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Analysis of pandemic closing-reopening cycles using rigorous homotopy continuation: a case study with Montreal COVID-19 data


Abstract and Figures

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.
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Analysis of pandemic closing-reopening cycles using rigorous
homotopy continuation: a case study with Montreal COVID-19
Kevin E. M. Church
McGill University, Department of Mathematics and Statistics
December 28, 2020
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
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
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
I(u)du ∈ {IR, IC},(1)
where IR< ICare reopening and closing levels, wis the time window for the rolling average and 0 α1
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?
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 Cwith 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
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:IRfor a closed interval I, then the function ftCis defined by ft(θ) = f(t+θ); it exists if
[tw, 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 AB(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:UVand B:XY, then C= diag(A, B) :
U×XV×Yis defined by C(u, x)=(A(u), B(x)). The symbol Br(x) denotes the open ball of radius
rcentered at xX. If x= [min(x),max(x)] is an interval vector with min(x)max(x) in the usual
partial order on Rnand t0, 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).
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
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 S0R+,φCand 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]) <infC[t, b))
1,sup(ΣC[0, t]) <infR[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.
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.
cwand pwc.
I(pw)ICand IRI(cw).
We say the cycle is sharp if the above inequalities are strict.
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 cwand pwc.
S.5 Iis monotone increasing on [0, c)and IRI(cw).
S.6 Iis monotone decreasing on [c, p)and I(pw)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
If t[0, c] then I0(t)>0. On the other hand, if t[w, c] then I(t)> I (tw), 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
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
[cw, c], k = 1
[c, p w], k = 2
[pw, c], k = 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.
1.4495 1.45 1.4505 1.451
1.4605 1.461 1.4615 1.462
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
Symbolically set S4=S0and I4=I0, and consider the following boundary-value problem.
Sk= Λ β(k)SkIkµSk, 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 [cw, c)
(S2(t), I2(t)), t [c, p w),
(S3(t), I3(t)), t [pw, p],
satisfies conditions S.1–S.4 of Lemma 1.
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
To complete the change of variables, write
(S0, I0)(t) = ( ˜
cw1,(S1I1)(t) = ( ˜
(S2, I2)(t) = ( ˜
pwc,(S3, I3) = ( ˜
for ( ˜
Ik) : [1,1] R2
+. The, define scaling factors zk=zk(c, p) according to
2, k = 0
2, k = 2
2, k ∈ {1,3}.
If one completes the change of variables, then, dropping the tildes, we get the boundary-value problem
R[Sk] = L[Sk+1]
R[Ik] = L[Ik+1]
0 = IC(1 α)R[I1]α
0 = IR(1 α)R[I3]α
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 zk0. If I0and I1are monotone increasing with IRI0(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
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
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
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
for Tnthe nth Chebyshev polynomial of the first kind. The coefficients fncan be computed using the
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
is finite. If we write ωn= 1 for n= 0 and ωn= 2νnfor n1, we can write it more compactly in the
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
then the coefficients fnand f0
nare related by the equation
n+1, n 1.(8)
Also, for n2 the Chebyshev polynomials admit the indefinite integrals
ZTn(t)dt =1
n+ 1 Tn1(t)
The Chebyshev polynomials satisfy the identities Tn(1) = 1 and Tn(1) = (1)nfor all n0. Finally,
in the scope of differentiation of Chebyshev series, the tridiagonal operator T:`1
νdefined by
T(a)n=0, n = 0
an+1 an1, n > 0(10)
will be quite useful.
Let `1
νdenote the normed vector space of sequences {an:nN}bounded with respect to the || · ||ν
norm. This is a Banach space. If a, b `1
ν, define their convolution abaccording to
It is a standard exercise to check that (`1
ν,) is a Banach algebra; that is, :`1
νis a continuous
bilinear map with ||ab||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
ak,nTn(t) (11)
Ik(t) = bk,0+ 2
In the following, the symbols akand bkwill refer to the sequences {ak,n :nN}and {bk,n :nZ}. The
products of Chebyshev series induce convolutions at the level of their coefficients. We have
Sk(t)Ik(t)=(akbk)0+ 2
where (akbk)n:= (ak,·bk,·)n. Substituting (11) and (12) into (6), using the relations (8) and (9), and
making the identification a4a0and b4b0, we get the following set of equations.
bk,n =(TΨ1
k(ak, bk, c, p))n
k(ak, bk, c, p))n, n 1,(13)
bk,0+ 2
n=1 ak,n
bk,n =ak+1,0
bk+1,0+ 2
bk+1,n (14)
0 = IC(1 α) b1,0+ 2
0 = IR(1 α) b3,0+ 2
where Ψk:`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,
k(a, b, c, p)n
k(a, b, c, p)n=Λnβ(k)(ab)nµan
β(k)(ab)n(µ+γ)bnzk(c, p) := Φ1
k(a, b)n
k(a, b)nzk(c, p),(17)
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
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
νwith one-dimensional range by H(a, b)n= 0 for n > 0 and
H(u, v)0=u0v0+ 2
(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
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 Pn1b1,n )αb1,0Pn2b1,n 1+(1)n
IR(1 α)(b3,0+ 2 Pn1b3,n )αb3,0Pn2b3,n 1+(1)n
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
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.
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
νaccording to
πN(a)n=an, n N
0n > N,
Then, define a projection πN: (`1
πN(a0, b0, . . . , a3, b3, c, p) = (πNa0, πNb0, . . . , πNa3, πNb3)
and a complementary projector π: (`1
ν)8via π=I(`1
ν)8πN. Define also ˜πN:XXto 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) =
and extend this to an isomorphism iN:XNR8(N+1)+2 via
iN(a0, b0, . . . , a3, 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
N:XNXN, 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×RXand FN:XN×RXNby
FN(x, s) = ˜πNF(˜πNx, α), F N(x, s) = iNFN(i1
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 FNFpointwise, it should be expected that numerical zeroes of FN
will, when embedded in XN, generate approximate zeroes of F.
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 1010). 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 cwand pwc. 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 1015.
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 102until 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 1015. 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
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, λ1R. Define the predictors
xs= (1 s)x0+sx1, λs= (1 s)λ0+1.
Let FCk(X ×R,Y)for some k1and assume there exist bounded linear operators AB(X,Y)and
AB(Y,X), with Ainjective, and Y0,Z0,Z1and Z2(r)0such that
||AF (xs, λs)k|XY0,s[0,1] (22)
||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+Z01)r+Y0.(26)
If there exists r0>0such that p(r0)<0, then there exists a Ckfunction
˜x: [0,1] [
such that Fx(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 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 Afor 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 Aand 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.
S.4 cs> w and psw > csuniformly for s[0,1].
S.5 IR< I0(1; s)and I2(1; s)< ICuniformly for s[0,1].
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 θsXsatisfies ||xsθs||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.
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→∞ ||ΘnΦ(·;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 nm. Decomposing Θnfor nmaccording to Proposition 2 and applying
the change of variables (4), let θnXdenote 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+ (tt0)∆znΛ + ZZt
2βrun(s)2+ (2µ+γ)un(s)ds
for 1t0t1. 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
2βrun(ω(s))2+ (2µ+γ)un(ω(s))ds.
Since ∆un0 and ∆zn0 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 Xνand that (θnxs)0 in Xν. Since
xsXν, we must have θnXνfor 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.
3.6.1 Definition of the operators Aand A
Suppose x0XNis an approximate (numerical) zero of FN(·, α0). Denote x0=i1
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
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:XYby
NAiN˜πN+ diag(L, 0R2π.(27)
It is straightforward to verify that Ais well-defined.
To construct A:YX, we first define a linear map L+:`1
ν, with
L+(a)n=0, n = 0
2nan, n > 0.(28)
Then, extend it to a map L+: (`1
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 ||IAA|| ≈ 0. We can now define
A:YXas follows:
NAiN˜πN+ diag(L+,0R2π.(29)
The following lemma is now a straightforward consequence of the definitions of Aand A.
Lemma 7. Aand 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)i1
Nx1, αs= (1 s)α0+1
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:XNXN. We also set
so that xs=x0+sxand αs=α0+sα.
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)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
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 Pn1b1,n ) + q2b1,0Pn2b1,n 1+(1)n
(1 q1)(b3,0+ 2 Pn1b3,n ) + q2b3,0Pn2b3,n 1+(1)n
and write I= [ ICIR]|. The reason for introducing the parameters q1and q2will be apparent soon.
Define Ψ : X(`1
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
k(a, b, c, p)h=zk(c, p)β(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2 (35)
k(a, b, c, p)=Φj
k(a, b, c, p)zk(36)
k(a, b, c, p)[h1, h2]=(1)jzk(c, p)β(k)(u1
k) (37)
k(a, b, c, p)h=zkβ(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2 (38)
k(a, b, c, p)[h1, h2]=(1)jzkβ(k)(u1
while all partial derivatives with respect to (c, p) of order 2 and above are zero. The gradient zkcan be
computed directly from (5).
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) = i1
NAF N(xs, αs) + diag(L+,0R2)˜πF(xs, αs)≡ Y(1)
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
ADmFN(x0, α0)x
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 xsXNand the post-multiplication with
πkills all modes lower than N. The result is that in the norm on X,
0= max
k,j Wj||L+TΨj
k(as,k, bs,k , cs, ps)||νmax
k,j Wj
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 n2N+ 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](c1c0), p =p0+ [1,0](p1p0),
then using the computational isomorphisms and the above discussion,
ADmFN(x0, α0)x
+ max
k,j Wj
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
0≤ ||AF N(x0, α0)||XN+Z1
ADF N(x0+tsx, x0+tsα)sx
≤ ||AF N(x0, α0)||XN+ sup
ADF N(x0+zx, α0+zα)x
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 IAAis straightforward and
the following bound on the operator norm is tight:
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˜πNDxFπNx0, α0), we can carefully compute
DxF(x0, α0)A= ˜πNH+T DxΨ(x0)
G(·, α0)π+ ˜πT DxΨ(x0)
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] = i1
NAiN˜πNH+T DxΨ(x0)
G(·, α0)π+ ˜πL+T DxΨ(x0)
0R2:= Z(1)
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
k(a, b, c, p)h=zk(c, p)β(k)(akvk+ukbk)λjuk, j = 1
β(k)(akvk+ukbk)λjvk, j = 2
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
1. Let hXsatisfy ||h|| ≤ 1. Define for k= 0,...,3 and j= 1,2
1) + λjW1
j+||∇zk||1· ||Φj
Na0,k, i1
Nb0,k, c0, p0)||ν.
k,j Wj
n=N+1 T DxΨj
k(a0,k, b0,k , c0, p0)hn
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
2(N+ 1) max
k,j Wjβ(k)(||a0,k ||νW1
1) + λjW1
j+||∇zk||1· ||Φj
k(a0,k, b0,k , c0, p0)||ν
2(N+ 1) max
k,j Z(2,k,j)
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+ν1,
which can be inferred from Appendix A and the definition of the norm on X.
Next, we need a uniform (in ||h||X1) 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),
k(a, b, c, p)h=ˆ
k(a, b, c, p)h+rj
k(c, p)h
k(a, b, c, p)h= (1)jzk(c, p)β(k)(akvk+ukbk),
k(c, p)h=zk(c, p)µuk, j = 1
(µ+γ)vk, j = 2
ψ(a, b, c, p)h= ( ˆ
0(a0, b0, c, p)h, ˆ
0(a0, b0, c, p)h, . . . , ˆ
3(a3, b3, c, p)h, ˆ
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. (45)
For brevity, set ˜πh=h= (u
0, v
0, . . . , u
3, v
3,0,0). We can then decompose Z(1)
1further as
ψ(x0)h+T D(c,p)Ψ(x0)h
NAiN˜πNH(h) + πNT r(c0, p0)h
G(h, α0)
:= Z(1,1)
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,
ak(m) = max
2νn|(iNa0,k)nm|,bk(m) = max
ck(m) = |zk(c0, p0)β(k)|(W1
2ak(m) + W1
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
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×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
νN+1 W1i1
NH, πNT r(c0, p0)h
νN+1 W1i1
G(h, α0)1
νN+1 W1i1
where ξ= (µ+γ) maxk|zk(c0, p0)|/2. It follows that
νN+1 ||AW1H||XN+||AW1G||XN+ξ||AW1r||XN,
where we have abused notation and identified W1with iNW1i1
N. Combining the previous results, we
conclude that
Z1= (2ν+ν1)||abs(A)h||XN+2ν+ν1
2(N+ 1) max
k,j Z(2,k,j)
νN+1 ||AW1H||XN+||AW1G||XN+ξ||AW1r||XN
is a suitable Z1bound.
3.6.6 Computation of the Z2bound
Let δBr(0) X. Then
DxF(xs+δ, αs)DxF(x0, α0) = T(DxΨ(xs+δ)DxΨ(x0))
G(·, αs) + G(·, α0):= "TZ(1)
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 hXsatisfy ||h||X1, 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 ]|
k,j +ξ(2)
k,j zk[u v ]|.
We can bound each of these quantities in turn. Taking into account ||h||X1, ||δ||Xrand the weights
in the space X, we ultimately get |∇zk[u v ]|| ≤ W1
k,j ||νβ(k)h||akbk||ν+||a0,k bk+b0,k ak||ν+rW1
1(||b0,k||ν+||bk||ν)· ··
k,j (r),(47)
k,j ||ν≤ |zk(c0, p0)|β(k)W1
2(rW 1
1+||ak||ν) + W1
1(rW 1
+ abs(zk)[ c+W1
1r). . .
k,j (r).(48)
Observe that each of the ˆ
k,j (r) for q= 1,2 can be interpreted as degree two polynomials in r. As for
2hR2, the following bound is straightforward and its derivation is omitted: for hXwith ||h|| ≤ 1,
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 ||νX
m,` +W1
m,`) + A(1,2)
k,j W1
2(N+ 1) (ˆ
k,j +W1
k,j )
ˆµk,j (r)
|µ,j | ≤ X
m,` +W1
m,`) + A(2,2)
,j W1
2(N+ 1) W1
ˆµ,j (r)
Where the constants Aare given by
k,j,m,` = max
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)
j,m,` = max
,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
k,j,m,`hm,` +A(1,2)
k,j hR2
(Ah),j =X
j,m,`hm,` +A(2,2)
,j hR2,
with A(1,1)
k,j,m,` :`1
k,j :R2`1
ν, and A(2,1)
j,m,` :`1
,j :R2R, 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+ν1)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
pw > 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 vRmare indexed as v= (v0, . . . , vm1).
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
<(b0)0+ 2
(b0)n,(b2)0+ 2
(b2)n< ICr0
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
W2||Ik(t;s)Ik(t;s)|| ≤ r0.
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
for all t[1,1] and s[0,1]. Making use of the uniform inclusion
Ik(t;s)(bk)0+ 2
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,
3[r0/2 0 r00]|
Nbk||ν) + W1
+ (µ+γ)(||i1
ηk=|zk(c, p)| · β(k)(W1
2r0) + W1
2(µ+γ)r0+ (∆z)kφk,
where akbkR2(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
dtIk(t;s)zk(c, p)· ηk,0+ 2
Tn(t)ηk,n!+ [1,1]∆ηkˆ
Consequently, the sharpness condition ˆ
S.6 is satisfied if inftˆ
k(t)>0for k= 0,1and suptˆ
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
as,k,nTn(t), I k(t;s) = bs,k,0+ 2
Sk(t;s) = ˜as,k,0+ 2
˜as,k,n Tn(t), Ik(t;s) = ˜
bs,k,0+ 2
for ˜as,k = ˜a0,k +sa1,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
dtIk(t;s) = Ψ2
kas,k ,˜
bs,k, cs, ps)0+ 2
kas,k ,˜
bs,k, cs, ps)nTn(t),
dtIk(t;s) = Ψ2
k(as,k, bs,k , cs, ps)0+ 2
k(as,k, bs,k , cs, ps)nTn(t)ηk,0+ 2
Let ∆Ψk= Ψ2
kas,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=zkcs,˜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||νr0/W1,||bs,k||νr0/W2, and one can verify directly that
|zk| ≤ W1
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
ki= min
m=1,...,M inf
m=1,...,M sup
By construction, ˆ
k(t)⊆ hˆ
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
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
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
zk(c, p)Z1
1 bk,0+ 2
bk,nTn(t)!dt =2
zk(c, p) bk,0
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 xsx0and λsα0— that is, we
are only interested in a single zero of the map Fat a fixed parameter α. We let
[I] = 1
zkc, ˜p)Z1
1 ˜
bk,0+ 2
be the average across the theoretically guaranteed cycle Ifor which, from the radii polynomial method,
we have ||˜
bkbk||νr0/W2,|˜cc| ≤ r0/W3and |˜pp| ≤ 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
pzk(r0) + W1
3r0|zk(c, p)|
Nbk||ν) (56)
Proof. Set bk,n = 0 for n > N and define ∆bk,n =˜
bk,n bk,n , where (˜a0,˜
b3,˜c, ˜p) is the exact
zero of the map Ffor the relevant convex parameter α. With a bit of algebra, we get
zk(c, p)Z1
k=0 zkc, ˜p)
˜pzk(c, p)
bk,0+ ∆bk,0+
(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
zkc, ˜p)
˜pzk(c, p)
pp|zkc, ˜p)zk(c, p)|+|(˜pp)zk(c, p)|
p˜ppzk(r0) + W1
3r0|zk(c, p)|
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]r0W1
2+b3,0+ 2