Preemptive periodic epidemic control reduces life and healthcare system costs
without aggravation of social and economic losses

Giuseppe De Nicolao1*, Marta Colaneri2, Alessandro Di Filippo2, Franco
Blanchini3‡, Paolo Bolzern4‡, Patrizio Colaneri4‡, Giulia Giordano5‡, Raffaele
Bruno2,6‡

1 Department of Electrical, Computer and Biomedical Engineering, University of
Pavia, Pavia, Italy

2 Division of Infectious Diseases I, Fondazione IRCCS Policlinico San Matteo,
Pavia

3 Department of Mathematics, Computer Science and Physics, University of
Udine, Udine, Italy

4 Department of Electronics, Information and Bioengineering, Politecnico di
Milano, Milan, Italy

5 Department of Industrial Engineering, University of Trento, Trento, Italy

6 Department of Clinical, Surgical, Diagnostic, and Pediatric Sciences,
University of Pavia, Pavia, Italy

These authors contributed equally to this work.

‡These authors also contributed equally to this work.

*<EMAIL_ADDRESS>
## Abstract

Many countries are managing COVID-19 epidemic by switching between lighter and
heavier restrictions. While an open-close and a close-open cycle have
comparable socio-economic costs, the former leads to a much heavier burden in
terms of deaths and pressure on the healthcare system. An empirical
demonstration of the toll ensuing from procrastination was recently observed
in Israel, where both cycles were enforced from late August to mid-December
2020, yielding some 1,600 deaths with open-close compared to 440 with close-
open.

## Author summary

Two periodic epidemic control strategies are compared: (i) keep open and close
only when the situation becomes too serious, (ii) close as soon as possible,
and reopen only when the number of cases is low enough. We demonstrate that an
anticipated action drastically reduces deaths and healthcare system costs
without any significant increase in the other socio-economic costs.

## Introduction

Non Pharmacological Interventions (NPI) implemented to contain the spread of
SARS-CoV-2 are mainly concerned with the closure of production, commercial,
educational and recreational activities, together with travel bans or
restrictions. These strict regulations have been effective in slowing down the
rate of contagion and possibly anticipating the peak of infection cases [1],
[2]. It is widely agreed that early timing in adopting restrictions is crucial
[3]. However, due to the far-reaching social and economic consequences, the
governments worldwide are facing an increasing pressure to ease restrictions.

In our analysis, we distinguish two broad categories of costs: health and
human life costs of the epidemic on the one hand and socio-economic costs of
NPIs on the other hand. The former ones include the impact of Covid-19 on
deaths and the health status of the population and the healthcare system
financial costs incurred to cure infected people. Examples of socio-economic
cost of NPIs are the financial losses due to the restrictions imposed on
industrial, commercial, and recreational activities, as well as the social
costs due to the reduced effectiveness of distance education and the general
deterioration of the quality of life.

Different policy responses are being taken by European governments to curb the
spread of the virus [4]. Although with varying intensity, a common strategy in
many countries has been to enforce harsher restrictions once the number of
infected individuals rises to an alarming level, followed by a release when
the number of infected has decreased and become manageable with the available
public health resources [5]. The closure trigger has been generally identified
with the strain on healthcare systems, particularly intensive care units
(ICUs). However, this widely-accepted strategy of closing only when
approaching saturation of the ICU capacity may not be the most appropriate.

Previous observational and modeling studies have suggested intermittent
distancing measures to avoid critical care capacity overload in the absence of
effective therapies and sustainable preventive measures for SARS-CoV-2
infection [6], [7],[8],[9]. Here, we consider a lock/release cycle driving the
system to the same initial infection condition, possibly periodically
repeated. Not only we show that the first-close-then-open approach has the
potential to dramaticaly reduce fatality and ICU stress with respect to the
miopic first-open-then-close strategy, but we also show that this comes at no
additional socio-economic cost.

## Results

### The average principle and its consequences on socio-economic costs

An ICU-capacity approach can be modelled as an _open-close_ (OC) scheme: the
reproduction number $R_{t}$ remains equal to its maximal value $r_{o}>1$
during a first _open phase_ of length $T_{o}$, followed by a _close phase_ of
length $T_{c}$, characterized by $R_{t}$ constant and equal to its minimal
value $r_{c}<1$. For the control strategy to be periodic, at the end of the
whole cycle, lasting $T=T_{o}+T_{c}$ days, the number of active cases returns
equal to the number $I_{0}$, observed at day zero.

By the Average principle (see Methods) periodicity is achieved only if the
reproduction number averages to one during the cycle, i.e. if

$\frac{r_{o}T_{o}+r_{c}T_{c}}{T}=1.$

When the average principle is satisfied, the periodic application of the OC
strategy yields a periodic profile of the active cases that returns to $I_{0}$
every $T$ days.

By just exchanging the open and close phases, another admissible periodic
control strategy is obtained, named _close-open_ (CO). In particular, the
reproduction number $R_{t}$ remains equal to $r_{c}<1$ during a first _close
phase_ of length $T_{c}$, followed by an _open phase_ of length $T_{o}$,
characterized by $R_{t}$ constant and equal to $r_{o}>1$.

According to the basic SIR model [10],

$R_{t}=\frac{\beta(t)}{\gamma}\frac{S(t)}{N}$

where $1/\gamma$ is the average duration of infectiousness, $\beta(t)$ is the
transmission parameter at time $t$, and $S(t)/N$ is the fraction of
susceptible subjects $S(t)$ within the overall population of size $N$,
consisting of susceptible, infected, and removed subjects. The transmission
parameter $\beta(t)$ depends on biological, epidemiological, environmental and
social factors and can be decreased by NPIs that reduce the rate at which
individuals have close contact with other individuals. Therefore, a lower
$\beta$ corresponds to larger socio-economic costs. We assume that during a
single period of the periodic control scheme, $S(t)/N$ does not undergo
significant changes. This assumption is particularly plausible when letting
the infection spread on a large scale would collapse the healthcare system.
Then, $\beta(t)$ is approximately proportional to $R_{t}$, whose value
essentially depends on the intensity of the NPIs. In particular, more
restrictive measures lead to a decrease in $R_{t}$.

The proportionality between $R_{t}$ and $\beta(t)$ implies that the socio-
economic costs of the OC and CO strategies are equivalent. In fact, in both
cycles the time lengths of the open and close phases are identical, the only
difference being the order. Possible differences may be due to seasonal
economic activities, e.g. related to tourism, that are sensitive to
translations of the bans. Nevertheless, for most activities and services,
economic and social losses depend mainly on the duration and stringency of the
restrictions enforced to keep $R_{t}$ within the desired limits. Having
established the socio-economic equivalence of the two strategies, now it
remains to compare their costs in health and human lives.

### Periodic control strategies compared: open-close vs close-open

A measure of the life and healthcare system costs of a periodic control
strategy is given by deaths, hospital and ICU occupancies. In turn, these are
proportional to the total number of new cases during one cycle, which equals
the area under the curve (AUC) of the daily new cases during one cycle, see
Methods. Under periodic control, this AUC is proportional to the AUC of the
active cases that can therefore be used as a measure of the healthcare system
costs (see Life and healthcare system costs in Methods). In particular, it
offers a visually intuitive comparison of the costs of alternative strategies,
see Fig. 1, where the AUCs are displayed of specular OC and CO cycles having
closure and opening sub-intervals of the same length, but with the order
exchanged.

Fig 1: Open-close (OC) vs close-open (CO). The AUCs are proportional to the
life and healthcare system costs. The ratio of OC’s cost to CO’s one is equal
to $I_{MAX}/I_{0}$.

As seen in Fig. 1, the life and healthcare system costs of the two strategies
are dramatically different because returning to the initial number of active
cases through the upper route, i.e. with an exponential growth followed by an
exponential decrease, yields a much greater AUC than following the lower
route, consisting of an exponential decrease followed by an exponential
growth. This kind of plot offers also a visually intuitive guideline to the
decision maker dealing with periodic epidemic control. Indeed, it is of
immediate understanding that to minimize its AUC you must keep the curve of
active cases as low as you can, consistent with the constraint of returning to
the starting value

As shown in the Methods (Cost of open-close vs close-open), the ratio between
the two AUCs obeys the formula

$\frac{\mathrm{AUC}_{OC}}{\mathrm{AUC}_{CO}}=K=\frac{I_{max}}{I_{0}},$

where $I_{max}$ denotes the maximum number of active cases reached during the
OC cycle. For instance, if the number of active cases quadruples at the top of
the open phase of OC, the associated life and healthcare system cost will be
four times larger than the cost of the preemptive CO scheme.

The comparison of open-close and close-open control strategies is further
illustrated in Panel A of Figure 2, displaying the simulated profile of active
cases under an open-close cycle followed by a close-open one, each of 54 days.
Both cycles consist of an open phase of duration $T_{o}=31$ days and a close
phase of duration $T_{c}=23$ days, but in reverse order.

The profile of the active cases follows Equation (8) during the OC cycle and
Equation (10)during the CO one, with $\alpha=0.0410$ day-1 and $\beta=0.0553$
day-1. Under these assumptions, at the end of each cycle, the active cases
return to the initial value $I_{0}=21,000$, see points P, Q, and R. By
Properties 5 and 6, the predicted life and healthcare system costs of the two
periodic strategies are proportional to the red and blue areas. The maximum
number of active cases reached during the OC cycle is
$I_{\mathrm{max}}=75,000$, so that $I_{\mathrm{MAX}}/I_{0}=3.6$. Therefore, in
this simulation the cost of the OC strategy is $3.6$ times larger than that of
the CO strategy.

Fig 2: Validation on Israel data. Periodic control strategies compared: in
accordance with model predictions, a more than 3-fold death toll was observed
in Israel after an open-close cycle followed by a close-open one.

### Validation on Israel data

The cost comparison presented in the previous section was empirically
validated on Israel data retrieved from the COVID-19 Data Repository by the
Center for Systems Science and Engineering at Johns Hopkins University [11].

During the 108-day period from August 30 to December 16, Israel’s active cases
(see panel B in Fig. 2) evolved very much like the simulation in panel A. In
view of the similarity of Israel’s active cases profile with a sequence of an
ideal OC cycle followed by a CO one, the actual deaths can be used to validate
our prediction (12) of the ratio between the OC and CO costs.

The first step is obtaining the total number of new cases occurring within the
two cycles. For this purpose, it suffices to intersect the starting and ending
dates of the cycles with the cumulative total cases, see panel C in Fig.2. It
is found that there were some 190,000 cases during the OC cycle and 52,000
during the CO one

The final step is to assess the life costs by multiplying the cases by the
estimated $\hat{C}FR=0.0085$, see Estimation of Israel’s CFR in Methods. By
properly scaling the axis of total deaths, this last conversion is visualized
in Panel D of Fig. 2. It turns out that the OC and CO cycles caused some 1,600
and 440 deaths, respectively. The ratio is 3.7, thus validating the formula
(12) that predicted a 3.6-fold cost of the open-close strategy with respect to
the close-open one.

The costs of hospital and ICU occupancy associated with the two strategies
could be assessed in a similar way. Indeed, for a given ratio of infected
subjects requiring hospitalization and ICU admission, these costs are again
proportional to the new cases and hence to the AUC of the active cases.

It should be noted that even when the use of Equation (12) is precluded
because changes in nonpharmaceutical interventions are too gradual to be
approximated via ideal OC and CO cycles, the cost assessment based on the AUCs
remains valid and is itself sufficient to show the advantage of preemptive
strategies over procrastinating ones.

## Discussion and Conclusion

We compared two periodic epidemic control strategies: an open-close strategy
(keep open and only close when the situation becomes too serious) and a close-
open one (close as soon as possible, and reopen only when the number of cases
is low enough). The duration of the closure and opening periods being equal in
both cases, the former strategy does not bring any socio-economic benefit,
while the latter saves lives and reduces healthcare system costs.

An immediate and simple quantitative assessment of the cost of procrastinating
intervention is given by the ratio $K$ between the maximum number of active
cases during the open-close cycle and the initial number of active cases: the
life and healthcare system costs of the procrastinating strategy are $K$ times
larger than those of the preemptive one. In view of our analysis, it appears
that an anticipated action is probably the most effective tool available to
decision makers to drastically cut the life and healthcare system costs of the
epidemic without a significant increase in the other socio-economic costs.

It is also interesting to note that a constant control strategy enforcing
$R_{t}=1$, though preferable to the OC one, is outperformed by CO periodic
control. In fact, it is reasonable to think that the overall socio-economic
costs would depend on the average $R_{t}$, being therefore equal to the ones
of the OC and CO strategies. On the other hand the health and human life costs
of the constant strategy, measured by the AUC of the active cases, are clearly
heavier than those of the CO strategy. Therefore, the epidemic is better
controlled by a lock/release cycle rather than by a steady containment.

Our analysis assumes that the ratio $S(t)/N$ does not change significantly
during a control period. If we consider its change, $R_{t}$, besides depending
through $\beta$ on the stringency of the restrictions, is also proportional to
the declining ratio of susceptible individuals. This means that, during the
close phase of the OC strategy, $R_{t}=r_{c}$ can be enforced with NPIs that
are less restrictive than those required to ensure $R_{t}=r_{c}$ during the
close phase of the CO strategy, meaning that a periodic $R_{t}$ is no more
associated with periodic NPIs. This observation, however, does not affect the
validity of our conclusions for two reasons. Before the start of vaccination,
the susceptible ratio cannot be allowed to grow quickly unless health
structures are overwhelmed. Second, the gap between the health and human life
costs of OC and CO is so large to be robust in the face of this approximation.

Perhaps surprisingly, the socio-economic costs of the periodic strategies
depend on the intensity and duration of NPIs, but are not affected by the
initial number of infected individuals. This means that, once it has been
decided to stabilize the epidemic by applying a periodic control, the socio-
economic costs are the same whether we start with 100 or 100,000 infected
people. In the latter case, however, the health and human life costs are 1,000
times larger: this constitutes a formidable argument in favor of a drastic
abatement of the prevalence followed by a periodic CO strategy until mass
vaccination finally makes it possible to decrease $R_{t}$ without having to
prolong stringent NPIs.

Our analysis does not accounts the effects of vaccination. Future work should
address the quantitative assessment of the potential advantages of preemption
during vaccination rollout. Although the analysis could somehow follow the
tracks of the present paper, it is expected to be more complex, because the
faster decline of the rate susceptible individuals prevents a simple use of a
periodic paradigm.

## Methods

### Notation

$t\in[0,T]$: time (days)

$I(t)$: stock of infected subjects at time $t$ also called _active cases_.

$n(t)$: daily rate of new infected cases at time $t$.

$\gamma$: removal rate of infected subjects.

$R_{t}$: reproduction number at time $t$.

$S_{R}(t)=\int_{0}^{t}R_{\tau}d\tau$: Area Under the Curve (AUC) of the
reproduction number.

$r_{c}<1$ minimal $R_{t}$ achieved with stronger nonpharmaceutical
interventions.

$r_{o}>1$: maximal value of $R_{t}$, achieved with milder nonpharmaceutical
interventions.

_Open-close (OC) cycle_ : the adoption of nonpharmaceutical interventions such
that

$R_{t}=r_{o},\quad 0\leq t<T_{o},\quad R_{t}=r_{c},\quad T_{o}\leq
t<T_{o}+T_{c}$

_Close-open (CO) cycle_ : the adoption of nonpharmaceutical interventions such
that

$R_{t}=r_{c},\quad 0\leq t<T_{c},\quad R_{t}=r_{o},\quad T_{c}\leq
t<T_{c}+T_{o}$

### Balance equation

The stock of infected is described as a single compartment obeying the balance
equation

$\dot{I}(t)=-\gamma I(t)+n(t),\qquad I(0)=I_{0}$ (1)

Assuming that $n(t)=R_{t}\gamma I(t)$, it follows that

$\dot{I}(t)=(R_{t}-1)\gamma I(t),\qquad I(0)=I_{0}$ (2)

whose solution is

$I(t)=I_{0}e^{\gamma\int_{0}^{t}(R_{\tau}-1)d\tau}=I_{0}e^{\gamma(S_{R}(t)-t)}$
(3)

### Periodic epidemic control: average principle

######  Definition 1.

$R_{t}$ is said to achieve periodic epidemic control of period $T$ if
$I(T)=I_{0}$.

In view of (3), $I(T)=I_{0}$ if and only if $S_{R}(T)=T$, so that the
following property holds.

######  Property 1.

If $R_{t}$ achieves periodic control of period $T$, then it satisfies the
_average principle_

$\frac{S_{R}(T)}{T}=1$

i.e. the average reproduction number $R_{t}$ in the interval $[0,T]$ is equal
to one.

Letting $T=T_{o}+T_{c}$, for both the OC and the CO cycle it holds that

$S_{R}(T)=r_{o}T_{o}+r_{c}T_{c}$ (4)

Therefore, the average principle implies that a cycle (either OC or CO)
achieves periodic control of period $T=T_{o}+T_{c}$ if

$\frac{r_{o}T_{o}+r_{c}T_{c}}{T}=1$

This means that, under periodic control with a given period $T$, the lengths
$T_{o}$ and $T_{c}$ of the open and close phases depend only on $r_{o}$ and
$r_{c}$, regardless of the order in which they are applied. In particular,

$T_{o}=\frac{1-r_{c}}{r_{o}-r_{c}}T,\quad T_{c}=\frac{r_{o}-1}{r_{o}-r_{c}}T$

### Life and healthcare system costs

The life and healthcare system costs are obviously proportional to the total
number of cases, i.e. to the AUC of $n(t)$ over the considered time interval
$T$. In this section, it is shown that under $T$-periodic control they are
also proportional to the AUC of the active cases $I(t)$ over the period $T$.
This property will be instrumental in obtaining an immediate comparison of OC
vs CO in terms of the maximum number of active cases reached during the OC
cycle.

Let $a(t)$ denote the solution of (1) with $I_{0}=0$ and define

$\mathrm{AUC}_{f}(t_{1},t_{2})=\int_{t_{1}}^{t_{2}}f(t)dt,\quad\mathrm{AUC}_{f}=\int_{0}^{\infty}f(t)dt$

######  Property 2.

$\mathrm{AUC}_{n}=\gamma\mathrm{AUC}_{a}$.

###### Proof.

In view of (1),

$a(t)=\int_{0}^{t}e^{-\gamma(t-\tau)}n(\tau)d\tau$

Then,

$\displaystyle\mathrm{AUC}_{a}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{0}^{t}e^{-\gamma(t-\tau)}n(\tau)d\tau
dt=\int_{0}^{\infty}n(\tau)\int_{\tau}^{\infty}e^{-\gamma(t-\tau)}dt\,d\tau$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}n(\tau)\int_{0}^{\infty}e^{-\gamma
s}ds\,d\tau=\mathrm{AUC}_{n}/\gamma$

∎

######  Property 3.

$\mathrm{AUC}_{n}(0,T)=\gamma\mathrm{AUC}_{a}(0,T)+a(T)$.

###### Proof.

Define

$\tilde{n}(t)=\left\\{\begin{array}[]{cc}n(t),&0\leq T\\\
0&t>T\end{array}\right.$

and let $\tilde{a}(t),t\geq 0$ be the solution of (1) with $I_{0}=0$ and
$n(t)=\tilde{n}(t)$. Then,

$\mathrm{AUC}_{n}(0,T)=\mathrm{AUC}_{\tilde{n}}=\gamma\mathrm{AUC}_{\tilde{a}}=\gamma\mathrm{AUC}_{a}(0,T)+\gamma\mathrm{AUC}_{\tilde{a}}(T,\infty)$

Observing that $\tilde{a}(t)=a(T)e^{-\gamma t},t>T$,

$\mathrm{AUC}_{\tilde{a}}(T,\infty)=a(T)\int_{T}^{\infty}e^{-\gamma
t}dt=\frac{a(T)}{\gamma}$

and the thesis follows. ∎

######  Property 4.

For a given $I_{0}$, let $n(t)$ be such that $I(T)=I_{0}$. Then,

$\mathrm{AUC}_{n}(0,T)=\gamma\mathrm{AUC}_{I}(0,T)$

###### Proof.

Since $I(t)=e^{-\gamma t}I_{0}+a(t)$, the periodicity constraint $I(T)=I_{0}$
implies that

$a(T)=(1-e^{-\gamma T})I_{0}$

Then,

$\displaystyle\mathrm{AUC}_{I}(0,T)$ $\displaystyle=$ $\displaystyle
I_{0}\int_{0}^{T}e^{-\gamma t}dt+\mathrm{AUC}_{a}(0,T)$ $\displaystyle=$
$\displaystyle\frac{I_{0}}{\gamma}(1-e^{-\gamma T})+\mathrm{AUC}_{a}(0,T)$
$\displaystyle=$ $\displaystyle\frac{a(T)}{\gamma}+\mathrm{AUC}_{a}(0,T)$
$\displaystyle=$ $\displaystyle\frac{\mathrm{AUC}_{n}(0,T)}{\gamma}$

thus proving the thesis. ∎

Since the life and healthcare system costs of different periodic control
strategies are proportional to the AUCs of the active cases,
$C=\mathrm{AUC}_{I}(0,T)$ can be used as a measure of such costs.

### Cost of open-close vs close-open

Let $\alpha=\gamma(r_{o}-1)$ and $\beta=\gamma(1-r_{c})$. Moreover, we denote
by $I_{\mathrm{MAX}}$ the maximum number of active cases reached at the end of
the open phase of the OC cycle:

$I_{\mathrm{MAX}}=I_{0}e^{\alpha T_{o}}$ (5)

######  Property 5.

The cost $C_{oc}$ of the periodic OC strategy is

$C_{oc}=\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(e^{\alpha
T_{o}}-1\right)I_{0}=\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(I_{\mathrm{MAX}}-I_{0}\right)$
(6)

where

$T_{o}=\frac{\beta T}{\alpha+\beta}$ (7)

###### Proof.

From (4) it follows that

$\alpha T_{o}-\beta T_{c}=0$

which in turn implies

$T_{o}=\frac{\beta T}{\alpha+\beta}$

From (2) it follows that

$I(t)=\left\\{\begin{array}[]{cc}I_{0}e^{\alpha t},&0\leq t\leq T_{o}\\\
I_{0}e^{\alpha T_{o}}e^{-\beta(t-T_{o})},&T_{o}\leq t\leq T\end{array}\right.$
(8)

Therefore, observing that $\exp(\alpha T_{o})\exp(-\beta(T-T_{o}))=1$,

$\displaystyle C_{oc}$ $\displaystyle=$
$\displaystyle\mathrm{AUC}_{I}(0,T)=\int_{0}^{T_{o}}I_{0}e^{\alpha
t}dt+\int_{T_{o}}^{T}I_{0}e^{\alpha T_{o}}e^{-\beta(t-T_{o})}dt$
$\displaystyle=$ $\displaystyle\frac{I_{0}}{\alpha}\left(e^{\alpha
T_{o}}-1\right)+\frac{I_{0}e^{\alpha
T_{o}}}{\beta}\left(1-e^{\beta(T-T_{o})}\right)$ $\displaystyle=$
$\displaystyle
I_{0}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(e^{\alpha
T_{o}}-1\right)$

and the thesis follows. ∎

######  Property 6.

The cost $C_{co}$ of the periodic CO strategy is

$C_{co}=\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(1-e^{-\alpha
T_{o}}\right)I_{0}=e^{-\alpha
T_{o}}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(I_{\mathrm{MAX}}-I_{0}\right)$
(9)

###### Proof.

From (2) it follows that

$I(t)=\left\\{\begin{array}[]{cc}I_{0}e^{-\beta t},&0\leq t\leq T_{c}\\\
I_{0}e^{-\beta T_{c}}e^{\alpha(t-T_{c})},&T_{c}\leq t\leq T\end{array}\right.$
(10)

Therefore, observing that $\exp(-\beta T_{c})\exp(\alpha(T-T_{c}))=1$,

$\displaystyle C_{co}$ $\displaystyle=$
$\displaystyle\mathrm{AUC}_{I}(0,T)=\int_{0}^{T_{c}}I_{0}e^{-\beta
t}dt+\int_{T_{c}}^{T}I_{0}e^{-\beta T_{c}}e^{\alpha(t-T_{c})}dt$
$\displaystyle=$ $\displaystyle\frac{I_{0}}{\beta}\left(1-e^{-\beta
T_{c}}\right)+\frac{I_{0}e^{-\beta
T_{c}}}{\alpha}\left(e^{\alpha(T-T_{c})}-1\right)$ $\displaystyle=$
$\displaystyle
I_{0}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(1-e^{-\beta
T_{c}}\right)$ $\displaystyle=$ $\displaystyle
e^{-\alpha(T-T_{c})}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(e^{\alpha(T-T_{c})}-1\right)I_{0}$
$\displaystyle=$ $\displaystyle e^{-\alpha
T_{o}}\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)\left(e^{\alpha
T_{o}}I_{0}-I_{0}\right)$

and the thesis follows. ∎

######  Property 7.

The cost $C_{co}$ of the CO strategy is less then the cost
$C_{\mathrm{const}}$ of a constant control strategy maintaining
$I(t)=I_{0},0\leq t\leq T$:

$C_{co}<C_{\mathrm{const}}=I_{0}T$

###### Proof.

Let

$\rho=\left(\frac{1}{\beta}+\frac{1}{\alpha}\right)^{-1}$

From (7) it follows that $\alpha T_{o}=\rho T$. Equation (9) can be rewritten
as

$C_{co}=\frac{I_{0}}{\rho}\left(1-e^{-\alpha
T_{o}}\right)=\frac{I_{0}}{\rho}\left(1-e^{-\rho T}\right)$ (11)

Then, letting $y=\rho T$, we have that

$\frac{C_{co}}{C_{\mathrm{const}}}=\frac{1-e^{-y}}{y}$

The right-hand side is always strictly less than one, $\forall y>0$, thus
proving the thesis. ∎

######  Property 8.

The ratio between the costs of the periodic open-close and close-open
strategies is equal to the maximum growth ratio of the active cases during the
open-close cycle:

$\frac{C_{oc}}{C_{co}}=\frac{I_{\mathrm{MAX}}}{I_{0}}$ (12)

###### Proof.

In view of (6) and (9), the proof follows directly from (5). ∎

### Israel data

The data were downloaded from the COVID-19 Data Repository by the Center for
Systems Science and Engineering at Johns Hopkins University [11]. The
downloaded time series included: deaths, confirmed cases, and recovered ones.
From these cumulative data, daily deaths and new cases, displayed in Fig. 3,
were obtained by differencing consecutive data. Active cases were computed by
subtracting deaths and recovered cases from the confirmed ones.

Fig 3: Israel data: time series of new cases (top) and daily deaths (bottom).

By inspecting the active cases, see Fig. 4, it is seen that on August 30 they
were equal to 20,876. Starting from that date, they had an upswing, reaching
71,114 on October 3, when a downswing started, leading to a nadir of 8,697
cases on November 16. On December 16, the active cases had risen to 20,791,
close to those of August 30. In other words, after an upper arc and a lower
one, each during about 54 days, the active cases returned to their initial
value.

Taking into account a physiological delay between interventions and published
data, the time course of the active cases can be associated to the
nonpharmaceutical interventions. Schools reopened on September 1, following
the Education Ministry’s Safe Learning Plan that provided for smaller class
sizes in the younger grades and at least two days in school from grades 5
through 12. Due to the growth of the cases, the Government resolved to impose
a total national lockdown effective from Friday September 25 for the duration
of 14 days, reserving the option to extend it as it actually happened. In the
first phase of exit, which came into effect on 18 October, beaches, nature
reserves and parks in Israel were reopened, as were commercial activities
without public reception and take-away services. Starting from November 1, the
lockdown was gradually eased. For instance, pupils of grades 1-4 were
readmitted to school and on November 8 shops reopened with 4-customers only
limit. Relaxation of restrictions in most of the country was followed by a
growth of the cases that has provoked a new national lockdown to begin on 27
December 2020.

Fig 4: Israel data: active cases. Circles, spaced 54 days apart on the time
axis starting from August 30, are placed in correspondence of 21,000 cases.

### Estimation of Israel’s CFR

In order to assess the life cost of new cases, an estimate of the CFR (Case
Fatality Rate) is needed. This parameter is affected by the testing protocol
and healthcare system reaction. Therefore, it was estimated using only data
posterior to June 1, 2020, so as to discard first-wave data.

It is assumed that the daily deaths $d(t)$ depend on the previous daily new
cases $n(t)$ according to the equation

$d(t)=\sum_{i=0}^{\infty}w(i)n(t-i)$ (13)

where the weight $w(i)$ denotes the fraction of infected subjects at time
$t-i$ that die at time $t$. The Case Fatality Rate is then given by

$\mathrm{CFR}=\sum_{i=0}^{\infty}w(i)$

In order to estimate the weights, a simple exponential model with delay was
assumed:

$w(i)=\left\\{\begin{array}[]{cc}0,&i<k\\\ ba^{i-k},&i\geq
k\end{array}\right.$

where $k$ is the delay and $a$ and $b$ are unknown parameters.

Fig 5: Daily deaths predicted by weighted average of new cases, with delay
equal to 3 and exponentially decaying weights.

As seen in Fig. 3, both the new cases and the daily deaths exhibit an apparent
weekly seasonality. In the estimation procedure, the original series were
therefore replaced by their 7-day moving average (red lines in Fig. 3).

For given values of $a,b,k$, Equation (13) can be used to predict daily
deaths.

Israel data from June 1 to December 29 were used to estimate the parameters
via least squares using the function oe.m of the MATLAB System Identification
Toolbox, version R202b. The estimation of $a$ and $b$ was repeated for all
delays $k$ ranging from $0$ to $15$ and the delay $k=3$ associated with the
best sum of squares was eventually selected. In spite of the simplicity of the
exponential-with-delay model, the time series of deaths is explained in a very
satisfactory way, see Fig. 5.

The estimated parameters and their percent coefficients of variation were:

$\left.\begin{array}[]{lll}\hat{a}=&0.943,&CV(\%)=0.34\\\
\hat{b}=&0.000485,&CV(\%)=5.17\end{array}\right.$

Then, recalling the formula for the sum of the harmonic series,

$\hat{C}FR=\sum_{i=0}^{\infty}\hat{w}(i)=\frac{\hat{b}}{1-\hat{a}}=0.0085$

The estimated $\hat{C}FR$ is in good agreement with the value 0.008, currently
(30.12.2020) reported in the COVID-19 Data Repository at Johns Hopkins
University [11].

## Acknowledgments

This work has been partially supported by the Italian Ministry for Research in
the framework of the 2017 Program for Research Projects of National Interest
(PRIN), Grant no. 2017YKXYXJ.

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