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Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias
Applied Mathematics Letters 100 (2020) 106052
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Applied Mathematics Letters www.elsevier.com/locate/aml
Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias Jinliang Wang a , Yuming Chen b ,∗ a b
School of Mathematical Science, Heilongjiang University, Harbin 150080, PR China Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5 Canada
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Article history: Received 13 July 2019 Accepted 9 September 2019 Available online 18 September 2019 Keywords: Spatial heterogeneity Global asymptotic stability Basic reproduction number Threshold dynamics
abstract In this paper, we formulate and investigate a vector-borne disease model with vector-bias in a bounded spatial domain. The main result is a threshold dynamics characterized in terms of the basic reproduction number ℜ0 or equivalently in terms of the principal eigenvalue of the linearized system. Roughly speaking, if ℜ0 < 1 then the semi-trivial equilibrium is globally asymptotically stable and the disease dies out; if ℜ0 > 1 then the disease persists. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Zika virus disease is caused by the Zika virus (ZIKV), a mosquito-borne flavivirus. ZIKV is primarily transmitted by the bite of an infected mosquito from the Aedes genus, mainly Aedes aegypti, in tropical and subtropical regions. The recent outbreak in Brazil (March 2015) has attracted the attention of many researchers. Reaction–diffusion models have been proposed to effectively understand the spread of ZIKV (see, for example, [1–8] and references therein). In particular, Webb and his collaborators [4,8] studied the outbreak of Zika in Rio De Janerio. Assume that Ω ⊂ Rn is a bounded domain with a smooth boundary ∂Ω . Let Hu (x, t), Hi (x, t), Vu (x, t), and Vi (x, t) be the densities of susceptible hosts, infected hosts, uninfected vectors, and infected vectors at location x and time t, respectively. They assumed Hu to be constant as for Zika in Rio De Janerio the number of infected is fairly small in comparison with the number of the total population (also see [9]). This leads a deterministic reaction–diffusion model involving Hi , Vu , and Vi . They established a threshold dynamics determined by the basic reproduction number. Their proof “combines arguments from monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semiflows”. Moreover, mosquitos may choose infectious human hosts more frequently than healthy ones to bite. This difference is called vector-bias and has been evidenced by the disease malaria. On the one hand, it is found ∗ Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Y. Chen).
https://doi.org/10.1016/j.aml.2019.106052 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
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J. Wang and Y. Chen / Applied Mathematics Letters 100 (2020) 106052
through experiment on malaria infection that infectious humans are more attractive mosquitos [10]. On the other hand, Daniel and Kingsolver used a mechanistic model to argue that mosquitos may maximize their rate of protein intake during feeding by choosing malaria infected hosts [11]. Zhao and his collaborators [7,12] have studied the impact of vector-bias on the epidemiology of malaria. To describe the phenomenon of vectorbias in space, they introduced p and l to denote the probabilities that a mosquito randomly arrives at a human and picks the human if he is infectious and susceptible, respectively. Borrowing the idea in [7,12], the numbers of newly infected hosts and newly infected mosquitos per unit time at position x and time pHi u (x) t are respectively cβ1 (x) pHlH Vi and bβ2 (x) pH +lH V , where β1 (x) is the biting rate of infectious u (x) u i +lHu (x) i mosquitos, c is the transmission probability from an infected mosquito to a susceptible host, β2 (x) is the biting rate of susceptible mosquitos, and b is the transmission probability from infectious host to susceptible mosquitos. The aim of this paper is to study the combined effect of spatial heterogeneity and vector-bias on epidemic dynamics. Precisely, based on the recent works in [4,7,8,12], we propose and investigate the following vector-biased vector-borne model with spatial structure, ⎧ cβ1 (x)lHu (x)Vi ∂Hi ⎪ ⎪ x ∈ Ω , t > 0, ⎪ ∂t − ∇ · d1 (x)∇Hi = −λ(x)Hi + pHi +lHu (x) , ⎪ ⎪ ⎪ ⎪ ⎪ ∂Vu − ∇ · d2 (x)∇Vu = α(x)(Vu + Vi ) − µ(x)(Vu + Vi )Vu − bβ2 (x)pHi Vu , x ∈ Ω , t > 0, ⎪ pHi +lHu (x) ⎨ ∂t bβ2 (x)pHi Vu ∂Vi (1.1) x ∈ Ω , t > 0, ⎪ ∂t − ∇ · d2 (x)∇Vi = pHi +lHu (x) − µ(x)(Vu + Vi )Vi , ⎪ ⎪ ⎪ ⎪ ∂Hi ∂Vi ∂Vu ⎪ x ∈ ∂Ω , t > 0, ⎪ ∂ν = ∂ν = ∂ν = 0, ⎪ ⎪ ⎩ 3 (Hi (x, 0), Vu (x, 0), Vi (x, 0)) = (Hi0 (x), Vu0 (x), Vi0 (x)) ∈ C(Ω ; R+ ), where d1 (x), d2 (x) ∈ C 1+α (Ω ) are respectively the diffusion rates of the infected hosts and vectors, which are assumed to be strictly positive, the functions Hu (x), λ(x), α(x), β1 (x), β2 (x), and µ(x) are strictly ∂ positive and belong to C(Ω ). Here ∂ν denotes the normal derivative with respect to ∂Ω . The initial functions Hi0 (·), Vu0 (·), Vi0 (·) are nonnegative continuous functions. The existence, uniqueness, and positivity of global classical solutions of (1.1) can be proved by using similar arguments as those in [8]. The remaining part of this paper is organized as follows. We first study the stability of the diseasefree equilibria. Then we derive the basic reproduction number. Followed is the main result of this paper, a threshold dynamics characterized by the basic reproduction number. 2. Stability of the disease-free equilibria Let V := Vu + Vi , be the total density of vectors. Then V (x, t) satisfies the diffusive logistic equation ⎧ ⎪ ⎪ ∂V 2 ⎪ ⎪ ⎨ ∂t − ∇ · d2 (x)∇V = α(x)V − µ(x)V , x ∈ Ω , t > 0, ∂V (2.1) x ∈ ∂Ω , t > 0, ⎪ ∂ν = 0, ⎪ ⎪ ⎪ ⎩ V (x, 0) = V0 ∈ C(Ω ; R+ ). The following result follows from [13]. Lemma 2.1. Assume that V0 ∈ C(Ω ; R+ ). Then system (2.1) admits a unique global classical solution V (x, t). Furthermore, for all (x, t) ∈ Ω × (0, ∞), we have V (x, t) > 0 and limt→∞ ∥V (·, t) − V (x)∥∞ = 0, where V (x) is the unique positive solution of the problem ⎧ ⎨ −∇ · d2 (x)∇V = α(x)V − µ(x)V 2 , x ∈ Ω , (2.2) ⎩ ∂V = 0, x ∈ ∂Ω . ∂ν
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System (1.1) always has the two disease-free equilibria: the trivial equilibrium E0 = (0, 0, 0) and the unique semi-trivial equilibrium E1 = (0, V , 0). Theorem 2.1. The trivial equilibrium E0 = (0, 0, 0) is unstable. Proof . Linearizing system (1.1) at E0 , we arrive at the following eigenvalue problem, ⎧ ⎪ ⎪kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, x ∈ Ω , ⎨ kϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ), x ∈ Ω, kψ − ∇ · d (x)∇ψ = 0, x ∈ Ω, ⎪ 2 ⎪ ⎩ ∂φ ∂ϕ ∂ψ = = = 0, x ∈ ∂Ω . ∂ν ∂ν ∂ν
(2.3)
Denote by ˜ k(d, f ) the principal eigenvalue associated with a positive eigenvector of the following eigenvalue problem { kϕ − ∇ · d(x)∇ϕ = f ϕ, x ∈ Ω , (2.4) ∂ϕ x ∈ ∂Ω , ∂ν = 0, where d(x) ∈ C 1 (Ω ) is strictly positive on Ω and f ∈ C(Ω ). It is easy to check that ˜ k(d, f ) is monotone in the ˜ ˜ ˜ sense that if f1 ≥ f2 then k(d, f1 ) > k(d, f2 ). Let ϕ be a positive eigenvector corresponding to the eigenvalue ˜ ˜ 0). k(d2 , α) of (2.4). Then obviously ˜ k(d2 , α) is an eigenvalue of (2.3) with a corresponding eigenvector (0, ϕ, As ˜ k(d2 , 0) > 0, it follows that E0 is linearly unstable. Now, linearizing system (1.1) at E1 and letting (Hi (x, t), Vu (x, t), Vi (x, t)) = (φ(x)ekt , ϕ(x)ekt , ψ(x)ekt ) yield ⎧ x ∈ Ω, ⎪kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, ⎪ ⎪ ⎨kϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ) − 2µ(x)V ϕ − µ(x)V ψ − bβ2 (x) pV φ, x ∈ Ω , lHu (x) (2.5) pV ⎪ kψ − ∇ · d (x)∇ψ = bβ (x) φ − µ(x)V ψ, x ∈ Ω, ⎪ 2 2 lHu (x) ⎪ ⎩ ∂φ ∂ϕ ∂ψ x ∈ ∂Ω . ∂ν = ∂ν = ∂ν = 0, It is necessary to consider the following problem ⎧ x ∈ Ω, ⎪ ⎨kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, pV kψ − ∇ · d2 (x)∇ψ = bβ2 (x) lHu (x) φ − µ(x)V ψ, x ∈ Ω , ⎪ ⎩ ∂φ ∂ψ x ∈ ∂Ω . ∂ν = ∂ν = 0,
(2.6)
Since (2.6) is cooperative, it follows from the standard Krein–Rutman theorem that the eigenvalue problem (2.6) has a principal eigenvalue k0 (d1 , d2 , V , V ) associated with a positive eigenvector (φ0 , ψ0 ). Theorem 2.2. The semi-trivial equilibrium E1 = (0, V , 0) is locally asymptotically stable if k0 (d1 , d2 , V , V ) < 0 and unstable if k0 (d1 , d2 , V , V ) > 0. Proof . Recall that V is the unique positive solution of the elliptic problem (2.2). It follows that ˜ k(d2 , α − µV ) = 0. From the monotonicity of ˜ k(d, f ), we have ˜ k(d2 , α − 2µV ) < 0. Denote by k the eigenvalue of (2.5). It follows that k is also an eigenvalue of either (2.6) or the following eigenvalue problem: { kϕ − ∇ · d2 (x)∇ϕ = α(x)ϕ − 2µ(x)V ϕ, x ∈ Ω , ∂ϕ ∂ν
= 0,
x ∈ ∂Ω .
Since k0 < 0 and ˜ k(d2 , α − 2µV ) < 0, we have k < 0. By the arbitrariness of k, E1 is linearly stable. While if k0 > 0, denote by (φ0 , ψ0 ) the positive eigenvector corresponding to k0 . It then follows from
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˜ k(d2 , α − 2µV ) < 0 and the Fredholm alternative that the following problem admits a unique solution ϕ0 , { k0 ϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ0 ) − 2µ(x)V ϕ − µ(x)V ψ0 − bβ2 (x) lHpV φ , x ∈ Ω, u (x) 0 ∂ϕ x ∈ ∂Ω . ∂ν = 0, Therefore, we have arrived at the conclusion that (2.5) has an eigenvector (φ0 , ϕ0 , ψ0 ) corresponding to an eigenvalue k0 > 0. Consequently, E1 is linearly unstable. 3. The basic reproduction number According to [14,15], we derive the basic reproduction number of (1.1). Let B : [C(Ω )]2 → [C(Ω )]2 be the operator ( )( ) ∇ · d1 (x)∇ − λ(x) cβ1 (x) φ B(φ, ψ) = , ψ 0 ∇ · d2 (x)∇ − µ(x)V } { ∂ψ where Dom(B) = (φ, ψ) ∈ C(Ω ; R2 ) : ∂φ ∂ν = ∂ν = 0, x ∈ ∂Ω . Define ( ) 0 0 C = . bβ2 (x) lHpV 0 u (x) Let A = B + C , which is regarded as a positive perturbation of B. It is easy to check from [14] that A and B are resolvent positive. On the other hand, the spectral bound of B is negative, i.e., s(B) < 0. Then we can apply the results in [14, Theorem 3.5] to conclude that k0 = s(A ) has the same sign with r(−C B −1 )−1, where r(−C B −1 ) is the spectral radius of −C B −1 . The basic reproduction number ℜ0 for (1.1) is defined by ℜ0 = r(−C B −1 ). The following result restates the above discussion. Lemma 3.1.
ℜ0 − 1 and k0 have the same sign.
4. A threshold dynamics The main result of this paper is that ℜ0 determines the disease extinction and persistence. We first consider the global dynamics of the model when ℜ0 < 1. Theorem 4.1. If ℜ0 < 1, then E1 is globally asymptotically stable in the sense that limt→∞ ∥ (Hi (x, t), Vu (x, t), Vi (x, t)) − E1 ∥∞ = 0 for any nonnegative initial condition (Hi0 , Vu0 , Vi0 ) ∈ C(Ω ; R3+ ) with Vu0 + Vi0 ̸= 0. Proof . By Theorem 2.2, E1 is locally asymptotically stable and k0 (d1 , d2 , V , V ) < 0. Choose ε > 0 small enough such that k1 := k0 (d1 , d2 , V + ε, V − ε) < 0 with a corresponding positive eigenvector (φε , ψε ). By Lemma 2.1, we know that Vu + Vi → V (x) uniformly on Ω as t → ∞. Hence there exists t0 > 0 such that V (x) − ε < Vu + Vi < V (x) + ε for x ∈ Ω and t > t0 . It then follows that the solution (Hi , Vi ) of (1.1) is a lower solution of the following problem ⎧ ⎪ ⎪ ∂ Hˇ i − ∇ · d1 (x)∇H ˇ i = −λ(x)H ˇ i + cβ1 (x)Vˇi , x ∈ Ω , t > t0 , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ Vˇi − ∇ · d (x)∇Vˇ = bβ (x) p(V +ε) H ˇ ˇ 2 i 2 ∂t lHu (x) i − µ(x)(V − ε)Vi , x ∈ Ω , t > t0 , (4.1) ⎪ ˇi ⎪ ∂H ∂ Vˇi ⎪ x ∈ ∂Ω , ⎪ ⎪ ∂ν = ∂ν = 0, ⎪ ⎪ ⎩ˇ Hi (x, t0 ) = M φε (x), Vˇi (x, t0 ) = M ψε (x), x ∈ Ω,
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where M is a large enough positive constant such that (Hi (x, t0 ), Vi (x, t0 )) ≤ (M φε (x), M ψε (x)). By the ˇ i (x, t), Vˇi (x, t)) comparison principle for cooperative systems, we can conclude that (Hi (x, t), Vi (x, t)) ≤ (H ˇ i (x, t), Vˇi (x, t)) = for all x ∈ Ω and t ≥ t0 . Since (4.1) is a linear system, it admits a unique solution (H k1 t k1 t ˇ ˇ (M φε (x)e , M ψε (x)e ). It follows from k1 < 0 that (Hi (x, t), Vi (x, t)) → (0, 0) uniformly for x ∈ Ω as t → ∞ and hence (Hi (x, t), Vi (x, t)) → (0, 0) uniformly for x ∈ Ω as t → ∞. Moreover, by Vu0 + Vi0 ̸= 0 and Lemma 2.1, we know that Vu (x, t) + Vi (x, t) → V (x) uniformly on Ω , which implies that Vu (x, t) → V (x) uniformly for x ∈ Ω as t → ∞. This completes the proof. The result below can be easily derived with a minor modification of the arguments in the proof of [4, Lemma 3.10]. Lemma 4.1. There exists Mmax > 0, independent of initial data, such that any solution (Hi , Vu , Vi ) of (1.1) satisfies 0 ≤ Hi (x, t), Vu (x, t), Vi (x, t) ≤ Mmax , x ∈ Ω , t ≥ t0 , where t0 is dependent on the initial data. Before proving the uniform persistence, we first consider weak persistence. To this end, we set X := C(Ω , R3 ), which is a Banach space equipped with the supremum norm ∥ · ∥∞ . Then (X, X+ ) is a strongly ordered space, where X+ = C(Ω , R3+ ). Let X0 := {ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ : ϕ1 + ϕ3 ̸= 0 and ϕ2 + ϕ3 ̸= 0} , ∂X0 := {ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ : ϕ1 + ϕ3 = 0 or ϕ2 + ϕ3 = 0} . By using these notations, we have X+ = X0 ∪ ∂X0 , X0 is relatively open with X0 = X+ , and ∂X0 is relatively closed in X+ . Denote by Φ(t) the solution semiflow generated by (1.1). From Lemma 4.1, Φ(t) is point dissipative. Furthermore, Φ(t) is compact for any t > 0. Denote by ω(ϕ) the omega limit set of the orbit O+ (ϕ) := {Φ(t)(ϕ) : t ≥ 0}. Lemma 4.2. If ℜ0 > 1, then E0 and E1 are uniform weak repellers for X0 in the sense that there exists a sufficiently small constant ε0 > 0 such that for any ϕ ∈ X0 , lim supt→∞ ∥Φ(t)ϕ − E0 ∥∞ ≥ ε0 and lim supt→∞ ∥Φ(t)ϕ − E1 ∥∞ ≥ ε0 . Proof . On the one hand, let ε1 < 21 minx∈Ω V (x). Then lim supt→∞ ∥Φ(t)ϕ − E0 ∥ ≥ ε1 for all ϕ ∈ X0 . Otherwise, there exists ϕ ∈ X0 such that lim supt→∞ ∥Φ(t)ϕ − E0 ∥ < ε1 . Then there exists t1 > 0 such that Vu (x, t) < εˆ1 and Vi (x, t) < εˆ1 for all x ∈ Ω and t ≥ t1 for some εˆ1 < ε1 . This will contradict with the fact that Vu (x, t) + Vi (x, t) → V (x) uniformly on Ω as t → ∞. On the other hand, it follows from Lemma 3.1 that ℜ0 > 1 implies that k0 (d1 , d2 , V , V ) > 0. Then there exists a sufficiently small ε2 > 0 such that k2 := k0 (d1 , d2 , V − ε2 , V + ε2 ) > 0 and it has a corresponding positive eigenvector (φε2 , ψε2 ). We claim that lim supt→∞ ∥Φ(t)ϕ − E1 ∥ ≥ ε2 for ϕ ∈ X0 . If this is not true, then there exists ϕ ∈ X0 such that lim supt→∞ ∥Φ(t)ϕ − E1 ∥∞ < ε2 . It follows that there exists a t2 > 0 such that Hi (x, t) < ε2 , V (x) − ε2 < Vu (x, t) < V (x) + ε2 , and Vi (x, t) < ε2 for all x ∈ Ω and t ≥ t2 . By (1.1), we get { ∂H
i
∂t ∂Vi ∂t
cβ1 (x)lHu (x) pε2 +lHu (x) Vi , bβ2 (x)p(V (x)−ε2 ) Hi − µ(x)(V pε2 +lHu (x)
− ∇ · d1 (x)∇Hi ≥ −λ(x)Hi + − ∇ · d2 (x)∇Vi ≥
x ∈ Ω , t > t2 , (x) + ε2 )Vi ,
x ∈ Ω , t > t2 .
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Since Hi (x, t) > 0 and Vi (x, t) > 0 for all x ∈ Ω and t > 0, there exists α > 0 such that (Hi (x, t0 ), Vi (x, t0 )) ≥ α (φε2 , ψε2 ). Recall that αek2 (t−t2 ) (φε2 , ψε2 ) is the solution of ⎧ ⎨ ∂H i − ∇ · d (x)∇H = −λ(x)H + cβ1 (x)lHu (x) V , x ∈ Ω , t > t2 , 1 i i ∂t pε2 +lHu (x) i bβ (x)p(V (x)−ε ) ∂V 2 ⎩ i − ∇ · d2 (x)∇V i = 2 Hi − µ(x)(V (x) + ε2 )V i , x ∈ Ω , t > t2 . ∂t
pε2 +lHu (x)
By the comparison principle, we obtain (Hi (x, t), Vi (x, t)) ≥ αek2 (t−t2 ) (φε0 , ψε0 ) for t ≥ t0 . Thus Hi (x, t) → ∞ and Vi (x, t) → ∞ as k2 > 0, which leads to a contradiction with the boundedness of (Hi (x, t), Vi (x, t)). This proves the claim. Letting ε0 = min{ε1 , ε2 } immediately completes the proof. Now we are ready to state and prove the main result of uniform persistence to conclude the paper, which also implies the existence of at least one positive equilibrium. Theorem 4.2. Suppose ℜ0 > 1. Then (1.1) is uniformly persistent in the sense that there exists ς > 0 such that, for any initial condition (Hi0 , Vu0 , Vi0 ) ∈ X0 , lim
inf ∥(Hi (·, t), Vu (·, t), Vi (·, t)) − ω∥∞ ≥ ς.
t→∞ ω∈∂X0
Moreover, (1.1) has at least one positive equilibrium EE = (Hi∗ (x), Vu∗ (x), Vi∗ (x)) ∈ X0 . Proof . Firstly, we prove that X0 is positively invariant with respect to Φ(t). Let w0 = (Hi0 , Vu0 , Vi0 ) ∈ X0 . Then Hi0 + Vi0 ̸= 0 and Vu0 + Vi0 ̸= 0. If Vi0 ̸= 0, it then follows from the maximum principle and ∂Vi ∂t − ∇ · d2 (x)∇Vi ≥ −µ(x)(Vu + Vi )Vi that Vi (x, t) > 0 for x ∈ Ω and t > 0. Now assume that Vi0 = 0. Then Hi0 ̸= 0 and Vu0 (x) ̸= 0. From the first equation of (1.1), we get ∂Hi − ∇ · d1 (x)∇Hi ≥ −λ(x)Hi . ∂t This combined with Hi0 ̸= 0 and the maximum principle implies that Hi (x, t) > 0 for x ∈ Ω and t > 0. Then it follows from the second equation of (1.1) that ∂Vu − ∇ · d2 (x)∇Vu ≥ [α(x) − µ(x)(Vu + Vi ) − bβ2 (x)]Vu . ∂t Again Vu (x, t) > 0 for x ∈ Ω and t > 0 from Vu0 ̸= 0 and the maximum principle. In summary, we have shown that Φ(t)w0 ∈ X0 for w0 ∈ X0 and t ≥ 0, that is, ∂X0 is positively invariant with respect to Φ(t). Next, we show that ∂X0 is invariant with respect to Φ(t). Suppose w0 = (Hi0 , Vu0 , Vi0 ) ∈ ∂X0 , namely, Hi0 + Vi0 = 0 or Vu0 + Vi0 = 0. If Hi0 + Vi0 = 0 and Vu0 ̸= 0, then it is easy to see from the first and third equations of (1.1) that Hi (·, t) = Vi (·, t) = 0 for t ≥ 0. Then Vu is governed by (2.1) and by Lemma 2.1 we get Vu (x, t) > 0 for x ∈ Ω and t > 0 and Vu (x, t) → V (x) uniformly on Ω as t → ∞. Thus ω (w0 ) = {E1 }. Now if Vu0 + Vi0 = 0, then by the second and third equations of (1.1), we can get Vu (·, t) = Vi (·, t) = 0 for t ≥ 0. Thus the first equation of (1.1) becomes ∂Hi − ∇ · d1 (x)∇Hi = −λ(x)Hi , ∂t which implies that Hi (x, t) → 0 uniformly in Ω as t → ∞. Therefore, ω(w0 ) = {E0 }. To summarize, we have shown that ∂X0 is positively invariant with respect to Φ(t). Denote Φ∂ (t) := Φ(t)|∂X0 , the restriction of Φ(t) on ∂X0 . From the above, we see that Φ∂ (t) admits a compact global attractor, which is {E0 , E1 }.
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Thirdly, from Lemma 4.2, we see that E0 and E1 are isolated in X0 with X0 ∩ W s (E0 ) = ∅ and X0 ∩ W s (E1 ) = ∅, where W s (E0 ) and W s (E1 ) are the stable manifolds of E0 and E1 , respectively. Since there is no cycle in ∂X0 from E0 to E1 , we can use [16, Theorem 3] to conclude the existence of a constant ς > 0 such that { } min ϕ∈ω(w0 )
min inf ϕi (x)
i=1,2,3 x∈Ω
>ς
for all u0 ∈ X0 .
This implies that lim inf t→∞ Hi (x, t; u0 ), Vu (x, t; u0 ), Vi (x, t; u0 ) > ς and thus the uniform persistence stated in the theorem holds. Finally, from [17, Theorem 3.7], Φ(t) : X0 → X0 admits a global attractor A0 that attracts every point in X+ . Then Φ(t) has an equilibrium EE = (Hi∗ , Vu∗ , Vi∗ ) ∈ X0 by [17, Theorem 4.7]. This completes the proof. Acknowledgments The research of Wang was supported by the National Natural Science Foundation of China (No. 11871179), Natural Science Foundation of Heilongjiang Province (No. LC2018002), and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems. The research of Chen was supported by NSERC of Canada. References [1] Y. Cai, Z. Ding, B. Yang, Z. Peng, W. Wang, Transmission dynamics of zika virus with spatial structure–a case study in rio de janeiro, Brazil, Physica A 514 (2019) 729–740. [2] Y. Cai, K. Wang, W. Wang, Global transmission dynamics of a zika virus model, Appl. Math. Lett. 92 (2019) 190–195. [3] D. Gao, Y. Lou, D. He, et al., Prevention and control of zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep. 6 (2016) 28070. [4] P. Magal, G. Webb, Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity 31 (2018) 5589–5614. [5] B. Tang, Y. Xiao, J. Wu, Implication of vaccination against dengue for zika outbreak, Sci. Rep. 6 (2016) 35623. [6] L. Wang, H. Zhao, Dynamics analysis of a zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Physica A 522 (2019) 248–273. [7] X. Wang, X.-Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math. 77 (2017) 181–201. [8] W.E. Fitzgibbon, J.J. Morgan, G.F. Webb, An outbreak vector-host epidemic model with spatial structure: the 2015-2016 zika outbreak in rio de janeiro, Theor. Biol. Med. Model. 14 (2017) 7. [9] D.A.M. Villela, L.S. Bastos, L.M. de Carvalho, et al., Zika in rio de janeiro: assessment of basic reproduction number and comparison with dengue outbreaks, Epidemiol. Inf. 145 (2016) 1649–1657. [10] R. Lacroix, W.R. Mukabana, L.C. Gouagna, J.C. Koella, Malaria infection increases attractiveness of humans to mosquitoes, PLoS Biol. 3 (2005) e298. [11] T.L. Daniel, J.G. Kingsolver, The feeding strategy and the mechanics of blood sucking in insects, J. Theoret. Biol. 105 (1983) 661–677. [12] Z. Bai, R. Peng, X.-Q. Zhao, A reaction–diffusion malaria model with seasonality and incubation period, J. Math. Biol. 77 (2018) 201–228. [13] R.S. Cantrell, C. Cosner, Spatial Ecology Via Reaction–Diffusion Equations, John Wiley & Sons, Ltd, Chichester, 2003. [14] H.R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math. 70 (2009) 188–211. [15] W. Wang, X.-Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst. 11 (2012) 1652–1673. [16] H.L. Smith, X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. 47 (2001) 6169–6179. [17] P. Magal, X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal. 37 (2005) 251–275.
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Applied Mathematics Letters www.elsevier.com/locate/aml
Threshold dynamics of a vector-borne disease model with spatial structure and vector-bias Jinliang Wang a , Yuming Chen b ,∗ a b
School of Mathematical Science, Heilongjiang University, Harbin 150080, PR China Department of Mathematics, Wilfrid Laurier University, Waterloo, ON N2L 3C5 Canada
article
info
Article history: Received 13 July 2019 Accepted 9 September 2019 Available online 18 September 2019 Keywords: Spatial heterogeneity Global asymptotic stability Basic reproduction number Threshold dynamics
abstract In this paper, we formulate and investigate a vector-borne disease model with vector-bias in a bounded spatial domain. The main result is a threshold dynamics characterized in terms of the basic reproduction number ℜ0 or equivalently in terms of the principal eigenvalue of the linearized system. Roughly speaking, if ℜ0 < 1 then the semi-trivial equilibrium is globally asymptotically stable and the disease dies out; if ℜ0 > 1 then the disease persists. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Zika virus disease is caused by the Zika virus (ZIKV), a mosquito-borne flavivirus. ZIKV is primarily transmitted by the bite of an infected mosquito from the Aedes genus, mainly Aedes aegypti, in tropical and subtropical regions. The recent outbreak in Brazil (March 2015) has attracted the attention of many researchers. Reaction–diffusion models have been proposed to effectively understand the spread of ZIKV (see, for example, [1–8] and references therein). In particular, Webb and his collaborators [4,8] studied the outbreak of Zika in Rio De Janerio. Assume that Ω ⊂ Rn is a bounded domain with a smooth boundary ∂Ω . Let Hu (x, t), Hi (x, t), Vu (x, t), and Vi (x, t) be the densities of susceptible hosts, infected hosts, uninfected vectors, and infected vectors at location x and time t, respectively. They assumed Hu to be constant as for Zika in Rio De Janerio the number of infected is fairly small in comparison with the number of the total population (also see [9]). This leads a deterministic reaction–diffusion model involving Hi , Vu , and Vi . They established a threshold dynamics determined by the basic reproduction number. Their proof “combines arguments from monotone dynamical system theory, persistence theory, and the theory of asymptotically autonomous semiflows”. Moreover, mosquitos may choose infectious human hosts more frequently than healthy ones to bite. This difference is called vector-bias and has been evidenced by the disease malaria. On the one hand, it is found ∗ Corresponding author. E-mail addresses: [email protected] (J. Wang), [email protected] (Y. Chen).
https://doi.org/10.1016/j.aml.2019.106052 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
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J. Wang and Y. Chen / Applied Mathematics Letters 100 (2020) 106052
through experiment on malaria infection that infectious humans are more attractive mosquitos [10]. On the other hand, Daniel and Kingsolver used a mechanistic model to argue that mosquitos may maximize their rate of protein intake during feeding by choosing malaria infected hosts [11]. Zhao and his collaborators [7,12] have studied the impact of vector-bias on the epidemiology of malaria. To describe the phenomenon of vectorbias in space, they introduced p and l to denote the probabilities that a mosquito randomly arrives at a human and picks the human if he is infectious and susceptible, respectively. Borrowing the idea in [7,12], the numbers of newly infected hosts and newly infected mosquitos per unit time at position x and time pHi u (x) t are respectively cβ1 (x) pHlH Vi and bβ2 (x) pH +lH V , where β1 (x) is the biting rate of infectious u (x) u i +lHu (x) i mosquitos, c is the transmission probability from an infected mosquito to a susceptible host, β2 (x) is the biting rate of susceptible mosquitos, and b is the transmission probability from infectious host to susceptible mosquitos. The aim of this paper is to study the combined effect of spatial heterogeneity and vector-bias on epidemic dynamics. Precisely, based on the recent works in [4,7,8,12], we propose and investigate the following vector-biased vector-borne model with spatial structure, ⎧ cβ1 (x)lHu (x)Vi ∂Hi ⎪ ⎪ x ∈ Ω , t > 0, ⎪ ∂t − ∇ · d1 (x)∇Hi = −λ(x)Hi + pHi +lHu (x) , ⎪ ⎪ ⎪ ⎪ ⎪ ∂Vu − ∇ · d2 (x)∇Vu = α(x)(Vu + Vi ) − µ(x)(Vu + Vi )Vu − bβ2 (x)pHi Vu , x ∈ Ω , t > 0, ⎪ pHi +lHu (x) ⎨ ∂t bβ2 (x)pHi Vu ∂Vi (1.1) x ∈ Ω , t > 0, ⎪ ∂t − ∇ · d2 (x)∇Vi = pHi +lHu (x) − µ(x)(Vu + Vi )Vi , ⎪ ⎪ ⎪ ⎪ ∂Hi ∂Vi ∂Vu ⎪ x ∈ ∂Ω , t > 0, ⎪ ∂ν = ∂ν = ∂ν = 0, ⎪ ⎪ ⎩ 3 (Hi (x, 0), Vu (x, 0), Vi (x, 0)) = (Hi0 (x), Vu0 (x), Vi0 (x)) ∈ C(Ω ; R+ ), where d1 (x), d2 (x) ∈ C 1+α (Ω ) are respectively the diffusion rates of the infected hosts and vectors, which are assumed to be strictly positive, the functions Hu (x), λ(x), α(x), β1 (x), β2 (x), and µ(x) are strictly ∂ positive and belong to C(Ω ). Here ∂ν denotes the normal derivative with respect to ∂Ω . The initial functions Hi0 (·), Vu0 (·), Vi0 (·) are nonnegative continuous functions. The existence, uniqueness, and positivity of global classical solutions of (1.1) can be proved by using similar arguments as those in [8]. The remaining part of this paper is organized as follows. We first study the stability of the diseasefree equilibria. Then we derive the basic reproduction number. Followed is the main result of this paper, a threshold dynamics characterized by the basic reproduction number. 2. Stability of the disease-free equilibria Let V := Vu + Vi , be the total density of vectors. Then V (x, t) satisfies the diffusive logistic equation ⎧ ⎪ ⎪ ∂V 2 ⎪ ⎪ ⎨ ∂t − ∇ · d2 (x)∇V = α(x)V − µ(x)V , x ∈ Ω , t > 0, ∂V (2.1) x ∈ ∂Ω , t > 0, ⎪ ∂ν = 0, ⎪ ⎪ ⎪ ⎩ V (x, 0) = V0 ∈ C(Ω ; R+ ). The following result follows from [13]. Lemma 2.1. Assume that V0 ∈ C(Ω ; R+ ). Then system (2.1) admits a unique global classical solution V (x, t). Furthermore, for all (x, t) ∈ Ω × (0, ∞), we have V (x, t) > 0 and limt→∞ ∥V (·, t) − V (x)∥∞ = 0, where V (x) is the unique positive solution of the problem ⎧ ⎨ −∇ · d2 (x)∇V = α(x)V − µ(x)V 2 , x ∈ Ω , (2.2) ⎩ ∂V = 0, x ∈ ∂Ω . ∂ν
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System (1.1) always has the two disease-free equilibria: the trivial equilibrium E0 = (0, 0, 0) and the unique semi-trivial equilibrium E1 = (0, V , 0). Theorem 2.1. The trivial equilibrium E0 = (0, 0, 0) is unstable. Proof . Linearizing system (1.1) at E0 , we arrive at the following eigenvalue problem, ⎧ ⎪ ⎪kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, x ∈ Ω , ⎨ kϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ), x ∈ Ω, kψ − ∇ · d (x)∇ψ = 0, x ∈ Ω, ⎪ 2 ⎪ ⎩ ∂φ ∂ϕ ∂ψ = = = 0, x ∈ ∂Ω . ∂ν ∂ν ∂ν
(2.3)
Denote by ˜ k(d, f ) the principal eigenvalue associated with a positive eigenvector of the following eigenvalue problem { kϕ − ∇ · d(x)∇ϕ = f ϕ, x ∈ Ω , (2.4) ∂ϕ x ∈ ∂Ω , ∂ν = 0, where d(x) ∈ C 1 (Ω ) is strictly positive on Ω and f ∈ C(Ω ). It is easy to check that ˜ k(d, f ) is monotone in the ˜ ˜ ˜ sense that if f1 ≥ f2 then k(d, f1 ) > k(d, f2 ). Let ϕ be a positive eigenvector corresponding to the eigenvalue ˜ ˜ 0). k(d2 , α) of (2.4). Then obviously ˜ k(d2 , α) is an eigenvalue of (2.3) with a corresponding eigenvector (0, ϕ, As ˜ k(d2 , 0) > 0, it follows that E0 is linearly unstable. Now, linearizing system (1.1) at E1 and letting (Hi (x, t), Vu (x, t), Vi (x, t)) = (φ(x)ekt , ϕ(x)ekt , ψ(x)ekt ) yield ⎧ x ∈ Ω, ⎪kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, ⎪ ⎪ ⎨kϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ) − 2µ(x)V ϕ − µ(x)V ψ − bβ2 (x) pV φ, x ∈ Ω , lHu (x) (2.5) pV ⎪ kψ − ∇ · d (x)∇ψ = bβ (x) φ − µ(x)V ψ, x ∈ Ω, ⎪ 2 2 lHu (x) ⎪ ⎩ ∂φ ∂ϕ ∂ψ x ∈ ∂Ω . ∂ν = ∂ν = ∂ν = 0, It is necessary to consider the following problem ⎧ x ∈ Ω, ⎪ ⎨kφ − ∇ · d1 (x)∇φ = −λ(x)φ + cβ1 (x)ψ, pV kψ − ∇ · d2 (x)∇ψ = bβ2 (x) lHu (x) φ − µ(x)V ψ, x ∈ Ω , ⎪ ⎩ ∂φ ∂ψ x ∈ ∂Ω . ∂ν = ∂ν = 0,
(2.6)
Since (2.6) is cooperative, it follows from the standard Krein–Rutman theorem that the eigenvalue problem (2.6) has a principal eigenvalue k0 (d1 , d2 , V , V ) associated with a positive eigenvector (φ0 , ψ0 ). Theorem 2.2. The semi-trivial equilibrium E1 = (0, V , 0) is locally asymptotically stable if k0 (d1 , d2 , V , V ) < 0 and unstable if k0 (d1 , d2 , V , V ) > 0. Proof . Recall that V is the unique positive solution of the elliptic problem (2.2). It follows that ˜ k(d2 , α − µV ) = 0. From the monotonicity of ˜ k(d, f ), we have ˜ k(d2 , α − 2µV ) < 0. Denote by k the eigenvalue of (2.5). It follows that k is also an eigenvalue of either (2.6) or the following eigenvalue problem: { kϕ − ∇ · d2 (x)∇ϕ = α(x)ϕ − 2µ(x)V ϕ, x ∈ Ω , ∂ϕ ∂ν
= 0,
x ∈ ∂Ω .
Since k0 < 0 and ˜ k(d2 , α − 2µV ) < 0, we have k < 0. By the arbitrariness of k, E1 is linearly stable. While if k0 > 0, denote by (φ0 , ψ0 ) the positive eigenvector corresponding to k0 . It then follows from
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˜ k(d2 , α − 2µV ) < 0 and the Fredholm alternative that the following problem admits a unique solution ϕ0 , { k0 ϕ − ∇ · d2 (x)∇ϕ = α(x)(ϕ + ψ0 ) − 2µ(x)V ϕ − µ(x)V ψ0 − bβ2 (x) lHpV φ , x ∈ Ω, u (x) 0 ∂ϕ x ∈ ∂Ω . ∂ν = 0, Therefore, we have arrived at the conclusion that (2.5) has an eigenvector (φ0 , ϕ0 , ψ0 ) corresponding to an eigenvalue k0 > 0. Consequently, E1 is linearly unstable. 3. The basic reproduction number According to [14,15], we derive the basic reproduction number of (1.1). Let B : [C(Ω )]2 → [C(Ω )]2 be the operator ( )( ) ∇ · d1 (x)∇ − λ(x) cβ1 (x) φ B(φ, ψ) = , ψ 0 ∇ · d2 (x)∇ − µ(x)V } { ∂ψ where Dom(B) = (φ, ψ) ∈ C(Ω ; R2 ) : ∂φ ∂ν = ∂ν = 0, x ∈ ∂Ω . Define ( ) 0 0 C = . bβ2 (x) lHpV 0 u (x) Let A = B + C , which is regarded as a positive perturbation of B. It is easy to check from [14] that A and B are resolvent positive. On the other hand, the spectral bound of B is negative, i.e., s(B) < 0. Then we can apply the results in [14, Theorem 3.5] to conclude that k0 = s(A ) has the same sign with r(−C B −1 )−1, where r(−C B −1 ) is the spectral radius of −C B −1 . The basic reproduction number ℜ0 for (1.1) is defined by ℜ0 = r(−C B −1 ). The following result restates the above discussion. Lemma 3.1.
ℜ0 − 1 and k0 have the same sign.
4. A threshold dynamics The main result of this paper is that ℜ0 determines the disease extinction and persistence. We first consider the global dynamics of the model when ℜ0 < 1. Theorem 4.1. If ℜ0 < 1, then E1 is globally asymptotically stable in the sense that limt→∞ ∥ (Hi (x, t), Vu (x, t), Vi (x, t)) − E1 ∥∞ = 0 for any nonnegative initial condition (Hi0 , Vu0 , Vi0 ) ∈ C(Ω ; R3+ ) with Vu0 + Vi0 ̸= 0. Proof . By Theorem 2.2, E1 is locally asymptotically stable and k0 (d1 , d2 , V , V ) < 0. Choose ε > 0 small enough such that k1 := k0 (d1 , d2 , V + ε, V − ε) < 0 with a corresponding positive eigenvector (φε , ψε ). By Lemma 2.1, we know that Vu + Vi → V (x) uniformly on Ω as t → ∞. Hence there exists t0 > 0 such that V (x) − ε < Vu + Vi < V (x) + ε for x ∈ Ω and t > t0 . It then follows that the solution (Hi , Vi ) of (1.1) is a lower solution of the following problem ⎧ ⎪ ⎪ ∂ Hˇ i − ∇ · d1 (x)∇H ˇ i = −λ(x)H ˇ i + cβ1 (x)Vˇi , x ∈ Ω , t > t0 , ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂ Vˇi − ∇ · d (x)∇Vˇ = bβ (x) p(V +ε) H ˇ ˇ 2 i 2 ∂t lHu (x) i − µ(x)(V − ε)Vi , x ∈ Ω , t > t0 , (4.1) ⎪ ˇi ⎪ ∂H ∂ Vˇi ⎪ x ∈ ∂Ω , ⎪ ⎪ ∂ν = ∂ν = 0, ⎪ ⎪ ⎩ˇ Hi (x, t0 ) = M φε (x), Vˇi (x, t0 ) = M ψε (x), x ∈ Ω,
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where M is a large enough positive constant such that (Hi (x, t0 ), Vi (x, t0 )) ≤ (M φε (x), M ψε (x)). By the ˇ i (x, t), Vˇi (x, t)) comparison principle for cooperative systems, we can conclude that (Hi (x, t), Vi (x, t)) ≤ (H ˇ i (x, t), Vˇi (x, t)) = for all x ∈ Ω and t ≥ t0 . Since (4.1) is a linear system, it admits a unique solution (H k1 t k1 t ˇ ˇ (M φε (x)e , M ψε (x)e ). It follows from k1 < 0 that (Hi (x, t), Vi (x, t)) → (0, 0) uniformly for x ∈ Ω as t → ∞ and hence (Hi (x, t), Vi (x, t)) → (0, 0) uniformly for x ∈ Ω as t → ∞. Moreover, by Vu0 + Vi0 ̸= 0 and Lemma 2.1, we know that Vu (x, t) + Vi (x, t) → V (x) uniformly on Ω , which implies that Vu (x, t) → V (x) uniformly for x ∈ Ω as t → ∞. This completes the proof. The result below can be easily derived with a minor modification of the arguments in the proof of [4, Lemma 3.10]. Lemma 4.1. There exists Mmax > 0, independent of initial data, such that any solution (Hi , Vu , Vi ) of (1.1) satisfies 0 ≤ Hi (x, t), Vu (x, t), Vi (x, t) ≤ Mmax , x ∈ Ω , t ≥ t0 , where t0 is dependent on the initial data. Before proving the uniform persistence, we first consider weak persistence. To this end, we set X := C(Ω , R3 ), which is a Banach space equipped with the supremum norm ∥ · ∥∞ . Then (X, X+ ) is a strongly ordered space, where X+ = C(Ω , R3+ ). Let X0 := {ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ : ϕ1 + ϕ3 ̸= 0 and ϕ2 + ϕ3 ̸= 0} , ∂X0 := {ϕ = (ϕ1 , ϕ2 , ϕ3 ) ∈ X+ : ϕ1 + ϕ3 = 0 or ϕ2 + ϕ3 = 0} . By using these notations, we have X+ = X0 ∪ ∂X0 , X0 is relatively open with X0 = X+ , and ∂X0 is relatively closed in X+ . Denote by Φ(t) the solution semiflow generated by (1.1). From Lemma 4.1, Φ(t) is point dissipative. Furthermore, Φ(t) is compact for any t > 0. Denote by ω(ϕ) the omega limit set of the orbit O+ (ϕ) := {Φ(t)(ϕ) : t ≥ 0}. Lemma 4.2. If ℜ0 > 1, then E0 and E1 are uniform weak repellers for X0 in the sense that there exists a sufficiently small constant ε0 > 0 such that for any ϕ ∈ X0 , lim supt→∞ ∥Φ(t)ϕ − E0 ∥∞ ≥ ε0 and lim supt→∞ ∥Φ(t)ϕ − E1 ∥∞ ≥ ε0 . Proof . On the one hand, let ε1 < 21 minx∈Ω V (x). Then lim supt→∞ ∥Φ(t)ϕ − E0 ∥ ≥ ε1 for all ϕ ∈ X0 . Otherwise, there exists ϕ ∈ X0 such that lim supt→∞ ∥Φ(t)ϕ − E0 ∥ < ε1 . Then there exists t1 > 0 such that Vu (x, t) < εˆ1 and Vi (x, t) < εˆ1 for all x ∈ Ω and t ≥ t1 for some εˆ1 < ε1 . This will contradict with the fact that Vu (x, t) + Vi (x, t) → V (x) uniformly on Ω as t → ∞. On the other hand, it follows from Lemma 3.1 that ℜ0 > 1 implies that k0 (d1 , d2 , V , V ) > 0. Then there exists a sufficiently small ε2 > 0 such that k2 := k0 (d1 , d2 , V − ε2 , V + ε2 ) > 0 and it has a corresponding positive eigenvector (φε2 , ψε2 ). We claim that lim supt→∞ ∥Φ(t)ϕ − E1 ∥ ≥ ε2 for ϕ ∈ X0 . If this is not true, then there exists ϕ ∈ X0 such that lim supt→∞ ∥Φ(t)ϕ − E1 ∥∞ < ε2 . It follows that there exists a t2 > 0 such that Hi (x, t) < ε2 , V (x) − ε2 < Vu (x, t) < V (x) + ε2 , and Vi (x, t) < ε2 for all x ∈ Ω and t ≥ t2 . By (1.1), we get { ∂H
i
∂t ∂Vi ∂t
cβ1 (x)lHu (x) pε2 +lHu (x) Vi , bβ2 (x)p(V (x)−ε2 ) Hi − µ(x)(V pε2 +lHu (x)
− ∇ · d1 (x)∇Hi ≥ −λ(x)Hi + − ∇ · d2 (x)∇Vi ≥
x ∈ Ω , t > t2 , (x) + ε2 )Vi ,
x ∈ Ω , t > t2 .
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Since Hi (x, t) > 0 and Vi (x, t) > 0 for all x ∈ Ω and t > 0, there exists α > 0 such that (Hi (x, t0 ), Vi (x, t0 )) ≥ α (φε2 , ψε2 ). Recall that αek2 (t−t2 ) (φε2 , ψε2 ) is the solution of ⎧ ⎨ ∂H i − ∇ · d (x)∇H = −λ(x)H + cβ1 (x)lHu (x) V , x ∈ Ω , t > t2 , 1 i i ∂t pε2 +lHu (x) i bβ (x)p(V (x)−ε ) ∂V 2 ⎩ i − ∇ · d2 (x)∇V i = 2 Hi − µ(x)(V (x) + ε2 )V i , x ∈ Ω , t > t2 . ∂t
pε2 +lHu (x)
By the comparison principle, we obtain (Hi (x, t), Vi (x, t)) ≥ αek2 (t−t2 ) (φε0 , ψε0 ) for t ≥ t0 . Thus Hi (x, t) → ∞ and Vi (x, t) → ∞ as k2 > 0, which leads to a contradiction with the boundedness of (Hi (x, t), Vi (x, t)). This proves the claim. Letting ε0 = min{ε1 , ε2 } immediately completes the proof. Now we are ready to state and prove the main result of uniform persistence to conclude the paper, which also implies the existence of at least one positive equilibrium. Theorem 4.2. Suppose ℜ0 > 1. Then (1.1) is uniformly persistent in the sense that there exists ς > 0 such that, for any initial condition (Hi0 , Vu0 , Vi0 ) ∈ X0 , lim
inf ∥(Hi (·, t), Vu (·, t), Vi (·, t)) − ω∥∞ ≥ ς.
t→∞ ω∈∂X0
Moreover, (1.1) has at least one positive equilibrium EE = (Hi∗ (x), Vu∗ (x), Vi∗ (x)) ∈ X0 . Proof . Firstly, we prove that X0 is positively invariant with respect to Φ(t). Let w0 = (Hi0 , Vu0 , Vi0 ) ∈ X0 . Then Hi0 + Vi0 ̸= 0 and Vu0 + Vi0 ̸= 0. If Vi0 ̸= 0, it then follows from the maximum principle and ∂Vi ∂t − ∇ · d2 (x)∇Vi ≥ −µ(x)(Vu + Vi )Vi that Vi (x, t) > 0 for x ∈ Ω and t > 0. Now assume that Vi0 = 0. Then Hi0 ̸= 0 and Vu0 (x) ̸= 0. From the first equation of (1.1), we get ∂Hi − ∇ · d1 (x)∇Hi ≥ −λ(x)Hi . ∂t This combined with Hi0 ̸= 0 and the maximum principle implies that Hi (x, t) > 0 for x ∈ Ω and t > 0. Then it follows from the second equation of (1.1) that ∂Vu − ∇ · d2 (x)∇Vu ≥ [α(x) − µ(x)(Vu + Vi ) − bβ2 (x)]Vu . ∂t Again Vu (x, t) > 0 for x ∈ Ω and t > 0 from Vu0 ̸= 0 and the maximum principle. In summary, we have shown that Φ(t)w0 ∈ X0 for w0 ∈ X0 and t ≥ 0, that is, ∂X0 is positively invariant with respect to Φ(t). Next, we show that ∂X0 is invariant with respect to Φ(t). Suppose w0 = (Hi0 , Vu0 , Vi0 ) ∈ ∂X0 , namely, Hi0 + Vi0 = 0 or Vu0 + Vi0 = 0. If Hi0 + Vi0 = 0 and Vu0 ̸= 0, then it is easy to see from the first and third equations of (1.1) that Hi (·, t) = Vi (·, t) = 0 for t ≥ 0. Then Vu is governed by (2.1) and by Lemma 2.1 we get Vu (x, t) > 0 for x ∈ Ω and t > 0 and Vu (x, t) → V (x) uniformly on Ω as t → ∞. Thus ω (w0 ) = {E1 }. Now if Vu0 + Vi0 = 0, then by the second and third equations of (1.1), we can get Vu (·, t) = Vi (·, t) = 0 for t ≥ 0. Thus the first equation of (1.1) becomes ∂Hi − ∇ · d1 (x)∇Hi = −λ(x)Hi , ∂t which implies that Hi (x, t) → 0 uniformly in Ω as t → ∞. Therefore, ω(w0 ) = {E0 }. To summarize, we have shown that ∂X0 is positively invariant with respect to Φ(t). Denote Φ∂ (t) := Φ(t)|∂X0 , the restriction of Φ(t) on ∂X0 . From the above, we see that Φ∂ (t) admits a compact global attractor, which is {E0 , E1 }.
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Thirdly, from Lemma 4.2, we see that E0 and E1 are isolated in X0 with X0 ∩ W s (E0 ) = ∅ and X0 ∩ W s (E1 ) = ∅, where W s (E0 ) and W s (E1 ) are the stable manifolds of E0 and E1 , respectively. Since there is no cycle in ∂X0 from E0 to E1 , we can use [16, Theorem 3] to conclude the existence of a constant ς > 0 such that { } min ϕ∈ω(w0 )
min inf ϕi (x)
i=1,2,3 x∈Ω
>ς
for all u0 ∈ X0 .
This implies that lim inf t→∞ Hi (x, t; u0 ), Vu (x, t; u0 ), Vi (x, t; u0 ) > ς and thus the uniform persistence stated in the theorem holds. Finally, from [17, Theorem 3.7], Φ(t) : X0 → X0 admits a global attractor A0 that attracts every point in X+ . Then Φ(t) has an equilibrium EE = (Hi∗ , Vu∗ , Vi∗ ) ∈ X0 by [17, Theorem 4.7]. This completes the proof. Acknowledgments The research of Wang was supported by the National Natural Science Foundation of China (No. 11871179), Natural Science Foundation of Heilongjiang Province (No. LC2018002), and Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems. The research of Chen was supported by NSERC of Canada. References [1] Y. Cai, Z. Ding, B. Yang, Z. Peng, W. Wang, Transmission dynamics of zika virus with spatial structure–a case study in rio de janeiro, Brazil, Physica A 514 (2019) 729–740. [2] Y. Cai, K. Wang, W. Wang, Global transmission dynamics of a zika virus model, Appl. Math. Lett. 92 (2019) 190–195. [3] D. Gao, Y. Lou, D. He, et al., Prevention and control of zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis, Sci. Rep. 6 (2016) 28070. [4] P. Magal, G. Webb, Y. Wu, On a vector-host epidemic model with spatial structure, Nonlinearity 31 (2018) 5589–5614. [5] B. Tang, Y. Xiao, J. Wu, Implication of vaccination against dengue for zika outbreak, Sci. Rep. 6 (2016) 35623. [6] L. Wang, H. Zhao, Dynamics analysis of a zika-dengue co-infection model with dengue vaccine and antibody-dependent enhancement, Physica A 522 (2019) 248–273. [7] X. Wang, X.-Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math. 77 (2017) 181–201. [8] W.E. Fitzgibbon, J.J. Morgan, G.F. Webb, An outbreak vector-host epidemic model with spatial structure: the 2015-2016 zika outbreak in rio de janeiro, Theor. Biol. Med. Model. 14 (2017) 7. [9] D.A.M. Villela, L.S. Bastos, L.M. de Carvalho, et al., Zika in rio de janeiro: assessment of basic reproduction number and comparison with dengue outbreaks, Epidemiol. Inf. 145 (2016) 1649–1657. [10] R. Lacroix, W.R. Mukabana, L.C. Gouagna, J.C. Koella, Malaria infection increases attractiveness of humans to mosquitoes, PLoS Biol. 3 (2005) e298. [11] T.L. Daniel, J.G. Kingsolver, The feeding strategy and the mechanics of blood sucking in insects, J. Theoret. Biol. 105 (1983) 661–677. [12] Z. Bai, R. Peng, X.-Q. Zhao, A reaction–diffusion malaria model with seasonality and incubation period, J. Math. Biol. 77 (2018) 201–228. [13] R.S. Cantrell, C. Cosner, Spatial Ecology Via Reaction–Diffusion Equations, John Wiley & Sons, Ltd, Chichester, 2003. [14] H.R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math. 70 (2009) 188–211. [15] W. Wang, X.-Q. Zhao, Basic reproduction numbers for reaction–diffusion epidemic models, SIAM J. Appl. Dyn. Syst. 11 (2012) 1652–1673. [16] H.L. Smith, X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. 47 (2001) 6169–6179. [17] P. Magal, X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal. 37 (2005) 251–275.