The spreading fronts of an infective environment in a man-environment-man epidemic model
aa r X i v : . [ m a t h . A P ] A ug The spreading fronts of an infective environmentin a man-environment-man epidemic model ∗ Inkyung Ahn a and Zhigui Lin b a Department of Mathematics, Korea University, Jochiwon,Chung-Nam 339-700, South Korea b School of Mathematical Science, Yangzhou University, Yangzhou 225002, China
Abstract.
A reaction-diffusion model is investigated to understandinfective environments in a man-environment-man epidemic model. Thefree boundary is introduced to describe the expanding front of an infectiveenvironment induced by fecally-orally transmitted disease. The basicreproduction number R F ( t ) for the free boundary problem is introduced,and the behavior of positive solutions to the reaction-diffusion systemis discussed. Sufficient conditions for the bacteria to vanish or spreadare given. We show that, if R ≤
1, the bacteria always vanish, and if R F ( t ) ≥ t ≥
0, the bacteria must spread, while if R F (0) < < R , the spreading or vanishing of the bacteria depends on the initialnumber of bacteria, the length of the initial habitat, the diffusion rate,and other factors. Moreover, some sharp criteria are given. MSC: primary: 35R35; secondary: 35K60
Keywords:
Reaction-diffusion systems; epidemic model; Free bound-ary; Spreading and vanishing
Recently, many mathematical models have been proposed to investigate thespatial spread of infectious diseases epidemics (see [1, 2, 3, 28]). To understand ∗ This work is supported by Collaborative Research Project under the NRF-NSFC Cooper-ative Program (11211140236) and the National Research Foundation of Korea (NRF) Grant(NRF-2012K1A2B1A03000598). ∂u ( x,t ) ∂t = d ∆ u ( x, t ) − a u ( x, t ) + a v ( x, t ) , ( x, t ) ∈ Ω × (0 , + ∞ ) , ∂v ( x,t ) ∂t = − a v ( x, t ) + G ( u ( x, t )) , ( x, t ) ∈ Ω × (0 , + ∞ ) , ∂u∂η + αu = 0 , ( x, t ) ∈ ∂ Ω × (0 , + ∞ ) ,u ( x,
0) = u ( x ) , v ( x,
0) = v ( x ) , x ∈ Ω , (1.1)where u ( x, t ) and v ( x, t ) represent the spatial densities of bacteria and the in-fective human population, respectively, at a point x in the habitat Ω ∈ R n , andat time t ≥
0, and ∂/∂η denotes the outward normal derivative. The positiveconstant d denotes the diffusion constant of the bacteria, 1 /a > − a u denotes the natu-ral growth rate of the bacterial population, 1 /a > − a v describes the natural damping ofthe infective population due to the finite mean duration of the infectiousnessof humans, a > a v is the contribution of the infectivehumans to the growth rate of the bacteria. The last term G ( u ) is the infec-tion rate of the humans under the assumption that the total susceptible humanpopulation is constant during the evolution of the epidemic. This kind of mech-anism is used to interpret other epidemics with oro-faecal transmission such astyphoid fever, infectious hepatitis, polyomelitis, and the like ; see [4, 5] and thereferences therein for more details.Assume that(A1) G ∈ C ([0 , ∞ )) , G (0) = 0 , G ′ ( z ) > , ∀ z ≥ G ( z ) z is decreasing and lim z → + ∞ G ( z ) z < a a a .An example is G ( z ) = a z z with a > ( du ( t ) dt = − a u ( t ) + a v ( t ) , t > , dv ( t ) dt = − a v ( t ) + G ( u ( t )) , t > , (1.2)linearization and spectrum analysis show that a threshold parameter R (:= G ′ (0) a a a ) exists such that if 0 < R <
1, then the epidemic always tends toextinction, while for R >
1, a nontrivial endemic level appears which is globallyasymptotically stable in the positive quadrant.2or problem (1.1), in which the bacteria diffuse but the infective humanpopulation does not, the authors in [4] introduced a threshold parameter R D (:= G ′ (0) a ( a + dλ ) a ) such that for 0 < R D <
1, the epidemic eventually tends to extinc-tion, while for R D > λ is the first eigenvalue of the bound-ary value problem − ∆ φ = λφ in Ω with ∂φ∂η + αφ = 0 on ∂ Ω . To understand the whole dynamical structure of solutions to (1.1) and itscorresponding reaction systems, traveling waves and entire solutions were widelystudied. The existence, uniqueness and stability of traveling waves were estab-lished in [21, 23, 24, 25, 26, 27]. Recently, Wu [25] considered entire solutionsof a bistable reaction-diffusion system (1.1) in the bistable case, and proved theexistence of entire solutions that behave like two monotone increasing travelingwave solutions propagating from both sides of the x -axis. The time-delayedand diffusive model has been considered in [23] and entire solutions have beengiven. It was shown that there exist a great diversity of different types of entiresolutions of reaction-diffusion equations, which are different from traveling wavesolutions.It must be pointed out that the solution of (1.1) in a fixed (bounded orunbounded) domain is always positive for any t > −∞ , ∞ ) is constant,and that the environment in g ( t ) < x < h ( t ) is infected by bacteria, the densityof which is denoted by u ( x, t ) with the infective human population denoted by v ( x, t ), and no bacteria or infective humans in the remaining portion of the en-vironment. The right spreading front of the infected environment is representedby the free boundary x = h ( t ). Assuming that h ( t ) grows at a rate proportionalto the bacteria population gradient at the front [17], the conditions on the rightfront (free boundary) are u ( h ( t ) , t ) = 0 , − µ ∂u∂x ( h ( t ) , t ) = h ′ ( t ) . Similarly, the conditions on the left front (free boundary) are u ( g ( t ) , t ) = 0 , − µ ∂u∂x ( g ( t ) , t ) = g ′ ( t ) . In such a case, we have the problem for u ( x, t ) and v ( x, t ) with free boundaries x = g ( t ) and x = h ( t ) such that ∂u ( x,t ) ∂t = d ∂ u ( x,t ) ∂x − a u ( x, t ) + a v ( x, t ) , g ( t ) < x < h ( t ) , t > , ∂v ( x,t ) ∂t = − a v ( x, t ) + G ( u ( x, t )) , g ( t ) < x < h ( t ) , t > ,u ( x, t ) = 0 , x = g ( t ) or x = h ( t ) , t > ,g (0) = − h , g ′ ( t ) = − µ ∂u∂x ( g ( t ) , t ) , t > ,h (0) = h , h ′ ( t ) = − µ ∂u∂x ( h ( t ) , t ) , t > ,u ( x,
0) = u ( x ) , v ( x,
0) = v ( x ) , − h ≤ x ≤ h , (1.3)where x = g ( t ) and x = h ( t ) are the moving left and right boundaries to bedetermined, h and µ are positive constants, and the initial functions u and v are nonnegative and satisfy (cid:26) u ∈ C ([ − h , h ]) , u ( ± h ) = 0 and 0 < u ( x ) , x ∈ ( − h , h ) ,v ∈ C ([ − h , h ]) , v ( ± h ) = 0 and 0 < v ( x ) , x ∈ ( − h , h ) . (1.4)The remainder of this paper is organized as follows. In the next section, theglobal existence and uniqueness of the solution to (1.3) are proved using a con-traction mapping theorem, and a comparison principle is presented. Section 3 isdevoted to sufficient conditions for the bacteria to vanish. Section 4 deals withthe case and conditions for the bacteria to expand and the whole environmentbecome infected. Finally, we give a brief discussion in Section 5.4 Existence and uniqueness
In this section, we first present the following local existence and uniquenessresult using the contraction mapping theorem and then show global existenceusing suitable estimates.
Theorem 2.1
For any given ( u , v ) satisfying (1.4) , and any α ∈ (0 , , thereis a T > such that problem (1.3) admits a unique solution ( u, v ; g, h ) ∈ [ C α, (1+ α ) / ( D T )] × [ C α/ ([0 , T ])] ; moreover, k u k C α, (1+ α ) / ( D T ) + k v k C α, (1+ α ) / ( D T ) + || g k C α/ ([0 ,T ]) + || h k C α/ ([0 ,T ]) ≤ C, (2.1) where D T = { ( x, t ) ∈ R : x ∈ [ g ( t ) , h ( t )] , t ∈ [0 , T ] } , C and T depend only on h , α, k u k C ([ − h ,h ]) and k v k C ([ − h ,h ]) . Proof:
As in [29], we first straighten the double free boundary fronts by makingthe following change of variable: y = 2 h xh ( t ) − g ( t ) − h ( h ( t ) + g ( t )) h ( t ) − g ( t ) , w ( y, t ) = u ( x, t ) , z ( y, t ) = v ( x, t ) . Then (1.3) can be transformed into w t = Aw y + Bw yy − a w ( y, t ) + a z ( y, t ) , t > , − h < y < h ,z t = Az y − a z ( y, t ) + G ( w ( y, t )) , t > , − h < y < h ,w = 0 , h ′ ( t ) = − h µh ( t ) − g ( t ) ∂w∂y , t > , y = h ,w = 0 , g ′ ( t ) = − h µh ( t ) − g ( t ) ∂w∂y , t > , y = − h ,h (0) = h , g (0) = − h ,w ( y,
0) = w ( y ) := u ( y ) , z ( y,
0) = z ( y ) := v ( y ) , − h ≤ y ≤ h , (2.2)where A = A ( h, g, y ) = y h ′ ( t ) − g ′ ( t ) h ( t ) − g ( t ) + h h ′ ( t )+ g ′ ( t ) h ( t ) − g ( t ) , and B = B ( h, g ) = h d ( h ( t ) − g ( t )) .This transformation changes the free boundaries x = h ( t ) and x = g ( t ) to thefixed lines y = h and y = − h respectively; therefore, the equations becomemore complex, because now the coefficients in the first and second equations of(2.2) contain unknown functions h ( t ) and g ( t ).The rest of the proof uses by the contraction mapping argument as in [10, 29]with suitable modifications; we omit it here. (cid:3) To show the global existence of the solution, we need the following estimate.5 emma 2.2
Let ( u, v ; g, h ) be a solution to (1.3) defined for t ∈ (0 , T ] for some T ∈ (0 , + ∞ ) . Then there exist constants C and C independent of T suchthat < u ( x, t ) ≤ C for g ( t ) < x < h ( t ) , t ∈ (0 , T ] , < v ( x, t ) ≤ C for g ( t ) < x < h ( t ) , t ∈ (0 , T ] . Proof:
The positivity of u and v are obvious, since the initial values are non-trivial and nonnegative and the system is quasi-increasing. Now let us considerits upper bounds. Note that lim z → + ∞ G ( z ) z < a a a by the assumption ( A C and C such that C ≥ u ( x ) , C ≥ v ( x ) in [ − h , h ] , − a C + a C < , − a C + G ( C ) < . Define M = M ( u, T ) = max [ g ( t ) ,h ( t )] × [0 ,T ] u ( x, t ) ,M = M ( G, C , T ) = max ≤ z ≤ max { C ,M } G ′ ( z )and let ( U ( x, t ) , V ( x, t )) = ( C − u, C − v ) e − ( a + M ) t , then we have G ( u ) = G ( C − U e ( a + M ) t )= G ( C ) − G ′ ( ξ ( x, t )) U e ( a + M ) t , where ξ ( x, t ) is between C and u ( x, t ), and therefore ( U, V ) satisfies ∂U ( x,t ) ∂t > d ∂ U ( x,t ) ∂x − ( a + a + M ) U ( x, t )+ a V ( x, t ) , g ( t ) < x < h ( t ) , < t ≤ T , ∂V ( x,t ) ∂t > − ( a + a + M ) V ( x, t )+ G ′ ( ξ ( x, t )) U ( x, t ) , g ( t ) < x < h ( t ) , < t ≤ T ,U ( x, t ) = C e − ( a + M ) t , x = g ( t ) or x = h ( t ) , t > ,V ( x, t ) = C e − ( a + M ) t , x = g ( t ) or x = h ( t ) , t > ,U ( x, ≥ , V ( x, ≥ , − h ≤ x ≤ h . (2.3)We now show that min { U ( x, t ) , V ( x, t ) } ≥ g ( t ) , h ( t )] × [0 , T ]. Other-wise, there exists ( x , t ) ∈ ( g ( t ) , h ( t )) × (0 , T ] such thatmin { U ( x , t ) , V ( x , t ) } = min [ g ( t ) ,h ( t )] × [0 ,T ] min { U ( x, t ) , V ( x, t ) } < . If U ( x , t ) = min { U ( x , t ) , V ( x , t ) } <
0, then U ( x, t ) attains its mini-mum in [ g ( t ) , h ( t )] × [0 , T ] at ( x , t ); therefore, ∂U ( x , t ) ∂t − d ∂ U ( x , t ) ∂x ≤ . − ( a + a + M ) U ( x , t ) + a V ( x , t ) ≥ − ( a + M ) U ( x , t ) > V ( x , t ) = min { U ( x , t ) , V ( x , t ) } <
0, then V ( x, t ) attainsits minimum in [ g ( t ) , h ( t )] × [0 , T ] at ( x , t ); therefore, ∂V ( x ,t ) ∂t ≤
0. How-ever, − ( a + a + M ) V ( x , t ) + G ′ ( ξ ) U ( x , t ) ≥ − ( a + a ) V ( x , t ) > { U ( x, t ) , V ( x, t ) } ≥ g ( t ) , h ( t )] × [0 , T ], or u ( x, t ) ≤ C and v ( x, t ) ≤ C in [ g ( t ) , h ( t )] × [0 , T ]. (cid:3) The next lemma shows that the left free boundary for (1.3) is strictly mono-tone decreasing and the right boundary is increasing.
Lemma 2.3
Let ( u, v ; g, h ) be a solution to (1.3) defined for t ∈ (0 , T ] for some T ∈ (0 , + ∞ ) . Then there exists a constant C independent of T such that < − g ′ ( t ) , h ′ ( t ) ≤ C for t ∈ (0 , T ] . Proof:
Applying the strong maximum principle to the equation of u gives u x ( h ( t ) , t ) < < t ≤ T . Hence h ′ ( t ) > t ∈ (0 , T ] by the free boundary condition in (1.3). Similarly, g ′ ( t ) < t ∈ (0 , T ].It remains to be shown that − g ′ ( t ) , h ′ ( t ) ≤ C for t ∈ (0 , T ] and some C .The proof is similar to that of Lemma 2.2 in [10] with C = 2 M C µ and M = max ( h , r a C dC , k u k C ([ − h ,h ]) C ) , we omit it here. (cid:3) Since u, v and g ′ ( t ) , h ′ ( t ) are bounded in ( g ( t ) , h ( t )) × (0 , T ] by constantsindependent of T , the global solution is guaranteed. Theorem 2.4
The solution of (1.3) exists and is unique for all t ∈ (0 , ∞ ) . In what follows, we exhibit the comparison principle, which can be provedsimilarly to a Lemma 3.5 in [10].
Lemma 2.5 (The Comparison Principle) Assume that g, h ∈ C ([0 , + ∞ )) , u ( x, t ) , v ( x, t ) ∈ C ([ g ( t ) , h ( t )] × [0 , + ∞ )) ∩ C , (( g ( t ) , h ( t )) × (0 , + ∞ )) , and ∂u∂t ≥ d ∂ u∂x − a u + a v, g ( t ) < x < h ( t ) , t > , ∂v∂t ≥ − a v + G ( u ) , g ( t ) < x < h ( t ) , t > ,u ( x, t ) = v ( x, t ) = 0 , x = g ( t ) or x = h ( t ) , t > ,g (0) ≤ − h , g ′ ( t ) ≤ − µ ∂u∂x ( g ( t ) , t ) , t > ,h (0) ≥ h , h ′ ( t ) ≥ − µ ∂u∂x ( h ( t ) , t ) , t > ,u ( x, ≥ u ( x ) , v ( x, ≥ v ( x ) , − h ≤ x ≤ h . hen the solution ( u, v ; g, h ) to the free boundary problem (1 . satisfies h ( t ) ≤ h ( t ) , g ( t ) ≥ g ( t ) , t ∈ [0 , + ∞ ) ,u ( x, t ) ≤ u ( x, t ) , v ( x, t ) ≤ v ( x, t ) , ( x, t ) ∈ [ g ( t ) , h ( t )] × [0 , + ∞ ) . Remark 2.1
The pair ( u, h ) in Lemma 2.5 is usually called an upper solutionof (1.3) . We can define a lower solution by reversing all of the inequalities inthe obvious places. Moreover, one can easily prove an analogue of Lemma 2.5for lower solutions. We next fix v , µ, a ij , let u = σφ ( x ) and examine the dependence of the so-lution on σ , writing ( u σ , v σ ; g σ , h σ ) to emphasize this dependence. As a corollaryof Lemma 2.5, we have the following monotonicity: Corollary 2.6
Let ( u , v ) = σ ( φ ( x ) , ψ ( x )) . For fixed φ ( x ) , ψ ( x ) , µ and a ij , if σ ≤ σ , then u σ ( x, t ) ≤ u σ ( x, t ) and v σ ( x, t ) ≤ v σ ( x, t ) in [ g σ ( t ) , h σ ( t )] × (0 , ∞ ) , g σ ( t ) ≥ g σ ( t ) and h σ ( t ) ≤ h σ ( t ) in (0 , ∞ ) . It follows from Lemma 2.3 that x = h ( t ) is monotonic increasing, x = g ( t ) ismonotonic decreasing and therefore there exist h ∞ , − g ∞ ∈ (0 , + ∞ ] such thatlim t → + ∞ h ( t ) = h ∞ and lim t → + ∞ g ( t ) = g ∞ . The next lemma shows that if h ∞ < ∞ , then − g ∞ < ∞ , and vice versa. That is, the double free boundaryfronts x = g ( t ) and x = h ( t ) are both finite or infinite simultaneously. Lemma 3.1
Let ( u, v ; g, h ) be a solution to (1 . defined for t ∈ [0 , + ∞ ) and x ∈ [ g ( t ) , h ( t )] . Then we have − h < g ( t ) + h ( t ) < h for t ∈ [0 , + ∞ ) . Proof:
By continuity we know g ( t ) + h ( t ) > − h holds for small t >
0. Define T := sup { s : g ( t ) + h ( t ) > − h for all t ∈ [0 , s ) } . As in [12], we claim that T = ∞ . Otherwise, 0 < T < ∞ and g ( t ) + h ( t ) > − h for t ∈ [0 , T ) , g ( T ) + h ( T ) = − h . Hence, g ′ ( T ) + h ′ ( T ) ≤ . (3.1)8o get a contradiction, we consider the functions w ( x, t ) := u ( x, t ) − u ( − x − h , t ) , z ( x, t ) := v ( x, t ) − v ( − x − h , t )over the region Λ := { ( x, t ) : x ∈ [ g ( t ) , − h ] , t ∈ [0 , T ] } . It is easy to check that the pair ( w, z ) is well-defined for ( x, t ) ∈ Λ since − h ≤− x − h ≤ − g ( t ) − h ≤ h ( t ), and the pair satisfies w t − dw xx = − a w + a z for g ( t ) < x < − h , < t ≤ T,z t = c ( x, t ) w − a z for g ( t ) < x < − h , < t ≤ T with 0 ≤ c := G ( u ( x,t )) − G ( u ( − x − h ,t )) u ( x,t ) − u ( − x − h ,t ) ∈ L ∞ (Λ), and w ( − h , t ) = z ( − h , t ) = 0 , w ( g ( t ) , t ) < , z ( g ( t ) , t ) < < t < T. Moreover, w ( g ( T ) , T ) = u ( g ( T ) , T ) − u ( − g ( T ) − h , T ) = u ( g ( T ) , T ) − u ( h ( T ) , T ) = 0 . Applying the proof for the strong maximum principle and the Hopf lemma, wededuce w ( x, t ) < , z ( x, t ) < g ( t ) , − h ) × (0 , T ] and w x ( g ( T ) , T ) < . However, w x ( g ( T ) , T ) = ∂u∂x ( g ( T ) , T ) + ∂u∂x ( h ( T ) , T ) = − [ g ′ ( T ) + h ′ ( T )] / ( µ ) , which implies g ′ ( T ) + h ′ ( T ) > , a contradiction to (3.1). Hence we have proven g ( t ) + h ( t ) > − h for all t > . Analogously, we can prove g ( t ) + h ( t ) < h for all t > W ( x, t ) := u ( x, t ) − u (2 h − x, t ) , Z ( x, t ) := v ( x, t ) − v (2 h − x, t )over the region Λ ′ := [ h , h ( t )] × [0 , T ′ ] with T ′ := sup { s : g ( t ) + h ( t ) < h for all t ∈ [0 , s ) } . The completes the proof. (cid:3) Next, we discuss the properties of the free boundary, because the transmis-sion of the bacteria depends on whether h ∞ − g ∞ = ∞ and lim sup t → + ∞ ( || u ( · , t ) || C ( g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) = 0. We then have the following definitions:9 efinition 3.1 The bacteria are vanishing if h ∞ − g ∞ < ∞ and lim t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) = 0 , and spreading if h ∞ − g ∞ = ∞ and lim sup t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) > . The next result shows that if h ∞ − g ∞ < ∞ , then vanishing occurs. Lemma 3.2 If h ∞ − g ∞ < ∞ , then lim t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) =0 . Proof:
We first prove that lim t → + ∞ || u ( · , t ) || C ([ g ( t ) ,h ( t )]) = 0. Assume thatlim sup t → + ∞ || u ( · , t ) || C ([ g ( t ) ,h ( t )]) = δ > x k , t k ) in ( g ( t ) , h ( t )) × (0 , ∞ )such that u ( x k , t k ) ≥ δ/ k ∈ N , and t k → ∞ as k → ∞ . Since −∞ < g ∞ < g ( t ) < x k < h ( t ) < h ∞ < ∞ , we then have that a subsequenceof { x n } converges to x ∈ ( g ∞ , h ∞ ). Without loss of generality, we assume x k → x as k → ∞ .Define W k ( x, t ) = u ( x, t k + t ) and Z k ( x, t ) = v ( x, t k + t ) for x ∈ ( g ( t k + t ) , h ( t k + t )) , t ∈ ( − t k , ∞ ). It follows from parabolic regularity that { ( W k , Z k ) } has a subsequence { ( W k i , Z k i ) } such that ( W k i , Z k i ) → ( ˜ W , ˜ Z ) as i → ∞ and( ˜ W , ˜ Z ) satisfies (cid:26) ˜ W t − d ˜ W xx = − a ˜ W + a ˜ Z, g ∞ < x < h ∞ , t ∈ ( −∞ , ∞ ) , ˜ Z t = − a ˜ Z + G ( ˜ W ) , g ∞ < x < h ∞ , t ∈ ( −∞ , ∞ ) . Note that ˜ W ( x , ≥ δ/
2; therefore, ˜
W > g ∞ , h ∞ ) × ( −∞ , ∞ ).Using a similar method to prove the Hopf lemma at the point ( h ∞ ,
0) yields˜ W x ( h ∞ , ≤ − σ for some σ > − g ( t ) and h ( t ) are increasing and bounded, it fol-lows from standard L p theory and then the Sobolev imbedding theorem ([16])that for any 0 < α <
1, there exists a constant ˜ C depending on α, h , k u k C [ − h ,h ] , k v k C [ − h ,h ] , and g ∞ , h ∞ such that k u k C α, (1+ α ) / ([ g ( t ) ,h ( t )] × [0 , ∞ )) + k h k C α/ ([0 , ∞ )) ≤ ˜ C. (3.2)Now, since k h k C α/ ([0 , ∞ )) ≤ ˜ C and h ( t ) is bounded, we then have h ′ ( t ) → t → ∞ , that is, ∂u∂x ( h ( t k ) , t k ) → t k → ∞ by the free boundary condition.10oreover, the fact that k u k C α, (1+ α ) / ([ g ( t ) ,h ( t )] × [0 , ∞ )) ≤ ˜ C gives ∂u∂x ( h ( t k ) , t k +0) = ( W k ) x ( h ( t k ) , → ˜ W x ( h ∞ ,
0) as k → ∞ , and then ˜ W x ( h ∞ ,
0) = 0,which leads to a contradiction to the fact that ˜ W x ( h ∞ , ≤ − σ <
0. Thuslim t → + ∞ || u ( · , t ) || C ([ g ( t ) ,h ( t )]) = 0.Note that v ( x, t ) satisfies ∂v ( x, t ) ∂t = − a v ( x, t ) + G ( u ( x, t )) , g ( t ) < x < h ( t ) , t > , and G ( u ( x, t )) → x ∈ [ g ( t ) , h ( t )] as t → ∞ ; therefore, we havelim t → + ∞ || v ( · , t ) || C ([ g ( t ) ,h ( t )]) = 0. (cid:3) In the introduction, a threshold R , usually called the basic reproductionnumber, is given to decide whether the bacteria described by (1.2) vanish. No-tice that the interval domain for free boundary problem (1.3) changes with t ;therefore, the basic reproduction number is not a constant and should changewith t .Now we introduce the basic reproduction number R F ( t ) for (1.3) by R F ( t ) := R D (( g ( t ) , h ( t ))) = G ′ (0) a a a + d ( πh ( t ) − g ( t ) ) , where we use R D (Ω) to denote the basic reproduction number for the corre-sponding problem in Ω with null Dirichlet boundary condition on ∂ Ω. Now, thefollowing result is obvious; see also Lemma 2.3 in [14].
Lemma 3.3 − R F ( t ) has the same sign as λ , where λ is the principal eigen-value of the problem (cid:26) − dψ xx = − a ψ + G ′ (0) a a ψ + λ ψ, x ∈ ( g ( t ) , h ( t )) ,ψ ( x ) = 0 , x = g ( t ) or x = h ( t ) . (3.3)In fact, here λ = a + d ( πh ( t ) − g ( t ) ) − G ′ (0) a a = [ a + d ( πh ( t ) − g ( t ) ) ](1 − R F ( t )) . With the above defined reproduction number, we also have
Lemma 3.4 R F ( t ) is strictly monotone increasing function of t , that is if t We first show that h ∞ − g ∞ < + ∞ . In fact, direct calculations yielddd t Z h ( t ) g ( t ) [ u ( x, t ) + a a v ( x, t )]d x = Z h ( t ) g ( t ) [ u t + a a v t ]( x, t )d x + h ′ ( t )[ u + a a v ]( h ( t ) , t ) − g ′ ( t )[ u + a a v ]( g ( t ) , t )= Z h ( t ) g ( t ) du xx d x + Z h ( t ) g ( t ) − a u ( x, t ) + a a G ( u ( x, t ))d x = − dµ ( h ′ ( t ) − g ′ ( t )) + Z h ( t ) g ( t ) − a u ( x, t ) + a a G ( u ( x, t ))d x. Integrating from 0 to t ( > 0) gives Z h ( t ) g ( t ) [ u + a a v ]( x, t )d x = Z h (0) g (0) [ u + a a v ]( x, x (3.4)+ dµ ( h (0) − g (0)) − dµ ( h ( t ) − g ( t ))+ Z t Z h ( s ) g ( s ) − a u ( x, t ) + a a G ( u ( x, t )) dxds, t ≥ . (3.5)Since G ( z ) z ≤ G ′ (0) by the assumption ( A R ≤ − a u ( x, t ) + a a G ( u ( x, t )) ≤ x ∈ [ g ( t ) , h ( t )] and t ≥ 0, we have dµ ( h ( t ) − g ( t )) ≤ Z h (0) g (0) [ u + a a v ]( x, x + dµ ( h (0) − g (0))for t ≥ 0, which in turn gives that h ∞ − g ∞ < ∞ . Therefore, the bacteria arevanishing as a consequence of Lemma 3.2. (cid:3) Theorem 3.6 If R F (0) < and || u ( x ) || C ([ − h ,h ]) , || v ( x ) || C ([ − h ,h ]) are suffi-ciently small. Then h ∞ − g ∞ < ∞ and lim t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) =0 . Proof: We construct a suitable upper solution for ( u, v ). Since R F (0) < 1, itfollows from Lemma 3.3 that there is a λ > < ψ ( x ) ≤ − h , h )such that (cid:26) − dψ xx = − a ψ + G ′ (0) a a ψ + λ ψ, − h < x < h ,ψ ( x ) = 0 , x = ± h . (3.6)12herefore, there exists a small δ > − δ + ( 1(1 + δ ) − | − a + G ′ (0) a a | + [ 1(1 + δ ) − 14 ] λ ≥ . Similarly as in [10], we set σ ( t ) = h (1 + δ − δ e − δt ) , t ≥ , and u ( x, t ) = εe − δt ψ ( xh /σ ( t )) , − σ ( t ) ≤ x ≤ σ ( t ) , t ≥ .v ( x, t ) = ( G ′ (0) a + λ a ) u ( x, t ) , − σ ( t ) ≤ x ≤ σ ( t ) , t ≥ . Direct computations yield u t − d ∂ u∂x + a u − a v = − δu − εe − δt ψ ′ xh σ ′ ( t ) σ ( t ) + ( h σ ( t ) ) [ − a + G ′ (0) a a + λ ] u +[ a − G ′ (0) a a − λ u ≥ u {− δ + ( 1(1 + δ ) − | − a + G ′ (0) a a | + [ 1(1 + δ ) − 14 ] λ } ≥ ,v t + a v − G ( u ( x, t ))= − δv − εe − δt ψ ′ xh σ ′ ( t ) σ ( t ) ( G ′ (0) a + λ a ) + a ( G ′ (0) a + λ a ) u ( x, t ) − G ( u ( x, t )) ≥ ( a − δ )( G ′ (0) a + λ a ) u ( x, t ) − G ( u ( x, t ))= ( a − δ )( G ′ (0) a + λ a ) u ( x, t ) − G ′ ( ξ ( x, t )) u ( x, t )= [ G ′ (0) − G ′ ( ξ ( x, t )) + ( a − δ ) λ a − G ′ (0) a δ ] u ( x, t )for all t > − σ ( t ) < x < σ ( t ), where ξ ∈ (0 , u ). Since u ≤ ε , if δ and ε aresufficiently small, then we have[ G ′ (0) − G ′ ( ξ ( x, t )) + ( a − δ ) λ a − G ′ (0) a δ ] ≥ . 13n the other hand, we have σ ′ ( t ) = h δ e − δt , − u x ( σ ( t ) , t ) = − ε h σ ( t ) ψ ′ ( h ) e − δt ,and − u x ( − σ ( t ) , t ) = − ε h σ ( t ) ψ ′ ( − h ) e − δt . Noticing that ψ ′ ( − h ) = − ψ ′ ( h ), wenow choose ε = − δ h (1+ δ )2 µψ ′ ( h ) such that ∂u∂t ≥ d ∂ u∂x − a u + a v, − σ ( t ) < x < σ ( t ) , t > ,∂v∂t ≥ − a v + G ( u ) , − σ ( t ) < x < σ ( t ) , t > ,u ( x, t ) = v ( x, t ) = 0 , x = ± σ ( t ) t > , − σ (0) < − h , − σ ′ ( t ) ≤ − µ ∂u∂x ( − σ ( t ) , t ) , t > ,σ (0) > h , σ ′ ( t ) ≥ − µ ∂u∂x ( σ ( t ) , t ) , t > . If || u || L ∞ ≤ εψ ( h δ/ ) and || v || L ∞ ≤ εψ ( h δ/ )( G ′ (0) a + λ a ), then u ( x ) ≤ εψ ( h δ/ ) ≤ u ( x, 0) = εψ ( x δ/ ) and v ( x ) ≤ εψ ( h δ/ )( G ′ (0) a + λ a ) ≤ v ( x, 0) for x ∈ [ − h , h ]. We can now apply Lemma 2.5 to conclude that g ( t ) ≥ − σ ( t ) and h ( t ) ≤ σ ( t ) for t > 0. It follows that h ∞ − g ∞ ≤ lim t →∞ σ ( t ) = 2 h (1 + δ ) < ∞ ,and lim t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) = 0 by Lemma 3.2. (cid:3) In this section, we give the sufficient conditions for the bacteria to be spreading.We first prove that if R F (0) ≥ 1, the bacteria are spreading. Theorem 4.1 If R F (0) ≥ , then h ∞ = − g ∞ = ∞ and lim inf t → + ∞ || u ( · , t ) || C ([0 ,h ( t )]) > , that is, spreading occurs. Proof: We first consider the case that R F (0) := R D (( − h , h )) > 1. In thiscase, we have that the eigenvalue problem (cid:26) − dψ xx = − a ψ + G ′ (0) a a ψ + λ ψ, − h < x < h ,ψ ( x ) = 0 , x = ± h (4.1)admits a positive solution ψ ( x ) with || ψ || L ∞ = 1, where λ is the principaleigenvalue. It follows from Lemma 3.3 that λ < u ( x, t ) = δψ ( x ) , v = ( G ′ (0) a + λ a ) δψ ( x )for − h ≤ x ≤ h , t ≥ 0, where δ is sufficiently small.14irect computations yield ∂u∂t − d ∂ u∂x + a u − a v = δψ ( x )( 34 λ ) ≤ ∂v∂t + a v − G ( u ) = δψ ( x )[ G ′ (0) − G ′ ( ξ ( x, t )) + a λ a ]for all t > − h < x < h , where ξ ∈ (0 , u ). Noting that λ < ≤ ξ ( x, t ) ≤ u ( x, t ) ≤ δ , we can chose δ sufficiently small such that ∂u∂t ≤ d ∂ u∂x − a u + a v, − h < x < h , t > ,∂v∂t ≤ − a v + G ( u ) , − h < x < h , t > ,u ( x, t ) = v ( x, t ) = 0 , x = ± h t > , − h ′ ≥ − µ ∂u∂x ( − h , t ) , t > , h ′ ≤ − µ ∂u∂x ( h , t ) , t > ,u ( x, ≤ u ( x ) , v ( x, ≤ v ( x ) , − h ≤ x ≤ h . Hence, applying Remark 2.1 yields that u ( x, t ) ≥ u ( x, t ) and v ( x, t ) ≥ v ( x, t ) in[ − h , h ] × [0 , ∞ ). It follows that lim inf t → + ∞ || u ( · , t ) || C ([ g ( t ) ,h ( t )]) ≥ δψ (0) > h ∞ − g ∞ = + ∞ by Lemma 3.2.If R F (0) = 1, then for any positive time t , we have g ( t ) < − h and h ( t ) > h ; therefore, R F ( t ) > R F (0) = 1 by the monotonicity in Lemma 3.4.Replacing the initial time 0 by the positive time t , we then have h ∞ − g ∞ = + ∞ as above. (cid:3) Remark 4.1 It follows from the above proof that spreading occurs, if there ex-ists t ≥ such that R F ( t ) ≥ . Theorem 3.6 shows if R F (0) < 1, vanishing occurs for small initial size ofinfected bacteria, and Theorem 3.5 implies that if R ≤ 1, vanishing alwaysoccurs for any initial values. The next result shows that spreading occurs forlarge values. Theorem 4.2 Suppose that R F (0) < < R . Then h ∞ − g ∞ = ∞ and lim inf t → + ∞ || u ( · , t ) || C ([0 ,h ( t )]) > if || u ( x ) || C ([ − h ,h ]) and || v ( x ) || C ([ − h ,h ]) aresufficiently large. Proof: We construct a vector ( u, v, h ) such that u ≥ u , v ≥ v in [ − h ( t ) , h ( t )] × [0 , T ], and also g ( t ) ≤ − h ( t ), h ( t ) ≥ h ( t ) in [0 , T ]. If we can choose T suchthat R D (( − h ( √ T ) , h ( √ T )) > 1, then h ∞ − g ∞ = ∞ .15e first consider the following eigenvalue problem (cid:26) − dψ ′′ − ψ ′ = µ ψ, < x < ,ψ (0) = ψ (1) = 0 . (4.2)It is well known that the principal eigenvalue µ of this problem is simple; thecorresponding eigenfunction ψ ( x ) can be chosen to be positive in [0 , 1) and || ψ || L ∞ = 1. It is also easy to see that µ > d and ψ ′ ( x ) < , ψ in [0 , 1] to an even function φ in [ − , 1] yields (cid:26) − dφ ′′ − s gn ( x )2 φ ′ = µ φ, − < x < ,φ ( − 1) = φ (1) = 0 . (4.3)We now construct a suitable lower solution to (1.3) and we define h ( t ) = √ t + δ, t ≥ ,u ( x, t ) = M ( t + δ ) k φ ( x √ t + δ ) , −√ t + δ ≤ x ≤ √ t + δ, t ≥ ,v ( x, t ) = M ( t + δ ) k φ ( x √ t + δ ) , −√ t + δ ≤ x ≤ √ t + δ, t ≥ , where δ, M, T , k are chosen as follows :0 < δ ≤ min { , h } , R D (( − p T , p T )) > ,k ≥ max { µ + a ( T + 1) , a ( T + 1) } , − µM φ ′ (1) > ( T + 1) k . Here we have used the assumption that R > R D (( −√ T , √ T )) → R as T → ∞ by Lemma 3.4.Direct computations yield ∂u∂t − d ∂ u∂x + a u − a v = − M ( t + δ ) k +1 [ dφ ′′ + x √ t + δ φ ′ + ( k + ( − a + a )( t + δ )) φ ] ≤ − M ( t + δ ) k +1 [ dφ ′′ + sgn ( x )2 φ ′ + µ φ ]= 0 , ∂v∂t + a v − G ( u ( x, t )) ≤ − M ( t + δ ) k +1 [( k + ( − a + G ( u ) /u )( t + δ )) φ ] ≤ − M ( t + δ ) k +1 [ k − a ( T + 1)] ≤ < t ≤ T and − h < x < h . h ′ ( t ) + µu x ( √ t + δ, t ) = 12 √ t + δ + µM ( t + δ ) k +1 / φ ′ (1) < . Then we have ∂u∂t ≤ d ∂ u∂x − a u + a v, − h < x < h, < t ≤ T ,∂v∂t ≤ − a v + G ( u ( x, t )) , − h < x < h, < t ≤ T ,u ( x, t ) = v ( x, t ) = 0 , x = ± h ( t ) , < t ≤ T , − h = −√ δ ≥ − h , < t ≤ T ,h = √ δ ≤ h , < t ≤ T h ′ ( t ) < − µu x ( √ t + δ, t ) , < t ≤ T , − h ′ ( t ) > − µu x ( −√ t + δ, t ) , < t ≤ T . If u ( x, 0) = Mδ k φ ( x √ δ ) < u ( x ) and v ( x, 0) = Mδ k φ ( x √ δ ) < v ( x ) in [0 , √ δ ], thenusing Lemma 2.5 yields h ( t ) ≥ h ( t ) and g ( t ) ≤ − h ( t ) in [0 , T ]. In particular, h ( T ) − g ( T ) ≥ h ( T ) = 2 √ T + δ ≥ √ T . Noting that R F ( T ) := R D (( g ( T ) , h ( T ))) ≥ R D (( − h ( T ) , h ( T ))) ≥ R D (( − p T , p T )) > , we then have h ∞ − g ∞ = + ∞ by Theorem 4.1. (cid:3) Theorem 4.3 (Sharp threshold) Suppose that R > , with fixed µ , h and ( φ, ψ ) satisfying (1 . . Let ( u, v ; g, h ) be a solution of (1 . with ( u , v ) =( σφ ( x ) , σψ ( x )) for some σ > . Then there exists σ ∗ = σ ∗ ( φ, ψ ) ∈ [0 , ∞ ) suchthat spreading occurs when σ > σ ∗ , and vanishing occurs when < σ ≤ σ ∗ . Proof: It follows from Theorem 4.1 that spreading always occurs if R F (0) ≥ σ ∗ ( φ, ψ ) = 0 for any φ and ψ .For the remaining case R F (0) < 1, define σ ∗ := sup { σ : h ∞ ( σφ, σψ ) < ∞ for σ ∈ (0 , σ ] } . By Theorem 3.6, we see that in this case vanishing occurs for all small σ > σ ∗ ∈ (0 , ∞ ]. On the other hand, it follows from Theorem 4.2 thatin this case spreading occurs for all large σ . Therefore, σ ∗ ∈ (0 , ∞ ), spreadingoccurs when σ > σ ∗ , and vanishing occurs when 0 < σ < σ ∗ by Corollary 2.6.We claim that vanishing occurs when σ = σ ∗ . Otherwise h ∞ − g ∞ = ∞ for σ = σ ∗ . Since R F ( t ) → R > t → ∞ , there exists T > R F ( T ) > 1. By the continuous dependence of ( u, v ; g, h ) on its initialvalues, we can find ǫ > u , v ) = ( σ ∗ − ǫ )( φ ( x ) , ψ ( x )), denoted by ( u ǫ , v ǫ ; g ǫ , h ǫ ) satisfies R F ( T ) > u ǫ , v ǫ ; g ǫ , h ǫ ), contradicting the definitionof σ ∗ . This completes the proof. (cid:3) Similarly, if we consider µ instead of u as a varying parameter, the followingresult holds; see also Theorem 4.4 in [11]. Theorem 4.4 (Sharp threshold) Suppose that R > , with fixed h , u and v . Then there exists µ ∗ ∈ [0 , ∞ ) such that spreading occurs when µ > µ ∗ , andvanishing occurs when < µ ≤ µ ∗ . Next, we consider the asymptotic behavior of the solution to (1.3) when thespreading occurs. Theorem 4.5 Suppose that R > . If spreading occurs, then the solution offree boundary problem (1.3) satisfies lim t → + ∞ ( u ( x, t ) , v ( x, t )) = ( u ∗ , v ∗ ) uni-formly in any bounded subset of ( −∞ , ∞ ) , where ( u ∗ , v ∗ ) is the unique positiveequilibrium of (1.2) . Proof: (1) The limit superior of the solutionWe recall that the comparison principle gives ( u ( x, t ) , v ( x, t )) ≤ ( u ( t ) , v ( t ))for ( x, t ) ∈ [ g ( t ) , h ( t )] × (0 , ∞ ), where ( u ( t ) , v ( t )) is the solution of the problem u ′ ( t ) = − a u ( t ) + a v ( t ) , t > ,v ′ ( t ) = − a v ( t ) + G ( u ( t )) , t > ,u (0) = || u || L ∞ [ − h ,h ] , v (0) = || v || L ∞ [ − h ,h ] . (4.4)Since R > 1, the unique positive equilibrium ( u ∗ , v ∗ ) is globally stable for theODE system (4.4) and lim t →∞ ( u ( t ) , v ( t )) = ( u ∗ , v ∗ ); therefore we deducelim sup t → + ∞ ( u ( x, t ) , v ( x, t )) ≤ ( u ∗ , v ∗ ) (4.5)uniformly for x ∈ ( −∞ , ∞ ).(2) The lower bound of the solution for a large time.Note that R > l →∞ G ′ (0) a a a + d ( π l ) = R > L such that G ′ (0) a a a + d ( π L ) > . This implies that the principaleigenvalue λ ∗ of (cid:26) − dψ xx = − a ψ + G ′ (0) a a ψ + λ ∗ ψ, x ∈ ( − L , L ) ,ψ ( x ) = 0 , x = ± L (4.6)18atisfies λ ∗ = a + d ( π L ) − G ′ (0) a a < . Since h ∞ − g ∞ = ∞ by assumption, h ∞ = g ∞ = ∞ by Lemma 3.1. Thus, forany L ≥ L , there exists t L > g ( t ) ≤ − L and h ( t ) ≥ L for t ≥ t L .Letting U = δψ and V = G ( U ) /a , we can choose δ sufficiently small suchthat ( U , V ) satisfies U t ≤ dU xx − a U + a V , − L < x < L , t > t L ,V t = − a V + G ( U ) , − L < x < L , t > t L ,U ( x, t ) = 0 , x = ± L , t > t L ,U ( x, t L ) ≤ u ( x, t L ) , V ( x, t L ) = v ( x, t L ) , − L ≤ x ≤ L , meaning that ( U , V ) is a lower solution of the solution ( u, v ) in [ − L , L ] × [ t L , ∞ ). We then have ( u, v ) ≥ ( δψ, G ( δψ ) /a ) in [ − L , L ] × [ t L , ∞ ), whichimplies that the solution can not decay to zero.(3) The limit inferior of the solution.We extend ψ ( x ) to ψ L ( x ) by defining ψ L ( x ) := ψ ( x ) for − L ≤ x ≤ L and ψ L ( x ) := 0 for x < − L or x > L . Now for L ≥ L , ( u, v ) satisfies u t = du xx − a u + a v, g ( t ) < x < h ( t ) , t > t L ,v t = − a v + G ( u ) , g ( t ) < x < h ( t ) , t > t L ,u ( x, t ) = 0 , x = g ( t ) or x = h ( t ) , t > t L ,u ( x, t L ) ≥ δψ L , v ( x, t L ) ≥ G ( δψ L ) /a , − L ≤ x ≤ L ; (4.7)therefore, we have ( u, v ) ≥ ( u, v ) in [ − L, L ] × [ t L , ∞ ), where ( u, v ) satisfies u t = du xx − a u + a v, − L < x < L, t > t L ,v t = − a v + G ( u ) , − L < x < L, t > t L ,u ( x, t ) = 0 , x = ± L, t > t L ,u ( x, t L ) = δψ L , v ( x, t L ) = G ( δψ L ) /a , − L ≤ x ≤ L. (4.8)The system (4.8) is quasimonotone increasing; therefore, it follows from theupper and lower solution method and the theory of monotone dynamical systems( [20] Corollary 3.6) that lim t → + ∞ ( u ( x, t ) , v ( x, t )) ≥ ( u L ( x ) , v L ( x )) uniformlyin [ − L, L ], where ( u L , v L ) satisfies − du ′′ L = − a u L + a v L , − L < x < L, − a v L + G ( u L ) = 0 , − L < x < L,u L ( x ) = 0 , x = ± L (4.9)and is the minimum upper solution ( δψ L , G ( δψ L ) /a ).19ow we give the monotonicity and show that if 0 < L < L , then u L ( x ) ≤ u L ( x ) in [ − L , L ]. The result is derived by comparing the boundary conditionsand initial conditions in (4.8) for L = L and L = L .Let L → ∞ . By classical elliptic regularity theory and a diagonal procedure,it follows that ( u L ( x ) , v L ( x )) converges uniformly on any compact subset of( −∞ , ∞ ) to ( u ∞ , v ∞ ) that is continuous on ( −∞ , ∞ ) and satisfies − du ′′∞ = − a u ∞ + a v ∞ , −∞ < x < ∞ , − a v ∞ + G ( u ∞ ) = 0 , −∞ < x < ∞ ,u ∞ ( x ) ≥ δψ L , v ∞ ( x ) ≥ G ( δψ L ) /a , −∞ < x < ∞ . Next, we observe that u ∞ ( x ) ≡ u ∗ and v ∞ ( x ) ≡ v ∗ , which can be derived byconsidering the problem − dw ′′ = − a w + a a G ( w ) . The uniqueness of the positive solution follows from the assumption on G andthe condition R > − M, M ] with M ≥ L , since that ( u L ( x ) , v L ( x )) → ( u ∗ , v ∗ ) uniformly in [ − M, M ], which is the compact subset of ( −∞ , ∞ ), as L → ∞ , we deduce that for any ε > 0, there exists L ∗ > L such that( u L ∗ ( x ) , v L ∗ ( x )) ≥ ( u ∗ − ε, v ∗ − ε ) in [ − M, M ]. As above, there is t L ∗ suchthat [ g ( t ) , h ( t )] ⊇ [ − L ∗ , L ∗ ] for t ≥ t L ∗ . Therefore,( u ( x, t ) , v ( x, t )) ≥ ( u ( x, t ) , v ( x, t )) in [ − L ∗ , L ∗ ] × [ t L ∗ , ∞ ) , and lim t → + ∞ ( u ( x, t ) , v ( x, t )) ≥ ( u L ∗ ( x ) , v L ∗ ( x )) in [ − L ∗ , L ∗ ] . Using the fact that ( u L ∗ ( x ) , v L ∗ ( x )) ≥ ( u ∗ − ε, v ∗ − ε ) in [ − M, M ] giveslim inf t → + ∞ ( u ( x, t ) , v ( x, t )) ≥ ( u ∗ − ε, v ∗ − ε ) in [ − M, M ] . Since ε > t → + ∞ u ( x, t ) ≥ u ∗ and lim inf t → + ∞ v ( x, t ) ≥ v ∗ uniformly in [ − M, M ], which together with (4.5) imply that lim t → + ∞ u ( x, t ) = u ∗ and lim t → + ∞ v ( x, t ) = v ∗ uniformly in any bounded subset of ( −∞ , ∞ ). (cid:3) Combining Remark 4.1, Theorem 4.2 and Theorem 4.5, we immediatelyobtain the following spreading-vanishing dichotomy: Theorem 4.6 Suppose that R > . Let ( u ( x, t ) , v ( x, t ); g ( t ) , h ( t )) be the solu-tion of free boundary problem (1.3) . Then, the following alternatives hold:Either i ) Spreading: h ∞ − g ∞ = + ∞ and lim t → + ∞ ( u ( x, t ) , v ( x, t )) = ( u ∗ , v ∗ ) uni-formly in any bounded subset of ( −∞ , ∞ ) ;or ( ii ) Vanishing: h ∞ − g ∞ ≤ h ∗ with G ′ (0) a a a + d ( πh ∗ ) = 1 and lim t → + ∞ ( || u ( · , t ) || C ([ g ( t ) ,h ( t )]) + || v ( · , t ) || C ([ g ( t ) ,h ( t )]) ) = 0 . In this paper, a free boundary problem is used to describe the expanding of bac-teria in a man-environment-man epidemic model in a one-dimensional habitat.We take into account the spreading and vanishing of the bacteria. Here, van-ishing implies not only that the bacteria disappear eventually, but also that theinfected habitat is limited, and spreading means the existence of the bacteria inthe long run with an uncontrollable infected environment. Sufficient conditionsfor the bacteria spreading or vanishing are given.Compared with existing work described by reaction-diffusion systems (1.1) in[4] or established by travelling waves and entire solutions in [21, 23, 24, 25, 26, 27], our model (1.3) provides a different way to understand the expanding process ofbacteria. It is well-known that for the ODE system (1.2), the basic reproductionnumber R (:= G ′ (0) a a a ) determines whether the bacteria die out ( R < 1) orremain endemic ( R > t ; therefore, we introduced the basic reproduction number R F ( t ) := G ′ (0) a a a + d ( πh ( t ) − g ( t ) ) , which depends on the habitat ( g ( t ) , h ( t )), the diffusion rate d and the coeffi-cients in (1.3). We showed that R F ( t ) ≤ R and R F ( t ) → R if ( g ( t ) , h ( t )) → ( −∞ , + ∞ ) as t → ∞ . Furthermore, if R ≤ 1, the bacteria are always vanish-ing (Theorem 3.5). The result is the same as that for the corresponding ODEsystem (1.2). However, if R F ( t ) ≥ t ≥ 0, the bacteria are spreading(Theorem 4.1 and Remark 4.1). For the case R F (0) < < R , the spreading orvanishing of the bacteria depends on the initial size of bacteria (Theorem 4.3 ),or the ratio ( µ ) of the expansion speed of the free boundary and the populationgradient at the expanding fronts (Theorem 4.4).Ecologically, our main results reveal that if the multiplicative factor of theinfectious bacteria is small, the bacteria will die out eventually and the humansare safe. Otherwise, the spreading or vanishing of the bacteria depends on theinitial infected habitat, the diffusion rate, and other factors. In particular, the21nitial number of bacteria plays a key role. A large initial number can inducethe spreading of bacteria easily. A similar result obtained for an invasive specieshas been supported by substantial empirical evidence; see [10]. 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