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Analysis of a SIR model with pulse vaccination and temporary

immunity: stability, bifurcation and a cylindrical attractor

Kevin E.M. Church and X. Liu

1. Introduction

Pulse vaccination is a disease control policy under which at certain times, a por-

tion of the population is vaccinated en-masse. It has been argued empirically and

veriﬁed analytically that pulse vaccination might be more eﬀective than continuous

vaccination in preventing epidemics that exhibit seasonality, such as measles [1, 15].

Since then, the impact on pulse vaccination has been studied in ever more complex

models of disease transmission. For instance, ﬁnite infectious periods [19], saturation

incidence with latent period and immune period [6], incubation period [13], force of

infection by distributed delay [7], nonlinear vaccination [22], quarantine measures [14]

and stochastic eﬀects [16] have been considered.

Dynamical anaysis of these pulsed vaccination models often include stability crite-

ria for the disease-free equilibrium or periodic orbit, eﬀectively providing a proxy for

the basic reproduction number. However, due to the presence of the impulse eﬀect,

establishing the existence of an endemic periodic orbit is much more diﬃcult. When

there are no delayed terms, methods of bifurcation theory for discrete time systems

have been used to prove the existence of endemic periodic orbits from bifurcations

at disease-free states; see [2, 17, 22, 21, 20] for some recent examples. In contrast,

when delays are present, most analytical studies prove only permanence when R0>1,

which means that the disease persists. Numerical simulations are needed to obtain

further detail, and this provides only a heuristic description of the orbit structure at

a possible bifurcation point. We refer the reader to [6, 13, 14, 19] for examples. It

would therefore appear that it is not for lack of interest that no authors have studied

bifurcations in pulsed vaccination models involving delays, but rather that bifurca-

tion theory techniques such as centre manifold reduction [3] have only recently been

developed for impulsive functional diﬀerential equations.

Restricting to SIR models without pulse vaccination speciﬁcally, there are many

papers that consider various forms of population dynamics and their interplay with

delayed eﬀects. Since endemic equilibrium points are often analytically available, Hopf

bifurcations can often be studied analytically without the aid of numerical methods.

One may consult [5, 9, 10, 12, 18] for some recent examples of this.

It is our goal to use centre manifold theory for impulsive delay diﬀerential equa-

tions [4, 3] to obtain more precise information about the orbit structure in a particu-

Preprint submitted to Elsevier December 3, 2018

lar pulsed SIR vaccination model involving delay. Our starting point is the model of

Kyrychko and Blyuss [11]:

˙

S=µ−µS −ηf (I(t))S(t) + γI(t−τ)e−µτ

˙

I=ηf (I(t))S(t)−(µ+γ)I(t)

˙

R=γI (t)−γI(t−τ)e−µτ −µR(t).

Here, f(I) is a general nonlinear incidence rate, infected individuals clear their

infection at rate γand acquire temporary immunity of length τ,ηis a recruit-

ment rate and µis a natural death rate, with birth rate scaled accordingly so that

N(t) = S(t) + I(t) + R(t) approaches unity as t→ ∞. The incidence rate is assumed

to satisfy the properties: f(0) = 0, f0(0) >0, f00(0) <0 and limI→∞ f(I) = c < ∞.

Kyrychko and Blyuss [11] proved global stability of the disease-free equilibrium when

R0<1 for arbitrary nonlinear incidence satisfying the previous conditions, and con-

sidered the existence and stability of an endemic equilibrium for the particular inci-

dence f(I) = I/(1 + I). They numerically observed Hopf bifurcations at this equi-

librium upon varying the immunity period τ. Soon after, Jiang and Wei [10] proved

that the endemic equilibrium may indeed undergo a Hopf bifurcation, by taking ηas

a bifurcation parameter.

We here extend the model of Kyrychko and Blyuss to include pulse vaccination.

To do this, we make the following assumptions.

1) At speciﬁc instants of time tkfor k∈Z, any individuals that received their

vaccine at time tk−τand are still alive lose their immunity and re-enter the

susceptible cohort, at which point a fraction v∈[0,1) of the the total susceptible

cohort is vaccinated.

2) Vaccinated individuals are immune to infection for a period τ(the same im-

munity period as having recovered from infection) and are subject to the same

natural death rate µ.

3) The sequence of vaccination times is periodic with shift of τ: there exists q > 0

such that tk+q=tk+τfor all k∈Z.

The interpretation of 3) is that the period of the pulse vaccination schedule is synchro-

nized with the immunity period. This seems reasonable for seasonal ﬂu epidemics,

for example, provided most of the pulse vaccination times are clustered around the

beginning of ﬂu season. With these assumptions in place, the pulse vaccination model

takes the following form, where we will ignore the recovered (R) component since it

is decoupled from the remaining equations:

˙

S=µ−µS −ηf (I(t))S(t) + γI(t−τ)e−µτ , t 6=tk(1)

˙

I=ηf (I(t))S(t)−(µ+γ)I(t), t 6=tk(2)

∆S=−vS(t−) + vS(t−τ)e−µτ , t =tk.(3)

2

A derivation of the impulse condition ∆Susing assumptions 1)–3) is available in

Appendix A.

It is known [10] that the model of Kyrychko and Blyuss can exhibit Hopf bifurca-

tion. Numerically, it appears as though the bifurcating periodic orbit may be globally

(excluding the other two equilibria) asymptotically stable. In the presence of impulse

eﬀects, Hopf points are known to generically lead to bifurcations to invariant cylinders

[3]. The ramiﬁcations of this result to the present model are that, in the presence of

pulse vaccination, we expect a bifurcation from an endemic periodic solution to an

invariant cylinder. Verifying this hypothesis is our primary goal.

The structure of the paper is as follows. Section 2 contains some necessary back-

ground material and notation that will be used throughout, as well as a reformulation

of the model that will be needed for some of the bifurcation analysis. We study the

disease-free periodic solution in Section 3. A numerical analysis of the cylinder bifur-

cation is completed in Section 4. We end with the concluding Section 5.

2. Background material and model reformulation

Here we recall some key theoretical aspects and notation inherent to impulsive

functional diﬀerential equations, as well as a reformulation of the model (1)–(3) that

will be needed later.

2.1. Impulsive functional diﬀerential equations

Abstractly, a semilinear impulsive (retarded) functional diﬀerential equation is a

dynamical system of the form

˙x=A(t)xt+f(t, xt), t 6=tk

∆x=Bkxt−+gk(xt−), t =tk,

for x∈Rn, where the notation ∆xat time t=tkshould be understood to mean

∆x=x(tk)−x(t−

k),and the latter term denotes the limit from the left. As is typical

with delay diﬀerential equtions, xt(θ) = x(t+θ) for θ∈[−r, 0] is the solution history.

To contrast, the jump condition (the second equation) contains xt−, which is given

by

xt−(θ) = x(t−), θ = 0

x(t+θ), θ < 0.

A classical solution of such a system is a function x: [α−r, β)→Rnfor β > α

that satisﬁes the diﬀerential equation except possibly at those times tk∈[α, β], while

also satisfying the jump condition at all tk∈(α, β]. Phase space considerations make

it necessary to consider mild solutions which are deﬁned by an abstract variation

3

of constants formula – see [4] for details. Speciﬁcally, to consider the above as an

impulsive semidynamical system, it is necessary to consider the phase space

RCR =φ: [−r, 0] →Rn:φis continuous from

the right and has limits on the left .

The symbol RCR comes from the observation that these functions are right-continuous

and regulated.RCR becomes a Banach space when equipped with the supremum

norm. The space RCR comes up at several points in the following sections. The ana-

logue of this space in delay diﬀerential equations without impulses is C, the continuous

functions φ: [−r, 0] →Rn.

Given a periodic linear system of period T,

˙x=A(t)xt, t 6=tk

∆x=Bkxt−, t =tk,(4)

one can guarantee that provided the operators A(t) and Bkare suﬃciently regular

(e.g. bounded, linear combinations of discrete and distributed delays with reasonable

kernels), the space RCR splits into an internal direct sum RCR =RCRs⊕ RCRc⊕

RCRuconsisting of a stable, centre and unstable ﬁber bundle, respectively. The

latter direct sum is a slight notational abuse: to be precise, each of RCRjis a

subset of R× RCR, and if we deﬁne the t-ﬁbers RCRj(t) = {φ: (t, φ)∈ RCR},

then RCR =RCRs(t)⊕ RCRc(t)⊕ RCRu(t) is an internal direct sum for each

t∈R. The t-ﬁbers are also periodic in the sense that RCRj(t+T) = RCRj(t).

These stable, centre and unstable ﬁber bundles play the role of the stable, centre and

unstable subspace from ordinary and delay diﬀerential equations without impulses.

All systems we consider in this paper are periodic, so this formalism is quite relevant.

In particular, the following stability result will be important later. It follows by the

same reasoning as the arguments of [Section 3.1.1, [3]].

Proposition 2.1.1. Let Λdenote the set of complex numbers λsuch that there exists

φ:R→Cnperiodic with period Tsuch that x(t) = φ(t)eλt is a solution of (4).Λ

is the Floquet spectrum, and its elements Floquet exponents.x= 0 is exponentially

stable if and only if Λ⊂ {z∈C:<(z)<0}.

We denote <(z) and =(z) the real and imaginary part of a complex number z.

We deﬁne χ0=χ{0}(·)IX, where IX:X→Xis the identity operator on X(a given

vector space) and χ{0}:I→Ris the indicator function on the set {0}, where I⊂R:

χ{0}(x) = 1, x = 0

0, x 6= 0.

The space Xand the interval Iwill be implied by context, but they will usually be

Rnand [−r, 0], respectively.

4

2.2. Vaccinated component formalism

We have indicated that it is our goal to complete a bifurcation analysis on the

system (1)–(3). However, there are some technical diﬃculties associated with this

endeavor because the overlap condition of Church and Liu [3] is not satisﬁed, since

each of tk−τis an impulse time and equation (3) contains delayed terms. While

the failure of the overlap condition does not complicate stability analysis, it does

complicate the bifurcation analysis. To remedy this, we will at times instead consider

the following modiﬁcation of model (1)–(3):

˙x=µ−µx −ηf (y(t))x(t) + γy(t−τ)e−µτ , t 6=tk(5)

˙y=ηf (y(t))x(t)−(µ+γ)y(t), t 6=tk(6)

˙

Vj= 0, t 6=tk,(7)

∆x=−vx(t−) + (1 −v)Vj(t−)e−µτ , t =tj+qk (8)

∆Vj=vx(t−)−(1 −ve−µτ )Vj(t−), t =tj+qk .(9)

In the above impulsive delay diﬀerential equation, jranges from 0 to q−1 where q

is the period of the sequence of impulse times as deﬁned in assumption 3). Taking

note that tj+qk =tj+q(k−1) +τand Vis constant except at impulse times where it is

continuous from the right, we see that for t=tj+qk,

(1 −v)Vj(t−)e−µτ = (1 −v)Vj(t−τ)e−µτ

= (1 −v)[vx((t−τ)−) + ve−µτ Vj((t−τ)−)]

=v[(1 −v)x((t−τ)−) + (1 −v)Vj((t−τ)−)e−µτ ]

=vx(t−τ).

Substituting the above into the jump condition for x, the result is

∆x=−vx(t−) + (1 −v)Vj(t−)e−µτ =−vx(t−) + vx(t−τ)e−µτ .

This is precisely the same functional form as the jump condition (3) for the original

model. Since the continuous-time dynamics are identical for both models, we can

analyze bifurcations in (1)–(3) by equivalently studying bifurcations in the model

(5)–(9) with explicit vaccinated components.

3. Disease-free periodic solution, stability and analytical bifurcation anal-

ysis

In this section we will complete a thorough investigation of the local properties of

disease-free states, namely their stability and bifurcations. For part of this section,

the number of vaccination moments qper period will remain an arbitrary natural

number. However, we will eventually specialize to the case where q= 1. Without

loss of generality, we will assume t0= 0.

5

3.1. Existence of the disease-free periodic solution

When there is no disease – that is, on the invariant subspace {(S, I) : I= 0}– the

nontrivial dynamics are determined solely by the linear, nonhomogeneous impulsive

system

˙z=−µz +µ, t 6=tk(10)

∆z=−vz(t−) + vz(t−τ)e−µτ , t =tk.(11)

By the variation of constants formula of Church and Liu [4], every solution z(t) passing

through an initial condition φ∈ RCR at time t= 0 can be written

zt=U(t, 0)φ+Zt

0

U(t, s)χ0µds,

where the integral is a weak integral and U(t, s) is the evolution family associated to

the homogeneous equation

˙w=−µw, t 6=tk(12)

∆w=−vw(t−) + vw(t−τ)e−µτ , t =tk.(13)

Lemma 3.1.1. Suppose the trivial solution of the homogeneous equation (12)–(13)

is exponentially stable. Then, the system (1)–(3) has a unique disease-free periodic

solution (˜

S, 0), with period τ.

Proof. From the variation of constants formula, deﬁne the linear operator

V:φ7→ U(τ, 0)φ+Zτ

0

U(τ, s)χ0µds

on RCR. If the trivial solution of (12)–(13) is exponentially stable, then [Theorem

7.2.1, [4]] implies ||U(t, s)|| ≤ Ke−α(t−s)for some α > 0 and K≥1, for all t≥s.

The periodicity U(t+τ, s +τ) = U(t, s) of the evolution family, it continuity and the

cocycle property U(t, s) = U(t, v)U(v, s) for s≤v≤timplies that the nth iterate of

Vsatisﬁes the inequality

||Vnφ−Vnψ|| ≤ Ke−ατn ||φ−ψ||.

Consequently, V:RCR → RCR is an eventual contraction and has a unique ﬁxed

point, which we denote φv. From the variation of constants formula, it follows that

with

t7→ ˜

S(t, v) = [U(t, 0)φv](0) + Zt

0

[U(t, s)χ0µ](0)ds,

(˜

S, 0) is the claimed disase-free periodic solution.

6

Lemma 3.1.2. The trivial solution of the homogeneous equation (12)–(13) is expo-

nentially stable.

Proof. To verify the exponential stability of the trivial solution, it is enough by Propo-

sition 2.1.1 for us to show that all Floquet exponents have negative real part. Let

w(t) = φ(t)eλt be a solution of (12)–(13) with φperiodic. Substituting this ansatz into

the dynamical system, using the periodicity condition φ(t) = φ(t−τ) and cancelling

exponentials, we arrive at the following impulsive diﬀerential equation for φ:

˙

φ+λφ =−µφ, t 6=tk(14)

∆φ=−vφ(t−) + vφ(t)e−(µ+λ)τ, t =tk.(15)

The second equation is an implicit jump conditon, but we can easily rearrange it to

obtain the explicit condition

φ(tk) = 1−v

1−ve−(µ+λ)τφ(t−

k).

Calculating the solution of the above impulsive diﬀerential equation at time τgiven

an initial condition at time t= 0, one obtains

φ(τ) = e−(µ+λ)τ1−v

1−ve−(µ+λ)τq

φ(0) := D(λ)φ(0).

φis periodic provided φ(τ) = φ(0), so we are left with describing the location of the

solutions of the transcedental equation D(λ) = 1. Deﬁning z=e−λτ , it follows that

λis a solution of D(λ) = 1 if and only if zis a solution of

0 = f(z) + g(z),

f(z) = 1,

g(z) = −ze−µτ 1−v

1−ve−µτ zq

.

We will show that |g(z)|<|f(z)|on the unit circle |z|= 1. We have

|g(z)|=e−µτ 1−v

|1−ve−µτ z|q

≤e−µτ 1−v

|1− |ve−µτ z|| q

=e−µτ 1−v

1−ve−µτ q

≤e−µτ <1 = |f(z)|,

as claimed. By Rouch´e’s theorem, the equation f(z) + g(z) = 0 has no solutions

satisfying |z| ≤ 1. Consequently, there are no Floquet exponents λsatisfying the

inequality |e−λτ | ≤ 1. We conclude that all Floquet exponents have negative real

part and the proof is complete.

7

As a consequence of Lemma 3.1.1 and Lemma 3.1.2, we are guaranteed a unique

disease-free periodic solution that, in the absence of infection, is globally exponentially

stable.

Corollary 3.1.0.1. The model (1)–(3) has a unique disease-free periodic solution

t7→ (˜

S(t, v),0) of period τ. Restricted to the disease-free subspace D0={(S, I) : I=

0}, this periodic solution is globally exponentially stable.

3.2. Stability

Introduce the basic reproduction number

R0=ηf 0(0)

τ(γ+µ)Zτ

0

˜

S(t, v)dt. (16)

Note that if one denotes the average of ˜

Sover the interval [0, τ] by [ ˜

S], then one can

equivalently write the basic reproduction number in the more suggestive form

R0=ηf 0(0)[ ˜

S]

γ+µ.

Then, the interpretaton is that R0is the product of the average number of suscepti-

bles, multiplied by the small-infection (i.e. near I= 0) incidence rate, divided by the

aggregate rate of leaving the infected class through death or clearance of the infection.

Lemma 3.2.1. R0= 1 is an epidemiological threshold: if R0<1, the disease-free

periodic solution is locally asymptotically stable, while if R0>1it is unstable.

Proof. The linearization at ( ˜

S, 0) produces the linear homogeneous impulsive system

˙u1=−µu1(t)−ηf 0(0) ˜

S(t, v)u2(t) + γe−µτ u2(t−τ), t 6=tk

˙u2=ηf 0(0) ˜

S(t, v)u2(t)−(γ+µ)u2(t), t 6=tk

∆u1=−vu1(t−) + vu1(t−τ)e−µτ , t =tk.

Notice that the second equation is decoupled from the ﬁrst. Taking an ansatz Floquet

eigensolution u(t) = φ(t)eλt, we can examine the second component independently.

Indeed, φ= [ φ1φ2]Tsatisﬁes

˙

φ2+λφ2=ηf 0(0) ˜

S(t, v)φ2−(γ+µ)φ2.

If φ26= 0, then as φis assumed to be periodic with period τ, the only possible Floquet

exponent in this case is

λ0=−(γ+µ) + ηf 0(0)

τZτ

0

˜

S(t, v)dt. (17)

8

Conversely, if φ2= 0, then φ1and λmust satisfy (14)–(15). But it is already known

that all Floquet exponents λassociated to this equation have negative real part;

see Lemma 3.1.2. Consequently, the Floquet spectrum includes the special Floquet

exponent λ0and the remainder with strictly negative real part. The equilibrium is

locally asymptotically stable provided all Floquet exponents have negative real part,

and is unstable if at least one has positive real part. Since λ0is real and the others

are guaranteed to have negative real part, we obtain the conclusion of the lemma

by noticing that λ0<0 is equivalent to R0<1 and that λ0>0 is equivalent to

R0>1.

3.3. Existence of a bifurcation point

Before we can study bifurcations, we must establish the existence of a bifurcation

point.

Lemma 3.3.1. Consider the critical Floquet exponent λ0=λ0(v)as deﬁned in equa-

tion (17).λ0is strictly decreasing. As consequence, if λ0(0)λ0(1) ≤0, there is a

unique v∗∈[0,1] such that λ0(v∗) = 0; that is, a critical vaccination coverage v∗at

which R0= 1.

Proof. Note that, given the explicit form of λ0, it is enough to prove that v7→ ˜

S(t, v)

is decreasing for all t∈[0, τ ]. To accomplish this, we recall that ˜

Sis the unique

periodic solution of (10)–(11). By a similar argument to the proof of Lemma 3.1.2,

we can show that the jump condition can be simpliﬁed, and that ˜

Sis in fact the

unique periodic solution of the impulsive diﬀerential equation without delay

˙z=µ−µz, t 6=tk

∆z= (ρ(v)−1)z(t−), t =tk,

where ρ(v) = (1 −v)/(1 −ve−µτ ). If one denotes t7→ z(t;z0, v) the unique solution

of the above impulsive diﬀerential equation for vaccination coverage vand initial

condition z(0; z0, v) = z0, it is not diﬃcult to show that dρ

dv <0 and, subsequently,

that ∂z

∂v ≤0 for all t≥0. Also, one has ∂z

∂z0>0 for all t≥0. Using the variation of

constants formula for impulsive diﬀerential equations, routine calculations yield

z(τ;z0, v) = e−µτ ρ(v)q"z0+

q

X

i=1

ρ(v)1−i(eµti−eµti−1)#,

from which we can compute the initial condition ˜

S(0, v) of the disease-free periodic

orbit, by solving the equation z(τ;˜

S(0, v), v) = ˜

S(0, v). The result is

˜

S(0, v) = e−µτ

1−ρ(v)qe−µτ

q

X

k=1

ρ(v)q+1−i(eµτi−eµτi−1),

9

which indeed satisﬁes d

dv ˜

S(0, v)<0. Since ˜

S(t, v) = z(t;˜

S(0, v), v), one may conclude

from the chain rule that d

dv ˜

S(t, v)<0 for all t∈[0, τ ], so v7→ λ0(v) is decreasing.

The conclusions about the critical vaccination coverage v∗follow by the intermediate

value theorem.

3.4. Transcritical bifurcation in terms of vaccine coverage at R0= 1 with one vacci-

nation pulse per period

There are several choices we can make for the bifurcation parameter. Mathemat-

ically the easiest ones to deal with are the model parameters γ,µand η, as these

appear linearly in the model and in the expression for the important Floquet exponent

λ0in equation (17). Biologically, a natural choice is the vaccine coverage, v, since

this is a parameter that can in principle be controlled. It is more diﬃcult to state

closed-form results for bifurcations in terms of the vaccine coverage, so for this reason

we will simplify matters and assume that q= 1, so there is one vaccination pulse per

period. That is, the sequence of impulse times is precisely tk=kτ for k∈Z. Then,

from the previous section, we can explicitly calculate

˜

S(t, v) = 1 −ve−µ[t]τ,(18)

which implies that ˜

S(τ−, v)=1−ve−µτ and ˜

S(0, v)=1−v. We can also explicitly

calculate the critical vaccination coverage where R0= 1. We ﬁnd

v∗=µτ

1−e−µτ 1−γ+µ

ηf 0(0) .(19)

As a consequence, we have the following preliminary stability result.

Lemma 3.4.1. If there is q= 1 vaccination pulse per period, the disease-free periodic

solution is locally asymptotically stable provided v > v∗, and unstable if v < v∗.

We will now pass to the equivalent system with vaccinated component (5)–(9).

Deﬁne the changes of variables and parameters

X+˜

S(·, v) = x, V +v˜

S(τ−, v)

1−ve−µτ =V0, Y =y, +v∗=v.

The result is the following system of impulsive delay diﬀerential equations:

˙

X=−µX(t) + ηf (Y)˜

S(t, v∗+) + X+γY (t−τ)e−µτ , t 6=kτ

˙

Y=ηf (Y)˜

S(t, v∗+) + X−(µ+γ)Y(t), t 6=kτ

˙

V= 0, t 6=kτ

˙= 0, t 6=kτ

∆X=−(v∗+)X(t−) + (1 −(v∗+))e−µτ V(t−), t =kτ

∆Y= 0, t =kτ

∆V= (v∗+)X(t−)−(1 −(v∗+)e−µτ )V(t−), t =kτ

∆= 0, t =kτ.

(20)

10

Notice that (X, Y, V, ) = (0,0,0, ) is an equilibrium whenever v∗+∈[0,1]. The

change of variables has had the eﬀect of translating the disease-free periodic solution

to the origin.

Following the centre manifold reduction of Church and Liu [3], the next step is to

linearize the above system at a candidate nonhyperbolic equilibrium. The origin is

expected to be nonhyperbolic with a pair of Floquet exponents with zero real part,

with the ﬁrst zero exponent resulting from the nonhyperbolicity of ˜

Sat the critical

vaccination coverage v=v∗, and the second zero exponent coming from the trivial

dynamics equation for the parameter . The result is

˙u1=−µu1(t)−ηf 0(0) ˜

S(t, v∗)u2(t) + γu2(t−τ)e−µτ , t 6=kτ

˙u2=ηf 0(0) ˜

S(t, v∗)u2(t)−(γ+µ)u2(t), t 6=kτ

˙u3= 0, t 6=kτ

˙u4= 0, t 6=kτ

∆u1=−v∗u1(t−) + (1 −v∗)e−µτ u3(t−), t =kτ

∆u2= 0, t =kτ

∆u3=v∗u1(t−)−(1 −v∗e−µτ )u3(t−), t =kτ

∆u4= 0, t =kτ.

(21)

3.4.1. Centre ﬁber bundle

Before we characterize the centre ﬁber bundle, we introduce a few convenience

functions that will be useful both in this and subsequent sections. Deﬁne

β(t, s;α) = exp Zt

s

(−γ−µ+ηf 0(0) ˜

S(u+α, v∗))du.

Then deﬁne the matrix Z1(t, s;z, α)∈C2×2for t≥sand z∈C\ {0}by

Z1(t, s;z, α) = e−µ(t−s)Rt

se−µ(t−u)(−ηf 0(0) ˜

S(u+α, v∗) + 1

zγe−µτ )β(u, s;α)du

0β(t, s;α).

Then, set Z(t, s;z, α) = diag(Z1(t, s;z, α), I2×2). Also deﬁne the matrix B∈R4×4:

B=

1−v∗0 (1 −v∗)e−µτ 0

0 1 0 0

v∗0v∗e−µτ 0

0 0 0 1

.

Finally, the function βsatisﬁes a few useful identities. They are clear from its deﬁni-

tion:

β(t, s;α) = β(t, s, [α]τ),

β(t, s;α) = β(t+τ, s +τ;α),

β(t, s;α) = β(t+α, s +α; 0).

11

For convenience, we abuse notation and write β(t, 0; 0) = β(t).

Since we have already determined that the dominant Floquet exponent of (1)–(3)

at the disease-free periodic solution must be real – see Lemma 3.2.1 – we take the

ansatz that u(t) is periodic with period τ. As a consequence, u2(t−τ) = u2(t), and

(21) reduces to an ordinary impulsive diﬀerential equation. If we denote X(t, s) the

Cauchy matrix of the resulting system, then M=X(τ, 0) is a monodromy matrix.

Speciﬁcally, M=BZ(τ, 0; 1,0);

M=

(1 −v∗)e−µτ (1 −v∗)κ(1 −v∗)e−µτ 0

0 1 0 0

v∗e−µτ v∗κ v∗e−µτ 0

0 0 0 1

, κ =eT

1Z1(τ, 0; 1,0)e2.

The eigenvalues are 1, 0 and e−µτ . The periodic solutions are generated by the two-

dimensional generalized eigenspace associated to the eigenvalue 1. The eigenvectors

are m1=(1 −v∗)κ1−e−µτ v∗κ0Tand m2=e4. As consequence, we can

competely describe the centre ﬁber bundle.

Lemma 3.4.2. The centre ﬁber bundle, RCRc, associated to the nonhyperbolic equi-

librium 0∈R4of the system (20), is two-dimensional. A basis matrix Φt, whose

columns form a basis for the t-ﬁber RCRc(t), is periodic with period τand is given

explicitly by

Φt(θ) = Z([t+θ]τ,0; 1,0)

(1 −v∗)κ0

1−e−µτ 0

v∗κ0

0 1

:= Φt,1(θ) 03×1

0 1 ,

where Φt,1(θ)∈R3.

3.4.2. Projection of χ0onto the centre ﬁber bundle

Another ingredient necessary in the centre manifold reduction concerns the pro-

jection of χ0onto the centre ﬁber bundle. Speciﬁcally, if Pc(t) : RCR → RCRc(t)

denotes the spectral projection, then there exists a unique Y(t)∈R2×4such that

Pc(t)χ0= ΦtY(t). It is characterized as the solution of the equation

ΦtY(t) = 1

2πi ZΓ1

(zI −Vt)−1χ0dz (22)

where Vtdenotes the monodromy operator associated to the linear delay impulsive

system (21), and Γ1is a simple closed contour in Csuch that 1 is the only eigenvalue

of Vtcontained in the closure of its interior. We must compute Y(t). Therefore, to

proceed we solve the equation

zy −Vty=χ0ξ(23)

12

for y∈ RCR, with ξ∈ {e1, e2, e3, e4}. Our ﬁrst task will be to obtain a representation

of Vty. We start by repartitioning the dynamics of (21) in terms of matrices. This

system can equivalently written

˙u=A(t)u(t) + f(t), t 6=kτ

∆u= (B−I)u(t−), t =kτ,

A(t) = −µ(E11 +E22) + ηf0(0) ˜

S(t, v∗)(−E21 +E12)−γE22 ,

with standard basis matrices Eij =eieT

j∈R4×4and f(t) = γe−µτ E12 u(t−τ). Note

that we have treated the delayed term as a nonhomogeneous forcing. If U0(t, s) de-

notes the Cauchy matrix associated to the (formally) homogeneous equation (without

delays), we can use the variation of constants formula to write

u(t) = U0(t, s)u(s) + Zt

s

U0(t, r)γe−µτ E12u(r−τ)dr.

Since Vty(θ) = u(t+τ+θ;t, y) where u(·;t, y) denote the solution with initial condition

(t, y)∈R× RCR, we obtain the representation

Vty(θ) = U0(t+τ+θ, t)y(0) + Zτ+θ

0

U0(t+τ+θ, t +r)γe−µτ E12y(r−τ)dr (24)

for θ∈[−τ, 0], after a few changes of variables.

Returning to equation (23), we notice that zy(θ) = Vty(θ) for θ < 0. From the

above representation, it follows that θ7→ Vty(θ) is diﬀerentiable except at times

θ∈(−τ, 0] where t+τ+θ=kτ for some k∈Z, where it is continuous from the right.

At θ= 0, there is an external discontinuity because of the χ0ξterm in (23). Taking

this into account, we can take derivatives in θon both sides of zy(θ) = Vty(θ), and

compute jumps at those times where θ=−[t]τ. We ﬁnd that y(θ) is a solution of

y0= [A(t+θ) + 1

zγe−µτ E12 ]y, θ 6=−[t]τ(25)

∆y= (B−I)y(θ−), θ =−[t]τ.(26)

for θ∈[−τ, 0). Using the convenience function Zfrom earlier, we can explicitly write

y(θ) = Z(θ, −τ;z, t)y(−τ), θ < −[t]τ

Z(θ, −[t]τ;z, t)BZ(−[t]τ,−τ;z, t)y(−τ), θ ≥ −[t]τ

(27)

Since y(−τ) appears linearly on the right-hand side of the above, we will write it as

a matrix product

y(θ) = H(θ;z, t)y(−τ) (28)

Next, from (23) we have zy(0) −Vty(0) = ξ. It is our goal to compute y(0),

and to facilitate this we consider two separate cases. If [t]τ= 0, then we have

13

Vty(0) = BVty(0−), as can be veriﬁed via equation (24). Since Vt(θ) = zy(θ) for

θ < 0, it then follows that Vty(0) = Bzy(0−). The equation zy(0) −Vty(0) = ξis

then equivalent to zy(0)−Bzy(0−) = ξ. A similar argument in the case where [t]τ6= 0

then implies that, in both cases, the end result is

y(0) = 1

zξ+H(0−;z, t)y(−τ) (29)

Our ﬁnal task is to solve for y(−τ). To do this, substitute (29) into (24) and set

θ=−τ. Since Vt(−τ) = zy(−τ), the result is

zy(−τ) = 1

zξ+H(0−;z, t)y(−τ).(30)

Lemma 3.4.3. z7→ (zI −H(0−;z, t))−1has a pole at z= 1. In particular, 1is an

eigenvalue of multiplicity two for H(0−;z, t).

Proof. The spectrum (as a multiset) is found to be

σH(0−;z, t)={e−µτ , β(0,−τ;t),0,1}.

The second eigenvalue in the list is, explicitly,

β(0,−τ;t) = exp Z0

−τ

(−γ−µ+ηf 0(0) ˜

S(u+t, v∗))du,

which is equal to 1 because the integrand is periodic with period τ, integrates to zero

on [0, τ ], and tacts as a translation parameter. The result follows.

We can now calculate y= (zI −Vt)−1χ0. Solving equation (30) and substituting

the result into (28), the following lemma is proven.

Lemma 3.4.4. (zI −Vt)−1χ0has the explicit form

(zI −Vt)−1χ0(θ) = 1

zH(θ;z, t)(zI −H(0−;z, t))−1.(31)

The next step is to explicitly calculate the contour integral in (22). The following

lemma provides just enough detail for later calculations.

Lemma 3.4.5. There exist real constants a, b such that

1

2πi ZΓ1

(zI −Vt)−1χ0=H(θ; 1, t)

0ab 0 0

0 1 0 0

0a0 0

0 0 0 1

.(32)

14

Proof. We provide only an outline of the proof of this lemma. To begin, perform the

diagonalization

(zI −H(0−;z, t)) = P(z)(zI −D)P(z)−1

where D= diag(0, e−µτ ,1,1) has the eigenvalues of H(0−;z, t) on the diagonal. With

this representation, both Pand P−1are holomorphic in a neighbourhood of z= 1.

After lengthy calculations, one can show that

P=

1 1 P13(z) 0

0 0 P23(z) 0

−eµ[t]τv∗

1−v∗eµ[t]τ1 0

0 0 0 1

,

for some P13 and P23, with P23 6= 0 in a neighbourhood of z= 1. Taking into account

(31), we have

1

2πi ZΓ1

(zI −Vt)−1χ0=1

2πi ZΓ1

1

zH(θ;z, t)P(z)(zI −D)−1P(z)−1dz

=H(θ; 1, t)P(1)diag(0,0,1,1)P(1)−1,

with the second line being a consequence of Cauchy’s integral formula. Explicitly

calculating the product P(1)diag(0,0,1,1)P(1)−1, the result is the matrix on the

right-hand side of (32) with a= 1/P23 (1) and b=P13(1).

Lemma 3.4.6. The matrix Y(t)appearing in the decomposition (22) is

Y(t) = 0 (1 −e−µτ )−1β(−[t]τ,−τ;t) 0 0

0 0 0 1 .(33)

Proof. Since the matrix Y(t) appearing in (22) is unique [Section 3.3, [3]] and therefore

independent of the argument θ∈[−τ, 0], we can evaluate both sides of the equation

at θ=−[t]τto simplify the computation. Using Lemma 3.4.5 and Lemma 3.4.6, the

result is the equation

(1 −v∗)κ0

1−e−µτ 0

v∗κ0

0 1

Y11 Y12 Y13 Y14

Y21 Y22 Y23 Y24 =H(−[t]τ; 1, t)

0ab 0 0

0 1 0 0

0a0 0

0 0 0 1

.

Explicitly calculating H(−[t]τ; 1, t), one immediately ﬁnds that the only nonzero en-

tries of Yare Y12 and Y24, the latter of which is Y24 = 1. The Y12 entry satisﬁes the

equation

Y12

(1 −v∗)κ

1−e−µτ

v∗κ

0

=H(−[t]τ; 1, t)

ab

1

a

0

.

Comparing the entries in the second row, we ﬁnd Y12 ·(1 −e−µτ ) = β(−[t]τ,−τ;t),

and the result follows.

15

3.4.3. Dynamics on the centre manifold and bifurcation

On the centre manifold, the time evolution is generally determined by an ordi-

nary impulsive diﬀerential equation. In this case particular, to quadratic order the

dynamics are actually a scalar ordinary diﬀerential equation.

Lemma 3.4.7. The coordinate dynamics on the two-dimensional parameter-dependent

centre manifold of the nonhyperbolic equilibrium 0∈R4of the impulsive delay diﬀer-

ential equation (20) are, for ||(w, )|| suﬃciently small,

˙w=η(1 −e−µτ )β(−[t]τ,−τ;t)g(t)w2+f0(0)∂v˜

S(t, v∗)w+R(t, w, ),

˙= 0,

g(t) = ˜

S(t, v∗)(1 −e−µτ )β(t) 1

2f00(0)(1 −e−µτ )β(t)

+f0(0) Z[t]τ

0

e−µ([t]τ−s)(γe−µτ −ηf0(0) ˜

S(s, v∗))β(s)ds!,

(34)

where R(t, w, )satisﬁes R(t, 0, )=0, is periodic and right-diﬀerentiable in its ﬁrst

argument, and is C∞in (w, )for ﬁxed t. On the centre manifold, the evolution in

the phase space RCR is determined by the time evolution rule

t7→ Φtw(t)

(t).(35)

Proof. From [3], the dynamics on the centre manifold are given by

˙z= Λz+eΛtY(t)F(Qtz+z(t, z, ·)), t 6=kτ

∆z=eΛtY(t)G(Qt−z+h(t−, z, ·)), t =kτ,

where Φt=QteΛtis a Floquet decomposition of the basis matrix for the centre ﬁber

bundle with t7→ Qtperiodic with period τ,h:R×R2×[−τ, 0] →R4is the Euclidean

space representation of the centre manifold, and F= (F1, F2, F3, F4) and Gcontain

all nonlinear terms from (20). In this instance, Λ = 0 and Φt=Qtin the Floquet

decomposition. Also, Y(t)G= 0 and Y(t)F= [ Y12(t)F20]T. Keeping only terms

of order two in z= (w, ), the result is (34). The smoothness properties of the

remainder follow from those of h: namely [Theorem 3.1, [3]], z7→ h(t, z, ·) is C∞and

t7→ h(t, z, θ) is diﬀerentiable from the right.

With this lemma in place, we can ﬁnally state and prove our bifurcation theorem.

Theorem 3.4.1. For a generic set of parameters, a transcritical bifurcation occurs

in the model (1)–(3) along the disease-free periodic solution as vcrosses through the

critical vaccination coverage level v∗. Speciﬁcally

`=Zτ

0

β(−[t]τ,−τ;t)˜

S(t, v∗)β(t)g(t)dt,

16

is nonzero on a generic subset of parameter space, and the following is satisﬁed for

|v−v∗|small enough and in a suﬃciently small neighbourhood of (S, I)=(˜

S(t, v∗),0).

•There are at most two periodic solutions: the disease-free solution and a second

solution t7→ ξ(t, v)that is exponentially stable when v < v∗, unstable when

v > v∗, and satisﬁes ξ(t, v∗)=(˜

S(t, v∗),0).

•The unique periodic solution is conditionally stable when v=v∗in some half-

space.

•ξ(·, v)is positive (in both components) if and only if (v−v∗)` > 0.

Proof. The time τ(Poincar´e) map associated to the ordinary diﬀerential equation

(34) is readily found to satisfy

w7→ w+η(1 −e−µτ )[`w2+mw] + h(w, )

7→ ,

where `is as in the statement of the theorem, mis given by

m=Zτ

0

β(−[t]τ,−τ;t)f0(0)∂v˜

S(t, v∗)dt,

and h(w, ) = Rτ

0R(t, w, )dt is a C∞remainder satisfying h(0, ) = 0 and containing

all terms of order 3 and above in (w, ). Note that the mixed w term, m, is strictly

negative because f0(0) >0, β > 0, and ∂v˜

S(t, v∗)<0. As for the quadratic term,

the equation `= 0 is unstable with respect to perturburbations in f00(0), as can be

veriﬁed by the functional form of g(t) appearing in (34). Consequently, on a generic

set of parameters we have `6= 0 and m < 0. From the transcritical bifurcation for

maps, there exists a unique C1nontrivial ﬁxed point (w(), ) for ||suﬃciently small,

satisfying w(0) = 0. From (35), we obtain the claimed nontrivial periodic solution.

The stability assertions follow by the reduction principle [Theorem 6.21, [4]].

To see that ξ(·, v) is positive only when (v−v∗)`=` > 0, we ﬁrst remark that

the ﬁxed point satisﬁes the estimate w() = −m

`+O(2).This follows because of the

properties of the remainder term h. Also, since ξ(t, v)→(˜

S(t, v∗),0) as v→v∗, it

suﬃces to consider only the sign of the second component. This is precisely

sign(ξ2(t, v∗+)) = sign −m

`eT

2Φt,1(0)

= sign `(1 −e−µτ )β(t)

= sign(`),

which is what was claimed.

Remark 3.4.1. We would typically expect ` < 0for biological reasons. Namely, ` < 0

would imply that increasing the vaccination coverage to the critical level v∗drives a

stable endemic (ie. positive) periodic solution toward the disease-free state. However,

proving that `is indeed negative on the entire parameter space for which v∗∈[0,1]

seems to be rather diﬃcult.

17

Parameter Numerical value/range

µ0.5

η50

γ25

τ1

v∗0.6227

v[0, v∗]

Table 1: Parameters used for the numerical bifurcation analysis.

4. Numerical bifurcation analysis

In the previous section we proved that in the event there is only one vaccination

pulse per period, the disease-free periodic orbit generically undergoes a transcritical

bifurcation when the vaccination coverage crosses a critical threshold. In the absence

of pulse vaccination, the model (1)–(3) reduces to the SIR model of Kyrychko and

Blyuss, and it is known that the endemic equilibrium can undergo Hopf bifurcation

[10]. In impulsive systems, Hopf points result in bifurcations to invariant cylinders

under generic parameter variation [3], so it is natural for us to track the bifurcating

endemic periodic orbit and search for Hopf points. In this section, we will use the

illustrative parameter choices provided in Table 1, and to keep results consistent with

the analysis appearing in [10, 11] we will use the indence rate f(x) = x

1+x.

4.1. Continuation of endemic perodic solution

As it is not possible to express the endemic periodic solution analytically, we will

need to resort to numerical methods. The ﬁrst step is to continue the bifurcating

endemic periodic solution. To simplify matters, we will take advantage of the conve-

nient fact that since we seek periodic solutions of period τ, it is not necessary to treat

the impulsive delay diﬀerential equation (1)–(3) directly. Speciﬁcally, every instance

of S(t−τ) and I(t−τ) can be replaced by S(t) and I(t), since we search explic-

itly for periodic solutions. Doing this and adjusting the jump condition accordingly,

the branch (S∗(t, v), I∗(t, v)) of endemic periodic solutions must satisfy the following

boundary-value problem:

d

dtS∗=µ−µS∗−ηf(I∗)S∗+γe−µτ I∗,

d

dtI∗=ηf (I∗)S∗−(µ+γ)I∗,

S∗(0) = ρ(v)S∗(τ−),

I∗(0) = I∗(τ−),

ρ(v) = (1 −v)(1 −ve−µτ )−1.

Following Theorem 3.4.1, we could use

S∗(t, v∗+) = ˜

S(t, v∗)−m

`eT

1Φt,1(0)

I∗(t, v∗+) = −m

`eT

2Φt,1(0)

(36)

18

Figure 1: Plots of the periodic solution obtained by the continuation scheme described in Section 4.1

for vaccine coverage v∈[0, v∗]. Dots indicate the “initial” points (S∗(0, v), I∗(0, v)) on each periodic

solution, followed by evolution along the corresponding curve at each level vwith time left implicit.

The periodic solution is constant in the Ivariable at v=v∗, and collapses to a ﬁxed point at v= 0.

To improve visibility, only fourteen vaccination coverages in the interval [0, v∗] are displayed.

as a linear-order guess for the ﬁrst point on the branch, for some ||suﬃciently small.

Under the assumption that ` < 0 – the more biologically expected case – we will do

continuation for < 0.

For the illustrative system parameters of Table 1, we solve this boundary value

problem using the bvp4c function in MATLAB R2018a. We take the solution for

perturbation parameter nas the initial guess for perturbation parameter n+1 < n,

except for the ﬁrst step 0where we use the linear guess (36). That is, we keep things

simple and implement natural continuation. We do not expect any turning points

along the branch (note, we expect a Hopf point, which will generically lead to an in-

variant cylinder on which there are no additional periodic solutions), so no diﬃculties

are anticipated. Figure 1 provides a sample of the periodic solutions generated by

the continuation scheme.

4.2. Floquet exponents

To test for additional bifurcation points, we will need to compute the dominant

Floquet exponents along the periodic solution continuation. While this can in prin-

ciple be done by solving an approriate boundary-value problem, our attempts to

accomplish this in MATLAB using built-in boundary-value problem solvers had se-

19

rious issues with convergence. Instead, we discretize the monodromy operator on

one of its well-behaved invariant subspaces and approximate the Floquet multipliers,

thereby granting an approximation of the Floquet exponents. The method is outlined

in the Appendix B.

The real part dominant Floquet exponent is plotted versus the vaccination cov-

erage in Figure 2. Numerically windowing the crossing of the imaginary axis, we see

that the real part crosses through zero for some v∈[0.4063,0.4068]. The approxi-

mate Floquet spectrum (i.e. the set of Floquet exponents) for v= 0.4068 is plotted in

Figure 3, where we see that, as expected, there is a pair of (approximately) imaginary

Floquet exponents. To the order of discretization used (200 mesh points), the pair

of Floquet exponents is simple, so we should expect [Theorem 5.2, [3]] a cylinder

bifurcation to occur at some v∗

c∈[0.4063,0.4068].

4.3. Cylinder bifurcation

We can easily check – at least to the level of numerical accuracy achieved – two

of the nondegeneracy conditions associated to the cylinder bifurcation theorem [The-

orem 5.2, [3]]. The theorem (and its generic corollary) are reproduced in Appendix

C. The ﬁrst condition G.1 states that we must have ekiω 6= 1 for k= 1,2,3,4, where

ωis the imaginary part of the dominant Floquet exponent. Our numerical estimate

(see Figure 3) is

ω= 1.9886.(37)

One can then verify that |ekiω −1| ≥ 0.316, and the ﬁrst nondegeneracy condition is

passed.

The second nondegeneracy condition G.2 pertains to the transversal crossing of

the Floquet exponents across the imaginary axis. Examining the plot of real part

of the dominant Floquet exponent in Figure 2, we see that the real part is strictly

decreasing and appears smooth in the critical interval [0.4063,0.4068], from which we

conclude that the second nondegeneracy condition is satisﬁed.

The third nondegeneracy condition requires one to calculate the quadratic term

of the centre manifold at the critical parameter and, following this, compute the ﬁrst

Lyapunov coeﬃcient. While this can indeed be accomplished using a similar numerical

scheme to the one from Appendix B in conjunction with the centre manifold approxi-

mation method described in [3], the beneﬁts are somewhat limited and do not greatly

aid in the exposition. We will therefore content ourselves with the ﬁrst two nondegen-

eracy conditions, knowing that in a generic sense (i.e. up to perturbation in quadratic

terms), a cylinder bifurcation does indeed occur at some v∗

c∈[0.4063,0.4068]. More-

over, because the real part of the dominant Floquet exponent is decreasing at v∗

c, we

obtain by Theorem C.0.1 that the periodic solution (S∗(t, v), I ∗(t, v)) is unstable for

v < v∗

cand locally asymptotically stable for v > v∗

c.

Remark 4.3.1. We brieﬂy comment that the theorem should be applied not to (1)–

(3), but rather to the system with vaccinated component introduced in Section 2.2,

20

Figure 2: A plot of the real part of the dominant Floquet exponent of the branch v7→

(S∗(·, v), I ∗(·, v)) of periodic solutions. The real part crosses the line <(λmax) = 0 in the window

v∈[0.4063,0.4068]. All computations were done using the discretization scheme from Appendix B

with N= 200 mesh points.

since the latter satisﬁes the overlap condition. All statements concerning the Floquet

exponents carry over, however, so our conclusions remain correct.

4.3.1. Time series

Provided the invariant cylinder is attracting in the parameter regime where it

exists, we should expect based on Theorem C.0.1 that the cylinder is attracting for

0v < v∗

c≈0.4063. In a time series, we would expect to see convergence to

an oscillatory but non-periodic solution. Figure 4 provides such a time series for a

selection of six vaccination thresholds v∈[0, v∗]. The solution converges to a clearly

deﬁned periodic solution for v= 0.6 and v= 0.45, while in the intermediate regime of

v∈(0, v∗

c] demonstrated by the second row of plots and the ﬁrst plot on the ﬁnal row,

the dynamics are eventually oscillatory with no discernable period. A clear periodic

solution is seen at v= 0.

4.3.2. Phase space plots

The geometry of the cylindrical attractor is more clearly seen if one plots S(t) and

I(t) together with t7→ f(xt) for some functional f, with t7→ xtthe solution in the

phase space RCR. One might think the sum of vaccinated and recovered components

as in Figure 4 to be a natural choice, but there is some transient linear dependence

between these and S(t) and I(t) that hides some of the geometry. Instead, we plot

t7→ (S(t), I(t), S(t−1)).

21

-5 -4 -3 -2 -1 0 1

-3

-2

-1

0

1

2

3

Figure 3: The approximate Floquet spectrum associated to the periodic solution t7→

(S∗(t, v), I ∗(t, v)) for v= 0.4063 restricted to the strip {z∈C:<(z)∈[−5,∞)}. All compu-

tations were done using the discretization scheme from Appendix B with N= 200 mesh points.

To illustrate the birth of the cylindrical attractor as the parameter vis varied

close to the bifurcation point v∗

c, we plot both the continuation periodic solution t7→

(S∗(t, v), I ∗(t, v)) and the foward time integration from the constant initial condition

(S(0), I(0)) = (0.5,0.5). The delayed state S(t−1) is used as a third spatial variable

to aid in visualization as described above. We integrate the solution for t∈[0,1300]

and plot only for t∈[300,1000]. The result is provided in Figure 5, where we clearly

see the cylindrical topology appearing at v= 0.395.

At v= 0.385 there appears to be phase locking, although the phase-locked regions

still appear to lie on a cylindrical structure. To compare, as vdecreases to 0.375, then

to 0.35 and 0.315 – see Figure 6 – the radius of the cylinder becomes more variable

variable along its length, the latter of which is contracted. The structure of the

attractor bears little resemblance to the periodic solution t7→ (S∗(t, v), I∗(t, v), S∗(t−

1, v)) it bifurcated from.

Further decreasing the vaccination coverage from v= 0.25 through to v= 0 shows

convergence of the attractor to the periodic orbit of the Kyrychko and Blyuss model.

Topologically, the cylinder contracts to a circle. This can be visualized in Figure 6.

5. Conclusions

The SIR model with temporary immunity of Kyrychko and Blyuss [11] was ex-

tended to include pulse vaccination. Motivated by the analytical proof of the Hopf

bifurcation in this system by Jiang and Wei [10] and the cylinder bifurcation theorem

22

0 2 4 6 8 10 12 14 16 18 20

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30 35 40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 5 10 15 20 25 30 35 40 45 50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 10 20 30 40 50 60

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Figure 4: Time series from the constant initial condition (S(0), I (0), R(0), V (0)) = (0.5,0.5,0,0) for

various vaccination coverages. Susceptible, infected and sum of recovered and vaccinated populations

are plotted, with a legend inset in the ﬁrst frame. The vaccinated population is governed by (39).

(Top row: v= 0.6, v= 0.45. Middle row: v= 0.395, v= 0.3. Bottom row: v= 0.15, v= 0.)

23

Figure 5: Plot of t7→ (S(t), I(t), S(t−1)) for t∈[300,1300] from a constant initial condition

of (S(0), I(0)) = (0.5,0.5), for v= 0.45 (top) and v= 0.395 (bottom). Purple corresponding

to arguments t=k∈Zand yellow to arguments t→k−. Inset: Plots of the image of t7→

(S∗(t, v), I ∗(t, v), S∗(t−1, v)).

24

of Church and Liu [3], we suspected that the model with pulse vaccination might

exhibit a bifurcation to an invariant cylinder if a Hopf point could be identiﬁed.

We began our analysis in Section 3 with the existence and stability of the disease-

free periodic solution. We proved analytically that this periodic solution exists and is

unique and derived the basic reproduction number. The basic reproduction number

was shown to be an epidemiological threshold and its biological interpretation was

discussed. These results were all shown to hold regardless of the number of vacci-

nation pulses per period. Also, we proved (Lemma 3.3.1) the existence of a unique

critical vaccination coverage v∗at which the disease-free periodic solution gains (re-

spectively, loses) its stability. Consequently, treating the vaccination coverage as a

bifurcation parameter, v=v∗is the unique parameter at which a bifurcation along

this periodic solution can occur. We subsequently proved that under generic condi-

tions, a transcritical bifurcation occurs as vcrosses through v∗, with the result being

the birth of an endemic periodic (i.e. positive) periodic solution in the regime v < v∗

(assuming the biologically sensible case ` < 0).

Since the endemic periodic solution born out of the transcritical bifurcation could

not be expressed analytically, we turned out attention to numerical methods in Section

4. Using a numerical continuation scheme, we continued the endemic periodic solution

into the parameter regime [0, v∗) for an illustrative set of parameters. Subsequently,

by implementing a monodromy operator discretization scheme we tracked the real part

of the dominant Floquet exponent along the branch of endemic periodic solutions and

identiﬁed a parameter window where a bifurcation point was located. It was found

that the real part crossed zero at some v∈[0.4063,0.4068]. Taking v∗

c= 0.4063 as

an approximate bifurcation point, we approximated the spectrum of the monodromy

operator and determined that it was a Hopf point, with λ=±1.9886ibeing a simple

pair of complex conjugate Floquet exponents.

We compared time series plots of the solution to phase plots in RCR, visualized

by projection onto the variables (S(t), I(t), S(t−1)). The time series plots were

consistent with an endemic periodic solution undergoing a bifurcation to an oscillatory

but aperiodic mode, followed by eventual collapse onto a periodic orbit as vdecreased

from v∗, through v∗

c< v∗and ﬁnally to zero. The phase plots revealed the expected

cylindrical attractor for v < v∗

c. Phase locking was observed at v= 0.385, followed

by a contraction of the cylinder along its length as vdecreased, with collapse to a

topological circle at v= 0.

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27

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A. Derivation of the impulse condition in equation (3)

Suppose Vkis the number of individuals that received a vaccine at time tk−τ. By

assumption 3), Vke−µτ of them are still alive at time tk. Thus, upon transferring into

the susceptble cohort by assumption 1), the total number of susceptible individuals

becomes Sk:= S(t−

k) + Vke−µτ .A fraction vof these individuals are vaccinated at

time tk, so there are (1 −v)Skremaining. We can write the latter as

S(tk) = (1 −v)Sk=S(t−

k)−vS(t−

k) + (1 −v)Vke−µτ .(38)

29

Now, if the total number of susceptible individuals (including those that lost their

immunity) at time tk−τis denoted S−

k, then by 3) we have S(tk−τ) = (1 −v)S−

k

and Vk=vS−

k, which together imply Vk=v

1−vS(tk−τ). Substituting into (38), we

have that at time t=tk,

∆S=S(tk)−S(t−

k)

=−vS(t−

k) + (1 −v)Vke−µτ

=−vS(t−

k) + vS(tk−τ)e−µτ ,

which is precisely equation (3). Similarly, the number of vaccinated individuals V

satisﬁes

˙

V=−µV, t 6=tk

∆V=vS(t−

k)−vS(tk−τ)e−µτ , t =tk.(39)

Note that this count of the number of vaccinated inviduals is diﬀerent than the one

appearing in Section 2.2. In the latter, the component Vjis introduced to circumvent

the overlap condition and it does not explicitly track deaths.

B. Monodromy operator discretization

Our goal is to compute the Floquet exponents associated to the linear system

˙x=A(t)x(t) + Bx(t−τ), t 6=kτ

∆x=Cx(t−) + Dx(t−τ), t =kτ (40)

with data A(t), B,Cand Dgiven by

A(t) = −µ−ηf (I∗(t, v)) −ηf 0(I∗(t, v))S∗(t, v)

ηf (I∗(t, v)) ηf 0(I∗(t, v))S∗(t, v)−γ−µ

B=0γe−µτ

0 0 , C =−v0

0 0 , D =ve−µτ 0

0 0 .

Recall that the Floquet exponents are independent of the choice of monodromy op-

erator [Theorem 7.1.2, [4]], so we may restrict our attention to V0:RCR → RCR.

If we denote X(t, s) for t≥sthe Cauchy matrix associated to the linear ordinary

diﬀerential equation ˙y=A(t)yand set X(t, s) = 0 for t<s, then the variation of

constants formula for ordinary diﬀerential equations and the method of steps implies

that the solution of (40) passing through the initial function φ∈ RCR at time 0 can

be written for t∈[0, τ ] as follows:

x(t) = X(t, 0)φ(0) + Zt

0

X(t, s)Bφ(s−τ)ds

+χ{τ}(t)(CX(τ , 0) + D)φ(0) + CZτ

0

X(τ, s)Bφ(s−τ)ds.

30

Evaluating at t=τ+θand taking θ∈[−τ, 0] as the argument, exploiting the

periodicity X(s1+τ, s2+τ) = X(s1, s2) and simplifying the expression using a few

changes of variables, we can express the monodromy operator as

V0φ(θ) = [X(τ+θ, 0) + χ0(θ) (CX(τ, 0) + D)] φ(0)

+Z0

−τ

[X(θ, s) + χ0(θ)CX(0, s)] Bφ(s)ds. (41)

Our next key observation is that if vis an eigenvector of V0, then vis in fact C2

on [−r, 0) with a ﬁnite jump discontinuity at zero. If we deﬁne

X={φ: [−r, 0] →R2:φ|[−r,0) ∈C2,|φ(0) −φ(0−)|<∞},

then Xis an invariant subspace of V0and it contains all of its eigenvectors. Therefore,

we may consider the restriction V0:X→Xinstead of the action of the monodromy

operator on the entire space RCR.

The representation (41) and the description of Xsuggests a decomposition of φ

into its part at φ= 0 and on [−τ, 0). To do this we introduce the product space

Y=C2([−τ, 0],R2)×R2. The function G:Y→Xdeﬁned by

G(y1, y2) = χ[−τ,0)y1+χ{0}y2

is an isomometry, so we will make the identiﬁcation X∼Y. Then, for (φ1, φ2)∈Y,

we can write

V0φ1

φ2="X(τ+·,0)φ2+R0

−τX(·, s)Bφ1(s)ds

((I+C)X(τ, 0) + D)φ2+R0

−τ(I+C)X(0, s)Bφ1(s)ds #.(42)

We are ready to discretize the monodromy operator. Let N≥1 denote the number

of mesh points, and let −τ < s1<· · · < sN<0 denote the Gaussian quadrature

points in the interval [−τ, 0]. Let w1, . . . , wNdenote the associated weights. Then,

we have the limit

lim

N→∞

N

X

k=1

f(sk)wk=Z0

−τ

f(s)ds

for any f∈C2. We approximate φ1∈C2([−τ, 0],R2) by the vector

ˆ

φ1= [ φ1(s1). . . φ1(sN)]T∈RN

and, taking into account the convergence of the Gaussian quadrature for C2integrands

and using the identity X(τ+x, τ +y) = X(x, y), we can make the approximation

V0φ≈V0,N [ˆ

φ1φ2]T, where

V0,N =

X(s1, s1)Bw1· · · X(s1, sN)BwNX(s1,−τ)

.

.

.....

.

..

.

.

X(sN, s1)Bw1· · · X(sN, sN)BwNX(sN,−τ)

(I+C)X(0, s1)Bw1· · · (I+C)X(0, sN)BwN(I+C)X(0,−τ) + D

.

(43)

31

Given the compactness [Lemma 7.1.1, [3]] of V0, we should expect that any eigenvalue

of V0is given by the limit of some eigenvalue of V0,N as N→ ∞ provided V0,N →V0

(after deﬁning V0,N on Xby an appropriate polynomial embedding, so as to make

this convergence sensical). Conseqently, the spectrum of V0,N may be seen as an

approximation of that of V0, so we may approximate the Floquet spectrum via

Λ≈1

τlog µ:µ∈σ(V0,N ),

if we recall [Section 3.1, [3]] that the Floquet exponents are precisely λ=1

τlog µ

where µis an eigenvalue of V0, and log denotes the principal logarithm. The proof

of the convergence and approximation claims are beyond the scope of the article and

deferred to future work. We refer the reader to [8], where a similar scheme is developed

for the discretization of linear periodic delay diﬀerential equations and convergence

results are proven.

When we use this method to approximate Floquet exponents, we use a numerically

computed Cauchy matrxix X(t, s) generated by MATLAB’s ode45 function. The

entries X(si, sj) appearing in the matrix V0,N are zero for i<jand are the identity

for i=j, so most superdiagonal entries do not need to be calculated. As for the

nonzero entries, we remark that

X(si, sj) = X(si,−τ)[X(sj,−τ)]−1,

so it is enough to compute X(sk,−τ) for k= 1, . . . , N mesh points and recycle these

matrices to compute all others. This speeds up the process of computing the matrix

V0,N considerably.

C. Cylinder bifurcation theorem

To help keep this document self-contained, we summarize the cylinder bifurcation

theorem of Church and Liu [3] together with the generic corollary. The statement of

the theorem is simpliﬁed somewhat.

Theorem C.0.1. Suppose the impulsive delay diﬀerential equation with real para-

mater

˙x=Ax(t) + Bx(t−r) + f(x(t), x(t−r), ), t 6=k∈Z

∆x=Bx(t−) + Cx(t−r) + g(x(t−), x(t−r), ), t =k∈Z,

satisﬁes the overlap condition, f(0,0,0) = g(0,0,0) = 0 and Df (0,0,0) = Dg(0,0,0) =

0, with real parameter . Let λ=±iω with ω > 0be Floquet exponents associated

to the linearization at parameter , let the centre ﬁber bundle be two-dimensional and

suppose there are no Floquet exponents with positive real part. For fand gboth C3, a

generic set of systems of this type exhibit the cylinder bifurcation as passes through

32

zero. That is, there is a neighbourhood Nof 0∈ RCR and a smooth invertible change

of parameters ν=ν()satifying ν(0) = 0 such that for ν > 0, there is an invariant

(topological) cylinder in S1×Nthat trivializes to the circle S1× {0}as η→0, to-

gether with a unique periodic solution in Nthat persists for all |η|suﬃciently small

and trivializes to zero (the equilibrium) as η→0. The generic conditions are

G.1 ekiω 6= 1 for k= 1,2,3,4;

G.2 the critical Floquet exponent 7→ λ()satisfying λ(0) = iω crosses the imaginary

axis transversally at = 0 — that is, λ0(0) 6= 0;

G.3 the ﬁrst Lyapunov coeﬃcient d(0) is nonzero (see [3] for details).

In this case, the cylinder exists for ||small enough such that d(0)λ0(0) < 0, and we

have the following (local) stability assertions:

•the periodic solution is asymptotically stable for λ0(0) < 0, stable for = 0, and

unstable for λ0(0) > 0, while the cylinder is attracting for γ(0) > 0, provided

d(0) <0;

•the periodic solution is asymptotically stable for λ0(0) < 0and unstable for

λ0(0)≥0, while the cylinder is unstable for λ0(0) < 0, provided d(0) >0.

33