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A time-delayed SIR model with general nonlinear incidence rate, pulse vaccination and temporary immunity is developed. The basic reproduction number is derived and it is shown that the disease-free periodic solution generically undergoes a transcritical bifurcation to an endemic periodic solution as the vaccination coverage drops below a critical level. Using numerical continuation and a monodromy operator discretization scheme, we track the bifurcating endemic periodic solution as the vaccination coverage is decreased and a Hopf point is detected. This leads to a bifurcation to an attracting, invariant cylinder. As the vaccination coverage is further decreased, the geometry of the cylinder contracts along its length until it finally collapses to a periodic orbit when the vaccination coverage goes to zero. In the intermediate regime, phase locking on the cylinder is observed.
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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=µµS ηf (I(t))S(t) + γI(tτ)eµτ
˙
I=ηf (I(t))S(t)(µ+γ)I(t)
˙
R=γI (t)γI(tτ)eµτ 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 kZ, any individuals that received their
vaccine at time tkτand 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 kZ.
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=µµS ηf (I(t))S(t) + γI(tτ)eµτ , t 6=tk(1)
˙
I=ηf (I(t))S(t)(µ+γ)I(t), t 6=tk(2)
S=vS(t) + vS(tτ)eµτ , 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 xRn, 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: [αr, 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
tR. 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:RCnperiodic 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⊂ {zC:<(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:XXis the identity operator on X(a given
vector space) and χ{0}:IRis the indicator function on the set {0}, where IR:
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 tkτis 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=µµx ηf (y(t))x(t) + γy(tτ)eµτ , 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)eµτ , t =tj+qk (8)
Vj=vx(t)(1 veµτ )Vj(t), t =tj+qk .(9)
In the above impulsive delay differential equation, jranges from 0 to q1 where q
is the period of the sequence of impulse times as defined in assumption 3). Taking
note that tj+qk =tj+q(k1) +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)eµτ = (1 v)Vj(tτ)eµτ
= (1 v)[vx((tτ)) + veµτ Vj((tτ))]
=v[(1 v)x((tτ)) + (1 v)Vj((tτ))eµτ ]
=vx(tτ).
Substituting the above into the jump condition for x, the result is
x=vx(t) + (1 v)Vj(t)eµτ =vx(t) + vx(tτ)eµτ .
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(tτ)eµτ , 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(tτ)eµτ , 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)|| ≤ Keα(ts)for some α > 0 and K1, for all ts.
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 svtimplies that the nth iterate of
Vsatisfies the inequality
||VnφVnψ|| ≤ Keατn ||φψ||.
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(tτ) and cancelling
exponentials, we arrive at the following impulsive differential equation for φ:
˙
φ+λφ =µφ, t 6=tk(14)
φ=(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) = 1v
1ve(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)τ1v
1ve(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=eλτ , 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) = zeµτ 1v
1veµτ zq
.
We will show that |g(z)|<|f(z)|on the unit circle |z|= 1. We have
|g(z)|=eµτ 1v
|1veµτ z|q
eµτ 1v
|1− |veµτ z|| q
=eµτ 1v
1veµτ q
eµτ <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 |eλτ | ≤ 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) + γeµτ u2(tτ), t 6=tk
˙u2=ηf 0(0) ˜
S(t, v)u2(t)(γ+µ)u2(t), t 6=tk
u1=vu1(t) + vu1(tτ)eµτ , 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 vat
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=µµz, t 6=tk
z= (ρ(v)1)z(t), t =tk,
where ρ(v) = (1 v)/(1 veµτ ). 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
dv <0 and, subsequently,
that ∂z
∂v 0 for all t0. Also, one has z
∂z0>0 for all t0. Using the variation of
constants formula for impulsive differential equations, routine calculations yield
z(τ;z0, v) = eµτ ρ(v)q"z0+
q
X
i=1
ρ(v)1i(eµtieµti1)#,
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) = eµτ
1ρ(v)qeµτ
q
X
k=1
ρ(v)q+1i(eµτieµτi1),
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 vfollow 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=for kZ. Then,
from the previous section, we can explicitly calculate
˜
S(t, v) = 1 veµ[t]τ,(18)
which implies that ˜
S(τ, v)=1veµτ and ˜
S(0, v)=1v. We can also explicitly
calculate the critical vaccination coverage where R0= 1. We find
v=µτ
1eµτ 1γ+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)
1veµτ =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 (tτ)eµτ , t 6=
˙
Y=ηf (Y)˜
S(t, v+) + X(µ+γ)Y(t), t 6=
˙
V= 0, t 6=
˙= 0, t 6=kτ
X=(v+)X(t) + (1 (v+))eµτ V(t), t =
Y= 0, t =
V= (v+)X(t)(1 (v+)eµτ )V(t), t =
= 0, t =.
(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(tτ)eµτ , t 6=
˙u2=ηf 0(0) ˜
S(t, v)u2(t)(γ+µ)u2(t), t 6=
˙u3= 0, t 6=
˙u4= 0, t 6=
u1=vu1(t) + (1 v)eµτ u3(t), t =
u2= 0, t =
u3=vu1(t)(1 veµτ )u3(t), t =
u4= 0, t =.
(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
(γµ+ηf 0(0) ˜
S(u+α, v))du.
Then define the matrix Z1(t, s;z, α)C2×2for tsand zC\ {0}by
Z1(t, s;z, α) = eµ(ts)Rt
seµ(tu)(ηf 0(0) ˜
S(u+α, v) + 1
zγeµτ )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 BR4×4:
B=
1v0 (1 v)eµτ 0
0 1 0 0
v0veµτ 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(tτ) = 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)eµτ (1 v)κ(1 v)eµτ 0
0 1 0 0
veµτ vκ veµτ 0
0 0 0 1
, κ =eT
1Z1(τ, 0; 1,0)e2.
The eigenvalues are 1, 0 and eµτ . The periodic solutions are generated by the two-
dimensional generalized eigenspace associated to the eigenvalue 1. The eigenvectors
are m1=(1 v)κ1eµτ vκ0Tand 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 0R4of 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
1eµτ 0
vκ0
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=
u= (BI)u(t), t =,
A(t) = µ(E11 +E22) + ηf0(0) ˜
S(t, v)(E21 +E12)γE22 ,
with standard basis matrices Eij =eieT
jR4×4and f(t) = γeµτ E12 u(tτ). 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)γeµτ E12u(rτ)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)γeµτ E12y(rτ)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 kZ, 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
zγeµτ E12 ]y, θ 6=[t]τ(25)
y= (BI)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)={eµτ , β(0,τ;t),0,1}.
The second eigenvalue in the list is, explicitly,
β(0,τ;t) = exp Z0
τ
(γµ+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, eµτ ,1,1) has the eigenvalues of H(0;z, t) on the diagonal. With
this representation, both Pand P1are 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
1veµ[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 eµτ )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
1eµτ 0
vκ0
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
1eµτ
vκ
0
=H([t]τ; 1, t)
ab
1
a
0
.
Comparing the entries in the second row, we find Y12 ·(1 eµτ ) = 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 0R4of the impulsive delay differ-
ential equation (20) are, for ||(w, )|| sufficiently small,
˙w=η(1 eµτ )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 eµτ )β(t) 1
2f00(0)(1 eµτ )β(t)
+f0(0) Z[t]τ
0
eµ([t]τs)(γeµτ 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 Cin (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=
z=eΛtY(t)G(Qtz+h(t, z, ·)), t =,
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 Cand
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
|vv|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=vin some half-
space.
ξ(·, v)is positive (in both components) if and only if (vv)` > 0.
Proof. The time τ(Poincar´e) map associated to the ordinary differential equation
(34) is readily found to satisfy
w7→ w+η(1 eµτ )[`w2+mw] + 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 Cremainder 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 (vv)`=` > 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 vv, 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 eµτ )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 vdrives 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
v0.6227
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(tτ) and I(tτ) 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=µµSηf(I)S+γeµτ I,
d
dtI=ηf (I)S(µ+γ)I,
S(0) = ρ(v)S(τ),
I(0) = I(τ),
ρ(v) = (1 v)(1 veµτ )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
c0.4063. 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(t1)).
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 {zC:<(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(t1) 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(t1)) 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=kZand yellow to arguments tk. Inset: Plots of the image of t7→
(S(t, v), I (t, v), S(t1, v)).
24
Figure 6: Plot of t7→ (S(t), I (t), S(t1)) for t[300,1300] from a constant initial condition of
(S(0), I(0)) = (0.5,0.5), for v= 0.385, v= 0.375 (top row), v= 0.35 and v= 0.315 (bottom row).
Colours and insets have the same interpretation as in Figure 5.
25
0
0.05
0.3
0.9
0.4
0.8
0.5
0.7
0.6
0.6
0.7
0.5 0.1
0.8
0.4 0.3
0.9
Figure 7: Plot of t7→ (S(t), I (t), S(t1)) for t[300,1300] from a constant initial condition of
(S(0), I(0)) = (0.5,0.5), for v= 0.25, v= 0.15 (top row), v= 0.05 and v= 0 (bottom row). Colours
have the same interpretation as in Figure 5.
26
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 vat which the disease-free periodic solution gains (re-
spectively, loses) its stability. Consequently, treating the vaccination coverage as a
bifurcation parameter, v=vis 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(t1)). 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< vand 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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A. Derivation of the impulse condition in equation (3)
Suppose Vkis the number of individuals that received a vaccine at time tkτ. By
assumption 3), Vkeµτ 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) + Vkeµτ .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)Vkeµτ .(38)
29
Now, if the total number of susceptible individuals (including those that lost their
immunity) at time tkτis denoted S
k, then by 3) we have S(tkτ) = (1 v)S
k
and Vk=vS
k, which together imply Vk=v
1vS(tkτ). Substituting into (38), we
have that at time t=tk,
S=S(tk)S(t
k)
=vS(t
k) + (1 v)Vkeµτ
=vS(t
k) + vS(tkτ)eµτ ,
which is precisely equation (3). Similarly, the number of vaccinated individuals V
satisfies
˙
V=µV, t 6=tk
V=vS(t
k)vS(tkτ)eµτ , 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(tτ), t 6=kτ
x=Cx(t) + Dx(tτ), t =(40)
with data A(t), B,Cand Dgiven by
A(t) = µηf (I(t, v)) ηf 0(I(t, v))S(t, v)
ηf (I(t, v)) ηf 0(I(t, v))S(t, v)γµ
B=0γeµτ
0 0 , C =v0
0 0 , D =veµτ 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 tsthe 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)(sτ)ds
+χ{τ}(t)(CX(τ , 0) + D)φ(0) + CZτ
0
X(τ, s)Bφ(sτ)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)] (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:XXinstead 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:YXdefined by
G(y1, y2) = χ[τ,0)y1+χ{0}y2
is an isomometry, so we will make the identification XY. Then, for (φ1, φ2)Y,
we can write
V0φ1
φ2="X(τ+·,0)φ2+R0
τX(·, s)1(s)ds
((I+C)X(τ, 0) + D)φ2+R0
τ(I+C)X(0, s)1(s)ds #.(42)
We are ready to discretize the monodromy operator. Let N1 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 fC2. We approximate φ1C2([τ, 0],R2) by the vector
ˆ
φ1= [ φ1(s1). . . φ1(sN)]TRN
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
V0φV0,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
Λ1
τlog µ:µσ(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(tr) + f(x(t), x(tr), ), t 6=kZ
x=Bx(t) + Cx(tr) + g(x(t), x(tr), ), t =kZ,
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=±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 η0, to-
gether with a unique periodic solution in Nthat persists for all |η|sufficiently small
and trivializes to zero (the equilibrium) as η0. 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) = 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
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