GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE … · The purpose of this paper is to study the global dynamics of a time-delayed dengue transmission model. In Section 2, we present the

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 20, Number 1, Spring 2012

GLOBAL DYNAMICS OF A TIME-DELAYED

DENGUE TRANSMISSION MODEL

Dedicated to Herb Freedman on the occasion of his 70th birthday

ZHEN WANG AND XIAO-QIANG ZHAO

ABSTRACT. In this paper, we present a time-delayed den-gue transmission model. We first introduce the basic reproduc-tion number for this model and then show that the disease

persists when R0 > 1. It is also shown that the disease willdie out if R0 < 1, provided that the invasion intensity is notstrong. We further establish a set of sufficient conditions for theglobal attractivity of the endemic equilibrium by the method offluctuations. Numerical simulations are performed to illustrateour analytic results.

1 Introduction Dengue fever is the most common viral diseasespread to humans by mosquitos, and has become an international pub-lic health concern. Dengue is caused by a group of four antigenicallydistinct flavivirus serotypes: DEN-1, DEN-2, DEN-3, and DEN-4; andis primarily transmitted by Aedes mosquitos, particularly A. aeqyptimosquitos. Dengue is found in tropical and subtropical regions aroundthe world, predominately in urban and peri-urban areas. The incidenceof dengue has grown dramatically around the world in recent decades.It is endemic in more than 110 countries in Africa, the Americas, theEastern Mediterranean, South-east Asia and the Western Pacific. Itinfects 50 to 100 million people worldwide a year, leading to 50 mil-lion hospitalizations, and approximately 12,500 to 25,000 deaths a year[3, 4, 11, 16].

The human is the main amplifying host of the virus, although stud-ies have shown that in some part of the world monkeys may become

Research supported in part by the NSERC of Canada and the MITACS ofCanada.2010 MSC: 34K60, 37L15, 92D30.

Keywords: Dengue transmission, vector-borne disease, basic reproduction num-ber, uniform persistence, global attractivity.

Copyright c©Applied Mathematics Institute, University of Alberta.

89

90 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, andA. aeqypti mosquitos may acquire the virus when they feed on an infec-tious individual. Much have been done in terms of modeling and anal-ysis of disease transmission with structured vector population. Wangand Zhao [15] proposed a nonlocal and time-delayed reaction-diffusionmodel of dengue fever, and established a threshold dynamics in termsof the basic reproduction number R0. Lou and Zhao [9] presented amalaria transmission model with structured vector population, and alsoestablished a threshold type result, which states that when R0 < 1 andthe disease invasion is not strong, the disease will die out; when R0 > 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 profoundimpact on the transmission of disease: first, the immature mosquitos donot fly and bite human, so they do not participate in the infection cycle;second, mature mosquitos are quite different from immature mosquitosfrom biological and epidemiological perspectives. In view of realisticconsideration, we take these different stages into account. We also in-clude the time delay to describe the incubation periods of mosquitosand the human populations, which is important because there are incu-bation realistically and the time period is not small. In fact, from theexpression of R0 in Section 3, we can see those delays reduce the valuesof R0. Therefore, the neglect of the delays overestimated the infectionrisk.

The purpose of this paper is to study the global dynamics of a time-delayed dengue transmission model. In Section 2, we present the modelsystem and prove its wellposedness. In Section 3, we first introduce thebasic reproduction number R0, and then show that the disease is uni-formly persistent when R0 > 1 by appealing to the theory developed in[2, 13]. Under certain conditions, we also obtain the nonlocal stabilityof the disease-free equilibrium when R0 < 1. In Section 4, we obtaina set of sufficient conditions for the endemic equilibrium to be glob-ally attractive by the method of fluctuations. In Section 5, we performnumerical simulations to illustrate our analytic results.

2 The model In this section, following the ideas in [15], we presentan age-structure dengue model with time delay for the cross infectionbetween mosquitos and human individuals. We divide the mosquitopopulation into two subclasses: aquatic population and winged pop-

A TIME-DELAYED DENGUE TRANSMISSION MODEL 91

ulation. Winged female A. aegypti mosquitoes lay eggs in unattendedwater. Eggs may develop into larvae from two days up to one week. Thelarvae spend up to three days to pass through four instars to enter thepupal stage. The pupa develops into an adult after about two days. Theimmature mosquitos live in aquatic habitats and mature mosquitos dis-perse to search for food. Let A denote the density of aquatic populationof mosquitos, W be the density of winged population of mosquitos, andτA be the length of immature stage of mosquitos. Following the modelto formulate a stage-structured population in Aiello and Freedman [1],we suppose the dynamics of mosquitos is described by

dA(t)

dt= B((W (t))W (t) − aA(t) − e−aτAB(W (t− τA))W (t− τA),

dW (t)

dt= e−aτAB(W (t− τA))W (t− τA) − µwW (t),

where B is the per capita birth rate of adult mosquitos, a is the percapita death rate of aquatic mosquitos, and µw is the death rate ofadult mosquitos. Following [15], we assume that the function B(W )Wis the logistic growth rate:

rW [1 −W/K]+ =

rW

[1 −

W

K

]if 0 ≤W ≤ K

0 if W > K,

For the dynamics of human population, we assume that the densityN of the human population obeys

dN

dt= H − µhN,

where H is a constant recruitment rate and µh is the death rate.To consider dengue transmission between mosquitos and human in-

dividuals, we let W1, We and W2 denote the density of susceptible, ex-posed, and infectious mosquitos of winged population, respectively; anddivide the human population into four compartments: susceptible (S),exposed (E), infectious (I) and recovered (R). Let τw be the incubationperiod of dengue virus within mosquitos and τh be the incubation periodof dengue virus within hosts. Following Chowell et al. [5], we supposethat the infection rates of susceptible mosquitos and susceptible humanindividuals are described by

bpI

NW1, bq

S

NW2,


respectively, where b is the mean rate of mosquito bites per mosquito, pis the probability that a bite by a susceptible mosquito to an infectioushost will cause infections, q is the probability that a bite by an infectiousmosquito 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 infectiousmosquito may have lower fecundity than a susceptible mosquito, we letσ ∈ [0, 1] denote the relative fecundity of an infected mosquito to asusceptible mosquito. Specifically, the infectious mosquito has the samereproduction rate as a susceptible mosquito if σ = 1, and have lowerreproduction rate if σ < 1. Then we have the following model:

dA(t)

dt= r

[1 −

W (t)

K

]

+

Wσ(t) − aA(t)

− re−aτA

[1 −

W (t− τA)

K

]

+

Wσ(t− τA),

dW1(t)

dt= re−aτA

[1 −

W (t− τA)

K

]

+

Wσ(t− τA)(2.1)

− µwW1(t) − βw

I(t)

N(t)W1(t),

We(t) =

∫ t

t−τw

e−µw(t−s)βw

I(s)

N(s)W1(s) ds,(2.2)

dW2(t)

dt= βwe

−µwτwI(t− τw)

N(t− τw)W1(t− τw) − (µw + εw)W2(t),(2.3)

dS(t)

dt= H − µhS(t) − βhW2(t)

S(t)

N(t),(2.4)

E(t) =

∫ t

t−τh

e−µh(t−s)βh

S(s)

N(s)W2(s) ds,(2.5)

dI(t)

dt= βhe

−µhτhW2(t− τh)S(t− τh)

N(t− τh)− (µh + εh + γ)I(t),(2.6)

dR(t)

dt= γI(t) − µhR(t),(2.7)

where

W (t) = W1(t) +We(t) +W2(t), Wσ(t) = W1(t) +We(t) + σW2(t),


βw = bp, βh = bq, γ is the recovery rate of infected human individuals,εw and εh are the infection-induced death rates of infected mosquitosand human individuals, respectively.

Note that the equation for aquatic population of mosquitos is de-coupled from the other equations. It then suffices to consider system(2.1)–(2.7) which is an integro-differential equation system. Differenti-ating (2.2) and (2.5) gives

dWe(t)

dt= βw

I(t)

N(t)W1(t) − µwWe(t)(2.8)

− βwe−µwτw

I(t− τw)

N(t− τw)W1(t− τw),

dE(t)

dt= βhW2(t)

S(t)

N(t)− µhE(t)(2.9)

− βhe−µhτhW2(t− τh)

S(t− τh)

N(t− τh).

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, wewill refer to this system as “the model system” in the rest of this paper.

Let τ = max{τA, τw, τh}, and define C := C([−τ, 0],R7). For φ =

(φ1, φ2, · · · , φ7) ∈ C, define ‖φ‖ =∑7

i=1 ‖φi‖∞, where

‖φi‖∞ = maxθ∈[−τ,0]

|φi(θ)|.

Then C is a Banach space. Define C+ = {φ ∈ C : φi(θ) ≥ 0, ∀1 ≤ i ≤7, θ ∈ [−τ, 0]}. Then C+ is a normal cone of C with nonempty interiorin C. For a continuous function u : [−τ, σφ) → R

7 with σφ > 0, wedefine ut ∈ C for each t ≥ 0 by ut(θ) = u(t+ θ), ∀θ ∈ [−τ, 0].

In view of (2.2) and (2.5), we choose the initial data for the modelsystem in Xδ , which is defined as

Xδ =

{φ ∈ C+ :

7∑

i=4

φi(s) ≥ δ, ∀s ∈ [−τ, 0],

φ2(0) =

∫ 0

−τw

eµwsβw

φ6(s)φ1(s)∑7i=4 φi(s)

ds,

φ5(0) =

∫ 0

−τh

eµhsβh

φ4(s)φ3(s)∑7i=4 φi(s)

ds

}


for small δ ∈(0, H/(εh + µh)

). The following result shows that the

model system is wellposed in Xδ, and the solution semiflow admits aglobal attractor on Xδ .

Theorem 2.1. For any φ ∈ Xδ, the model system has a unique non-

negative solution u(t, φ) satisfying u0 = φ. Furthermore, the solution

semiflow Φ(t) = ut(·) : Xδ → Xδ has a compact global attractor.

Proof. Given φ ∈ Xδ , define

G(φ) := (G1(φ), G2(φ), G3(φ), G4(φ), G5(φ), G6(φ), G7(φ)),

where

G1(φ) = re−aτA

[1 −

∑3i=1 φi(−τA)

K

]

+

(φ1(−τA) + φ2(−τA)

+ σφ3(−τA)) − µwφ1(0) − βw

φ6(0)∑7i=4 φi(0)

φ1(0),

G2(φ) = βw

φ6(0)∑7i=4 φi(0)

φ1(0) − µwφ2(0)

− βwe−µwτw

φ6(−τw)∑7i=4 φi(−τw)

φ1(−τw),

G3(φ) = βwe−µwτw

φ6(−τw)∑7i=4 φi(−τw)

φ1(−τw) − (µw + εw)φ3(0),

G4(φ) = H − µhφ4(0) − βh

φ4(0)∑7i=4 φi(0)

φ3(0),

G5(φ) = βh

φ4(0)∑7i=4 φi(0)

φ3(0) − µhφ5(0)

− βhe−µhτh

φ4(−τh)∑7i=4 φi(−τh)

φ3(−τh),

G6(φ) = βhe−µhτh

φ4(−τh)∑7i=4 φi(−τh)

φ3(−τh) − (µh + εh + γ)φ6(0),

G7(φ) = γφ6(0) − µhφ7(0).

Note that Xδ is closed in C, and for all φ ∈ Xδ, G(φ) is continuous andLipschitz in φ in each compact set in R × Xδ. By [6, Theorem 2.3],


it then follows that for any φ ∈ Xδ, there is an unique solution of themodel system through (0, φ) on its maximal interval [0, σφ) of existence.

Since Gi(φ) ≥ 0 whenever φ ∈ Xδ with φi(0) = 0, [12, Theorem 5.2.1]implies that the solutions of the model system are nonnegative for allt ∈ [0, σφ). Note that the total host population satisfies

dN(t)

dt= H − µhN(t) − εhI(t) ≥ H − (µh + εh)N(t).

For the system dy/dt = H − (µh + εh) y(t), the equilibrium H/(µh + εh)is globally asymptotically stable. For any 0 < δ < H

µh+εh, dy/dt|y=δ =

H − (µh + εh) δ > 0. So if y(0) ≥ δ, then y(t) ≥ δ, for any t ≥ 0. From(2.8), we get

eµwt(W ′

e(t) + µwWe(t)) = eµwt

(βw

I(t)

N(t)W1(t)

− βwe−µwτw

I(t− τw)

N(t− τw)W1(t− τw)

)

By integrating on both sides, we obtain

eµwtWe(t) −We(0)

=

∫ t

0

eµwsβw

I(s)

N(s)W1(s) ds

−

∫ t

0

eµw(s−τw)βw

I(s− τw)

N(s− τw)W1(s− τw) ds

=

∫ t

0

eµwsβw

I(s)

N(s)W1(s) ds−

∫ t−τw

−τw

eµwsβw

I(s)

N(s)W1(s) ds

=

∫ t

t−τw

eµwsβw

I(s)

N(s)W1(s) ds−

∫ 0

−τw

eµwsβw

I(s)

N(s)W1(s) ds.

Therefore, if We(0) =∫ 0

−τweµwsβw

I(s)N(s)W1(s) ds is satisfied, then

We(t) =

∫ t

t−τw

e−µw(t−s)βw

I(s)

N(s)W1(s) ds.

Similarly, if E(0) =∫ 0

−τheµhsβh

W2(s)S(s)N(s) ds is satisfied, then

E(t) =

∫ t

t−τh

e−µh(t−s)βh

S(s)

N(s)W2(s) ds.


This implies that ut ∈ Xδ , ∀t ∈ [0, σφ).Note that

dN(t)

dt= H − µhN(t) − εhI(t) ≤ H − µhN(t).

For the system

(2.10)dN(t)

dt= H − µhN(t),

the equilibrium N∗ = H/µh is globally asymptotically stable. By thecomparison principle, it follows that

(2.11) lim supt→∞

N(t) ≤ N∗.

Regarding the total vector population, we have

dW (t)

dt= re−aτA

[1 −

W (t− τA)

K

]

+

Wσ(t− τA) − µwW (t) − εwW2(t)

≤ re−aτA

[1 −

W (t− τA)

K

]

+

W (t− τA) − µwW (t)

≤ re−aτAK

4− µwW (t).

For the system dy/dt = re−aτA(K/4)−µwy(t), the equilibrium re−aτA K4µw

is globally asymptotically stable. By the comparison principle, it followsthat

(2.12) lim supt→∞

W (t) ≤re−aτAK

4µw

.

By (2.11) and (2.12), it follows that σφ = ∞, all the solutions existglobally, and are ultimately bounded. Moreover, when N(t) > max{ H

µh,

re−aτAK4µw

} and W (t) > max{ Hµh, re−aτA K

4µw}, we have

dN(t)

dt< 0,

dW (t)

dt< 0

which implies that all solutions are uniformly bounded. Therefore , thesolution semiflow Φ(t) = ut(·) : Xδ → Xδ 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 Xδ .


3 Threshold dynamics In this section, we establish the thresh-old dynamics for the model system in terms of the basic reproductionnumber.

We define the “diseased classes” as the mosquito and human popu-lations that are either exposed or infectious, i.e., We, W2, E and I . Toget the disease-free equilibrium, letting We = W2 = E = I = 0, we thenget R = 0 and

dW1(t)

dt= re−aτA

[1 −

W1(t− τA)

K

]

+

W1(t− τA) − µwW1(t),(3.1)

dS(t)

dt= H − µhS(t).(3.2)

There are two disease free equilibria, E0 = (0, 0, 0, N∗, 0, 0, 0) and E1 =

(W ∗, 0, 0, N∗, 0, 0, 0), where W ∗ = K(re−aτA−µw)re−aτA

. By [20, Proposition4.1], for system (3.1), the equilibrium W ∗ is globally asymptoticallystable if the following condition is satisfied

(H1) µw < re−aτA ≤ 3µw.

Linearizing the model system at the disease free equilibrium E1, weobtain the following system (here we only write down the equations forthe diseased classes):

dWe(t)

dt= βw

W ∗

N∗I(t) − µwWe(t) − βwe

−µwτwW ∗

N∗I(t− τw),

dW2(t)

dt= βwe

−µwτwW ∗

N∗I(t− τw) − (µw + εw)W2(t),

dE(t)

dt= βhW2(t) − µhE(t) − βhe

−µhτhW2(t− τh),

dI(t)

dt= βhe

−µhτhW2(t− τh) − (µh + εh + γ)I(t).

Following the idea in [17], we introduce the basic reproduction numberfor the model system. Denote x1, x2, x3 and x4 be the number ofeach diseased class at time t = 0, and x1(t), x2(t), x3(t) and x4(t) bethe remaining populations of each class at time t, respectively, then weobtain

x2(t) = x2e−(µw+εw)t,

x4(t) = x4e−(µh+εh+γ)t.


The total number of newly infected in each diseased class is

x1 =

∫∞

0

βwW∗

N∗x4(t) dt =

βwW∗

N∗(µh + εh + γ)x4,

x2 =

∫∞

τw

βwe−µwτwW ∗

N∗x4(t− τw) dt =

βwe−µwτwW ∗

N∗(µh + εh + γ)x4,

x3 =

∫∞

0

βhx2(t) dt =βh

µw + εw

x2,

x4 =

∫∞

τh

βhe−µhτhx2(t− τh) dt =

βhe−µhτh

µw + εw

x2.

Since

x1

x2

x3

x4

=

0 0 0βwW

∗

N∗(µh + εh + γ)

0 0 0βwe

−µwτwW ∗


0βh

µw + εw

0 0

0βhe

−µhτh

µw + εw

0 0

·

x1

x2

x3

x4

,

we can see that the 4 × 4 matrix:

M0 =

0 0 0βwW

∗


0 0 0βwe

−µwτwW ∗


0βh

µw + εw

0 0

0βhe

−µhτh

µw + εw

0 0

is the next infection operator. As usual, we define the spectral radiusof the matrix M0 as the basic reproduction number R0 for the modelsystem. It then follows that

R0 =

√βwβhe−(µwτw+µhτh)W ∗

N∗(µw + εw)(µh + εh + γ).

Our first result shows that the disease is uniformly persistent if R0 > 1.


Theorem 3.1. Let (H1) hold. If R0 > 1, then there is an η > 0 such

that any solution u(t, φ) of the model system with φ ∈ Xδ, φ3(0) 6= 0 and

φ6(0) 6= 0 satisfies

lim inft→∞

(W2(t), I(t)) ≥ (η, η).

Proof. Define

X0 = {φ = (φ1, φ2, . . . , φ7) ∈ Xδ : φ3(0) 6= 0, and φ6(0) 6= 0}.

Clearly, we have

∂X0 = Xδ \X0 = {φ ∈ Xδ : φ3(0) = 0, or φ6(0) = 0}.

Define

M∂ = {φ ∈ Xδ : Φ(t)φ ∈ ∂X0, ∀t ≥ 0}.

Claim 1. There exists a δ1 > 0, such that for any φ ∈ X0,

lim supt→∞

‖ Φ(t)φ−E0 ‖≥ δ1.

Since µw < re−aτA , we can choose ε0 > 0 and δ1 > 0 sufficientlysmall, such that

x3

x1 + x2 + x3 + x4< ε0, ∀ |(x1, x2, x3, x4) − (N∗, 0, 0, 0)| < δ1,(3.3)

µw + βwε0 < re−aτA

(1 −

3δ1K

).(3.4)

For any φ ∈ Xδ, since φ3(0) 6= 0, and φ6(0) 6= 0, it follows from [12,

Theorem 5.2.1],

(3.5) W2(t) > 0, I(t) > 0, ∀t > 0.

Next we show that there exists a t0 ≥ 0 such that W1(t0, φ) > 0 for allφ ∈ X0. Otherwise, there exists ψ ∈ X0 such that W1(t, ψ) = 0 for allt ≥ 0. From (2.2), we get We(t) ≡ 0 for all t ≥ τw, then from (2.1), weget W2(t) ≡ 0, for all t ≥ τw, a contradiction with (3.5). Then, by [12,

Theorem 5.2.1], W1(t) > 0 for all t ≥ t0.


Suppose, by contradiction, that lim supt→∞‖ Φ(t)ψ − E0 ‖< δ1 for

some ψ ∈ X0. Thus, ‖ Φ(t)ψ − E0 ‖< δ1 holds for all large t. Then wecan choose large number t1 > t0 such that for all t ≥ t1, there holds that

dW1(t)

dt≥ re−aτA

(1 −

3δ1K

)W1(t− τA) − (µw + βwε0)W1(t).

Consider the next linear and monotone time-delayed system

(3.6)dw(t)

dt= re−aτA

(1 −

3δ1K

)w(t− τA) − (µw + βwε0)w(t).

Let λ0 be the principal eigenvalue of the corresponding eigenvalue prob-lem of equation (3.6). By [12, Corollary 5.5.2] and (3.4), it follows thatλ0 > 0. We can choose l > 0 small enough such that leλ0t ≤W1(t), ∀t ∈[t1, t1 + τA]. Clearly, leλ0t satisfies (3.6) for all t ≥ t1. Then by the com-parison principle, we get

leλ0t ≤W1(t), ∀t ≥ t1 + τA,

Since λ0 > 0 and l > 0, leλ0t → ∞ as t→ ∞. Thus, limt→∞W1(t) = ∞,a contradiction.

Claim 2. There exists a δ2 > 0, such that for any φ ∈ X0,

lim supt→∞

‖ Φ(t)φ−E1 ‖≥ δ2.

First we consider the following linear cooperative system

(3.7)

dW2(t)

dt= βwe

−µwτw

(W ∗

N∗− ε

)I(t− τw) − (µw + εw)W2(t),

dI(t)

dt= βhe

−µhτh(1 − ε)W2(t− τh) − (µh + εh + γ)I(t).

For sufficiently small ε > 0, let λ1(ε) be the principle eigenvalue ofsystem (3.7). Since R0 > 1, it is easy to see from [12, Corollary 5.5.2]that λ1(0) > 0. Thus, we can restrict ε small enough such that λ1(ε) > 0.For this small ε, there exists δ2 = δ2(ε) > 0 such that

x4

x4 + x5 + x6 + x7> 1 − ε > 0 and

x1 + x2 + x3

x4 + x5 + x6 + x7>W ∗

N∗− ε > 0, ∀ |(x1, x2, · · · , x7) −E1| < δ2.


Assume, by contradiction, that lim supt→∞‖Φ(t)φ − E1‖ < δ2 for

some φ ∈ X0. Then there exists a large number t2, such that for allt ≥ t2,

‖Φ(t)φ− (W ∗, 0, 0, N∗, 0, 0, 0)‖ < δ2.

For any ε > 0, we can further choose t3 > t2 large enough, such that forall t ≥ t3,

W (t)

N(t)≥W ∗

N∗− ε,

S(t)

N(t)≥ 1 − ε.

That is, when t ≥ t3, we have

dW2(t)

dt≥ βwe

−µwτw

(W ∗

N∗− ε

)I(t− τw) − (µw + εw)W2(t),

dI

dt≥ βhe

−µhτh(1 − ε)W2(t− τh) − (µh + εh + γ)I(t).

Let v = (v1, v2)T be the positive right eigenvector associated with

λ1(ε) for system (3.7), choose l > 0 small enough such that

lv1eλ1(ε)t ≤W2(t), ∀t ∈ [t3, t3 + τ ],

lv2eλ1(ε)t ≤ I(t), ∀t ∈ [t3, t3 + τ ],

Clearly, leλ1(ε)t(v1, v2)T satisfies (3.7) for t ≥ t3. Then by the compari-

son principle, we get

(W2(t), I(t)) ≥ leλ1(ε)t(v1, v2), ∀t ≥ t3 + τ.

Since λ1(ε) > 0, letting t→ ∞, we obtain

limt→∞

W2(t) = ∞, limt→∞

I(t) = ∞,

a contradiction.Let ω(φ) be the omega limit set of the orbit of Φ(t) through φ ∈ Xδ .

Claim 3.⋃

φ∈M∂ω(φ) = E0 ∪ E1.

For any φ ∈ M∂ , i.e., Φ(t)φ ∈ ∂X0, we have W2(t, φ) ≡ 0, orI(t, φ) ≡ 0. If W2(t, φ) ≡ 0, then from the equations of S, E and I , wehave limt→∞ S(t, φ) = N∗, limt→∞E(t, φ) = 0 and limt→∞ I(t, φ) = 0.By the theory of asymptotically autonomous semiflows (see, e.g., [14]),


it follows that limt→∞W1(t, φ) = W ∗ or 0, limt→∞We(t, φ) = 0 andlimt→∞R(t, φ) = 0.

If W2(t, φ) 6≡ 0, then there exists t0 ≥ 0, such that W2(t0, φ) >0. We then obtain that W2(t, φ) > 0 for all t ≥ t0, and I(t, φ) ≡0. From the equations of We, W2 and R, we have limt→∞We(t, φ) =0, limt→∞W2(t, φ) = 0, and limt→∞R(t, φ) = 0. By the theory ofasymptotically autonomous semiflows, we get limt→∞W1(t, φ) = W ∗ or0, limt→∞E(t, φ) = 0 and limt→∞ S(t, φ) = N∗. Consequently, we have⋃

φ∈M∂ω(φ) = E0 ∪ E1.

Define a continuous function p : Xδ → R+ by

p(φ) = min{φ3(0), φ6(0)}, ∀φ ∈ Xδ .

Clearly, p−1(0,∞) ⊂ X0. It follows from (3.5) that p has the propertythat if either p(φ) = 0 and φ ∈ X0, or p(φ) > 0, then p(Φ(t)φ) > 0,for all t > 0. Thus p is a generalized distance function for the semiflowΦ(t) : Xδ → Xδ . By Claim 3, we get that any forward orbit of Φ(t) in M∂

converges to E0 or E1, by Claims 1 and 2, we conclude that E0 and E1

are two isolated invariant sets in Xδ , and (W s(E0)∪Ws(E1))∩X0 = ∅.

Moreover, it is easy to see that no subset of {E0, E1} forms a cycle in∂X0. By [13, Theorem 3], it then follows that there exists η > 0 suchthat lim inft→∞ p(Φ(t)φ) ≥ η for all φ ∈ X0, which implies the uniformpersistence stated in the theorem.

The subsequent result shows that the disease dies out if R0 < 1,provided there is only a small invasion in the W2 and I classes. For anygiven M > 0, denote

XMδ =

{φ ∈ C([−τ, 0], [0,M ]7) :

7∑

i=4

φi(s) ≥ δ, ∀s ∈ [−τ, 0],

φ2(0) =

∫ 0

−τw

eµwsβw

φ6(s)φ1(s)∑7i=4 φi(s)

ds,

φ5(0) =

∫ 0

−τh

eµhsβh

φ4(s)φ3(s)∑7i=4 φi(s)

ds

}.

Then we have the following result.

Theorem 3.2. Let (H1) hold. If R0 < 1, then for every M > max{

Hµh,

re−aτAK4µw

}, there exists a ζ = ζ(M) > 0 such that for any φ ∈ XM

δ \E0

with (φ3(s), φ6(s)) ∈ [0, ζ]2 for all s ∈ [−τ, 0], the solution u(t, φ) of the

model system through φ satisfies limt→∞ ‖u(t, φ) −E1‖ = 0.


Proof. Let M > max{ Hµh, re−aτA K

4µw} be given. From the prove of Theo-

rem 2.1, we see that XMδ is positively invariant for the solution semiflow

of the model system. We then have

u(t, φ) ∈ [0,M ]7, ∀t ≥ 0, φ ∈ XMδ .

Consider the following linear and monotone system

(3.8)

dW2(t)

dt= βwe

−µwτw

(W ∗ + ε

N∗ − ε

)I(t− τw) − (µw + εw)W2(t),

dI(t)

dt= βhe

−µhτhW2(t− τh) − (µh + εh + γ)I(t).

For sufficiently small ε > 0, let λ2(ε) be the principle eigenvalue of thiseigenvalue problem. Since R0 < 1, it is easy to see from [12, Corollary5.5.2] that λ2(0) < 0. Thus, we can restrict ε small enough such thatλ2(ε) < 0. Let (e1, e2)

T be positive right eigenvector associated withλ2(ε).

Now we consider the following equations:

dW

dt= re−aτA

[1 −

W (t− τA)

K

]

+

W (t− τA) − µwW (t) − εwξ1,(3.9)

dN

dt= H − µhN(t) − εhξ1.(3.10)

Choose small ξ1 > 0 and large T = T (M) > 0 such that for any solutionsof (W (t, φ), N (t, φ)). We then have

W (t) < W ∗ + ε, N(t) > N∗ − ε, ∀t ≥ T.

Denote the solution of system (3.8) by u(t, φ) = (W2(t), I(t)) with re-spect to initial data φ = (φ1, φ2) ∈ C([−τ, 0], [0,M ]2). Then for system(3.8), for ξ1 > 0, there exists ξ2 > 0, such that if (ξ2e1, ξ2e2) � (ξ1, ξ1).Since λ2(ε) < 0, we get that

(3.11) (ξ2e1eλ2(ε)t, ξ2e2e

λ2(ε)t) � (ξ1, ξ1)), ∀t ≥ 0.

For every solution of the model system through φ, there exists a ζ =ζ(M) > 0 such that

(3.12) (W2(t, φ), I(t, φ)) � (ξ1, ξ1), ∀t ∈ [0, T1]


provided that (φ3(s), φ6(s)) < (ζ, ζ).We further claim that (3.12) holds for all t ≥ 0. Suppose, by contra-

diction, that the claim is not true. Then there exists a T2 = T2(φ) > T1

such that (W2(t, φ), I(t, φ)) � (ξ1, ξ1), for all t ∈ [0, T2), andW2(T2, φ) =ξ1 or I(T2, φ) = ξ1. By the comparison principle, for t ∈ [T1, T2], we have

(3.13) (W2(t, φ), I(t, φ)) ≤ (W2(t, φ), I(t, φ)) � (ξ1, ξ1)

a contradiction. So (3.12) holds for all t ≥ 0. From (3.11) and (3.13), wesee that limt→∞(W2(t, φ), I(t, φ)) = (0, 0). By the theory of asymptot-ically autonomous semiflows (see [14]), it follows that limt→∞ u(t, φ) =(W ∗, 0, 0, N∗, 0, 0, 0).

4 Global attractivity In this section, we study the global attrac-tivity in the model system in the case where the disease-induced deathrates of infected mosquitos and human individuals are zero, and the fe-cundity of infected mosquitos is the same as the susceptible mosquitos.In this case, the model system becomes

(4.1)

dW1(t)

dt= re−aτA

[1 −

W (t− τA)

K

]

+

W (t− τA)

− µwW1(t) − βw

I(t)

N(t)W1(t)

dWe(t)

dt= βw

I(t)

N(t)W1(t) − µwWe(t)

− βwe−µwτw

I(t− τw)

N(t− τw)W1(t− τw),

dW2(t)

dt= βwe

−µwτwI(t− τw)

N(t− τw)W1(t− τw) − µwW2(t),

dS(t)

dt= H − µhS(t) − βhW2(t)

S(t)

N(t),

dE(t)

dt= βhW2(t)

S(t)

N(t)− µhE(t)

− βhe−µhτhW2(t− τh)

S(t− τh)

N(t− τh).

dI(t)

dt= βhe

−µhτhW2(t− τh)S(t− τh)

N(t− τh)− (µh + γ)I(t),

dR(t)

dt= γI(t) − µhR(t).


It is clear that when R0 < 0, system (4.1) has only two equilibriaE0 and E1. However, system (4.1) admits a unique positive equilibriumE∗ := (W ∗

1 ,W∗

e ,W∗

2 , S∗, E∗, I∗, R∗) when R0 > 1, where

S∗ =H(µhβw + µw(µh + γ)eµhτh)

µh(µhβw + µw(µh + γ)eµhτhR20),

I∗ =Hµw(R2

0 − 1)

µhβw + µw(µh + γ)eµhτhR20

,

and

W ∗

2 =(µh + γ)N∗eµhτhI∗

βhS∗,

W ∗

1 =µwN

∗eµwτwW ∗

2

βwI∗,

W ∗

e = (eµwτw − 1)W ∗

2 ,

E∗ =(eµhτh − 1)(µh + γ)I∗

µh

,

R∗ =γI∗

µh

.

The following two results show the global attractivity of system (4.1).

Theorem 4.1. Let (H1) hold and assume that σ = 1 and εw = εh =0. If R0 < 1, then the disease-free equilibrium of the model system is

globally attractive in Xδ \E0.

Proof. If σ = 1 and εw = εh = 0, then the whole mosquitos and humanpopulations satisfy the following two equations:

dW (t)

dt= re−aτA

[1 −

W (t− τA)

K

]

+

W (t− τA) − µwW (t),

dN(t)

dt= H − µhN(t).

Since W ∗ and N∗ is globally asymptotically stable for the above twoequations, respectively, there exists T = T (ε) > 0 such that

W (t) ≤W ∗ + ε, N(t) ≥ N∗ − ε, ∀t ≥ T.


Thus, when t ≥ T , we have

dW2(t)

dt≤ βwe

−µwτw

(W ∗ + ε

N∗ − ε

)I(t− τw) − µwW2(t),

dI

dt≤ βhe

−µhτhW2(t− τh) − (µh + γ)I(t).

When R0 < 1, ε small enough, by the analysis of system (3.8) and thecomparison principle, we then have

limt→∞

(W2(t), I(t)) = (0, 0).

It then follows from the theory of asymptotically semiflows (see [14])that

limt→∞

(W1(t),We(t), S(t), E(t), R(t)) = (W ∗, 0, N∗, 0, 0).

This completes the proof.

To obtain the global attractivity of the endemic equilibrium, we needthe following additional assumption:

(H2) βwµh ≥ µw(µh + γ)eτhµh .

Theorem 4.2. Let (H1) and (H2) hold and assume that σ = 1 and

εw = εh = 0. If R0 > 1, then for any φ ∈ Xδ with φ3(0) 6= 0, φ6(0) 6= 0,we have limt→∞ u(t, φ) = E∗.

Proof. When εw = εh = 0, σ = 1, we have

(4.2)

dW (t)

dt= re−aτA

[1 −

W (t− τA)

K

]

+

W (t− τA) − µwW (t),

dN(t)

dt= H − µhN(t).

When (H1) holds, (W ∗, N∗) is globally asymptotically stable for sys-


tem (4.2). Hence, we have the following limiting system:

(4.3)

dW1(t)

dt= A− µwW1(t) − β′

wI(t)W1(t),

dWe(t)

dt= β′

wI(t)W1(t) − µwWe(t)

− β′

we−µwτwI(t− τw)W1(t− τw),

dW2(t)

dt= β′

we−µwτwI(t− τw)W1(t− τw) − µwW2(t),

dS(t)

dt= H − µhS(t) − β′

hW2(t)S(t),

dE(t)

dt= β′

hW2(t)S(t) − µhE(t)

− β′

he−µhτhW2(t− τh)S(t− τh),

dI(t)

dt= β′

he−µhτhW2(t− τh)S(t− τh) − (µh + γ)I(t),

dR(t)

dt= γI(t) − µhR(t),

where A = W ∗µw, β′

w = βw/N∗, β′

h = βh/N∗.

Let g(t − τw) = W1(t − τw) + eµwτwW2(t), that is, g(t) = W1(t) +eµwτwW2(t+ τw). It follows that

g′(t) = W ′

1(t) + eµwτwW ′

2(t+ τw)

= A− µw(W1(t) + eµwτwW2(t+ τw))

= A− µwg(t).

Then the equilibrium A/µw = W ∗ is globally asymptotically stable. Forsystem (4.3), we then consider the following limiting system:

dW 2(t)

dt= β′

we−µwτw I(t− τw)(W ∗ − eµwτwW 2(t)) − µwW 2(t),(4.4)

dS(t)

dt= H − µhS(t) − β′

hS(t)W 2(t),(4.5)

dI(t)

dt= β′

he−µhτh S(t− τh)W 2(t− τh) − (µh + γ)I(t).(4.6)


Claim 4. The set D := C([−τ, 0], [0,W ∗e−µwτw ]×R2+) is positively in-

variant for system (4.4)–(4.6).

To prove this claim, we define

F (ψ) :=

β′

we−µwτwψ3(−τw)(W ∗ − eµwτwψ1(0)) − µwψ1(0)

H − µhψ2(0) − β′

hψ1(0)ψ2(0)

β′

he−µhτhψ1(−τh)ψ2(−τh)) − (µh + γ)ψ3(0)

, ∀ψ ∈ D.

Note that D is relatively closed in C([−τ, 0],R3), and F (ψ) is continuousand 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 (0, ψ) on its maximal interval of existence. Since Fi(ψ) ≥ 0whenever ψ ∈ D with ψi(0) = 0, [12, Theorem 5.2.1] implies that thesolution of (4.4)–(4.6) are nonnegative for all t in its maximal intervalof existence. Furthermore, if ψ1(0) = W ∗e−µwτw , then F1(ψ) ≤ 0. Itfollows by [12, Remark 5.2.1] that W2(t, ψ) ≤ W ∗e−µwτw for all t > 0.Thus, D is positively invariant.

By the arguments similar to those in Theorem 3.1, it easily followsthat system (4.4)–(4.6) is uniformly persistent in the sense that thereexists a η1 > 0 such that for any given ψ = (ψ1, ψ2, ψ3) ∈ D withψ1(0) 6= 0, ψ3(0) 6= 0, the solution (W 2(t, ψ), S(t, ψ), I(t, ψ)) of (4.4)–(4.6) satisfies

lim inft→∞

(W 2(t, ψ), I(t, ψ)) ≥ (η1, η1).

For any given ψ ∈ D with ψ1(0) 6= 0 and ψ3(0) 6= 0, let (W 2(t), S(t),I(t)) = (W 2(t, ψ), S(t, ψ), I(t, ψ)). In order to use the method of fluctu-ations (see, e.g., [8, 18]) for system (4.4)–(4.6), we define

W∞

2 = lim supt→∞

W 2(t), W 2∞ = lim inft→∞

W 2(t);

S∞ = lim supt→∞

S(t), S∞ = lim inft→∞

S(t);

I∞ = lim supt→∞

I(t), I∞ = lim inft→∞

I(t).

Clearly, W∞

2 ≥ W 2∞ ≥ η1 > 0, S∞ ≥ S∞ and I∞ ≥ I∞ ≥ η1 > 0.Further, there exist sequences tin → ∞ and σi

n → ∞, i = 1, 2, 3, such


that

limn→∞

W 2(t1n) = W

∞

2 , W′

2(t1n) = 0, ∀n ≥ 1;

limn→∞

W 2(σ1n) = W 2∞, W

′

2(σ1n) = 0, ∀n ≥ 1;

limn→∞

S(t2n) = S∞, S′(t2n) = 0, ∀n ≥ 1;

limn→∞

S(σ2n) = S∞, S′(σ2

n) = 0, ∀n ≥ 1;

limn→∞

I(t3n) = I∞, I ′(t3n) = 0, ∀n ≥ 1;

limn→∞

I(σ3n) = I∞, I ′(σ3

n) = 0, ∀n ≥ 1.

Let m1 = βwe−µwτwW ∗/N∗ and m2 = βhe

−µhτh/N∗. It then followsfrom (4.4) and the above claim that

I∞(m1 − β′

wW∞

2 ) − µwW∞

2 ≥ 0 ≥ I∞(m1 − β′

wW∞

2 ) − µwW∞

2 ,

I∞(m1 − β′

wW 2∞) − µwW 2∞ ≥ 0 ≥ I∞(m1 − β′

wW 2∞) − µwW 2∞,

and hence,

I∞ ≥µwW

∞

2

m1 − β′wW

∞

2

≥µwW 2∞

m1 − β′wW 2∞

≥ I∞.(4.7)

By (4.5), we have

H − S∞(µh + β′

hW 2∞) ≥ 0 ≥ H − S∞(µh + β′

hW∞

2 ),

H − S∞(µh + β′

hW 2∞) ≥ 0 ≥ H − S∞(µh + β′

hW∞

2 ),

which implies that

H

µh + β′

hW 2∞

≥ S∞ ≥ S∞ ≥H

µh + β′

hW∞

2

.(4.8)

In view of (4.6), we obtain

m2S∞W

∞

2 − (µh + γ)I∞ ≥ 0 ≥ m2S∞W 2∞ − (µh + γ)I∞,

m2S∞W

∞

2 − (µh + γ)I∞ ≥ 0 ≥ m2S∞W 2∞ − (µh + γ)I∞,


and hence,

m2S∞W

∞

2

µh + γ≥ I∞ ≥ I∞ ≥

m2S∞W 2∞

µh + γ.(4.9)

Therefore, combining (4.8) and (4.9) together, we get

H

µh + β′

hW 2∞

m2W∞

2

µh + γ≥ I∞ ≥ I∞ ≥

H

µh + β′

hW∞

2

m2W 2∞

µh + γ.(4.10)

Comparing (4.7) with (4.10), we obtain

H

µh + β′

hW 2∞

m2W∞

2

µh + γ≥

µwW∞

2

m1 − β′wW

∞

2

;

H

µh + β′

hW∞

2

m2W 2∞

µh + γ≤

µwW 2∞

m1 − β′wW 2∞

.

Simplifying the above two inequalities, we get

βwµh(W∞

2 −W 2∞) ≤ µw(µh + γ)e−τhµh(W∞

2 −W 2∞)

Since condition (H2) holds, we have W∞

2 = W2∞. By (4.8) and (4.10),we get S∞ = S∞ and I∞ = I∞. It follows that limt→∞(W 2(t), S(t),I(t)) = (W ∗

2 , S∗, I∗) for any ψ ∈ D with ψ1(0) 6= 0 and ψ3(0) 6= 0.

By the theory of chain transitive sets (see, e.g., [19, Section 1.2.1])and the arguments similar to [10, Appendix A] (see also [17, Theorem2.1]), we can lift the global attractivity for system (4.4)-(4.6) to themodel system. It follows that limt→∞ u(t, φ) = E∗, for any φ ∈ Xδ withφ3(0) 6= 0 and φ6(0) 6= 0.

5 Numerical simulations In this section, we carry out numericalsimulations to illustrate our analytic results.

In view of [15], we fix τA = 10, τw = 10, τh = 5, and then take threesets of values of other parameters to perform the numerical simulations.

First, we take βw = 0.06, βh = 0.15, r = 1, a = 0.2, γ = 0.15,µw = 0.1, µh = 0.0001, H = 0.001, K = 10, σ = 0.8, εw = 0.01, εh =0.0001. It is easy to verify that condition (H1) holds, and R0 = 0.174,W ∗ = 2.61, N∗ = 10. It follows from Theorem 3.2 that when W2(s) andI(s), s ∈ [−τ, 0], are small, the disease will die out (see Figure 1).


0 100 200 300 4000

2

4

a: W1

0 50 1000

0.5

1x 10−3 b: W

e

0 50 1000

1

2x 10−4 c: W

2

0 2 4 6

x 104

6

8

10d: S

0 50 1000

0.5

1x 10−4 e: E

0 50 1000

1

2x 10−4 f: I

FIGURE 1: Long-term behavior of the population of each class whenR0 < 1 and the invasion is small.

0 100 200 3000

2

4

a: W1

0 100 200 3000

2

4

b: We

0 100 200 3000

2

4

c: W2

0 100 200 3000

2

4d: S

0 100 200 3000

1

2e: E

0 100 200 3000

2

4f: I

FIGURE 2: E∗ is globally asymptotically attractive when R0 > 1 and

conditions (H1) and (H2) hold.


0 100 200 300 400 5000

1

2

3

a: W2

0 100 200 300 400 5000

1

2

3b: I

FIGURE 3: Persistence of infected mosquitos and human individuals.

Second, we take βw = 0.9, βh = 0.5, r = 1, a = 0.4, γ = 0.05, µw =0.01, µh = 0.001, H = 0.1, K = 10, σ = 1, εw = εh = 0. Then we getthat (H1) and (H2) hold, and R0 = 17.509, W ∗ = 4.540, N∗ = 10. ByTheorem 4.2, we obtain E∗ = (1.655, 0.275, 2.610, 0.076, 0.049, 0.1936,9.681) is globally attractive (see Figure 2).

Third, by taking βw = 0.9, βh = 0.5, r = 1, a = 0.4, γ = 0.05,µw = 0.01, µh = 0.001, H = 0.1, K = 10, σ = 0.8, εw = 0.01, εh =0.0001, we get R0 = 11.462. We can see that the disease is uniformpersistence. Figure 3 indicates the behavior of the infectious mosquitosand infectious human population.

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17. Z. Xu and X.-Q. Zhao, A vector-bias malaria model with incubation period anddiffusion, Discrete Contin. Dyn. Syst. Ser. B 17 (2012), 2015–2034.

18. Y. Yuan and X.-Q. Zhao, Global stability for non-monotone delay equations(with application to a model of blood cell production), J. Differential Equations252 (2012), 2189–2209.

19. X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, NewYork, 2003.

20. X.-Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusionequations with time delay, Canad. Appl. Math. Quart. 17 (2009), 271–281.

Department of Mathematics and Statistics,

Memorial University of Newfoundland,

St. John’s, NL A1C 5S7, Canada.

E-mail address: [email protected]

E-mail address: [email protected]

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GLOBAL DYNAMICS OF A TIME-DELAYED DENGUE … · The purpose of this paper is to study the global dynamics of a time-delayed dengue transmission model. In Section 2, we present the