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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∗

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.

Key words. rigorous numerics, periodic orbit, SIR model, moving average, COVID-19, contact

rate

AMS subject classiﬁcations. 37M15, 37N25, 93C30

1. Introduction. Beginning in the mid-second quarter of 2020, governments

worldwide began eﬀorts to reopen their economies [10,12,29] in response to slow-

ing 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 re-

strictions (i.e., reopening) are applied [22,26]. Since the formalism involves a delayed

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

vector ﬁeld, and the end result is one typically observes oscillation about the thresh-

old. An advantage of this formalism is that natural time lags between collection of

∗McGill University, Department of Mathematics and Statistics, Montreal, Quebec, Canada.

(kevin.church@mcgill.ca).

1

2KEVIN E. M. CHURCH

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 mechanisms.

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 epi-

demic intervention thresholds, we consider properties of limit cycles (referred to as

closing-reopening cycles in this publication) 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 pub-

lication, 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},(1.1)

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 num-

ber of active cases of the closing-reopening cycle compare in the cases α= 0

PANDEMIC CLOSING-REOPENING CYCLES 3

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?

We will from here on refer to the threshold condition (1.1) 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 us-

ing 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 (1.1) 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 X

is 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 (1.1) 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 ques-

tions 2–4 by numerical quadratures and zero-ﬁnding. However, these results are not

rigorous because we have not proven that a true closing-reopening cycle exists, and

4KEVIN E. M. CHURCH

we do not have precise information about how good an approximation the numerical

branch of solutions {(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 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 an-

swer 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 (1.1) 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 Sec-

tion 2, the model is explicitly formulated and some elementary properties of closing-

reopening cycles are stated and proven, including a boundary-value problem that

such cycles must satisfy. Section 3concerns 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 over-

shooting 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 5and Section 6.

PANDEMIC CLOSING-REOPENING CYCLES 5

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 v

are two intervals, we write u<vif and only if sup(v)<inf(u).

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.

(2.1)

Sand Irepresent the number of susceptible and infected humans, respectively. Λ

is a constant recruitment rate, µthe per capita death rate, βthe combined con-

tact/infection rate in units of 1/(humans ·time) and γthe combined death and recov-

ery 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 βsimply 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] →R

according 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< βr

be 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 (2.1) together with a rule that states when the contact rate

parameter βswitches. Speciﬁcally, the contact rate switches whenever the relationship

(1.1) 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 1may aid in visualization.

Definition 2.1. 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},

6KEVIN E. M. CHURCH

Fig. 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.

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< β0

and convex parameter αif it is a solution of the piecewise-continuous system (2.1)

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 2.2. In Deﬁnition 2.1, we have opted to label the respective contact rates

βrand βcinstead as β0and β1. This is to avoid confusion later in Deﬁnition 2.3 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.

We will not discuss details such as existence, uniqueness and continuability of

solutions for the opening/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 continued 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 (2.1) are ultimately unformly

bounded for β=βσ(t)for any switching signal σ.

Definition 2.3. A solution (S, I)the relay model is a closing-reopening cycle

with period pif S,Iand σare p-periodic, and the associated switching function has

exactly two discontinuities in the interval [0, p]. A closing-reopening cycle is normal-

ized if σ(0) = 0,σ(c)=1for 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

PANDEMIC CLOSING-REOPENING CYCLES 7

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 2.4. 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.

The following lemma demonstrates how slow closing-reopening cycles can be construc-

ted using special solutions of (2.1) with a piecewise-constant parameter β..

Lemma 2.5. 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 (2.1)with parameter βron [0, c), and satisﬁes (2.1)with pa-

rameter 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 deﬁ-

nition of slow closing-reopening cycle and conditions S.1–S.3 of Lemma 2.5 deﬁne a

8KEVIN E. M. CHURCH

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

Fig. 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 (2.2)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 4.1, and each of them is sharp according to Deﬁnition 2.4.

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.

(2.2)

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 2for a visual-

ization.

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.

PANDEMIC CLOSING-REOPENING CYCLES 9

Proposition 2.6. If (Sk, Ik)for k= 0,1,2,3have the domains (2.2)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 2.5.

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 2.5. 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,

(2.3)

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}.

(2.4)

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.

(2.5)

The following lemma is a direct consequence of the previous derivation, Lemma 2.5

and Proposition 2.6.

Lemma 2.7. Suppose (Sk, Ik):[−1,1] →R2

+for k= 0,1,2,3is a solution of the

boundary-value problem (2.5)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.

Remark 2.8. 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,

10 KEVIN E. M. CHURCH

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 (2.5) into a zero-ﬁnding problem in an inﬁnite-dimensional function

space. We then determine a ﬁnite-dimensional projection and discuss how to com-

pute 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 (2.5) 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|(3.1)

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.

PANDEMIC CLOSING-REOPENING CYCLES 11

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.(3.2)

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.(3.3)

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

(3.4)

will be quite useful.

Let `1

νdenote the normed vector space of sequences {an:n∈N}bounded with

respect to the ||·||νnorm. 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 (2.5) as a Chebyshev series. Speciﬁcally, make the expansions

Sk(t) = ak,0+ 2

∞

X

n=1

ak,nTn(t)(3.5)

Ik(t) = bk,0+ 2

∞

X

n=1

bk,nTn(t).(3.6)

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 (3.5) and (3.6) into (2.5), using the

relations (3.2) and (3.3), and making the identiﬁcation a4≡a0and b4≡b0, we get

12 KEVIN E. M. CHURCH

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,

(3.7)

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

(3.8)

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

(3.9)

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

(3.10)

where Ψk:`1

ν×`1

ν×R2→`1

ν×`1

νis the representation of the right-hand side of the

ODE in (2.5) 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),

(3.11)

with Λ0= Λ and Λn= 0 for n > 0. To be precise, (3.7) corresponds to the diﬀerential

equations, (3.8) to the R-Lboundary conditions, and (3.9) and (3.10) the convex

threshold conditions from the BVP (2.5).

We can transform (3.7)–(3.10) 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

3.1) by

L(a)n= 2nan.(3.12)

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 3.5). The norm on

Xmust be chosen carefully, as it is typical for the Sand Icomponents of solutions

to the BVP (2.5) 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|}.

(3.13)

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

PANDEMIC CLOSING-REOPENING CYCLES 13

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

(3.14)

The boundary conditions (3.8) have been encoded into the linear map H. By con-

struction, 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 (3.7)–(3.10) for closing time cand period p.

Subsequently, if the monotonicity requirements of Lemma 2.7 can be checked, this will

uniquely deﬁne a slow closing-reopening cycle through the identiﬁcations (3.5)–(3.6)

and Lemma 2.7. 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 3.1. Let ν > 1be ﬁxed and consider the norm || · ||ωon real-valued se-

quences 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 (2.5)by way of the

equivalence (3.5)–(3.6), and vice-versa.

Remark 3.2. A zero of F(equivalently, a solution of the BVP (2.5)) does not

necessarily deﬁne a closing-reopening cycle. Indeed, up to the identiﬁcation with the

Chebyshev series (3.5)–(3.6), it only deﬁnes a solution that satisﬁes S.1–S.3 of Lemma

2.5. 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 pro jection and numerical continuation. Let N >

0 be a ﬁxed integer. Deﬁne a projection 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)

14 KEVIN E. M. CHURCH

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→X

and FN:XN×R→XNby

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

Nx, α).(3.15)

FNis the nonlinear map Ffrom (3.14) truncated to NChebyshev modes, while

FNis its representation in the ﬁnite-dimensional space XN. By Lemma 3.1, each of

these maps is C∞. Since FNis a nonlinear map on XN=R8(N+1)+2 and FN→F

pointwise, it should be expected that numerical zeroes of FNwill, when embedded in

XN, generate approximate zeroes of F.

3.3.1. Numerical computation of zeroes. In practice, we compute such nu-

merical 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 coeﬃ-

cients 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 c

and 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.

PANDEMIC CLOSING-REOPENING CYCLES 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 α

over 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 3.3. 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](3.16)

||IX−AA†||B(X)≤Z0

(3.17)

||A[DxF(x0, λ0)−A†]||B(X)≤Z1

(3.18)

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

(3.19)

Deﬁne the radii polynomial

p(r) = Z2(r)r+ (Z1+Z0−1)r+Y0.(3.20)

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).

In the above theorem, the idea is that x0and x1are numerical zeroes of Fat param-

eters 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 r0around (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 3.4. Theorem 3.3 is a rigorous continuation result for two numerical ze-

roes 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.

16 KEVIN E. M. CHURCH

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

map Ffrom (3.14). 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 (2.1) for a switching signal σα, such that conditions S.1 through S.3 of

Lemma 2.5 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 3.5. Suppose Theorem 3.3 successfully proves the existence of a branch

of zeroes (xs, αs)of F:X×[0,1] →Yfor the map in (3.14)parameterized by the

continuation parameter s∈[0,1]. Let t7→ Ik(t;s)for k= 0,1,2,3denote func-

tions generated by the identiﬁcation (3.6)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 2.6 after inverting the transformation (2.3), 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 normal-

ized, 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 > 0

such that sups∈[0,1] ||Φ(t;s)−Θ(t;s)||∞> δ.

Proof. Clearly, the sharpness conditions imply S.4–S.6 from Lemma 2.5, which

together with Theorem 3.3 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 2.6 and applying the

change of variables (2.3), 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

PANDEMIC CLOSING-REOPENING CYCLES 17

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 3.6. Analogously to Remark 3.4, this result can be globalized. If the

sharpness conditions of Theorem 3.5 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 3.5 are satisﬁed, the closing-reopening cycle is iso-

lated 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 2.4) 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 3.3.

3.6.1. Deﬁnition of the operators Aand A†.Suppose x0∈XNis an ap-

proximate (numerical) zero of FN(·, α0). Denote x0=i−1

N(x0); this symbol has the

same interpretation as x0in Theorem 3.3. Deﬁne the ﬁnite-dimensional linear map

(interpreted as a matrix)

A†=DxFN(x0, α0).

18 KEVIN E. M. CHURCH

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 (3.12). Then deﬁne A†:X→Y

by

A†=i−1

N◦A†◦iN◦˜πN+ diag(L, 0R2)˜π∞.(3.21)

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.

(3.22)

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∞.(3.23)

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

and A.

Lemma 3.7. A†and Aare well-deﬁned and bounded, and Ais injective provided

Ahas maximal rank.

3.6.2. The Yand Zbounds: preparation. With Lemma 3.7 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).(3.24)

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 XNobtained by cutting oﬀ the zero modes with the

operator iN:XN→XN. We also set

∆x=x1−x0,∆α=α1−α0,

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.

PANDEMIC CLOSING-REOPENING CYCLES 19

To facilitate the computation of partial derivatives of the map F, we will make a

few additional deﬁnitions 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 (3.4). Then, deﬁne H:X→(`1

ν)8by

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)),(3.25)

where we recall the components Ψj

kare deﬁned in (3.11). We can then compactly

write F:X×[0,1] →Yin the form

F(x, α) = L(x) + H(x) + TΨ(x)

I − G(x, α, α).(3.26)

The expression (3.26) allows us to simplify the expression for the diﬀerential

DxF(x, α). Each of L,Hand Gis linear in x(for αﬁxed). In block operator form,

we get

DxF(x, α) = L+H+T DxΨ(x)

−G(·, α, α).(3.27)

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

DαF(x, α) = 0

−G(x, 2,1) .(3.28)

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 (3.25) 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

(3.29)

D(c,p)Ψj

k(a, b, c, p)=Φj

k(a, b, c, p)∇zk

(3.30)

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)

(3.31)

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

(3.32)

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),

(3.33)

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

gradient ∇zkcan be computed directly from (2.4).

20 KEVIN E. M. CHURCH

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

.(3.34)

For implementation, these are computed using the the explicit formulas from (3.27),

(3.28) and (3.29)–(3.33).

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 ),(3.35)

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|

(3.36)

is a bound satisfying (3.16). The expression for (3.36) requires only a ﬁnite number

of computations and can be rigorously bounded above using interval arithmetic.

Remark 3.8. 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 (3.34).

PANDEMIC CLOSING-REOPENING CYCLES 21

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).(3.37)

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

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.

(3.38)

To proceed further we will need to compute the diﬀerential DxΨ(x). For cleanli-

ness 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 (2.4). 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 (3.24). To get the bound we used the Banach

22 KEVIN E. M. CHURCH

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).

We can therefore write

D(a,b)Ψ(a, b, c, p)h=ˆ

ψ(a, b, c, p)h+r(c, p)h.(3.39)

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 en

denote 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,

PANDEMIC CLOSING-REOPENING CYCLES 23

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

(3.40)

+1

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

is a suitable Z1bound.

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∈X

satisfy ||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||1

and

||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),

(3.41)

||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).

(3.42)

Observe that each of the ˆ

ξ(q)

k,j (r) for q= 1,2 can be interpreted as degree two polynomi-

als in r. As for Z(2)

2h∈R2, the following bound is straightforward and its derivation

24 KEVIN E. M. CHURCH

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