STABILITY OF A DELAYED SYSTEM MODELLING …...STABILITY OF A DELAYED SYSTEM MODELLING HOST-PARASITE ASSOCIATIONS SANDOR KOVACS AND SHABAN A. H. ALY ABSTRACT. This paper deals with

CANADIAN APPLIED

MATHEMATICS QUARTERLY

Volume 18, Number 1, Spring 2010

STABILITY OF A DELAYED SYSTEM

MODELLING HOST-PARASITE ASSOCIATIONS

SANDOR KOVACS AND SHABAN A. H. ALY

ABSTRACT. This paper deals with stability analysis of adelayed SIS epidemiological model with disease-induced mortal-ity and nonlinear incidence rate. Conditions are derived underwhich there can be no change in stability. Using the discretetime delay as a bifurcation parameter it is found that Hopf bi-furcation occurs when the delay passes through a critical value.In case of the single delay a formula for determining the stabilityof the periodic solutions is given by using the centre manifoldtheory and the normal form method. Results are verified bycomputer simulations.

1 Introduction In modelling the spread of infections the popula-tion is usually considered to be subdivided into disjoint epidemiologicalclasses (or compartments) of individuals in relation to the infectiousdisease: susceptible individuals, S, exposed individuals, E, infectiousindividuals, I and removed individuals, R. The development of the in-fection is represented by transitions between these classes. The numberof compartments included depends on the disease being modelled. If wetake into account that the acquired immunity to reinfection is virtuallynon existent and hence recovered individuals pass directly back to thecorresponding susceptible class then we deal with so called SIS models.These situations can be described, at least up to a crude first approxi-mation, by a simple system of first order ordinary differential equationsfor the rates of transfer from one compartment to another. Studies ofthe dynamical properties of such models usually consist of finding con-stant equilibrium solutions, and then conducting a linearized analysis todetermine their stability with respect to small disturbances.

AMS subject classification: 92D30 (34C23, 37G15).Keywords: Epidemic model, time delay, characteristic equation, Hopf bifurca-

tion.Copyright c©Applied Mathematics Institute, University of Alberta.

59

60 S. KOVACS AND S. A. H. ALY

In order to have more realism, it is often necessary taking into con-sideration that the present dynamics, the present rate of change of thestate variables depends not only on the present state of the processesbut also on the history of the phenomenon, on past values of the statevariables. Thus, for some disease transmission models the incorporationof time delay effect is necessary: in case of, e.g., tuberculosis, influenza,measles, on adequate contact with an infective, a susceptible individualbecomes infected but is not yet infective (cf. [12]). However, there areother aspects which motivate the incorporation of time delay effect intothe system describing disease transmission, namely, it can also be used tomodel a period of temporal immunity (in a model including R), a latentperiod (in a model including E) or a maturation period (cf. [15] and thereferences therein). If delay effects must be considered then these mod-els are formulated as a system of a functional differential and/or integralequations. Delays in epidemic models can destabilize the originally sta-ble equilibrium, so that periodic solutions arise by Hopf bifurcation. Fora survey of epidemiological models with delays we refer to [14].

The time evolution model, we are dealing with was proposed in [1].This is a SIS epidemiological model with disease-induced mortality andbilinear incidence rate, in which the population is assumed to be dividedin two interacting classes of individuals, namely susceptibles S and in-fectives I . This model has, contrary to the most classical models (cf.[17]), a variable demographic structure: the total population, the sum ofthe numbers in all compartments was here assumed to be not constant.

Our paper is organized as follows. In Section 2, we introduce themodel. In Section 3 we summarize the results about the model withoutdelay regarded the existence and global stability of possible equilibria.The main result of this paper is given in Section 4. We examine thestability of the constant equilibria of the delayed system and the occur-rence of periodic orbits. The stability of these orbits in a special caseis determined by constructing a centre manifold and by applying thenormal form method. In Section 5 we give a short discussion.

2 The model Microparasites can have pronounced effects on thegrowth characteristics of their host population. The work [1] focuses onmicroparasitic infections that are directly transmitted among inverte-brate hosts and deals among others with the following first order ODE

AN SIS EPIDEMIC MODEL WITH DELAY 61

system

(1)S = fS(S, I) := a(S + I) − bS − βSI + γI,

I = fI(S, I) := βSI − (α + b+ γ)I.

Here, the dot means differentiation with respect to time t; S(t) and I(t)are the densities of the populations of mature susceptible and infectivehosts at time t. a > 0 represents the birth rate of the population (itis assumed that all newborns are susceptible and the birth rate is inde-pendent of whether or not the host is infected); b > 0 and α > 0 areparameters associated with the death rate of susceptible and infectiveclasses; γ > 0 is the rate at which infected individuals recover and againbecome susceptible to re-infection; β > 0 denotes the per capita hor-izontal transfer rate between a single susceptible and a single infectedindividual. A schematic representation of the model is shown in Figure 1.

The rate of change of the total population of hosts, N := S + I , isobtained by adding equations in (1) to give

(2) N = rN − αI

where r := a − b is the intrinsic growth rate of the disease-free hostpopulation. Thus, the total population is not a constant, but rather adynamic variable (cf. [7, 10, 13]). Models with variable population sizeare interesting from several reasons. Instead of one threshold given bythe basic reproduction number, these models can involve several thresh-olds that determine the asymptotic behaviour (cf. [23]).

In order to have a realistic model, we need to take into account thatin SIS epidemiologic models, susceptibles become infectious after a suf-ficient contact with an infective. Therefore, it is reasonable to assumethat the migration of the individuals from the class of susceptibles intothe one of infected is subject to delay. We assume furthermore that theoffspring of susceptible and infective parents are immune to the diseasefor a period, after which they become susceptible, hence the inflow ofnewborns into the susceptible class is also subject to time delay. Thus,the following system will be considered

(3)S = a(S(· − τ2) + I(· − τ2)) − bS − βSI + γI,

I = βS(· − τ1)I(· − τ1) − (α+ b+ γ)I,

with initial conditions

S0(ϑ) = ϕ1(ϑ) ≥ 0, I0(ϑ) = ϕ2(ϑ) ≥ 0 (ϑ ∈ [−τ, 0]), S0(0) > 0, I0(0) > 0


6

?

-infection

- INFECTEDS ISUSCEPTIBLES S

recovery

γ

β

?

death

b α+ b

death

?

6a

birth

? 6a

FIGURE 1: The schematic representation of (1).

where ϕ = (ϕ1, ϕ2) belongs to the Banach space C := C([−τ, 0] ,

(R

+0

)2 )

equipped with the norm ‖ϕ‖ := max |ϕ(ϑ)| : ϑ ∈ [−τ, 0] where | · | isany norm in R2 and τ := maxτ1, τ2.

To the authors’ knowledge, (1) has not yet been studied with thedelay as it is in (3), still it was subject of delay by many papers. In [6]the delayed epidemic model

(4)

dS

dt= −rS + bS(t− τ) + pb′I(t− τ) − kS(t)I(t)

dI

dt= −r′I(t) + qb′I(t− τ) + kS(t)I(t)

(t > 0)

with b = be−r∗τ , b′ = b′e−r∗′τ and S(t) = S0(t), I(t) = I0(t) (−τ ≤t ≤ 0), S0, I0 ∈ C has been investigated. A global stability analysis isgiven for the model when the maturation time is zero and the epidemi-ological effects of vertical transmission are discussed. Special cases ofmodels with maturation delays, incubation delays and spatial diffusion


are analyzed. In [27] the delayed model of the form

(5)

dS

dt= (b− r)S(t) + pb′I(t− τ1) −KS(t)I(t),

dI

dt= −r′I(t) + qb′I(t− τ2) +KS(t)I(t),

with S(t) = S0(t), I(t) = I0(t) (−τ ≤ t ≤ 0), S0, I0 ∈ C has beenconsidered. Using the Nyquist criterion on the characteristic equation,an estimate on the length of delays is given for which a system that isstable in the absence of delays remains stable. Further, conditions arederived under which there can be no change of stability.

3 The no-delay case In this section we summarize the resultsconcerning existence and stability of equilibria. For a general systemof autonomous ODEs this can be determined by using the saddle-nodeand Andronov-Hopf curves. A general method for finding these curvesis the parametric representation method (cf. [32]). Here the systemwithout delay is relatively simple hence the equilibria can be determinedexplicitly.

It is easy to see that system (1) has the trivial equilibrium (0, 0) forall parameter values (in case of r < 0 this is the unique steady state);disease free equilibria (boundary equilibria) (K, 0) (with optionalK > 0,i.e., the whole S-axis consists equilibria) as long as the birth and deathrates of susceptibles are identical: a = b, i.e., the intrinsic growth rateof the host population is zero: r = 0; and one equilibrium with positivecoordinates (endemic equilibrium)

(6)(S, I

):=

T

β· (α+ b− a, a− b) with T :=

α+ b+ γ

α+ b− a

provided that

(7) R0 > 1

holds, where we have defined the threshold R0 := α/r; R0 is the basicreproductive rate of the parasite. Clearly, the inequality (7) impliesr > 0. As we will see in the next paragraph, the proportion to 0 and 1of the numbers r and R0, respectively, determines the local asymptoticstability of these equilibria. If the parasite is sufficiently pathogenetic,i.e., R0 > 1, it regulates the host population at a stable equilibrium level


(S, I

). Conversely, if R0 < 1 then the disease is not able to regulate the

host population to a stable level.If the birth and death rates of susceptibles are identical, i.e., a = b

holds, the slope of the trajectories is given by

dI

dS=I

S=βS − (α+ a+ γ)

a+ γ − βS.

Thus, if S > (α + a + γ)/β, then S < 0 and I > 0, if (a + γ)/β <S < (α + a + γ)/β, then S < 0 and I < 0, and if S < (a + γ)/β, thenS > 0 and I < 0. Furthermore, the maximum number of infectivesoccurs when S = (α+ a+ γ)/β. A standard stability analysis based onthe Jacobian

J(S, I) :=

[−βI + a− b −βS + (a+ γ)

βI βS − (α+ b+ γ)

]

for the equilibria shows that in case of r 6= 0

• the origin is asymptotically stable if and only if the intrinsic growthrate of the disease-free host population is negative, i.e., r < 0 holds:

the eigenvalues of the matrix J(0, 0) =[

a−b a+γ

0 −(α+b+γ)

]are a− b and

−(α+ b+ γ);• the endemic equilibrium is asymptotically stable if it exists, i.e., ifR0 > 1, the characteristic polynomial of the matrix

J(S, I) =

(b− a)a+ γ

α+ b− a−(α+ b− a)

(a− b)T 0

is

(8) z2 + (a− b)a+ γ

α+ b− a· z + (a− b)(α+ b+ γ)

which is clearly stable if inequality (7) holds.

The phase space portraits for different values of the parameters are pre-sented in Figure 2.

It can be shown that in case of R0 > 1 the endemic equilibrium isalso globally asymptotically stable. To prove this, one has to find asuitable Lyapunov function (e.g., using the so called Beretta-Capassoapproach (cf. [10])) or to show that any positive orbit in the positive


1

2

3

4

I

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

S

0

2

4

6

8

I

5 10 15 20 25 30

S

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

I

12 14 16 18 20

S

FIGURE 2: Phase portraits of the system (1) for r < 0 (top), r = 0(middle) and r > 0 with R0 > 1 (bottom).


quadrant of the phase space Ω is bounded (cf. [30]), and to rule outthe existence of periodic orbits in Ω. The latter result is easy to prove,because h(S, I) := 1/SI (S, I > 0) is a Dulac-function for system (1)(cf. [16]): for f := (fS , fI)

div (hf) =∂(hfS)

∂S+∂(hfI)

∂I= −

γ + a

S2< 0 (S > 0, I > 0)

holds.

4 The model with delay In this section we focus on investigat-ing the stability of equilibria and Hopf bifurcation from the endemicequilibrium of the system (3).

Using, e.g., step method (cf. [20]) it is easy to see that for everyinitial function ϕ there exists a unique solution to equations (3). Ap-plying Theorem 1.2 from [4], we conclude that the solutions to (3) arenonnegative for any nonnegative initial condition.

4.1 Stability of equilibria Clearly, the equilibria (0, 0), (K, 0) (withK > 0) and (S, I) of (1) are steady states of (3), too. Now, we are goingto determine the stability of equilibria (0, 0) and (S, I) for (3). The

variational system of (3) with respect to the solution (S, I) (cf. [26])assumes the form

[uv

]=

[−b− βI γ − βS

0 −(α+ b+ γ)

]·

[uv

]

+

[0 0

βI βS

]·

[u(· − τ1)v(· − τ1)

]+

[a a0 0

]·

[u(· − τ2)v(· − τ2)

].

Thus, the linearized system

• at (0, 0) is

[uv

]=

[−b γ0 −(α+ b+ γ)

]·

[uv

]+

[a a0 0

]·

[u(· − τ2)v(· − τ2)

],

• at (S, I) it has the form

[uv

]=

[−b− βI γ − βS

0 −(α+ b+ γ)

]·

[uv

]

+

[0 0βI βS

]·

[u(· − τ1)v(· − τ1)

][a a0 0

]·

[u(· − τ2)v(· − τ2)

].


The characteristic function of equation (3) can be obtained by substi-tuting the trial solution (u(·), v(·)) := (cu, cv) exp(z·) into the linearizedsystem (cf. [21]):

• at (0, 0)

∆(0,0)(z; τ1, τ2) :≡ det

[−b+ ae−zτ2 − z γ + ae−zτ2

0 −(α+ β + γ) − z

]

≡ ∆(1)(0,0)(z) · ∆

(2)(0,0)(z; τ2)

:≡ (z + α+ β + γ) ·(z − ae−zτ2 + b

),

and• at (S, I):

∆(S,I)(z; τ1, τ2)

:≡ det

[−b− (a− b)T + ae−zτ2 − z −(α+ b) + ae−zτ2

(a− b)Te−zτ1 (α+ b+ γ) (e−zτ1 − 1) − z

]

≡ z2 +Az +B + Ce−zτ1 +De−zτ2 +Eze−zτ1 + Fze−zτ2 +Ge−z(τ1+τ2)

where

A :=(α+ b)2 − ab+ αγ

α+ b− a, D := −a(α+ b+ γ),

B := [αa+ γ(a− b)]T, E := −(α+ b+ γ),

C := − [γ(a− b) + b(α+ b− a)]T, F := −a,

G := a(α+ 2b− 2a)T.

4.1.1 The stability of the trivial equlibrium Clearly, the characteristic

equation ∆(0,0)(z; τ1, τ2) = 0 is stable if and only if ∆(2)(0,0)(·; τ2) is stable,

because the ∆(1)(0,0)(·) is stable for all parameter values. For τ1 ∈ [0,+∞),

τ2 = 0 the function ∆(0,0)(·; τ1, τ2) is identical with the characteristicfunction of J(0, 0). The behaviour of characteristic function of type

∆(2)(0,0)(·; τ2) has been described in detail e.g., in [22] (cf. [3], [20, p. 339]),

[18]. It is easy to examine its stability due to its simple structure. Thus,we can prove the following.


Theorem 4.1. If a 6= b, then delay does not change the stability of the

trivial equilibrium, i.e.,

1. if a > b, then (0, 0) is an unstable equilibrium of system (1) and is

unstable for (3), too;

2. if a < b, then (0, 0) is a stable equilibrium of system (1) and remains

stable for (3).

Proof. Step 1 If a > b then (0, 0) is an unstable equilibrium of (3)for τ1 = τ2 = 0. It remains unstable for τ1 ≥ 0, τ2 > 0 because,

as a consequence of the Bolzano Theorem, ∆(2)(0,0)(·; τ2) has a positive

root: ∆(2)(0,0)(0; τ2) = b − a < 0, ∆

(2)(0,0)(·; τ2) : R → R is a continuous

function with limz→+∞ ∆(2)(0,0)(z; τ2) = +∞.

Step 2 If b > a then (0, 0) is a stable equilibrium of (3) for τ1 = τ2 = 0.

Clearly, if τ2 > 0 and ∆(2)(0,0)(·; τ2) has purely imaginary roots ±ıω

(ω > 0), then

b− a cos(ωτ2) + ı[ω + a sin(ωτ2)] = 0,

i.e., ω2 = a2 − b2 ≤ 0, which is a contradiction.

4.1.2 The stability of the equilibrium (S, I) We turn now to the charac-teristic equation ∆(S,I)(z; τ1, τ2) = 0. Equation of this type has already

been studied by many authors. Using the Nyquist-plot technique (cf.[29, 35]) in [19] was proved that (S, I) is an asymptotically stable equi-librium of (3) if

(9) B + C +D +G > 0 and τ1 + τ2 <A+E + F

|C| + |D| + |G|

holds, furthermore conditions where derived that imply no change instability. In [33] an estimate of the delays was given for which stabilitywill persist:

(10)B + C +D +G > 0 and

A+E + F > (G+ 0.22|C|)τ1 + (G+ 0.22|D|)τ2.

In order to examine the stability of ∆(S,I)(·; τ1, τ2) we deal with thefollowing two cases:


1. to fix one of the two delays at zero, we consider the other one as aparameter and find intervals for the delay where the characteristicequation is stable;

2. to make the two delays equal: τ := τ1 = τ2 and examine the reducedcharacteristic equation from point of view of stability.

Clearly, in this section, the condition (7) will be assumed everywherewhich implies that

(11)A+ E + F =

(a− b)(a+ γ)

α+ b− a> 0,

B + C +D +G = (a− b)(α+ b+ γ) > 0

hold. Hence, for τ1 = τ2 = 0, the equilibrium (0, 0) is unstable, thepolynomial ∆(S,I)(z; 0, 0), i.e., (8) is stable and, as a consequence, (S, I)

is asymptotically stable as an equilibrium of system (3). Furthermore,for all τ1, τ2 > 0, z = 0 is not a root of ∆(S,I)(z; τ1, τ2) = 0.

In the first case the characteristic equation ∆(S,I)(z; τ1, τ2) = 0 re-duces to

∆(S,I)(z; τ, 0) :≡ z2 + (A+ F +Ee−zτ )z

+B +D + (C +G)e−zτ = 0 (τ2 = 0),

(12)

respectively,

∆(S,I)(z; 0, τ) :≡ z2 + (A+E + Fe−zτ )z +B + C

+ (D +G)e−zτ = 0 (τ1 = 0),

(13)

and in the second case,

(14) ∆(S,I)(z, τ) :≡ z2+Az+B+(C+D)e−zτ +(E+F )ze−zτ +Ge−2zτ .

The case of the single delay Now, in the first case using the Mikhailovcriterion we are going to estimate the domains of the delay τ for whichthe endemic equilibrium is stable. In order to find stability regions forthe quasipolynomial ∆(·, τ) it is enough to investigate the change of theargument of ∆(ıω, τ) as ω increases from 0 to +∞ which is formulatedin detail in the following


Lemma 4.1 (Mikhailov (cf. [25, 26]). Let P and Q be polynomials of

degree m and n, respectively, with m > n, τ > 0, and assume that the

quasi-polynomial

(15) ∆(z, τ) ≡ P (z) +Q(z) exp(−zτ)

has no roots on the imaginary axis. Then ∆(·, τ) is stable, i.e., all of its

roots have negative real part if and only if

[arg ∆(ıω, τ)]ω=+∞

ω=0 =π

2· degP (ıω),

i.e., the argument of ∆(ıω, τ) increases mπ/2 as ω increases from 0 to

+∞.

Thus, we are able to prove the following

Theorem 4.2. If

• τ2 = 0,

(16) γ 6= α+ b− 2a and τ1 < τs1:=

a+ γ

(α+ b+ γ)|2a− α− b+ γ|

respectively• τ1 = 0,

(17) τ2 < τs2:=

a+ γ

a(α+ b+ γ),

then (S, I) is a stable equilibrium of (3).

Proof. Equations ∆(S,I)(z; τ, 0) = 0 and ∆(S,I)(z; 0, τ) = 0 imply incase of τ1 = τ , τ2 = 0

P1(z) ≡ z2 + (A+ F )z +B +D, Q1(z) ≡ C +G+Ez,

and in case of τ1 = 0, τ2 = τ

P2(z) ≡ z2 + (A+E)z +B + C, Q2(z) ≡ D +G+ Fz,

respectively. Thus, ∆(S,I)(·; τ, 0) or ∆(S,I)(·; 0, τ) are stable if and onlyif

[arg∆(S,I)(ıω; τ, 0)

]ω=+∞

ω=0= π or

[arg ∆(S,I)(ıω; 0, τ)

]ω=+∞

ω=0= π,


respectively. We have

(18)

<(∆(S,I)(ıω; τ, 0)

)= −ω2 +B +D

+ (C +G) cos(ωτ) +Eω sin(ωτ)

=(∆(S,I)(ıω; τ, 0)

)= (A+ F )ω

− (C +G) sin(ωτ) +Eω cos(ωτ),

respectively.

(19)

<(∆(S,I)(ıω; 0, τ)

)= −ω2 +B + C

+ (D +G) cos(ωτ) + Fω sin(ωτ)

=(∆(S,I)(ıω; 0, τ)

)= (A+E)ω

− (D +G) sin(ωτ) + Fω cos(ωτ).

Moreover,

<(∆(S,I)(0; τ, 0)

)= <

(∆(S,I)(0; 0, τ)

)

= B +D + C +G = (a− b)(α+ b+ γ) > 0,

=(∆(S,I)(0; τ, 0)

)= =

(∆(S,I)(0; 0, τ)

)= 0.

Hence, if =(∆(S,I)(ıω; τ, 0)

)> 0 (ω > 0) respectively =

(∆(S,I)(ıω; 0, τ)

)

> 0 (ω > 0), then because of limω→+∞ <(∆(S,I)(ıω; τ, 0)

)= −∞, re-

spectively limω→+∞ <(∆(S,I)(ıω; 0, τ)

)= −∞, the change of the ar-

gument of ∆(S,I)(ıω; τ, 0), respectively of ∆(S,I)(ıω; 0, τ) is equal to π.Namely, in this case

sin (arg∆(ıω, τ)) == (∆(ıω, τ))√

<2 (∆(ıω, τ)) + =2 (∆(ıω, τ))→ 0 (ω → +∞),

cos (arg∆(ıω, τ)) =< (∆(ıω, τ))√

<2 (∆(ıω, τ)) + =2 (∆(ıω, τ))→ −1 (ω → +∞)

with ∆(ıω, τ) ∈∆(S,I)(ıω; τ, 0),∆(S,I)(ıω; 0, τ)

.


Substituting w := ωτ in (18), respectively (19) and multiplying theresult by τ , we obtain

τ=(∆(S,I)

( ıwτ

; τ, 0))

= (A+ F )w − (C +G)τ sin(w) +Ew cos(w),

respectively,

τ=(∆(S,I)

( ıwτ

; 0, τ))

= (A+E)w − (D +G)τ sin(w) + Fw cos(w).

For every w ≥ 0, we have

Ew cos(w) ≥ −|E|w,

(C +G)τ sin(w) ≤ τ |C +G|w,

respectively,

Fw cos(w) ≥ −|F |w,

(D +G)τ sin(w) ≤ τ |D +G|w.

Therefore,

τ=(∆(S,I)

( ıwτ

; τ, 0))

≥ w (A+ F − |E| − τ |C +G|) ,

respectively,

τ=(∆(S,I)

( ıwτ

; 0, τ))

≥ w (A+E − |F | − τ |D +G|) .

It is easy to calculate that

A+ F − |E| =(a− b)(a+ γ)

α+ b− a> 0

and

|C +G| = (a− b) |2a− α− b+ γ|T ,

respectively,

A+E − |F | =(a− b)(a+ γ)

α+ b− a> 0 and |D +G| = a(a− b)T.


Thus, if

τ1 <A+ F − |E|

|C +G|=

a+ γ

(α+ b+ γ)|2a− α− b+ γ|(γ 6= α+ b− 2a),

respectively,

τ2 <A+E − |F |

|D +G|=

a+ γ

a(α+ b+ γ),

then all roots of ∆(S,I)(·; τ1, 0) respectively of ∆(S,I)(·; 0, τ2) have nega-tive real part which means stability.

Remark 4.1. In certain cases our estimates for the stable intervals ofτi (i ∈ 1, 2) are better than the ones in [19], respectively in [33]. Forexample, for the parameter values a = 0.0600, b := 0.0080, β := 0.0056,γ := 0.0200 and α = 0.3200 (9), respectively (10) imply

τ1 + τ2 < 0.3710 respectively τ1 < 0.8751, τ2 < 0.7246

while (16), respectively (17) gives the following values

τ1 < τs1= 1.2228, τ2 < τs2

= 3.8314 .

The case of the multiple delay Now we are going to examinethe stability of the characteristic equation in the second case where∆(S,I)(·, τ) in (14) is not yet a quasipolynomial. For easier reference

we quote here the simplified version of Corollary 3.3. from [36] aboutthe stability of the characteristic function of the form

(20) ∆(z, τ) ≡ p(z) + q(z)e−zτ + r(z)e−2zτ ,

where p, q and r are polynomials for which deg(r) < deg(q) holds, is tobe used.

Lemma 4.2. For y ∈ R define

a(y) := <(r(ıy)) + <(p(ıy)), b(y) := =(r(ıy)) −=(p(ıy)),

c(y) := =(r(ıy)) + =(p(ıy)), d(y) := <(p(ıy)) −<(r(ıy)),

e(y) := <(q(ıy)), f(y) := =(q(ıy))


and suppose that in (20) the polynomial ∆(·, 0) is stable. If

(21) D(y) := a(y)d(y) − b(y)c(y) 6= 0 (0 6= y ∈ R),

then ∆(·, τ) is stable—i.e., ∆(·, τ) has no root ıω (ω > 0)—for any given

delay τ if and only if there is no y > 0 for which

(22) [e(y)d(y) − f(y)b(y)]2

+ [e(y)c(y) − f(y)a(y)]2 − [D(y)]

2= 0

holds.

In what follows, the role of the polynomials p, q and r will be playedby

(23) p(z) ≡ z2 +Az +B, q(z) ≡ (E + F )z + C +D, r(z) ≡ G.

Hence, using these notations (14) takes the form of (20).Now we are able to prove the following.

Theorem 4.3. If conditions A2 > 2B and B2 > G2 are fulfilled, more-

over, F3 has a positive root ω where

(24) F3(y) := y8 + c6y6 + c4y

4 + c2y2 + c0 (y ∈ R)

with coefficients defined as

c6 := 2(A2 − 2B) − (E + F )2,

c4 := A4 − (C +D)2 + 2B(3B + (E + F )2) −A2(4B + (E + F )2)

+ 2G((E + F )2 −G),

c2 := 2(B2 −G2)(A2 − 2B) + 2(C +D)(C +D −A(E + F ))(B −G)

− (A(C +D) − (E + F )(B +G))2,

c0 := (B −G)2(B − C −D +G)(B + C +D +G),

then the endemic equilibrium (S, I) may lose its stability and eventually

undergo a Hopf bifurcation as τ increases and passes through a critical

value, i.e., there may occur a small amplitude periodic solution with

period approximately equal to 2π/ω.


Proof. Clearly,

a(y) = G+B − y2, b(y) = −Ay,

c(y) = Aω, d(y) = B −G− y2, (y ∈ R)

e(y) = C +D, f(y) = (E + F )y.

Therefore,

D(y) = a(y)d(y) − b(y)c(y)

= y4 + (A2 − 2B)y2 +B2 −G2 (y ∈ R).

Thus, A2 > 2B and B2 > G2 imply that D(y) > 0 for all y ∈ R.Denoting the left-hand side in (22) by F3(y) for y ∈ R a straightforwardcomputation shows that (S, I) may lose its stability only if F3 has apositive root.

4.2 Hopf bifurcation from the equilibrium (S, I) In this sectionwe apply the Hopf bifurcation theorem to show the existence of nontrivialperiodic solutions to (3). We use the delay as a parameter of bifurcation.Similarly to stability investigations we deal with the two cases: first, itis assumed that one of the two delays is equal to zero, second, equatingthe two delays we show the existence of a limit cycle for (3). In the caseof the single delay the stability of the bifurcating periodic solution willbe also examined.

4.2.1 The case of single delay To obtain stability switch one needs tohave an imaginary root of ∆(S,I)(·; τ, 0), respectively ∆(S,I)(·; 0, τ). Let

z = ıω (ω > 0). Then

∆(S,I)(ıω; τ, 0) = 0, respectively ∆(S,I)(ıω; 0, τ) = 0

implies

(25) |P1(ıω)| = |Q1(ıω)| , respectively |P2(ıω)| = |Q2(ıω)| .

Thus, (25) determines a set of possible values of ω. We define the aux-iliary function

Fi(y) := |Pi(ıy)|2 − |Qi(ıy)|

2 (y ∈ R) for i ∈ 1; 2

and look for stability switches that may occur when Fi(ωi) = 0 for someωi > 0 (i = 1 respectively i = 2). But before we proceed we quote herea lemma for easier reference about the stability of the characteristicfunction of the form in (15).


Lemma 4.3. (cf. [5, 11]) Let p and q be analytic functions in a right

half-plane <(z) > −c (c > 0) which satisfy the following conditions:

• p and q have no common imaginary root;• p(−ıy) = p(ıy), q(−ıy) = q(ıy);• p(0) + q(0) 6= 0;• lim sup |q(z)/p(z)| : |z| → ∞,<(z) ≥ 0 < 1;• for all y ∈ R F(y) := |p(ıy)|2 − |q(ıy)|2 has at most a finite number

of real zeros, furthermore, let ∆(z, τ) :≡ p(z) + q(z) exp(−zτ).

Then the following statements are true.

1. If F has no positive roots, then no stability switch may occur, i.e., if

∆(·, τ) is stable at τ = 0, it remains stable for all τ ≥ 0, whereas if

it is unstable at τ = 0, it remains unstable for all τ ≥ 0.2. If F has at least one positive root and each of them is simple, then

as τ increases, a finite number of stability switches may occur, and

eventually ∆(·, τ) becomes unstable, i.e., there is a τ ∗ > 0 such that

for all τ > τ∗ ∆(·, τ) is unstable.

Clearly, Pi and Qi (∈ 1; 2) are trivially analytic functions in a righthalf-plane<(z) > −c (c > 0) (they are polynomials) satisfying the conditionsof Lemma 4.3 and for i ∈ 1, 2

Fi(y) = y4 + κiy2 + ζi (y ∈ R)

where the coefficients are

κ1 := (A+ F )2 − 2(B +D) −E2 =(a− b)2(a+ γ)2

(α+ b− a)2> 0,

ζ1 := (B +D + C +G)(B +D − C −G)

= (a− b)2(α+ b− a)T 2(3a− α− b+ 2γ),

respectively

κ2 := (A+E)2 − 2(B + C) − F 2

=(a− b)

(α+ b− a)2× [a2(2α+ b) + a

(2(α+ b)(α+ b+ γ) + γ2

)

− a3 − 2(α+ b)2(α+ b+ γ) − bγ2],


ζ2 := (B + C +D +G)(B + C −D −G)

= (a− b)T 2[(α+ b)2 − a2] > 0.

Thus, we are able to prove the following.

Theorem 4.4. If τ2 = 0 and α > 3a − b + 2γ, then there exists a

pair τcrit1 , ω1 > 0 such that (S, I) undergoes a Poincare-Andronov-Hopf

bifurcation as τ1 increases and passes through τcrit1 , i.e., (S, I) loses its

stability and there occurs a small amplitude periodic solution with period

approximately 2π/ω1.

Proof. Clearly, in this case ζ1 < 0. Therefore F1 has unique positiveroot

ω1 =

√√κ2

1 − 4ζ1 − κ1

2

and |Q1(ıω1)|2

= (C +G)2 +E2ω21 > 0. Thus, z1(τcrit1) := ıω1 is a root

of ∆(S,I)(·; τcrit1 , 0). Separating the real and imaginary parts, it followsthat

cos(ω1τcrit1) = −<(P1(ıω1))<(Q1(ıω1)) + =(P1(ıω1))=(Q1(ıω1))

|Q1(ıω1)|2

=(C +G)(ω2

1 − B −D) −E(A+ F )ω21

(C +G)2 +E2ω21

,

respectively

sin(ω1τcrit1) ==(P1(ıω1)<(Q1(ıω1)) −<(P1(ıω1))=(Q1(ıω1))

|Q1(ıω1)|2

=Eω3

1 + [((A+ F )(C +G) −E(B +D)]ω1

(C +G)2 +E2ω21

.

Thus, at the critical value

τcrit1 =1

ω1cos−1

((C +G)(ω2

1 −B −D) −E(A+ F )ω21

(C +G)2 +E2ω21

)

of the delay τ = τ1 the endemic equilibrium may lose its stability.To check if the equilibrium (S, I) loses its stability, one has to deter-

mine the sign of the derivative with respect to τ of the real part of the


smooth extension of the root z1(τcrit1). Let us denote by z1(τ) the rootof ∆(S,I)(·; τ, 0) that assumes the value ıω1 at τcrit1 and by

(26) D(z, τ) :≡ P1(z) +Q1(z) exp(−zτ),

the characteristic quasipolynomial in (12) as a function of the delayparameter τ . Since D (z1(τcrit1), τcrit1) = D (ıω1, τcrit1) = 0 and ıω1 is asimple root of the quasipolynomial D (·, τcrit1), the smooth function z1is uniquely determined by D (z1(τ), τ) ≡ 0, z1(τcrit1) = ıω1. From theimplicit function theorem it follows that

z′1(τcrit1) = −∂τD (ıω1, τcrit1)

∂zD (ıω1, τcrit1)

=ıω1Q1(ıω1)

P ′

1(ıω1) exp(ıω1τcrit1) +Q′

1(ıω1) − τcrit1Q1(ıω1).

Due to a result in [11], we have

sgn (< (z′1(τcrit1))) = sgn (F ′

1(ω1)) = sgn(4ω3

1 + 2κ1ω1

)

= sgn(2ω2

1 + κ1

)= sgn

(√κ2

1 − 4ζ1

)= 1,

which completes the proof.

Remark 4.2. Clearly, for all parameter values ζ2 > 0 therefore in caseof τ1 = 0, τ2 > 0 instability occurs only if F2 has positive root(s), i.e., if

(27) κ2 < −2√ζ2

holds. Comparing the formula for κ2 and the formula for

κ22 − 4ζ2 = (a− b)2

(− 4T 2[(α+ b)2 − a2] +

1

(α + b− a)4

× [a2 + 2(α+ b)3 − a2(2α+ b) + 2(α+ b)2γ

+ bγ2 − a(2(α+ b)(α+ b+ γ) + γ2]2),

it seems hopeless to find such parameters for which (27) holds. There-fore, this delay per se causes most likely no instability (cf. Figure 3)which means biologically that the delay in birth causes no changes inthe qualitative behaviour of the system.


FIGURE 3: Time evolution of system (3) with τ1 = 0, τ2 > 0.

The stability of the bifurcating periodic solution depends on the non-linearity of the system. In the remainder of this section, we alwaysassume that the assumptions of the last theorem hold with τ0 := τcrit1 ,ω0 := ω1 and study the stability of these periodic solutions that is wedetermine whether the Hopf bifurcation is supercritical (stable limit cy-cle exists around the unstable equilibrium) or subcritical (unstable limitcycle exists around the stable equilibrium).

Moving the interior equilibrium (S, I) to the origin by the coordinatetransformation x1 := S−S, x2 := I − I and separating the linear termsfrom the nonlinear terms, we get system (3) in the form

(28) x = Ax +Bx(· − τ) + f(x, x(· − τ))


where x := (x1, x2) and

A :=

[r(1 − T ) −Γ/T

0 −Γ

], B :=

[0 0rT Γ

],

f(x, x(· − τ)) :=

[−βx1x2

βx1(· − τ)x2(· − τ)

].

We shall investigate the stability of the periodic solutions of (28) at thecritical value τ0. Our aim is to reduce this investigation for the ordinarydifferential equation of the form

(29)

[ξη

]=

[0 ω0

−ω0 0

]·

[ξη

]+

[F (ξ, η)G(ξ, η)

]

with F (0, 0) = G(0, 0) = 0, F ′(0, 0) = G′(0, 0) = 0 and to apply thefollowing result (cf. [2]).

Lemma 4.4. The trivial solution of (29) is attractive or is a repellor

if δ < 0 or δ > 0, respectively, where

δ :=1

8ω0· [(Fξξ(0, 0) + Fηη(0, 0))(Gξξ(0, 0)

− Fξη(0, 0) −Gηη(0, 0)) + (Gξξ(0, 0) +Gηη(0, 0))

· (Fξξ(0, 0) − Fηη(0, 0) +Gξη(0, 0))]

+1

8· [3Fξξξ(0, 0) + Fξηη(0, 0) +Gξξη(0, 0) + 3Gηηη(0, 0)].

(30)

This reduction includes four steps as follows.The first step is to transform (28) into the operator differential equa-

tion

(31) xt = Lµxt + Nµ(xt)

where

xt(ϑ) := x(t+ϑ), x : [−τ, 0] → R2, ϑ ∈ [−τ, 0], µ := τ −τ0 (µ ∈ R)

and the linear operator Lµ assumes the form

(32) (Lµu)(ϑ) :=

u′(ϑ), ϑ ∈ [−τ, 0),

Au(0) +Bu(−τ), ϑ = 0(u ∈ B)


while the nonlinear operator N can be written as

(33) (Nµ(u))(ϑ) :=

0, ϑ ∈ [−τ, 0),

f(u(0), u(−τ)), ϑ = 0(u ∈ B).

Here, dot still refers to differentiation with respect to time t, while primestands for differentiation with respect to ϑ, B denotes the Banach spaceof continuously differentiable mappings from [−τ, 0] into K2 (K standsfor R or C) and the dependence on the bifurcation parameter µ is alsoemphasized. Equation (31) is just the trivial equation xt = x′t whenϑ ∈ [−τ, 0), and it becomes (28) when ϑ = 0.

The second step is to determine the normalized eigenfunctions of thelinear operator L0 in order to describe the centre manifold of (31) later.For this purpose, it is useful to define the adjoint operator L∗

µ of Lµ

acting on the adjoint space B∗ := C1([0, τ ],K2

)as follows

L∗

µv(σ) =

−v′(σ), σ ∈ (0, τ ],

AT v(0) +BT v(τ), σ = 0(v ∈ B∗)

and the bilinear form

〈ψ, ϕ〉 := ψ∗(0)ϕ(0) +

∫ 0

−τ

ψ∗(ξ + τ)Bϕ(ξ) dξ

for ψ ∈ C([0, τ ],K2) and ϕ ∈ C([−τ, 0],K2) (cf. [34]) where ∗ denoteseither adjoint operator or transposed conjugate vector.

Clearly, the operator L0 has the same characteristic roots as the linearpart of the delay-differential equation (28):

Ker(L0 − zI) 6= 0 ⇐⇒ det(A+Be−zτ0 − zI) = 0,

and the corresponding two characteristic exponents are also the same:λ1,2(τ0) = ±ıω0 (cf. [31]). Now, it is easy to calculate the eigenfunctionp ∈ B of L0 corresponding to the eigenvalue ıω0 and the eigenfunctionq ∈ B∗ of L∗

0 corresponding to the eigenvalue −ıω0. These eigenfunc-tions satisfy the boundary value problems

(34)(L0p)(ϑ) = ıω0p(ϑ), ϑ ∈ [−τ0, 0],

(L∗

0q)(σ) = −ıω0q(σ), σ ∈ [0, τ0],


that is,

(35)

p′(ϑ) = ıω0p(ϑ), ϑ ∈ [−τ0, 0),

(A− ıω0I)p(0) +Bp(−τ0) = 0, ϑ = 0,

respectively

(36)

q′(σ) = ıω0q(σ), σ ∈ (0, τ0],(AT + ıω0I

)q(0) +BT q(τ0) = 0, σ = 0.

The solutions of these BVPs are

p(ϑ) = p(0) exp(ıω0ϑ), ϑ ∈ [−τ0, 0) with p(0) = (1, π)

and

q(σ) = q(0) exp(ıω0σ), σ ∈ (0, τ0] with q(0) = ρ(1, κ),

respectively, where

π :=rT (1 − T )− ıω0T

Γ, κ :=

r(1 − T ) + ıω0

rT exp(ıω0τ0)),

ρ :=1

1 + κπ + κτ0 exp(ıω0τ0) [rT + πΓ].

Here, we are taking into account the orthonormality condition 〈q, p〉 = 1.Of course, 〈q, p〉 = 0, since ıω0 is a simple eigenvalue of L0:

−ıω0 〈q, p〉 = 〈q,−ıω0p〉 = 〈q,L0p〉 = 〈L∗

0q, p〉

= 〈−ıω0q, p〉 = ıω0 〈q, p〉

(cf. [24]).The third step is to compute a two-dimensional invariant manifold,

the centre manifold. For such a purpose, the phase space C is decom-posed as C = P ⊕ Q, where P is a two-dimensional subspace spannedby eigenfunctions of operator L0: P = span <(p),=(q) and Q is thecomplementary space of P. P and Q are invariant under the flow asso-ciated with the linear part of equation (28). The long-term behaviour of


solutions to the equation (28) is well approximated by the flow on thismanifold (cf. [21]). The centre manifold for equation (31) is given by

Mf :=φ ∈ C : φ = Φz + h(z, f), z in a neighbourhood of zero in R

2.

The flow on this centre manifold is

xt = Φz + h(z, f)

where h ∈ Q and z satisfies the ordinary differential equation

(37) z = Jz + Ψ(0)f(Φz)

with

Φ := (<(p),=(p)), J = Φ−1Φ′ and Ψ := (<(q),=(q))

(cf. [8, 9]). A straightforward calculation shows that

Φ =

[cos(ω0ϑ) sin(ω0ϑ)

<(π) cos(ω0ϑ) + =(π) sin(ω0ϑ) <(π) sin(ω0ϑ) −=(π) cos(ω0ϑ)

]

and

J =

[0 ω0

−ω0 0

], Ψ(0) =

[ψ11 ψ12

ψ21 ψ22

],

where

ψ11 := Γr2[Γ − T (r2(T − 2)(T − 1)2 + ω2

0T )τ0]/δ,

ψ12 := ΓTrτ0ω0

[ω2

0 + r2(T − 2)(T − 1)]/δ,

ψ21 := T (Γ(−Γr2(T − 1) + (ω20 + r2(T − 2)(T − 1))

× (ω20 + r2(T − 1)2)Tτ0) · cos(ω0τ0)

+ Γω0r(Γ − (ω20 + r2Tτ0(T − 1)2)) sin(ω0τ0))/δT

2,

ψ22 := TΓ(ω0r(Γ − (ω20 + r2(T − 1)2)Tτ0) cos(ω0τ0)

+ (Γr2(T − 1) − (ω20 + r2(T − 2)(T − 1))

× (ω20 + r2(T − 1)2)Tτ0) sin(ω0τ0))/δT

2

with


δ := ω20

[ω2

0 + r2(T − 3)(T − 1))]2T 2τ2

0

+ r2[Γ − T

(r2(T − 2)(T − 1)2 + ω2

0T)τ0

]2.

The nonlinear function f in equation (37) is then given as

(38) f(Φz) =

[f111ξ2 + f1

12ξη + f122η2 + f1

111ξ3 + f1112ξ2η + f1

122ξη2 + f1222η3

f211ξ2 + f2

12ξη + f222η2 + f2

111ξ3 + f2112ξ2η + f2

122ξη2 + f2222η3

]

where z = [ξ, η]T and

f111 = −β<(π), f1

12 = β=(π),

f122 = f1

111 = f1112 = f1

122 = f1222 = 0,

f211 = β cos(ω0τ0) [=(π) cos(ω0τ0) −<(π) sin(ω0τ0)] ,

f212 =

β

2<(π) −=(π) − (<(π) + =(π)) [cos(2ω0τ0) + sin(2ω0τ0)] ,

f222 = β sin(ω0τ0) [=(π) cos(ω0τ0) + <(π) sin(ω0τ0)] ,

f2111 = f2

112 = f2122 = f2

222 = 0.

Substituting (38) into (37) yields the dynamical system

(39)

ξ = ω0η +(ψ11f

111 + ψ12f

211

)ξ2

+(ψ11f

112 + ψ12f

212

)ξη + ψ12f

222η

2,

η = −ω0ξ +(ψ21f

111 + ψ22f

211

)ξ2

+(ψ21f

112 + ψ22f

212

)ξη + ψ22f

222η

2.

Using formula (30) we can calculate the Poincare-Lyapunov constant δfrom (39)

δ =1

8ω0[(ψ11f

111 + ψ12f

211 + ψ22f

222)

· (ψ21f111 + ψ22f

211 − ψ11f

112 + ψ12f

212 − ψ22f

222)

+ (ψ21f111 + ψ22f

211 + ψ22f

222)

· (ψ11f111 + ψ12f

211 − ψ12f

222 + ψ21f

112 + ψ22f

212)].

The sign of δ determines the direction and stability of the Hopf bifurca-tion which is subcritical or supercritical if δ > 0 or δ < 0, respectively.


Example 4.1. Set a = 0.0600, b := 0.0080, β := 0.005600, γ :=0.020000 and α = 0.320000. The unique endemic equilibrium of (3) is(S, I) = (62.1429, 12.0576). We can see that if τ2 = 0 and τ1 < τs1

thenthere is no stability switch, i.e, (S, I) remains stable. The polynomialF1 has one positive root: ω0(= ω1) = 0.106600. The critical value of thedelay τ1 at which Hopf bifurcation takes place is τ0(= τcrit1) = 1.659400(cf. Figure 4) which is supercritical because δ = −0.000022.

FIGURE 4: Time evolution of system (3) with τ2 = 0, τ < τs1and

τ > τcrit1 .

4.2.2 The case of the multiple delay To obtain the value of the delayat which stability switch may occur one needs to find a positive solutionω3 of the equation F3(y) = 0 and then calculate the value of τcrit3 from∆(S,I)(ıω3, τ) = 0.

Clearly, if z = ıω (ω > 0) is a root, then


p(ıω) + q(ıω)e−ıωτ + r(ıω)e−2ıωτ

= −ω2 +Aωı+B + (C +D + (E + F )ωı) e−ıωτ +Ge−2ıωτ = 0

holds. Multiplying with the factor eıωτ and separating into real andimaginary parts, we obtain

(40) <(∆(S,I)(ıω, τ)

)= (B +G− ω2) cos(ωτ)

−Aω sin(ωτ) + C +D = 0

and

(41) =(∆(S,I)(ıω, τ)

)= (B −G− ω2) sin(ωτ)

+Aω cos(ωτ) + (E + F )ω = 0.

Solving (40) (say) for τ we get the formula

(42) τcrit3 =1

ω3

[sin−1

(−

C +D√(B +G− ω2

3)2 +A2ω2

3

)− ϕ

]

where tan(ϕ) = −Aω3/(B +G− ω2

3

)and F3(ω3) = 0.

Let us denote the root of ∆(S,I)(·; τ) that assumes the value ıω3 at

τcrit3 by z3(τ) and the characteristic function ∆(S,I)(·; τ) as a functionof the parameter τ by

F(µ, τ) :≡ p(z) + q(z)e−zτ + r(z)e−2zτ .

We are going to determine the derivative of the implicit function z3 atτ0:

z′3(τcrit3)

= −∂τF(ıω3, τcrit3)

∂zF(ıω3, τcrit3)

= −−zq(z)e−zτ − 2zr(z)e−2zτ

p′(z) + [q′(z) − τq(z)]e−zτ + [r′(z) − 2τr(z)]e−2zτ

∣∣∣∣z=ıω3, τ=τcrit3

=zq(z) + 2zr(z)e−zτ

p′(z)ezτ + [q′(z) − τq(z)] + [r′(z) − 2zr(z)]e−zτ

∣∣∣∣z=ıω3, τ=τcrit3


Thus,

z′3(τcrit3) = ıω3(C +D + (E + F )ıω3) − 2ıω3G exp(−ıω3τcrit3)

× (2ıω3 +A) exp(ıω3τcrit3) +E + F

− τcrit3(C +D + (E + F )ıω3)

− 2τcrit3G exp(−ıω3τcrit3)−1

=Aen(ω3, τcrit3) + Ben(ω3, τcrit3)ı

Aden(ω3, τcrit3) + Bden(ω3, τcrit3)ı

where

Aen(ω3, τcrit3) := ω3 [2G sin(ω3τcrit3) − (E + F )ω3] ,

Ben(ω3, τcrit3) := ω3 [C +D + 2G cos(ω3τcrit3)] ,

Aden(ω3, τcrit3) := E + F − τcrit3(C +D) − 2ω3 sin(ω3τcrit3)

+ (A− 2τcrit3G) cos(ω3τcrit3),

Bden(ω3, τcrit3) := 2ω3 cos(ω3τcrit3) − τcrit3(E + F )ω3

+ (2τcrit3G+A) sin(ω3τcrit3).

Therefore, Hopf bifurcation occurs if

sgn (<(z′3(τcrit3))) = sgn (Aen(ω3, τcrit3)Aden(ω3, τcrit3)

+ Ben(ω3, τcrit3)Bden(ω3, τcrit3))

= sgn (−(E + F )2ω3 + [2(C +D)

−A(E + F )]ω3 cos(ω3τcrit3)

+ 4Gω3 cos(2ω3τcrit3) + [A(C +D) + 2(E + F )

× (G+ ω23) + 4AG cos(ω3τcrit3)] sin(ω3τcrit3))

= ±1

holds.

Example 4.2. Set a = 0.0500, b := 0.0040, β := 0.0056, γ := 0.0200and α = 0.4200. The unique endemic equilibrium is (S, I) = (81.0714, 9.9714).The polynomial F3 has two positive roots: ω3 := 0.0352 and ω3 :=


0.1204 with possible critical values of the delay: τcrit3 := 18.1173 andτcrit3 := 0.7953. Then the critical value of the delay at which Hopfbifurcation takes place is τcrit3 , because sgn (< (z′3(τcrit3))) = 1 andsgn (< (z′3(τ ))) = −1. In the second case, the endemic equilibrium be-comes again stable (c.f. Figure 5).

FIGURE 5: Time evolution of system (3) when A2

> 2B and B2

> G2

hold and F3 has two positive roots, for τ1 = τ2 =: τ < τcrit3 in the firstgraph, τcrit3 < τ < eτcrit3 in the middle and eτcrit3 < τ in the last graph.


5 Summary In this paper, a detailed study of the effect of timedelays on the dynamics of a system modelling host-parasite associationswas presented. This delays were introduced in both birth and transmis-sion terms. By analyzing the associated characteristic equation we haveobtained some sufficient conditions for the stability of the system. Us-ing the delays as the bifurcation parameter, we have shown that a Hopfbifurcation occurs when this parameter passes through a critical value;i.e., a family of periodic orbits bifurcates from the endemic equilibrium.In case of a single delay, when the one in the birth term is fixed to zeroand the other one in the transmission therm is used as a parameter, thedirection of Hopf bifurcation and the stability of the bifurcating periodicorbits were also discussed.

Acknowledgments The authors thank professor Peter L. Simonfor useful discussions on the topics investigated in this paper. The workof S. Aly was partially supported by the Hungarian Scholarship Board.

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Corresponding author: Sandor KovacsEotvos Lorand University, Department of Numerical Analysis,Budapest H-1117, Pazmany Peter setany 1/C, HungaryE-mail address: [email protected]

A. H. Aly,University of Al-Azhar, Department of Mathematics,Faculty of Science, Assiut 71511, Egypt

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