Global asymptotic stability of an SIS epidemic model with variable population size and a delay
Jaafar El Karkri
∗and Khadija Niri
Department of Mathematics and Computer Science, Laboratory MACS,
Faculty of Sciences Ain Choq, Hassan II University,
Km 8 Route d’El Jadida, B.P 5366 Maarif,
Casablanca 20100, Morocco
Email: professionnelcpge@gmail.com Email: khniri@yahoo.fr
∗Corresponding author
Abstract: An SIS epidemiological model with an exponential demographic structure and a delay corresponding to the infectious period is studied. We derive sufficient conditions for the global asymptotic stability of the infected steady state. The study is mainly based on very recent results of the monotone dynamical systems theory rarely used in mathematical epidemiology. We propose an improved and practical version of an important result in the theory.
The obtained results are a partial - but important - resolution of the open problem proposed by Hethcote and van den Driessche in 1995. Numerical simulations are conducted to demonstrate our theoretical results.
Keywords: delay differential equation; essentially strongly order-preserving semi-flows; global asymptotic stability; monotone dynamical systems; SIS epidemic model.
Referenceto this paper should be made as follows: El Karkri, J. and Niri, K.
(2017) ‘Global asymptotic stability of an SIS epidemic model with variable population size and a delay’, Int. J. Dynamical Systems and Differential Equations, Vol. 7, No. 4, pp.289–300.
Biographical notes:Jaafar EL Karkri is a Professor of Mathematics at “CRMEF of Rabat” and a PhD student at the University of Hassan II. His research interests lie in the area of monotone dynamical systems theory and mathematical epidemiology.
Khadija Niri is a Professor for Dynamical systems at the University of Hassan II.
Her research interest lies in the analysis of delay differential equations, especially of monotone type, and of epidemiological models that are cooperative.
1 Introduction
In recent years, many studies have been conducted on the asymptotic behaviour of delayed systems arising from compartmental epidemiological models. The basic approach for the global asymptotic stability analysis of a delayed system is based on the well-known Copyright c2017 Inderscience Enterprises Ltd.
Lyapunov direct method (Ma and Li, 2009). The advantage of Lyapunov direct method is that sharp conditions may be found in many cases despite its difficult applicability on many models. In these cases, the monotone dynamical systems theory can be very helpful.
Rarely used in mathematical epidemiology, monotone approach is a powerful and efficient tool to analyse the dynamics of mathematical models governed by delay differential equations. This approach has been early applied to ordinary differential equations in the pioneering works of Muller (1927) and Kamke (1932), then in an excellent series of articles entitled “Systems of differential equations that are competitive or cooperative”, parts I through VI, Hirsch introduced the main tools of what is now often referred to as monotone dynamical systems theory (Hirsch, 1982, 1985, 1984, 1988a, 1988b, 1989, 1991). In 1984, Matano developed in Matano (1984) the key concept of strongly order preserving semi-flows which would play a crucial role in the infinite dimensional theory of monotone systems. Then, in 1990, a synthesis of Hirch and Matano’s approaches had been especially covered in a very fundamental work of Smith and Thieme (1990a, 1990b, 1991a, 1991b). There, the ideas of Matano and Hirch were simplified, streamlined and enhanced. In brief, under an additional compactness assumption (T), the typical orbit of a strongly order preserving semi-flow converges to the set of equilibriums.
Assumption (T) is in many situations difficult to be proved (see El Karkri and Niri (2014) for further details).
Recently, a very practical approach for monotone systems has been developed by Yi and Huang (2006, 2008), Yi and Zou (2009a, 2009b). Yi studies the class of essentially strongly order preserving semi-flows. This condition is more flexible than the idea of strongly order preserving semi-flows used by Smith and Thieme. Furthermore, this innovative method does not necessitate the “delicate” compactness assumption (T) used in the classical approach.
The SIS epidemiological model, under this study, was suggested in 1995 by Hethecote and van den Driessche (1995); the study involves both a variable population size and a delay corresponding to the infectious period. The model’s dynamics is described by a scalar differential equation with constant delay. Recently, have proved in Iggidr et al.
(2010) that under the quasi-monotonicity hypothesis, the model presents slow fluctuations around the endemic disease equilibrium. Our goal in this paper is to solve partially the open problem of the endemic equilibrium’s global asymptotic stability proposed at the end of Hethecote and van den Driessche (1995). To the best of our knowledge, since 1995, no sufficient conditions for global stability of the endemic equilibrium have been proposed for this epidemiological model. The prime outcome of the article is that when θ >2, the global asymptotic stability of the endemic equilibrium holds onΩ. We hope that our work contributes to the construction of a framework to apply monotone dynamical systems theory in Mathematical Epidemiology.
Our paper is organised as follows. In the next section, the main tools of essentially strongly order preserving semi-flows for differential equations with constant delay are set out. Principal properties of the model are presented in Section 3. The focus of Section 4 is on the main result of the paper, sufficient condition of global asymptotic stability for the endemic disease equilibrium whenθ >2. In Section 5, numerical simulations are given.
We end the paper with a summary and a discussion.
2 Preliminaries
For ω >0, let C=C0([−ω,0],R) denote the Banach space of continuous function mapping[−ω,0]intoRwith the maximum norm.
Let X be an open subset of C and f :X →Rcontinuous and lipschitzian in every compact subset of X .
Consider the autonomous functional differential equation (x′(t) =f(xt) t≥0
x(t) =ϕ(t) −ω≤t≤0 (2.1)
whereϕ∈Cis the initial condition and for allt≥0and−ω≤θ≤0:xt(θ) =x(t+θ).
For the existence and uniqueness of solutions, we have the Theorem bellow taken from Hale and Verduyn Lunel (1993) where detailed proofs are given.
Theorem 2.1: Let X be an open subset ofCand suppose thatf :X →Rbe continuous and lipschitzian in every compact subset of X. Ifϕ∈X, then problem (2.1) has a unique solutionx(., ϕ) =xϕdefined fort≥0.
Definition 2.1: x∗∈Ris said to be an equilibrium of the delay differential equation (2.1) if the constant functionˆx:t7−→x∗is a solution of (2.1) .
It is clear thatx∗is an equilibrium of (2.1) if and only if[˜x:t∈[−ω,0]7−→x∗]∈Dand f(x) = 0.e
The set of equilibriums of (2.1) will be denoted by E.
Consider the coneK=R+inR.
The coneC([−ω,0], K)generates a partial order on C.
LetQandCdenote the sets of quasi convergent and convergent points, respectively.
Throughout the remainder of this paper, we consider the autonomous delay differential equation (2.1), but we assume thatf : Ω→Ris Lipschitz andΩ⊂Cis the closure of an open subsetΠ⊆C.
We assume in all this section - as pointed out in Yi and Huang (2008)- that a solution xϕof the delay differential equation (2.1) withϕ∈Ωexists fort∈R+and is unique.
For readers’ convenience, we begin by recalling the three important assumptions defined in Yi and Huang (2008)(Section 2.3) adapted to our context( n = 1 and K=R+).
We haveK∗={λ∈R/(∀x∈K=R+)λ(x)≥0}= [0,+∞[.
Assumption (P) “For λ∈K∗= [0,+∞[ there exists a continuous mapping αλ: Ω×Ω−→Rsuch thatλ(f(ψ)−f(ϕ))≥αλ(ϕ, ψ)λ(ψ−ϕ) (0)”.
Assumption (I) “Suppose that ψ, ϕ∈Ω with ψ≥ϕ. Denote D=
λ∈K∗/ λ(ψ(θ)−φ(θ))>0, θ∈[−ω,0] .
If D6=∅ andD6=K∗\ {0}, then there exists λ∈K∗\D such that either λ(ψ(0)− φ(0))>0orλ(f(ψ)−f(φ))>0”.
Assumption (T D1) “f maps bounded subsets of Ωto bounded subsets of R, and the positive semi-orbit of every solution of (2.1) is bounded”’.
We recall the fundamental result of Section 3.2 in Yi and Huang (2008).
Theorem 2.2: Assume thatf satisfies the assumptions (P),(I) and (T D1). ThenΩ contains an open and dense subset of stable quasi-convergent points.
Our next goal is to provide a new, simpler and practical version of Theorem 2.2 which will be useful for our main result. To this end, we first introduce the following assumptions.
Assumption(P′)“Forλ≥0, there exists a continuous mappingαλ: Ω×Ω−→Rsuch that for allψ≥ϕinΩλ.(f(ψ)−f(ϕ))≥αλ(ϕ, ψ)λ.(ψ−ϕ) (0)”
Assumption(SD)“There exists a continuous mappingα: Ω×Ω−→Rsuch that for all ψ≥ϕinΩ (f(ψ)−f(ϕ))≥α(ϕ, ψ) (ψ−ϕ) (0)”.
The main result of this section is the following.
Theorem 2.3: Iff satisfies the assumptions(SD)and(T D1), thenΩcontains an open and dense subset of stable convergent points.
If (2.1) has a unique equilibrium, then it attracts all solutions.
If (2.1) has two equilibria, then each solution tends to one of these.
Proof: Recall that n=1 andK=R+.
A careful reader will no doubt observe thatAssumption(P)has been used in the proof of Theorem 3.4. in Yi and Huang (2008) just whenψ≥ϕ, and hence(P)can easily be replaced in Theorem 2.2 by(P′).
Sinceλ≥0we have,
(f(ψ)−f(ϕ))≥α(ϕ, ψ) (ψ−ϕ) (0)⇐⇒(∀λ≥0) λ(f(ψ)−f(ϕ))≥α(ϕ, ψ)λ(ψ−ϕ) (0)
Then(P′)⇐⇒(SD).
Hence, when n=1 and K=R+, in Theorem 2.2, one can replace condition (P)by condition(SD).
On the other hand, in assumption (I), we haveD=
λ≥0/λ(ψ(θ)−φ(θ))>0, θ∈[−ω,0] .
IfD6=∅, thenD=]0,+∞[=K∗.
Thus either we haveD=∅orD=]0,+∞[=K∗− {0}.
Thus Hypothesis “D6=∅andD6=K∗− {0}” is always false, and then assumption (I)is always fulfilled in our context(n= 1and K =R+).
Sincef satisfies the assumptions(SD)and(T D1), conditions(P′),(I)and(T D1) hold, and consequently Theorem 2.2 (after replacing (P) by (P′)) applies and then Ω contains an open and dense subset of stable quasi-convergent points.
From Remark 4.2. of Chapter 1 in Smith (1995), for scalar delay differential equations, all quasi-convergent points are convergent , consequently - under hypothesis of Theorem 2.3-Ωcontains an open and dense subset of stable convergent points.
The last two results follow in the familiar way from Theorems 2.3.1 and 2.3.2 in Smith
(1995).
3 The model
The dynamics of the epidemiological model studied here are described schematically in Figure 1. The reader is referred to Hethecote and van den Driessche (1995), Iggidr et al.
(2010) for further details.
The total population of sizeN(t)is divided into two disjoint classes that are susceptible and infective with class sizes denoted at time t byS(t)andI(t),respectively.
The population has an exponential demographic structure.
That isN(t) =N(0) exp{(b−d)t}wherebandddenote, respectively, birth rate and natural death rate.
β >0is the contact rate, it is the average number of adequate contacts of a susceptible per unit time.
ThenβSIN individual leaves the susceptible class per unit time.
We assume that the mean length of infection is equal to a constantω >0andP(t)is a step function, namely,P(t) = 1on[0, ω]and vanishes elsewhere.
We normalize the model by introducing, respectively, the proportions of susceptible and infective individuals in the populations(t) = N(t)S(t) andi(t) = N(t)I(t).
Figure 1 The model (see online version for colours)
Hethecote and van den Driessche (1995) proved that the disease transmission is governed by the autonomous delay differential equation
i′(t) = [β−b−βi(t)]i(t)−β[1−i(t−ω)]i(t−ω) exp{−bω} (3.1)
(3.1) is equivalent to
x′(t) =F(xt) t≥0
x(t) =ϕ(t) −ω≤t≤0 (3.2)
where F(ϕ) =βg(ϕ(0))−βg(ϕ(−ω)) exp{−bω} −bϕ(0) for all ϕ∈C([−ω,0],R) withg(x) =x(1−x)for allx∈R.
An equilibrium of an epidemic system is said to be endemic if it corresponds to a state with infective individuals.
For allϕ∈C([−ω,0],[0,1])Eq. (3.2) has a unique continuous solution fort≥0, by arguments similar to those used in the proof of Theorem 3.1 in El Karkri and Niri (2014).
In Niri and El Karkri (2015), a detailed local asymptotic stability analysis of the model based on the monotone dynamical systems theory leads to the theorem below.
Theorem 3.1: Putθ=β(1−e−
bω)
b .
Equation (3.1) has always a disease-free equilibrium.
Ifθ >1, then (3.1) has an endemic disease equilibriumi∗= 1−1θ.
Ifθ <1, then the disease-free equilibrium of Eq. (3.1) is locally asymptotically stable.
Ifθ >1, then the disease-free equilibrium of Eq. (3.1) is unstable , and the endemic disease equilibriumi∗is locally asymptotically stable.
It is proved in Hethecote and van den Driessche (1995) that whenθ <1, the disease-free equilibrium of (3.1) is globally asymptotically stable.
Recently, Niri et al. proved in Iggidr et al. (2010) the following proposition and theorem.
Proposition 3.1: Assume thatθ >2.
1 The endemic equilibriumi∗satisfies 12 < i∗<1.
2 For allϕ∈Csuch that 12 ≤ϕ≤1, we have(∀t≥0) 12 ≤x(t, ϕ)≤1.
PutΠ =C [−ω,0],1
2,1 .
As a consequence, under the assumptionθ >2,Ω = Πis a positively invariant subset of C.
Theorem 3.2: Ifθ >2, then the system (3.1) is monotone onΩ.
That is, for each pair of initial conditions ϕand ψ inΩ, ifϕ(t)≤ψ(t)for every t∈[−ω,0], thenxϕ(t)≤xψ(t)for allt≥0.
4 Global asymptotic stability of the endemic disease equilibriumi∗onΩ =D Now we consider again the delay differential equation (3.2) studied in Section 3.
In all this section we assume thatθ=β(1−be−bω)>2.
SinceΩ = Πis positively invariant (Proposition 3.1), we can consider the system x′(t) =F(xt) t≥0
x(t) =ϕ(t) −ω≤t≤0 W ith ϕ∈Ω = Π =C [−ω,0],1
2,1 (4.1) We take nowF : Ω −→R.
Lemma 4.1: Suppose thatθ >2 LetGbe a bounded subset ofΩ = Π.
Then, there existα, β∈R, such that(∀ϕ∈G)α≤F(ϕ)≤β
Proof: We have F(ϕ) =βg(ϕ(0))−βg(ϕ(−ω)) exp{−bω} −bϕ(0) for all ϕ∈C withg(x) =x(1−x)for allx∈[0,1].
We can writeF(ϕ) =H(ϕ(0), ϕ(−ω)), withH(x, y) =βg(x)−bx−βg(y)e−bω SinceGis bounded, there existsL >0such that(∀ϕ∈G)kϕk∞≤L.
Then we have(∀ϕ∈G)|ϕ(0)|≤L, |ϕ(−ω)|≤L That is
(∀ϕ∈G) (ϕ(0), ϕ(−ω))∈[−L, L]2 (4.2)
H is a continuous function on the compact subset[−L, L]2ofR2, then∃α, β∈Rsuch that ∀(x, y)∈[−L, L]2
α≤H(x, y) =βg(x)−bx−βg(y)e−bω≤β.
Finally, from (4.2) we have,(∀ϕ∈G)α≤H(ϕ(0), ϕ(−ω))≤β.
Thus(∀ϕ∈G)α≤F(ϕ)≤β.
Proposition 4.1: Fsatisfies the assumption(T D1).
Proof: By Lemma 4.1, F transforms bounded subsets ofΩto bounded subsets ofR.
Take ϕ∈Ω, the positive semi-orbit of xϕ is o+(ϕ) ={xt(ϕ)/t≥0} ⊂ Ω=C [−ω,0],[12,1]
.
SinceΩis bounded in(C,kk∞),o+(ϕ) ={xt(ϕ)/t≥0}is bounded.
The semi-orbit of every solution of (4.1) is bounded.
Finally,Fsatisfies the assumption(T D1).
Proposition 4.2: Fsatisfies the assumption(SD).
Proof: SinceΩis order convex and F is continuously differentiable, it follows that for all ψ, ϕ∈Ωwithϕ≤ψwe have
F(ψ)−F(ϕ) = Z 1
0
dF(tϕ+ (1−t)ψ) (ψ−ϕ)dt
= Z 1
0
β(1−2 (tϕ(0) + (1−t)ψ(0))−b) (ψ−ϕ) (0)−βe−bω
×(1−2 (tϕ(−ω) + (1−t)ψ(−ω))) (ψ−ϕ) (0)dt
Sinceϕ, ψ∈C [−ω,0],[12,1]
,we have for allt∈[0,1] (tϕ(−ω) + (1−t)ψ(−ω))≥12. Then (1−2(tϕ(−ω) + (1−t)ψ(−ω)))≤0 and then −βe−bω(1−2(tϕ(−ω) + (1−t)ψ(−ω)))(ψ−ϕ)(0)≥0.
Consequently, for all t∈[0,1], we have F(ψ)−F(ϕ)≥R1
0 β(1−2(tϕ(0) + (1−t)ψ(0))−b)(ψ−ϕ)(0)dt
Asϕ(0)≤ψ(0), for allt∈[0,1]we havetϕ(0) + (1−t)ψ(0)≤ψ(0).
Thenβ(1−2 (tϕ(0)+(1−t)ψ(0))−b) (ψ−ϕ) (0)≥β(1−2ψ(0)−b) (ψ−ϕ) (0).
ThusF(ψ)−F(ϕ)≥β(1−2ψ(0)−b) (ψ−ϕ) (0).
Putα(ϕ, ψ) =β(1−2ψ(0)−b)for allϕ, ψ∈Ω, αis clearly continuous.
As a result, for allψ, ϕ∈Ωwithϕ≤ψ,F(ψ)−F(ϕ)≥α(ϕ, ψ) (ψ−ϕ) (0).
FinallyF satisfies the assumption(SD).
Theorem 4.1: Ω =D contains an open and dense subset of stable convergent points of (4.1). All solutions of (4.1) converge to the endemic disease equilibriumi∗ andi∗ is globally asymptotically stable onΩ =D.
Proof: By Propositions 4.1 and 4.2, assumptions(SD)and(T D1)are fulfilled, then from Theorem 2.3, we have the global convergence toi∗onΩ =D.
Then the local asymptotic stability ofi∗ (see Niri and El Karkri (2015)) implies its
global asymptotic stability onΩ =D.
5 Numerical simulations
In order to illustrate our analysis, we present numerical simulations with the parameters in table below.
The parameter The value
The birth rate b= 0.01/year The mean length of infection ω= 10years The effective contact rate β= 0.51/year
The thresholdθ 4.85329168 The endemic equilibrium i∗≃0.7939542756
Thus individuals leave the susceptible class at a rate0.51SIN per year.
The threshold quantity isθ= β(1−e−
bω)
b ≃4.85329168>2.
The endemic disease equilibrium isi∗= 1−1θ ≃0.7939542756.
Asθ >2, from Theorem 4.1 ,i∗= 1−1θ ≃0.7939542756is globally asymptotically stable, and all solutions converge to the endemic disease equilibrium.
Indeed, consider the initial valuesϕ1,ϕ2,ϕ3andϕ4defined by (∀t∈[−ω,0]) ϕ1(t) = 0.5 + 0.49exp(t)
(∀t∈[−ω,0]) ϕ2(t) = 0.65
(∀t∈[−ω,0]) ϕ3(t) = 1−0.49exp(t)and, (∀t∈[−ω,0]) ϕ4(t) = 0.6 + 0.22exp(10t)
We clearly have{ϕ1, ϕ2, ϕ3, ϕ4} ⊂D=C [−10,0],]12,1[
Numerical simulations show in Figures 2–5 that solutionsxϕ1(t),xϕ2(t),xϕ3(t)and xϕ4(t)converge to the endemic equilibriumi∗≃0.7939542756.
Figure 2 The solution i(t) for the initial value 0.5+0.49.exp(t) (see online version for colours)
Figure 3 The solution i(t) for the initial value 0.65 (see online version for colours)
Figure 4 The solution i(t) for the initial value 1-0.49.exp(t) (see online version for colours)
Figure 5 The solution i(t) for the initial value 0.6+0.22.exp(10.t) (see online version for colours)
6 Summary and discussion
The chief objective of this work is to apply essentially strongly order preserving semi- flows principle with the aim to study the global asymptotic stability of the endemic disease equilibrium of an SIS model with variable population size and a delay.
Theorem 2.3 states that condition (P) in Theorem 2.2 can be replaced by condition (SD), which is simpler and more practical. Then, after recalling principal hypothesis and properties of the model, we proved that assumptions (T D1) and(SD)are fulfilled on Ω = Π. Consequently, Theorem 2.3 applies and the global convergence to the endemic equilibrium holds on Ω = Π. Since it is locally asymptotically stable, it is globally asymptotically stable. Numerical simulations illustrate the obtained theoretical results.
The presented analysis could be further applied to many other compartmental epidemic systems. Our paper demonstrates that monotone approach can significantly contribute in the stability analysis of delayed epidemiological models and enhance the tools utilised in identifying stability conditions. We think that our partial resolution of the delicate open problem “Global stability ofi∗” can be generalised to a wider subset of initial conditions.
We also think that the ideas of Iggidr et al. (2010), Niri (1988, 1994, 2003), combined with our approach, will give valuable results concerning the oscillatory behaviour of the solutions. Another important perspective is to use our method in the study of the same model but with a disease related death rate.
References
El Karkri, J. and Niri, K. (2014) ‘Global stability of an epidemiological model with relapse and delay’,Applied Mathematical Science, Vol. 8, No. 73, pp.3619–3631.
Hale, J.K. and Verduyn Lunel, S.M. (1993) Introduction to Functional Differential Equations, Springer Verlag, Berlin.
Hethecote, H.W. and van den Driessche, P. (1995) ‘An SIS epidemic model with variable population size and a delay’,Jouranl of Mathematical Biology, Vol. 34, pp.177–194.
Hirsch, M. (1982) ‘Systems of differential equations which are competitive or cooperative 1: limit sets’,SIAM Journal on Applied Mathematics, Vol. 13, pp.167–179.
Hirsch, M. (1985) ‘Systems of differential equations which are competitive or cooperative II: convergence almost everywhere’, SIAM Journal on Mathematical Analysis, Vol. 16, pp.423–439.
Hirsch, M. (1984) ‘The dynamical systems approach to differential equations’, Bulletin of the American Mathematical Society, Vol. 11, pp.1–64.
Hirsch, M. (1988a) ‘Systems of differential equations which are competitive or cooperative III:
competing species’,Nonlinearity, Vol. 1, pp.51–71.
Hirsch, M. (1988b) ‘Stability and convergence in strongly monotone dynamical systems’,Journal fur die Reine und Angewandte Mathematik, Vol. 383, pp.1–53.
Hirsch, M. (1989) ‘Systems of differential equations that are competitive or cooperative V:
convergence in 3-dimensional systems’,Journal of Differential Equations, Vol. 80, pp.94–106.
Hirsch, M. (1990) ‘Systems of differential equations that are competitive or cooperative IV: structural stability in three dimensional systems’,SIAM Journal on Mathematical Analysis, Vol. 21, pp.1225–1234.
Hirsch, M. (1991) ‘Systems of differential equations that are competitive or cooperative. VI: a local Crclosing lemma for 3-dimensional systems’,Ergodic Theory and Dynamical Systems, Vol. 11, pp.443–454.
Iggidr, A., Niri, K. and Ould Moulay Ely, E. (2010) ‘Fluctuations in a SIS epidemic model with variable size population’,Applied Mathematics and Computation, Vol. 217, pp.55–64.
Kamke, E. (1932) ‘Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II. (German)’, Acta Mathematica, Vol. 58, No. 1, pp.57–85.
Ma, Z. and Li, J. (2009)Dynamical Modeling and Analysis of Epidemics, World Scientific Publishing Co. Pte. Ltd, Singapore.
Muller, M. (1927) ‘Uber das Fundamentaltheorem in der Theorie der gewohnlichen Differential- gleichungen (German)’,Mathematische Zeitschrift, Vol. 26, No. 1, pp.619–645.
Matano, M. (1984) ‘Existence of nontrivial unstable sets for equilibriums of strongly order preserving systems’,Journal of the Faculty of Science, the University of Tokyo, Vol. 30, pp.645–673.
Niri, K. (1988) Studies on Oscillatory Properties of Monotone Delay Differential Systems, PhD thesis, Pau University.
Niri, K. and El Karkri, J. (2015) ‘Local asymptotic stability of an SIS epidemic model with variable population size and a delay’,Applied Mathematical Science, Vol. 9, No. 34, pp.3165–3180.
Niri, K. (1994). Theoretical and numerical Studies on oscillations for delay differential systems, Doctoral thesis.
Niri, K. (2003) ‘Oscillations in differential equations with state-dependant delay’, Nonlinear Oscillations (N. Y.), Vol. 6, pp.250–257.
Smith, H.L. (1995)Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Mathematical Surveys and Monographs, vol. 41, Amer. Math. Soc., Providence.
Smith, H.L. and Thieme, H. (1990a) ‘Quasi convergence for strongly ordered preserving semiflows’, SIAM Journal on Mathematical Analysis, Vol. 21, pp.673–692.
Smith, H.L. and Thieme, H. (1990b) ‘Monotone semiflows in scalar non-quasi-monotone functional differential equations’,Journal of Mathematical Analysis and Applications, Vol. 150, pp.289–306.
Smith, H.L. and Thieme, H. (1991a) ‘Convergence for strongly ordered preserving semiflows’,SIAM Journal on Mathematical Analysis, Vol. 22, pp.1081–1101.
Smith, H.L. and Thieme, H. (1991b) ‘Strongly order preserving semiflows generated by functional differential equations’,Journal of Differential Equations, Vol. 93, pp.332–363.
Yi, T.S. and Huang, L.H. (2006) ‘Convergence and stability for essentially strongly order-preserving semiflows’,Journal of Differential Equations, Vol. 221, pp.36–57.
Yi, T.S. and Huang, L.H. (2008) ‘Dynamics of smooth essentially strongly order-preserving semiflows with application to delay differential equations’,Journal of Mathematical Analysis and Applications, Vol. 338, pp.1329–1339.
Yi, T.S. and Zou, X. (2009) ‘New generic quasi-convergence principles with applications’,Journal of Mathematical Analysis and Applications, Vol. 353, pp.178–185.
Yi, T.S. and Zou, X. (2009) ‘Generic quasi-convergence for essentially strongly order-preserving semiflows’,Canadian Mathematical Bulletin, Vol. 52, No. 2, pp.315–320.
