GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 2, Number 1, Spring 212 GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE TRANSMISSION MODEL Dedicated to Herb Freedman on the occasion of his 7th birthday ZHEN WANG AND XIAO-QIANG ZHAO ABSTRACT. In this paper, we present a time-delayed dengue transmission model. We first introduce the basic reproduction number for this model and then show that the disease persists when R > 1. It is also shown that the disease will die out if R < 1, provided that the invasion intensity is not strong. We further establish a set of sufficient conditions for the global attractivity of the endemic equilibrium by the method of fluctuations. Numerical simulations are performed to illustrate our analytic results. 1 Introduction Dengue fever is the most common viral disease spread to humans by mosquitos, and has become an international public health concern. Dengue is caused by a group of four antigenically distinct flavivirus serotypes: DEN-1, DEN-2, DEN-3, and DEN-4; and is primarily transmitted by Aedes mosquitos, particularly A. aeqypti mosquitos. Dengue is found in tropical and subtropical regions around the world, predominately in urban and peri-urban areas. The incidence of dengue has grown dramatically around the world in recent decades. It is endemic in more than 11 countries in Africa, the Americas, the Eastern Mediterranean, South-east Asia and the Western Pacific. It infects 5 to 1 million people worldwide a year, leading to 5 million hospitalizations, and approximately 12,5 to 25, deaths a year [3, 4, 11, 16]. The human is the main amplifying host of the virus, although studies have shown that in some part of the world monkeys may become Research supported in part by the NSERC of Canada and the MITACS of Canada. 21 MSC: 346, 37L15, 92D3. eywords: Dengue transmission, vector-borne disease, basic reproduction number, uniform persistence, global attractivity. Copyright c Applied Mathematics Institute, University of Alberta. 89

2 9 Z. WANG AND X-Q. ZHAO infected and perhaps serve as a source of virus for uninfected mosquitos [3]. Human may get infected by a bite from the infected mosquitos, and A. aeqypti mosquitos may acquire the virus when they feed on an infectious individual. Much have been done in terms of modeling and analysis of disease transmission with structured vector population. Wang and Zhao [15] proposed a nonlocal and time-delayed reaction-diffusion model of dengue fever, and established a threshold dynamics in terms of the basic reproduction number R. Lou and Zhao [9] presented a malaria transmission model with structured vector population, and also established a threshold type result, which states that when R < 1 and the disease invasion is not strong, the disease will die out; when R > 1, the disease will persist. In this paper, we incorporate the stage structure of mosquitos (see, e.g., [9]), since the development stages of mosquitos have a profound impact on the transmission of disease: first, the immature mosquitos do not fly and bite human, so they do not participate in the infection cycle; second, mature mosquitos are quite different from immature mosquitos from biological and epidemiological perspectives. In view of realistic consideration, we take these different stages into account. We also include the time delay to describe the incubation periods of mosquitos and the human populations, which is important because there are incubation realistically and the time period is not small. In fact, from the expression of R in Section 3, we can see those delays reduce the values of R. Therefore, the neglect of the delays overestimated the infection risk. The purpose of this paper is to study the global dynamics of a timedelayed dengue transmission model. In Section 2, we present the model system and prove its wellposedness. In Section 3, we first introduce the basic reproduction number R, and then show that the disease is uniformly persistent when R > 1 by appealing to the theory developed in [2, 13]. Under certain conditions, we also obtain the nonlocal stability of the disease-free equilibrium when R < 1. In Section 4, we obtain a set of sufficient conditions for the endemic equilibrium to be globally attractive by the method of fluctuations. In Section 5, we perform numerical simulations to illustrate our analytic results. 2 The model In this section, following the ideas in [15], we present an age-structure dengue model with time delay for the cross infection between mosquitos and human individuals. We divide the mosquito population into two subclasses: aquatic population and winged pop-

3 A TIME-DELAYED DENGUE TRANSMISSION MODEL 91 ulation. Winged female A. aegypti mosquitoes lay eggs in unattended water. Eggs may develop into larvae from two days up to one week. The larvae spend up to three days to pass through four instars to enter the pupal stage. The pupa develops into an adult after about two days. The immature mosquitos live in aquatic habitats and mature mosquitos disperse to search for food. Let A denote the density of aquatic population of mosquitos, W be the density of winged population of mosquitos, and τ A be the length of immature stage of mosquitos. Following the model to formulate a stage-structured population in Aiello and Freedman [1], we suppose the dynamics of mosquitos is described by da(t) dw (t) = B((W (t))w (t) aa(t) e aτa B(W (t τ A ))W (t τ A ), = e aτa B(W (t τ A ))W (t τ A ) µ w W (t), where B is the per capita birth rate of adult mosquitos, a is the per capita death rate of aquatic mosquitos, and µ w is the death rate of adult mosquitos. Following [15], we assume that the function B(W )W is the logistic growth rate: [ rw 1 W ] if W rw [1 W/] + = if W >, For the dynamics of human population, we assume that the density N of the human population obeys dn = H µ hn, where H is a constant recruitment rate and µ h is the death rate. To consider dengue transmission between mosquitos and human individuals, we let W 1, W e and W 2 denote the density of susceptible, exposed, and infectious mosquitos of winged population, respectively; and divide the human population into four compartments: susceptible (S), exposed (E), infectious (I) and recovered (R). Let τ w be the incubation period of dengue virus within mosquitos and τ h be the incubation period of dengue virus within hosts. Following Chowell et al. [5], we suppose that the infection rates of susceptible mosquitos and susceptible human individuals are described by bp I N W 1, bq S N W 2,

4 92 Z. WANG AND X-Q. ZHAO respectively, where b is the mean rate of mosquito bites per mosquito, p is the probability that a bite by a susceptible mosquito to an infectious host will cause infections, q is the probability that a bite by an infectious mosquito to a susceptible host will cause infection to the host, and N = S+E+I+R is the total density of human population. Since an infectious mosquito may have lower fecundity than a susceptible mosquito, we let σ [, 1] denote the relative fecundity of an infected mosquito to a susceptible mosquito. Specifically, the infectious mosquito has the same reproduction rate as a susceptible mosquito if σ = 1, and have lower reproduction rate if σ < 1. Then we have the following model: (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) da(t) dw 1 (t) W e (t) = dw 2 (t) ds(t) E(t) = di(t) dr(t) [ = r 1 W (t) ] W σ (t) aa(t) + [ re aτa 1 W (t τ A) = re aτa [ 1 W (t τ A) t ] ] W σ (t τ A ), + W σ (t τ A ) + µ w W 1 (t) β w I(t) N(t) W 1(t), t τ w e µw(t s) β w I(s) N(s) W 1(s) ds, = β w e µwτw I(t τ w) N(t τ w ) W 1(t τ w ) (µ w + ε w )W 2 (t), = H µ h S(t) β h W 2 (t) S(t) N(t), t t τ h e µ h(t s) β h S(s) N(s) W 2(s) ds, = β h e µ hτ h W 2 (t τ h ) S(t τ h) N(t τ h ) (µ h + ε h + γ)i(t), = γi(t) µ h R(t), where W (t) = W 1 (t) + W e (t) + W 2 (t), W σ (t) = W 1 (t) + W e (t) + σw 2 (t),

5 A TIME-DELAYED DENGUE TRANSMISSION MODEL 93 β w = bp, β h = bq, γ is the recovery rate of infected human individuals, ε w and ε h are the infection-induced death rates of infected mosquitos and human individuals, respectively. Note that the equation for aquatic population of mosquitos is decoupled from the other equations. It then suffices to consider system (2.1) (2.7) which is an integro-differential equation system. Differentiating (2.2) and (2.5) gives (2.8) (2.9) dw e (t) de(t) = β w I(t) N(t) W 1(t) µ w W e (t) β w e µwτw I(t τ w) N(t τ w ) W 1(t τ w ), = β h W 2 (t) S(t) N(t) µ he(t) β h e µ hτ h W 2 (t τ h ) S(t τ h) N(t τ h ). The system consisting of (2.1), (2.8), (2.3), (2.4), (2.9), (2.6) and (2.7) is an ordinary differential system with time delays. For simplicity, we will refer to this system as the model system in the rest of this paper. Let τ = max{τ A, τ w, τ h }, and define C := C([ τ, ], R 7 ). For φ = (φ 1, φ 2,, φ 7 ) C, define φ = 7 i=1 φ i, where φ i = max φ i(θ). θ [ τ,] Then C is a Banach space. Define C + = {φ C : φ i (θ), 1 i 7, θ [ τ, ]}. Then C + is a normal cone of C with nonempty interior in C. For a continuous function u : [ τ, σ φ ) R 7 with σ φ >, we define u t C for each t by u t (θ) = u(t + θ), θ [ τ, ]. In view of (2.2) and (2.5), we choose the initial data for the model system in X δ, which is defined as X δ = { φ C + : φ 2 () = φ 5 () = 7 φ i (s) δ, s [ τ, ], i=4 e µws φ 6 (s)φ 1 (s) β w 7 τ w i=4 φ i(s) ds, } e µ hs φ 4 (s)φ 3 (s) β h 7 τ h i=4 φ i(s) ds

6 94 Z. WANG AND X-Q. ZHAO for small δ (, H/(ε h + µ h ) ). The following result shows that the model system is wellposed in X δ, and the solution semiflow admits a global attractor on X δ. Theorem 2.1. For any φ X δ, the model system has a unique nonnegative solution u(t, φ) satisfying u = φ. Furthermore, the solution semiflow Φ(t) = u t ( ) : X δ X δ has a compact global attractor. Proof. Given φ X δ, define G(φ) := (G 1 (φ), G 2 (φ), G 3 (φ), G 4 (φ), G 5 (φ), G 6 (φ), G 7 (φ)), where 3 G 1 (φ) = re [1 aτa i=1 φ ] i( τ A ) (φ 1 ( τ A ) + φ 2 ( τ A ) + + σφ 3 ( τ A )) µ w φ 1 () β w φ 6 () 7 i=4 φ i() φ 1(), G 2 (φ) = β w φ 6 () 7 i=4 φ i() φ 1() µ w φ 2 () β w e µwτw φ 6 ( τ w ) 7 i=4 φ i( τ w ) φ 1( τ w ), G 3 (φ) = β w e µwτw φ 6 ( τ w ) 7 i=4 φ i( τ w ) φ 1( τ w ) (µ w + ε w )φ 3 (), G 4 (φ) = H µ h φ 4 () β h φ 4 () 7 i=4 φ i() φ 3(), G 5 (φ) = β h φ 4 () 7 i=4 φ i() φ 3() µ h φ 5 () β h e µ hτ h φ 4 ( τ h ) 7 i=4 φ i( τ h ) φ 3( τ h ), G 6 (φ) = β h e µ hτ h φ 4 ( τ h ) 7 i=4 φ i( τ h ) φ 3( τ h ) (µ h + ε h + γ)φ 6 (), G 7 (φ) = γφ 6 () µ h φ 7 (). Note that X δ is closed in C, and for all φ X δ, G(φ) is continuous and Lipschitz in φ in each compact set in R X δ. By [6, Theorem 2.3],

7 A TIME-DELAYED DENGUE TRANSMISSION MODEL 95 it then follows that for any φ X δ, there is an unique solution of the model system through (, φ) on its maximal interval [, σ φ ) of existence. Since G i (φ) whenever φ X δ with φ i () =, [12, Theorem 5.2.1] implies that the solutions of the model system are nonnegative for all t [, σ φ ). Note that the total host population satisfies dn(t) = H µ h N(t) ε h I(t) H (µ h + ε h )N(t). For the system dy/ = H (µ h + ε h ) y(t), the equilibrium H/(µ h + ε h ) is globally asymptotically stable. For any < δ < H µ h +ε h, dy/ y=δ = H (µ h + ε h ) δ >. So if y() δ, then y(t) δ, for any t. From (2.8), we get ( e µwt (W e(t) + µ w W e (t)) = e µwt I(t) β w N(t) W 1(t) β w e I(t τ ) w) µwτw N(t τ w ) W 1(t τ w ) By integrating on both sides, we obtain e µwt W e (t) W e () = = = t t t e µws I(s) β w N(s) W 1(s) ds t e µw(s τw) β w I(s τ w ) N(s τ w ) W 1(s τ w ) ds e µws β w I(s) N(s) W 1(s) ds t τ w e µws β w I(s) N(s) W 1(s) ds t τw τ w e µws β w I(s) N(s) W 1(s) ds τ w e µws β w I(s) N(s) W 1(s) ds. Therefore, if W e () = τ w e µws I(s) β w N(s) W 1(s) ds is satisfied, then W e (t) = t t τ w e µw(t s) β w I(s) N(s) W 1(s) ds. Similarly, if E() = τ h e µ hs W β 2(s)S(s) h N(s) ds is satisfied, then E(t) = t t τ h e µ h(t s) β h S(s) N(s) W 2(s) ds.

8 96 Z. WANG AND X-Q. ZHAO This implies that u t X δ, t [, σ φ ). Note that For the system (2.1) dn(t) = H µ h N(t) ε h I(t) H µ h N(t). dn(t) = H µ h N(t), the equilibrium N = H/µ h is globally asymptotically stable. By the comparison principle, it follows that (2.11) lim sup N(t) N. Regarding the total vector population, we have [ dw (t) = re aτa 1 W (t τ ] A) re aτa [ 1 W (t τ A) re aτa 4 µ ww (t). ] W σ (t τ A ) µ w W (t) ε w W 2 (t) + W (t τ A ) µ w W (t) + For the system dy/ = re aτa (/4) µ w y(t), the equilibrium re aτ A 4µ w is globally asymptotically stable. By the comparison principle, it follows that (2.12) lim sup W (t) re aτa. 4µ w By (2.11) and (2.12), it follows that σ φ =, all the solutions exist globally, and are ultimately bounded. Moreover, when N(t) > max{ H µ h, re aτ A 4µ w } and W (t) > max{ H µ h, re aτ A 4µ w }, we have dn(t) <, dw (t) which implies that all solutions are uniformly bounded. Therefore, the solution semiflow Φ(t) = u t ( ) : X δ X δ is point dissipative. By [6, Theorem 3.6.1], Φ(t) is compact for any t > τ. Thus, [7, Theorem 3.4.8] implies that Φ(t) has a compact global attractor in X δ. <

9 A TIME-DELAYED DENGUE TRANSMISSION MODEL 97 3 Threshold dynamics In this section, we establish the threshold dynamics for the model system in terms of the basic reproduction number. We define the diseased classes as the mosquito and human populations that are either exposed or infectious, i.e., W e, W 2, E and I. To get the disease-free equilibrium, letting W e = W 2 = E = I =, we then get R = and (3.1) (3.2) dw 1 (t) ds(t) [ = re aτa 1 W ] 1(t τ A ) W 1 (t τ A ) µ w W 1 (t), + = H µ h S(t). There are two disease free equilibria, E = (,,, N,,, ) and E 1 = (W,,, N,,, ), where W = (re aτ A µ w). By [2, Proposition re aτ A 4.1], for system (3.1), the equilibrium W is globally asymptotically stable if the following condition is satisfied (H1) µ w < re aτa 3µ w. Linearizing the model system at the disease free equilibrium E 1, we obtain the following system (here we only write down the equations for the diseased classes): dw e (t) dw 2 (t) de(t) di(t) = β w W N I(t) µ ww e (t) β w e µwτw W = β w e µwτw W N I(t τ w), N I(t τ w) (µ w + ε w )W 2 (t), = β h W 2 (t) µ h E(t) β h e µ hτ h W 2 (t τ h ), = β h e µ hτ h W 2 (t τ h ) (µ h + ε h + γ)i(t). Following the idea in [17], we introduce the basic reproduction number for the model system. Denote x 1, x 2, x 3 and x 4 be the number of each diseased class at time t =, and x 1 (t), x 2 (t), x 3 (t) and x 4 (t) be the remaining populations of each class at time t, respectively, then we obtain x 2 (t) = x 2 e (µw+εw)t, x 4 (t) = x 4 e (µ h+ε h +γ)t.

10 98 Z. WANG AND X-Q. ZHAO The total number of newly infected in each diseased class is x 1 = x 2 = x 3 = x 4 = τ w τ h β w W N x 4 (t) = β w e µwτw W β w W N (µ h + ε h + γ) x 4, N x 4 (t τ w ) = β we µwτw W N (µ h + ε h + γ) x 4, β h x 2 (t) = β h µ w + ε w x 2, β h e µ hτ h x 2 (t τ h ) = β he µhτh µ w + ε w x 2. Since β w W x 1 N (µ h + ε h + γ) β w e µwτw W x 2 = N (µ h + ε h + γ) x 3 β h µ w + ε w x 4 β h e µ hτ h µ w + ε w x 1 x 2 x 3 x 4, we can see that the 4 4 matrix: β w W N (µ h + ε h + γ) β w e µwτw W M = N (µ h + ε h + γ) β h µ w + ε w β h e µ hτ h µ w + ε w is the next infection operator. As usual, we define the spectral radius of the matrix M as the basic reproduction number R for the model system. It then follows that β w β h e R = (µwτw+µ hτ h) W N (µ w + ε w )(µ h + ε h + γ). Our first result shows that the disease is uniformly persistent if R > 1.

11 A TIME-DELAYED DENGUE TRANSMISSION MODEL 99 Theorem 3.1. Let (H1) hold. If R > 1, then there is an η > such that any solution u(t, φ) of the model system with φ X δ, φ 3 () and φ 6 () satisfies Proof. Define lim inf (W 2(t), I(t)) (η, η). X = {φ = (φ 1, φ 2,..., φ 7 ) X δ : φ 3 (), and φ 6 () }. Clearly, we have X = X δ \ X = {φ X δ : φ 3 () =, or φ 6 () = }. Define M = {φ X δ : Φ(t)φ X, t }. Claim 1. There exists a δ 1 >, such that for any φ X, lim sup Φ(t)φ E δ 1. Since µ w < re aτa, we can choose ε > and δ 1 > sufficiently small, such that (3.3) (3.4) x 3 < ε, (x 1, x 2, x 3, x 4 ) (N,,, ) < δ 1, x 1 + x 2 + x 3 + x 4 ( µ w + β w ε < re aτa 1 3δ ) 1. For any φ X δ, since φ 3 (), and φ 6 (), it follows from [12, Theorem 5.2.1], (3.5) W 2 (t) >, I(t) >, t >. Next we show that there exists a t such that W 1 (t, φ) > for all φ X. Otherwise, there exists ψ X such that W 1 (t, ψ) = for all t. From (2.2), we get W e (t) for all t τ w, then from (2.1), we get W 2 (t), for all t τ w, a contradiction with (3.5). Then, by [12, Theorem 5.2.1], W 1 (t) > for all t t.

12 1 Z. WANG AND X-Q. ZHAO Suppose, by contradiction, that lim sup Φ(t)ψ E < δ 1 for some ψ X. Thus, Φ(t)ψ E < δ 1 holds for all large t. Then we can choose large number t 1 > t such that for all t t 1, there holds that dw 1 (t) re aτa ( 1 3δ 1 ) W 1 (t τ A ) (µ w + β w ε )W 1 (t). Consider the next linear and monotone time-delayed system ( dw(t) (3.6) = re aτa 1 3δ ) 1 w(t τ A ) (µ w + β w ε )w(t). Let λ be the principal eigenvalue of the corresponding eigenvalue problem of equation (3.6). By [12, Corollary 5.5.2] and (3.4), it follows that λ >. We can choose l > small enough such that le λt W 1 (t), t [t 1, t 1 + τ A ]. Clearly, le λt satisfies (3.6) for all t t 1. Then by the comparison principle, we get le λt W 1 (t), t t 1 + τ A, Since λ > and l >, le λt as t. Thus, lim W 1 (t) =, a contradiction. Claim 2. There exists a δ 2 >, such that for any φ X, lim sup Φ(t)φ E 1 δ 2. First we consider the following linear cooperative system ( ) dŵ2(t) W = β w e µwτw N (3.7) ε Î(t τ w ) (µ w + ε w )Ŵ 2 (t), dî(t) = β h e µ hτ h (1 ε)ŵ2(t τ h ) (µ h + ε h + γ)î(t). For sufficiently small ε >, let λ 1 (ε) be the principle eigenvalue of system (3.7). Since R > 1, it is easy to see from [12, Corollary 5.5.2] that λ 1 () >. Thus, we can restrict ε small enough such that λ 1 (ε) >. For this small ε, there exists δ 2 = δ 2 (ε) > such that x 4 x 4 + x 5 + x 6 + x 7 > 1 ε > and x 1 + x 2 + x 3 x 4 + x 5 + x 6 + x 7 > W N ε >, (x 1, x 2,, x 7 ) E 1 < δ 2.

13 A TIME-DELAYED DENGUE TRANSMISSION MODEL 11 Assume, by contradiction, that lim sup Φ(t)φ E 1 < δ 2 for some φ X. Then there exists a large number t 2, such that for all t t 2, Φ(t)φ (W,,, N,,, ) < δ 2. For any ε >, we can further choose t 3 > t 2 large enough, such that for all t t 3, W (t) N(t) W N ε, S(t) N(t) 1 ε. That is, when t t 3, we have dw 2 (t) ( ) W β w e µwτw N ε I(t τ w ) (µ w + ε w )W 2 (t), di β he µ hτ h (1 ε)w 2 (t τ h ) (µ h + ε h + γ)i(t). Let v = (v 1, v 2 ) T be the positive right eigenvector associated with λ 1 (ε) for system (3.7), choose l > small enough such that lv 1 e λ1(ε)t W 2 (t), lv 2 e λ1(ε)t I(t), t [t 3, t 3 + τ], t [t 3, t 3 + τ], Clearly, le λ1(ε)t (v 1, v 2 ) T satisfies (3.7) for t t 3. Then by the comparison principle, we get (W 2 (t), I(t)) le λ1(ε)t (v 1, v 2 ), t t 3 + τ. Since λ 1 (ε) >, letting t, we obtain lim W 2(t) =, lim I(t) =, a contradiction. Let ω(φ) be the omega limit set of the orbit of Φ(t) through φ X δ. Claim 3. φ M ω(φ) = E E 1. For any φ M, i.e., Φ(t)φ X, we have W 2 (t, φ), or I(t, φ). If W 2 (t, φ), then from the equations of S, E and I, we have lim S(t, φ) = N, lim E(t, φ) = and lim I(t, φ) =. By the theory of asymptotically autonomous semiflows (see, e.g., [14]),

14 12 Z. WANG AND X-Q. ZHAO it follows that lim W 1 (t, φ) = W or, lim W e (t, φ) = and lim R(t, φ) =. If W 2 (t, φ), then there exists t, such that W 2 (t, φ) >. We then obtain that W 2 (t, φ) > for all t t, and I(t, φ). From the equations of W e, W 2 and R, we have lim W e (t, φ) =, lim W 2 (t, φ) =, and lim R(t, φ) =. By the theory of asymptotically autonomous semiflows, we get lim W 1 (t, φ) = W or, lim E(t, φ) = and lim S(t, φ) = N. Consequently, we have φ M ω(φ) = E E 1. Define a continuous function p : X δ R + by p(φ) = min{φ 3 (), φ 6 ()}, φ X δ. Clearly, p 1 (, ) X. It follows from (3.5) that p has the property that if either p(φ) = and φ X, or p(φ) >, then p(φ(t)φ) >, for all t >. Thus p is a generalized distance function for the semiflow Φ(t) : X δ X δ. By Claim 3, we get that any forward orbit of Φ(t) in M converges to E or E 1, by Claims 1 and 2, we conclude that E and E 1 are two isolated invariant sets in X δ, and (W s (E ) W s (E 1 )) X =. Moreover, it is easy to see that no subset of {E, E 1 } forms a cycle in X. By [13, Theorem 3], it then follows that there exists η > such that lim inf p(φ(t)φ) η for all φ X, which implies the uniform persistence stated in the theorem. The subsequent result shows that the disease dies out if R < 1, provided there is only a small invasion in the W 2 and I classes. For any given M >, denote { 7 Xδ M = φ C([ τ, ], [, M] 7 ) : φ i (s) δ, s [ τ, ], φ 2 () = φ 5 () = i=4 e µws φ 6 (s)φ 1 (s) β w 7 τ w i=4 φ i(s) ds, Then we have the following result. e µ hs φ 4 (s)φ 3 (s) β h 7 }. τ h i=4 φ i(s) ds Theorem 3.2. Let (H1) hold. If R < 1, then for every M > max { H µ h, re aτ A 4µ w }, there exists a ζ = ζ(m) > such that for any φ X M δ \ E with (φ 3 (s), φ 6 (s)) [, ζ] 2 for all s [ τ, ], the solution u(t, φ) of the model system through φ satisfies lim u(t, φ) E 1 =.

15 A TIME-DELAYED DENGUE TRANSMISSION MODEL 13 Proof. Let M > max{ H µ h, re aτ A 4µ w } be given. From the prove of Theorem 2.1, we see that Xδ M is positively invariant for the solution semiflow of the model system. We then have u(t, φ) [, M] 7, t, φ X M δ. Consider the following linear and monotone system (3.8) d W 2 (t) dĩ(t) ( W ) + ε = β w e µwτw N Ĩ(t τ w ) (µ w + ε w ) W 2 (t), ε = β h e µ hτ h W2 (t τ h ) (µ h + ε h + γ)ĩ(t). For sufficiently small ε >, let λ 2 (ε) be the principle eigenvalue of this eigenvalue problem. Since R < 1, it is easy to see from [12, Corollary 5.5.2] that λ 2 () <. Thus, we can restrict ε small enough such that λ 2 (ε) <. Let (e 1, e 2 ) T be positive right eigenvector associated with λ 2 (ε). Now we consider the following equations: (3.9) (3.1) d ˇW [ = re aτa 1 ˇW ] (t τ A ) ˇW (t τ A ) µ w ˇW (t) εw ξ 1, + dň = H µ hň(t) ε hξ 1. Choose small ξ 1 > and large T = T (M) > such that for any solutions of ( ˇW (t, φ), Ň(t, φ)). We then have ˇW (t) < W + ε, Ň(t) > N ε, t T. Denote the solution of system (3.8) by ũ(t, φ) = ( W 2 (t), Ĩ(t)) with respect to initial data φ = ( φ 1, φ 2 ) C([ τ, ], [, M] 2 ). Then for system (3.8), for ξ 1 >, there exists ξ 2 >, such that if (ξ 2 e 1, ξ 2 e 2 ) (ξ 1, ξ 1 ). Since λ 2 (ε) <, we get that (3.11) (ξ 2 e 1 e λ2(ε)t, ξ 2 e 2 e λ2(ε)t ) (ξ 1, ξ 1 )), t. For every solution of the model system through φ, there exists a ζ = ζ(m) > such that (3.12) (W 2 (t, φ), I(t, φ)) (ξ 1, ξ 1 ), t [, T 1 ]

16 14 Z. WANG AND X-Q. ZHAO provided that (φ 3 (s), φ 6 (s)) < (ζ, ζ). We further claim that (3.12) holds for all t. Suppose, by contradiction, that the claim is not true. Then there exists a T 2 = T 2 (φ) > T 1 such that (W 2 (t, φ), I(t, φ)) (ξ 1, ξ 1 ), for all t [, T 2 ), and W 2 (T 2, φ) = ξ 1 or I(T 2, φ) = ξ 1. By the comparison principle, for t [T 1, T 2 ], we have (3.13) (W 2 (t, φ), I(t, φ)) ( W 2 (t, φ), Ĩ(t, φ)) (ξ 1, ξ 1 ) a contradiction. So (3.12) holds for all t. From (3.11) and (3.13), we see that lim (W 2 (t, φ), I(t, φ)) = (, ). By the theory of asymptotically autonomous semiflows (see [14]), it follows that lim u(t, φ) = (W,,, N,,, ). 4 Global attractivity In this section, we study the global attractivity in the model system in the case where the disease-induced death rates of infected mosquitos and human individuals are zero, and the fecundity of infected mosquitos is the same as the susceptible mosquitos. In this case, the model system becomes (4.1) dw 1 (t) dw e (t) dw 2 (t) ds(t) de(t) di(t) dr(t) = re aτa [ 1 W (t τ A) ] W (t τ A ) + µ w W 1 (t) β w I(t) N(t) W 1(t) = β w I(t) N(t) W 1(t) µ w W e (t) β w e µwτw I(t τ w) N(t τ w ) W 1(t τ w ), = β w e µwτw I(t τ w) N(t τ w ) W 1(t τ w ) µ w W 2 (t), = H µ h S(t) β h W 2 (t) S(t) N(t), = β h W 2 (t) S(t) N(t) µ he(t) β h e µ hτ h W 2 (t τ h ) S(t τ h) N(t τ h ). = β h e µ hτ h W 2 (t τ h ) S(t τ h) N(t τ h ) (µ h + γ)i(t), = γi(t) µ h R(t).

17 A TIME-DELAYED DENGUE TRANSMISSION MODEL 15 It is clear that when R <, system (4.1) has only two equilibria E and E 1. However, system (4.1) admits a unique positive equilibrium E := (W 1, W e, W 2, S, E, I, R ) when R > 1, where and S = H(µ hβ w + µ w (µ h + γ)e µ hτ h ) µ h (µ h β w + µ w (µ h + γ)e µ hτ hr 2 ), I = Hµ w (R 2 1) µ h β w + µ w (µ h + γ)e µ hτ hr 2, W 2 = (µ h + γ)n e µ hτ h I β h S, W 1 = µ wn e µwτw W 2 β w I, W e = (e µwτw 1)W 2, E = (eµ hτ h 1)(µ h + γ)i µ h, R = γi µ h. The following two results show the global attractivity of system (4.1). Theorem 4.1. Let (H1) hold and assume that σ = 1 and ε w = ε h =. If R < 1, then the disease-free equilibrium of the model system is globally attractive in X δ \ E. Proof. If σ = 1 and ε w = ε h =, then the whole mosquitos and human populations satisfy the following two equations: [ dw (t) = re aτa 1 W (t τ ] A) W (t τ A ) µ w W (t), + dn(t) = H µ h N(t). Since W and N is globally asymptotically stable for the above two equations, respectively, there exists T = T (ε) > such that W (t) W + ε, N(t) N ε, t T.

18 16 Z. WANG AND X-Q. ZHAO Thus, when t T, we have dw 2 (t) ( W ) + ε β w e µwτw N I(t τ w ) µ w W 2 (t), ε di β he µ hτ h W 2 (t τ h ) (µ h + γ)i(t). When R < 1, ε small enough, by the analysis of system (3.8) and the comparison principle, we then have lim (W 2(t), I(t)) = (, ). It then follows from the theory of asymptotically semiflows (see [14]) that lim (W 1(t), W e (t), S(t), E(t), R(t)) = (W,, N,, ). This completes the proof. To obtain the global attractivity of the endemic equilibrium, we need the following additional assumption: (H2) β w µ h µ w (µ h + γ)e τ hµ h. Theorem 4.2. Let (H1) and (H2) hold and assume that σ = 1 and ε w = ε h =. If R > 1, then for any φ X δ with φ 3 (), φ 6 (), we have lim u(t, φ) = E. Proof. When ε w = ε h =, σ = 1, we have (4.2) dw (t) dn(t) [ = re aτa 1 W (t τ ] A) W (t τ A ) µ w W (t), + = H µ h N(t). When (H1) holds, (W, N ) is globally asymptotically stable for sys-

19 A TIME-DELAYED DENGUE TRANSMISSION MODEL 17 tem (4.2). Hence, we have the following limiting system: (4.3) dw 1 (t) dw e (t) dw 2 (t) ds(t) de(t) di(t) dr(t) = A µ w W 1 (t) β w I(t)W 1(t), = β wi(t)w 1 (t) µ w W e (t) β w e µwτw I(t τ w )W 1 (t τ w ), = β we µwτw I(t τ w )W 1 (t τ w ) µ w W 2 (t), = H µ h S(t) β h W 2(t)S(t), = β hw 2 (t)s(t) µ h E(t) β he µ hτ h W 2 (t τ h )S(t τ h ), = β h e µ hτ h W 2 (t τ h )S(t τ h ) (µ h + γ)i(t), = γi(t) µ h R(t), where A = W µ w, β w = β w /N, β h = β h/n. Let g(t τ w ) = W 1 (t τ w ) + e µwτw W 2 (t), that is, g(t) = W 1 (t) + e µwτw W 2 (t + τ w ). It follows that g (t) = W 1 (t) + eµwτw W 2 (t + τ w) = A µ w (W 1 (t) + e µwτw W 2 (t + τ w )) = A µ w g(t). Then the equilibrium A/µ w = W is globally asymptotically stable. For system (4.3), we then consider the following limiting system: (4.4) (4.5) (4.6) dw 2 (t) d S(t) dī(t) = β we µwτwī(t τ w)(w e µwτw W 2 (t)) µ w W 2 (t), = H µ h S(t) β h S(t)W 2 (t), = β h e µ hτ h S(t τh )W 2 (t τ h ) (µ h + γ)ī(t).

20 18 Z. WANG AND X-Q. ZHAO Claim 4. The set D := C([ τ, ], [, W e µwτw ] R 2 +) is positively invariant for system (4.4) (4.6). To prove this claim, we define β we µwτw ψ 3 ( τ w )(W e µwτw ψ 1 ()) µ w ψ 1 () F (ψ) := H µ h ψ 2 () β h ψ 1()ψ 2 (), ψ D. β h e µ hτ h ψ 1 ( τ h )ψ 2 ( τ h )) (µ h + γ)ψ 3 () Note that D is relatively closed in C([ τ, ], R 3 ), and F (ψ) is continuous and Lipschitz in ψ in each compact set in R D. By [6, Theorem 2.3], it follows that for all ψ D, there is an unique solution of system (4.4) (4.6) through (, ψ) on its maximal interval of existence. Since F i (ψ) whenever ψ D with ψ i () =, [12, Theorem 5.2.1] implies that the solution of (4.4) (4.6) are nonnegative for all t in its maximal interval of existence. Furthermore, if ψ 1 () = W e µwτw, then F 1 (ψ). It follows by [12, Remark 5.2.1] that W 2 (t, ψ) W e µwτw for all t >. Thus, D is positively invariant. By the arguments similar to those in Theorem 3.1, it easily follows that system (4.4) (4.6) is uniformly persistent in the sense that there exists a η 1 > such that for any given ψ = (ψ 1, ψ 2, ψ 3 ) D with ψ 1 (), ψ 3 (), the solution (W 2 (t, ψ), S(t, ψ), Ī(t, ψ)) of (4.4) (4.6) satisfies lim inf (W 2(t, ψ), Ī(t, ψ)) (η 1, η 1 ). For any given ψ D with ψ 1 () and ψ 3 (), let (W 2 (t), S(t), Ī(t)) = (W 2 (t, ψ), S(t, ψ), Ī(t, ψ)). In order to use the method of fluctuations (see, e.g., [8, 18]) for system (4.4) (4.6), we define W 2 = lim sup W 2 (t), S = lim sup S(t), Ī = lim sup Ī(t), W 2 = lim inf S = lim inf Ī = lim inf W 2(t); S(t); Ī(t). Clearly, W 2 W 2 η 1 >, S S and Ī Ī η 1 >. Further, there exist sequences t i n and σi n, i = 1, 2, 3, such

21 A TIME-DELAYED DENGUE TRANSMISSION MODEL 19 that lim n W 2(t 1 n) = W 2, W 2(t 1 n) =, n 1; lim W 2(σ 1 n n ) = W 2, W 2 (σ1 n ) =, n 1; lim S(t 2 n n ) = S, S (t 2 n ) =, n 1; lim S(σ 2 n n ) = S, S (σn 2 ) =, n 1; lim Ī(t 3 n n ) = Ī, Ī (t 3 n ) =, n 1; lim Ī(σ 3 n n ) = Ī, Ī (σn 3 ) =, n 1. Let m 1 = β w e µwτw W /N and m 2 = β h e µ hτ h /N. It then follows from (4.4) and the above claim that Ī (m 1 β ww 2 ) µ w W 2 Ī (m 1 β ww 2 ) µ w W 2, Ī (m 1 β ww 2 ) µ w W 2 Ī (m 1 β ww 2 ) µ w W 2, and hence, (4.7) Ī µ w W 2 m 1 β w W 2 µ w W 2 m 1 β ww 2 Ī. By (4.5), we have H S (µ h + β hw 2 ) H S (µ h + β hw 2 ), H S (µ h + β h W 2 ) H S (µ h + β h W 2 ), which implies that (4.8) H µ h + β h W 2 S S H µ h + β h W 2. In view of (4.6), we obtain m 2 S W 2 (µ h + γ)ī m 2 S W 2 (µ h + γ)ī, m 2 S W 2 (µ h + γ)ī m 2 S W 2 (µ h + γ)ī,

22 11 Z. WANG AND X-Q. ZHAO and hence, (4.9) m 2 S W 2 µ h + γ Ī Ī m 2 S W 2. µ h + γ Therefore, combining (4.8) and (4.9) together, we get (4.1) H µ h + β h W 2 m 2 W 2 µ h + γ Ī Ī H µ h + β h W 2 m 2 W 2 µ h + γ. Comparing (4.7) with (4.1), we obtain H µ h + β h W 2 m 2 W 2 µ h + γ µ w W 2 m 1 β w W 2 ; H µ h + β h W 2 m 2 W 2 µ h + γ µ w W 2 m 1 β ww 2. Simplifying the above two inequalities, we get β w µ h (W 2 W 2 ) µ w (µ h + γ)e τ hµ h (W 2 W 2 ) Since condition (H2) holds, we have W 2 = W 2. By (4.8) and (4.1), we get S = S and I = I. It follows that lim (W 2 (t), S(t), Ī(t)) = (W 2, S, I ) for any ψ D with ψ 1 () and ψ 3 (). By the theory of chain transitive sets (see, e.g., [19, Section 1.2.1]) and the arguments similar to [1, Appendix A] (see also [17, Theorem 2.1]), we can lift the global attractivity for system (4.4)-(4.6) to the model system. It follows that lim u(t, φ) = E, for any φ X δ with φ 3 () and φ 6 (). 5 Numerical simulations In this section, we carry out numerical simulations to illustrate our analytic results. In view of [15], we fix τ A = 1, τ w = 1, τ h = 5, and then take three sets of values of other parameters to perform the numerical simulations. First, we take β w =.6, β h =.15, r = 1, a =.2, γ =.15, µ w =.1, µ h =.1, H =.1, = 1, σ =.8, ε w =.1, ε h =.1. It is easy to verify that condition (H1) holds, and R =.174, W = 2.61, N = 1. It follows from Theorem 3.2 that when W 2 (s) and I(s), s [ τ, ], are small, the disease will die out (see Figure 1).

23 A TIME-DELAYED DENGUE TRANSMISSION MODEL a: W x 1 3 b: W e x 1 4 c: W d: S x 1 4 e: E x 1 4 f: I 1 x FIGURE 1: Long-term behavior of the population of each class when R < 1 and the invasion is small. a: W 1 b: W e c: W 2 4 d: S e: E 4 f: I FIGURE 2: E is globally asymptotically attractive when R > 1 and conditions (H1) and (H2) hold.

24 112 Z. WANG AND X-Q. ZHAO 3 a: W b: I FIGURE 3: Persistence of infected mosquitos and human individuals. Second, we take β w =.9, β h =.5, r = 1, a =.4, γ =.5, µ w =.1, µ h =.1, H =.1, = 1, σ = 1, ε w = ε h =. Then we get that (H1) and (H2) hold, and R = 17.59, W = 4.54, N = 1. By Theorem 4.2, we obtain E = (1.655,.275, 2.61,.76,.49,.1936, 9.681) is globally attractive (see Figure 2). Third, by taking β w =.9, β h =.5, r = 1, a =.4, γ =.5, µ w =.1, µ h =.1, H =.1, = 1, σ =.8, ε w =.1, ε h =.1, we get R = We can see that the disease is uniform persistence. Figure 3 indicates the behavior of the infectious mosquitos and infectious human population. REFERENCES 1. W. G. Aiello and H. I. Freedman, A time delay model of single-species growth with stage structure, Math. Biosci. 11 (199), G. Butler, H. I. Freedman and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96 (1986), Canada Communicalbe Disease Report (29), publicat/ccdr-rmtc/9vol35/acs-dcc-2/index-eng.php. 4. Centers for Disease Control and Prevention, dengue/.

25 A TIME-DELAYED DENGUE TRANSMISSION MODEL G. Chowell, P. Diaz-Duenas, J. C. Miller, A. Alcazar-Velazco, J. M. Hyman, P. W. Fenimore, and C. Castillo-Chavez, Estimation of the reproduction number of dengue fever from spatial epidemic data, Math. Biosci. 28 (27), J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, J. Hale, Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence, RI, W.M. Hirsch, H. Hanisch and J-P. Gabriel, Differential Equation Models of Some Parasitic Infections: Methods for the Study of Asymptotic Behavior, Comm. Pure Appl. Math. 38 (1985), Y. Lou and X.-Q. Zhao, A climate-based malaria transmission model with structured vector population, SIAM J. Appl. Math. 7 (21), Y. Lou and X.-Q. Zhao, Modelling Malaria control by introduction of larvivorous fish, Bull. Math. Biol. 73 (211), Public Health Agency of Canada (29), /tmppmv/info/dengue-eng.php. 12. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math. Surveys Monogr. 41, American Mathematical Society, Providence, RI, H. L. Smith and X.-Q. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. 47 (21), H. R. Thieme, Convergence results and Poincare-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol. 3 (1992), W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction diffusion model of dengue transmission, SIAM J. Appl. Math. 71 (211), World Health Organization (29), Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period and diffusion, Discrete Contin. Dyn. Syst. Ser. B 17 (212), Y. Yuan and X.-Q. Zhao, Global stability for non-monotone delay equations (with application to a model of blood cell production), J. Differential Equations 252 (212), X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Canad. Appl. Math. Quart. 17 (29), Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John s, NL A1C 5S7, Canada. address: zhenw@mun.ca address: zhao@mun.ca

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