Stability analysis of a delayed SIS epidemiological model
J. El Karkri
∗and K. Niri
Laboratory MACS,
Department of Mathematics and Computer Science, Faculty of sciences Ain Chock (FSAC),
Hassan II University of Casablanca,
Km 8 Route d’El Jadida B.P 5366 Maarif Casablanca 20100, Morocco
Email: professionnelcpge@gmail.com Email: khniri@yahoo.fr
∗Corresponding author
Abstract:In this paper, we investigate global stability of the endemic steady state of the SIS epidemic model studied recently in terms of fluctuations.
The epidemiological model has an exponential demographic structure, disease- related deaths and a delay corresponding to the infectious period. The disease spread is governed by a scalar delay differential equation. Our study is mainly based on the monotone dynamical systems theory. We begin by simplifying the classical framework of stability analysis for non-quasi-monotone scalar autonomous delay differential equations by using recent results on essentially strongly order-preserving semiflows. Under certain conditions, it is proved that the endemic equilibrium is globally asymptotically stable on a closed subset of the phase space C. To our knowledge, this is the first study proving the endemic steady state’s global stability for this model. Numerical simulations which illustrate the results are carried out.
Keywords: essentially strongly order-preserving semiflow; exponential ordering; global asymptotic stability; non-quasi-monotone delay differential equation; SIS epidemic system.
Referenceto this paper should be made as follows: Karkri, J.E. and Niri, K.
(2016) ‘Stability analysis of a delayed SIS epidemiological model’, Int. J.
Dynamical Systems and Differential Equations, Vol. 6, No. 2, pp.173–185.
Biographical notes:J. EL Karkri is a Professor of Mathematics at “Centre of preparatory classes for engineering schools in Mohammedia” and a PhD student at the University of Hassan II. His research interest lies in the area of monotone dynamical systems theory and mathematical epidemiology.
K. 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.
Copyright c2016 Inderscience Enterprises Ltd.
Introduction
Besides the Lyapunov direct method, it is well known that the monotone dynamical systems theory has always been one of the main tools for stability analysis of delay differential equations. A classical method based on this theory has been used in the study of many models arising from ecological and biological systems in Karkri and Niri (2014);
Iggidr, Niri and Ould Moulay Ely (2010); Niri (1988); Niri and El Karkri (2015); Niri (1994); Pituk (2003); Smith (1995). In Chapter 6 of Smith (1995), H.L.Smith has proved that assumptions (T) and(SMµ)are sufficient conditions for the set of convergent points in D to contain an open and dense subset. However, it is easily seen that hypothesis (T) is difficult to verify for a wide class of delay differential equations with constant delay.
Thus, it will be very interesting to reduce and simplify this assumption in the context of non-quasi-monotone systems.
The past decade has witnessed a large development in monotone approach for delay differential equations. In a series of valuable articles Yi and Huang (2006, 2008); Yi and Zou (2009a,b) Yi et al introduced the concept of essentially strongly order-preserving semiflows and proved in Yi and Zou (2009a) that the (T) assumption can be reduced to a simpler hypothesis (TD1) for essentially cooperative delay differential equations. In this paper, we prove that (TD1) can replace (T) for non-quasi-monotone delay differential equations studied in Chapter 6 of Smith (1995).
The improved result will be applied to the delay differential equation arising from the SIS epidemiological model with variable population size, a delay corresponding to the infectious period and disease-related death rate presented in Hethecote and van den Driessche (1995).
As pointed out in Hethecote and van den Driessche (1995); Iggidr, Niri and Ould Moulay Ely (2010), under the conditionθ >1the model has two steady states: 0 andi∗, and 0 is unstable. Note that there are two types of steady states for epidemiological models:
An equilibrium is said to be free if it corresponds to a steady state of the system without infection, otherwise it is said to be endemic.
Our goal is to prove that for certain values of this model’s parameters the endemic equilibrium is globally asymptotically stable on a subsetD of the ‘phase space’ C. The model has been recently studied by Niri et al in terms of fluctuations Iggidr, Niri and Ould Moulay Ely (2010), where a monotone approach provides sufficient conditions on the parameters of the model for the solutions to present slow fluctuations around the equilibrium.
Our study is mainly based on the exponential ordering introduced by Smith and Thieme in Smith and Thieme (1990) but used earlier in 1977 by Hadeler and Tomiuk to show the existence of non-trivial periodic solutions of delay differential equations using fixed point methods Hadeler and Tomiuk (1977). We prove that the semiflow associated with the delay differential equation is strongly order preserving onD, then we show that the improved condition (TD1) holds, and finally we deduce that the endemic equilibrium is globally asymptotically stable onD.
Our paper is organised as follows: in Section 1, the main tools of generic convergence for scalar autonomous non-quasi-monotone delay differential equations are presented.
Principal properties of the model are given in Section 2. In Section 3, we provide sufficient conditions for local asymptotic stability of the endemic equilibrium. The focus of Section 4 is on the main result of the paper, which includes sufficient condition of global asymptotic stability for the endemic disease equilibrium whenθ >2. Finally, numerical simulations, summary and discussion are given in Section 5.
1 Generic convergence for scalar autonomous non-quasi-monotone delay differential equations
For ω >0, let C=C0([−ω,0],R) denote the Banach space of continuous functions mapping[−ω,0]intoRwith the maximum norm.
Consider the following autonomous delay differential equation:
x′(t) =f(xt) t≥0
x(t) =ϕ(t) t∈[−ω,0] withϕ∈Ω (1.1)
wheref : Ω→Ris Lipschitz andΩ⊂Cis the closure of an open subsetΠ⊆C.
Forµ≥0, put:Cµ ={φ∈C; φ≥0and φ(s)eµsis non−decreasing on [−ω,0]}. Cµis a cone in C generating a partial order denoted by≤µand defined as follows:
(∀φ, ϕ∈C) : φ≤µψ⇐⇒φ−ψ∈Cµ
That is,φ≤µψ⇐⇒φ≤ψ and(φ(s)−ψ(s))eµsis non-increasing on[−ω,0].
We write φ <µψ if φ≤µψandφ6=ψ.
The reader is referred to Pituk (2003); Smith (1995) for further details on the partial order≤µcalled exponential ordering.
The delay differential equation (1.1) is said to satisfy the(SMµ)property onΩif there existsµ >0such that for allφ, ψ∈Ω:
φ <µψ =⇒ µ(ψ(0)−φ(0)) +f(ψ)−f(φ)>0 Pituk(2003); Smith(1995) A bounded linear functionalL:C−→Ris said to verify the property(Mµ)if for some µ≥0:
(Mµ) : L(φ) +µφ(0)≥0 wheneverφ∈C andφ≥µ0 Smith(1995) Consider the autonomous linear delay differential equation:
x′(t) =L(xt) =λx(t) +ηx(t−ω) +γ Z t
t−ω
x(s)ds (1.2)
The following theorem provides sufficient conditions for the operator L defined in the special linear equation (1.2) to satisfy condition(Mµ).
Also we say that equationx′(t) =L(xt)verifies(Mµ).
Theorem 1.1: Hypothesis(Mµ)holds for the operatorLdefined in the special equation (1.2) if one of the following holds:
a λ+η−+γ−ω >0or
b i λ+η−+γ−ω <0, and ii ω|η−+γ−ω|<1and iii ωλ−ln(ω|η−+γ−ω|)>1 whereη− = min(0, η)andγ− = min(0, γ).
Proof: For allϕ∈Cwe haveL(ϕ) =λϕ(0) +ηϕ(−ω) +γR0
−ωϕ Considerϕ∈Csuch that0≤µ ϕ.
Sinces7−→ϕ(s)eµsis non-decreasing on[−ω,0], (∀s∈[−ω,0])ϕ(−ω)e−µω≤ϕ(s)eµsds≤ϕ(0) Then we have
Z0
−ω
ϕ(s)eµs≤ωeµωϕ(0)andϕ(−ω)e−µω≤ϕ(0)
Thus
(∀µ≥0) (∀ϕ≥µ0)µϕ(0) +L(ϕ) = (µ+λ)ϕ(0) +ηϕ(−ω) +γ Z 0
−ω
ϕ≥h(µ)ϕ(0) whereh(µ) =µ+λ+ (η−+γ−ω)eµωfor allµ≥0.
(Mµ)holds for (1.2) if and only if there existsµ≥0such thath(µ)≥0 Condition (a) implies thath(0)>0.
Condition (b) implies thath(µ)increases fromh(0) =λ+ (η−+γ−ω), reaches its maximum at (µ∗, h(µ∗) = (−lnω|η−ω+γ−ω|,ωλ−ln(ω|ηω−+γ−ω|−1) and decreases to −∞
withµ∗>0 and h(µ∗)>0.
Hence, if (a) or (b) holds, then(∃µ >0) h(µ)≥0and then(Mµ)holds.
Consider the functionf˜defined by
f˜(u) =f(ˆu(t)) (1.3)
withu(t) =ˆ ufor allt,uˆis a constant function.
The following theorem established in Smith and Thieme (1990) will be important for our study:
Theorem 1.2Consider the delay differential equation (1.1): Letf˜defined by (1.3).
Suppose thatf is continuously differentiable in a neighbourhood of an equilibriumx∗ of (1.1) anddf(x∗)verifies(Mµ)for someµ≥0, then:
i Iff˜′(x∗)<0thenx∗is locally asymptotically stable.
ii Iff˜′(x∗)>0thenx∗is unstable.
For reader’s convenience, we begin by recalling the assumptions and notations below all taken from Yi and Huang (2008) (Section 2):
Assume that X is an ordered complete metric space with a metric d.
φdenotes a semiflow in X.
(A1): φis an essentially strongly order-preserving semiflow on X.
(A2): There existst1>0such thatφt1is a conditionally set-condensing map on X.
(A3): For eachx∈X,O(x)is a bounded subset of X.
(A4): X is an ordered bounded space.
(A5): X is a normally ordered space.
Edenotes the set of equilibriums ofφ.
Sis the set of stable points.
Ais the set of asymptotically stable points.
Cis the set of convergent points.
X+={x∈X/x can be essentially approximated f rom above by a sequence of X} X−={x∈X/x can be essentially approximated f rom below by a sequence of X} The reader is referred to Yi and Huang (2006, 2008); Yi and Zou (2009a,b) for further details.
Assumption(T D1): “f maps bounded subsets ofΩto bounded subsets ofR, and the positive semiorbit of every solution of (1.1) is bounded”.
We recall the following fundamental statement (Theorem 2.1 in Yi and Zou (2009a)):
Theorem 1.3Assume that (A1), (A2), (A3) and (A4) hold: If there exists an open and dense subset Y of X such that Y ⊂X+∪X− then Y ⊂int(Q)∪int(C) and hence X=int(Q).
The main result of this section is the following theorem which provides a sufficient condition for the convergence of the generic solution of (1.1).
Theorem 1.4Consider the delay differential Equation (1.1): Assume thatf satisfies the assumptions(SMµ)and(T D1). ThenΩcontains an open and dense subset of convergent points. If E consists of a single point, then it attracts all solutions of (1.1). IfΩis order convex and E consists of two points, then each solution of (1.1) converges to one of these.
Proof: It is evident thatΩis a complete metric space.
Since f satisfies (SMµ) by Smith (1995)( Chapter 6, Theorem 2.3), the semiflow φ associated with the delay differential equation (1.1) is strongly order preserving inΩ, and consequently it is essentially strongly order preserving. Thus, assumption (A1) in Theorem 1.3 holds.
As pointed out in Yi and Huang (2008), by the assumption (TD1), Theorem 3.6.1 in Hale and Verduyn Lunel (1993) implies thatφω+1is a conditionally completely continuous map and henceφω+1is a conditionally set-condensing map, assumption (A2) is fulfilled.
(A3) is clearly implied by(T D1).
Let prove (A4).
From Yi and Huang (2008), C andΩare ordered bounded spaces for the partial order
≤generated by the cone:
C+={φ∈C:φ(θ)≥0, −ω≤θ≤0}. For allϕ, ψ∈Ω, put[ϕ, ψ]µ ={ν∈C/ϕ≤µν≤µψ} and[ϕ, ψ] ={ν ∈C/ϕ≤ν≤ψ}.
SinceΩis an ordered bounded space for the partial order≤, for allϕ, ψ∈Ωthe subset [ϕ, ψ] is bounded.
and as [ϕ, ψ]µ ⊂[ϕ, ψ] it follows that [ϕ, ψ]µ is bounded. Thus ((Ω, d),≤µ) is an ordered bounded space.
Finally, (A4) holds.
From Smith (1995) ( Chapter 6, proof of Theorem 3.1), every point of Π can be approximated from below and from above in((Ω, d),≤µ).
Finally, Theorem 1.3 applies, and Ω contains an open and dense subset of quasi- convergent points.
From Remark 4.2. of Chapter 1 in Smith (1995) we have Q=C for scalar delay differential equations, consequentlyΩcontains an open and dense subset of convergent points.
By an argument similar to that used in Theorem 3.1 Chapter 6 of Smith (1995), it is easily seen that if E consists of a single point, then it attracts all solutions of (1.1) and if Ωis order convex and E consists of two points, then each solution converges to one of
these.
2 Model
In this section, we recall principal properties of the model studied here, the reader is referred to Hethecote and van den Driessche (1995) for a detailed and rigorous presentation of the system.
As described in Figure 1, the population of size N(t) is divided into two disjoint compartments that are susceptible and infective with sizes denoted at time t byS(t)and I(t), respectively.
Figure 1 The model
The population is assumed to have an exponential demographic structure, that isN(t) = N(0) exp{(b−d)t}, wherebandddenote, respectively, birth rate and natural death rate.
Furthermore, a disease-related death rate denotedδis taken into consideration.
The mean length of infection is R∞
0 P(u)du=ω, where the probability P(t) is assumed to be non-negative and non-increasing withP(0+) = 1.
we also assume that the force of infection isβI/N, whereβ >0is the effective contact rate of an infective individual.
It was proved by Hethecote and van den Driessche (1995) that the disease dynamic was described by the functional equation:
I(t) =I0(t) + Z t
0
βS(u)I(u)
N(u) P(t−u) exp{−(d+δ)(t−u)}du (2.1)
N′(t) = (b−d)N(t)−δI(t), (2.2)
I0(t)denotes the number of initially infected individuals that are still infected at timet.
Puti(t) = N(t)I(t),s(t) =NS(t)(t), Eq. (2.2) gives
N′(t) = (b−d−δi(t))N(t), (2.3)
As proved by Hethecote and van den Driessche (1995), the dynamic of the system is governed by the following delay differential equation:
i′(t) = [β−(b+δ)−(β−δ)i(t)]i(t)−β[1−i(t−ω)]i(t−ω)
(2.4) exp
−(b+δ)ω+δ Z t
t−ω
i(p)dp
We can write (2.4) as
x′(t) =F(xt) t≥0
x(t) =ϕ(t) −ω≤t≤0 (2.5)
whereF :C−→R, such that for allϕ∈C F(ϕ) = (β−δ)g(ϕ(0))−βg(ϕ(−ω)) exp
−(b+δ)ω+δ Z 0
−ω
ϕ(p)dp
(2.6)
−bϕ(0)
with g(x) =x(1−x) for allx∈[0,1].
Putθ= β(1−exp(−(b+δ)ω))
b+δ .
The following theorem has been proved in Hethecote and van den Driessche (1995) (Theorem 3.1):
Theorem 2.1: The model has always a disease-free equilibrium.
1 Ifθ≤1, then the disease-free equilibrium is locally asymptotically stable.
2 Ifθ >1, then the model has an endemic equilibriumi∗.
It is also proved in Hethecote and van den Driessche (1995) (Theorem 4.2) that the disease- free equilibrium is globally asymptotically stable, ifδω≤θ≤1.
Let(H)denotes the assumption:(H) : β
h1−e−bω b+δ2
i
>2.
(H)impliesθ >2and then the model has two equilibriums: free and endemic Iggidr, Niri and Ould Moulay Ely (2010).
An other fundamental result has been recently proved in Iggidr, Niri and Ould Moulay Ely (2010) (Proposition 3.1):
Proposition 2.1:
1 If(H)holds, then the endemic equilibrium satisfies 12 < i∗<1.
2 If(H)holds, then(∀ϕ∈C):12 ≤ϕ≤1 =⇒ ∀t≥0 12 ≤x(t, ϕ)≤1.
That is,
ϕ∈C/ (∀t∈[−ω,0]) 12≤ϕ(t)≤1 is positively invariant for the semiflow associated with the delay differential equation ( 2.5).
3 Local asymptotic stability of i
∗In this section, we are concerned with the local asymptotic stability of the endemic equilibriumi∗.
We first recall that in Hethecote and van den Driessche (1995) the stability of i∗ has been declared as an open problem, when δ >0 : “Global stability of the endemic equilibrium is an open problem” Hethecote and van den Driessche (1995). In the presence of term in the delay differential equation (2.5), due toδ >0 the characteristic equation becomes more complicated, and application of the classical method of the local stability analysis based on roots of the characteristic equation is consequently more difficult. In this subsection, we investigate the asymptotic behaviour ofi∗by applying results of monotone dynamical systems theory especially Theorem 1.2. Sufficient practical conditions for local asymptotic stability ofi∗are obtained.
The precise statement is formulated in the following theorem:
Theorem 3.1: Assume thati∗> 12and
ln(δω2[(β−δ)(1−i∗)−b]i∗) + [(2i∗−1)(β−δ) +b]ω+ 1<0 (3.1) Then the endemic disease equilibriumi∗is locally asymptotically stable.
Proof: With notations of (1.3), we write:
(∀u∈[0,1]) F(u) = [(β˜ −δ)(1−u)−b]u−β[1−u]ue−(b+δ)ωeδωu Therefore(∀u∈[0,1])
F˜′(u) = (β−δ)(1−2u)−β(1−2u)e−(b+δ)ueδωu−βδω(1−u)ue−(b+δ)ωeδωu−b Then
F˜′(i∗) = (β−δ)(1−2i∗)−β(1−2i∗)e−(b+δ)ωeδωi∗−βδω(1−i∗)i∗e−(b+δ)ωeδωi∗ −b.
Sincei∗is an equilibrium,F˜(i∗) = 0and then we have:
βe−(b+δ)ωeδωi∗ =β−δ− b
1−i∗ (3.2)
Thus,F˜′(i∗) =−b1i∗
−i∗ −βδωi∗(1−i∗)e−(b+δ)ωi∗e−(b+δ)ω<0 We have:
dF(i∗)(ϕ) = ((β−δ)(1−2i∗)−b)ϕ(0)−βe−(b+δ)ω(1−2i∗)eδωi∗ϕ(−ω)
−βδe−(b+δ)ω(1−i∗)i∗eδωi∗ Z 0
−ω
ϕ
This may be rewritten with notations of Theorem 1.1 as:
dF(i∗)(ϕ) =λϕ(0) +ηϕ(−ω) +γ Z 0
−ω
ϕ
whereλ= (β−δ)(1−2i∗)−b, η=−βe−(b+δ)ω(1−2i∗)eδωi∗ andγ=−βδe−(b+δ)ω(1−i∗)i∗eδωi∗.
It is evident thatλ+η−+γ−ω <0 (observe that (3.2) implies thatβ−δ >0).
Thus, condition (b:(i)) of Theorem 1.1 holds.
Sinceln(βδe−(b+δ)ω(1−i∗)i∗eδωi∗ω2) + ((2i∗−1)(β−δ) +b)ω+ 1
=ln(δω2[(β−δ)(1−i∗)−b]i∗) + [(2i∗−1)(β−δ) +b]ω+ 1<0,
we haveln(βδe−(b+δ)ω(1−i∗)i∗eδωi∗ω2)<−((2i∗−1)(β−δ) +b)ω−1<0.
Then, βδe−(b+δ)ω(1−i∗)i∗eδωi∗ω2<1, that is, ω|η−+γ−ω|<1 (note that η−= 0becausei∗> 12) and then assumption (b:(ii)) of Theorem 1.1 holds.
Furthermore, condition (3.1) impliesωλ−ln(ω|η−+γ−ω|)>1.
Finally condition (b:(iii)) of Theorem 1.1 holds, and consequently hypothesis (Mµ) holds for the linear applicationdF(i∗).
Summarising(Mµ)holds fordF(i∗)andF˜′(i∗)<0. Therefore, Theorem 1.2 applies
andi∗is locally asymptotically stable.
4 Global stability of the endemic equilibrium on D =
ϕ ∈ C/ ( ∀ t ∈ [ − ω, 0])
12≤ ϕ(t) ≤ 1
PutD=ϕ∈C/ (∀t∈[−ω,0]) 12 < ϕ(t)<1 . We haveD=
ϕ∈C/ (∀t∈[−ω,0]) 12 ≤ϕ(t)≤1 . Let(S)denotes the assumption: ln
4 βδeω2
>(β−δ)ω
Theorem 4.1: If (H) and (S) hold, then the(SMµ)property holds for (2.5) onD.
Proof: Letµ >0andϕ, ψ∈D.
ConsiderG:R3−→R, such that
∀(x, y, z)∈R3 G(x, y, z) = (β−δ)g(x)−βg(y)e−(b+δ)ωeδz−bx (we recall that g(r) =r(1−r)for allr∈[0,1]).
We have∀ϕ∈C F(ϕ) =G(ϕ(0), ϕ(−ω),R0
−ωϕ) Suppose thatϕ <µψ.
F(ψ)−F(ϕ) =G(ψ(0), ψ(−ω), Z 0
−ω
ψ)−G(ϕ(0), ϕ(−ω), Z 0
−ω
ϕ)
∃λ0∈[0,1], such that G(ψ(0), ψ(−ω),
Z 0
−ω
ψ)−G(ϕ(0), ϕ(−ω), Z 0
−ω
ϕ) =
∇~G((1−λ0)X+λ0X′)|X′−X
WithX′ = (ψ(0), ψ(−ω),R0
−ωψ)andX = (ϕ(0), ϕ(−ω),R0
−ωϕ) Put(x, y, ωz) = (1−λ0)X+λ0X′ withx, y, z∈]12,1[
F(ψ)−F(ϕ) = ((β−δ)(1−2x)−b) (ψ−ϕ)(0)
−β(1−2y)e−(b+δ)ωeδωz(ψ−ϕ)(−ω)−βδg(y)e−(b+δ)ωeδωz Z 0
−ω
(ψ−ϕ) From ϕ <µψ, we have (ψ−ϕ)(0)>0 , ((ψ−ϕ)(−ω)≤eµω(ψ−ϕ)(0) and
R0
−ω(ψ−ϕ)≤eµωω(ψ−ϕ)(0).
Using x, y, z∈1
2,1
, g(x), g(y)∈ 0,14
and β−δ >0 (arising from (3.2)), it follows that
F(ψ)−F(ϕ) +µ(ψ−ϕ)(0)≥(µ−β+δ−b−βδω
4 e−bωeµω))(ψ−ϕ)(0).
The function L : µ7→µ−β+δ−b−βδω4 e−bωeµω reaches its maximum at (µ∗, L(µ∗))withL′(µ∗) = 0.
Thus,eµ∗ω=βδω4ebω2.
It is easily seen that condition (S) impliesµ∗>0 and L(µ∗)>0.
Therefore, F(ψ)−F(ϕ) +µ(ψ−ϕ)(0)≥L(µ∗)(ψ−ϕ)(0)>0, whenever ϕ, ψ∈Dwithϕ <µ∗ ψ.
Finally,(SMµ)property holds for (2.5) onD.
Proposition 4.1: Assumption(T D1)holds for the delay differential equation (2.5) onD.
Proof: Let B be a bounded subset ofD.
There existsM >0, such that(∀ϕ∈B)kϕk∞≤M. Then we have
(∀s∈[−ω,0]) |ϕ(s)|≤M.
Thus,(∀ϕ∈B) |R0
−ωϕ(s)ds|≤ωM ,|ϕ(0)|≤M and|ϕ(−ω)|≤M. PutB′=
(x, y, z)∈R3/|x|≤M, |y|≤M, |z|≤ωM .
With notations of proof of Theorem 4.1, G is continuous and thenG(B′)is a bounded subset ofR.
AsF(B)⊂G(B′),F(B)is bounded. It follows thatF maps bounded subsets ofD to bounded subsets ofR.
On the other hand, we conclude from Proposition 2.1 that the positive semiorbit of every solution of (2.5) is bounded.
Hence,(T D1)holds for the delay differential equation (2.5) onD.
Proposition 4.2: If(H)and(S)hold, then the endemic disease equilibriumi∗is locally asymptotically stable.
Proof: In order to apply Theorem 3.1 , note that(H)implies by Proposition 2.1 that θ >2, then the endemic equilibriumi∗exists and 12 ≤i∗≤1.
On the other hand, by equation (3.2) we have:
ln(δω2[(β−δ)(1−i∗)−b]i∗) + [(2i∗−1)(β−δ) +b]ω+ 1
=ln
βδω2(1−i∗)i∗e−(b+δ)ωeδωi∗
+ [(2i∗−1)(β−δ) +b]ω+ 1.
=ln βδω2(1−i∗)i∗
+ (2i∗−1)βω−δωi∗+ 1
=ln βδω2g(i∗)
+ (2i∗−1)βω−δωi∗+ 1.
Besides that12 ≤i∗≤1, then(2i∗−1)βω−δωi∗+ 1≤βω+ 1.
Hence
ln(δω2[(β−δ)(1−i∗)−b]i∗) + [(2i∗−1)(β−δ) +b]ω+ 1≤ln βδω2
4
+βω+ 1
That is
ln δω2[(β−δ)(1−i∗)−b]i∗
+ [(2i∗−1)(β−δ) +b]
ω+ 1≤ −
ln 4
eβδω2
−βω
.
As hypothesis(S)holds,− ln
4 eβδω2
−βω
<0.
It follows that
ln(δω2[(β−δ)(1−i∗)−b]i∗) + [(2i∗−1)(β−δ) +b]ω+ 1<0
Theorem 3.1 applies and the endemic disease equilibrium i∗ is locally asymptotically
stable.
The main result of this paper can now be stated. It gives sufficient conditions for global asymptotic stability of the endemic equilibriumi∗onD.
Theorem 4.2: If (H) and (S) hold, then the endemic disease equilibriumi∗ is globally asymptotically stable for the delay differential equation (2.5) onD.
Proof: Since (H) and (S) are satisfied , by Theorem 4.1, the(SMµ)property holds for (2.5) onD. Moreover,(T D1)holds for (2.5) by Proposition 4.1.
The semiflow associated with the delay differential on D equation has a unique equilibriumi∗, then the global convergence toi∗onDfollows from Theorem 1.4.
On the other hand, by Proposition 4.2,i∗is locally asymptotically stable.
Finally, global convergence toi∗onDand its local asymptotic stability give the global
asymptotic stability ofi∗, completing the proof.
5 Simulations, summary and discussion
It is of interest to illustrate the analytical results by numerical simulations.
Take b=0.1 ,β = 4.5,δ= 0.002andω= 1.
We havei∗≃0.8921379227positive solution ofF(x) = 0.
We also have β
h1−e−bω b+δ2
i
= 4.239917020>2, consequently (H) holds.
ln 4
βδω2e
−βω= 0.596825063>0.
Thus, condition(S) : ln
4 βδω2e
>(β−δ)ωholds.
According to Theorem 4.2, all solutions starting from D converge to the endemic disease equilibriumi∗≃0.8921379227.
Lets see this in simulations for different initial conditions inD.
As shown in Figures 2 and 3, solutions corresponding to initial conditionsϕ1andϕ2
converge to the endemic equilibriumi∗≃0.8921379227.
Figure 2 Convergence ofi(t)to the EDEi∗for the initial conditionϕ1(t) = 0.2.t+ 0.8 (see online version for colours)
Figure 3 Convergence ofi(t)to the EDEi∗for the initial conditionϕ2(t) = 0.75(see online version for colours)
We have extended the technics of Yi and Zou (2009a), developed in the context of essentially cooperative systems, to the class of non-quasi-monotone delay differential equations described in Chapter 6 of Smith (1995). A practical and simplified framework has been obtained. The improved method has been applied to the delay differential equation proposed in Hethecote and van den Driessche (1995) and leads to valuable conditions of global stability of the endemic disease equilibrium. It is of interest to compare the
classical method used in Karkri and Niri (2014) with the improved framework presented in this paper. Our study has shown that the monotone dynamical systems theory can solve problems that Lyapunov method could not do. Monotone approach does not replace Lyapunove method but it enriches the tools for determining the stability conditions. We believe that our partial resolution of the delicate open problem ‘global stability ofi∗’ can be extended to a wider subset of initial values. Our approach combined with the ideas of Iggidr, Niri and Ould Moulay Ely (2010); Niri (1988, 1994) can surely give significant results concerning oscillations of solutions. An other important perspective is to extend this method to models described by the systems with two or three equations.
References
Hadeler, K. and Tomiuk, J. (1977) ‘Periodic solutions to difference-differential equations’, Arch.
Rational Mech. Anal., Vol. 65, pp.87–95.
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’,J. Math. Biol.,Vol. 34, pp.177–194.
Iggidr, A., Niri, K. and Ould Moulay Ely, E. (2010) ‘Fluctuations in a SIS epidemic model with variable size population’,Appl. Math. Comput.,Vol. 217, pp.55–64.
Karkri, J.El and Niri, K. (2014) ‘Global stability of an epidemiological model with relapse and delay’, Appl. Math. Sci.,Vol. 8, No. 73, pp.3619–3631.
Niri, K. (1988) Studies on Oscillatory Properties of Monotone Delay Differential Systems, PhD Thesis, Pau University.
Niri, K. (1994) Theoretical and Numerical Studies on Oscillations for Delay Differential Systems, Doctoral Thesis: Mohammed V University.
Niri, K. and Karkri, J.El (2015) ‘Local asymptotic stability of an SIS epidemic model with variable population size and a delay’,Appl. Math. Sci.,Vol. 9, No. 34, pp.3165–3180.
Pituk, M. (2003) ‘Convergence to equilibria in scalar nonquasimonotone functional differential equations’,J. Differ. Equat.,Vol. 193, pp.95–130.
Smith, H.L. (1995) ‘Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems’, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Providence, RI.
Smith, H.L. and Thieme, H.R. (1990) ‘Monotone semiflows in scalar non-quasi-monotone functional differential equations’,J. Math. Anal. Appl.,Vol. 150, pp.289–306.
Yi, T.S. and Huang, L.H. (2006) ‘Convergence and stability for essentially strongly order-preserving semiflows’, J. Differ. Equat. 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’, J. Math. Anal. Appl., Vol. 338, pp.1329–1339.
Yi, T.S. and Zou, X. (2009a) ‘New generic quasi-convergence principles with applications’,J. Math.
Anal. Appl.,Vol. 353, pp.178–185.
Yi, T.S. and Zou, X. (2009b) ‘Generic quasi-convergence for essentially strongly order-preserving semiflows’, Canad. Math. Bull., Vol. 52, No. 2, pp.315–320.
