hand, as h_0 < h_* , we can obtain
\bar{R}_t-d\bar{R}_{xx}-\gamma \bar{I}+\delta\bar{R} \geq \iota e^{-\varepsilon_2 t} \left(-\varepsilon_2 -\gamma +\frac{d}{8h_{0}^{2}}\right)\ge \iota e^{-\varepsilon_2t}\left(-2\gamma+\frac{d}{8h_0^2}\right) \geq 0. |
Moreover,
-\mu\bar{I}_x (\vartheta(t), t) = -\mu \iota e^{-\varepsilon_2 t}\tilde{\phi}_x(h_0)\frac{h_0}{\vartheta (t)}. |
If we choose 0 < \iota\leq-\varepsilon_2^2h_0(1+\frac{\varepsilon_2}{2})/2\mu\tilde{\phi}_x(h_0) , then
\vartheta'(t)\geq-\mu\bar{I}_x(\vartheta(t), t) |
since \tilde{\phi}_x(h_0) < 0 . Obviously, \bar{S}(x, 0)\geq\left\| S_0\right\| _\infty . If \|I_0\|_{\infty}\le \iota\phi(\frac{x}{1+\frac{\varepsilon_2}{2}}) and \|R_0\|_{\infty}\le V(\frac{x}{h_0(1+\frac{\varepsilon_2}{2})}) for x\in [0, h_0] , then I_0(x)\leq\bar{I}(x, 0) and R_0(x)\leq\bar{R}(x, 0) for x > 0 . This proves that (\bar{S}, \bar{I}, \bar{R}; \vartheta(t)) is an upper solution of (1.5). Thus, similalrly as in the proof of Theorem 3.2, we can get h(t)\leq\vartheta (t) , which yields h_\infty < \lim\limits_{t\rightarrow+\infty}\vartheta(t) = h_0(1+\varepsilon_2) < +\infty . This completes the proof.
We provide a sufficient condition on disease spreading to conclude this subsection.
Theorem 3.4. If \mathcal{R}_0 > 1 and h_0 > h^{*}: = \sqrt{\frac{d\delta\pi^{2}}{4\beta\Lambda(1-\frac{1}{\mathcal{R}_{0}})}} , then h_\infty = +\infty .
Proof. By way of contradiction, we assume that h_\infty < +\infty . It follows from Lemma 3.1 that \lim\limits_{t\rightarrow +\infty} \left\| I(\cdot, t)\right\|_{C([0, h(t)])} = 0 . Moreover, \lim\limits_{t\rightarrow +\infty} S(x, t) = \frac{\Lambda}{\delta} uniformly in any bounded subset of [0, +\infty) .
Since h_0 > h^{*} and \mathcal{R}_0 > 1 , we have \lambda_1 < 0 , where \lambda_1 is the principal eigenvalue of the eigenvalue problem (3.2). Choose \iota > 0 such that \lambda_1+\beta\iota < 0 and \mathcal{R}_0 > 1+\frac{\beta \iota}{\alpha+\delta+\gamma} (which implies that \beta(\frac{\Lambda}{\delta}-\iota)-\delta-\alpha-\gamma > 0 ). For this \iota , there exists T^* > 0 such that S(x, t)\geq\frac{\Lambda}{\delta}-\iota and I(x, t) < 1 for x\in [0, h(t)] and t > T^* . Then I(x, t) satisfies
\begin{equation*} \begin{cases} I_t-d I_{xx}\geq\left[\beta (\frac{\Lambda}{\delta}-\iota)-\delta-\alpha-\gamma\right]I(1-I), \quad &0 \lt x \lt h_0, t \gt T^*, \\ I_x(0, t) = 0, \quad I(h_0, t)\geq0, &t \gt T^*, \\ I(x, T^{*}) \gt 0, &0 \leq x \lt h_0. \end{cases} \end{equation*} |
It is easy to see that I(x, t)\ge \underline{I}(x, t) , where \underline{I}(x, t) satisfies
\begin{equation} \begin{cases} \underline{I}_t-d \underline{I}_{xx} = \left[\beta (\frac{\Lambda}{\delta}-\iota)-\delta-\alpha-\gamma\right]\underline{I}(1-\underline{I}), \quad &0 \lt x \lt h_0, t \gt T^*, \\ \underline{I}_x(0, t) = 0, \quad \underline{I}(h_0, t) = 0, &t \gt T^*, \\ \underline{I}(x, T^{*}) = I(x, T^*), &0 \leq x \lt h_0. \end{cases} \end{equation} | (3.3) |
Consider the following eigenvalue problem
\begin{equation*} \begin{cases} d\phi_{xx}+\left[\beta (\frac{\Lambda}{\delta}-\iota)-\delta-\alpha-\gamma\right]\phi+ \lambda\phi = 0, \quad 0 \lt x \lt h_0, \\ \phi_x(0) = \phi(h_0) = 0, \end{cases} \end{equation*} |
whose principal eigenvalue is
\widehat{\lambda}_1 = \frac{d\pi^2}{4h_0^2}- \left[\beta \left(\frac{\Lambda}{\delta}-\iota\right)-\delta-\alpha-\gamma\right] = \lambda_1+\beta\iota \lt 0. |
Employing Proposition 3.2 and Proposition 3.3 of [16], we obtain that \lim\limits_{t\rightarrow +\infty}\underline{I}(t, x) = \underline{I}(x) uniformly in x\in [0, h_0] , where \underline{I}(x) > 0 satisfies
\begin{equation*} \begin{cases} -d \underline{I}_{xx} = \left[\beta (\frac{\Lambda}{\delta}-\iota)-\delta-\alpha-\gamma\right]\underline{I}(1-\underline{I}), \quad &0 \lt x \lt h_0, \\ \underline{I}_x(0) = 0, \quad \underline{I}(h_0) = 0. \end{cases} \end{equation*} |
It follows that \liminf\limits_{t\rightarrow +\infty} I(x, t)\geq\lim\limits_{t\rightarrow +\infty} \underline{I}(x, t) = \underline{I}(x) > 0 uniformly in x\in [0, h_0] , which contradicts with \lim\limits_{t\rightarrow +\infty} \left\| I(\cdot, t)\right\|_{C([0, h(t)])} = 0 . Therefore, we have proved h_{\infty} = +\infty .
Remark 3.1. Obviously, h_0 > h^* is equivalent to d < d^*\triangleq \frac{4h_{0}^2\beta \Lambda(1-\frac{1}{\mathcal{R}_{0}})}{\delta\pi^{2}} . As a result, if \mathcal{R}_0 > 1 and 0 < d < d^{*} , then h_\infty = +\infty .
In this case, the basic reproduction number \widetilde{\mathcal{R}}_0 of problem (1.3) is given by
\widetilde{\mathcal{R}}_{0} = \frac{\beta\Lambda(\delta+\eta)}{\delta[\gamma\delta+(\delta+\eta)(\alpha+\delta)]}. |
As in the case where \eta = 0 , we start with disease vanishing.
Theorem 3.5. If \widetilde{\mathcal{R}}_0\leq1 , then \lim\limits_{t\rightarrow +\infty} \left\| I(\cdot, t)\right\|_{C([0, h(t)])} = 0 . Moreover, \lim\limits_{t\rightarrow +\infty} \left\| R(\cdot, t)\right\|_{C([0, h(t)])} = 0 and \lim\limits_{t\rightarrow +\infty} S(x, t) = \frac{\Lambda}{\delta} uniformly in any bounded subset of [0, +\infty) .
Proof. Consider the following system of ordinary differential equations,
\begin{equation} \begin{cases} \frac{d S(t)}{d t} = \Lambda-\beta S(t)I(t)-\delta S(t), \\ \frac{d I(t)}{d t} = \beta S(t)I(t)-(\alpha+\gamma+\delta)I(t)+\eta R(t), \\ \frac{d R(t)}{d t} = \gamma I(t)-(\delta+\eta)R(t), \end{cases} \end{equation} | (3.4) |
with (S(0), I(0), R(0)) = (\left\| S_0\right\|_{\infty}, \left\| I_0\right\|_{\infty}, \left\| R_0\right\|_{\infty}) . As in the proof of Theorem 3.2, a result similar as [12, Lemma 4.1] implies that S(x, t)\leq S(t) for (x, t)\in[0, +\infty)\times(0, +\infty) , and I(x, t)\leq I(t) and R(x, t)\leq R(t) for (x, t)\in\{(x, t):x\in [0, h(t)], t\in (0, +\infty)\} .
Obviously, (3.4) has a disease-free equilibrium E_{0} = (\frac{\Lambda}{\delta}, 0, 0) , which is globally asymptotically stable. Indeed, consider V:\mathbb{R}^{3}_+\rightarrow\mathbb{R} defined by
\begin{equation} V(S, I, R) = (\delta+\eta)\left (S-S^0-S^0\ln\frac{S}{S^0}\right)+(\delta+\eta)I+\eta R. \end{equation} | (3.5) |
It is clear that V (S, I, R) reaches its global minimum in \mathbb{R}^{3}_+ only at E_0 . Moreover, the derivative of (3.5) with respect to t along solutions of (3.4) is
\begin{eqnarray*} \frac{d}{dt}V (S, I, R)& = & (\delta+\eta)\frac{S-S^0}{S}\frac{d S}{d t}+(\delta+\eta)\frac{d I}{d t}+\eta\frac{d R}{d t} \\ & = & (\delta+\eta)\frac{S-S^0}{S}(\Lambda -\beta SI -\delta S) \\ & &+(\delta+\eta)[\beta SI -(\alpha + \gamma +\delta)I+\eta R]+\eta[\gamma I -(\delta+\eta)R] \\ & = & (\delta+\eta)\frac{S-S^0}{S}(\Lambda -\beta SI -\delta S)\\ & & +(\delta+\eta)[\beta SI -(\alpha + \gamma +\delta)I +\eta R]+\eta[\gamma I -(\delta+\eta)R]. \end{eqnarray*} |
Using the expression
\beta SI\frac{(S-S^0)}{S^0} = \beta I\frac{(S-S^0)^2}{S^0}+\beta I(S-S^0), |
we obtain
\begin{eqnarray*} \frac{d}{dt}V (S, I, R)& = & (\delta+\eta)\frac{S-S^0}{S}(\Lambda -\beta SI -\delta S) \\ & &+(\delta+\eta)[\beta SI -(\alpha + \gamma +\delta)I +\eta R]+\eta[\gamma I -(\delta+\eta)R] \\ & = & -(\delta+\eta)\frac{(S-S^{0})^{2}}{S} \\ & & +[\gamma \delta+(\delta+\eta)(\alpha+\delta)]I\left[\frac{(\delta+\eta)S^{0}\beta}{(\gamma \delta+\delta+\eta)(\alpha+\delta)}-1\right] \\ & = & -(\eta+\delta)\frac{(S-S^{0})^{2}}{S}-[\gamma \delta+(\delta+\eta)(\alpha+\delta)]I(1-\widetilde{\mathcal{R}}_{0}). \end{eqnarray*} |
Since \widetilde{\mathcal{R}}_0\le 1 , we have \frac{d}{dt}V(S, I, R)\le 0 for S > 0 . Moreover, if \frac{d}{dt}V(S, I, R) = 0 holds then S = S^0 . It is easy to verify from this that the disease-free equilibrium E_0 is the largest invariant set in the set where \frac{d}{dt}V(S, I, R) = 0 . Therefore, by LaSalle's invariance principle [17], E_0 is globally asymptotically stable. This, combined with the above estimates, gives us
\begin{eqnarray*} && \limsup\limits_{t\rightarrow +\infty}S(x, t)\leq \lim\limits_{t\rightarrow +\infty} S(t) = \frac{\Lambda}{\delta}\quad \mbox{uniformly for $x\in [0, +\infty)$, } \\ && \limsup\limits_{t\to\infty} I(x, t)\le \lim\limits_{t\to\infty}I(t) = 0\quad \mbox{uniformly in any bounded subset of $[0, h_\infty)$, } \\ && \limsup\limits_{t\to\infty} R(x, t)\le \lim\limits_{t\to\infty} R(t) = 0 \quad \mbox{uniformly in any bounded subset of $[0, h_\infty)$, } \end{eqnarray*} |
which implies that
\lim\limits_{t\rightarrow +\infty}\left\| I(\cdot, t)\right\| _{C([0, h(t)])} = \lim\limits_{t\rightarrow +\infty}\left\| R(\cdot, t)\right\|_{C([0, h(t)])} = 0. |
Then it follows from Lemma 3.1 that \lim\limits_{t\to+\infty}S(x, t) = \frac{\Lambda}{\delta} uniformly in any bounded subset of [0, +\infty) and this completes the proof.
Now we provide a sufficient condition on disease spreading.
Theorem 3.6. If \widetilde{\mathcal{R}}_0 > \mathcal{R}_0 > 1 and h_0 > h^{*}: = \sqrt{\frac{d\delta\pi^{2}}{4\beta\Lambda(1-\frac{1}{\mathcal{R}_{0}})}} , then h_\infty = +\infty .
Proof. We know that (S(x, t), I(x, t), R(x, t);h(t)) satisfies
\begin{equation*} \begin{cases} S_t(x, t) = d S_{xx}(x, t)+\Lambda-\beta S(x, t)I(x, t)-\delta S(x, t), & x \gt 0, t \gt 0, \\ I_t(x, t) \geq d I_{xx}(x, t)+\beta S(x, t)I(x, t)-(\alpha+\gamma+\delta)I(x, t), \qquad &0 \lt x \lt h(t), t \gt 0, \\ R_t(x, t) = d R_{xx}(x, t)+\gamma I(x, t)-(\delta+\eta)R(x, t), &0 \lt x \lt h(t), t \gt 0, \\ S_{x}(0, t) = I_{x}(0, t) = R_{x}(0, t) = 0, &t \gt 0, \\ I(h(t), t) = R(h(t), t) = 0, &x\geq h(t), \; t \gt 0, \\ h^{'}(t) = -\mu I_{x}(h(t), t), &t \gt 0, \\ h(0) = h_{0}, \\ S(x, 0) = S_{0}(x)\geq0, \; I(x, 0) = I_{0}(x)\geq0, \; R(x, 0) = R_{0}(x)\geq0, &x\geq0. \end{cases} \end{equation*} |
A result similar as [12, Lemma 4.1] for lower solutions gives S(x, t)\geq\underline{S}(x, t) for 0 < x < +\infty and t > 0 ; I(x, t)\geq\underline{I}(x, t) and R(x, t)\geq\underline{R}(x, t) for 0 < x < \underline{h}(t) and t > 0 ; and h(t)\geq\underline{h}(t) for t > 0 , where (\underline{S}(x, t), \underline{I}(x, t), \underline{R}(x, t);\underline{h}(t)) satisfies
\begin{equation*} \begin{cases} \underline{S}_t(x, t) = d \underline{S}_{xx}+\Lambda-\beta \underline{S}(x, t)\underline{I}(x, t)-\delta \underline{S}(x, t), & x \gt 0, t \gt 0, \\ \underline{I}_t(x, t) = d \underline{I}_{xx}+\beta \underline{S}(x, t)\underline{I}(x, t)-(\alpha+\gamma+\delta)\underline{I}(x, t), \qquad &0 \lt x \lt \underline{h}(t), t \gt 0, \\ \underline{R}_t(x, t) = d \underline{R}_{xx}+\gamma \underline{I}(x, t)-(\delta+\eta)\underline{R}(x, t), &0 \lt x \lt \underline{h}(t), t \gt 0, \\ \underline{S}_{x}(0, t) = \underline{I}_{x}(0, t) = \underline{R}_{x}(0, t) = 0, &t \gt 0, \\ \underline{I}(h(t), t) = \underline{R}(h(t), t) = 0, &x\geq \underline{h}(t), t \gt 0, \\ \underline{h}^{'}(t) = -\mu I_{x}(\underline{h}(t), t), &t \gt 0, \\ \underline{h}(0) = h_{0}, \\ \underline{S}(x, 0) = S_{0}(x)\geq0, \; \underline{I}(x, 0) = I_{0}(x)\geq0, \underline{R}(x, 0) = R_{0}(x)\geq0, &x\geq0. \end{cases} \end{equation*} |
It follows from Theorem 3.4 that if \widetilde{\mathcal{R}}_0 > \mathcal{R}_0 > 1 and h_0 > h^{*} then \underline{h}_\infty = +\infty , which implies h_\infty = +\infty .
In this paper, we proposed and analyzed a free boundary problem of a reaction-diffusion SIRI model with the bilinear incidence rate. We first obtained the existence and uniqueness of global solutions. Then we established several criteria on disease vanishing and spreading. Roughly speaking, for the case without relapse, the disease will vanish if one of the following three conditions holds. (a) The basic reproduction number \mathcal{R}_0 < 1 ; (b) \mathcal{R}_0 > 1 and the initial infected area h_0 and the boundary moving rate \mu are small enough; (c) \mathcal{R}_0 > 1 together with the initial values \|I_0\|_{\infty} , \|R_0\|_{\infty} , and h_0 being small enough. The disease will spread to the whole area if \mathcal{R}_0 > 1 and either h_0 is large enough or the diffusion rate d is small enough. For the case with relapse, the disease will die out if the basic reproduction number \widetilde{\mathcal{R}}_0\le 1 whereas the disease will spread to the whole area if \widetilde{\mathcal{R}}_0 > \mathcal{R}_0 > 1 and h_0 is large enough. Unfortunately, we have not considered the case where \widetilde {R}_0 > 1 > \mathcal{R}_0 . In this case, the disease transmission is complex, which we are working on. Moreover, when the free boundaries can extend to the whole area, we also gave an estimate on the spreading speed.
Compared with the ordinary differential equation model (1.1), the model we studied with free boundary allows more reasonable sufficient conditions on the disease spreading and vanishing. With the main results obtained, we can better understand the phenomenon of relapse. To illustrate this, we demonstrate how the basic reproduction numbers rely on the relapse rate \eta . For system (1.3), fix other parameters except \eta , we see that \mathcal{R}^{*}_{0}(\eta) = \widetilde{\mathcal{R}}_0 = \frac{\beta\Lambda(\delta+\eta)}{\delta(\gamma\delta+(\delta+\eta)(\alpha+\delta))} , which is a strictly increasing function of \eta . Thus there exists an \eta^*\in [0, +\infty) such that \mathcal{R}^*_0(\eta)\ge 1 when \eta\geq\eta^* and \mathcal{R}^{*}_{0}(\eta) < 1 when \eta < \eta^* . Then the relapse rate \eta plays an important role in \mathcal{R}^{*}_{0}(\eta) . In other words, when \eta varies, disease spreading and vanishing will change. Since \mathcal{R}^*_0(\eta) > \mathcal{R}_0 always holds, with relapse the disease will be more easily spread to the whole area than without relapse.
The authors would like to thank the two anonymous reviewers for their valuable suggestions and comments, which greatly improve the presentation of the paper. QD, YL, and ZG were supported by the National Natural Science Foundation of China (No. 11771104) and by the Program for Chang Jiang Scholars and Innovative Research Team in University (IRT-16R16). YC was supported partially by NSERC.
All authors declare no conflicts of interest in this paper.
[1] |
Ho WJ, Hsiao KY, Hu CH, et al. (2017) Characterized plasmonic effects of various metallic nanoparticles on silicon solar cells using the same anodic aluminum oxide mask for film deposition. Thin Solid Films 631: 64–71. doi: 10.1016/j.tsf.2017.04.016
![]() |
[2] |
Starowicz Z, Kędra A, Berent K, et al. (2017) Influence of Ag nanoparticles microstructure on their optical and plasmonic properties for photovoltaic applications. Sol Energy 158: 610–616. doi: 10.1016/j.solener.2017.10.020
![]() |
[3] | Taylor G (1969) Electrically Driven Jets. Proc R Soc Lond A 313: 453–475. |
[4] | Melcher JR (1963) Field-Couple Surface Waves: A Comparative Study of Surface Coupled Electrohydrodynamic and Magnetohydrodynamic Systems, Cambridge, Massachusetts: The MIT Press, 1–63. |
[5] |
VillaVelázquez-Mendoza CI, Mendoza-Barraza SS, Rodriguez JL, et al. (2016) Simultaneous synthesis of β-Si3N4 nanofibers and pea-pods and hand-fan like Si2N2O nanostructures by the CVD method. Mater Lett 175: 139–142. doi: 10.1016/j.matlet.2016.04.028
![]() |
[6] |
Maldonado JR, Peckerar M (2016) X-Ray lithography: Some history, current status and future prospects. Microelectron Eng 161: 87–93. doi: 10.1016/j.mee.2016.03.052
![]() |
[7] |
Stoychev GV, Okhrimenko DV, Appelhans D, et al. (2016) Electron beam-induced formation of crystalline nanoparticle chains from amorphous cadmium hydroxide nanofibers. J Colloid Interf Sci 461: 122–127. doi: 10.1016/j.jcis.2015.09.023
![]() |
[8] |
Subbiah T, Bhat GS, Tock RW, et al. (2005) Electrospinning of nanofibers. J Appl Polym Sci 96: 557–559. doi: 10.1002/app.21481
![]() |
[9] |
Gan YX, Chen AD, Gan RN, et al. (2017) Energy conversion behaviors of antimony telluride particle loaded partially carbonized nanofiber composite mat manufactured by electrohydrodynamic casting. Microelectron Eng 181: 16–21. doi: 10.1016/j.mee.2017.06.009
![]() |
[10] |
Gan YX, Draper CW, Gan JB (2017) Carbon nanofiber network made by electrohydrodynamic casting immiscible fluids. Mater Today Commun 13: 248–254. doi: 10.1016/j.mtcomm.2017.10.008
![]() |
[11] |
Han Y, Wei C, Dong J (2015) Droplet formation and settlement of phase-change ink in high resolution electrohydrodynamic (EHD) 3D printing. J Manuf Process 20: 485–491. doi: 10.1016/j.jmapro.2015.06.019
![]() |
[12] |
Han Y, Dong J (2017) High-resolution electrohydrodynamic (EHD) direct printing of molten metal. Procedia Manuf 10: 845–850. doi: 10.1016/j.promfg.2017.07.070
![]() |
[13] |
Zhang Y, Huang ZM, Xu X, et al. (2004) Preparation of core-shell structured PCL-r-gelatin bi-component nanofibers by coaxial electrospinning. Chem Mater 16: 3406–3409. doi: 10.1021/cm049580f
![]() |
[14] |
Loscertales IG, Barrero A, Guerrero I, et al. (2002) Micro/nano encapsulation via electrified coaxial liquid jets. Science 295: 1695–1698. doi: 10.1126/science.1067595
![]() |
[15] |
Kurban Z, Lovell A, Bennington SM, et al. (2010) A solution selection model for coaxial electrospinning and its application to nanostructured hydrogen storage materials. J Phys Chem C 114: 21201–21213. doi: 10.1021/jp107871v
![]() |
[16] |
Wang C, Yan KW, Lin YD, et al. (2010) Biodegradable core/shell fibers by coaxial electrospinning: Processing, fiber characterization, and its application in sustained drug release. Macromolecules 43: 6389–6397. doi: 10.1021/ma100423x
![]() |
[17] |
Zhang YZ, Wang X, Feng Y, et al. (2006) Coaxial electrospinning of (fluorescein isothiocyanate-conjugated bovine serum albumin)-encapsulated poly(ɛ-caprolactone) nanofibers for sustained release. Biomacromolecules 7: 1049–1057. doi: 10.1021/bm050743i
![]() |
[18] |
Zhang H, Zhao CG, Zhao YH, et al. (2010) Electrospinning of ultrafine core/shell fibers for biomedical applications. Sci China Chem 53: 1246–1254. doi: 10.1007/s11426-010-3180-3
![]() |
[19] | Li F, Zhao Y, Song Y (2010) Core-shell nanofibers: Nano channel and capsule by coaxial electrospinning, In: Kumar A, Nanofibers, Croatia: InTech, 419–438. |
[20] |
Chan KHK, Kotaki M (2009) Fabrication and morphology control of poly(methyl methacrylate) hollow structures via coaxial electrospinning. J Appl Polym Sci 111: 408–416. doi: 10.1002/app.28994
![]() |
[21] |
Chen H, Wang N, Di J, et al. (2010) Nanowire-in-microtube structured core/shell fibers via multifluidic coaxial electrospinning. Langmuir 26: 11291–11296. doi: 10.1021/la100611f
![]() |
[22] |
Yu JH, Fridrikh SV, Rutledge GC (2004) Production of submicron diameter fibers by two-fluids electrospinning. Adv Mater 16: 1562–1566. doi: 10.1002/adma.200306644
![]() |
[23] | Gan YX, Chen AD, Gan JB, et al. (2018) Electrohydrodynamic casting bismuth telluride micro particle loaded carbon nanofiber composite material with multiple sensing functions. J Micro Nano-Manuf 6: 011005. |
[24] |
Sun B, Long YZ, Zhang HD, et al. (2014) Advances in three-dimensional nanofibrous macrostructures via electrospinning. Prog Polym Sci 39: 862–890. doi: 10.1016/j.progpolymsci.2013.06.002
![]() |
[25] | Zhang Y, Tse C, Rouholamin D, et al. (2012) Scaffolds for tissue engineering produced by inkjet printing. Cent Eur J Eng 2: 325–335. |
[26] |
Park TH, Shuler ML (2003) Integration of cell culture and microfabrication technology. Biotechnol Progr 19: 243–253. doi: 10.1021/bp020143k
![]() |
[27] |
Lee M, Kim HY (2014) Toward nanoscale three-dimensional printing: Nanowalls built of electrospun nanofibers. Langmuir 30: 1210–1214. doi: 10.1021/la404704z
![]() |
[28] |
Mandrycky C, Wang Z, Kim K, et al. (2016) 3D bioprinting for engineering complex tissues. Biotechnol Adv 34: 422–434. doi: 10.1016/j.biotechadv.2015.12.011
![]() |
[29] |
Huang C, Jian G, DeLisio JB, et al. (2015) Electrospray deposition of energetic polymer nanocomposites with high mass particle loadings: A prelude to 3D printing of rocket motors. Adv Eng Mater 17: 95–101. doi: 10.1002/adem.201400151
![]() |
[30] | Liu Y, Pollaor S, Wu Y (2015) Electrohydrodynamic processing of p-type transparent conducting oxides. J Nanomater 2015: 423157. |
[31] |
Sun J, Zhou W, Huang D, et al. (2015) An overview of 3D printing technologies for food fabrication. Food Bioprocess Tech 8: 1605–1615. doi: 10.1007/s11947-015-1528-6
![]() |
[32] |
Mironov V, Trusk T, Kasyanov V, et al. (2009) Biofabrication: A 21st century manufacturing paradigm. Biofabrication 1: 022001. doi: 10.1088/1758-5082/1/2/022001
![]() |
[33] |
Visser J, Peters B, Burger TJ, et al. (2013) Biofabrication of multi-material anatomically shaped tissue constructs. Biofabrication 5: 035007. doi: 10.1088/1758-5082/5/3/035007
![]() |
[34] |
Mittal A, Negi P, Garkhal K, et al. (2010) Integration of porosity and bio-functionalization to form a 3D scaffold: Cell culture studies and in Vitro degradation. Biomed Mater 5: 045001. doi: 10.1088/1748-6041/5/4/045001
![]() |
[35] |
Ozbolat I, Yu Y (2013) Bioprinting towards organ fabrication: Challenges and future trends. IEEE T Biomed Eng 60: 691–699. doi: 10.1109/TBME.2013.2243912
![]() |
[36] |
Mironov V, Rels N, Derby B (2006) Bioprinting: A beginning. Tissue Eng 12: 631–634. doi: 10.1089/ten.2006.12.631
![]() |
[37] | Catros S, Guillemot F, Nandakumar A, et al. (2011) Layer-by-layer tissue microfabrication supports cell proliferation in vitro and in vivo. Tissue Eng 18: 1–9. |
[38] | Vozzi G, Tirella A, Ahluwalia A (2012) Rapid prototyping composite and complex scaffolds with PAM2, In: Liebschner M, Computer-Aided Tissue Engineering. Methods in Molecular Biology (Methods and Protocols), Totowa, NJ: Humana Press, 868: 59–70. |
[39] |
Shim JH, Yoon MC, Jeong CM, et al. (2014) Efficacy of rhBMP-2 loaded PCL/PLGA/β-TCP guided bone regeneration membrane fabricated by 3D printing technology for reconstruction of calvaria defects in rabbit. Biomed Mater 9: 065006. doi: 10.1088/1748-6041/9/6/065006
![]() |
[40] | Laudenslager MJ, Sigmund WM (2011) Developments in electrohydrodynamic forming: Fabricating nanomaterials from charged liquids via electrospinning and electrospraying. Am Ceram Soc Bull 90: 23–27. |
[41] |
Martins A, Chung S, Pedro AJ, et al. (2009) Hierarchical starch-based fibrous scaffold for bone tissue engineering applications. J Tissue Eng Regen M 3: 37–42. doi: 10.1002/term.132
![]() |
[42] |
Zhu W, Masood F, O'Brien J, et al. (2015) Highly aligned nanocomposite scaffolds by electrospinning and electrospraying for neural tissue regeneration. Nanomed-Nanotechnol 11: 693–704. doi: 10.1016/j.nano.2014.12.001
![]() |
[43] |
Erisken C, Kalyon DM, Wang H (2008) Functionally graded electrospun polycaprolactone and b-tricalcium phosphate nanocomposites for tissue engineering applications. Biomaterials 29: 4065–4073. doi: 10.1016/j.biomaterials.2008.06.022
![]() |
[44] |
Giannitelli SM, Mozetic P, Trombetta M, et al. (2015) Combined additive manufacturing approaches in tissue engineering. Acta Biomater 24: 1–11. doi: 10.1016/j.actbio.2015.06.032
![]() |
[45] | Xu T, Binder KW, Albanna MZ, et al. (2013) Hybrid printing of mechanically and biologically improved constructs for cartilage tissue engineering applications. Biofabrication 5: 015001. |
[46] |
Nam J, Huang Y, Agarwal S, et al. (2007) Improved cellular infiltration in electrospun fiber via engineered porosity. Tissue Eng 13: 2249–2257. doi: 10.1089/ten.2006.0306
![]() |
[47] |
Abdelaal OAM, Darwish SMH (2013) Review of rapid prototyping techniques for tissue engineering scaffolds fabrication, In: Öchsner A, da Silva L, Altenbach H, Characterization and Development of Biosystems and Biomaterials. Advanced Structured Materials, Berlin, Heidelberg: Springer, 29: 33–54. doi: 10.1007/978-3-642-31470-4_3
![]() |
[48] |
Zhu W, O'Brien C, O'Brien JR, et al. (2014) 3D nano/microfabrication techniques and nanobiomaterials for neural tissue regeneration. Nanomedicine 9: 859–875. doi: 10.2217/nnm.14.36
![]() |
[49] |
Karande TS, Ong JL, Agrawal CM (2004) Diffusion in musculoskeletal tissue engineering scaffolds: Design issues related to porosity, permeability, architecture, and nutrient mixing. Ann Biomed Eng 32: 1728–1743. doi: 10.1007/s10439-004-7825-2
![]() |
[50] |
Jung JW, Lee H, Hong JM, et al. (2015) A new method of fabricating a blend scaffold using an indirect three dimensional printing technique. Biofabrication 7: 045003. doi: 10.1088/1758-5090/7/4/045003
![]() |
[51] |
Kim JT, Seol SK, Pyo J, et al. (2011) Three-dimensional writing of conducting polymer nanowire arrays by meniscus-guided polymerization. Adv Mater 23: 1968–1970. doi: 10.1002/adma.201004528
![]() |
[52] | Pham QP, Sharma U, Mikos AG (2006) Electrospinning of polymeric nanofibers for tissue engineering applications: A review. Tissue Eng 12: 2249–2257. |
[53] |
Rosenthal T, Welzmiller S, Neudert L, et al. (2014) Novel superstructure of the rock salt type and element distribution in germanium tin antimony tellurides. J Solid State Chem 219: 108–117. doi: 10.1016/j.jssc.2014.07.014
![]() |
[54] |
Kitagawa H, Takimura K, Ido S, et al. (2017) Thermoelectric properties of crystal-aligned bismuth antimony tellurides prepared by pulse-current sintering under cyclic uniaxial pressure. J Alloy Compd 692: 388–394. doi: 10.1016/j.jallcom.2016.09.054
![]() |
[55] |
Hatsuta N, Takemori D, Takashiri M (2016) Effect of thermal annealing on the structural and thermoelectric properties of electrodeposited antimony telluride thin films. J Alloy Compd 685: 147–152. doi: 10.1016/j.jallcom.2016.05.268
![]() |
[56] |
Sasaki Y, Takashiri M (2016) Effects of Cr interlayer thickness on adhesive, structural, and thermoelectric properties of antimony telluride thin films deposited by radio-frequency magnetron sputtering. Thin Solid Films 619: 195–201. doi: 10.1016/j.tsf.2016.10.069
![]() |
[57] |
Takashiri M, Hamada J (2016) Bismuth antimony telluride thin films with unique crystal orientation by two-step method. J Alloy Compd 683: 276–281. doi: 10.1016/j.jallcom.2016.05.058
![]() |
[58] |
Catrangiu AS, Sin I, Prioteasa P, et al. (2016) Studies of antimony telluride and copper telluride films electrodeposition from choline chloride containing ionic liquids. Thin Solid Films 611: 88–100. doi: 10.1016/j.tsf.2016.04.030
![]() |
[59] |
Masayuki K, Takashiri M (2015) Investigation of the effects of compressive and tensile strain on n-type bismuth telluride and p-type antimony telluride nanocrystalline thin films for use in flexible thermoelectric generators. J Alloy Compd 653: 480–485. doi: 10.1016/j.jallcom.2015.09.039
![]() |
[60] |
Catlin GC, Tripathi R, Nunes G, et al. (2017) An additive approach to low temperature zero pressure sintering of bismuth antimony telluride thermoelectric materials. J Power Sources 343: 316–321. doi: 10.1016/j.jpowsour.2016.12.092
![]() |
[61] |
Urban P, Schneider MN, Oeckler O (2015) Temperature dependent ordering phenomena in single crystals of germanium antimony tellurides. J Solid State Chem 227: 223–231. doi: 10.1016/j.jssc.2015.04.007
![]() |
[62] |
Hu LP, Zhu TJ, Yue XQ, et al. (2015) Enhanced figure of merit in antimony telluride thermoelectric materials by In–Ag Co-alloying for mid-temperature power generation. Acta Mater 85: 270–278. doi: 10.1016/j.actamat.2014.11.023
![]() |
[63] |
Lee WY, Park NW, Hong JE, et al. (2015) Effect of electronic contribution on temperature-dependent thermal transport of antimony telluride thin film. J Alloy Compd 620: 120–124. doi: 10.1016/j.jallcom.2014.09.053
![]() |
[64] |
Rosalbino F, Carlini R, Zanicchi G, et al. (2013) Microstructural characterization and corrosion behavior of lead, bismuth and antimony tellurides prepared by melting. J Alloy Compd 567: 26–32. doi: 10.1016/j.jallcom.2013.03.071
![]() |
[65] |
Kim DH, Kwon IH, Kim C, et al. (2013) Tellurium-evaporation-annealing for p-type bismuth-antimony-telluride thermoelectric materials. J Alloy Compd 548: 126–132. doi: 10.1016/j.jallcom.2012.08.130
![]() |
[66] |
Bochentyn B, Miruszewski T, Karczewski J, et al. (2016) Thermoelectric properties of bismuth-antimony-telluride alloys obtained by reduction of oxide reagents. Mater Chem Phys 177: 353–359. doi: 10.1016/j.matchemphys.2016.04.039
![]() |
[67] |
Qiu W, Yang S, Zhao X (2011) Effect of hot-press treatment on electrochemically deposited antimony telluride film. Thin Solid Films 519: 6399–6402. doi: 10.1016/j.tsf.2011.04.106
![]() |
[68] |
Takashiri M, Tanaka S, Miyazaki K (2010) Improved thermoelectric performance of highly-oriented nanocrystalline bismuth antimony telluride thin films. Thin Solid Films 519: 619–624. doi: 10.1016/j.tsf.2010.08.013
![]() |
[69] |
Takashiri M, Tanaka S, Hagino H, et al. (2014) Strain and grain size effects on thermal transport in highly-oriented nanocrystalline bismuth antimony telluride thin films. Int J Heat Mass Tran 76: 376–384. doi: 10.1016/j.ijheatmasstransfer.2014.04.048
![]() |
[70] |
Lim SK, Kim MY, Oh TS (2009) Thermoelectric properties of the bismuth-antimony-telluride and the antimony-telluride films processed by electrodeposition for micro-device applications. Thin Solid Films 517: 4199–4203. doi: 10.1016/j.tsf.2009.02.005
![]() |
[71] |
Jung H, Myung NV (2011) Electrodeposition of antimony telluride thin films from acidic nitrate-tartrate baths. Electrochim Acta 56: 5611–5615. doi: 10.1016/j.electacta.2011.04.010
![]() |
[72] |
Fan P, Chen T, Zheng Z, et al. (2013) The influence of Bi doping in the thermoelectric properties of Co-sputtering deposited bismuth antimony telluride thin films. Mater Res Bull 48: 333–336. doi: 10.1016/j.materresbull.2012.10.026
![]() |
[73] |
Lensch-Falk JL, Banga D, Hopkins PE, et al. (2012) Electrodeposition and characterization of nano-crystalline antimony telluride thin films. Thin Solid Films 520: 6109–6117. doi: 10.1016/j.tsf.2012.05.078
![]() |
[74] |
Takashiri M, Tanaka S, Miyazaki K (2013) Growth of single-crystalline bismuth antimony telluride nanoplates on the surface of nanoparticle thin films. J Cryst Growth 372: 199–204. doi: 10.1016/j.jcrysgro.2013.03.028
![]() |
[75] |
Kim BG, Choi SM, Lee MH, et al. (2015) Facile fabrication of silicon and aluminum oxide nanotubes using antimony telluride nanowires as templates. Ceram Int 41: 12246–12252. doi: 10.1016/j.ceramint.2015.06.047
![]() |
[76] |
Ganguly S, Zhou C, Morelli D, et al. (2011) Synthesis and evaluation of lead telluride/bismuth antimony telluride nanocomposites for thermoelectric applications. J Solid State Chem 184: 3195–3201. doi: 10.1016/j.jssc.2011.09.031
![]() |
[77] | Li J, Chen Z, Wang X, et al. (1997) A novel two-dimensional mercury antimony telluride: Low temperature synthesis and characterization of RbHgSbTe3. J Alloy Compd 262–263: 28–33. |
[78] | Baba S, Sato H, Huang L, et al. (2014) Formation and characterization of polyethylene terephthalate-based (Bi0.15Sb0.85)2Te3 thermoelectric modules with CoSb3 adhesion layer by aerosol deposition. J Alloy Compd 589: 56–60. |
[79] |
Bark H, Kim JS, Kim H, et al. (2013) Effect of multiwalled carbon nanotubes on the thermoelectric properties of a bismuth telluride matrix. Curr Appl Phys 13: S111–S114. doi: 10.1016/j.cap.2013.01.019
![]() |
[80] |
Zhang HT, Luo XG, Wang CH, et al. (2004) Characterization of nanocrystalline bismuth telluride (Bi2Te3) synthesized by a hydrothermal method. J Cryst Growth 265: 558–562. doi: 10.1016/j.jcrysgro.2004.02.097
![]() |
[81] |
Sun Y, Cheng H, Gao S, et al. (2012) Atomically thick bismuth selenide freestanding single layers achieving enhanced thermoelectric energy harvesting. J Am Chem Soc 134: 20294–20297. doi: 10.1021/ja3102049
![]() |
[82] |
Prieto AL, Sander MS, Martin-Gonzalez MS, et al. (2001) Electrodeposition of ordered Bi2Te3 nanowire arrays. J Am Chem Soc 123: 7160–7161. doi: 10.1021/ja015989j
![]() |
[83] |
Borca-Tasciuc DA, Chen G, Prieto A, et al. (2004) Thermal properties of electrodeposited bismuth telluride nanowires embedded in amorphous alumina. Appl Phys Lett 85: 6001–6003. doi: 10.1063/1.1834991
![]() |
[84] |
Pang H, Piao YY, Tan YQ, et al. (2013) Thermoelectric behavior of segregated conductive polymer composites with hybrid fillers of carbon nanotube and bismuth telluride. Mater Lett 107: 150–153. doi: 10.1016/j.matlet.2013.06.008
![]() |
[85] |
Chatterjee K, Suresh A, Ganguly S, et al. (2009) Synthesis and characterization of an electro-deposited polyaniline-bismuth telluride nanocomposite-A novel thermoelectric material. Mater Charact 60: 1597–1601. doi: 10.1016/j.matchar.2009.09.012
![]() |
[86] |
Li JF, Liu J (2006) Effect of nano-SiC dispersion on thermoelectric properties of Bi2Te3 polycrystals. Phys Status Solidi A 203: 3768–3773. doi: 10.1002/pssa.200622011
![]() |
[87] |
Kim KT, Choi SY, Shin EH, et al. (2013) The influence of CNTs on the thermoelectric properties of a CNT/Bi2Te3 composite. Carbon 52: 541–549. doi: 10.1016/j.carbon.2012.10.008
![]() |
[88] |
Lu W, Ding Y, Chen Y, et al. (2005) Bismuth telluride hexagonal nanoplatelets and their two-step epitaxial growth. J Am Chem Soc 127: 10112–10116. doi: 10.1021/ja052286j
![]() |
[89] | Sumithra S, Takas NJ, Misra DK, et al. (2011) Enhancement in thermoelectric figure of merit in nanostructured Bi2Te3 with semimetal nanoinclusions. Adv Energy Mater 1: 1–7. |
[90] |
Zhao XB, Ji XH, Zhang YH, et al. (2005) Bismuth telluride nanotubes and the effects on the thermoelectric properties of nanotube-containing nanocomposites. Appl Phys Lett 86: 062111. doi: 10.1063/1.1863440
![]() |
[91] |
Chen CL, Chen YY, Lin SJ, et al. (2010) Fabrication and characterization of electrodeposited bismuth telluride films and nanowires. J Phys Chem C 114: 3385–3389. doi: 10.1021/jp909926z
![]() |
[92] |
Toprak M, Zhang Y, Muhammed M (2003) Chemical alloying and characterization of nanocrystalline bismuth telluride. Mater Lett 57: 3976–3982. doi: 10.1016/S0167-577X(03)00250-7
![]() |
[93] |
Kim KT, Koo HY, Lee GG, et al. (2012) Synthesis of alumina nanoparticle-embedded-bismuth telluride matrix thermoelectric composite powders. Mater Lett 82: 141–144. doi: 10.1016/j.matlet.2012.05.053
![]() |
[94] |
Chávez-Ángel E, Reparaz JS, Gomis-Bresco J, et al. (2014) Reduction of the thermal conductivity in free-standing silicon nano-membranes investigated by non-invasive Raman thermometry. APL Mater 2: 012113. doi: 10.1063/1.4861796
![]() |
[95] | Liang B, Song Z, Wang M, et al. (2013) Fabrication and thermoelectric properties of graphene/Bi2Te3 composite materials. J Nanomater 2013: 210767. |
[96] | Goldsmid HJ (2014) Bismuth telluride and its alloys as materials for thermoelectric generation. Materials 2014: 2577–2592. |
[97] |
Keshavarz MK, Vasilevskiy D, Masut RA, et al. (2013) p-Type bismuth telluride-based composite thermoelectric materials produced by mechanical alloying and hot extrusion. J Electron Mater 42: 1429–1435. doi: 10.1007/s11664-012-2284-2
![]() |
[98] | Chang HC, Chen CH (2011) Self-assembled bismuth telluride films with well-aligned zero-to three-dimensional nanoblocks for thermoelectric applications. CrystEngComm 13: 5956–5962. |
[99] |
Deng Y, Nan CW, Wei GD, et al. (2003) Organic-assisted growth of bismuth telluride nanocrystals. Chem Phys Lett 374: 410–415. doi: 10.1016/S0009-2614(03)00783-8
![]() |
[100] |
Liao CN, She TH (2007) Preparation of bismuth telluride thin films through interfacial reaction. Thin Solid Films 515: 8059–8064. doi: 10.1016/j.tsf.2007.03.086
![]() |
[101] |
Sokolova OB, Skipidarova SY, Duvankova NI, et al. (2004) Chemical reactions on the Bi2Te3-Bi2Se3 section in the process of crystal growth. J Cryst Growth 262: 442–448. doi: 10.1016/j.jcrysgro.2003.10.073
![]() |
[102] | Kim KT, Ha GH (2013) Fabrication and enhanced thermoelectric properties of alumina nanoparticle-dispersed Bi0.5Sb1.5Te3 matrix composites. J Nanomater 2013: 821657. |
[103] |
Gothard N, Wilks G, Tritt TM, et al. (2010) Effect of processing route on the microstructure and thermoelectric properties of bismuth telluride-based alloys. J Electron Mater 39: 1909–1913. doi: 10.1007/s11664-009-1051-5
![]() |
[104] |
Thiebaud L, Legeai S, Ghanbaja J, et al. (2018) Synthesis of Te-Bi core-shell nanowires by two-step electrodeposition in ionic liquids. Electrochem Commun 86: 30–33. doi: 10.1016/j.elecom.2017.11.010
![]() |
[105] |
Kim J, Lee JY, Lim JH, et al. (2016) Optimization of thermoelectric properties of p-type AgSbTe2 thin films via electrochemical synthesis. Electrochim Acta 196: 579–586. doi: 10.1016/j.electacta.2016.02.206
![]() |
[106] | Suzuki M, Tsuchiya T, Akedo J (2017) Effect of starting powder morphology on film texture for bismuth layer-structured ferroelectrics prepared by aerosol deposition method. Jpn J Appl Phys 56: 06GH02. |
[107] |
Chu F, Zhang Q, Zhou Z, et al. (2018) Enhanced thermoelectric and mechanical properties of Na-doped polycrystalline SnSe thermoelectric materials via CNTs dispersion. J Alloy Compd 741: 756–764. doi: 10.1016/j.jallcom.2018.01.178
![]() |
[108] |
Chung DDL (2017) Processing-structure-property relationships of continuous carbon fiber polymer-matrix composites. Mater Sci Eng R 113: 1–29. doi: 10.1016/j.mser.2017.01.002
![]() |
[109] |
Mahmoud L, Alhwarai M, Samad YA, et al. (2015) Characterization of a graphene-based thermoelectric generator using a cost-effective fabrication process. Energy Procedia 75: 615–620. doi: 10.1016/j.egypro.2015.07.466
![]() |
[110] |
Lee S, Kim J, Ku BC, et al. (2012) Structural evolution of polyacrylonitrile fibers in stabilization and carbonization. Adv Chem Eng Sci 2: 275–282. doi: 10.4236/aces.2012.22032
![]() |
[111] |
Saha B, Schatz GC (2012) Carbonization in polyacrylonitrile (PAN) based carbon fibers studied by ReaxFF molecular dynamics simulations. J Phys Chem B 116: 4684–4692. doi: 10.1021/jp300581b
![]() |
[112] |
Ma Q, Gao A, Tong Y, et al. (2016) The densification mechanism of polyacrylonitrile carbon fibers during carbonization. New Carbon Mater 31: 550–554. doi: 10.1016/S1872-5805(16)60031-8
![]() |
[113] |
Hameed N, Sharp J, Nunna S, et al. (2016) Structural transformation of polyacrylonitrile fibers during stabilization and low temperature carbonization. Polym Degrad Stabil 128: 39–45. doi: 10.1016/j.polymdegradstab.2016.02.029
![]() |
[114] |
Liu J, Wang PH, Li RY (1994) Continuous carbonization of polyacrylonitrile-based oxidized fibers: Aspects on mechanical properties and morphological structure. J Appl Polym Sci 52: 945–950. doi: 10.1002/app.1994.070520712
![]() |
[115] | Wang H, Zhang X, Zhang Y, et al. (2016) Study of carbonization behavior of polyacrylonitrile/tin salt as anode material for lithium-ion batteries. J Appl Polym Sci 2016: 43914. |
[116] |
Sun J, Wu G, Wang Q (2005) The effects of carbonization temperature on the properties and structure of PAN-based activated carbon hollow fiber. J Appl Polym Sci 97: 2155–2160. doi: 10.1002/app.21955
![]() |
[117] |
Rahaman MSA, Ismail AF, Mustafa A (2007) A review of heat treatment on polyacrylonitrile fiber. Polym Degrad Stabil 92: 1421–1432. doi: 10.1016/j.polymdegradstab.2007.03.023
![]() |
[118] |
Zhao LR, Jang BZ, Zhou JN (1998) Effect of polymeric precursors on properties of semiconducting carbon/carbon composites. J Mater Sci 33: 1809–1817. doi: 10.1023/A:1004392919018
![]() |
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