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Analysis of pandemic closing-reopening cycles using rigorous

homotopy continuation: a case study with Montreal COVID-19

data

Kevin E. M. Church

McGill University, Department of Mathematics and Statistics

December 28, 2020

Abstract

Moving averages and other functional forecasting models are used to inform policy in pandemic

response. In this paper, we analyze an infectious disease model in which the contact rate switches

between two levels when the moving average of active cases crosses one of two thresholds. The

switching mechanism naturally forces the existence of periodic orbits. In order to make unbiased

comparisons between periodic orbits in this model and a traditional one where the contact rate

switches based on more simplistic pointwise evaluations of active cases, we use a rigorous homotopy

continuation method. We develop computer-assisted proofs that can validate the continuation and

prove that the branch of periodic orbits has no folds and is isolated in the space of periodic solutions.

This allows a direct, rigorous comparison between the geometric and quantitative properties of the

cycles with a moving average threshold and a pointwise threshold. We demonstrate the eﬀectiveness

of the method on a sample problem modeled oﬀ of the COVID-19 pandemic in the City of Montreal.

1 Introduction

Beginning in the mid-second quarter of 2020, governments worldwide began eﬀorts to reopen their

economies [10, 12, 29] in response to slowing of new COVID-19 cases. The various contagion management

policies employed worldwide measurably slowed the spread of the novel coronavirus [16], but threats of

a second wave [20, 23, 30, 36] continue to loom as cases have been globally trending upwards [3]. Until

such time as a vaccine becomes in widespread use, closing and reopening could become a regular part of

the ebb and ﬂow of daily life.

Mathematical models can provide some insight into how the closing and reopening procedure might

have an impact on the progression of a pandemic. There exist several mathematical constructions that

are suitable for describing the closing and reopening of the economy based on active case numbers or

other such metrics, but here we will focus primarily on relays. A class of switched system — see [2, 5, 13]

for background — relay systems involve a partitioning of the phase space into multiple disjoint regions

with switching of the underlying vector ﬁeld when an orbit enters or exits a region. In modeling of closing

and reopening, two thresholds [8] can be deﬁned that represent upper and lower bounds on the number

of active cases. When the number of active cases reaches the upper bound, interventions are applied (i.e.,

closing) that decrease contact rates. If or when the number of active cases reaches the lower bound, the

interventions are softened or removed (i.e., reopening). The process then repeats itself.

Delayed relays [33] involve a switching of a vector ﬁeld when a delayed argument crosses through a

given threshold. In this way, the phase space can be partitioned into two disjoint regions where stronger

interventions (i.e., closing) and more lax restrictions (i.e., reopening) are applied [22, 26]. Since the for-

malism involves a delayed argument, the crossing of the threshold does not immediately cause a switching

1

of the vector ﬁeld, and the end result is one typically observes oscillation about the threshold. An ad-

vantage of this formalism is that natural time lags between collection of data and enforcing distancing

measures can be incorporated into the model.

In reality, epidemiological data is volatile in large part due to reporting errors [17]. Moving averages

are used to smooth out this volatility and are also used in forecasting [15, 18]. They have also been used

by health oﬃcials to state targets for control of the COVID-19 pandemic — for example, health oﬃcials

in the Canadian province of Quebec have stated [24] they want to avoid passing 20 daily new cases per

million people, where the metric of new cases is in fact a 7-day moving average. Since the previous two

relay model constructions make use of pointwise evaluations, be they delayed (as in the delayed relay)

or not, they can not fully take into account policy changes based on moving averages (also called rolling

averages) of active cases over longer periods of time. On a related note, ARIMA models [1] and other

forecasting models have seen much application in forecasting of the COVID-19 pandemic [4, 6, 7, 31],

and the predictions they provide make use of nonlocal properties of the time series data, weighted and

processed over long windows of observations. We therefore argue it is more realistic and applicable to

study relay models that incorporate nonlocal functional dependence in vector ﬁeld switching mechanism.

1.1 Closing-reopening cycles

Recently, a global analysis of a relay SIR model [8] with switching of contact rates and two thresholds

was completed, and conditions for the existence a a stable limit cycle with two switches per period were

determined. Therein, the intervention thresholds are based on pointwise numbers of active cases (i.e.,

number of active cases at a single time instant). Formally, there are two thresholds: Inat < Iint , and if

I(t) denotes the number (or density) of infected individuals, the contact rate βswitches whenever the

relation

I(t)∈ {Inat, Iint }

is satisﬁed. Reaching the larger of the two thresholds causes the contact rate to decrease to βint, the

intervention contact rate, while reaching the lower threshold causes it to switch to βnat, the natural

contact rate.

Motivated by our interest in examining nonlocal functional dependence on epidemic intervention

thresholds, we consider properties of limit cycles (referred to as closing-reopening cycles in this publi-

cation) if the threshold crossing criteria is based on a weighting between a (simple) moving average of

active cases over several days and a pointwise estimate of active cases on the current day. As in the

former publication, we consider the intervention to be one that reduces the contact rate. More formally,

we consider the relation above replaced with

(1 −α)I(t) + α

wZt

t−w

I(u)du ∈ {IR, IC},(1)

where IR< ICare reopening and closing levels, wis the time window for the rolling average and 0 ≤α≤1

is a parameter controlling the weight of the pointwise vs. rolling average computation in the threshold.

When α= 0 we get the pure pointwise threshold of [8], while with α= 1 only the rolling average is used.

The kinds of questions we want to answer include the following.

1. (Existence) Under what conditions does a closing-reopening cycle exist for α∈[0,1]?

2. (Comparison) Is there an objective way to compare closing-reopening cycles for various weights α?

3. (Average and extrema) How does the average, maximum and minumum number of active cases of

the closing-reopening cycle compare in the cases α= 0 and α= 1?

4. (Overshooting) To what extent can a closing-reopening cycle overshoot its two thresholds if rolling

averages are used (i.e., α= 1) and for how long?

2

We will from here on refer to the threshold condition (1) with α= 0 as the pointwise threshold and α= 1

as the rolling average threshold.

To give as broad an answer as possible to these questions, we will focus exclusively on the SIR

model. We will answer these questions by computing closing-reopening cycles, numerically continuing

in the parameter α, and using a rigorous a posteriori veriﬁcation scheme to prove that the numerically

computed cycles and connecting branches exist, rather than attempting a detailed technical study of

the equations using analytical means. Speciﬁcally, existence and comparison are covered in Section 3.4

through to Section 3.7, while averages will be covered in Section 3.8 and overshooting in Section 3.9.

Before continuining, we emphasize that it is inappropriate to try and answer these questions solely

by a non-rigorous numerical exploration at distinct values of the parameter α. For example, suppose we

integrate a classical SIR model with contact rate βthat switches whenever (1) is satisﬁed. Assume we do

this at α= 0 and α= 1, and we numerically observe a limit cycle for each parameter. There is no reason

to suspect that these two limit cycles are in any way related, yet we might want to compare such statistics

as their averages or extrema. Computing other limit cycles for intermediate values of α∈(0,1) might

give more conﬁdence if the cycles appear to vary continuously with the parameter. However, we can still

not be absolutely certain that our comparison of these limit cycles is objective. Moreover, depending on

the numerical method used, it might not be provably true that the observed limit cycles really exist. The

following section outlines our idea to complete this exploration in a mathematically rigorous way.

1.2 A rigorous homotopy continuation approach to closing-reopening cycles

If a unique (up to phase shift) closing-reopening cycle exists in the base model of [8], we ﬁrst compute

this orbit and represent it using truncated Chebyshev series. For the purposes of this introduction, let

us refer to the truncated orbit as PN

0and the “true” orbit as P0. Since such a cycle P0consists of a

Lipschitz (in fact, piecewise C∞with a only two points where the derivative is discontinuous) curve, the

representation of PN

0exists by classical results of approximation theory [34] and the coeﬃcients of the

series exhibit geometric decay when restricted to the two smooth segments. We then use a Chebyshev

spectral collocation procedure to transform the problem of ﬁnding a closing-reopening cycle into one of

computing a zero of a nonlinear function fin an inﬁnite-dimensional sequence space. Formally, we write

f:X×[0,1] →X, where Xis the sequence space and f(P, α) = 0 means that Pis a closing-reopening

cycle for the system with thresholds deﬁned informally by the relation (1) where IR< ICare reopening and

closing levels, wis the amount of time in the rolling average and tis time. By taking a ﬁnite-dimensional

projection XNof Xby truncating the number of modes, we can get a ﬁnite-dimensional projection

fN:XN×[0,1] →XNsuch that fN(PN

0,0) = 0. We can then use numerical continuation to compute a

discrete branch (PN

α, α) for α∈ {α0, . . . , αM}for α0= 0 and αM= 1 such that fN(PN

αi, αi) = 0.

If the numerical method converges, then it seems likely that a closing-reopening cycle exists for the

convex parameters αisince PN

αishould be a “good approximation” to a true closing-reopening cycle. We

can then provide approximate answers to questions 2–4 by numerical quadratures and zero-ﬁnding. How-

ever, these results are not rigorous because we have not proven that a true closing-reopening cycle exists,

and we do not have precise information about how good an approximation the numerical branch of solu-

tions {(PN

αi, αi), i = 1, . . . , M }is. This means that any answers to questions 2–4 are subject to unknown

numerical error. These errors are nontrivial to quantify because although we have accurately computed

the branch of closing-reopening cycles for the ﬁnite-dimensional projection, we can only interpret this as

an approximate solution by embedding it into the inﬁnite-dimensional space X.

To resolve this, we make use of validated continuation methods based on the radii polynomial approach

[19]. This is a computationally explicit variant of the Newton-Kantorovich theorem that is amenable to

computer-assisted proof. By performing all of the associated computations in interval arithmetic using

the INTLAB library [32] in MATLAB, we rigorously control roundoﬀ error and are able to prove that

the numerically computed branch is proximal to a true branch (in X) of closing-reopening cycles, with

explicitly computable error bounds. These error bounds can then be propagated to the associated solutions

3

of questions 2–4, giving answers with rigorous error bounds. Uniqueness of the branch is a convenient

side product of the method, so we are able to make faithful comparisons between the orbits in the active

cases threshold and the convex combination cases in the sense that there is a unique continuation from

one to the other with respect to the convex parameter α. More succinctly, we prove the existence of a

smooth homotopy continuation from a closing-reopening cycle at α= 0 to one at α= 1.

To summarize, we therefore propose validated numerics as a tool to directly answer the questions from

the previous section concerning existence and comparison of closing-reopening cycles under the diﬀerent

threshold deﬁnitions, as outlined in the previous paragraph. We will be able to answer the average and

overshooting question using post-processing of the data from this method.

From a mathematical perspective, our approach to parameterizing these periodic orbits has elements

in common with the work of Gameiro, Lessard and Ricaud [11] for crossing periodic orbits in Filippov

systems. The main diﬀerence here is that we have a relay (i.e., switched) system, and the switching rule

(1) involves dependence on past values of the solution. The switching manifolds (there are two: one for

each of the thresholds ICand IR) are in fact codimension one submanifolds of an inﬁnite-dimensional

function space rather than a subset of Rn.

1.3 Outline of the paper

The structure of the paper is as follows. In Section 2, the model is explicitly formulated and some elemen-

tary properties of closing-reopening cycles are stated and proven, including a boundary-value problem

that such cycles must satisfy. Section 3 concerns the rigorous numerics, and it is split into several parts.

We provide necessary background on Chebyshev series in Section 3.1 before converting the boundary-

value problem for closing-reopening cycles into a zero-ﬁnding problem in Section 3.2. The associated

ﬁnite-dimensional projection derived in Section 3.3. The radii polynomial approach, which is used to

prove branches of zeroes, is stated in Section 3.4. In Section 3.5, we prove suﬃcient conditions — what

we call sharpness conditions — that guarantee a correspondence between zeroes of the nonlinear map and

closing-reopening cycles. The bounds for the radii polynomial method are computed in Section 3.6, and a

computational approach to the sharpness conditions is presented in Section 3.7. The analysis of extrema,

averages and overshooting along closing-reopening cycles is completed in Section 3.8 and Section 3.9. In

Section 4, we apply our results to the COVID-19 pandemic in the City of Montreal. A discussion and

conclusion follow in Section 5 and Section 6.

1.4 Notation

For a ﬁxed w > 0, we denote C([−w, 0],R+) the vector space of nonnegative real-valued continuous

functions deﬁned on the interval [−w, 0]. We will often write it simply as Cwhen the explicit dependence

on wis understood.

If f:I→Rfor a closed interval I, then the function ft∈Cis deﬁned by ft(θ) = f(t+θ); it exists if

[t−w, t]⊆I. This is a standard convention in functional diﬀerential equations [14].

For a normed vector space X, the symbol B(X) refers to the set of bounded linear operators on X.

If A∈B(X), then ||A||B(X)denotes its operator norm. The symbol IXdenotes the identity operator on

a vector space X, and 0Xdenotes the zero map. If A:U→Vand B:X→Y, then C= diag(A, B) :

U×X→V×Yis deﬁned by C(u, x)=(A(u), B(x)). The symbol Br(x) denotes the open ball of radius

rcentered at x∈X. If x= [min(x),max(x)] is an interval vector with min(x)≤max(x) in the usual

partial order on Rnand t≥0, we deﬁne tx = [tmin(x), t max(x)]. If uand vare two intervals, we write

u < v if and only if sup(v)<inf(u).

4

2 Formulation of the model and elementary properties

As stated in the introduction, we will focus on a relay-like SIR model with two thresholds deﬁned by

convex relationships between pointwise and rolling average active cases. At all times, the dynamics (in

the sense of right-derivatives) follow the SIR model

˙

S= Λ −βSI −µS

˙

I=βSI −(µ+γ)I. (2)

Sand Irepresent the number of susceptible and infected humans, respectively. Λ is a constant recruitment

rate, µthe per capita death rate, βthe combined contact/infection rate in units of 1/(humans ·time) and

γthe combined death and recovery rate. We assume immunity is permanent, so the removed (recovered

or dead due to infection) class is decoupled and has not been included. For brevity, we will refer to β

simply as a contact rate, and γthe removal rate. All of these parameters are assumed positive. The

parameter βcan change according to a rule we will introduce shortly.

Let some w > 0 be ﬁxed and deﬁne a parameterized functional g:C×[0,1] →Raccording to

g(φ, α) = (1 −α)φ(0) + α

wZ0

−w

φ(u)du.

By deﬁnition, g(φ, α) is a convex combination of the value of the function φat zero and its average.

With this deﬁnition, we can introduce the relay dynamics. Let βc< βrbe two contact rates, and let

0< IR< ICrepresent thresholds of reopening (IR) and closing (IC). Informally, the “closing/reopening

relay model” is the ordinary diﬀerential equation (2) together with a rule that states when the contact

rate parameter βswitches. Speciﬁcally, the contact rate switches whenever the relationship (1) is satisﬁed.

When g(It, s) crosses (or is tangent to) IC, closing of the economy begins (or continues if it was already

in this state) and the contact rate switches to βc. Conversely, when g(It, s) crosses (or is tangent to)

IR, reopening begins and the contact rate switches to βr. More formally, we deﬁne solutions of the relay

model as follows. Figure 1 may aid in visualization.

Deﬁnition 1. Let S0∈R+,φ∈Cand σ0∈ {0,1}. Let (S, I)be continuous functions S: [0, b)→R+

and I: [−w, b)→R+with S(0) = S0and I0=φ. Deﬁne the switching sets

ΣR={t∈(0, b) : g(It, α) = IR},ΣC={t∈(0, b) : g(It, α) = IC},

and switching function σ: [0, b)→ {0,1}as follows:

σ(t) =

σ0, t = 0

0,sup(ΣR∩[0, t]) <inf(ΣC∩[t, b))

1,sup(ΣC∩[0, t]) <inf(ΣR∩[t, b)).

(S, I)is a solution of the relay model for thresholds IR< IC, contact rates β1< β0and convex parameter

αif it is a solution of the piecewise-continuous system (2) for t∈[0, b)with β=βσ(t). That is, it

diﬀerentiable and satisﬁes the ODE except at those times where σhas a discontinuity. The data (S0, φ, σ0)

is the initial condition.

Remark 1. In Deﬁnition 1, we have opted to label the respective contact rates βrand βcinstead as β0

and β1. This is to avoid confusion later in Deﬁnition 2 and Section 2.1. Also, since Iis continuous, gis

a continuous functional and IR< IC, the switching function is indeed well-deﬁned and continuous from

the right except possibly at t= 0.

5

Figure 1: Schematic drawing of a solution of the relay model in the case α= 0. The two vector ﬁelds

are drawn simultaneously, corresponding respectively to β0(green arrows) and β1(blue arrows). The

upper and lower black lines respectively represent I=ICand I=IR. On black dashed lines the solution

satisﬁes σ= 0 and on the dotted line it satisﬁes σ= 1. Between each switching, the solution follows the

vector ﬁeld corresponding to the value of σ.

We will not discuss details such as existence, uniqueness and continuability of solutions for the open-

ing/closing relay model. The latter follows under the category of switched systems with state-dependent

[21, 37] switching. Suﬃce it to say, given an initial condition, a unique solution exists and can be con-

tinued to b=∞. That solutions can be continued to the whole positive real line is a consequence of the

ﬁnite separation IR< ICand the fact that solutions of (2) are ultimately unformly bounded for β=βσ(t)

for any switching signal σ.

Deﬁnition 2. A solution (S, I)the relay model is a closing-reopening cycle with period pif S,Iand σ

are p-periodic, and the associated switching function has exactly two discontinuities in the interval [0, p].

A closing-reopening cycle is normalized if σ(0) = 0,σ(c) = 1 for some c∈(0, p), and the discontinuities

of the switching function in [0, p]are precisely at cand p. In this case, cis the closing time and p

(equivalently, zero) is the reopening time.

By deﬁnition, if a closing-reopening cycle exists, then it can always be normalized in a unique way.

We will typically assume that our closing-reopening cycles are normalized, and this can always be done

without loss of generality by an appropriate phase shift.

2.1 Slow closing-reopening cycles

Some closing-reopening cycles are easier to ﬁnd (and visualize) than others, and in the sections that follow

we will devise a numerical method to compute and continue branches of such cycles.

Deﬁnition 3. A normalized closing-reopening cycle is slow if the following are satisﬁed.

•c≥wand p−w≥c.

•I(p−w)≤ICand IR≤I(c−w).

We say the cycle is sharp if the above inequalities are strict.

6

The following lemma demonstrates how slow closing-reopening cycles can be constructed using special

solutions of (2) with a piecewise-constant parameter β..

Lemma 1. Let p, c ∈Rand (S, I) : [0, p]→R2

+be a continuous, piecewise-diﬀerentiable function.

Suppose the following are satisﬁed.

S.1 (S, I)satisﬁes (2) with parameter βron [0, c), and satisﬁes (2) with parameter βcon [c, p)

S.2 g(Ic1, α) = ICand g(Ip, α) = IR.

S.3 S(p) = S(0) and I(p) = I(0).

S.4 c≥wand p−w≥c.

S.5 Iis monotone increasing on [0, c)and IR≤I(c−w).

S.6 Iis monotone decreasing on [c, p)and I(p−w)≤IC.

The image of (S, I)coincides with the image of a slow closing-reopening cycle, and it is sharp if S.4–S.6

are satisﬁed with strict inequalities, Iis strictly increasing on [0, c)and is strictly decreasing on [c, p).

Proof. Extend (S, I) to a periodic function on R. Write G(t) = g(It, α) and consider the derivative

G0(t) = d

dtg(It, α) = (1 −α)I0(t) + α

w(I(t)−I(t−w)).

If t∈[0, c] then I0(t)>0. On the other hand, if t∈[w, c] then I(t)> I (t−w), so we can be certain that

G0(t)>0 on [w, c]. As for the restriction to [0, w], we claim that G(t)< IC. To see why, observe ﬁrst

that for t∈(−w, w], we I(t)< IC; verifying this is straightforward and uses the monotonicity properties

of I. Consequently, for t∈(0, w],

G(t) = (1 −α)I(t) + α

wZt

t−w

I(u)du < (1 −α)IC+αIC=IC,

while G(0) = IR< IC. We have therefore shown that G(t)< ICon [0, w] and Gis strictly increasing

on [w, c], from which it follows that the switching sets associated to (S, I) satisfy ΣC∩(0, c] = {c}and

ΣR∩(0, c] = ∅. By a symmetric argument, one can show that ΣC∩(c, p] = ∅and ΣR∩(c, p] = {p}.Let

σ0= 0 and consider the switching function σassociated to (S, I) with the initial point σ0. The associated

switching function σtherefore has exactly two discontinuities cand pin the interval [0, p], with σ(c) = 1

and σ(0) = 0. (S, I ) therefore deﬁnes a slow, normalized closing-reopening cycle.

2.2 A boundary-value problem for closing-reopening cycles

The deﬁnition of slow closing-reopening cycle and conditions S.1–S.3 of Lemma 1 deﬁne a natural

boundary-value problem (BVP). Introduce a family of R2

+-valued functions (Sk, Ik) for k= 0,1,2,3.

The domain of these functions are as follows:

dom(Sk, Ik) =

[0, c −w], k = 0

[c−w, c], k = 1

[c, p −w], k = 2

[p−w, c], k = 3.

(3)

These functions will deﬁne the restrictions of a candidate (S, I) for a slow closing-reopening cycle to each

of the above subintervals of [0, p]. See Figure 2 for a visualization.

7

1.4495 1.45 1.4505 1.451

106

400

600

800

1000

1200

1400

1600

1.4605 1.461 1.4615 1.462

106

400

600

800

1000

1200

1400

1600

1800

Figure 2: Two cycles with diﬀerent convex parameter α. On the left, α= 0 and the relay model can

be identiﬁed with a ﬁnite-dimensional switched ODE system with state-dependent switching. On the

right, α= 1 and the state space must be considered as inﬁnite-dimensional. The segments labeled 0,1,2,3

in equation (3) correspond respectively to the green, cyan (dashed), red, and blue (dashed) curves, and

they are also labeled in the ﬁgure. Arrows indicate time orientation. The black horizontal dashed lines

correspond to I∈ {IC, IR}. The cycles in this ﬁgure are the extremal points of the branch proven in

Theorem 12, and each of them is sharp according to Deﬁnition 3.

Deﬁne β(k) = βrfor k= 1,2 and β(k) = βcfor k= 3,4. Deﬁne three boundary functionals

L[f] = f(inf(dom(f))), R[f] = f(sup(dom(f))), G[f] = (1 −α)R[f] + α

wZdom(f)

f(u)du.

Symbolically set S4=S0and I4=I0, and consider the following boundary-value problem.

˙

Sk= Λ −β(k)SkIk−µSk, k = 0,...,3

˙

Ik=β(k)SkIk−(µ+γ)Ik, k = 0,...,3,

R[Sk] = L[Sk+1], k = 0,...,3,

R[Ik] = L[Ik+1], k = 0,...,3,

G[I1] = IC,

G[I3] = IR.

The following proposition holds.

Proposition 2. If (Sk, Ik)for k= 0,1,2,3have the domains (3) and satisfy the previous boundary-value

problem, then the function (S, I) : [0, p]→R2

+deﬁned by

(S, I)(t) =

(S0(t), I0(t)), t ∈[0, c −w),

(S1(t), I1(t)), t ∈[c−w, c)

(S2(t), I2(t)), t ∈[c, p −w),

(S3(t), I3(t)), t ∈[p−w, p],

satisﬁes conditions S.1–S.4 of Lemma 1.

8

If a solution of the boundary-value problem can be computed, then one can verify that its image is a

closing-reopening cycle by subsequently checking conditions S.5 and S.6 of Lemma 1. This is a separate

problem that we will solve on the computer in Section 3.7. At present, we are more interested in the BVP.

Before moving on, we will perform changes of variables to transform the domains of each of (Sk, Ik) to

the interval [−1,1]. This transformation will facilitate the conversion to a zero-ﬁnding problem in Section

3.2.

To complete the change of variables, write

(S0, I0)(t) = ( ˜

S0,˜

I0)2t

c−w−1,(S1I1)(t) = ( ˜

S1,˜

I1)2t

w−2c−w

w,

(S2, I2)(t) = ( ˜

S2,˜

I2)2t

p−w−c−c+p−w

p−w−c,(S3, I3) = ( ˜

S3,˜

I3)2t

w−2p−w

w,

(4)

for ( ˜

Sk,˜

Ik) : [−1,1] →R2

+. The, deﬁne scaling factors zk=zk(c, p) according to

zk=

c−w

2, k = 0

p−w−c

2, k = 2

w

2, k ∈ {1,3}.

(5)

If one completes the change of variables, then, dropping the tildes, we get the boundary-value problem

˙

Sk=zk(Λ −β(k)SkIk−µSk),

˙

Ik=zk(β(k)SkIk−(µ+γ)Ik),

R[Sk] = L[Sk+1]

R[Ik] = L[Ik+1]

0 = IC−(1 −α)R[I1]−α

2Z1

−1

I1(u)du

0 = IR−(1 −α)R[I3]−α

2Z1

−1

I3(u)du.

(6)

The following lemma is a direct consequence of the previous derivation, Lemma 1 and Lemma 2.

Lemma 3. Suppose (Sk, Ik):[−1,1] →R2

+for k= 0,1,2,3is a solution of the boundary-value problem

(6) with zk≥0. If I0and I1are monotone increasing with IR≤I0(1) and I2and I3are monotone

decreasing with I2(1) ≤IC, then the union of the images of (Sk, Ik)for k= 0,1,2,3coincides with

the image of a slow closing-reopening cycle. The cycle is sharp if the inequalities are strict and the

monotonicity is strict.

Remarks 2. One might ask why we represent a candidate closing-reopening cycle using four smooth

segments instead of two. The reason is because with four segments, the boundary conditions are very

simple since the bounds of the integrals are always the same. However, if two segments are used, the

boundary conditions involve the variables cand pand they become non-polynomial. For example, the

analogue of the ﬁrst condition is

0 = IC−(1 −α)R[I1]−αc

2wZ1

1−2wc−1

I1(u)du

for I1now representing the (infected component) segment of the cycle that runs from time t= 0 to time

t=c. When we move to computer-assisted proofs, this boundary condition is much harder to work with. It

is for this reason that we allow ourselves to suﬀer the extra cost in dimension aﬀorded by a representation

in four smooth segments.

3 Rigorous numerics for closing-reopening cycles

This section contains the main theoretical details concerning computation, continuation and computer-

assisted validation of closing-reopening cycles. We start by converting the boundary-value problem (6) into

9

a zero-ﬁnding problem in an inﬁnite-dimensional function space. We then determine a ﬁnite-dimensional

projection and discuss how to compute its numerical zeroes. Then, we outline the radii polynomial method,

which is used for rigorously proving branches of zeroes of the inﬁnite-dimensional problem based on a

numerical branch. Next, we determine a posteriori sharpness conditions that can be uniformly checked

along a numerically validated branch to prove that the zeroes uniquely determine closing-reopening cycles.

We then demonstrate explicitly (i.e., by deriving the appropriate bounds) how to implement these checks

in a computer. We conclude with an analysis of how to rigorously compute extrema, averages and

overshooting times along numerically computed closing-reopening cycles.

3.1 Chebyshev series and the space `1

ν

As suggested in the introduction, we will be solving the boundary-value problem (6) using Chebyshev

series expansions. The following background is contained in [34]. Recall that any suﬃciently smooth

(speciﬁcally, Lipschitz continuous) function f: [−1,1] →Rcan be represented as a uniformly convergent

Chebyshev series

f(t) = f0+ 2

∞

X

n=1

fnTn(t)

for Tnthe nth Chebyshev polynomial of the ﬁrst kind. The coeﬃcients fncan be computed using the

formula

fn=1

πZ1

−1

Tn(x)f(x)

√1−x2dx.

If fis real-analytic on [−1,1] and (some C-analytic continuation) is bounded on the Bernstein ν-ellipse

in the complex plane — that is, the closed ellipse with foci at ±1 and sum of semimajor and semiminor

axes equal to ν— then the quantity

||f||ν:= |f0|+ 2

∞

X

n=1

νn|fn|(7)

is ﬁnite. If we write ωn= 1 for n= 0 and ωn= 2νnfor n≥1, we can write it more compactly in the

form

||f||ν=

∞

X

n=0 |fn|ωn.

If fis real-analytic on [−1,1] then there necessarily exists some ν > 0 such that the above is true. If

f0(t) is written as a Chebyshev series

f0(t) = f0

0+ 2

∞

X

n=1

f0

nTn(t),

then the coeﬃcients fnand f0

nare related by the equation

2nfn=f0

n−1−f0

n+1, n ≥1.(8)

Also, for n≥2 the Chebyshev polynomials admit the indeﬁnite integrals

ZTn(t)dt =1

2Tn+1(t)

n+ 1 −Tn−1(t)

n−1.(9)

10

The Chebyshev polynomials satisfy the identities Tn(1) = 1 and Tn(−1) = (−1)nfor all n≥0. Finally,

in the scope of diﬀerentiation of Chebyshev series, the tridiagonal operator T:`1

ν→`1

νdeﬁned by

T(a)n=0, n = 0

an+1 −an−1, n > 0(10)

will be quite useful.

Let `1

νdenote the normed vector space of sequences {an:n∈N}bounded with respect to the || · ||ν

norm. This is a Banach space. If a, b ∈`1

ν, deﬁne their convolution a∗baccording to

(a∗b)n=X

k∈Z

a|k|b|n−k|.

It is a standard exercise to check that (`1

ν,∗) is a Banach algebra; that is, ∗:`1

ν×`1

ν→`1

νis a continuous

bilinear map with ||a∗b||ν≤ ||a||ν||b||ν.

3.2 Conversion from BVP to zero-ﬁnding problem

Write each of the functions Skand Ikfrom (6) as a Chebyshev series. Speciﬁcally, make the expansions

Sk(t) = ak,0+ 2

∞

X

n=1

ak,nTn(t) (11)

Ik(t) = bk,0+ 2

∞

X

n=1

bk,nTn(t).(12)

In the following, the symbols akand bkwill refer to the sequences {ak,n :n∈N}and {bk,n :n∈Z}. The

products of Chebyshev series induce convolutions at the level of their coeﬃcients. We have

Sk(t)Ik(t)=(ak∗bk)0+ 2

∞

X

n=1

(ak∗bk)nTn(t),

where (ak∗bk)n:= (ak,·∗bk,·)n. Substituting (11) and (12) into (6), using the relations (8) and (9), and

making the identiﬁcation a4≡a0and b4≡b0, we get the following set of equations.

2nak,n

bk,n =−(TΨ1

k(ak, bk, c, p))n

(TΨ2

k(ak, bk, c, p))n, n ≥1,(13)

ak,0

bk,0+ 2

∞

X

n=1 ak,n

bk,n =ak+1,0

bk+1,0+ 2

∞

X

n=1

(−1)nak+1,n

bk+1,n (14)

0 = IC−(1 −α) b1,0+ 2

∞

X

n=1

b1,n!−α

b1,0−X

n≥2

b1,n

1+(−1)n

n2−1

(15)

0 = IR−(1 −α) b3,0+ 2

∞

X

n=1

b3,n!−α

b3,0−X

n≥2

b3,n

1+(−1)n

n2−1

(16)

where Ψk:`1

ν×`1

ν×R2→`1

ν×`1

νis the representation of the right-hand side of the ODE in (6) on the

Chebyshev coeﬃcients, depending on the unknown closing time cand period p. In coordinates,

Ψ1

k(a, b, c, p)n

Ψ2

k(a, b, c, p)n=Λn−β(k)(a∗b)n−µan

β(k)(a∗b)n−(µ+γ)bnzk(c, p) := Φ1

k(a, b)n

Φ2

k(a, b)nzk(c, p),(17)

11

with Λ0= Λ and Λn= 0 for n > 0. To be precise, (13) corresponds to the diﬀerential equations, (14) to

the R-Lboundary conditions, and (15) and (16) the convex threshold conditions from the BVP (6).

We can transform (13)–(16) into a zero-ﬁnding problem on an appropriate Banach space. Deﬁne

L:`1

ν→`1

ξ(for `1

ξa Banach space to be introduced in Lemma 4) by

L(a)n= 2nan.(18)

Next, deﬁne a linear map H:`1

ν×`1

ν→`1

νwith one-dimensional range by H(a, b)n= 0 for n > 0 and

H(u, v)0=u0−v0+ 2

∞

X

n=1

(un+ (−1)n+1vn).

Finally, we set X= (`1

ν)8×R×R. We will sometimes write it as Xνwhen we want to emphasize

the choice of ν(see in particular the proof of Theorem 6). The norm on Xmust be chosen carefully,

as it is typical for the Sand Icomponents of solutions to the BVP (6) to diﬀer by several orders of

magnitude, which can result in poor conditioning. To facilitate this, we let W= (W1, W2, W3)∈R3

+

denote a weight vector. This will be explicitly chosen whenever we want to do a computer-assisted proof.

For φ= (a0, b0, . . . , a3, b3, c, p)∈X, we deﬁne the norm

||φ||X= max{W1max{||a0||ν,...,||a3||ν}, W2max{||b0||ν,...,||b3||ν}, W3|c|, W3|p|}.(19)

The role of the weights will be further elaborated upon in Section 4.

We consider a formal nonlinear map F:X×[0,1] →Ywith Ya Banach space that we will subsequently

identify:

F(a0, b0, . . . , a3, b3, c, p, α) =

L(a0) + H(a0, a1) + TΨ1

0(a0, b0, c, p)

L(b0) + H(b0, b1) + TΨ2

0(a0, b0, c, p)

L(a1) + H(a1, a2) + TΨ1

1(a1, b1, c, p)

L(b1) + H(b1, b2) + TΨ2

1(a1, b1, c, p)

L(a2) + H(a2, a3) + TΨ1

2(a2, b2, c, p)

L(b2) + H(b2, b3) + TΨ2

2(a2, b2, c, p)

L(a3) + H(a3, a1) + TΨ1

3(a3, b3, c, p)

L(b3) + H(b3, b1) + TΨ2

3(a3, b3, c, p)

IC−(1 −α)(b1,0+ 2 Pn≥1b1,n )−αb1,0−Pn≥2b1,n 1+(−1)n

n2−1

IR−(1 −α)(b3,0+ 2 Pn≥1b3,n )−αb3,0−Pn≥2b3,n 1+(−1)n

n2−1

(20)

The boundary conditions (14) have been encoded into the linear map H. By construction, if it so happens

that (a0, b0, . . . , a3, b3, c, p) is a zero of Ffor parameter α, then this (a0, b0, . . . , a3, b3) will satisfy (13)–

(16) for closing time cand period p. Subsequently, if the monotonicity requirements of Lemma 3 can be

checked, this will uniquely deﬁne a slow closing-reopening cycle through the identiﬁcations (11)–(12) and

Lemma 3. It can also be veriﬁed that Fis also smooth. The following lemma summarizes this fact and

characterizes an appropriate codomain for F. The proof is simple and is omitted.

Lemma 4. Let ν > 1be ﬁxed and consider the norm || · ||ωon real-valued sequences deﬁned as follows:

||a||ξ=|a0|+ 2

∞

X

n=1

νn

2n|an|.

Let `1

ξdenote the vector space of real-valued sequences for which the norm ||·||ξis ﬁnite. This is a Banach

space, and with Y= (`1

ξ)8×R×Requipped with the induced max norm, the map F:X×[0,1] →Yis

well-deﬁned and C∞. If F(a0, b0, . . . , c, p, α)=0, then this data deﬁnes a solution of the boundary-value

problem (6) by way of the equivalence (11)–(12), and vice-versa.

12

Remark 3. A zero of F(equivalently, a solution of the BVP (6)) does not necessarily deﬁne a closing-

reopening cycle. Indeed, up to the identiﬁcation with the Chebyshev series (11)–(12), it only deﬁnes a

solution that satisﬁes S.1–S.3 of Lemma 1. The conditions S.4–S.6 need to be checked after the fact. We

show how this can be accomplished numerically with rigorous error bounds in Section 3.7. Unsurprisingly,

S.4 requires the least eﬀort.

3.3 Finite-dimensional projection and numerical continuation

Let N > 0 be a ﬁxed integer. Deﬁne a pro jection map πN:`1

ν→`1

νaccording to

πN(a)n=an, n ≤N

0n > N,

Then, deﬁne a projection πN: (`1

ν)8→(`1

ν)8by

πN(a0, b0, . . . , a3, b3, c, p) = (πNa0, πNb0, . . . , πNa3, πNb3)

and a complementary projector π∞: (`1

ν)8→(`1

ν)8via π∞=I(`1

ν)8−πN. Deﬁne also ˜πN:X→Xto be

the projection operator ˜πN= diag(πN, IR2), with ˜π∞=IX−˜πN.

Let XN=πN(X)×R2. We will write XN

νif we want to emphasize the value of ν. Introduce the

“computational isomorphism” iNon π(`1

ν) by

iN(a) =

a0

.

.

.

aN

∈RN+1

and extend this to an isomorphism iN:XN→R8(N+1)+2 via

iN(a0, b0, . . . , a3, b3, c, p) =

iN(a0)

iN(b0)

.

.

.

iN(a3)

iN(b3)

c

p

.

Whenever we want to think of an element of Xwith zero tail (i.e. all zero Chebyshev ceoﬃcients above

mode N) as being a vector in some ﬁnite-dimensional space, we can apply the isomorphism iN. Similarly,

we can apply the inverse

i−1

N:XN→XN, XN:= R8(N+1)+2

to embed a ﬁnite-dimensional vector object of appopriate dimension into XN.

In what follows, we will use bars to denote “numerical” objects (i.e., objects that in practice will be

represented or computed with ﬁnite-dimensional vectors) while quantities without bars will typically be

analytical. Deﬁne the maps FN:X×R→Xand FN:XN×R→XNby

FN(x, s) = ˜πNF(˜πNx, α), F N(x, s) = iNFN(i−1

Nx, α).(21)

FNis the nonlinear map Ffrom (20) truncated to NChebyshev modes, while FNis its representation

in the ﬁnite-dimensional space XN. By Lemma 4, each of these maps is C∞. Since FNis a nonlinear

map on XN=R8(N+1)+2 and FN→Fpointwise, it should be expected that numerical zeroes of FN

will, when embedded in XN, generate approximate zeroes of F.

13

3.3.1 Numerical computation of zeroes

In practice, we compute such numerical zeroes ﬁrst at convex parameter α= 0 by implementing Newton’s

method for FNin double arithmetic. We initialize the method at a random guess with coeﬃcients

uniformly distributed over an appropriate hypercube and run for 300 iterations or until blowup (defect

greater than 103) or numerical convergence (defect less than 10−10). If a solution is not found, another

random guess is taken. This process of random searches generally converges to a numerical zero in less

than a second on modern hardware. We then manually inspect the output closing time and period cand

p, ensuring that they satisfy c≥wand p−w≥c. If this is true, the numerical zero is held as a candidate

and reﬁned further with Newton’s method until the defect is less than 10−15.

3.3.2 Numerical continuation of zeroes

Once a candidate zero for α= 0 has been computed and reﬁned, we implement natural parameter

continuation in αover a mesh of size 10−2until α= 1. The previous zero is used as the predictor for the

next zero, and we use Newton’s method to correct it until the defect is less than 10−15. Tangent predictors

might yield faster convergence, but even with this crude implementation the branch computation is not

too lengthy. We do not expect folds in the solution branch, and so do not implement a pseudo-arclength

continuation.

3.4 The radii polynomial approach

Once a branch (xαi, αi) of zeroes for FNhas been computed for a mesh {αi:i= 1, . . . , M }with α0= 0

and αM= 1, we would like to obtain rigorous results about the embedded branch in XN. We accomplish

this with validated continuation and a method that is sometimes called the radii polynomial approach.

The relevant theorem quoted below is a summary of the validated continuation result in [19].

Theorem 5. Let Xand Ybe Banach spaces. Let x0, x1∈ X and λ0, λ1∈R. Deﬁne the predictors

xs= (1 −s)x0+sx1, λs= (1 −s)λ0+sλ1.

Let F∈Ck(X ×R,Y)for some k≥1and assume there exist bounded linear operators A†∈B(X,Y)and

A∈B(Y,X), with Ainjective, and Y0,Z0,Z1and Z2(r)≥0such that

||AF (xs, λs)k|X≤Y0,∀s∈[0,1] (22)

||IX−AA†||B(X)≤Z0(23)

||A[DxF(x0, λ0)−A†]||B(X)≤Z1(24)

||A[DxF(xs+δ, λs)−DxF(x0, λ0)]||B(X)≤Z2(r),∀s∈[0,1], δ ∈Br(0) ⊂ X.(25)

Deﬁne the radii polynomial

p(r) = Z2(r)r+ (Z1+Z0−1)r+Y0.(26)

If there exists r0>0such that p(r0)<0, then there exists a Ckfunction

˜x: [0,1] →[

s∈[0,1]

Br0(xs)

such that F(˜x(s), λs)=0. Furthermore, these are the only zeroes of Fin the tube Ss∈[0,1] Br0(xs).

14

In the above theorem, the idea is that x0and x1are numerical zeroes of Fat parameters λ0and λ1.

Here, the term “numerical” is somewhat arbitrary. If the assertions of the theorem hold, the line segment

(xs, λs) for s∈[0,1] is an approximate branch of zeroes of Fin the sense that the tube of radius r0

around (xs, λs) contains exactly one zero for each value of s. In this way, the number r0gives a precise

statement about how far away the numerical zeroes can be from a true zero.

Remark 4. Theorem 5 is a rigorous continuation result for two numerical zeroes x0and x1. The theory

in [19] guarantees that if the radii polynomial theorem proves a branch from (x0, λ0)to (x1, λ1), and then

separately from (x1, λ1)to (x2, λ2), then then there is a Ckbranch from parameters λ0to λ2that is

r0-close to the union of the two predictor branches, with r0being the smallest radius from the two proofs.

In this way, one can obtain global continuation of branches of zeroes.

To apply the theorem we will need to ﬁrst construct the operators Aand A†for our map Ffrom

(20). After that, we will construct the Yand Zbounds analytically. In practice, they will be veriﬁed

along a numerical branch of zeroes by computing them with the interval arithmetic package INTLAB.

This will allow for a rigorous control of roundoﬀ error. If all of this can be done, then we will have proven

the existence of a numerical branch of zeroes of F, which in turn corresponds to a branch of solutions

(Sα, Iα) of (2) for a switching signal σα, such that conditions S.1 through S.3 of Lemma 1 are satisﬁed.

The operators Aand A†and the bound will be constructed in Section 3.6. To check that these truly

coincide with slow closing-reopening cycles, conditions S.4–S.6 must also be checked. We show how to do

this in Section 3.7.

3.5 A posteriori sharpness conditions and hyperbolicity

Here we show that if the radii polynomial proves the existence of a branch of zeroes of Fand some a

posteriori checks can be successfully completed, then the associated branch of closing-reopening cycles is

hyperbolic in the sense of static bifurcation theory. We brieﬂy mention a deﬁnition: we say the inequality

f(s)< g(s) for real-valued functions f, g : [0,1] →Rholds uniformly if there exists h > 0 such that

h < g(s)−f(s) for all s∈[0,1].

Theorem 6. Suppose Theorem 5 successfully proves the existence of a branch of zeroes (xs, αs)of F:

X×[0,1] →Yfor the map in (20) parameterized by the continuation parameter s∈[0,1]. Let t7→ Ik(t;s)

for k= 0,1,2,3denote functions generated by the identiﬁcation (12) for each continuation parameter s,

and let (cs, ps)be the crossing time and period. Suppose the following sharpness conditions are satisﬁed.

ˆ

S.4 cs> w and ps−w > csuniformly for s∈[0,1].

ˆ

S.5 IR< I0(1; s)and I2(1; s)< ICuniformly for s∈[0,1].

ˆ

S.6 d

dt I0(t;s)>0,d

dt I1(t;s)>0,d

dt I2(t;s)<0and d

dt I3(t;s)<0for t∈(−1,1), uniformly for s∈[0,1].

Let t7→ Φ(t;s) := (S(t;s), I(t;s)) denote the branch of functions for convex parameter αsdeﬁned by

Proposition 2 after inverting the transformation (4), and extend it to t∈[−w, ∞)by periodicity. For

each s,t7→ Φ(t;s)is a sharp closing-reopening cycle for convex parameter αs. Also, the following isolation

properties hold.

•Isolation in X: There exists r > 0such that if t7→ Θ(t;s)is another normalized, sharp closing-

reopening cycle for convex parameter αs, its representative θs∈Xsatisﬁes ||xs−θs||X> r.

•Isolation in C: If t7→ Θ(t;s)is any other branch of (not necessarily slow or sharp) closing-reopening

cycles for convex parameter αs, there exists δ > 0such that sups∈[0,1] ||Φ(t;s)−Θ(t;s)||∞> δ.

15

Proof. Clearly, the sharpness conditions imply S.4–S.6 from Lemma 1, which together with Theorem 5

prove t7→ Φ(t;s) deﬁnes a sharp closing-reopening cycle for convex parameter αsfor each s. We will prove

only isolation in C, since the result on isolation in Xis similar (and easier). By way of contradiction,

suppose Θn: [−w, ∞)→R2is a sequence of closing-reopening cycles and sn∈[0,1] is a sequence with

limn→∞ ||Θn−Φ(·;sn)||∞= 0. Without loss of generality, we may assume Θnis normalized. Since

Φ(·;s) is sharp and slow uniformly in sby the assumptions of the theorem, it follows that Θnis sharp

for nsuﬃciently large, say n≥m. Decomposing Θnfor n≥maccording to Proposition 2 and applying

the change of variables (4), let θn∈Xdenote the associated coordinates in the Banach space X. It

follows that F(θn, αs) = 0. From the radii polynomial, we know that for some r > 0, the ball Br(xs)

contains exactly one zero in Xνfor some ν > 1. However, limn→∞ θn=xsin Xν. To see this, ﬁrst

observe that each segment k= 0,1,2,3 of t7→ Φ(t;s) and t7→ Θn(t) can be identiﬁed with the solution

of a polynomial ordinary diﬀerential equation. In particular, since Θn(t)→Ψ(t;s) uniformly, there exist

positive constants Zand sequences ∆unand ∆znsatisfying limn→∞ ∆un= 0, limn→∞ ∆zn= 0, such that

the norms un(t) of the segments (deﬁned for t∈[−1,1]) for k= 0,1,2,3 of the diﬀerence Ψ(t;s)−Θn(t)

satisfy the integral inequality

un(t)≤∆un+ (t−t0)∆znΛ + ZZt

t0

2βrun(s)2+ (2µ+γ)un(s)ds

for −1≤t0≤t≤1. Recall that βc< βr. Extending to complex arguments, let ω(t) be a path in the

complex plane (parameterized by real t) satisfying ω(0) = t0. The solutions of the complexiﬁed ODE that

coincide on the real axis with one of the aforementioned solutions then satisfy the inequality

un(ω(t)) ≤∆un+ (1 + |ω(t)|)∆znΛ + ZZt

0

2βrun(ω(s))2+ (2µ+γ)un(ω(s))ds.

Since ∆un→0 and ∆zn→0 as n→ ∞, Gronwall’s inequality can be used to obtain uniform bounds

(in kand t0∈[−1,1]) for z7→ un(z) on the Bernstein ν∗-ellipse, for any ν∗>1 and nlarge enough. It

follows [34] that the segments of Φ(t;s)−Θn(t) are elements of Xν∗and that (θn−xs)→0 in Xν∗. Since

xs∈Xν, we must have θn∈Xνfor nlarge enough, and limn→∞ θn=xsin Xν. This is a contradiction,

since it implies the existence of two distinct zeroes of F(·, αs) in the ball Br(xs).

Remark 5. Analogously to Remark 4, this result can be globalized. If the sharpness conditions of Theorem

6 are satisﬁed separately along each segment of a global branch of zeroes of F, then the global branch of

closing-reopening cycles is hyperbolic.

If the conditions of Theorem 6 are satisﬁed, the closing-reopening cycle is isolated in a tube of radius

r > 0 with respect to the topology on X, up to identiﬁcation by the map (which we have implicitly deﬁned)

that sends a cycle from Cinto Xby way of normalization (Deﬁnition 3) and the segment decomposition.

In the space C, we have the slightly weaker result that no other branch of closing-reopening cycles can

ever intersect the branch, but we do not have a uniform tube enclosure.

If the entire branch is hyperbolic for the range α∈[0,1], then there are no folds along the branch

nor are there solution-crossings from α= 0 through to α= 1. As a consequence, we will have a unique,

continuous correspondence between the closing-reopening cycle at α= 0 and α= 1 by way of the rigorous

continuation, thereby giving a conditional (based on the success of the computer-assisted proof) answer

to the questions of existence and comparison from the introductory section.

3.6 Implementation of the radii polynomial approach

In this section we deﬁne the operators Aand A†, as well as construct the Yand Zbounds necessary to

apply Theorem 5.

16

3.6.1 Deﬁnition of the operators Aand A†

Suppose x0∈XNis an approximate (numerical) zero of FN(·, α0). Denote x0=i−1

N(x0); this symbol

has the same interpretation as x0in Theorem 5. Deﬁne the ﬁnite-dimensional linear map (interpreted as

a matrix)

A†=DxFN(x0, α0).

Next, by an abuse of notation we deﬁne L: (`1

ν)8→(`1

ξ)8according to

L(a0, b0, . . . , a3, b3)=(La0, Lb0, . . . , La3, Lb3),

where on the right-hand side Lis the operator from (18). Then deﬁne A†:X→Yby

A†=i−1

N◦A†◦iN◦˜πN+ diag(L, 0R2)˜π∞.(27)

It is straightforward to verify that A†is well-deﬁned.

To construct A:Y→X, we ﬁrst deﬁne a linear map L+:`1

ξ→`1

ν, with

L+(a)n=0, n = 0

1

2nan, n > 0.(28)

Then, extend it to a map L+: (`1

ξ)8→(`1

ν)8via

L+(a0, b0, . . . , a3, b3) = (L+a0, L+b0, . . . , L+a3, L+b3).

Next, let Abe a numerical inverse of A†; that is, a matrix such that ||I−AA†|| ≈ 0. We can now deﬁne

A:Y→Xas follows:

A=i−1

N◦A◦iN◦˜πN+ diag(L+,0R2)˜π∞.(29)

The following lemma is now a straightforward consequence of the deﬁnitions of A†and A.

Lemma 7. A†and Aare well-deﬁned and bounded, and Ais injective provided Ahas maximal rank.

3.6.2 The Yand Zbounds: preparation

With Lemma 7 at our disposal, we can move on to the computation of the bounds Yand Z. First, some

preparation. In what follows, we denote (xs, αs)∈XN×Rthe predictors and parameters

xs= (1 −s)i−1

Nx0+si−1

Nx1, αs= (1 −s)α0+sα1

for s∈[0,1], given numerical zeroes (x0, α0) and (x1, α1) of FN. Whenever we need to extract individual

components, we will write

xs= (as,0, bs,0, . . . , as,3, bs,3, cs, ps).(30)

We will now suppress the use of bars on the xterms; it will be understood that xsrepresents both

the inﬁnite-dimensional object in XNwith zeroes in its tail, and the ﬁnite-dimensional object in XN

obtained by cutting oﬀ the zero modes with the operator iN:XN→XN. We also set

∆x=x1−x0,∆α=α1−α0,

17

so that xs=x0+s∆xand αs=α0+s∆α.

We will deﬁne a norm on XNby way of the computational isomorphism: ||x||XN=||iNx||X. The

induced operator norm can then be evaluated as needed using one of the technical lemmas that appears

in Appendix A.

To facilitate the computation of partial derivatives of the map F, we will make a few additional deﬁni-

tions now. By an abuse of notation deﬁne T: (`1

ν)8→(`1

ν)8by T(a0, b0, . . . , a3, b3)=(T a0, T b0, . . . , T a3, T b3),

where on the right-hand side Tis the usual tridiagonal operator from (10). Then, deﬁne H:X→(`1

ν)8

by

H(x)=(H(a0, a1), H (b0, b1), H(a1, a2), H (b1, b2), H(a2, a3)H(b2, b3), H (a3, a1), H(b3, b1)).

The range of this operator is 8-dimensional, as each component has range in π0(`1

ν)∼R. Next, deﬁne

G:X×[0,1] ×[0,1] →R2by

G(x, q1, q2) =

(1 −q1)(b1,0+ 2 Pn≥1b1,n ) + q2b1,0−Pn≥2b1,n 1+(−1)n

n2−1

(1 −q1)(b3,0+ 2 Pn≥1b3,n ) + q2b3,0−Pn≥2b3,n 1+(−1)n

n2−1

and write I= [ ICIR]|. The reason for introducing the parameters q1and q2will be apparent soon.

Deﬁne Ψ : X→(`1

ν)8by

Ψ(x) = (Ψ1

0(a0, b0, c, p),Ψ2

0(a0, b0, c, p),...,Ψ1

3(a3, b3, c, p),Ψ2

3(a3, b3, c, p)),(31)

where we recall the components Ψj

kare deﬁned in (17). We can then compactly write F:X×[0,1] →Y

in the form

F(x, α) = L(x) + H(x) + TΨ(x)

I − G(x, α, α).(32)

The expression (32) allows us to simplify the expression for the diﬀerential DxF(x, α). Each of L,H

and Gis linear in x(for αﬁxed). In block operator form, we get

DxF(x, α) = L+H+T DxΨ(x)

−G(·, α, α).(33)

Similarly, we can write the diﬀerential DαF(x, α) in block form:

DαF(x, α) = 0

−G(x, 2,1) .(34)

We will also compute terms of Fup to order three (note that Fis indeed cubic). The only terms that

require a detailed look are those involving the functions Ψ. Because of the structure (31) of the operator

Ψ, it suﬃces to compute the diﬀerentials of the individual Ψj

k, which only involve four variables. Let

h= (u, v)∈(`1

ν)2. For brevity we will deﬁne (λ1, λ2) = (µ, µ +γ). Then

D(a,b)Ψj

k(a, b, c, p)h=zk(c, p)−β(k)(ak∗vk+uk∗bk)−λjuk, j = 1

β(k)(ak∗vk+uk∗bk)−λjvk, j = 2 (35)

D(c,p)Ψj

k(a, b, c, p)=Φj

k(a, b, c, p)∇zk(36)

D2

(a,b)Ψj

k(a, b, c, p)[h1, h2]=(−1)jzk(c, p)β(k)(u1

k∗v2

k+u2

k∗v1

k) (37)

D(c,p)D(a,b)Ψj

k(a, b, c, p)h=∇zk−β(k)(ak∗vk+uk∗bk)−λjuk, j = 1

β(k)(ak∗vk+uk∗bk)−λjvk, j = 2 (38)

D(c,p)D2

(a,b)Ψj

k(a, b, c, p)[h1, h2]=(−1)j∇zkβ(k)(u1

k∗v2

k+u2

k∗v1

k),(39)

while all partial derivatives with respect to (c, p) of order 2 and above are zero. The gradient ∇zkcan be

computed directly from (5).

18

3.6.3 Computation of the Y0bound

The ﬁrst bound, Y0, is a proxy for the numerical defect. Using the deﬁnition of FNand A, we can write

AF (xs, αs) = i−1

NAF N(xs, αs) + diag(L+,0R2)˜π∞F(xs, αs)≡ Y(1)

0+Y(2)

0.

To control the Y(1)

0term along the numerical branch, we expand FN(xs, αs) with respect to sas a

third-order Taylor expansion at s= 0. Since (xs, αs) = (x0, α0) + s(∆x, ∆α) and |s| ≤ 1, we get

Y(1)

0≤

3

X

m=0

1

m!

ADmFN(x0, α0)∆x

∆αmXN

.(40)

For implementation, these are computed using the the explicit formulas from (33), (34) and (35)–(39).

For Y2

0, many of the terms coming from Fcancel because xs∈XNand the post-multiplication with

π∞kills all modes lower than N. The result is that in the norm on X,

Y(2)

0= max

k,j Wj||L+TΨj

k(as,k, bs,k , cs, ps)||ν≤max

k,j Wj

∞

X

n=N+1

1

2nνn|(TΨj

k(as,k, bs,k , cs, ps))n|,

where the max runs for j= 1,2 and k= 0,1,2,3. Ψj

kcontains convolutions and linear terms, and since

their inputs are elements of πN(`1

ν), all modes above n= 2N+ 1 vanish. Thus, Ψj

k(as,k, bs,k , cs, ps)n= 0

for n≥2N+ 2. Taking into account the tridiagonal operator, the inﬁnite sum above terminates after

n= 2N+ 2. If we deﬁne the intervals

ak=a0,k + [1,0](a1,k −a0,k), bk=b0,k + [1,0](b1,k −b0,k ),(41)

c=c0+ [1,0](c1−c0), p =p0+ [1,0](p1−p0),

then using the computational isomorphisms and the above discussion,

Y0=

3

X

m=0

1

m!

ADmFN(x0, α0)∆x

∆αmXN

+ max

k,j Wj

2N+2

X

n=N+1

1

2nνn|(TΨj

k(i−1

Nak, iNbk, c, p))n|(42)

is a bound satisfying (22). The expression for (42) requires only a ﬁnite number of computations and can

be rigorously bounded above using interval arithmetic.

Remark 6. For our particular parameter choices, it was not necessary to use the third-order expansion

for the Y(1)

0term. Using the mean-value theorem for integrals in Banach space, we can also get

Y(1)

0≤ ||AF N(x0, α0)||XN+Z1

0

ADF N(x0+ts∆x, x0+ts∆α)s∆x

s∆αXN

dt

≤ ||AF N(x0, α0)||XN+ sup

z∈[0,1]

ADF N(x0+z∆x, α0+z∆α)∆x

∆αXN

.

In our implementation, this bound was used instead of the one in (40).

3.6.4 Computation of the Z0bound

Z0measures the quality of Aas an approximate inverse of A†. Because of the structure of Aand A†, the

error here is entirely due to numerical inversion of A†. The calculation of I−AA†is straightforward and

the following bound on the operator norm is tight:

Z0=||I−AA†||B(XN).(43)

An explicit formula for the operator norm || · ||B(XN)is provided in Appendix A.

19

3.6.5 Computation of the Z1bound

Making use of the identity A†=iN˜πNDxF(˜πNx0, α0), we can carefully compute

DxF(x0, α0)−A†= ˜πNH+T DxΨ(x0)

−G(·, α0)π∞+ ˜π∞T DxΨ(x0)

0R2.

We have deliberately partitioned the result as a sum of a map with range in XNand another with range

in ˜π∞(X). Applying Aon the left, we get

A[DxF(x0, α0)−A†] = i−1

NAiN˜πNH+T DxΨ(x0)

−G(·, α0)π∞+ ˜π∞L+T DxΨ(x0)

0R2:= Z(1)

1+Z(2)

1.

(44)

To proceed further we will need to compute the diﬀerential DxΨ(x). For cleanliness of presentation,

it suﬃces for us to calculate partial derivatives of the functions Ψj

k. Let h= (u, v)∈(`1

ν)2. Then

D(a,b)Ψj

k(a, b, c, p)h=zk(c, p)−β(k)(ak∗vk+uk∗bk)−λjuk, j = 1

β(k)(ak∗vk+uk∗bk)−λjvk, j = 2

D(c,p)Ψj

k(a, b, c, p)=Φj

k(a, b, c, p)∇zk

The gradient ∇zkcan be computed directly from (5). With this done, we will ﬁrst compute a bound for

Z(2)

1. Let h∈Xsatisfy ||h|| ≤ 1. Deﬁne for k= 0,...,3 and j= 1,2

Z(2,k,j)

1=Wjβ(k)(||iNa0,k||νW−1

2+||iNb0,k||νW−1

1) + λjW−1

j+||∇zk||1· ||Φj

k(i−1

Na0,k, i−1

Nb0,k, c0, p0)||ν.

Then

||Z(2)

1h||X≤max

k,j Wj

∞

X

n=N+1 T DxΨj

k(a0,k, b0,k , c0, p0)hn

νn

2n

≤2ν+ν−1

2(N+ 1) max

k,j Wj||D(a,b)Ψj

k(a0,k, b0,k , c0, p0)h||ν+||D(c,p)Ψj

k(a0,k, b0,k , c0, p0)h||ν

≤2ν+ν−1

2(N+ 1) max

k,j Wjβ(k)(||a0,k ||νW−1

2+||b0,k||νW−1

1) + λjW−1

j+||∇zk||1· ||Φj

k(a0,k, b0,k , c0, p0)||ν

=2ν+ν−1

2(N+ 1) max

k,j Z(2,k,j)

1

To understand the meaning of the subscripts, recall the indexing convention for our points xsalong the

predictor branch (30). To get the bound we used the Banach algebra and the bound ||T||B(X)≤2ν+ν−1,

which can be inferred from Appendix A and the deﬁnition of the norm on X.

Next, we need a uniform (in ||h||X≤1) bound for Z(1)

1h. To facilitate this, we will further decompose

the partial derivative D(a,b)Ψ as follows: with h= (u0, v0, . . . , u3, v3, d, q),

D(a,b)Ψj

k(a, b, c, p)h=ˆ

ψj

k(a, b, c, p)h+rj

k(c, p)h

ˆ

ψj

k(a, b, c, p)h= (−1)jzk(c, p)β(k)(ak∗vk+uk∗bk),

rj

k(c, p)h=−zk(c, p)µuk, j = 1

(µ+γ)vk, j = 2

ˆ

ψ(a, b, c, p)h= ( ˆ

ψ1

0(a0, b0, c, p)h, ˆ

ψ2

0(a0, b0, c, p)h, . . . , ˆ

ψ1

3(a3, b3, c, p)h, ˆ

ψ2

3(a3, b3, c, p)h,

r(c, p)h= (r1

0(c, p)h, r2

0(c, p)h, . . . , r1

3(c, p)h, r2

3(c, p)h).

20

We can therefore write

D(a,b)Ψ(a, b, c, p)h=ˆ

ψ(a, b, c, p)h+r(c, p)h. (45)

For brevity, set ˜π∞h=h∞= (u∞

0, v∞

0, . . . , u∞

3, v∞

3,0,0). We can then decompose Z(1)

1further as

Z(1)

1h=i−1

NAiN˜πNTˆ

ψ(x0)h∞+T D(c,p)Ψ(x0)h∞

0+i−1

NAiN˜πNH(h∞) + πNT r(c0, p0)h∞

−G(h∞, α0)

:= Z(1,1)

1+Z(1,2)

1.

The additional factor of πNwe have added in front of the T r(c0, p0) term is not superﬂuous (and is valid,

because of the post-composition with ˜πN) and will be used later. For k= 0,1,2,3 and m= 0, . . . , N + 1,

deﬁne

ak(m) = max

n=N+1,...,N+m

1

2νn|(iNa0,k)n−m|,bk(m) = max

n=N+1,...,N+m

1

2νn|(iNb0,k)n−m|,

ck(m) = |zk(c0, p0)β(k)|(W−1

2ak(m) + W−1

1bk(m)),h= (c0,c0,...,c3,c3,0,0).

Using the technical estimates from Appendix A, the (weighted) norm on X, the triangle inequality and

the fact D(c,p)Ψ(x0)h∞= 0, we can get the bound

||Z(1,1)

1h||X≤(2ν+ν−1)||abs(A)h||XN.

To get a bound for Z(1,2)

1h, we ﬁrst deﬁne a few interval vectors in XN. Let endenote the nth standard

basis vector in RN+1 and set

H= [ e|

1[−1,1] ··· e|

1[−1,1] 0 0 ]|, G = [ 0··· 0 [−1,1] [−1,1] ]|

r= [ e|

N+1[−1,1] · ·· e|

N+1[−1,1] 0 0 ]|.

Let W: (`1

ν)8×R2→(`1

ν)8×R2be the diagonal operator deﬁned by

W(a0, b0, . . . , a3, b3, c, p) = (W1a0, W2, b0, . . . , W1a3, W2, b3, W3c, W3p).

One can then verify

H(h∞)

0∈1

νN+1 W−1i−1

NH, πNT r(c0, p0)h∞

0∈ξ

νN+1 W−1i−1

Nr,

0

G(h∞, α0)∈1

νN+1 W−1i−1

NG,

where ξ= (µ+γ) maxk|zk(c0, p0)|/2. It follows that

||Z(1,2)

1h||X≤1

νN+1 ||AW−1H||XN+||AW−1G||XN+ξ||AW−1r||XN,

where we have abused notation and identiﬁed W−1with iNW−1i−1

N. Combining the previous results, we

conclude that

Z1= (2ν+ν−1)||abs(A)h||XN+2ν+ν−1

2(N+ 1) max

k,j Z(2,k,j)

1(46)

+1

νN+1 ||AW−1H||XN+||AW−1G||XN+ξ||AW−1r||XN

is a suitable Z1bound.

21

3.6.6 Computation of the Z2bound

Let δ∈Br(0) ⊂X. Then

DxF(xs+δ, αs)−DxF(x0, α0) = T(DxΨ(xs+δ)−DxΨ(x0))

−G(·, αs) + G(·, α0):= "TZ(1)

2

Z(2)

2#

To avoid excessive use of indices, we will abuse notation and refer to each component (in `1

νor R) of δby

also using the symbol δ. This should not cause too much confusion. We bound the diﬀerence of the DxΨ

ﬁrst, as this is the most tedious. Let h∈Xsatisfy ||h||X≤1, and set Z(1)

2h= (ξ0,1, ξ0,2, . . . , ξ3,1, ξ3,2)

for ξk,j ∈`1

ν. If we set h= ((hk,j ), u, v) for u, v ∈Rand hk,j ∈`1

νfor indices k= 0,...,3 and j= 1,2, we

can write

ξk,j = (D(a,b)Ψj

k(as,k +δ, bs,k +δ, cs+δ, ps+δ)−D(a,b)Ψj

k(a0,k, b0,k , c0, p0))[hk,1, hk,2]

+ (Φj

k(as,k +δ, bs,k +δ)−Φj

k(a0, b0))∇zk[u v ]|

≡ξ(1)

k,j +ξ(2)

k,j ∇zk[u v ]|.

We can bound each of these quantities in turn. Taking into account ||h||X≤1, ||δ||X≤rand the weights

in the space X, we ultimately get |∇zk[u v ]|| ≤ W−1

3||∇zk||1and

||ξ(2)

k,j ||ν≤β(k)h||∆ak∗∆bk||ν+||a0,k ∗∆bk+b0,k ∗∆ak||ν+rW−1

1(||b0,k||ν+||∆bk||ν)· ··

+rW−1

2(||a0,k||ν+||∆ak||ν)+r2W−1

1W−1

2i+rλjW−1

j

≡ˆ

ξ(2)

k,j (r),(47)

||ξ(1)

k,j ||ν≤ |zk(c0, p0)|β(k)W−1

2(rW −1

1+||∆ak||ν) + W−1

1(rW −1

2+||∆bk||ν)

+ abs(∇zk)[ ∆c+W−1

3r∆p+W−1

3r]|β(k)W−1

2(||a0,k||ν+||∆ak||ν+W−1

1r). . .

+W−1

1(||b0,k||ν+||∆bk||ν+W−1

2r)+λjW−1

j

≡ˆ

ξ(1)

k,j (r).(48)

Observe that each of the ˆ

ξ(q)

k,j (r) for q= 1,2 can be interpreted as degree two polynomials in r. As for

Z(2)

2h∈R2, the following bound is straightforward and its derivation is omitted: for h∈Xwith ||h|| ≤ 1,

||Z(2)

2h||∞≤W−1

2|∆α|.

Now, let

µ=A[DxF(xs+δ, αs)−DxF(x0, α0)]h= ((µk,j ), µ∞)

for µ∞= (µ∞,1, µ∞,2)∈R2. Combining the previous two estimates, the deﬁnition of Aand using the

technical bounds from Appendix A, we can get

||µk,j ||ν≤X

m,`

A(1,1)

k,j,m,`(ˆ

ξ(1)

m,` +W−1

3||∇zm||1ˆ

ξ(2)

m,`) + A(1,2)

k,j W−1

2|∆α|+1

2(N+ 1) (ˆ

ξ(1)

k,j +W−1

3||∇zk||1ˆ

ξ(2)

k,j )

≡ˆµk,j (r)

|µ∞,j | ≤ X

m,`

A(2,1)

j,m,`(ˆ

ξ(1)

m,` +W−1

3||∇zm||1ˆ

ξ(2)

m,`) + A(2,2)

∞,j W−1

2|∆α|+1

2(N+ 1) W−1

2|∆a|

≡ˆµ∞,j (r)

22

Where the constants Aare given by

A(1,1)

k,j,m,` = max

n=0,...,N

1

ωn

N

X

q=0 |(A(1,1)

k,j,m,`)q ,n|ωq,A(1,2)

m,` =||(A(1,2)

m,` )1||ν+||(A(1,2)

m,` )2||ν(49)

A(2,1)

j,m,` = max

n=0,...,N

1

ωn|(A(2,1)

j,m,`)n|,A(2,2)

∞,j =||A(2,2)

∞,j ||1,(50)

for a representation of the operator Ain block form: with h= (h0,1, h0,2, . . . , h3,1, h3,2, hR2)∈X,

(Ah)k,j =X

m,`

A(1,1)

k,j,m,`hm,` +A(1,2)

k,j hR2

(Ah)∞,j =X

m,`

A(2,1)

j,m,`hm,` +A(2,2)

∞,j hR2,

with A(1,1)

k,j,m,` :`1

ν→`1

ν,A(1,2)

k,j :R2→`1

ν, and A(2,1)

j,m,` :`1

ν→R,A(2,2)

∞,j :R2→R, and the associated

ﬁnite-dimensional projections Adeﬁned analogously. By deﬁnition of the norm on X, we can ﬁnally

obtain the Z2bound:

Z2(r) = max

k,j {(2ν+ν−1)Wjˆµk,j (r), W3ˆµ∞,j(r)}.(51)

3.7 A posteriori veriﬁcation of sharpness conditions

Here we will make use of the same notation convention taken at the beginning of Section 3.6.2, so that

xs= (as,0, bs,0, . . . , as,3, bs,3, cs, ps) denotes the predictor. We will also write

xs=iNxs= (as,0, bs,0, . . . , as,3, bs,3, cs, ps).

Note that for each sﬁxed, this is an element of XN. We also use the same intervals appearing in (41).

Our objective in this section is to demonstrate how the sharpness conditions ˆ

S.4– ˆ

S.6 of Theorem 6,

can be veriﬁed. It should come as no surprise that verifying ˆ

S.4 is the easiest task. In practice, since all

of our computations are done using interval arithmetic, we need only check

p−w > c > w. (52)

The following two lemmas can be used to verify the other sharpness conditions. Here we will be using

the convention that (interval) vectors v∈Rmare indexed as v= (v0, . . . , vm−1).

Lemma 8. Let r0>0be a radius for which the radii polynomial satisﬁes p(r0)<0. The sharpness

condition ˆ

S.5 of Theorem 6 is satisﬁed provided

IR+r0

W2

<(b0)0+ 2

N

X

n=1

(b0)n,(b2)0+ 2

N

X

n=1

(b2)n< IC−r0

W2

.

Proof. Let t7→ Ik(t;s) be theoretically guaranteed (by the radii polynomial approach) branches of the

infected components of the solution of the BVP (6) for convex parameter αs, and let Ik(t;s) be the

numerical approximations generated by (12) for Chebyshev coeﬃcients in XN. One can check that that

sup

t∈[−1,1]

W2||Ik(t;s)−Ik(t;s)|| ≤ r0.

23

This is a consequence of fact that for a function frepresented in Chebyshev series, we have ||f||∞≤ ||f||ν,

where on the right-hand side fis identiﬁed with its Chebyshev series coeﬃcients. It follows that

Ik(t;s)∈Ik(t;s)+[−1,1] r0

W2

for all t∈[−1,1] and s∈[0,1]. Making use of the uniform inclusion

Ik(t;s)∈(bk)0+ 2

N

X

n=1

(bk)nTn(t)

and evaluating at t= 1, we get the conclusion of the lemma since the inequalities imply IR< I0(1; s) and

I2(1; s)< ICuniformly.

Lemma 9. Let r0>0be a radius for which the radii polynomial satisﬁes p(r0)<0. For k= 0,1,2,3,

deﬁne

ηk=β(k)ak∗bk−(µ+γ)bk

0∈R2(N+1)

∆z=W−1

3[r0/2 0 r00]|

φk=β(k)||i−1

Nak∗bk||ν+r0(W−1

2||i−1

Nak||ν+W−1

1||i−1

Nbk||ν) + W−1

1W−1

2r2

0

+ (µ+γ)(||i−1

Nbk||ν+W−1

2r0)

∆ηk=|zk(c, p)| · β(k)(W−1

2||i−1

Nak||ν+W−1

1||i−1

Nbk||ν+W−1

1W−1

2r0) + W−1

2(µ+γ)r0+ (∆z)kφk,

where ak∗bk∈R2(N+1) is discrete convolution of akwith bk, and 0is the (N×1) ×1vector of zeroes.

For all t∈[−1,1], the functions Ik(t;s)of Theorem 6 satisfy

d

dtIk(t;s)∈zk(c, p)· ηk,0+ 2

2N+1

X

n=1

Tn(t)ηk,n!+ [−1,1]∆ηk≡ˆ

I0

k(t).(53)

Consequently, the sharpness condition ˆ

S.6 is satisﬁed if inftˆ

I0

k(t)>0for k= 0,1and suptˆ

I0

k(t)<0for

k= 2,3.

Proof. Let t7→ (Sk(t;s), Ik(t;s)) be theoretically guaranteed (by the radii polynomial approach) branches

of the solution of the BVP (6) for convex parameter αs, and let (Sk(t;s), Ik(t;s)) be the numerical

approximations generated by (11) and (12) for Chebyshev coeﬃcients in XN. Then

Sk(t;s) = as,k,0+ 2

∞

X

n=1

as,k,nTn(t), I k(t;s) = bs,k,0+ 2

∞

X

n=1

bs,k,nTn(t),

Sk(t;s) = ˜as,k,0+ 2

∞

X

n=1

˜as,k,n Tn(t), Ik(t;s) = ˜

bs,k,0+ 2

∞

X

n=1

˜

bs,k,nTn(t)

for ˜as,k = ˜a0,k +s(˜a1,k −˜a0,k ) and ˜

bs,k =˜

b0,k +s(˜

b1,k −˜

b0,k). From the derivations in Section 3.2 and

properties of quadratic convolutions, we know that

d

dtIk(t;s) = Ψ2

k(˜as,k ,˜

bs,k, cs, ps)0+ 2

∞

X

n=1

Ψ2

k(˜as,k ,˜

bs,k, cs, ps)nTn(t),

d

dtIk(t;s) = Ψ2

k(as,k, bs,k , cs, ps)0+ 2

∞

X

n=1

Ψ2

k(as,k, bs,k , cs, ps)nTn(t)∈ηk,0+ 2

2N+1

X

n=1

ηk,nTn(t).

24

Let ∆Ψk= Ψ2

k(˜as,k ,˜

bs,k, cs, ps)−Ψ2

k(as,k, bs,k , cs, ps). We have

∆Ψk,s =zk(cs, ps)β(k)(as,k ∗∆bs,k + ∆as,k ∗bs,k + ∆as,k ∗∆bs,k )−(µ+γ)∆bs,k

+ ∆zkβ(k)(as,k ∗bs,k +as,k ∗∆bs,k + ∆as,k ∗bs,k + ∆as,k ∗∆bs,k)−(µ+γ)(bs,k + ∆bs,k ),

where ∆zk=zk(˜cs,˜ps)−zk(cs, ps), ∆as,k = ˜as,k −as,k and ∆bs,k =˜

bs,k −bs,k. From the radii

polynomial, we know that ||∆as,k||ν≤r0/W1,||∆bs,k||ν≤r0/W2, and one can verify directly that

|∆zk| ≤ W−1

3[r0/2 0 r00]|. The convolution terms can be bounded using the Banach algebra. It

can then be veriﬁed that

∆Ψk,s ∈[−1,1]∆ηk

for all s, from which the inclusion (53) follows.

In practice, Lemma 9 is veriﬁed by evaluating the right-hand side of (53) over a ﬁne mesh of subintervals

of t∈[−1,1] and computing over-estimates. Speciﬁcally, let t0< t1<··· < tMbe a mesh of [−1,1]

with t0=−1 and tM= 1. We then explicitly compute (with interval arithmetic) an over-estimate for the

interval

hˆ

I0

ki= min

m=1,...,M inf

t∈[tm−1,tm]

ˆ

I0

k(t),max

m=1,...,M sup

t∈[tm−1,tm]

ˆ

I0

k(t)!.(54)

By construction, ˆ

I0

k(t)⊆ hˆ

I0

kifor all t∈[−1,1], and as the mesh becomes ﬁner the enclosure gets tighter.

As the mesh width goes to zero, the width of the interval enclosure (54) approaches 2∆ηk. As the latter

is strongly inﬂuenced by the radius r0, obtaining a successful proof of the sharpness condition ˆ

S.6 along

the numerical branch of zeroes of the map Ffrom (20) generally requires small step sizes to be made

in the continuation to keep r0small enough. As for the veriﬁcation of the conditions of Lemma 8 for

ˆ

S.5, the relevant inequality is already stated in terms of intervals, so we simply check them using interval

arithmetic. The same is true for (52), which is relevant for the sharpness check ˆ

S.4.

3.8 Average and extrema active cases over a closing-reopening cycle

Computing the average number of active cases over a closing-reopening cycle is fairly straightforward.

Given a numerical zero x0= (a0, b0, . . . , a3, b3, c, p)∈XNof the nonlinear map Ffor convex parameter

α0, a ﬁrst numerical approximation of the average over the numerical cycle I= (I0, . . . , I3) is

IN=1

p

3

X

k=0

zk(c, p)Z1

−1 bk,0+ 2

N

X

n=1

bk,nTn(t)!dt =2

p

3

X

k=0

zk(c, p) bk,0−

N

X

n=2

bk,n

1+(−1)n

n2−1!.(55)

To propagate error from the radii polynomial, let r0>0 be a radius for which the radii polynomial

satisﬁes p(r0)<0. To be clear, we are now applying Theorem 5 with xs≡x0and λs≡α0— that is, we

are only interested in a single zero of the map Fat a ﬁxed parameter α. We let

[I] = 1

˜p

3

X

k=0

zk(˜c, ˜p)Z1

−1 ˜

bk,0+ 2

∞

X

n=1

˜

bk,nTn(t)dt!dt

be the average across the theoretically guaranteed cycle Ifor which, from the radii polynomial method,

we have ||˜

bk−bk||ν≤r0/W2,|˜c−c| ≤ r0/W3and |˜p−p| ≤ r0/W3.

Lemma 10. Deﬁne ∆z= ∆z(r0)as in Lemma 9. Provided p > r0, the average of active cases [I]over

the closing-reopening cycle satisﬁes the inequality

[I]−[I]N≤W−1

2r0+

3

X

k=0

p∆zk(r0) + W−1

3r0|zk(c, p)|

p(p−W−1

3r0)(W−1

2r0+||i−1

Nbk||ν) (56)

25

Proof. Set bk,n = 0 for n > N and deﬁne ∆bk,n =˜

bk,n −bk,n , where (˜a0,˜

b0,...,˜a3,˜

b3,˜c, ˜p) is the exact

zero of the map Ffor the relevant convex parameter α. With a bit of algebra, we get

[I]−[I]N=1

p

3

X

k=0

zk(c, p)Z1

−1

∆bk,0+

∞

X

n=1

∆bk,nTn(t)dt

+

3

X

k=0 zk(˜c, ˜p)

˜p−zk(c, p)

p"Z1

−1

bk,0+ ∆bk,0+

∞

X

n=1

(bk,n + ∆bk,n )Tn(t)dt#.

One can then check that the quantity in the ﬁrst line is bounded above by r0/W2, while

zk(˜c, ˜p)

˜p−zk(c, p)

p≤p|zk(˜c, ˜p)−zk(c, p)|+|(˜p−p)zk(c, p)|

p˜p≤p∆zk(r0) + W−1

3r0|zk(c, p)|

p(p−r0W−1

3),

with the ﬁnal inequality being a consequence of the radii polynomial and two applications of the mean

value theorem for integrals. The result then follows by explicit computation of the integrals, the radii

polynomial and the deﬁnition of the ν-norm of bk.

When r0is computed in the scope of validation of a speciﬁc zero rather than a branch of a continuation

(that is, one ﬁxes x0=x1and λ0=λ1in Theorem 5), it is not diﬃcult to get r0close to a few multiples

of machine precision. The bounds of the lemma can therefore be made very tight. See later Section 4 for

numerical results.

Computing extrema is easier. One can show (and we omit the details) that the extrema Imin and

Imax over a slow, sharp, normalized closing-reopening cycle Ican be computed from the numerical zero

x0as follows:

Imin ∈[−1,1]r0W−1

2+b3,0+ 2

N

X

n=1

b3,n, Imax ∈[−1,1]r0W−1

2+b1,0+ 2

N