Fluctuations in a SIS epidemic model with variable size population

Fluctuations in a SIS epidemic model with variable size population

A. Iggidr

^a

, K. Niri

^b,*

, E. Ould Moulay Ely

^b

aINRIA Nancy – Grand Est and LMAM UMR CNRS 7122, I.S.G.M.P. Bat. A, Ile du Saulcy, University of Metz, 57045 Metz Cedex 01, France

bUniversité Hassan II, Faculté des Sciences Aı¨n chock, Département de Mathématiques et Informatique, Casablanca, Morocco

a r t i c l e i n f o

Keywords:

Retarded differential equation Oscillation

Equilibrium

Basic reproduction number

a b s t r a c t

In an epidemiological model, time spent in one compartment is often modeled by a delay in the model. In general the presence of delay in differential equations can change the stabil- ity of an equilibrium to instability and causes the appearance of oscillatory solutions.

In this paper we consider a SIS epidemiological model with demographic effects: birth, mortality and mortality caused by infection. The delay is the period of infection. We define the concept of oscillation in the sense that solutions of the model studied fluctuate around a steady state. Our goal is to show that in this model, there are oscillating solutions for cer- tain parameters values. We determine a large set of initial data for which solutions of this model are slowly oscillating.

Published by Elsevier Inc.

1. Introduction

In an epidemiological model, time spent in one compartment is often modeled by a delay in the model.

Often the presence of delay in differential equations can change the stability of an equilibrium to instability and causes the appearance of fluctuating solutions. The best known (especially among physicists) are periodic solutions.

The existence of periodic solutions in the epidemiological models whose formulation contains a delay has been the sub- ject of several works[2,7,11,12].

However, it appears that little work in mathematical epidemiology has used the concept of oscillation in the sense that the solutions of the model studied fluctuate around a steady state without necessarily being periodic.

The oscillatory properties in the case of delay differential equations have been the subject of several publications. Without being exhaustive, we can cite[1,3–6,13–15,17,18]. An important reference in this field is the book of Gyori and Ladas[10].

In this paper we consider a SIS epidemiological model with demographic effects: birth, natural mortality and mortality due to infection. The total size of the population is not constant. The formulation of the model leads to a delay differential equation in which the delay corresponds to the duration of infection.

Our goal is to show first, for certain values of the parameters of this model, there are solutions oscillating around the en- demic equilibrium independently of the choice of initial data.

Secondly, we will make a qualitative analysis of the global behavior of the model. It appears that this model is monotone;

that is the order between the initial data is preserved by the solutions corresponding to them. This monotonicity property can already move us towards the type of initial data set for which the solutions are oscillating: initial data located above or below the equilibrium gives a solution which is also in the same situation. Therefore, initial data crossing the endemic equi- librium once during an interval of time equal to the duration of infection, can give a solution oscillating around the equilib- rium. In this case, we calculate the first peak and the first zero of fluctuation. We show that the distance between two consecutive zeros is greater than the duration of infection, this is called slowly oscillating solution.

0096-3003/$ - see front matter Published by Elsevier Inc.

doi:10.1016/j.amc.2010.03.040

*Corresponding author.

E-mail addresses:iggidr@loria.fr,iggidr@univ-metz.fr(A. Iggidr),khniri@yahoo.fr(K. Niri).

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2. Formulation of the model

We study in this section the SIS model described in[12]where a population of sizeN(t) is divided into susceptible and infective individuals, with numbers denoted bySðtÞ;IðtÞ, respectively. Thus

NðtÞ ¼ SðtÞ þ IðtÞ: ð2:1Þ

All new born are assumed to be susceptible. The birth rate as well as the natural death and the disease-induced rate are all assumed to be constant, and are, respectively, denotedb;d, andd. The death outflow from the susceptible class is then given bydSand death outflow from the infective class isðdþdÞI. The force of infection isbI=Nwithb>0 being the effective con- tact rate of an infective individual. Thus the individuals leave the susceptible class at ratebSI=N. It is assumed that the dis- ease confers no immunity, so that infective individuals return to the susceptible class after recovery. Therefore we shall use an SIS model. FortP0, letPðtÞbe the probability for an individual to remain in the infective class at leastttime units before returning to the susceptible class. The probabilityPðtÞis assumed to be nonnegative, nonincreasing withPð0^þÞ ¼1 and R1

0 PðuÞdu¼

x

, the mean length of infection, which is assumed to be positive and finite. The SIS model is formulated with a generalPðtÞsatisfying the above assumptions.

FortP0, the integral equation describing the evolution of the number of infective individuals is

IðtÞ ¼ I

0

ðtÞ þ

Z t 0

b

SðuÞIðuÞ

NðuÞ Pðt uÞexpfðd þ

dÞðt

uÞgdu: ð2:2Þ

The quantityI0ðtÞrepresents the number of initially infected individuals that are still infected at timet. The integral in(2.2)is the summation up to timetof individuals who became infected at timeuand who have neither recovered back to suscep- tible status nor died. The total population varies according to the differential equation

N

⁰

ðtÞ ¼ ðb dÞNðtÞ

dIðtÞ;

ð2:3Þ

Defining, respectively, the proportions of susceptible and infective individuals in the population by sðtÞ ¼ SðtÞ=NðtÞ;iðtÞ ¼IðtÞ=NðtÞ, Eq.(2.1)gives

sðtÞ þ iðtÞ ¼

1:

ð2:4Þ

Eq.(2.3)gives

N

⁰

ðtÞ ¼ ðb d

diðtÞÞNðtÞ;

ð2:5Þ

which integrates to

NðtÞ ¼ Nð0Þ

exp

ðb dÞt

d Z t

0

iðpÞdp

:

ð2:6Þ

Using the expression(2.6)withi0ðtÞ ¼I0ðtÞ=NðtÞ, Eq.(2.2)gives

iðtÞ ¼ i

0

ðtÞ þ

Z t 0

b½1

iðuÞiðuÞPðt uÞ

exp

ðd þ

dÞðt

uÞ þ

d Z t

u

iðpÞdp

du: ð2:7Þ

This is an integral equation foriðtÞ. The local existence, uniqueness and continuation were proved in[18]. It was shown, in [17], that the solution remains in the interval [0, 1] so it exists for alltP0.

In the case when the length of infection is equal to a constant

x

>0, thenPðtÞis a step function, namely

PðtÞ ¼

1 on

½0; x

0 on

ð x

;

1Þ:

For this step functionPðtÞ, the initial infective individuals must have recovered by time

x

, soi0ðtÞis zero fortP

x

. The inte- gral Eq.(2.7)gives

iðtÞ ¼

Z t

tx

b½1

iðuÞiðuÞ

exp

ðb þ

dÞðt

uÞ þ

d Z t

u

iðpÞdp

du: ð2:8Þ

FortP

x

the integral Eq.(2.8)is equivalent to the following delay-integro-differential equation:

i

⁰

ðtÞ ¼

b½1

iðtÞiðtÞ

b½1

iðt x Þiðt x Þexp ðb þ

dÞ

x þ

d Zt

tx

iðpÞdp

ðb þ

dÞiðtÞ þdi²

ðtÞ: ð2:9Þ

Using the notationh¼b^{1expððbþdÞ}_bþd ^xÞ, it has been proved in[17]that the disease free stateðs;iÞ ¼ ð1;0Þis always an equilib- rium for Eq.(2.9). This equilibrium is stable ifh<1, unstable ifh>1 and globally asymptotically stable ifd

x

⁶h61. A un- ique endemic equilibrium exists if and only ifh>1.

Whend¼0 (i.e., there is no disease-related death), the Eq.(2.9)is a delay-differential-equation. So, forh>1, the endemic equilibrium is explicitly given byðs;iÞ ¼ ð1=h;11=hÞ, and it is asymptotically stable ([8]).

3. Monotonicity of the model and existence of oscillating solutions

3.1. Monotonicity of the equation

In this section, we shall show that Eq.(2.9)is monotone: the order established between two initial functions is preserved by the corresponding solutions. We then use this to prove afterward that Eq.(2.9)has oscillating solutions about the ende- mic equilibrium. The notion of oscillations defined here is linked to the monotonicity in the following manner: we shall say that a functionxðtÞoscillates aboutyðtÞifxðtÞandyðtÞare never comparable in the sense of the usual order inCð½

x

;0;R^þÞ.

In particular, a solutionxðtÞof(2.9)is oscillating aboutiif we do not havexðtÞ6inorxðtÞPifort>t0for somet0. Cooperative systems and many compartmental models satisfy the monotonicity property.

The oscillating properties of delay scalar differential equation solutions have been used by Mallet-Paret in the elaboration of Morse’s theory for delay equation[16](see also works of Cao[9]).

In the case of differential delay equation the study of the monotonicity properties has been done, first by Arino and Segu- ier[5], then by Arino and Seguier[6]. More recently, Smith has developed the theory of monotone dynamics systems for de- lay differential equations[19–21].

The study of oscillations for monotone systems has also been the topic of the thesis[18]and has been considered in[3].

Here, we are interested in the behavior of solutions of(2.9)in a neighborhood of the steady solutionxðtÞ i. The follow- ing three assumptions on the parameters will be considered:

½H

1

:b 1

e

^ðbþd=2Þx

b þ

d=2

<2;

½H

2

:b 1

e

^ðbþd=2Þx

b þ

d=2

>2;

½H

3

:b 1

e

^bx

b þ

d=2

>2

Proposition 3.1. ConsiderEq.(2.9)and let iðtÞbe the solution of(2.9)corresponding to an initial function

u

. 1. Suppose that½H1is verified. Then the endemic equilibrium satisfies i<1=2.

2. Suppose that½H2is verified. Then the endemic equilibrium satisfies i>1=2.

3. Suppose that½H3is verified. Then,¹₂6

u

ðtÞ61)¹₂6iðtÞ61.

Remark. ½H3implies½H2.

Proof. We recall the expression of the thresholdh:

h

¼

b1

expððb

þ

dÞ

x Þ

b þ

d :

First, one can notice thath¼hðdÞis a decreasing function ofd.

1. Suppose½H1is satisfied. The endemic equilibrium satisfies the equation:

Fði

Þ ¼

0;

ð3:1Þ

where

FðiÞ ¼ 1 þ

bð1

iÞð1 e

^{ðbþð1iÞdÞx}

Þ

b þ ð1 iÞd

:

ð3:2Þ

The mapi#FðiÞis decreasing. Moreover we haveFð0Þ ¼ 1þð^1eðbdÞxÞ^b

bþd ¼ 1þh>0;Fð1=2Þ ¼ 1þ ^1e

ðbd 2Þx

b

2ðbþ^d₂Þ <0 thanks to½H1, andFð1Þ ¼ 1<0. Thus there is a unique nontrivial equilibriumiand it satisfiesi<1=2.

2. The proof is similar to caseH1.

3. Suppose that the condition½H3is satisfied. We assume that¹₂<

u

ðtÞ<1for allt2 ½

x

;0and we prove that¹₂<iðtÞ<1 for alltP0.

Suppose there exists a timetsuch thatiðtÞ ¼¹₂. Lettbe the first time for whichiðtÞ ¼¹₂. The timetsatisfiesiðtÞ>¹₂if t<t andiðtÞ<¹₂ift<t<tþ

g

, for some

g

>0. In particular, we would haveiðt

x

Þ>¹₂andi⁰ðtÞ60. But, from Eq. (2.9), we have i⁰ðtÞP^b₄^b₄e^bxbþd₂ þd=4Pð2bþdÞh^b₂ð^1e_bþd=2^bxÞ 1i

>0, which is in contradiction with i⁰ðtÞ60.

ThusiðtÞP1=2 for alltP0. h

The following theorem shows that the order established between two initial functions is preserved by the corresponding solutions of Eq.(2.9)whend¼0.

Theorem 3.1. Supposed¼0and½H3is fulfilled. Then, the Eq.(2.9)is monotone; that is, for each pair of initial conditions

u

andw in Cð½

x

;0;½1=2;1Þ, with

u

ðtÞ6wðtÞ, for every t2 ½

x

;0, if xðtÞ(resp. y(t)) is the solution of(2.9)verifying x0¼

u

(resp.

y₀¼w) then xðtÞ6yðtÞ, for tP0.

Proof.Eq.(2.9)can be written in the following condensed form:

i

⁰

ðtÞ ¼ gðiðtÞÞ

bfðt;

i

t

Þ

for

t

P0

i

0

ðtÞ ¼ u ðtÞ

for

x

⁶

t

60;

(

ð3:3Þ

wheregðuÞ ¼bð1uÞubu.

itis the function defined on½

x

;0byitðhÞ ¼iðtþhÞfor everyt.

fðt; uÞ ¼ ½1 uð x Þuð x Þ

expðb

x Þ

Let

u

andwbe two initial functions inCð½

x

;0;½1=2;1Þ, and suppose that

u

ðtÞ6wðtÞ. Letx(respectivelyy) be the corre- sponding solution with initial functions

u

ðtÞ(respectivelywðtÞ). We will show that for alltP0, one hasxðtÞ6yðtÞ. Thanks to Assumption½H3andProposition 3.1, it is actually sufficient to comparexðtÞandyðtÞon½0;

x

.

It is convenient, for the comparison, to substitute fory, the solution of a perturbed problem which keeps the same properties as the initial problem, but which makes it possible to obtain strict comparisons. For

>0, we denote byy

the solution of perturbed equation:

dy

dt

¼ gðy

ðtÞÞ

bf t;

y

^e_t

þ

for

t

P0

y

ðtÞ ¼

wðtÞ for

x

⁶

t

60:

8<

:

ð3:4Þ

Eq.(2.9)is the non perturbed equation andy⁰¼y. We shall use the following lemma relating the solutions of(3.4)to those of (2.9). h

Lemma 3.1. There exists

e

0>0, such that for

e

;06

e

⁶

e

0, and for T>0;y^eexists on½0;T. Moreover y^econverges uniformly on

½0;Tto y, solution of(2.9).

The proof of this lemma is in ([18, p. 33]).

Let us now comparey^eandxon½0;T. We have:

ðy

^e

Þ

⁰

ðtÞ x

⁰

ðtÞ ¼ gðy

^e

ðtÞÞ

bf

ðt; y

_t

Þ gðxðtÞÞ þ

bf

ðt; x

t

Þ þ ¼ gðy

^e

ðtÞÞ gðxðtÞÞ þ

b

fðt; x

t

Þ f t; y

_t

þ

: We introduce the following functions:

AðtÞ ¼ gðy

^e

ðtÞÞ gðxðtÞÞ:

BðtÞ ¼

b½f

ðt; x

t

ÞÞ fðt; y

_t

Þ:

We haveBðtÞP0 fort2 ½0;

x

thanks to½H3and to the fact that the map

v

^#ð1

v

Þ

v

is decreasing on½1=2;1. On the other hand, we can writeAðtÞ ¼ ½y^eðtÞ xðtÞhðtÞ. We then have:

d

dt ðy

^e

ðtÞ xðtÞÞ ½y

^e

ðtÞ xðtÞhðtÞ ¼ BðtÞ þ e

;

d

dt ½y

^e

ðtÞ xðtÞ

exp

Z t

0

hðsÞds

¼ ½BðtÞ þ e

exp

Z t

0

hðsÞds

: We definezðtÞ ¼ ½y^eðtÞ xðtÞexp Rt

0hðsÞds

,zðtÞverifies the equation:

d

dt zðtÞ ¼ ½BðtÞ þ e

exp

Z t

0

hðsÞds

:

We prove thatzðtÞ>0 for allt. Suppose that this is not true. Lett₀be the first value oftsuch that the inequalityzðtÞ>0 ceases being satisfied. Then, zðt0Þ ¼0;zðtÞ>0 for t<t0, and zðtÞ<0 if t0<t<t0þ

g

, for some

g

>0. We have

d

dtzðtÞ ¼ ½BðtÞ þ

e

exp Rt 0hðsÞds

>0.

So,z⁰ðt₀Þ>0, and sincezðt₀Þ ¼0, we must havezðtÞ>0 fort₀<t<t₀þ

g

⁰with

g

⁰>0. Hence we reach a contradiction.

We deduce thaty^eðtÞ>xðtÞ

8

tP0.Lemma 3.1allows to finish the proof ofTheorem 3.1.

As a consequence ofTheorem 3.1, we have the following result which characterizes the global behavior of solutions of (2.9).

Proposition 3.2. Consider Eq.(2.9)withd¼0, and suppose that½H3is fulfilled. Let

u

be an initial function in C½

x

;0;¹₂;1 , then:

1. if

u

ðtÞPi, then the corresponding solution i(t) is such that iðtÞPifor all tP0;

2. if¹₂6

u

ðtÞ6i, then the corresponding solution iðtÞis such that¹₂6iðtÞ6ifor all tP0.

4. Existence of oscillating solutions

4.1. Existence of oscillating solutions

We give now some definitions of the oscillation concept.

Definition 4.1. Letxbe a continuous function defined on some infinite interval½a;1Þ. The functionxis said to oscillate or to be oscillatory about zero ifxhas arbitrarily large zeros. That is, for everyb>athere exists a pointc>bsuch thatxðcÞ ¼0.

Otherwisexis called non-oscillatory.

Definition 4.2. Letxbe a continuous function defined on some infinite interval½a;1Þ. The functionxis said to oscillate or to be oscillatory about a steady statexifxxoscillates about zero in the sense ofDefinition 4.1.

The monotonicity of Eq.(2.9)defined in Section3.1plays an important role in the existence of oscillating solutions about the endemic equilibrium. The following theorem illustrates the relation between these two concepts.

Theorem 4.1. Consider Eq. (2.9) with d¼0, and suppose that ½H3 is fulfilled. Choose

u

and w two initial functions in Cð½

x

;0;R^þÞsuch that¹₂6

u

⁶i6w<1;

u

–i;w–i;

u

andwbeing linearly independent.

Then, among the family of convex combinations of

u

andw;ð1kÞ

u

þkw, there is at least onek;06k61, such that the solution of(2.9)starting fromð1kÞ

u

þkwis oscillating about the endemic equilibrium i

Proof. A similar proof for a particular case of Eq.(2.9)has been made in[3].

We denote byiðt;wÞa solution of(2.9)with initial functionw. Let us define the sets :

K

^þ

¼ f0

6k61:

iðt; ð1

kÞ

u þ

kwÞP

i

;

iðt; ð1

kÞ

u þ

kwÞX

i

g:

K

¼ f0

6k61:

iðt; ð1

kÞ

u þ

kwÞ6

i

;

iðt; ð1

kÞ

u þ

kwÞX

i

g

:

We observe readily that 12K^þ;02K, and by monotonicity of the Eq.(2.9),K^þandKare intervals.

IfK^þ[Kis strictly contained in [0, 1], the conclusion of the theorem is verified. So, let us consider the situation where K^þ[K¼ ½0;1. Noting thatK^þ\K¼ ;, except if we hadiðtÞ ¼ifortlarge which is excluded in the definition ofK^þand K. We then have two possibilities either,K¼ ½0;aÞandK^þ¼ ½a;1, orK¼ ½0;aandK^þ¼ ða;1. In both cases we have 0<a<1.

Let us study the first case. The other one will follow the same line. Set

zðtÞ ¼ iðt; ð1 aÞ u þ awÞ; a 2 K

^þ:

By definition ofK^þ, we havezðtÞ>ifortP0. But the continuity of the solution implies that, for some

q

>0, and forksuch thata

q

<k6a, we still haveiðt;ð1kÞ

u

þkwÞPi;–i, for alltP0. This implies thata

q

;a K^þ, in contradiction with the definition ofa. So,K^þ[K–½0;1, which yields the desired conclusion h

4.2. Sufficient conditions of existence of oscillating solutions

Our goal in this section is to show that under sufficient conditions for certain values of the parameters of this model, there are solutions oscillating around the endemic equilibrium independently of the choice of initial data.

Recall the Eq.(2.9):

i

⁰

ðtÞ ¼

b½1

iðtÞiðtÞ

b½1

iðt x Þiðt x Þ

exp

ðb þ

dÞ

x þ

d Z t

tx

iðpÞdp

ðb þ

dÞiðtÞ þdi²

ðtÞ

:

ð4:1Þ

Eq.(4.1)can be written in the form:

i

⁰

ðtÞ ¼ a ½K iðtÞiðtÞ

bexpððb

þ

dÞ

x Þ½1 iðt x Þiðt x Þ

exp d Zt

tx

iðpÞdp

ð4:2Þ

with

a

¼bd;K¼1_bd^b.

Theorem 4.2. Consider the Eq.(4.2)and suppose that iis asymptotically stable.

Then, if

a

>0, the condition

K

2<

i

<1 2

e

^ðbþdÞx

2be

x

is sufficient so that all solutions ofEq.(4.2)are oscillating.

Proof.The principle of this proof is to compare our model to a linear differential equation with delay for which necessary and sufficient conditions for oscillations are known. We particularly use the following famous results whose complete proofs are done in ([10]: Theorems 2.2.3 and 3.2.2):

(1) The equation y⁰ðtÞ þpyðt

x

Þ ¼0is oscillating iff p

x

>¹_e. (2) For p>0;

x

>0, the following statements are equivalent:

The delay differential equation y⁰ðtÞ þpyðt

x

Þ ¼0has a positive solution.

The delay differential inequality y⁰ðtÞ þpyðt

x

Þ60has a positive solution.

AssumeiðtÞis a solution of(4.2)such thatiðtÞ>iand let

v

^{ðtÞ ¼iðtÞ i>0. Then 0<}

v

^{ðtÞ<1, and}

v

^ðtÞis solution of

v

⁰

^{ðtÞ ¼ a ½K}

2

i

v ^ðtÞ v ^ðtÞ

^b1

½1

2i

v ^ðt x Þ v ^ðt x Þ

exp d Z t

tx

v ^ðpÞdp

þ a ½K i

i

b₁

½1 i

i

exp d Z t

tx

v ^ðpÞdp

Withb₁¼be^ðbþdÞxe^dxi. This can be written

0

¼ v

⁰

^ðtÞ a ½K

2

i

v ^ðtÞ v ^{ðtÞ þ}

^b1

½1

2i

v ^ðt x Þ v ^ðt x Þexp

d Z t

tx

v ^ðpÞdp

a ½K i

i

þ

b₁

½1 i

i

exp d Zt

tx

v ^ðpÞdp

:

Sinceiis an equilibrium of Eq.(4.2), we have 0¼

a

½Kiib₁½1ii. So

a

½Ki>0, and

a ½K i

i

þ

b1

½1 i

i

exp d

Z t tx

v ^ðpÞdp

¼ a ½K i

i

exp d Z t

tx

v ^ðpÞdp

1

>0:

Since

a

>0 andi>^K₂we have

a

½K2i

v

^ðtÞ

v

^ðtÞ<0.

So we deduce

0>

v

⁰

^{ðtÞ þ}

^b1

½1

2i

v ^ðt x Þ v ^ðt x Þ

exp d Z t

tx

v ^ðpÞdp

On the other hand we have 0<

v

^ðtÞ<1 which implies b₁

½1

2i

v ^ðt x Þ v ^ðt x Þ

exp d

Zt tx

v ^ðpÞdp

>b₁

½1

2i

v ^ðt x Þ v ^ðt x Þ:

Therefore,

0>

v

⁰

^{ðtÞ þ}

^b1

½1

2i

v ^ðt x Þ v ^ðt x Þ

:

On the other hand ifiis asymptotically stable thenlimt!1

v

^{ðtÞ ¼0. So}

8 e

>0;

9A

>0:

8 t

>

A þ x

;1

2i

v ^ðt x Þ

>1

2i

e

and then,

0P

v

⁰

^{ðtÞ þ}

^b1

½1

2i

e v ^ðt x Þ;

For 0<

e

<12i_xeb¹

1, we deduce that inequality

v

^{0ðtÞ þb}1½12i

e v

^ðt

^x

^Þ<0 has a positive solution.

But the condition^K₂<i<¹₂^eðbþdÞ_xeb^ximplies

x

b₁½12i

e

>¹_e.

This is contradictory with the result cited at the beginning of our proof.

Therefore, the assumptioniðtÞ>iis false. h

4.3. Existence of slowly oscillating solutions when there is no disease-related death

Our aim in this section is to define a notion of slow oscillations and to prove in the cased¼0 the existence of a suffi- ciently large set of initial functions which are transformed into slowly oscillating solutions by the flow.

Definition 4.3.Letxbe a function defined on some interval½t0;þ1Þ. We will say thatxisslowly oscillatingifxis oscillating in the sense ofDefinition 4.1,xis alternatively positive and negative between its zeros and the distance between two successive zeros is not less than

x

.

Definition 4.3eliminates the case where the solution is vanishing as from a certain time.

LetCbe the cone defined by

C ¼ f u 2 Cð½ x

;0;R^þ

Þ

; such that

9 c 2 ½ x

;0;

u ð c Þ ¼ i

;

u ðsÞ

<

i

for

s

<

c

and

i

6

u ðsÞ

6

u ð0Þ

for

c

<

s

60

g

: Whend¼0, Eq.(2.9)becomes:

i

⁰

ðtÞ ¼

b½K

iðtÞiðtÞ

bexpðb

x Þ½1 iðt x Þiðt x Þ ð4:3Þ

withK¼1^b_b

In the following theorem we prove that there exist slowly oscillating solutions of(4.3). More precisely we state that solu- tions of(4.3)corresponding to initial functions

u

in the coneCreturn toCat a sequence of timetkwhose distance is larger than the delay 2

x

.

Theorem 4.3. Let i be a solution of Eq.(4.3),with

u

as initial function. Assume that

u

2C. Then, the solution i is slowly oscillating about the endemic equilibrium.

To proveTheorem 4.3, we need to prove first some technical lemmas that we give hereafter. We begin by doing two changes of variable:

The first change of variable isiðtÞ ¼xðtÞ þi. The mapt!xðtÞis then a solution of:

x

⁰

ðtÞ ¼

bxðtÞ½K0

xðtÞ

bexp

ð ðb x Þ Þ½K

1

xðt x Þxðt x Þ

;

ð4:4Þ

whereK₀¼K2iandK₁¼12i.

The second change of variable isyðtÞ ¼xðtÞe^bRt 0ðK0xðsÞÞds

¼xðtÞpðtÞ, where

pðtÞ ¼ e

^bR^t

0ðK0xðsÞÞds

: So we have:

iðtÞ ¼ i

() xðtÞ ¼

0

() yðtÞ ¼

0:

We will study the slowly oscillations ofyabout zero which is equivalent to studying the slow oscillations ofiabouti. The mapt!yðtÞis the solution of:

y

⁰

ðtÞ ¼ b e

^bx

pðtÞ yðt x Þ

pðt x Þ K

1

yðt x Þ pðt x Þ

¼ be

^bx

pðtÞ

pðt x Þ yðt x Þ K

1

yðt x Þ pðt x Þ

:

This can be written in a condensed form

y

⁰

ðtÞ ¼ r PðtÞQ ðt x

;

yðt x ÞÞ; ð4:5Þ

where:

r ¼

bexpðb

x Þ;

PðtÞ ¼ pðtÞ pðt x Þ

;

Q ðt; uÞ ¼ u K

1

u

pðtÞ

:

Lemma 4.1. the functions P and Q satisfy

1.

8

t; 0<PðtÞ<e^{ðrð1iÞþbÞx}, 2. Qðt;0Þ ¼0

8

t,

3. Qðt;uÞ<0 foru<0;K1>0 and

8

t,

4. Qðt;uÞ>0 foru>0;K1>0;u<K1pðtÞand

8

t, 5. Qðt;uÞ<uforu>0 and

8

t.

Proof. (2)–(4) are obvious.

1. We have

K i

16

K

0

x

6

K

0

þ i

¼ K i

; isatisfies

bðK

i

Þ ¼

be^bx

ð1 i

Þ ¼ r ð1 i

Þ:

So

r ð1 i

Þ

6

bðK

0

xÞ

6

r ð1 i

Þ þ

b;

e

^rð1iÞx6

e

^bR^t

txðK₀xðsÞÞds

6

e

^{ðrð1iÞþbÞx};

e

^rð1iÞx6

PðtÞ

6

e

^{ðrð1iÞþbÞx}6

e

^bx:

5. Foru>0, we have^Qðt;uÞ_u ¼K1_pðtÞ^u ¼12i_pðtÞ^u <1. h

Lemma 4.2.Suppose that be^bx>band i<¹₂.

Let

u

an initial function and yðtÞthe corresponding solution ofEq.(4.5).

If

u

<K1e^bx then yðtÞ<K1pðtÞ.

Proof.

i

<1

2

) K

1>0:

We suppose that

u

<K1and we prove that the solution of(4.4)verifiesxðtÞ<K1. We deduce thatyðtÞ<K1pðtÞ.

Suppose there exists a first timet0such thatxðt0Þ ¼K1. Then,xðtÞ<K1ift<t0andxðt0Þ>K1ift0<t<t0þ

g

, for some

g

>0. In particular, we would havexðt0

x

Þ<K1andx⁰ðt0Þ>0 . But, from Eq.(4.4), we have

x

⁰

ðt

0

Þ ¼

bK1

½K

0

K

1

r xðt

0

x Þ½K

1

xðt

0

x Þ ¼

bK1

½K

0

K

1

r K

1

xðt

0

x Þ þ r x

²

ðt

0

x Þ

¼

bK1

b

b

r K

1

xðt

0

x Þ þ r x

²

ðt x Þ

<

b K

1

þ r K

²₁

¼ ðb þ r K

1

ÞK

1

¼ ðb þ r ½1

2

i

Þ K

1<

ðb þ r Þ K

1

¼ K

1

½b þ

bexpðb

x Þ

<0:

Since we are studying the slowly oscillations ofyabout zero and because the properties in theLemma 4.2, we consider the following coneC0instead of the coneC:

C

0

¼ u 2 Cð½ x

;0;R^þ

Þ;

such that

9 c 2 ½ x

;0;

u ð c Þ ¼

0;

u ðsÞ

<0 for

s

<

c

; and 06

u ðsÞ

6

K

1

e

^bxfor

c

⁶

s

60 : We prove the following preliminary lemmas.

We give inLemma 4.3the first peak above the endemic equilibrium of the solution, inLemma 4.4the first zero of the solution and inLemma 4.5the first peak of the solution below the endemic equilibrium.

Lemma 4.3.Let

u

2C0, and t₀¼

c

þ

x

.

The solution y(t) of the Eq.(4.5)corresponding to the initial function

u

, is increasing on the interval0;t₀

, non-increasing on an interval on the right of t₀and y⁰ t₀ ¼0.

Proof.At the pointt₀, we havey⁰ t₀ ¼

r

P t₀ Qð

c

;yð

c

ÞÞ ¼

r

P t₀ Qð

c

;

u

ð

c

ÞÞ ¼

r

P t₀ Qð

c

;0Þ ¼0.

For 0<t<t₀, we have

x

⁶t

x

<

c

, hence

u

ðt

x

Þ<0.

Lemma 4.1impliesQðt

x

;

u

ðt

x

ÞÞ<0.

And soy⁰ðtÞ ¼

r

Pðt

x

ÞQðt

x

;yðt

x

ÞÞ ¼

r

Pðt

x

ÞQðt

x

;

u

ðt

x

ÞÞ>0.

Fort₀<t<

x

, we have

c

⁶t

x

<0. thus

u

ðt

x

Þ>0.

u

2C0)

u

ðt

x

Þ6K1e^bx)Qðt

x

;

u

ðt

x

ÞÞ>0 theny⁰ðtÞ<0.

We deduce thatyðt₀Þis the maximum of the values ofyover an interval containingt₀. h

Lemma 4.4. Let yðtÞ be the solution of the Eq. (4.5) corresponding to the initial function

u

2C0 and suppose

K

2¼¹₂1^b_b

6i<¹₂. Then, there exists a first finite positive time t1Pt₀þ

x

such that yðt1Þ ¼0. Moreover, yðtÞis non-increasing on the interval½t₀;t1.

Proof.We prove that yðtÞ is non-increasing on t₀;t1þ

x

. Indeed, for t2t₀;t1þ

x

, we have

c

⁶t

x

⁶t1. So, yðt

x

ÞÞP0 for eacht2t₀;t1þ

x

.

Since

u

2C0)yðt

x

Þ6K1pðt

x

ÞbyLemma 4.2. We havey⁰ðtÞ ¼

r

PðtÞQðt

x

;yðt

x

ÞÞÞ60 .

We show thatt1is finite: Ad Absurdum, suppose thatyðtÞ>0 for alltPt₀or equivalentlyiðtÞ>ifor alltPt₀. On the other hand we haveiðtÞ>ifor all

c

<t6t₀thanks toLemma 4.3. Thus, fortPt₀we havep⁰ðtÞ ¼ bðK0xðtÞÞpðtÞ.

K0xðtÞ ¼K2i ðiðtÞ iÞ ¼K ðiðtÞ þiÞ<K2i60 sincei>^K₂. Therefore the functiont#pðtÞis increasing, and hencepðtÞ>pðt

x

Þfor alltPt₀. This impliesPðtÞ>1 for alltPt₀.

Nowy⁰ðtÞ ¼

r

PðtÞQðt

x

;yðt

x

ÞÞ<

r

Qðt

x

;yðt

x

ÞÞ.

Qðt

x

;yðt

x

ÞÞ ¼ ðK1xðt

x

ÞÞyðt

x

Þ.xðt

x

Þ !0 ast! 1. Soxðt

x

Þ6K1_{r x}¹, fort

x

PTfor someT>0.

So, fortPTþ

x

,y⁰ðtÞ<_x¹yðt

x

Þ. Sinceyis decreasing, we haveyðTþ

x

Þ6yðt

x

Þ6yðTÞforTþ

x

⁶t6Tþ2

x

. Hence, y⁰ðtÞ<_x¹yðTþ

x

Þ forTþ

x

⁶t6Tþ2

x

. Integrating, we get yðTþ2

x

Þ yðTþ

x

Þ<yðTþ

x

Þwhich implies yðTþ2

x

Þ<0 which is in contradiction to the assumptionyðtÞ>0 fort>t₀.

On the other hand, using (1), (5) ofLemma 4.1, and the fact thaty t₀ is the maximum ofyont₀;t1þ

x

, we have:

y

⁰

ðtÞ ¼ r PðtÞQ ðt x

;

yðt x ÞÞ

>

be

^rð1iÞx

yðt x Þ

>

be

^rð1iÞx

yðt

₀

Þ 8 t 2

t

₀;

t

1

þ x

: Integrating on the intervalt₀;t

ðt2t₀;t1þ

x

Þyields:

ðyðtÞ

P

y t

₀ h1

be^rð1iÞx

t t

₀i :

Fort¼t1, we deduce that 0P1be^rð1iÞxt1t₀

, which implies thatt1t₀P ¹

berð1i Þx>

x

. Thent1Pt₀þ

x

. h

Lemma 4.5. Let t₁¼t1þ

x

, the solution y(t) of the Eq.(4.1)is non-increasing on t1;t₁;

and y⁰ t₁ ¼0.

Proof. Fort2t1;t₁

;t1

x

⁶t

x

⁶t1.

In view of the inequality inLemma 4.4,t₀6t

x

⁶t1, so:yðt

x

Þ>0 fort2t1;t₁

and then,

y

⁰

ðtÞ ¼ r KPðtÞQ ðt x

;

yðt x ÞÞ

<0

8 t 2 t

1;

t

₁

y t

₁

x

¼ yðt

1

Þ ¼

0

) y

⁰

t

₁

¼

0:

Proof ofTheorem 4.3: Lett₁be as defined inLemma 4.4and definewðtÞ ¼yðtþt₁þ

c

Þfort2 ½

x

;0.

Then, the functionwhas a zero on the interval½

x

;0.

Precisely, we havewð

c

Þ ¼yðt1Þ ¼0. So,w2 C.

We can see that the same technique that we used inLemmas 4.1–4.4 and 4.5 works if we assume

u

2 C, with the changes: non-decreasing to non-increasing, and non-increasing to non-decreasing. If

u

2 C, we havew2C.

So, starting from

u

2C, the solution y is non-increasing ont₀;t₁

, then it is non-decreasing on t₁;t2

, etc.

We obtain a sequencet_j;t_j, verifyingt_j t_j1P

x

;t_j

x

⁶t_j6t_j, such thatð1Þ^jyðtÞis non-decreasing onht_j1;t_ji and y⁰ðt_jÞ ¼0¼yðtjþ1Þ, forjP0.

The distance between two successive zeros is not less than

x

.

The behavior of

u

andyover the interval½

x

;t2is illustrated inFig. 1.

The following lemma shows that the solution does not degenerate and the peaks do not collapse.

Lemma 4.6. Let y be a solution of Eq.(4.5),with

u

as initial function.

(a) Suppose that

u

is such that:

u

ð

c

Þ ¼0for all

c

2 ½

x

;0. Then, yðtÞ ¼0, for all tP0.

(b) Suppose that

u

2C. Suppose that there exists t0P0such that yðtÞ ¼0for tPt0. Then, yðtÞ ¼0for all tP0, and

u

is such that

u

ð

c

Þ ¼0for all

c

2 ½

x

;0.

(c) Suppose that

u

2C, and consider the sequence t_{i iP0}as inLemma4.3.

Then, we have y t _i

>0for all iP0.

Proof. The properties (a) an (b) are obvious. If the delay

x

in Eq.(2.9)is not constant, then there exist other solutions than y¼0 which die out eventually, see[17].

(c) Att₀, we have,

u

2C)

u

ð

x

Þ<0)y⁰ð0^þÞ>0)yðt₀Þ>0.

Att₁, we haveyðt₁Þ<0.

Fig. 1.Behaviour of an initial functionuand its corresponding solutiony.

Otherwiseyðt₁Þ ¼0, this implies thatyis constant ont1;t₁

, thereforey¼0 ont1;t₁ .

We can deduce by the form of Eq.(4.5)thatyðtÞ ¼0 fortPt1. So by property (b)yðtÞ ¼0 for allt, in contradiction to

u

2C.

We complete the proof of the theorem as inLemma 4.6by defining a new initial datawðtÞ ¼y t t₁

. We obtainy t₂ >0 andy t₃ <0 and so on. h

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