Optimal, Near-Optimal, and Robust Epidemic Control

Optimal, near-optimal, and robust epidemic control

Dylan H. Morris1,3*, Fernando W. Rossine1*, Joshua B. Plotkin2, and Simon A. Levin1

1Department of Ecology & Evolutionary Biology, Princeton University, 106A Guyot Hall, Princeton, NJ 08544, USA 2Department of Biology & Department of Mathematics, The University of Pennsylvania, 433 S University Ave, Philadelphia, PA 19104, USA 3Present address: Department of Ecology & Evolutionary Biology, University of California Los Angeles, Terasaki Life Sciences Building, 610 Charles E. Young Dr South, Los Angeles, CA 90095, USA *these authors contributed equally; correspondence to [email protected], [email protected].

Abstract

In the absence of drugs and vaccines, policymakers use non-pharmaceutical interventions such as social distancing to decrease rates of disease-causing contact, with the aim of reducing or delaying the epidemic peak. These measures carry social and economic costs, so societies may be unable to maintain them for more than a short period of time. Intervention policy design often relies on numerical simulations of epidemic models, but comparing policies and assessing their robustness demands clear principles that apply across strategies. Here we derive the theoretically optimal strategy for using a time-limited intervention to reduce the peak prevalence of

arXiv:2004.02209v3 [q-bio.PE] 3 Mar 2021 a novel disease in the classic Susceptible-Infectious-Recovered epidemic model. We show that broad classes of easier-to-implement strategies can perform nearly as well as the theoretically optimal strategy. But neither the optimal strategy nor any of these near-optimal strategies is robust to implementation error: small errors in timing the intervention produce large increases in peak prevalence. Our results reveal fundamental principles of non-pharmaceutical disease control and expose their potential fragility. For robust control, an intervention must be strong, early, and ideally sustained.

1 Introduction

New human pathogens routinely emerge via zoonotic spillover from other species. Examples include ebolaviruses1, inﬂuenza viruses2, and most recently the sarbecoronavirus SARS-CoV- 23. Many of these emerging pathogens are antigenically novel: the human population possesses little or no preexisting immunity4. This not only can increase disease severity5 but also leads to explosive epidemic spread4. If left unchecked, that explosive spread can result in a large proportion of the population becoming synchronously infected, as occurred during the COVID- 19 global pandemic. Policymakers initially have limited tools for controlling a novel pathogen epidemic; it can take months to develop drugs, and years to develop and distribute vaccines6. If disease symptoms are severe, healthcare systems may be strained to the breaking point as the epidemic approaches its peak7.

In the absence of drugs and vaccines, mitigation efforts to reduce or delay the peak (ﬂattening the curve7,8) rely on non-pharmaceutical interventions9 such as social distancing10 that decrease rates of disease-transmitting contact. These measures carry social and economic costs, and so societies may be unable to maintain them for more than a short period of time.

Policy design for allocating non-pharmaceutical resources during the COVID-19 pandemic re- lied heavily on numerical simulations of epidemic models10,11, but it is difﬁcult to compare pre- dictions or assess robustness without broad principles that apply across strategies. Since models are not reality, robust model-based policy requires not only generating a desired outcome but also understanding what elements of the model are producing it. This typically requires analyt- ical and theoretical understanding.

Relatively little is known about globally optimal strategies for epidemic control, regardless of principal aim12,13 or intervention duration. One result establishes that time-limited interventions to reduce peak prevalence should start earlier than time-limited interventions to reduce the ﬁnal size14. A number of studies of COVID-19 have used optimal control theory—an approach that relies on numerical optimization to study continuous error-correction15,16.

But the optimal time-limited strategy to reduce peak prevalence is not known. Without an ana- lytical understanding of epidemic peak reduction, policy design based on numerical simulation may fail in unexpected ways; policies may be inefﬁcient, non-robust, or both.

The early months of the COVID-19 pandemic demonstrated how difﬁcult real-time epidemio-

2 logical modeling, inference, and response can be. Large numbers of asymptomatic and mildly symptomatic cases17, as well as difﬁculties with testing, particularly in the United States18, left policymakers with substantial uncertainty regarding the virus’s epidemiological parameters and the case numbers in many locations. Countries including the United States and United Kingdom waited to intervene until transmission was widespread; retrospective analyses later claimed that even slightly earlier intervention would have saved many lives19,20.

Epidemiological uncertainties mean that no policymaker can intervene at precisely the optimal time. To understand how this limitation can hinder policymaking, we assess the robustness of optimized interventions to timing error: what is the cost of intervening too early or too late?

We derive the theoretically optimal strategy for using a time-limited intervention to reduce the peak prevalence of a novel disease in the classic Susceptible-Infectious-Recovered (SIR) epidemic model21,22. We show that broad classes of strategies that are easier to implement can perform nearly as well as this theoretically optimal strategy. But we show that neither the theoretically optimal strategy nor any of these near-optimal strategies is robust to implementation error: small errors in timing of the intervention produce large increases in peak prevalence. To prevent disastrous outcomes due to imperfect implementation, a strong, early, and ideally sustained non-pharmaceutical response is required.

Results

Epidemic model and interventions

We consider the standard susceptible-infectious-recovered (SIR) epidemic model21, which de- scribes the fractions of susceptible S(t), infectious I(t), and recovered R(t) individuals in the population at time t22. New infections occur proportional to S(t)I(t) at a rate β, and infectious β individuals recover at a rate γ. The model has a basic reproduction number R0 = γ and an β max effective reproduction number Re = γ S(t). We denote the peak prevalence by I . We consider interventions that reduce the effective rate of disease-transmitting contacts, β(t) below its value in the absence of intervention β, which we treat as ﬁxed. In the context of COVID-19, this includes non-pharmaceutical interventions such as social distancing10, as well as some pharmaceutical ones, such as antivirals that might reduce virus shedding.

3 We permit interventions to operate on β only for a limited duration of time, τ. We impose this constraint in light of political, social, and economic impediments to maintaining aggressive intervention indeﬁnitely. That is, we treat costs of intervention implicitly, subsuming them in τ. We describe such an intervention by deﬁning a transmission reduction function b(t) such that:

dS = b(t) βSI dt − ∗ dI = b(t) βSI γI (1) dt ∗ − dR = γI dt

If the intervention is initiated at some time t = ti, it must stop at time t = ti + τ. So necessarily b(t) = 1 if t < ti or t > ti + τ. During the intervention (i.e. when ti t ti + τ), b(t) is an ≤ ≤ arbitrary function, possibly discontinuous, with range in [0,1]. This restriction assumes that we cannot intervene increase transmission above what occurs in the absence of intervention, but also optimistically assumes that b(t) can be adjusted instantaneously, and that the effective reproduction number Re can be reduced all the way to zero, at least for a limited time.

The optimal intervention

We pose the following optimization problem: given the epidemiological parameters R0 and γ and the ﬁnite duration τ, what is the optimal intervention b(t) that minimizes the epidemic peak Imax? The optimal intervention is of interest for two reasons. It provides a reference point for evaluating alternative interventions. And it will allow us to analyze the inherent risks and shortcomings of time-limited interventions, even in the best-case scenario.

We prove (Supplementary Note1, Theorem1) that for any R0, recovery rate γ, and duration opt τ, there is a unique globally optimal intervention b(t) that starts at an optimal time ti and is given by:

 γ  βS , t [ti,ti + f τ) bopt(t) = ∈ (2) 0, t [ti + f τ,ti + τ] ∈ Where f takes a speciﬁc value in [0,1]. This solution says that the optimal strategy is to maintain

4 and then suppress. The intervention spends a fraction f of the total duration τ in a maintain phase, with b(t) chosen so that Re = 1. This maintains the epidemic at a constant number of opt opt infectious individuals equal to I(ti ), while susceptibles are depleted at a rate γI(ti ). The intervention then spends the remaining fraction 1 f of the total duration in a suppress phase, − setting Re = 0 so that infectious individuals are depleted at a rate γI (Fig.1a,d).

The effectiveness of the optimal intervention, when it should commence, and the balance between maintenance and suppression all depend on the total allowed duration of the intervention, τ. The longer we can intervene (larger τ), the more we can reduce the peak (smaller Imax), the opt earlier we should optimally act (smaller ti ), and the longer we should spend maintaining versus suppressing (larger f ) (Fig.1a,g,h, Supplementary Note1, Theorem1, Lemma7, and

Corollary4). Notably, there is a critical duration τcrit such that if the intervention is shorter than τcrit, then the optimal strategy is to suppress for the entire duration of the intervention (Fig.1g,h, Supplementary Fig.1, Supplementary Note1, Theorem4).

5 ekprevalence peak a Optimal b Fixed control c Full suppression g Eﬀect of τ ) t 0.3 τ (days) 0.3 (

I 14 0.2 28 0.2 56 0.1 0.1 I max prevalence 0 0 0 90 180 270 0 90 180 270 0 90 180 270 0 30 60 90

d e f h time optimal ) t

( 1 95 b 0.75 90 0.5 85 0.25 80 t i opt

intervention 0 75 0 90 180 270 0 90 180 270 0 90 180 270 0 30 60 90 time (days) duration τ (days)

Fig. 1 Interventions reduce peak infection prevalence. a–f, Timecourses of epidemics under optimal| (a), fixed control (b), and full suppression (c) interventions with three different values of intervention duration τ: 14 days (dark lines), 28 days (intermediate lines), and 56 days (light lines), and their respective intervention functions b(t) (d–f). At time t, the basic reproduction number R0 is reduced to some fraction of its maximum (no intervention) value; b(t) gives that fraction. Dashed black line shows timecourse in the absence of intervention. g, h, Effect of the duration τ on the infectious prevalence peak (g) and intervention starting times (h) for the optimal intervention (green), the optimized fixed control intervention (blue), and the optimized full suppression intervention (red). Dotted red line shows the critical value τcrit below 1 which full suppression is the globally optimal intervention. Plotted parameters: R0 = 3, γ = 14 1 days− . See Table1 for parameter justifications.

6 Near-optimal interventions

Although theoretically enlightening, the optimal intervention described above is not feasible in practice. Implementing it would require policies ﬂexible enough to ﬁne-tune transmission rates continuously, imposing ever changing social behaviors. It would also require instantaneous and perfect information about the current state of the epidemic in the population, information that will clearly not be available even with greatly improved testing and contact-tracing.

We therefore also consider other families of potential interventions, and study how they perform compared to the optimal intervention.

Real-world interventions typically consist of simple rules that are ﬁxed for some period of time (quarantines, restaurant closures, physical distancing). We model such ﬁxed control strategies (previously investigated by others13,14) as interventions of the form:

bﬁx(t) = σ for t [ti,ti + τ] (3) ∈

Fixed control interventions are determined by two parameters: the starting time ti and the strictness σ [0,1]. For any intervention duration τ we can numerically optimize ti and σ to mini- ∈ mize peak prevalence Imax (Fig.1b,e)

For a given R0, γ, and τ, an optimized fixed control intervention yields an epidemic time course that is remarkably similar to the one obtained under the globally optimal intervention strategy (Fig.1a,b). Peak prevalence Imax is only slightly lower in the optimal intervention than in an optimized fixed control intervention (Fig.1a,b,g). The effectiveness of a fixed control intervention depends on τ in a similar manner to that of the optimal intervention: longer interventions are more effective, should start earlier, and are less strict (Fig.1e,g,h).

The similarities between fixed control interventions and optimal interventions can be understood by considering the time course of Re during the intervention. At first, the fixed control intervention mainly depletes the susceptible fraction. As susceptible hosts are depleted, Re falls, so the intervention naturally begins to deplete the infectious fraction. And so fixed control interventions are qualitatively similar to the maintain, then suppress optimal intervention.

Optimizing a ﬁxed control strategy has the effect of choosing a σ and ti that emulate—and thus perform nearly as well as—an optimal intervention (Fig.1a,b,g).

7 As τ becomes small, the optimal σ for a fixed control intervention also becomes small, mim- icking the suppression phase of the optimal intervention. For small enough τ, the optimal σ is equal to 0: the intervention consists entirely of suppression (Supplementary Fig.1a). As τ increases, the optimal σ also increases, producing interventions with longer and longer susceptible-depleting maintenance-like periods (Supplementary Fig.1a). The epidemic tra- jectory becomes increasingly flat, eventually approximating a pure maintenance intervention. Note that achieving this approximate flatness with a fixed control intervention requires allowing some initial growth of the infected class (Re > 1 at ti), albeit at a reduced rate compared to no intervention. Growth of the infectious class never occurs during optimal interventions.

We also analyze a third class of interventions: the full suppression interventions deﬁned by

b0(t) = 0, t [ti,ti + τ]. (4) ∈ These interventions emulate extremely strict quarantines (Fig.1c,f). They are characterized by the complete absence of susceptible depletion. Such interventions are fully determined by the starting time ti, which can be optimized given the total allowable duration τ.

Note that full suppression interventions are a limiting case both of maintain-suppress interventions (with no maintenance phase, f = 0) and of fixed control interventions (with maximal strictness, σ = 0). Accordingly, the optimized full suppression intervention performs similarly to the optimal intervention and to the optimized fixed control intervention for short durations, when those favor a relatively short maintenance phase and high strictness (Fig.1c). For longer interventions, the effectiveness of full suppression rapidly plateaus (Fig.1g, Supplementary Note1, Corollary6). Accordingly, the optimal time to initiate a full suppression intervention plateaus with increasing τ: there is no benefit in fully suppressing too early (Fig.1h, Supple- mentary Note1, Corollary6).

Taken together, these results show that the most efﬁcient way for long interventions to decrease peak prevalence Imax is to cause susceptible depletion while limiting how much the number of infectious individuals can grow. For short interventions, by contrast, it is most efﬁcient sim- ply to reduce the number of infectious individuals. Optimizing an intervention trades off cases now against cases later. We prove (Supplementary Note1, Theorems2,3) that optimal and near-optimal interventions cause the epidemic to achieve the peak prevalence exactly twice: once during the intervention and once strictly afterward. All optimized maintain-suppress in-

8 terventions of duration τ and maintenance fraction f have this twin-peak property, including the globally optimal intervention, the optimized full suppression intervention, and also the optimized ﬁxed control intervention for any duration τ. Note that this means that the optimized interventions always end before herd immunity is reached. In practice this means that if an intervention continues until herd immunity is reached, then an earlier intervention of the same duration could have produced a lower epidemic peak while permitting a small rebound. The absence of a second peak is not a sign of policy success; it is a sign that policymakers acted too late.

Mistimed interventions

The optimal and near-optimal interventions are extremely powerful. For COVID-like epidemic parameters, the 28-day optimal or ﬁxed control interventions reduce peak prevalence from about 30% of the population to under 15%. Even the full suppression intervention reduces peak prevalence to well under 20%. These are massive and potentially health system-saving reductions.

In practice, however, interventions are not automatically triggered at a certain number of infectious individuals or at a certain point in time. They are introduced by policymakers, who must estimate the current quantity of infectious individuals I(t), often from very limited data, must begin roll-out with an uncertain period of preparation, and must also estimate the epidemiological parameters R0 and γ. These tasks are difﬁcult, and so policymakers may fail to intervene at opt the optimal moment ti . We consider timing errors in both directions, though in practice acting late might be more common than acting early.

How costly is mistiming a time-limited intervention? We ﬁnd that even a single week of separa- opt tion between the time of intervention and ti can be enormously costly, for realistic COVID-19 epidemic parameters (Fig.2a–c, g–i).

While the optimal intervention achieves a dramatic reduction in the height of the peak, mistiming such an intervention can be disastrous. Intervening too early produces a resurgent peak, but it is even worse to intervene too late. For example, if the intervention is initiated one week later than the optimal time, then Imax is barely reduced compared to the absence of any intervention whatsoever, particularly for a full suppression intervention (Fig.2a–c,g–i).

opt The extreme costs of mistiming arise from the steepness of the I(t) curve at ti . Optimized interventions permit some cases now in order to reduce cases later. Both infectious depletion

9 and susceptible depletion require currently infectious individuals in order to be effective at reducing peak prevalence. This means that, except for interventions of very long duration τ, the opt optimal start time ti occurs during a period of rapid, near-exponential growth in the fraction infectious I(t).

The practical problem with this approach is intuitive: because S(t) and I(t) are so steep at opt opt S(ti ),I(ti ), small errors in timing produce large errors in terms of S(ti),I(ti) (Fig.2a–c). Indeed, for epidemics that have faster dynamics, the consequences of mistiming interventions are increasingly stark (Fig.2a–f, Fig.3b).

It is also clear why being late is costlier than being early (Fig.2g–i, Fig.3a–d). An early intervention is followed by a large resurgent second peak, but the resurgence is slower and smaller than the uncontrolled initial surge permitted by a late intervention, thanks to the susceptible depletion that occurs during the early intervention (Fig.2a–c,g–i, Fig.3a–d).

Importantly, Supplementary Note1 Theorems2 and3 imply that late implementation of an optimized intervention results in a maximal epidemic peak at the time the intervention is initiated; whereas early implementation postpones the maximal peak (Fig.2a–c). In practice this means that early interventions allow for course corrections.

10 a Time-limited d Sustained g Costs of mistiming 0.3 no intervention sustained control 0.3 week early only optimal time 0.2 week late 0.2 0.1 0.1 0.0 0.0

b e h prevalence peak

) 0.3 0.3 t ( I 0.2 0.2 0.1 0.1 I max prevalence 0.0 0.0

c f i 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 90 180 270 0 90 180 270 14 7 0 7 14 time (days) time (days) − − opt oﬀset from ti (days)

Fig. 2 Mistiming an intervention reduces its effectiveness. a-f, Timecourses of epidemics under optimal| a, fixed control b, and full suppression c, interventions that are possibly mistimed: a week late (dark lines), optimally timed (intermediate lines), and a week early (light lines). Dashed black line shows timecourse in the absence of intervention. d-f, Timecourses of epidemics with a sustained control that reduces the basic reproduction number R0 by 25%, combined with the effects of a possibly mistimed optimal d, fixed control b, and full suppression f interventions, with line lightness as before. Dashed grey line shows timecourse with only sustained control and no additional intervention. g-i, Effect of offset of intervention time ti from opt max optimal intervention time ti on epidemic peak prevalence I without (solid lines) and with (dotted lines) sustained control for optimal g, fixed control h, and full suppression i interventions. Dashed black and grey lines show Imax in the absence of intervention, without and with 1 1 sustained control, respectively. Plotted parameters: R0 = 3, γ = 14 days− . See Table1 for parameter justifications.

11 a b 10 0.30

0.25 5 prevalance peak 0.20 (days) opt i t 0 0.15 I

0.10 max

oﬀset from 5 − 0.05

10 0.00 − c d

5 ( Asymmetry 0.12 0 R 0.10 4

0.08 I max

3 0.06 late − I

0.04 max 2 0.02 early) basic reproduction number

10 30 60 90 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.00 1 duration τ (days) recovery rate γ (days− )

Fig. 3 Effect of parameter variation on peak reduction and robustness to mistiming. a, | max opt b, Peak prevalence I as a function of offset from optimal intervention initiation time ti and (a) intervention duration τ,(b) recovery rate γ. c, d, Asymmetry between intervening early and intervening late, quantiﬁed as Imax for a 7-day late intervention minus Imax for a 7-day early intervention, as a function of the basic reproduction number R0 and (c) duration τ,(d) recovery 1 1 rate γ. Unless otherwise stated, R0 = 3, γ = 14 days− , τ = 28 days. See Table1 for parameter justiﬁcations.

12 Sustained interventions

Are there any sustained measures that can improve the robustness of optimized interventions? Because the severity of mistiming is governed by the steepness of I(t), measures that reduce steepness should alleviate the impact of mistiming. We propose using weak measures of long duration to achieve this desired outcome. Even though these measures by themselves may have little effect on Imax, they can buffer timing mistakes when used in combination with stronger, time-limited interventions.

We consider sustained weak interventions, modeled as a constant reduction of R0 throughout the entire epidemic, both on their own and combined with optimized time-limited interventions (Fig.2d–f). If perfectly timed, an optimized time-limited intervention outperforms a sustained intervention. Moreover, adding a sustained intervention to a perfectly-timed time-limited intervention provides little extra beneﬁt (Fig.2g–i). However, even a slightly mistimed time-limited intervention is far worse than a sustained intervention. And, most importantly, if both sustained and time-limited interventions are adopted, the time-sensitivity of the time-limited intervention is reduced. In particular, the otherwise disastrous cost of intervening too late is reduced (Fig.2d–i).

13 Discussion

COVID-19 has thrown into relief the importance of ﬂattening the curve. Our analysis establishes the optimal strategy to minimize peak prevalence in the SIR epidemic model, given an intervention of limited duration. Simpler interventions can closely approximate the optimal outcome.

Deriving the optimal strategy highlights the fundamental materials—susceptible depletion and infectious depletion—of any epidemic mitigation strategy, and it provides a yardstick against which to measure all other strategies.

But it would be unwise to attempt an optimal or near-optimal intervention in practice. The inevitable errors in timing that arise from uncertainty in inference and delays in implementation will produce disaster.

It is particularly costly to act too late. This causes an elevated peak immediately before the intervention even starts. By contrast, a premature intervention leads to a substantially delayed second wave23 after the relaxation of controls. Such a delay may sometimes be more desirable than peak reduction itself. A second wave is less challenging to control and manage than a ﬁrst wave: healthcare capacity can be increased in the interim, pharmaceutical interventions such as antivirals may become available, epidemiological parameters will be better known, and accumulated population immunity will reduce exponential growth rate even in the absence of intervention. Moreover, our results apply to any interventions that policymakers might be able to take during a second wave.

Our analysis has many limitations, and it leaves a large body of important questions unresolved. We have generously assumed that policymakers possess complete information about epidemic parameters and about the current epidemic state. Future studies can work to address the problem of disease control under uncertainty, by coupling an understanding of optimal interventions with an analysis of epidemiological inference. Errors of inference are likely to be magniﬁed by subsequent intervention mistiming. Moreover, our analysis makes the unrealistic assumption that intervention strength b(t) can be tuned at will. But in practice b(t) can be tuned only coarsely, and even ﬁxed control interventions are not truly enforceable in their idealized form.

We have on time-limited interventions. This time limit captures the social and economic costs of intervention. It also reﬂects the possibility that non-compliance with measures may rise over

14 time, a concern that some policymakers had when planning COVID-19 responses24.

We have focused our analysis on the epidemic peak. This quantity is critical because it is the point at which health services will be most strained. An overwhelmed health system can dramat- ically increase infection-fatality rates, direct morbidity, and medical complications10,11. Peak epidemic prevalence is a good proxy for demands on the healthcare system, but an even better metric is the total person-days with prevalence exceeding the maximum healthcare capacity. If the peak prevalence cannot be reduced below this capacity, then the strategy that minimizes the peak will not necessarily minimize the cumulative impact above capacity. It might be preferable to permit a slightly higher peak, and then move to full suppression.

A policymaker also seeks to reduce the total cases during the epidemic. But for an explosively spreading novel pathogen, this consideration may be secondary to reducing peak prevalence and avoiding healthcare system collapse. Moreover, interventions that reduce the epidemic peak almost necessarily reduce the total case count (or final size) of the epidemic, though they may not do so as efficiently as interventions targeted directly at final size reduction14. A policymaker may also wish to delay the epidemic peak to allow time for healthcare capacity to build and pharmaceuticals to be developed; this too produces different policy trade-offs14.

In general, however, when healthcare capacity is larger, the problem of minimizing the cumulative impact above capacity becomes almost identical to the problem of minimizing peak prevalence. Whereas when capacity is smaller, the problem of minimizing cumulative impact becomes the same as minimizing the ﬁnal size of the outbreak14.

We use one of the simplest possible models of disease transmission: the SIR model in a homo- geneously mixing population, and in the large population limit in which differential equations are appropriate. This is by design. We show that even in a simple setting, without the other confounding factors of real world disease spread, such as population structure, stochasticity, time varying transmission rates, or partial immunity, time-limited non-pharmaceutical control is not robust to implementation error.

Had our model included those complicating factors, one might have conjectured that the danger of mistiming is linked to those modeled factors chosen, and could therefore be controlled by carefully accounting for them. Had we studied an intervention that was not provably optimal, one could have argued that a more optimal intervention would not carry such risks. Instead, the simplicity of our model and the provable optimality of the strategies studied demonstrate

15 that these risks of mistiming are a fundamental feature of the initial exponential growth of an epidemic, regardless of the optimality of the intervention or the putative realism of the model used.

That said, time-limited non-pharmaceutical control in less idealized circumstances is worthy of study. di Lauro and colleagues provide a numerical treatment of a metapopulation of in- ternally well-mixed SIR demes14. Countries such as Vietnam25, Taiwan26, and New Zealand27 successfully adopted non-pharmaceutical strategies aimed at eliminating of SARS-CoV-2 transmission locally and then suppressing reintroductions. Metapopulation models could reveal the conditions under which a local elimination with a time-limited intervention is viable and robust. Similarly, heterogeneities in a single population—for instance in individual susceptibility or in degree of social connectedness—can alter disease dynamics28,29. Future studies could assess the impact of realistic network topologies on the optimality and robustness of interventions.

Still, our simple analysis offers several clear, practical principles for policymakers. First, act early. There is a striking asymmetry in the costs of acting too early versus too late. Second, work to slow things down. Slowing the growth curve makes interventions more robust, and also makes inference of epidemiological parameters more accurate. Third, when in doubt, bear down. Even the crude policy of full suppression is remarkably successful at reducing peaks and delaying excess prevalence. And if policymakers are very late to act then full suppression is, in fact, optimal.

Naive optimization is dangerous. Real-world policy must emphasize robustness, not efﬁciency.

16 Methods

Model parameters

Table1, below, gives definitions of model parameters, their units, default values plotted in figures unless otherwise stated, and justifications for those choices.

Table 1 Model parameters, default values, and sources/justiﬁcations |

Parameter Meaning Units Value Source or justiﬁcation

R0 basic reproduction number unitless 3 Estimates for COVID range from 2 to 3.517,30

1 γ recovery rate 1/days 14 Infectious period for COVID of approximately 1–2 weeks31

β rate of disease-causing contact 1/days R0γ calculated

τ duration of a time-limited in- days 28 Approximately one month tervention

17 State-tuned and time-tuned maintain-suppress interventions

When we ask what it means for a maintain-suppress intervention to be mistimed, we need a model of how the intervention is implemented. One possibility is that the policymaker di- γ rectly observes S(t) throughout the intervention and chooses b(t) = βS(t) during the maintenance phase based on the directly observed S(t). We call this a state-tuned intervention.

Alternatively, the policymaker plans to intervene at some S(t) value Si predicted to occur at a time ti. The policymaker knows that during a successful maintenance phase, S(t) = Si − γIi(t ti). The policymaker then chooses the maintenance phase values of b(t) according to this − predicted S(t). We call this a time-tuned intervention.

When we study mistimed interventions, we use time-tuned interventions. Since instantaneous epidemiological observation is not possible, time-tuned interventions are a more realistic model of how a maintain-suppress intervention, if possible at all, would in fact be implemented. If instantaneous epidemiological observation were possible during interventions, timing errors could be mitigated better than either state-tuning or time-tuning allows. Policymakers could observe the true (Si,Ii) at the moment of intervention ti and then employ whichever intervention of duration τ is optimal given that it begins at (Si,Ii).

Indeed, it can be seen that time-tuned interventions are in fact slightly more robust to mistiming than state-tuned interventions. They are partially self-correcting where the state-tuned interventions are not.

max Late interventions have I = Ii. But since a late time-tuned intervention has higher initial max strictness than a late state-tuned intervention, it avoids unnecessary time spent at I = Ii.

Early time-tuned interventions achieve lower Imax than equivalent early state-tuned interventions. This is true even—in fact, especially—for very fast epidemics. During the maintain phase of a maintain-suppress intervention, the strictness decreases in time (see Supplementary Note1, equation 11). If the intervention is too early, this allows for some initial growth of the infectious fraction and thus improved depletion of the susceptible fraction (Fig.2a,d) relative to maintaining at Ii (as in a state-tuned intervention). The ensuing second peak is therefore reduced.

An intriguing side effect of this automatic course-correction is that for relatively fast epidemics, some premature interventions outperform others that are less early (note the branching

18 in Fig.3b).

References

1. Leroy, E. M. et al. Fruit bats as reservoirs of Ebola virus. Nature 438, 575–576 (2005). 2. Taubenberger, J. K. & Morens, D. M. Influenza: the once and future pandemic. Public Health Reports 125, 15–26 (2010). 3. Zhou, P. et al. A pneumonia outbreak associated with a new coronavirus of probable bat origin. Nature 579, 270–273 (2020). 4. Lipsitch, M., Grad, Y. H., Sette, A. & Crotty, S. Cross-reactive memory T cells and herd immunity to SARS-CoV-2. Nature Reviews Immunology 20, 709–713 (2020). 5. Dan, J. M. et al. Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection. Science (2021). 6. Graham, B. S., Mascola, J. R. & Fauci, A. S. Novel vaccine technologies: essential compo- nents of an adequate response to emerging viral diseases. Journal of the American Medical Association 319, 1431–1432 (2018). 7. Anderson, R. M., Heesterbeek, H., Klinkenberg, D. & Hollingsworth, T. D. How will country-based mitigation measures influence the course of the COVID-19 epidemic? The Lancet 395, 931–934 (2020). 8. Branswell, H. Why ‘flattening the curve’ may be the world’s best bet to slow the coronavirus. STAT News. https://www.statnews.com/2020/03/11/flattening-curve- coronavirus/ (Mar. 2020). 9. WHO Writing Group. Nonpharmaceutical interventions for pandemic influenza, national and community measures. Emerging Infectious Diseases 12, 88 (2006). 10. Kissler, S. M., Tedijanto, C., Lipsitch, M. & Grad, Y. H. Social distancing strategies for curbing the COVID-19 epidemic. medRxiv (Mar. 2020). 11. Ferguson, N. M. et al. Impact of non-pharmaceutical interventions (NPIs) to reduce COVID- 19 mortality and healthcare demand (Imperial College, London, Mar. 2020). https:// www.imperial.ac.uk/mrc-global-infectious-disease-analysis/covid-19/. 12. Feng, Z. Final and peak epidemic sizes for SEIR models with quarantine and isolation. Mathematical Biosciences & Engineering 4, 675 (2007).

19 13. Hollingsworth, T. D., Klinkenberg, D., Heesterbeek, H. & Anderson, R. M. Mitigation strategies for pandemic influenza A: balancing conflicting policy objectives. PLoS Com- putational Biology 7 (2011). 14. Di Lauro, F., Kiss, I. Z. & Miller, J. Optimal timing of one-shot interventions for epidemic control. medRxiv (2020). 15. Perkins, T. A. & Espana,˜ G. Optimal Control of the COVID-19 Pandemic with Non- pharmaceutical Interventions. Bulletin of Mathematical Biology 82, 118 (2020). 16. Di Lauro, F., Kiss, I. Z., Rus, D. & Della Santina, C. Covid-19 and Flattening the Curve: a Feedback Control Perspective. IEEE Control Systems Letters (2020). 17. Li, R. et al. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV2). Science (2020). 18. Shear, M. D. et al. The Lost Month: How a Failure to Test Blinded the U.S. to Covid-19. The New York Times, 1. https://www.nytimes.com/2020/03/28/us/testing- coronavirus-pandemic.html (Mar. 2020). 19. Pei, S., Kandula, S. & Shaman, J. Differential effects of intervention timing on COVID-19 spread in the United States. Science Advances 6, eabd6370 (2020). 20. Knock, E. S. et al. The 2020 SARS-CoV-2 epidemic in England: key epidemiological drivers and impact of interventions. medRxiv (2021). 21. Kermack, W. O. & McKendrick, A. G. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character 115, 700–721 (1927). 22. Weiss, H. The SIR model and the foundations of public health. MATerials MATematics,` 0001–17 (3 2013). 23. Leung, K., Wu, J. T., Liu, D. & Leung, G. M. First-wave COVID-19 transmissibility and severity in China outside Hubei after control measures, and second-wave scenario planning: a modelling impact assessment. The Lancet (2020). 24. Harvey, N. Behavioral Fatigue: Real Phenomenon, Na¨ıve Construct, or Policy Contrivance? Frontiers in Psychology 11 (2020). 25. Lee, A., Thornley, S., Morris, A. J. & Sundborn, G. Should countries aim for elimination in the COVID-19 pandemic? BMJ 370 (2020). 26. Summers, J. et al. Potential lessons from the Taiwan and New Zealand health responses to the COVID-19 pandemic. The Lancet Regional Health-Western Pacific, 100044 (2020).

20 27. Baker, M. G., Wilson, N. & Anglemyer, A. Successful elimination of Covid-19 transmission in New Zealand. New England Journal of Medicine 383, e56 (2020). 28. Miller, J. C. Spread of infectious disease through clustered populations. Journal of the Royal Society Interface 6, 1121–1134 (2009). 29. Volz, E. M., Miller, J. C., Galvani, A. & Meyers, L. A. Effects of heterogeneous and clustered contact patterns on infectious disease dynamics. PLoS Computational Biology 7, e1002042 (2011). 30. Park, S. W. et al. Reconciling early-outbreak estimates of the basic reproductive number and its uncertainty: framework and applications to the novel coronavirus (SARS-CoV-2) outbreak. Journal of the Royal Society Interface 17, 20200144 (2020). 31. Zou, L. et al. SARS-CoV-2 viral load in upper respiratory specimens of infected patients. New England Journal of Medicine 382, 1177–1179 (2020). 32. Van Rossum, G. & Drake, F. L. Python 3 Reference Manual ISBN: 1441412697 (CreateS- pace, Scotts Valley, CA, 2009). 33. Harris, C. R. et al. Array programming with NumPy. Nature 585, 357–362. https:// doi.org/10.1038/s41586-020-2649-2 (Sept. 2020). 34. Jones, E., Oliphant, T., Peterson, P., et al. SciPy: Open source scientiﬁc tools for Python 2001. http://www.scipy.org/. 35. Hunter, J. D. Matplotlib: A 2D graphics environment. Computing in Science & Engineer- ing 9, 90–95 (2007).

Acknowledgements

We thank Ada W. Yan, Amandine Gamble, Corina E. Tarnita, Elizabeth N. Blackmore, James O. Lloyd-Smith, and Judith Miller for helpful comments on previous versions of this work. We thank Juan Bonachela for helpful discussions. DHM and SAL gratefully acknowledge ﬁnancial support from NSF grant CCF 1917819 and a C3.ai DTI Research Award from C3.ai Inc. and Microsoft Corporation. SAL gratefully acknowledges ﬁnancial support in the form of a gift from Google, LLC. for work on COVID-19.

21 Author contributions

DHM conceived the study. DHM and FWR designed and analyzed the mathematical model, with support and proof verification from JBP and SAL. FWR proved key theorems, with support from DHM. DHM conducted numerical analysis and produced figures, with support from FWR. DHM, FWR, and JBP wrote the first draft of the manuscript, which all authors edited.

Competing interests

We have no competing interests to declare.

Additional information

Supplementary Information

Supplementary information for this paper is available online.

Correspondence

Correspondence and requests for materials should be addressed to Dylan H. Morris and Fer- nando W. Rossine.

Data availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study. Code to reproduce numerical model analysis is provided (see Code availability below).

Code availability

All code needed to reproduce numerical results and ﬁgures is archived on Github (https: //github.com/dylanhmorris/optimal-sir-intervention) and on OSF (https://osf. io/rq5ct/), and licensed for reuse, with appropriate attribution/citation, under a BSD 3-Clause Revised License.

22 We wrote numerical analysis and figure generation code in Python 332, using numerical solvers provided in NumPy33 and SciPy34, and produced figures using Matplotlib35. Parameter choices for numerical analysis are stated in figure captions and in Supplementary Table1.

23 Supplementary Information for Optimal, near-optimal, and robust epidemic control

Dylan H. Morris1,3*, Fernando W. Rossine1*, Joshua B. Plotkin2, and Simon A. Levin1

1 Supplementary Figures

a b 1 45 0.75 30

0.5 crit τ f, σ 0.25 15 0 0 0 20 40 60 80 0 5 10 15 20 τ (days) R0 c Optimal d Fixed control 0.3 τ (days) 0.3 )

t τ1 32

( ≈ 80 I 0.2 160 0.2

0.1 0.1 prevalence 0 0 0 90 180 270 0 90 180 270 time (days) time (days)

Supplementary Fig. 1 Effect of τ and R0 on the optimal and optimized fixed control strategies. a, Optimal maintenance| fraction f for the optimal strategy (green line) and optimized strictness σ for the fixed control intervention (blue line) as a function of intervention duration τ. Red dotted vertical line shows the critical value τcrit below which full suppression ( f = 0) is the globally optimal intervention. Blue dotted vertical line shows τ1, the point at which Re = 1 at the start of the optimized fixed control intervention. b, Full suppression critical value τcrit as a function of basic reproduction number R0. c, d, Example timecourses for τ τ1, under the (c) optimal and (d) fixed control interventions. Dashed black line shows timecourse≥ in the absence of intervention. Parameters as in Table1 of the main text unless otherwise stated.

2 1 Supplementary Note 1: Theorems and Proofs

1.1 Useful notation

We deﬁne Scrit for an SIR model to be the critical fraction susceptible at which Re = 1 and dI 1 = 0 (in the absence of intervention), i.e. Scrit . dt ≡ R0

1.2 Peak prevalence in an SIR model

Deﬁne Imax(t) for an SIR system as the maximum value of I(x) achieved during the interval x [t,∞). Notice that Imax(0) = Imax, where Imax is the global maximum value of I(x), which ∈ we are seeking to minimize with our intervention.

A known result that is immediate from the original work of Kermack and McKendrick1 (see 2 Weiss for an explicit derivation) holds that for S(t) Scrit: ≥ 1 1 1 1 Imax(t) = I(t) + S(t) log S(t) + log (5) − R0 − R0 R0 R0

Remark 1: Continuity of Imax. Imax is a continuous function Imax : (0,1]2 [0,1] of v = 7→max (S(t),I(t)) for v [Scrit,1] (0,1], and therefore also a continuous function of t, I : R [0,1] ∈ × 7→ for t ( ∞,tcrit], where tcrit is the time such that S(tcrit) = Scrit ∈ − This is immediate from the fact that Imax(t) is a linear combination of univariate functions of

S(t) and I(t) that are themselves continuous on [Scrit,1] and [0,1], respectively. And since S(t) max and I(t) are continuous functions of t, I is a continuous function of t for t ( ∞,tcrit]. ∈ − Lemma 1: The more ﬁre, the bigger the blaze. If 0 Ix(tx) Iy(ty) and Sx(tx) = Sy(ty) for two ≤ ≤ SIR systems x and y with identical parameters R0 and γ at possibly distinct times tx,ty [0,∞] max max ∈ then I (tx) I (ty), with equality only if Ix(tx) = Iy(ty). x ≤ y Proof. There are three cases.

max max max Case 1: Ix(tx) = 0. In this case, I (tx) = 0 Iy(ty) I (ty), with equality only if I (ty) = x ≤ ≤ y y 0, which can only occur if Iy(ty) = 0.

max max Case 2: Sx(tx) = Sy(ty) < Scrit. In this case, Ix (tx) = Ix(tx) and Iy (ty) = Iy(ty), so our result

3 holds.

max Case 3: Sx(tx) = Sy(ty) > Scrit. In this case, we can apply equation5. Fixing S(t) > Scrit, I (t) is an increasing function of I(t) (there is a single, positive I(t) term in the sum), so the result must hold.

Lemma 2: The more fuel, the bigger the blaze. If Ix(tx) = Iy(ty) > 0 and Sx(ty) Sy(ty) ≤ for two SIR systems x and y with identical parameters R0 and γ at possibly distinct times max max tx,ty [0,∞] then I I , with equality only if Sx(tx) = Sy(ty) or Sy(ty) Scrit ∈ tx ≤ ty ≤ Proof. There are three cases.

Case 1: Sy(ty) Scrit. If Sy(ty) Scrit, then Sx(tx) Scrit, and neither epidemic will grow after ≤ ≤ max ≤max tx or ty, respectively. It follows that Itx = Ix(ty) = Ity = Iy(ty) S t > S S t < S Imax I t Imax > I t dIy > Case 2: y( y) crit, x( x) crit. In this case, tx = x( y) but ty y( y), since dt 0 if max max Iy(t) > 0 and Sy(t) > Scrit. So we have Itx < Ity

Case 3: Sx(tx),Sy(ty) > Scrit. The result in this case follows immediately from the fact that, max ﬁxing I(t), I (t) is an increasing function of S(t) when S(t) > Scrit. We can see that it is by taking the partial derivative with respect to S of the expression in equation5:

∂ 1 Imax(t) = 1 (6) ∂S − R0S

If S > S , 1 < 1, so ∂ Imax(t) > 0, and Imax(t) is an increasing function of S(t). crit R0S ∂S

1.3 Intervention function

We say that a right-continuous function with ﬁnite discontinuity points b(t) : R [0,1] is an S 7→ intervention beginning at ti with duration τ if b(t) 1 for all t ( ∞,ti) (ti + τ,∞). The SIR ≡ ∈ − model under such an intervention will then take the form

4 dS = b(t) βSI dt − ∗ dI = b(t) βSI γI (7) dt ∗ − dR = γI dt

We wish to show that for every τ there exists an intervention that minimizes Imax and that such an optimal intervention must be identical except at a ﬁnite set of times t j t j (ti,ti + τ) to { | ∈ } one of the form:  γ  βS , t [ti,ti + f τ) bopt(t) = ∈ (8) 0, t [ti + f τ,ti + τ] ∈ For some values of ti R and f (0,1]. We divide the proof into a series of lemmas. ∈ ∈ Lemma 3: More ﬁre, less fuel. Let bx(t) and by(t) be two interventions beginning at the same ti and lasting until t f = ti +τ. Denote the infectious and susceptible fractions for bx(t) and by(t), respectively, by Ix(t),Sx(t) and Iy(t),Sy(t). If Ix(t) Iy(t) for all t [ti,t f ], and Ix(t) > Iy(t) for ≥ ∈ some t (ti,t f ), then Sx(t f ) < Sy(t f ). ∈

d∆Rxy Proof. The difference ∆Rxy(t) = Rx(t) Ry(t) obeys = γ(Ix Iy), which is non-negative − dt − for all t [ti,t f ]. Therefore ∆Rxy(t) is non-decreasing during any time interval contained in ∈ [ti,t f ]. Take t¯ (ti,t f ) such that Ix(t¯) Iy(t¯) > 0. Because the intervention b(t) allows only a ∈ − ﬁnite number of discontinuities, the resulting I(t) is continuous. And so there is an ε such that

Ix(t) Iy(t) > 0 for all t [t¯ ε,t¯+ ε]. Therefore ∆Rxy(t) must be strictly increasing during − ∈ − [t¯ ε,t¯+ ε]. − Because ∆Rxy(t) never decreases and sometimes increases during [ti,t f ], it follows that ∆Rxy(ti) < ∆Rxy(t f ), but ∆Rxy(ti) = 0 so ∆Rxy(t f ) > 0 and Rx(t f ) > Ry(t f ). But Ix(t f ) Iy(t f ) and S = ≥ 1 (I + R), so Sx(t f ) < Sy(t f ). − We now wish to prove that an epidemic left alone—without any intervention—will reach any given level of prevalence faster than it would have if any transmission-reducing intervention had taken place. In other words, an intervention always slows down the epidemic.

Lemma 4: Do nothing to burn faster and brighter. Let b1(t) 1 denote the null intervention max ≡ and take tmax such that I1(tmax) = I1 . Let bx(t) be an intervention that starts at time ti, has

5 duration τ, and satisﬁes bx(t) < 1 for almost all t [t¯ ε,t¯+ ε] for some ε > 0 and some ∈ − t¯ < tmax. Then Ix(t) < I1(t) for all t [t¯,tmax]. ∈

Proof. Let the time of divergence of the interventions b1 and bx be given by td = inf t { ∈ (ti,tmax) bx(t) < 1 . Note that this infimum must exist by completeness, because it is taken | } over a bounded and non-empty set. It is easy to see that bx(t) = 1 and that Ix(t) = I1(t) for all t < td. Also, by the right continuity of bx, there exists an interval (td,td +ε) such that I1 > Ix and b1βS1I1 > bxβSxIx. Suppose there exists a minimal te (td,tmax) such that Ix(te) = I1(te). We ∈ will find a contradiction. Note that Ix < I1 in (td,te). Because I1 is monotonic and continuous in 1 ( ∞,tmax), we can invert I1 and define tˆ = I− (Ix(t)) for all t ( ∞,tmax). − 1 ∈ − dI1 We can see that dt (tˆ) is the growth rate of the infectious class under the null intervention dI1 dIx when I1 = Ix(t). It follows that dt (tˆ) < dt (t) for some value of t, otherwise I1 would always dI1 dIx dominate Ix, and te would not exist. Let tinf = inf t (td,te) (tˆ) < (t) . This implies { ∈ | dt dt } that S1(tˆ) Sx(t) for some interval including tinf that can be maximally extended to the left as ≤ (tmin,tinf]. By continuity of S, S1(tˆmin) = Sx(tmin).

If tmin = td, then tmin > tˆmin, and R1(tˆmin) = Rx(tmin), but this is impossible because by construc- 6 dI1 dIx tion (tˆ) > (t) for all t (td,tmin), and therefore R1(tˆmin) < Rx(tmin). dt dt ∈ If tmin = td, then tmin = tˆmin. Also, R1(tˆ) > Rx(t) for all t (td,tinf). But that is impossible ∈ dR1(tˆd) b(td)βSI γI dRx(td) ˆ because dt = βSI γ−I γI < γI = dt , and trivially R1(td) = Rx(td). − max Lemma 5: Wait, maintain, suppress. Let bx be any intervention, and I = max Ix(t) . x { } There exists an intervention by with same beginning and duration as bx of the form

 1 t [ti,ti + gτ)  ∈ b (t) = γ (9) y βS , t [ti + gτ,ti + f τ)  ∈ 0, t [ti + f τ,ti + τ] ∈ For some g < f and g, f (0,1], such that Imax Imax. ∈ y ≤ x τ Proof. Deﬁne Ix = max Ix(t) t [ti,t f ] . Following Lemma4, take g such that ti + gτ = { | ∈ } τ 1 τ 1 Ix I− (I ). Now take f such that (1 f )τ = log . From Lemma4 it is clear that Iy(t) 1 x γ Ix(t f ) − τ ≥ Ix(t) for t [tiτ,ti + gτ). By construction Iy(t) = I for t [ti + gτ,ti + f τ), and therefore the ∈ x ∈

6 inequality remains true. Finally, for t [ti + f τ,ti + τ], because Iy(t) decays faster than Ix(t), ∈ then if Iy(t) < Ix(t) at any time, then Iy(t f ) < Ix(t f ), but by construction Iy(t f ) = Ix(t f ). By max max Lemma3, Sy(t f ) < Sx(t f ), and by Lemma2, Iy < Ix .

crit It is possible however that this proposed by is not a viable intervention if Sy(ty ) = Scrit for crit some ty [ti + gτ,ti + f τ), as this would require by to assume values larger than 1. If that is ∈ crit the case we can deﬁne by¯, by taking f such that ti + f τ = ty . Note that because by the end of the intervention by¯ the infectious class will monotonically decrease because Sy¯ Scrit, this ≤ implies that Imax = Iτ = Iτ Imax. y¯ y¯ x ≤ x max max Theorem 1: Maintain, then suppress. An intervention bopt(t) such that I I for any opt ≤ x intervention bx(t) must take the form

 γ  βS , t [ti,ti + f τ) bopt(t) = ∈ (10) 0, t [ti + f τ,ti + τ] ∈ for some value of ti R and f (0,1]. ∈ ∈

Proof. If bx(t) is an optimal intervention, then Lemma5 ensures that bx(t) is of the form given by equation9. Now consider the strategy bopt(t) of the form given by equation 10 (main text opt x opt x equation2) with t = t +gτ and f = f g. This new intervention functions exactly like bx i i − for the entire duration of bx, but then bopt is held at 0 for a little longer further reducing Iopt. This opt x opt x max max means that Sopt(t ) = Sx(t ) and Iopt(t ) Ix(t ) and therefore by Lemma1, I I . f f f ≤ f opt ≤ x

Remark 2: The strategy is set at the start. Because during an intervention bopt the susceptible class is depleted at a constant rate for all t [ti,ti + f τ) , bopt can be written as ∈  γ  , t [ti,ti + f τ) β St It γ(t ti) ∈ bopt(t) = i − i − (11) 0, t [ti + f τ,ti + τ] ∈

For Sti = S(ti) and Iti = I(ti).

7 1.4 Results of an optimal intervention

It follows that, given an optimal intervention b of duration τ and depletion fraction f begun at time ti and ending at t f = ti + τ:

S(t f ) = S(ti) γτ f I(ti) (12) −

I(t f ) = I(ti)exp[ γτ(1 f )] (13) − −

In an SIR system without intervention that begins with a wholly susceptible population, we have the relation:

1 I(S) = 1 S + log(S) (14) − R0

And so: 1 I(ti) = 1 S(ti) + log(S(ti)) (15) − R0

And we can likewise write S(t f ) and I(t f ) in terms of S(ti):

1 S(t f ) = S(ti) γτ f 1 S(ti) + log(S(ti)) (16) − − R0

1 I(t f ) = 1 S(ti) + log(S(ti)) exp[ γτ(1 f )] (17) − R0 − −

max This in turn allows us to express I (t f ) purely in terms of S(ti) and the parameters, though the expression is long:

8 max 1 I (t f ) = 1 S(ti) + log(S(ti)) exp[ γτ(1 f )] − R0 − − 1 + S(ti) γτ f 1 S(ti) + log(S(ti)) − − R 0 (18) 1 1 log S(ti) γτ f 1 S(ti) + log(S(ti)) − R0 − − R0 1 1 1 + log − R0 R0 R0

max max Remark 3: Continuity of I(ti),I (t f ). Notice that both I(ti) and I (t f ) are continuous functions of S(ti), since they are both linear combinations of continuous functions of S(ti) and since S(ti) is continuous in ti, they are continuous functions of ti.

max Lemma 6: Don’t be late. Let tp be the inﬁmum of times such that I(t) = I , and let ti be the max start of an optimal intervention. Then tp ti. That is, I cannot occur before the intervention ≥ begins for an optimal intervention.

Proof. Since b(t) = 1 for t < ti tp < ti necessarily implies that S(ti) < Scrit, that is, the epidemic is already declining when the intervention begins and will never grow again, regardless of the intervention approach (since b(t) 1, we cannot force a declining epidemic to grow). That in ≤ turn implies Imax = Ipeak, which we can with certainty improve upon. So Imax must occur during or after the intervention.

Corollary 1: Start with fuel. It is immediate that S(ti) > Scrit.

Corollary 2: Peak early. Since I(t) I(ti) for t [ti,ti + τ] during an optimal intervention, if max ≤max ∈ I occurs during the intervention, I = I(ti)

max Theorem 2: Twin Peaks. Let bx(t) be an optimal intervention. Then Ix(t) Ix for all t max ≤ ∈ ( ∞,t f ], with equality for t [ti,ti + τ f ], and furthermore Ix(tp) = I for some tp [t f ,∞) − ∈ x ∈ with tp = t f only if f = 1.

That is, if 0 < f < 1, there will be a plateau during the intervention followed by a peak of equal height that occurs strictly after the intervention ﬁnishes. If f = 0, there will be two peaks of equal height, one at the start of the intervention and one strictly after it ﬁnishes, and if f = 1 there will be a plateau during the intervention with no subsequent peak.

9 max Proof. Let bx be an optimal intervention of the form given by equation 10. By Lemma6, Ix max must occur during or after the intervention. First let us assume that the Ix occurs during the intervention and is never again attained after the intervention. If f = 1, this implies that the whole intervention is a plateau and no further peaks occur. For f < 1 we can build a new intervention by of the same form as bx but that starts at ti ε rather than at ti. From the continuity max − max of I in ti for any intervention, it follows that for some ε small enough, Iy (t f ) (the post- τ intervention maximum of I(t)) must still be smaller than I = I(ti ε) (the maximum during y − the intervention), but because I(ti ε) < I(ti), Corollary2 implies that our new by outperforms − bx, which contradicts the optimality of bx.

max Now let us assume that Ix occurs after the intervention and is larger than any value of Ix during the intervention. If f > 0, we can once again build a new intervention function by

 1 t [ti,ti + ε)  ∈ b (t) = γ (19) y βS , t [ti + ε,ti + fyτ)  ∈ 0, t [ti + fyτ,ti + τ] ∈

1 Ix(ti+ε) max With fy chosen such that (1 fy)τ = log . For sufﬁciently small ε, Iy(ti + ε) < I . γ Ix(t f ) x −max max Also, following Lemma5, Iy < Ix and therefore by outperforms bx, which once again contradicts the optimality of bx.

Finally, if f = 0, we build yet another by of the same form as bx but that starts at ti + ε rather than at ti. Because intervention by starts out with a smaller susceptible fraction than intervention bx, and an increase in the infected fraction that is smaller than the susceptible fraction decrease, max max it follows from equation5 that Iy < Ix and therefore by outperforms bx, which once again contradicts the optimality of bx.

opt Corollary 3: The longer the intervention, the earlier it starts. If ti (τ) is the starting time opt opt opt of the optimal intervention with duration τ, let Si (τ) = S(ti (τ)). Then Si (τ) is a non- decreasing function of τ.

Proof. Trivially a longer optimal intervention must outperform a shorter optimal intervention. opt max opt But by Theorem2 Ii = I for any optimal intervention, which implies that Ii (τ) is a non- opt increasing function of τ and Si (τ) is a non-decreasing function of τ.

10 Corollary 4: If you don’t have much gunpowder, don’t shoot until you see the whites of τ their eyes. Given the optimal intervention of duration τ, bopt, denote the initial time of that τ τ intervention by t . As τ 0, t tcrit. i → i → max Proof. From equation 18 we can see that Iopt is a continuous function of τ, and that as τ 0, max max → the effect of intervention deﬁned as I1 Iopt 0. We know from Theorem2 that Iopt(ti) = max −max → τ I , and therefore, as τ 0, Iopt(ti) I , which implies t tcrit. opt → → 1 i →

Corollary 5: No need to burn all the fuel. After an optimal intervention, S(t f ) Scrit, with ≥ equality if and only if f = 1.

max max max Proof. By Theorem2, I(ti) = I (t f ) = I . If f < 1, we must have I(t f ) < I(ti) = I , so max max in order for the epidemic to reach I (t f ) = I , we must have S(t f ) > Scrit. If f = 1, then max I(t f ) = I(ti) = I , so then must have S(t f ) = Scrit, otherwise the epidemic would grow to a peak above I(ti) = I(t f ), which would be a contradiction of Theorem2.

Corollary 6: Putting out existing ﬁre can only do so much. Consider a full suppression intervention of duration τ deﬁned by b0(t) = 0 for all t [ti,ti + τ]. For every τ there is a ti ∈ that minimizes Imax. Consider these optimized full suppression interventions. Then, as τ 0 → ∞, the maximum infectious prevalence Imax 1 + 1 log 1 1 . In other words, full 0 → 2 2R0 R0 − suppression interventions have a limit in how much they can reduce Imax.

Proof. From the proof of Theorem2 for f = 0, it follows that full suppression interventions max have an optimal start time ti, and that I0(ti) = I0 . Also, because no new infections occur during a full suppression intervention, S0(ti) = S0(t f ) so substituting into equation5

max 1 1 1 1 I0 (ti) = I0(t f ) + S0(ti) log S0(ti) + log (20) − R0 − R0 R0 R0

We the further use equation 14 to substitute S0(ti) and take τ ∞ such that I0(t f ) 0 which → → ﬁnally yields

max 1 1 1 I0 = + log 1 (21) 2 2R0 R0 −

11 Corollary 7: But when in doubt, put out the ﬁre. Let b0 be the optimized full suppression 0 intervention with duration τ and starting time ti . For a full suppression intervention b0ˆ that 0ˆ 0 also has duration τ but that has a starting time t [t ,tcrit], the infected fraction peak is i ∈ i Imax = I (t0ˆ ), moreover no intervention starting at that same time can attain a lower peak. 0ˆ 0ˆ i

Proof. It sufﬁces to show that delaying a full suppression intervention by an inﬁnitesimal ε diminishes the secondary peak. From equation 25 with f = 0, it is clear that

∂ max S0(ti) γτ I (t f ) = + e− I0(ti) + S0(ti) ∂ti −R0Si ∂ max γτ I (t f ) = γIi + e− (βIiSi γIi) βIiSi (22) ∂ti − − ∂ max γτ I (t f ) = (e− 1)(I0(ti)) ∂ti − γτ But I (ti) > 0 for ti < tcrit and (e 1) < 0 which means that delaying the full suppression 0 − − decreases the post-intervention peak. Trivially, no intervention can attain a global Imax value lower than its initial prevalence I(ti), which concludes the proof.

Theorem 3: Twin peaks for fixed control interventions. Let bx be an optimized fixed control intervention of duration τ that starts at some time ti and has strictness σ. That is, bx outperforms max any other fixed control intervention of duration τ regardless of their ti or σ. Then Ix = Ix at exactly two time points, one during the intervention (t [ti,ti +τ]) and one after the intervention ∈ (t (ti + τ,∞)). ∈ Proof. Because during the intervention and after the intervention, the time course is that of an

SIR (albeit with a modiﬁed R0 during the intervention), it is clear that there can be at most one local I maximum during the intervention and at most one after the intervention.

First let us assume that the infectious peak during the intervention is higher than any peak after the intervention. By continuity it is possible to construct a new ﬁxed control intervention that starts a little earlier and is slightly stricter such that the peak during the intervention is reduced but the peak after the intervention is still below the intervention peak. This new intervention outperforms bx which contradicts the optimality of bx.

Now let us assume that the infectious peak during the intervention is lower than any peak after the intervention. We will show that a small change in the ti of the intervention will lower the

12 post-intervention peak. It sufﬁces to show that ∂ Imax = 0. For a ﬁxed control intervention we ∂ti 6 have

∂ max (1 σ) ∂ I = − Sx(t f ) (23) ∂ti σR0Sx(t f ) ∗ ∂ti

∂ Which when set to 0 implies Sx(t f ) = 0. This in turn can be substituted into the equation that ∂ti relates I to S under a ﬁxed control intervention

1 1 I(t f ) = S(t f ) + log S(t f ) + I(ti) + S(ti) log(S(ti)) (24) − σR0 − R0

∂ ∂ Which leads to Ix(t f ) = 0 and ﬁnally Rx(t f ) = 0 but this is impossible because the frac- ∂ti ∂ti tion of recovered individuals by the end of a ﬁxed control intervention must decrease as the intervention starts earlier. This completes the proof.

1.5 First-order conditions for ti, f

max Applying equations 12, 13, the partial derivatives of I (t f ) with respect to f and ti are given by: ∂ max (γτIi) (1 f )γτ I (t f ) = + γτIie− − γτIi ∂ f − R0 Si γτ f Ii − (25)

∂ max S0(ti) f γτI0(ti) (1 f )γτ I (t f ) = − + e− − I0(ti) f γτI0(ti) + S0(ti) ∂ti − − R0 Si f γτIi −

dS dI Substituting the values of dt , dt :

∂ max ( βSiIi) f γτ (βSiIi γIi) I (t f ) = − − − ∂ti − R0 (Si f γτIi) − (1 f )γτ (26) + e− − (βSiIi γIi) − f γτ (βSiIi γIi) βSiIi − − −

13 Setting equal to zero and simplifying:

∂ max 1 (1 f )γτ I (t f ) = + γτe− − 1 = 0 ∂ f − (27) R0 Si f γτIi −

∂ max ( βSi) f γτ (βSi γ) (1 f )γτ I (t f ) = − − − + e− − (βSi γ) ∂ti − R0 (Si f γτIi) − (28) − f γτ (βSi γ) βSi = 0 − − −

While these ﬁrst-order conditions do not yield a simple closed form for the optimal f and ti, we use them in the proofs above and below to establish properties of the optimal strategy.

τ Lemma 7: If you have lots of gunpowder, shoot early. For an optimal intervention bopt τ acting on a SIR with full susceptibility as its initial condition, as τ ∞,S (ti) 1; that is, → opt → we intervene almost immediately, when almost all the population is initially susceptible and a long intervention is possible.

τ 1 Scrit Proof. Deﬁne an auxiliary intervention bx with τ > γ−Imax such that

 γ τ  βS , if t > tx and S > Scrit bx (t) = (29) 1, elsewhere

1 1 Scrit with tx = I1− ( −γτ ). It is clear that such an intervention has a duration of τ or less, and that max 1 Scrit by the end of such an intervention Sx Scrit. Therefore I = Ix(tx) = − , but by deﬁnition ≤ x γτ max max τ 1 Scrit τ Iopt Ix . Combining with Corollary2, Iopt(ti) −γτ so as τ ∞ both Iopt(ti) 0 and τ ≤ ≤ → → S (ti) 1. opt → Theorem 4: Sometimes maintain, always suppress. For an SIR with full susceptibility as its initial condition, it is the case that any optimal intervention with positive duration as deﬁned by equation 10 has f < 1. In other words, the optimal intervention for an emerging pathogen

(S(0) 1,I(0) 0) always has a total suppression phase. Also, for any R0 (1,∞) there → → ∈ is a τcrit such that the optimal intervention is always a full suppression intervention ( f = 0) if

τ < τcrit. Moreover, limR0 1 τcrit = ∞ and limR0 ∞ τcrit = 0. → →

14 Proof. Suppose f = 1. From equation 27,

1 + γτ 1 = 0 − (30) R0 Si γτIi −

But following Corollary5, Si γτIi = Scrit, which substituting in the previous equation implies − that γτ = 0, which is impossible. Therefore f cannot be 1.

Now let us investigate if f = 0 can be a local minimum. Once again, from the ﬁrst order condition expressed in equation 27 we require that

1 Si = R0x (31) γτ x = 1 e− − The second order condition for an extremum to be a minimum is given by

∂ 2 (γτI )2 max 2 γτ(1 f ) i (32) 2 I = Ii(γτ) e− − + 2 > 0 ∂ f R0(Si γτ f Ii) − Note that the condition is always satisﬁed, and therefore any extremum is always a minimum. This also ensures that any minimum must be unique and that Imax grows monotonically as f moves away from f opt.

max From Theorem2 we have Ii = I . Substituting this and equations 31 and 14 into equation 18, we get an implicit function for x (and therefore for τ) such that f = 0 is a minimum:

0 = x(log(x) + xlog(R0x) R0x) + 1 (33) −

We now show that R0 (1,∞) implies x (0,1) which in turn implies a τcrit that satisﬁes the ∈ ∈ relation.

From equation 33 it is easy to see that as R0 1, then x 1, and as R0 ∞, then x 0. So → → → → showing that x(R0) is a continuous, decreasing function of R0 on R0 (1,∞) will establish the ∈ existence of τcrit. By implicit differentiation we have that

15 3 R0 1 ∂x x − R0 (34) = 2 ∂R0 x + x xlog(x) 2 − − For positive x the numerator is positive and the denominator is always negative if x (0,1). ∈ Therefore, for R0 (1,∞) there exists x (0,1) that satisﬁes equation 33. ∈ ∈ To complete the proof, it sufﬁces to show that f opt is an increasing function of τ when f = 0, opt opt as this implies that f (τcrit) = 0 and cannot go above 0 for τ < τcrit. We start by letting f , opt opt Si , and Ii be functions of τ, applying equation 27, implicitly differentiating with respect to τ, and setting f = 0. After a bit of algebra we obtain

opt γτ τIi ∂ opt γτ 1 ∂ opt τe− + f = e− + S (35) opt 2 opt 2 i R0(Si ) ∗ ∂τ R0γ(Si ) ∗ ∂τ

∂ opt Note that ∂τ f is being multiplied by a strictly positive quantity on the left-hand side of the γτ equation. On the right-hand side of the equation there is a positive quantity (e− ) added to ∂ Sopt 1 ∂ Sopt > ∂τ i multiplied by another positive quantity ( opt 2 ), but by Corollary3, ∂τ i 0. It R0(Si ) ∂ opt opt follows that ∂τ f > 0 when f = 0 which completes the proof.

1.6 A general classiﬁcation of interventions

From equation5 it is clear that there are two fundamental methods of reducing the peak of an epidemic: depleting the infected fraction and depleting the susceptible fraction. We have shown that different interventions achieve peak reduction with different combinations of those methods. Our optimal intervention, for example, is characterized by a pure susceptible depletion phase followed by a pure infected depletion phase. Full suppression interventions, in contrast, operate solely by depleting the infected fraction. We observe that interventions can be classi- ﬁed in terms of how much they rely on depleting the susceptible fraction versus depleting the infected fraction.

The effect of an intervention can be understood as the infectious peak if no intervention were to take place minus the infectious peak given the intervention. In a more formal notation, an max max max max intervention bx has an effect I I = I (Ix(ti),Sx(ti)) I (Ix(t f ),Sx(t f )) = ∆x(t f ). By − x −

16 applying equation5 we obtain

h i h i ∆x(t f ) = Ix(t f ) Ix(ti) G Sx(t f ) G Sx(ti) − − − − 1 (36) G(S) = S log(S) − R0 h i h i Then if Ix(t f ) Ix(ti) > G Sx(t f ) G Sx(ti) we can say that the intervention bx − − − − overall relies more on infected depletion, whereas if the opposite is true we can say that it relies more on susceptible depletion. Moreover, by applying the fundamental theorem of calculus, we obtain

Z t f Z t f 1 ∆x0 (t)dt = Ix0(t) + Sx0 (t) 1 dt (37) ti − ti − R0Sx(t)

Which allows us to look at a certain time t [ti,t f ] and say that if ∈ 1 Ix0(t) < Sx0 (t) 1 (38) − R0Sx(t)

Then at that moment t, the intervention bx acts more by depleting the infected fraction than by depleting the susceptible fraction. The condition can be simpliﬁed to

bx(t) 2R0Sx(t) 1 < 1 (39) −

Supplementary References

1. Kermack, W. O. & McKendrick, A. G. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character 115, 700–721 (1927). 2. Weiss, H. The SIR model and the foundations of public health. MATerials MATematics,` 0001–17 (3 2013).