The previous section updated results from a model based on Caulkins et al. (2020) that sought to determine the optimal start and length of a lockdown when the intensity of that lockdown was given exogenously. We next present an extension of the model as given in Caulkins et al. (2021) that allows the intensity of the lockdown to vary continuously over time.
As in Caulkins et al. (2020), we define \(\gamma (t)\) to be the share of potential workers who are employed at time t. However, now we model \(\gamma (t)\) as a state variable that can be altered continuously via a control u(t):
$$\begin{aligned} \dot{\gamma }(t)=u(t),\quad \gamma (0)=1, \end{aligned}$$
We set \(\gamma (0) = 1.0\) because the planning horizon begins when COVID- 19 first arrives, and so before there is any lockdown. We include a state constraint \(\gamma (t)\le 1\) for \(0\le t\le T\), since employment cannot exceed 100%. This formulation allows for multiple lockdowns, takes into account that employment takes time to adjust, and it recognises that changing employment levels induces adjustment costs, which we allow to be asymmetric, with it being harder to restart the economy than it is to shut it down.
Since public approval of lockdowns may wane the longer a lockdown lasts, we introduce a further state variable that models this “lockdown fatigue” z(t):
$$\begin{aligned} \dot{z}(t)=\kappa _1\left( 1-\gamma (t)\right) -\kappa _2 z(t), \end{aligned}$$
where \(\kappa _1\) governs the rate of accumulation of fatigue and \(\kappa _2\) measures its rate of decay. Note that if the worst imaginable lockdown (\(\gamma (t)=0\)) lasted forever then z(t) would grow to its maximum possible value of \(z_{\max } = \kappa _1 / \kappa _2\).
We use an epidemiological model based on an open-population \({\text {SIR}}\)Footnote 2 model with a birth rate \(\nu \) and extra mortality for individuals who are infected (\(\mu _I\)) above and beyond that for those who are susceptible or recovered (\(\mu \)). In addition, we allow a backflow of recovered individuals back into the susceptible state at a rate \(\phi \). How long immunity will last with SARS-CoV-2 virus is not known at the time of this writing, but immunity to other corona viruses often lasts 3–5 years, so we set \(\phi \) to 0.001 per day in our base case, which corresponds to a mean duration of immunity of \(1000/365 = 2.74\) years.
The state dynamics in our extended model can then be written as
$$\begin{aligned}&\dot{S}(t)=\nu N(t)-\beta (\gamma (t),z(t))\frac{S(t)I(t)}{N(t)}-\mu S(t)+\phi R(t)\\&\dot{I}(t)=\beta (\gamma (t),z(t)))\frac{S(t)I(t)}{N(t)}- (\alpha +\mu +\mu _I)I(t)\\&\dot{R}(t)=\alpha I(t)-\mu R(t)-\phi R(t)\\&\dot{\gamma }(t)=u(t),\quad \gamma (0)=1\\&\dot{z}(t)=\kappa _1(1-\gamma (t))-\kappa _2z(t),\quad z(0)=0\\&\gamma (t)\le 1,\quad 0\le t\le T \end{aligned}$$
where \(N(t) = S(t) + I(t) + R(t)\) is the total population. As before, the factor \(\beta (\gamma ,z)\) captures the number of interactions and the likelihood that an interaction produces an infection. It is assumed to depend on both the intensity of the lockdown \(\gamma \) and the level of lockdown fatigue z in the following way:
$$\begin{aligned} \beta (\gamma ,z):=\beta _1+\beta _2\left( \gamma ^\theta +f\frac{\kappa _ 2}{\kappa _1}z(1-\gamma ^\theta )\right) \end{aligned}$$
This expression can be interpreted as follows. In the absence of lockdown fatigue, we might model \(\beta (\gamma ,0)\) as some minimum level of infection risk \(\beta _1\) that is produced just by essential activities plus an increment \(\beta _2\) that is proportional to \(\gamma \) raised to an exponent \(\theta >1\). Having \(\theta \) greater than 1 is consistent with locking down first the parts of the economy that generate the most infections per unit of economic activity (perhaps concerts and live sporting events) and shutting down last industries with high economic output per unit of social interaction (perhaps highly automate manufacturing and mining).
The term \((\kappa _2/\kappa _1) z\) is the lockdown fatigue expressed as a percentage of its maximum possible value. So if \(f=1\) and z reached its maximum value, then all of the potential benefits of locking down would be negated. Lockdown fatigue will not actually reach that maximum because the planning horizon is relatively short. Also, we choose a relatively small value of \(f=0.05\), so this lockdown fatigue has only a modest effect. Nonetheless, including this term at least acknowledges this human dimension of the public’s response to lockdowns.
The objective function includes health costs (due to deaths from COVID- 19), economic loss (due to locking down), and the adjustment costs of changing the employment level \(\gamma \). We assume these adjustment costs to be quadratic in the control u and allow for them to be asymmetric with different constants for shutting down businesses \(c_l\) and reopening them \(c_r\), with an extra penalty for reopening after an extended shut down so that
$$\begin{aligned} V_u(u,\gamma ):={\left\{ \begin{array}{ll} c_lu^2 &{} u\le 0\\ c_r(z+1)u^2 &{} u>0. \end{array}\right. } \end{aligned}$$
The resulting optimal control model and the base case parameter values are summarised in Appedix 2 and Table 9.4 Appendix 3.
9.3.1 Results
9.3.1.1 The Effect of Increased Infectivity
Our main interest is in how a mutated strain that is more contagious alters what strategies are optimal. That is perhaps best captured in Fig. 9.3, which has two panels. The one on the left corresponds to the old reproduction number of \(R_0=2.5\); the one on the right corresponds to the new, higher number of \(R_0=4\). Both are similar to the right panel of Fig. 9.1 in that they show how the value function depends on the parameter M describing the cost per premature death.
This value function can be thought of as the “score” that a social planner “earns” when he or she follows the optimal strategy. Naturally in both panels the value function slopes down. The greater the penalty the social planner “pays” for each premature death, the lower the score. On the left side of each panel the value function slopes down steeply because there isn’t much locking down so there are a lot of deaths; thus, a given increment in the cost per death gets “paid” many times. On the right side of each panel, the optimal strategy involves an extended lockdown, so there are fewer deaths and the same increment in the cost per death reduces the social planner’s score by less.
There are, though, two noteworthy differences between the value functions across the two panels. First, the kink in the curve, indicating the point at which an extended lockdown becomes preferred, occurs at a larger value of M in the right-hand panel. That is because when the reproduction number is larger, it takes a more determined lockdown to pull off the extended lockdown strategy, making it more costly and less appealing unless the penalty per premature death is larger. The difference is not enormous though, with valuations equivalent to about (\(M=11,560\)) 32 times GDP per capita in the right panel and (\(M=10,140\)) 28 times GDP per capita in the left panel.
The second difference is that—at least with all other parameters at their base case values—increasing \(R_0\) increased the number of different types of strategies that can be optimal. With \(R_0=4\) there are five distinguishable types of lockdown strategies that can be optimal, not just two.
Here is how to interpret the labels of the four regions Ia, Ib, IIa, and IIb. The Roman numeral I or II refers to whether there are one or two lockdowns. The ‘b’ versus ‘a’ roughly indicates whether there is a substantial lockdown later in the planning horizon to prevent a rebound epidemic. (A rebound may be possible after an appreciable number of previously infected individuals have lost their immunity and returned to the susceptible state S via the backflow.)
Figure 9.4 shows example control trajectories for all five regions. The vertical axis is \(\gamma \), the proportion of workers who are allowed to work, so any dip below 1.0 indicates a lockdown. If the social planner places a very low value on preventing COVID-19 deaths (e.g., \(M=1500\) in panel a), then there is only a small, short early lockdown which does little except to take a bit of the edge off the initial spike in infections. Such a small effort does not prevent many people from getting infected, but it shifts a few infections to later, when hospitals are less overwhelmed. When M is a little larger (specifically \(M=3200\) in panel c), then there is also a similarly small lockdown later, to take a bit of the edge off of the rebound epidemic. But in neither of those cases is there much locking down or much reduction in infections.
When M is still larger (\(M=5000\) in panel b) the later lockdown gets considerably larger—large enough to essentially prevent the rebound epidemic. Curiously, at this point the initial lockdown disappears, but it wasn’t very big to begin with, so this qualitative change is not actually a very big difference substantively. When M increases further (\(M=11,000\)) the initial lockdown reappears, albeit as a very small blip.
Then rather abruptly when M crosses the Skiba curve separating type I and II strategies from type III strategies it becomes optimal to use a very large and sustained lockdown to reduce infections and deaths dramatically. Panel e shows the particular optimal lockdown trajectory when \(M=13,000\), which is equivalent to valuing a premature death at 35 times GDP per capita. That sustained lockdown averts most of the infections and deaths, but at the considerable cost of almost 50% unemployment for about a year and a half.
Thus, when the lockdown intensity is allowed to vary continuously, many nuances emerge, but the overall character still boils down to an almost binary choice. If M is high enough, then use a sustained and forceful lockdown to largely preempt the epidemic despite massive levels of economic dislocation. Otherwise, lockdowns are too blunt and expensive to employ as the primary response to the epidemic. Thus, the model prescribes an almost all-or-nothing approach to economic lockdowns.
For certain combinations of parameter values (e.g., Fig. 9.4 panels b and d corresponding to \(M=5,000\) and \(M=11,000\)) it can be optimal to act fairly decisively against the rebound epidemic even if all one does in response to the first epidemic is a bit of curve flattening. It may seem odd to lock down more aggressively in response to the second, smaller epidemic, but the reason is eminently practical. When the reproduction number is high enough, it is very hard to prevent the epidemic from exploding if everyone is susceptible. But there is already an appreciable degree of herd immunity when the second, rebound epidemic threatens, so a less severe lockdown can be sufficient to preempt it.
9.3.1.2 Interpreting the Types of Lockdown Strategies that Can be Optimal
Table 9.1 summarises the nature and performance of each of the strategies in the right-hand panel of Fig. 9.3. Its columns merit some discussion. The lockdowns’ start and end times are self-explanatory except to note that with strategies IIa and IIb, there are two separate lockdowns, so there are two separate start and end times. The intensity of the lockdown measures the amount of unemployment that the lockdown creates on a scale where 365 corresponds to no one in the population working for an entire year.
Table 9.1 shows that even when lockdown intensity and duration are allowed to vary continuously, there are basically only three sizes that emerge as optimal: very small (less than 1.04), modest (around 30–35, or the equivalent of the economy giving up one month of economic output), and large (around 360, or the equivalent of the economy giving up a full year of economic output).
The levels of deaths also fall into basically three levels. High (around 2.9% of the population) goes with small lockdowns. Medium-high deaths (around 2.4%) goes with modest lockdowns. Small deaths (around 0.2%) goes with large lockdowns. It would be nice to have a small number of deaths despite only imposing a small lockdown, but that just isn’t possible.
In sum, there are basically three strategies: (1) Do very little locking down and suffer deaths both from the initial epidemic and also the rebound epidemic as people lose immunity, (2) Only do a bit of curve flattening during the first epidemic but use a modest sized lockdown later on to prevent the rebound epidemic and so have a medium-high number of deaths, or (3) Lockdown forcefully more or less throughout the entire planning horizon in order to avert most of the deaths altogether.
Figure 9.2 provides the corresponding information when \(R_0=2.5\). It shows that when the virus is less contagious the large lockdown does not need to be quite as large (size of 257 or about 8.5 months of lost output, not a full year) in order to hold the number of deaths down to low levels. Perhaps surprisingly, the minimalist strategies (Ia) are less minimalist when \(R_0=2.5\); when \(R_0=4.0\) the epidemic is just so powerful that it is not even worth doing as much curve flattening as it is when \(R_0=2.5\).
Table 9.1 \(R_0=4\): Data characterising the optimal solutions for the different regimes of Fig. 9.4. The size of the lockdown is defined as \(\int _{\tau _s}^{\tau _e}(1-\gamma (t))\mathrm {d}t\), where \(\tau _s\) is the starting and \(\tau _e\) the exit time of the lockdown
Table 9.2 \(R_0=2.5\). The optimal solutions for \(R_0=2.5\) evaluated at the same M values as for \(R_0=4\)
9.3.1.3 The Effects of Lockdown Fatigue
One feature of the current model is its recognition of lockdown fatigue. Recall that fatigue means that the infection-preventing benefits of an economic lockdown may be eroded over time by the public becoming less compliant, e.g., because the economic suffering produces pushback. The results above used parameter values that meant the power of that fatigue was fairly modest. In this subsection we explore how greater tendencies to fatigue can influence what strategy is optimal.
The tool again is a bifurcation diagram with the horizontal axis denoting M, the value the social planner places on preventing a premature death. (See Fig. 9.5.) Now, though, the vertical axis measures the strength of the fatigue effect, running from 0 (no effect) up to 1.0. The units of this fatigue effect are difficult to interpret, but roughly speaking, over the time horizons contemplated here, if \(f=1.0\) then when employing the sustained lockdown strategies, the lockdowns lose about half of their effectiveness by the time they are relaxed.
Figure 9.5 shows the results. When that fatigue parameter is small (lower parts of Fig. 9.5), the march across the various strategies with increasing M is the same as that depicted in Fig. 9.3. With large values (top of Fig. 9.5), there are two differences. First, Region Ib disappears but Region IIb remains, meaning if it is ever optimal to use a moderately strong lockdown to forestall a rebound epidemic, then one also does at least something in response to the first epidemic. Second, Region IIIa gives way to Region IIIb in which some degree of lockdown is maintained for an extended time, but it is relaxed somewhat between the first and rebound epidemics in order to let levels of fatigue dissipate somewhat.
The still more important lesson though pertains to the curve separating regions where some major lockdown is optimal (whether that is of type IIIa or IIIb) and regions where only small or moderate sized lockdowns are optimal (Regions Ia, Ib, IIa, or IIb). That boundary slopes upward and to the right, meaning that the greater the tendency of the public to fatigue, the higher the cost per premature death (M) has to be in order for a very strong and sustained lockdown to be optimal. That makes sense. If fatigue will undermine part of the effectiveness of a large lockdown, then the valuation of the lockdown’s benefits has to be greater in order to justify its considerable costs.
This suggests that those advocating for very long lockdowns might want to think about whether there are ways of making that lockdown more palatable in order to minimise fatigue. For example, some Canadian provinces tempered their policies limiting social interaction to people within a household bubble so that people living alone were permitted to meet with up to two other people, to avoid the mental health harms of total isolation.
9.3.1.4 Illustrating Skiba Trajectories
One key finding here is that for certain sets of parameter values, two—or sometimes even three—very different strategies can produce exactly the same net value for the social planner. We close by illustrating this phenomenon in greater detail.
Returning to Fig. 9.3b, with the higher level of infectivity believed to pertain for the UK variant of the virus, as the valuation placed on preventing a premature death (M) increases, one crosses two Skiba thresholds, one at \(M=3395\) separating Regions IIa and Ib and another at \(M=11,560\) separating Regions IIb and IIIa. These thresholds are denoted in Fig. 9.3b by solid vertical lines. They can also be seen in Fig. 9.5 by moving left to right at the bottom level (\(f=0\)).
Figure 9.6 shows the two alternate strategies, in terms of \(\gamma \), the proportion of employees who are allowed to work. The left panel shows the two equally good strategies when \(M=3395\); the right-hand panel shows the two strategies that are equally good when \(M=11,560\). We have already discussed their nature. On the left side one is choosing between two very small lockdowns and one moderately large lockdown later. On the right side one is choosing between a pair of lockdowns (very small early and moderately large later) and one very deep and sustained lockdown.
The observation to stress for present purposes is just how different the trajectories are in each pairing. When one crosses a Skiba threshold, what is optimal can change quite radically. Likewise, when one is standing exactly at that Skiba threshold, one has two equally good options, but those options are radically different.
That means that when two people advocate very different lockdown strategies in response to COVID-19, one cannot presume that they have very different understandings of the science or very different value systems. They might actually share very similar or indeed even identical worldviews, but still favour radically different policies.
Table 9.3 illustrates how this can be so. Its first column summarises the outcomes (costs) when there is no control. Health costs are enormous because more or less everyone gets infected and 2.9% of the population dies; the numbers are on a scale such that 365 is one year’s GDP, so the health cost of 335.7 is almost as bad as losing an entire year’s economic output. There are also some economic losses from losing the productivity of those who die prematurely, producing a total cost of 353.9.
The second column shows that modest deployment of lockdowns only reduces health costs by 20%, to 271.4, whereas a severe and sustained lockdown reduces them by 92.5%, to 25.3. However, the severe and sustained lockdown multiplies costs of lost labor fifteenfold, to 270.1, and creates an additional cost equivalent to 13.6 days of output from forcing businesses to adjust to changing lockdown policies. Summing across all three types of costs produces the same total of 309.1 for both types of lockdown strategies.
Thus the two lockdown strategies produce the same aggregate performance (309.1), but with very different compositions. The moderate lockdown strategy creates smaller economic costs but only reduces health costs by 20%. The severe and sustained lockdown eliminates most of the healthcare costs but creates very large economic dislocation.
What is quite sobering is that either optimal policy only reduces total social cost by 13%, from 353.9 to 309.14. The COVID-19 pandemic is truly horrible; at least within this model, even responding to it optimally alleviates only a modest share of the suffering. Lockdowns can convert health harms to economic harms, but they cannot do much to reduce the total amount of harm.
Table 9.3 Data on costs for Skiba point at \(M=11,560\)