http://dx.doi.org/10.12988/ams.2015.5116
Local Asymptotic Stability of an SIS Epidemic Model with Variable Population Size and a Delay
Khadija Niri and Jaafar El Karkri Laboratory MACS
Department of Mathematics and Computer Science Faculty of Sciences, University Hassan II Ain Chock
B.P. 5366-Maarif, Casablanca 20100 Morocco
Copyright©2015 Khadija Niri and Jaafar El Karkri. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We study a SIS epidemic model with an exponential demographic structure and a delay corresponding to the infectious period. The disease spread is de- scribed by a delay differential equation. Equilibriums and the basic repro- duction number θ are identified. Using the monotone dynamical systems the- ory, local asymptotic stability of the two steady states is completely deter- mined. Numerical simulations are carried out to illustrate the theoretical results.
Keywords: Delay differential equation; Equilibrium; Exponential monotonic- ity; Local asymptotic stability; Monotone dynamical systems; SIS epidemio- logical model
Introduction
For many epidemiological models, it’s necessary to consider time spent in compartments. For example, the incubation period for HIV could reach ten years [42]. It was shown in many studies that the introduction of the delay in a dynamical system can change stability of an equilibrium to instability [20, 42].
Moreover, in retarded models, fluctuating solutions are often observed [20].
Hence, the presence of delays in epidemic systems changes completely the asymptotic behavior of solutions.
In recent years, there have been many works dealing with dynamical properties of epidemiological models described by delay differential equations[8, 5, 6, 20, 41, 42]. The basic classical method for the local stability analysis of an epidemic delayed system, is to consider roots of the characteristic equation of the linearized system at an equilibrium. The equilibrium is locally asymptotically stable if and only if all roots have negative real parts [12, 22, 43]. On the other hand, the traditional framework to analyze the global stability is the well known: Liapunov direct method (the reader may consult [7, 29] for more details).
The present work provides a typical local stability analysis based on the mono- tone dynamical systems theory for a scalar delay differential equation. Most of applications of this theory were devoted to competitive systems which arise naturally from models in biological population dynamics[10, 24, 25, 40, 44].
However and in spite of their practicality and efficiency, very little appli- cations of this theory in the mathematical epidemiology area have been developed [8, 20, 30, 31, 41]. Our previous article [8] has shown the useful- ness of the implementation of monotone approaches in epidemiological studies.
The delay differential equation studied here arises from an SIS epidemic model with variable population size and a delay corresponding to the infectious period. This model has been proposed and analyzed in 1995 by Herbert W.
Hethcote et al.[13] and recently by K.Niri et al. in 2010 [20], where it was proved that under the quasi-monotonicity hypothesis, the model presents slow fluctuations around the endemic disease equilibrium. In the present paper, an approach to study the dynamic of the model using the exponential ordering is presented. With this method, the local asymptotic behavior around the two equilibriums is determined, the basic reproduction number is identified and its threshold property is established.
Our study is centred on the exponential ordering which was introduced for dynamical systems by H.L.Smith and H.R. Thieme in [35, 37], but used ear- lier by K.Hadeler and J.Tomiuk to show the existence of nontrivial periodic solutions of delay differential equations using fixed point methods (1977)[9].
We recall in details via the preliminary section the definition of the exponential ordering.
Let us just know that the choice of an order ≤ on IR determines an order induced on the spaceC of continuous functions with values inIR.
Let G be a semi group on the space C. We say that G is monotone (with respect to the order) if:
For everyϕ, ψ ∈C,ϕψ =⇒G(t)ϕG(t)ψ for all t≥0.
If we limit ourselves to a semi-linear group, this notion is equivalent to:
”G(ϕ)≥0 whenever ϕ0 ”.
Monotone proprieties of dynamical systems have received the earlier attention for ordinary differential equations. For this case, there have been many contributions. Without any attempt to be complete, we can quote notably [21, 23, 26]. The important part of this theory is due to M.Hirsch who developed the well-known ”Monotone dynamical systems theory” through many papers entitled ”Systems of differential equations which are competitive or cooperative” [14, 15, 16, 17, 18, 19].
For the infinite dimensional theory of monotone systems case, several gen- eralizations were undertaken, notably by H.Matano[26] who introduced the notion of a strongly order preserving semi-flow, more flexible than the strong monotonicity used by M.Hirsch (see also H.Matano [27, 28]) .
For discrete delay equations, the study of properties due to monotonicity was made first by P.Seguier [2] and later by O.Arino et al. [3, 4] and K.Niri et al.
[1, 30, 31, 32].
Recently, employing the ideas of M.Hirsch and H.Matano, H.L.Smith and H.R.
Thieme have extended the applicability of monotone methods to functional differential equations [34, 35, 36, 37].
However, monotonicity is not usually verified and its conditions are very restrictive. An other type of monotonicity defined by a partial order relation called exponential ordering was introduced by H.L.Smith [38, 39].
The notion of monotonicity associated with this order is more flexible than that defined by the usual order, often called quasi-monotonicity.
The organization of this paper is as follows: In Section 1, we present some main tools of monotone dynamical systems theory for scalar differential equations with constant delay. Basic mathematical properties of the model are given in Section 2. In Sections 3 and 4 , we study the local asymptotic stability of the disease free equilibrium and the endemic equilibrium respectively. In Section 4, the results obtained are illustrated by numerical simulations.
1 Preliminaries:
Letω >0 and put: C =C0([−ω,0],R). Consider the scalar autonomous delay differential equation :
x0(t) =f(xt) t≥0
x(t) =ϕ(t) −ω≤t ≤0 (1.1)
Where : ϕ∈C is the initial condition, and for allt≥0 and all −ω ≤θ ≤0 : xt(θ) =x(t+θ).
Equation (1.1) has a unique solution, denoted x(., ϕ), ( or also : x(ϕ)(.)) on an interval [0, T) if, for example, f is Lipschitz on compact subsets of C.
More precisely we have the theorem:
Theorem 1.1. [11]
Let D be an open subset of C and suppose that f :D →R be continuous and lipschitzian in every compact subset of D. If ϕ∈D, then problem (1.1) has a unique solution defined for t≥ −ω.
Hale’s book [11] is a standard reference for functional differential equations.
Let now define the linearised equation around an equilibrium .
Definition 1.1. Let f : C −→ R a differentiable function. Let x∗ an equilibrium of (1.1) and L = df(x∗), the linear Delay differential equation:
y0(t) = L(yt) is called the linearized system of (1.1) around the equilibriumx∗. H.L.Smith and H.R. Thieme introduced the exponential ordering generated by the closed cone defined for some µ≥0 by:
Cµ={φ∈C; φ≥0and φ(s)eµsis nondecreasing on [−ω,0]}
This order will be denoted by≤µ .
That is :φ≤µψ ⇐⇒φ−ψ ∈Cµ. ie: φ ≤µ ψ ⇐⇒φ≤ψ and(φ(s)−ψ(s))eµs is non-increasing on [−ω,0]. If φ and ψ are differentiable functions then:
φ≤µψ ⇐⇒φ ≤ψ and (ψ−φ)0+µ(ψ−φ)≥0 We write φ <µψ if φ≤µ ψ and φ6=ψ.
A bounded linear functional L :C −→ R is said to verify the property (Mµ) if , for someµ≥0 we have:
(Mµ) : L(φ) +µφ(0)≥0 whenever φ∈C and φ≥µ0
Consider the special linear equation:
x0(t) =L(xt) =λx(t) +ηx(t−ω) (1.2)
From, [38] page 104 : Proposition 1.3, we have:
Theorem 1.2. Hypothesis (Mµ) holds for the operator L defined in the the special equation (1.2) if one of the following holds:
(a) η≥0 or
(b) η <0 and λ+η≥0 or (c) i) η <0 , λ+η <0, and
ii) ω |η|<1 and iii) ωλ−ln(ω |η|)≥1
One of the important results of the theory developed by H.L.Smith and H.R.
Thieme in [35, 37] is that if (Mµ) holds for the linearised equation (1.1) around the equilibrium x∗, then the stability of this equilibrium is similar as for the following ordinary differential equation obtained by ignoring the delay.
x0(t) = ˆf(x(t)) (1.3)
where ˆf is defined by
fˆ(u) =f(ˆu(t)) (1.4)
with ˆu(t) =u for all t ≥0.
The precise statement of the above concept is formulated in the following theorem established in [33, 38] :
Theorem 1.3. Consider the equation (1.1) . Let fˆdefined by (1.4). Suppose thatf is continuously differentiable in a neighbourhood of an equilibrium x∗ of (1.1) and df(x∗) verifies (Mµ) for some µ≥0. Then :
i) If fˆ0(x∗)<0 then x∗ is asymptotically stable.
ii) If fˆ0(x∗)>0 then x∗ is unstable.
Figure 1: The model
2 The model:
The model to be studied is schematically presented in Figure 1.
S(t), I(t) and N(t) denote the numbers of susceptible, infective, and total population size at time t, respectively.
β, b,and ddenote respectively contact rate, birth rate and natural death rate.
Put: i(t) = N(t)I(t) ,s(t) = N(t)S(t).
We suppose that the length of infection is equal to a constantω >0, thenP(t) is a step function, namely,
P(t) = 1 0≤t≤ω
P(t) = 0 t > ω (2.1)
It has been proven in [13] and [20] that the disease spread is governed by the equations below [13]:
i0(t) = [β−b−βi(t)]i(t)−β[1−i(t−ω)]i(t−ω) exp{−bω} (2.2)
Finally to study the asymptotic behavior of (2.2) we consider the delay differ- ential equation :
x0(t) =f(xt) t≥0
W here f or all φ in C:
f(φ) = [β−b−βφ(0)]φ(0)−β[1−φ(−ω)]φ(−ω) exp{−bω}
(2.3) For the existence of solutions we have the statement:
Theorem 2.1. For everyϕ∈C([−ω,0],]0,1[) the equation (2.3) has a unique solution x(., ϕ) : [−ω,+∞[−→]0,1[
Proof of Theorem (2.1) :
f is trivially aC∞ function from C([−ω,0],]0,1[) toR . Thus, it is Lipschitz on any compact subset ofC([−ω,0],]0,1[) , and then by Theorem 1.1 for each initial condition ϕ ∈ D the equation (2.3) has a unique solution . Moreover it has been proved in Theorem 2.1 of [13] that for all t ≥ 0 we have: 0<
x(t, ϕ) =xϕ(t)<1.
Equilibriums and linearised equation of the model:
To simplify notation, we will identifyx∈R with ˆx∈C the constant function equal tox for all values of its argument.
Letθ = β(1−eb−bω).
It is proved in [13] that Equation (2.3) has always a disease free equilibrium.
An endemic equilibrium i∗ exists if θ >1 and it is given by: i∗ = 1−1θ[13].
The linearized equation around the equilibrium x∗ associated to the delay differential equation (2.3) is given by:
x0(t) = λx(t) +ηx(t−ω) λ=β(1−2x∗)−b, η=−βe−bω(1−2x∗)
(2.4)
The function ˆf described in the ordinary equation associated with equation (2.3) is defined by:
f(u) = [β(1ˆ −u)−b]u−β[1−u]ue−bω (2.5) and its derivative is:
fˆ0(u) =β(1−2u)−β(1−2u)e−bu−b (2.6)
3 Local asymptotic stability of the disease free equilibrium:
Forx∗ = 0 , the linearised equation (2.4) reduces to :
x0(t) = (β−b)x(t)−βe−bωx(t−ω) (3.1) or
x0(t) = λx(t) +ηx(t−ω) (3.2) with λ= (β−b) and η =−βe−bω.
Theorem 3.1. The disease free equilibrium of equation (2.3) is locally lo asymptotically stable if θ <1 and unstable if θ >1.
The proof of this theorem is a direct consequence of the following two propo- sitions:
Proposition 3.1. Suppose equation (3.1) is (Mµ), then the free disease equi- librium is stable if θ <1 and unstable if θ >1.
Proposition 3.2. Equation (3.1) is (Mµ) for all θ >0.
Proof of Proposition 3.1:
By virtue Theorem 1.3, the study of stability returns to establish the sign of fˆ0(0).
According to equation (1.4), fˆ0(0) =β(1−e−bω)−b=b(θ−1)
It follows that: θ <1⇒fˆ0(0) <0 and θ >1⇒fˆ0(0)>0 Proof of Proposition 3.2:
We will prove that:
i) Ifθ < 1, thenc) of Theorem 1.2 is verified.
ii) If θ >1, then b) of Theorem 1.2 is verified.
Suppose that : θ >1. We have : η =−βe−bω <0 and η+λ =β(θ−1)>0, since θ >1.
So the condition (b) of Theorem 1.2 holds.
Suppose now 0< θ <1 and show that condition (c) of Theorem 1.2 holds.
i) We have here : η=−βe−bω <0 and η+λ=β(θ−1)<0.
ii) Since : ω|η|=βωe−bω = 1−eθ−bω bω
e−bω, by the finite increments formula there exists: 0≤α≤b, such that : 1−ebω−bω =e−αω.
Sinceθ < 1 andα < b, then we have : ω|η|=βωe−bω =θe(α−b)ω <1.
iii)ωλ−ln(ω|η |)−1 = βω−ln(βω)−1 > 0, this is always true because:
∀x∈R∗+− {1}
x−ln(x)−1>0.
We conclude that ifθ <1 then the condition (c) of Theorem 1.2 holds.
4 Local asymptotic stability of the endemic disease equilibrium :
For this casex∗ =i∗ = 1−β(1−eb−bω) = 1−1θ and the linearised equation around i∗ is:
x0(t) = [β(1−2i∗)−b]x(t)−βe−bω(1−2i∗)x(t−ω) (4.1) By takingi∗ = 1−β(1−eb−bω) = 1− 1θ in equation (4.1), we have:
x0(t) = (β(2−θ)
θ −b)x(t)−β(2−θ)
θ e−bωx(t−ω) (4.2)
Put: λ= β(2−θ)θ −b and η=−β(2−θ)θ e−bω.
Theorem 4.1. If θ > 1, then the endemic disease equilibrium of equation (2.3) is locally asymptotically stable .
Let us first prove the two propositions below :
Proposition 4.1.Suppose equation (4.2) is (Mµ) andθ > 1, then the endemic disease equilibrium is locally asymptotically stable.
Proposition 4.2. Equation (4.2) is (Mµ) for all θ >1.
Proof of Proposition (4.1):
The function ˆf defined by (1.4) is such that:
f(u) =ˆ β[1−u)]u−β[1−u]ue−bω−bu and ˆf0(u) = β(1−2u)(1−e−bω)−b.
Thus, ˆf0(i∗) =β(1−2i∗)(1−e−bω)−b=b[(1−2i∗)θ−1] =b[(2−θθ )θ−1] = b(1−θ) Then ˆf0(i∗)<0 for 1< θ.
Proof of Proposition 4.2:
We distinguish two cases:
1. Case 1< θ <2.
We will prove that the three conditions of c) in Theorem 1.2 hold:
i)It’s clear that: η=−β(2−θ)θ e−bω <0 and λ+η=b(1−θ)<0 ii) We have :
(s): ω |η|=βωe−bω2−θθ = 1−ee−bω−bω bω
(2−θ)
From the finite increments formula there exists 0 < α < b such that:
e−αω = 1−ebω−bω. Then , from (s) we have : ω | η | =e(α−b)ω(2−θ) As : 1< θ <2 and 0< α < b we have : 0<2−θ <1 and e(α−b)ω <1.
Finally : e(α−b)ω(2−θ)<1.
Thus : ω|η|<1
iii)ωλ−ln(ω | η |) > 1 ⇐⇒ ωβ(2−θ)θ −bω −lnωβ(2−θ)θ e−bω > 1 ⇐⇒
ωβ(2−θ)θ e−bω < eωβ(2−θ)θ e−bωe−1 ⇐⇒ωβ(2−θ)θ e−ωβ(2−θ)θ < e−1.
This is always true because xe−x < e−1 for allx 6= 1. We conclude that if 1< θ <2, then the condition (c) of Theorem (1.2) holds.
2. Caseθ > 2: We have η >0, condition (a) of Theorem 1.2 is verified.
Proof of Theorem 4.1:
Combining Proposition 4.1 , Proposition 4.2 and Theorem 1.3, we complete the proof of Theorem 4.1.
5 Numerical simulations:
In order to illustrate the results, we present the following numerical simula- tions.
Let take: ω= 10, b= 0.01 and β = 0.51.
Thus we have: θ'4.85329168>2 and i∗ '0.7939542756.
For different initial conditions, numerical simulations show that solutions con- verge to the endemic equilibrium i∗ ' 0.7939542756. See graphes (a),(b),(c) and (d) in Figure 2.
(a) ϕ(t) = 1+exp(t)2 (b)ϕ(t) = 0.5 + exp(3t)30
(c)ϕ(t) = 0.3 (d)ϕ(t) = 0.5 + 0.3exp(t)
Figure 2: Convergence to the endemic equilibrium for 4 different initial condi- tions ϕ∈C.
ω= 10, b= 0.01 and β = 0.51.
Conclusion :
The main goal of this paper was to apply monotone dynamical systems theory to study the local asymptotic stability of a SIS epidemiological model incorporating both an infectious period and exponential demographic
structure.
Via this approach, the local asymptotic behavior of the two equilibriums has been completely determined. This theory is not called nor developed in this work.
The study of the asymptotic stability of this model is mainly based on Theorem 1.2 cited in the preliminary section. Necessary and sufficient conditions for the linearized equation around an equilibrium to satisfy the property (Mµ) are given. In combination with the very important Theorem 1.3, which reflects the similarity between the stability of the model to that of the correspondent ordinary differential equation obtained by neglecting the delay, we establish necessary and sufficient conditions for the asymptotic stability.
Our method is elegant, very satisfactory and practical. We validated the paper’s results numerically with various numerical simulations (see Figure 2).
Using monotone approach, we find in Theorem 3.1 and Theorem 4.1 the same stability conditions established by Herbert W. Hethcote et al. in [13] via the characteristic equation’s roots method.
One of our perspectives is to extend this method to models described by systems with two or three equations. It is our belief that the main result of this work can give significan’t results concerning the oscillations and periodicity for that system.
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Received: January 18, 2015; Published: April 16, 2015
