DDid the Indian lockdown avert deaths?
Suvrat Raju
International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Shivakote,Bengaluru 560089, India.
E-mail: [email protected]
Abstract:
Within the context of SEIR models, we consider a lockdown that is both imposedand lifted at an early stage of an epidemic. We show that, in these models, although sucha lockdown may delay deaths, it eventually does not avert a significant number of fatalities.Therefore, in these models, the efficacy of a lockdown cannot be gauged by simply comparingfigures for the deaths at the end of the lockdown with the projected figure for deaths bythe same date without the lockdown. We provide a simple but robust heuristic argumentto explain why this conclusion should generalize to more elaborate compartmental models.We qualitatively discuss some important effects of a lockdown, which go beyond the scopeof simple models, but could cause it to increase or decrease an epidemic’s final toll. Giventhe significance of these effects in India, and the limitations of currently available data, weconclude that simple epidemiological models cannot be used to reliably quantify the impactof the Indian lockdown on fatalities caused by the COVID-19 pandemic. a r X i v : . [ q - b i o . P E ] A ug ontents India entered a nationwide lockdown on 24 March 2020, in response to the Coronavirusdisease (COVID-19) pandemic [1]. After an initial lockdown period of 21 days, restrictionswere gradually eased [2]. The lockdown failed to prevent a rapid growth in the number ofcases and deaths [3].On 22 May, the Indian government presented the results of numerous studies that claimedthat the lockdown had averted a significant number of deaths. One study by the BostonConsulting Group (BCG), which reportedly used a Susceptible-Exposed-Infectious-Recovered(SEIR) model, claimed that the lockdown had saved between 1,20,000 – 2,10,000 lives [4].Subsequently, the modelling subgroup of the Indian Scientists Response to COVID-19 alsoannounced that it had found, using its model, called “INDSCI-SIM”, that the lockdown hadaverted between 8,000 – 32,000 deaths by 15 May [5].These claims deserve scrutiny due to the wide publicity that they received and alsobecause they were later invoked in parliamentary documents and public statements by seniormembers of the Indian government [6, 7].The purpose of this paper is to explain why, even within the context of simple SEIRepidemiological models, and their generalizations, such claims are misleading. This is becausesuch models uniformly predict that lockdowns that are implemented early in the epidemic—and fail to quash the epidemic—have a negligible impact on the final death toll. While alockdown may delay deaths, the models themselves suggests that the fatalities in a “lockdownscenario” will rapidly catch up with fatalities in a “no lockdown scenario.”Therefore, within the framework of these models, the comparison of fatalities betweentwo scenarios by an arbitrarily chosen fixed date — in this case, 15 May — is an absurdmetric to gauge the efficacy of a lockdown. This conclusion is described in section 2.– 1 –he conclusions of epidemiological models are known to be often sensitive to the choiceof parameters [8]. However, the specific property of the models described above is very robust and insensitive to the parameters chosen. Moreover, while we use a SEIR model to obtainspecific analytic results, we explain in section 3 that our results should qualitatively generalizeto more complicated models. Of course, one may choose to eschew models altogether [9]. Butto the extent that we rely on epidemiological models at all, one of the few reliable lessonswe should take away is that lockdowns should not be evaluated by comparing fatalities by afixed date.In the real world, lockdowns may have a positive or negative impact on the epidemic thatmay be difficult to capture in the model. For instance, if the lockdown quashes the epidemic,or can be used to greatly ramp up healthcare capacity, it may reduce the final toll of theepidemic significantly. On the other hand, if the lockdown engenders economic insecurity,and makes it harder for people to implement long-term precautions like physical distancing,then it may have a negative impact on the eventual toll of the epidemic. We list these effects qualitatively in section 4. Nevertheless, we explain how these effects are much more important for the final toll of the epidemic than the direct effect of the lockdown itself. In the Appendixwe discuss the INDSCI-SIM model, as an example of an extended SEIR model that confirmsour general expectations.The conclusion of this paper is that simple epidemiological models cannot be used toreliably determine the impact of the Indian lockdown on fatalities due to the COVID-19pandemic. Rather, an analysis of whether the lockdown has been successful or not mustrely on balancing the qualitative factors described in 4. It seems apparent to us that aconsideration of these factors suggests that the Indian lockdown has contributed to worsening the final toll of the epidemic.We would like to state an important caveat at the outset. At various points in thisanalysis, we use assumptions that are appropriate for the lockdown in India but may notapply to other countries. So the analysis here should not be construed as a comment onlockdowns in other parts of the world.
We start by considering the simplest epidemiological models — the SEIR models. It is possibleto obtain simple analytic results in these models. A SEIR model was reportedly used in theBCG study [10]. Moreover, the INDSCI-SIM model can also be thought of as an extendedSEIR model.However, we will explain in section 3 why the essential elements of our discussion arerobust and model independent, and why we expect them to generalize to other models aswell. – 2 – .1 Setting
The SEIR model starts by dividing the population into four different compartments. Thepopulation is divided into a susceptible group (S), an exposed group (E), an infected group(I) and a removed group (R), which includes those who have recovered as well as thosedeceased. The differential equations that govern the model are very simple [11]. dSdt = − β SIN ; dEdt = β SIN − β E ; dIdt = β E − β I ; dRdt = β I. (2.1)We ignore natural births and deaths since they are not relevant for this analysis.All the variables above are functions of time. But, as above, instead of writing S ( t ), wewill generally suppress this dependence by writing S . We only display the time-dependence ifwe wish to refer to a specific time. For instance, S (0) is the number of susceptible individualsat time 0, taken to be the start of the epidemic, and similar conventions hold for the othervariables.Here, N is the total population size at the start of the epidemic. We assume that S (0) + E (0) + I (0) + R (0) = N . This is clearly conserved by the equations.In analyzing the model, it is convenient to define a variable F = I + E , which satisfies dFdt = I (cid:18) β SN − β (cid:19) . (2.2)Moreover, S is itself a monotonically decreasing function in this model, which is clear fromthe first equation above. It is useful, especially for the analysis of asymptotic properties,to reparameterize the time-variable in terms of the instantaneous value of the susceptiblepopulation. This leads to dFdS = (cid:18) β Nβ S (cid:19) − . (2.3)This can be easily integrated to yield the path of the epidemic in S − F space, which is SN − β β log (cid:0) SN (cid:1) + FN = K . (2.4)Here we have introduced a new constant K , which identifies the trajectory of the epidemic.We may set K using the initial conditions. For instance, if we assume that the epidemicis seeded by a single exposed case, F (0) = 1 and that the entire population, except for thisindividual, is initially susceptible: S (0) = N −
1. Then we find that K = 1 − β β log(1 − N ) forthe initial conditions above. Since, for a large population, N (cid:28)
1, we may simply approximate
K ≈ K is important because it allows us to compute the size of the epidemic asfollows. At late times, the epidemic ends and so F ( ∞ ) = 0. Therefore the final susceptible– 3 –raction, s ∞ = S ( ∞ ) /N is just given by solving s ∞ − β β log s ∞ = K . (2.5)We will be interested in the case where β > β , which is when the epidemic grows exponen-tially at the start. Then (2.5) has a single root in the range s ∞ ∈ (0 , S (0) N ), which is the onewe are interested in.To compute fatalities, we assume a fatality rate, µ . This means that a fraction µ of allthose infected do not recover but instead succumb to the infection. Note that this number isstill counted in the variable R and the fatalities, at any point of time, are just µR . Then thefinal number of fatalities, which we denote by D ∞ , is given by D ∞ = µN (1 − s ∞ ) , where s ∞ solves the equation (2.5) above.We will compare multiple scenarios below. The parameter that we vary in these scenariosis β and the quantity we examine is s ∞ . To avoid confusion, we adopt the convention thatwhenever these variables appear without a subscript, eg. β or s ∞ , they refer to the genericvariable and not a particular value. On the other hand, expressions like β ( a )1 or s ( l ) ∞ refer tothe values of these variables in a specific scenario. In this paper, we are concerned with the effect of a lockdown both imposed and lifted early in the epidemic. We make these terms precise below and compare this scenario with a no-lockdown scenario.
Lockdown scenario
In the lockdown scenario, we consider an epidemic that starts with some initial value of F (0) and S (0) = N − F (0). The epidemic starts to evolve freely with an initial value of β = β ( b )1 . At 1 − SN = (cid:15) (cid:28)
1, the value of β is abruptly changed to β ( l )1 where β ( l )1 < β ( b )1 .Then the epidemic evolves with these new parameters until 1 − SN = (cid:15) (cid:28)
1. Note that wealways have (cid:15) > (cid:15) . Then β is changed again to β ( a )1 such that β ( l )1 < β ( a )1 < β ( b )1 , and thenthe epidemic evolves freely till its end.The interpretation of the scenario above should be obvious. At the beginning, the epi-demic evolves with some initial parameters. A lockdown enforces physical distancing andbrings down the contact rate between individuals. It does not change the biological param-eters of the infection, including its incubation period or time to recovery, and therefore it ismodelled by a decrease in β . When the lockdown ends, the contact rate does not necessarilyreturn to its initial value. Since several measures, including some measure of physical distanc-ing and perhaps behavioural alterations and other precautions (such as mask-wearing) remainin place after the lockdown, the model allows for a third value of β in the post-lockdown– 4 –hase. This third value is expected to be in-between the pre-lockdown value, and the valueduring the lockdown.The fact that the lockdown is imposed early and ends early during the epidemic is indi-cated by (cid:15) (cid:28) (cid:15) (cid:28) No-lockdown scenario
In the no-lockdown scenario, we consider an epidemic that starts with the same value of F (0) as above and again take S (0) = N − F (0). The epidemic again initially evolves with β = β ( b )1 . For 1 − SN ∈ ( (cid:15) , (cid:15) ), we take β to vary as some monotonic function β (1 − SN ) withthe condition that β ( (cid:15) ) = β ( b )1 and β ( (cid:15) ) = β ( a )1 . The details of this monotonic functionwill be unimportant for our purpose. Beyond 1 − SN = (cid:15) , the epidemic evolves with β = β ( a )1 until its end.The interpretation of this scenario should also be obvious. At the beginning of theepidemic, it evolves with some initial parameters. At some point, behavioural changes areintroduced through physical distancing and other precautions. These are the same changesand precautions that set β = β ( a )1 in the lockdown scenario. These changes can often beimplemented rapidly and, in fact, they can be implemented instead of the lockdown. So itwould be quite reasonable to approximate the function β (1 − SN ) by a step function witha term proportional to θ (1 − SN − (cid:15) ). Nevertheless, we consider a more general scenariowhere these behavioural changes and precautions are implemented more gradually. But wedo assume that they are in place by the time the epidemic has reached the stage, 1 − SN = (cid:15) ,which is the stage of susceptibility where the lockdown ends in the “lockdown scenario”.However, note, in particular, that the rate, β , in the lockdown scenario is always loweror equal to its value in the no-lockdown scenario. We now compare final fatalities in the lockdown and no-lockdown scenarios.
Fatalities in the lockdown scenario
The change in parameters, upon the imposition of the lockdown, also leads to a change inthe value of K by δ K , which is given by δ K = (cid:32) β ( b )1 − β ( l )1 (cid:33) β log(1 − (cid:15) ) . (2.6)This equation is derived by noting that S and F remain continuous when the lockdown isimposed, and the change in K comes only from the change in the value of β . When thelockdown is lifted, there is an additional change in this constant, δ K , which is given by δ K = (cid:32) β ( l )1 − β ( a )1 (cid:33) β log(1 − (cid:15) ) . (2.7)– 5 –he net change in K can be expanded out to first order in (cid:15) and (cid:15) as δ K = δ K + δ K = − β (cid:32) (cid:15) − (cid:15) β ( l )1 + (cid:15) β ( b )1 − (cid:15) β ( a )1 (cid:33) + S , (2.8)where we have clubbed the higher order terms into S , since they will be largely irrelevantfor our analysis. S = − β (cid:16) (cid:15) β ( a )1 ( β ( l )1 − β ( b )1 ) + β ( b )1 (cid:15) ( β ( a )1 − β ( l )1 ) (cid:17) β ( l )1 β ( b )1 β ( a )1 + O (cid:0) (cid:15) (cid:1) . (2.9)It is convenient to compare the final toll of the epidemic by comparing it with a “reference”epidemic, where the value of β = β ( a )1 throughout the epidemic, and where the epidemic startswith the same initial value of K as the epidemic above. If we define the solution of (2.5) forthis reference case as s ref ∞ and if we denote the value of s ∞ in the lockdown scenario by s ( l ) ∞ ,then we can then use equation (2.5) to compute the difference between these two quantities.To first order, this is controlled by ds ∞ d K = − (cid:32) β β ( a )1 s ∞ − (cid:33) − , (2.10)where the quantity in brackets is positive. And therefore s ( l ) ∞ − s ref ∞ = (cid:32) β β ( a )1 s ref ∞ − (cid:33) − β (cid:32) (cid:15) − (cid:15) β ( l )1 + (cid:15) β ( b )1 − (cid:15) β ( a )1 (cid:33) + O (cid:0) (cid:15) (cid:1) . (2.11) Fatalities in the no-lockdown scenario
In the “no lockdown” scenario, we have an unknown function β (1 − SN ) which controlshow fast the long-term equilibrium value of β = β ( a )1 is achieved. Even though this functionis unknown, under the assumption that it is monotonic, we can bound the asymptotic valueof s ∞ both above and below by considering two extreme subcases within this scenario.First consider the case where the value of β changes abruptly to β ( a )1 when 1 − SN = (cid:15) .In this case, the epidemic shifts to a new trajectory where K changes by δ K = β log(1 − (cid:15) ) (cid:32) β ( b )1 − β ( a )1 (cid:33) = − β (cid:15) (cid:32) β ( b )1 − β ( a )1 (cid:33) + O (cid:0) (cid:15) (cid:1) . (2.12)The second sub-scenario is the case where the value of β stays constant until 1 − SN = (cid:15) and then changes abruptly to β ( a )1 . In this case, the value of K changes by δ K = β log(1 − (cid:15) ) (cid:32) β ( b )1 − β ( a )1 (cid:33) = − β (cid:15) (cid:32) β ( b )1 − β ( a )1 (cid:33) + O (cid:0) (cid:15) (cid:1) . (2.13)The reason to consider these two sub-scenarios is that the final value of s ∞ in the “nolockdown” scenario is bounded on both sides by its value in the cases above. In particular,– 6 –enote the final value of the susceptibility in an arbitrary no-lockdown scenario (as specifiedabove) by s ( n ) ∞ . Then, we find that independent of the details of the variation of β (1 − SN )in the range 1 − SN ∈ ( (cid:15) , (cid:15) ) in this scenario, we have s ref ∞ − s ( n ) ∞ ≥ s ref ∞ − s ( n ) ∞ = (cid:32) β β ( a )1 s ref ∞ − (cid:33) − β (cid:15) (cid:32) β ( a )1 − β ( b )1 (cid:33) . (2.14)We also have s ref ∞ − s ( n ) ∞ ≤ s ref ∞ − s ( n ) ∞ = (cid:32) β β ( a )1 s ref ∞ − (cid:33) − β (cid:15) (cid:32) β ( a )1 − β ( b )1 (cid:33) . (2.15)Here s ( n ) ∞ and s ( n ) ∞ are the final susceptible fractions in the two extreme no-lockdown sce-narios. Difference in fatalities
The difference in fatalities in the two scenarios is proportional to this difference in s ∞ inthe two scenarios. We have δD = µN ( s ( l ) ∞ − s ( n ) ∞ ) . (2.16)Even though we have an unknown interpolating function in the no-lockdown scenario we can bound δD both from above and below. We find that the factors conspire to give the simpleresult µN β (cid:18) β β ( a )1 s ref ∞ − (cid:19) ( (cid:15) − (cid:15) ) (cid:32) β ( l )1 − β ( a )1 (cid:33) ≤ δD ≤ µN β (cid:18) β β ( a )1 s ref ∞ − (cid:19) ( (cid:15) − (cid:15) ) (cid:32) β ( l )1 − β ( b )1 (cid:33) . (2.17)We will use the notation ( δD ) ( n ) for the lower bound above and ( δD ) ( n ) for the upperbound. These correspond to the difference in deaths between the lockdown scenario and thetwo extreme no-lockdown scenarios detailed above.The reason that this effect is small is because it is proportional to (cid:15) and (cid:15) and, byassumption, these numbers are small for the “early lockdown” scenario that we consider. Deaths averted and deaths during lockdown
In fact, a useful estimate of the figure of deaths averted is given by the number of deathsthat take place during the lockdown.Note that the fatalities observed during the lockdown can be computed from (2.4) to be δD ( l ) = µN β β ( l )1 δ (log( S )) ≈ µN β β ( l )1 ( (cid:15) − (cid:15) ) + O (cid:0) (cid:15) (cid:1) . (2.18)In particular we see that 1 − β ( l )1 β ( a )1 (cid:18) β β ( a )1 s ref ∞ − (cid:19) ≤ δDδD ( l ) ≤ − β ( l )1 β ( b )1 (cid:18) β β ( a )1 s ref ∞ − (cid:19) . (2.19)– 7 –ote that several uncertain parameters — including the fatality rate, µ , and even (cid:15) and (cid:15) — have dropped out of the formula above.The O (1) factors that remain above depend, in detail, on the various constants in theproblem. In fact, for several reasonable values of the parameters, these factors are smallerthan 1 meaning that deaths averted are smaller than the deaths that have taken place duringthe lockdown. But, in any case, an immediate implication is that one should be skeptical ofestimates that suggest that deaths averted are much higher than deaths during the lockdown.In India’s case, since 2743 deaths had been recorded during the lockdown, from 24 March to15 May [12], this immediately tells us that the significantly higher figures described in theIntroduction are absurd, even within the context of epidemiological models.Here, we would like emphasize an important general point. In the SEIR model, the figurefor deaths averted by the lockdown is entirely separate from the figure for the deaths thattake place in the same time period in the no-lockdown scenarios. If one considers an “effectivelockdown” that is imposed early, then even after a long time (in units set by β − , β − , β − )the value of (cid:15) may remain very small. In the meantime, the deaths in the no-lockdownscenario in the same time period will diverge exponentially from the deaths in the lockdownscenario. It is this exponentially large difference that gives rise to the figures described in theIntroduction. But, within the model, this large number is irrelevant for the final figure of thedeaths averted. The results above might seem puzzling, since “common sense” suggests that a lockdownshould “avert deaths”. Before we explain them heuristically, we provide a simple numericalexample.The point of this example is not to make realistic predictions for the Indian epidemic. Themodel is too simple to do that, and moreover the available data—including that of the totalnumber of cases and fatalities—is too flawed. Our point is only to illustrate the theoreticaleffect described above. For this reason, we deliberately do not consider the fatality data forIndia.Instead we consider a hypothetical epidemic in a population of N = 10 . The epidemic isseeded by a small set of exposed cases: E (0) = 10 , I (0) = 0 , S (0) = N −
10. The fatality rateis taken to be µ = .
005 or 0 . β = (day) − and β = (day) − . In ourhypothetical example, before the lockdown, the epidemic grows rapidly with β ( b )1 = (day) − .In the lockdown scenario, a lockdown of 70 days is imposed on day 20 of the epidemic andlifted on day 90 of the epidemic. The lockdown brings β down to β ( l )1 = (day) − . Thislockdown corresponds to the values of (cid:15) = 4 . × − and (cid:15) = 4 . × − and thereforeour assumption of an “early lockdown” is met. At the end of the lockdown, β rises againto β ( a )1 = (day) − . It does not rise to its original value, since some sustainable behaviouralchanges are made and some long-term precautions are put in place.We compare this lockdown scenario with the two “no lockdown” scenarios describedabove. In one scenario, denoted by “NL1” in the graphs below, the long-term precautions– 8 –hanges are implemented, instead of the lockdown on day 20. In the other scenario, denotedby “NL2”, the epidemic continues to evolve freely past day 20. The long-term precautionsare implemented only when 1 − SN = (cid:15) . In terms of time, this value of (cid:15) is reached at 44 . β down from β ( b )1 to β ( a )1 are put in place 25 days after the lockdown is imposedin scenario “L”. As explained above, if we take a more general variation of β we expect theasymptotic fatalities to be bounded above and below by these two scenarios.The fatalities as a function of time in the three scenarios are shown in Figure 1. This
50 100 150 200 250 300 350 40010000200003000040000
LNL1NL2
Figure 1 : Total deaths as a function of time in the three scenarios. “L” is the lockdownfollowed by long-term precautions; NL1 implements only the long-term precautions instead ofthe lockdown; NL2 implements long-term precautions after a delay. figure perfectly bears out the analysis above. When one compares the lockdown scenario toNL2 scenario on day 90, the no-lockdown scenario has had 11,722 excess deaths. However,it is absurd to conclude from this figure that the lockdown has “averted more than 11,000deaths”. This is because, at the end of 365 days, deaths in the lockdown scenario have caughtup. In fact, in the model, there are less than 56 excess deaths after 365 days compared to thelockdown scenario.The comparison with the NL1 scenario is similar. Here, in the model, there are 2078 excessdeaths compared to the lockdown scenario by day 90. But after 365 days, the difference indeaths drops below 28.The difference in deaths between the lockdown scenario and the NL1 and NL2 scenariosis also plotted in Figure 2. This graph shows that the difference in deaths appears to becomevery large at intermediate times, but then drops to a very small number by the end of theepidemic.The example also provides us with an opportunity to check our first order analyticalformulas. The match between the numerics and the first order approximations is very good– 9 –
NL1 - LNL2 - L Figure 2 : Death delayed by the lockdown. The graph shows the difference in fatalities betweenthe lockdown scenario and the no-lockdown scenarios as a function of time. By the end ofthe epidemic, deaths in the lockdown scenario catch up almost entirely with the no-lockdownscenarios. as shown in the table below.Quantity Numerical( N ) First-order Approx( A ) Error (cid:16) | N − A | N (cid:17) δ K − . × − − . × − . × − δ K . × − . × − . × − δ K . × − . × − . × − δD ( l ) .
64 27 .
51 4 . × − δD ( n ) .
92 27 .
78 4 . × − δD ( n ) .
70 55 .
57 2 . × − We also note that for these parameters, the lower and upper bounds that appear in (2.19)are 0.17 and 0.34 respectively. In this model, the deaths that take place during the lockdownare given by δD ( l ) = 162 .
75. As expected, the two extreme no-lockdown solutions saturatethe lower and upper bound of (2.19). So, in this example, the deaths averted are significantly smaller than the deaths that take place during the lockdown.The reader might wonder, from Figure 1 whether, in the lockdown scenario, the peakof infections is “flattened”. However, even this does not happen. For the parameters above,although the peaks occur at significantly different times, the actual peak values of I in theNL2, NL1 and L scenarios are similar and given by I ( n )max = 9 . × ; I ( n )max = 9 . × ; I ( l )max = 9 . × . (2.20)So, we have I ( n )max − I ( l )max N = 1 . × − ; I ( n )max − I ( l )max N = 5 . × − . (2.21)This is entirely consistent with the understanding that we develop in section 3. Althoughthe focus of this note is on “deaths averted” in fact, in reasonable models, all the intrinsic – 10 – roperties of the trajectory with an early lockdown are very close to those of the trajectorywithout the lockdown. Epidemiological models are often unreliable and subject to sensitivity in the parameters. Infact, this author is of the opinion that the only predictions of models that should be regardedas robust and taken seriously are those that can also be reproduced by a simple heuristicargument. So in this section we provide a heuristic argument to explain the result above,which states that a lockdown has only a small impact on the final number of deaths. We thenuse our heuristic explanation to generalize our result to extensions of the SEIR model.
The formula (2.17) has two aspects. The first aspect, which is the important one qualitatively,is that the result for deaths averted is O ( (cid:15) ). This means that, in the limit, where we take thelockdown to be imposed and lifted earlier and earlier in the epidemic, its effect on the finalnumber of fatalities vanishes. This is easy to understand. In such a situation, the lockdownonly translates the entire curve of infections to the right, without having any appreciableimpact on its shape. In particular, at leading order, an early lockdown does not avert fatalities at all.This is a robust argument, and independent of the details of the model. Therefore weexpect that, even in more complicated models, lockdowns, by themselves, should be ineffectivein reducing deaths.The formula (2.17) also has an O ( (cid:15) ) piece and this term can also be understood through aslightly more elaborate heuristic argument. The key point is to track the epidemic’s progressin terms of the fraction of the population that remains susceptible, rather than directly usethe time-variable. Even in more complicated models, with a number of compartments, aslong as recovery from infection grants immunity, the fraction of the population that remainssusceptible is a monotonic function and so it can be used to track the progress of the epidemic.Let us use this to understand how the results obtained above for the SEIR model canbe understood heuristically. In this model, at the start when 1 − SN ≈
0, each individualeffectively infects β ( b )1 β other individuals. In one of the extreme no-lockdown scenarios above,when 1 − SN = (cid:15) , this number is lowered to β ( a )1 β . The epidemic grows until SN = β β ( a )1 . Beyondthis threshold, the epidemic starts to decay but it does not end. Importantly the final toll ofthe epidemic is controlled by the threshold where it starts to decay but also the number ofexposed and infected individuals at this threshold. These individuals cause an “overshoot”beyond the “herd immunity” threshold and increase the final toll of the epidemic.Now the lockdown alters this trajectory as follows. First note that since it takes longerfor the population to reach the level 1 − SN = (cid:15) , the lockdown tends to advance recoveries overthe no-lockdown scenario. This effect is clearly proportional to (cid:15) − (cid:15) . But this implies that– 11 –he total number of infected and exposed individuals are smaller in the lockdown scenariothan in the no-lockdown scenario, when the two epidemics are compared at the same valueof 1 − SN . This effect is proportional to ( (cid:15) − (cid:15) ) since R + F + S = N and so the change in F is precisely the change in S . This effect causes the epidemic to end at a lower value of SN than without the lockdown. Since the final number of fatalities is controlled by the end-pointof the epidemic, one sees that the lockdown lowers fatalities by a term that is proportionalto (cid:15) − (cid:15) .The comparison of the lockdown scenario with the other extreme no-lockdown scenario— where behavioural changes are implemented instead of the lockdown and at the same timeas the lockdown— is similar. One again finds that fatalities in the lockdown scenario arelower by a factor that is proportional to (cid:15) − (cid:15) . We now explain why we expect the heuristic argument above to continue to apply to moregeneral compartmental models.In more complicated models, the population is often divided into additional compart-ments. For instance, instead of simply considering “infected individuals” one may dividethem into a group that is “asymptomatic” and another that is “symptomatic”. The detailsof these compartments are not important for our argument and say that we have divided thepopulation into Q -compartments, whose occupancy we denote by C i , with i = 1 . . . Q .The dynamics associated with the additional compartments will also be different fromthe SEIR model. For instance, “asymptomatic” and “symptomatic” individuals might passon the infection to susceptible individuals at different rates. The details of these dynamicsare also not important for the argument, and say that the trajectory of the epidemic withoutthe lockdown follows a path d C ( n ) i dS = F i ( SN , C ( n ) i N ) , (3.1)for some set of functions F i . These functions are O (1) functions of O (1) quantities andnot expected to scale with N since the left hand side does not scale with N either. Thisrepresents the fact that in such models, the important quantity is the fraction of the totalpopulation that is in a compartment and not the absolute number for the occupancy of thatcompartment.The framework described by (3.1) is quite general. For instance, it can be used toeasily model a fatality rate that is a function of the instantaneous number of infections orthe occupancy of another compartment in the model. To account for this possible effect,we introduce a compartment for fatalities, which we call C D . In the SEIR model, with aconstant death-rate, this would always just be proportional to recoveries, but if the fatalityrate depends on other variables, it might be different.Note also that, in writing the equations in the form (3.1), where the trajectory is pa-rameterized by the susceptibility, we have assumed that there is no explicit time-dependencein the parameters that govern the model. This assumption may be false if, for instance, the– 12 –ealthcare system undergoes a rapid improvement over a short period of time. We return tothis issue in section 4.Now, a lockdown alters the dynamics of the epidemic for a limited period. This meansthat the lockdown changes the trajectory of the epidemic to one that satisfies d C ( l ) i dS = F i ( SN , C ( l ) i N ) + θ (1 − SN − (cid:15) ) θ ( (cid:15) − SN ) G i ( SN , C ( l ) i N ) . (3.2)Here, during the lockdown, which starts when the fraction of the susceptible population is1 − (cid:15) and lasts till it becomes 1 − (cid:15) , we have assumed that the trajectory is controlled bysome new evolution functions that differ from the old ones by G i .One may choose to set the start and end-points of the lockdown in terms of time, ratherthan susceptibility, but equation (3.2) still applies with (cid:15) and (cid:15) merely being set by thevalue of 1 − SN at the starting and ending points of the lockdown.Since the impact of the lockdown is constrained to the interval 1 − SN ∈ ( (cid:15) , (cid:15) ), thetrajectories (3.1) and (3.2) satisfy the same differential equation both before and after thisinterval. In particular, if we compare the trajectories of the two scenarios after the lockdown,then it is only the initial conditions for these two trajectories that are different. Moreover,these initial conditions differ by only a small amount. This can be written as1 N (cid:12)(cid:12)(cid:12) C ( l ) i ( S = N (1 − (cid:15) )) − C ( n ) i ( S = N (1 − (cid:15) )) (cid:12)(cid:12)(cid:12) = ( (cid:15) − (cid:15) ) ρ i , (3.3)where ρ i is some set of model-dependent O (1) constants controlled by the manner in whichthe lockdown modifies dynamics.In the SEIR model the “termination condition” for the epidemic was simply F = 0. Saythat this termination condition generalizes to T ( S, C i ) = 0 , (3.4)in the more elaborate model. We are interested in the value of S , at which (3.4) is satisfiedand the expected fatalities can then be computed as C D ( S ). But as a result of (3.3), weexpect that if (3.4) is solved by some value S ( l ) for the lockdown trajectory and some othervalue S ( n ) for the no-lockdown trajectory then C D ( S ( n ) ) N − C D ( S ( l ) ) N = ( (cid:15) − (cid:15) ) σ, (3.5)where σ is again some O (1) constant that depends on the model.The assumption here is that the differential equations that govern the epidemic, (3.1), are not chaotic so that a small difference in the initial conditions, (3.3), leads to a small differencein the asymptotic value of SN . This is a reasonable assumption to make about epidemiologicalmodels. As discussed in section 4 there are some exceptions to this assumption. For instance, it may happen thatwhen the C i are below some threshold, the healthcare system can keep them below that threshold indefinitelyand not otherwise. In such a case, small differences in initial conditions could make a significant differencein the final outcome. These exceptions, which apply when the epidemic is quashed by the lockdown, do notapply to the Indian lockdown as explained in section 4. – 13 –o summarize: while the precise factors in (2.17) are specific to the simple SEIR model,in more complicated models, we still expect that the number of deaths averted by a lockdownimposed and lifted early in the epidemic will be O ( (cid:15) )While we have paid special attention to the question of fatalities averted, the argumentabove is general enough that it can be applied to any intrinsic property of the trajectory. Forinstance, one may introduce a compartment corresponding to hospitalized patients, and askabout the peak occupancy of this compartment. But the argument above tells us that, whenparameterized by the susceptible population, the entire trajectory of the lockdown scenariois separated from the no-lockdown scenario by a difference proportional to ( (cid:15) − (cid:15) ).Therefore, the argument implies that the difference between this peak, in the lockdownand no-lockdown scenarios, will also be proportional to ( (cid:15) − (cid:15) ). An early lockdown may delay the peak in terms of time, but it will not bring down its height. In Appendix A weverify some of these predictions for one example of an extended SEIR model.In this section, we have heuristically argued and verified that even in general compart-mental models, the effect of a lockdown is small. The reader might correctly point out thatif the population is large then even an O ( (cid:15) ) fraction might translate into a large number,in terms of actual lives. However, it is important to understand that these O ( (cid:15) ) predictionsmade by models are not reliable. As we explain in section 4, the real-world impact of alockdown is controlled by different factors, which are not accounted for in simple models, butcan change the outcome of the epidemic much more significantly than these O ( (cid:15) ) effects. The results above apply in an idealized setting, and so we now describe several real-worldcaveats to the results. In reality, a lockdown may have both negative and positive impactson the final toll of an epidemic. We list some of the factors that could lead to this. We areespecially interested in those factors that might be relevant in India’s case.
A lockdown may have a number of negative long-term impacts.1.
Lockdown leading to larger long-term values of β We described above how the eventual toll of the epidemic is almost entirely controlledby its rate of spread after the lockdown ends; we denoted the relevant parameter by β ( a )1 above. In reality, β ( a )1 depends on long-term sustainable precautions that individuals cantake. A lockdown that fails to emphasize welfare — as has been the case in India —may make it harder for people to implement such precautions.For instance, measures such as reducing the number of working days in a week, or im-plementing staggered hours for markets and shops to reduce crowding can contribute toa reduction of β ( a )1 . However, such precautions almost invariably have an economic cost.If the lockdown has engendered economic insecurity, and chipped away at the economic– 14 –nd social reserves of people, it may make it harder to implement such precautionssustainably in the long term.Therefore, the lockdown, by itself, may cause an increase in the long term value of β ,by making it harder for people to implement sustainable physical-distancing measures.This can easily overwhelm the very small O ( (cid:15) ) direct gains in deaths averted due to thelockdown.2. Lockdown leading to a humanitarian crisis
More than 90% of India’s workforce is employed in the unorganized sector, or infor-mally employed in the organized sector [13]. Consequently, the economic impact of thelockdown has been severe. The Center for Monitoring Indian Economy estimated that,despite a slightly improvement in the month of May compared to the month of April,“over a 100 million people were still out of jobs compared to employment in 2019-20”[14]. Another glimpse of the extent of the crisis is provided by the observation made bythe Stranded Workers Action Network (SWAN). SWAN contacted thousands of workersbut found that “90% . . . did not get paid by their employers” and “96% . . . did not getrations from the government” [15].This humanitarian crisis is a direct consequence of the lockdown, and constitutes thesingle most significant impact that the lockdown has had on India’s people.3.
Lockdown reducing access to healthcare
A lockdown may make it harder for people to access healthcare for other conditions. Itis known that tuberculosis (TB), by itself, claims about 450,000 lives every year in India[16]. But, as a direct consequence of the lockdown, reports suggest that the number ofnew TB cases notified in government healthcare centers fell very sharply in April 2020when compared with the previous year [17]. If the Indian lockdown causes the rate ofTB deaths to rise by even a fraction, it could again easily overwhelm the O ( (cid:15) ) gainsmade in combating COVID-19.
A lockdown may possibly have positive long-term effects although it is not clear if such effectsare directly visible in India’s case.1.
Possibility of quashing the epidemic
A lockdown may succeed in quashing the epidemic or reducing the epidemic to a verylow level so that, through testing, tracing and other interventions, its rate of spreadcan subsequently be kept low for a long period. When the number of cases is very low,the dynamics of an epidemic cannot be reliably understood through compartmentalmodels. For instance, at the time of writing, the lockdown in New Zealand is believedto have successfully limited COVID-19 transmission there [18]. Similarly, the lockdownin Hubei that provided the template for other lockdowns, brought the epidemic down– 15 –o such a low level that testing-and-tracing has subsequently succeeded in keeping thenumber of new infections very limited [19].This is irrelevant for the discussion in India, where the epidemic has evidently not beenquashed by the lockdown.2.
Late lockdowns
If a lockdown is imposed late in the epidemic, when the number of new infections isclose to its peak, then even a simple lockdown may reduce the number of fatalitiesby reducing the “overshoot” beyond herd immunity. In this situation, the analysis ofsection 2 would not be valid since (cid:15) , (cid:15) would not be small.3. Preparing the healthcare system
A lockdown may reduce the absolute number of cases, and so reduce the immediatestress on the healthcare system. It may also provide time for the state to ramp uphealth facilities. We use this term to include the ramping-up of testing and quarantinefacilities.We note that the observation that the lockdown has currently prevented healthcarefacilities from being overwhelmed does not, by itself, provide a compelling rationale forthe lockdown. This involves the same logical error that we described above in the caseof fatalities. If the lockdown merely delays the date on which the healthcare system isoverwhelmed, then this implies that its long-term positive-effects are limited.Therefore, the question that must be asked is whether the lockdown has been used toramp up health facilities so that they are better-prepared to confront the epidemic inthe long run.4.
Behavioural changes
It is sometimes suggested that the lockdown was necessary to promote behaviouralchange and to persuade individuals to adopt precautions [20]. We list this factor forcompleteness but we do not believe that this constitutes a genuinely long-term impact.For instance, while wearing masks in public or hand-washing where possible are bothimportant precautions, it appears clear that the these precautions could have beenencouraged even without the lockdown.5.
A Vaccine
If a vaccine for a disease is already available, the delay induced by a lockdown can,in principle, be used to vaccinate a large number of people. However, in the case ofthe COVID-19 pandemic, in spite of early encouraging results [21] it appears clearthat a vaccine will not be available at least for several months and perhaps for longer.Moreover, even after a safe vaccine becomes available, it will take a considerable amountof time to actually vaccinate a significant fraction of the population in India. Since thesetime-scales are so much longer than the time-scale of the delay induced by the Indian– 16 –ockdown, the potential availability of a vaccine in the future cannot be used to evaluatethe lockdown in India.The objective of this paper is not to enter into a detailed discussion of these qualitativeeffects. Here, we would merely like to point out that the factors above are far more im-portant than meaningless measures of deaths-delayed by a fixed date. This is because evensmall changes in the factors above — either in the positive or the negative direction — cancompletely overwhelm the O ( (cid:15) ) fatalities averted by the lockdown. So the lockdown mustbe gauged by evaluating whether the positive long-term effects above outweigh its negativelong-term effects.In principle, some of the effects above can be included as parameters in models. Forinstance one could include a “back-reaction effect” of the lockdown accounting for its impacton the long term value of β . But, the available data in India does not allow for a quantitativeanalysis of any of these factors, and we are not aware of any such analysis having beenperformed. So, at the moment, the comparison of the positive and negative effects of thelockdown must necessarily rely on a qualitative, and not a quantitative, analysis. An analysis of simple epidemiological models, performed in section 2 and section 3, wouldsuggest that the Indian lockdown has largely delayed deaths, and averted only a small numberof deaths.Nevertheless, some groups have attempted to claim that the lockdown has averted asignificant number of deaths by using SEIR-models or more-general compartmental models.These estimates appear to compare fatalities between scenarios by an arbitrarily chosen fixeddate.We showed that such claims are misleading and appear to arise from a lack of appreci-ation of some elementary aspects of the dynamics of such models. We emphasize that whileour precise formulas were derived in the context of SEIR models in section 2, our heuristicarguments from section 3 are robust and also apply to more complicated models.Simple epidemiological models do not account for several important real-world effects ofthe lockdown. This means that question of whether the Indian lockdown has really averteddeaths must rely on a comparison of the effects described in section 4. A quantitative analysisof these effects is not possible given the extant data.So the factors listed in section 4 must be compared qualitatively. In this paper, we do notprovide a detailed qualitative analysis of these factors. Nevertheless, to this author, it appearsclear that the Indian lockdown has not led to any dramatic improvement in the ability of thehealthcare system to address the COVID-19 pandemic. And the steps that have been takento ramp up the production of Personal Protective Equipment (PPE) [22], or increase testing[23] appear to be far from sufficient to offset the humanitarian and healthcare crises causedby the lockdown, or even the negative impact that the lockdown has had on the ability of– 17 –eople to sustainably maintain physical-distancing measures in the long run. Therefore, thisauthor is of the opinion that the lockdown of the country has exacerbated the challenges ofthe COVID-19 pandemic and created the risk of excess deaths, rather than averting deaths.Viable alternatives to a nationwide lockdown, which were advocated by several publichealth experts [22], may have included early testing and tracing, localized lockdowns, and apromotion of more sustainable physical-distancing measures, together with a strong focus onthe welfare of people affected by restrictions.
Acknowledgments.
I am grateful to Pinaki Chaudhuri, Sourendu Gupta, Alok Laddha,Gautam Menon, Subroto Mukerjee, Madhusudhan Raman, Joseph Samuel, Srikanth Sastry,Ramachandran Shankar, Rahul Siddharthan and Supurna Sinha for comments on a draft ofthis manuscript. I am also grateful to members of the Indian Scientists Response to COVID-19collective and the Politically Mathematics collective for several discussions on related issues.
AppendixA The INDSCI-SIM model
In this appendix, we discuss the INDSCI-SIM model [24], since this is an extended compart-mental model and it was also used to generate one of the figures mentioned in the Introduction.Although the predictions of INDSCI-SIM have been discussed in the popular media, asof the time this paper was written, the modellers had released neither a detailed preprintdescribing the model and related calculations, nor the computer code for the model. In thediscussion below, we are forced to rely on an online tool that the modellers created, and somesummary slides of documentation that they have put up. Both of these are accessible throughthe model’s website. We found this information insufficient to recreate, in detail, the varioussteps that led to the final figure for “deaths averted” due to the lockdown that the modellersannounced. Therefore we have been forced to make plausible inferences in what follows.
Some generalities
The INDSCI-SIM model differs from the SEIR model, since it has 9 compartments insteadof 4, and additionally envisions a division of the Indian population into state-level populationsthat interact with each other. For instance, there are compartments corresponding to thosewho are “asymptomatic”, “pre-symptomatic”, “mildly symptomatic”, “hospitalized” etc.While the dynamics of this model are more complicated , it would be erroneous to concludethat they are a more accurate representation of reality as we now explain.Even for a single sub-population, the model appears to have at least 14 parameters.Some of these are clinical: a parameter for the fraction of asymptomatic individuals, anotherparameter for the fraction for those who are pre-symptomatic but have mild symptoms etc.Some other parameters correspond to social dynamics: a parameter controlling the rate ofcontacts between asymptomatic and susceptible individuals, another controlling the same rate– 18 –or mildly symptomatic individuals etc. If one consider multiple sub-populations, then thereare additional parameters that control the contacts between these populations.What is common to all these parameters is that they are poorly understood and havecertainly not been reliably quantified. As is well known to modellers, given a surfeit ofparameters, it is possible to fit a variety of data. Moreover, by tweaking these parameters, itis often easy to obtain a variety of predictions.Nevertheless, as we explained above, the result that “a lockdown does not avert deaths,but only delays them,” is rather robust. This result constitutes one of the few robust predic-tions made by models like INDSCI-SIM. This result is also mentioned in the documentationfor the model (slide 19 in v1.0).We used the model’s online tool on 4 June 2020 to verify this property of the model.We maintained the tool’s default settings, which seed the epidemic only in Maharashtra.(We emphasize that these default parameters are not realistic and used only for purposes ofillustration.)We set the epidemic to start on 1 March, and allowed it to run for 300 days. Wecompared the following two scenarios. In the first scenario, no intervention was introduced.In the second scenario, we set the intervention to what the tool calls a “simple lockdown”beginning on day 20 and continuing till day 90.We found that, without the lockdown, the number of deaths in Maharashtra by 25 De-cember was 31,372. With the lockdown, the number of deaths by that date was 31,302. So,in this example, the INDSCI-SIM model itself “predicts” that a simple lockdown of 70 dayswill save 70 lives asymptotically.On the other hand, if one were to do something arbitrary like comparing deaths by15 May, then one would find in this example that, without the lockdown, there would be8,400 deaths whereas the lockdown would bring this number down to 217. But, of course, itwould be extremely misleading to conclude that the lockdown has “averted more than 8,000deaths”. Even in this model, it has only delayed these deaths and they rapidly catch up withthe no-lockdown scenario.
Perspective on lockdowns
In spite of this, the modellers suggest that a lockdown may be effective in the long term.(Slides 19 and 20 in v1.0.)This is because, apart from the parameters above, the model introduces additional param-eters corresponding to an exponentially increasing ability of the state to test and quarantineinfected individuals. Mathematically, this is done by inserting an exponentially decaying in-tensity of contacts between infected and susceptible individuals. If γ i are the parameters thatcontrol this intensity (which are generalizations of the parameter β in our discussion of theSEIR model), the INDSCI-SIM model sets γ i ( t ) = γ i (0) e − tτi (A.1)with different time-constants, τ i , for the different categories of infected individuals.– 19 –ince the parameters in the model now have an explicit time-dependence, the analysisof section 3.2 is no longer formally valid. In words, the assumption is that the period of thelockdown is used to exponentially improve the ability of the healthcare system to test andquarantine. And as we explained in section 4, it is true that, in principle , such an exponentialimprovement would lead to a long-term positive effect.There are two important issues. The first is that the model does not include parametersfor any of the negative long-term impacts of the lockdown, but only this possible positiveimpact. So the benefits of a lockdown are almost inbuilt into this model unless one sets τ i → ∞ .But a more serious problem is that there is no good independent measure of how muchthe healthcare system (including testing) has improved during the lockdown. Therefore thereis no good way to quantify the parameters τ i that enter above. Prediction for deaths averted
We now turn to the INDSCI-SIM claim that the Indian lockdown averted between 8,000to 32,000 deaths and explain why this conclusion is baseless.The recorded fatality data [12] for India is shown from 11 March (the data of the firstfatality) to 15 May (the time frame for the analysis under consideration) in Figure 3.
10 20 30 40 50 60 Days2468Log [ deaths ] Figure 3 : Logarithm of the number of recorded deaths in India from 11 March to 15 May.The start of the lockdown is marked with a dashed line.
The reader will immediately note that there are very few data points before the lockdown.In fact, the number of data points is smaller than the number of parameters for a single sub-population in the INDSCI-SIM model.Consequently, while the rate of growth of fatalities clearly slowed in April, there is no way to separate a number of effects above: the effect of the enforced physical distancing, the effectof increased testing-quarantining, the effect of behavioural changes and other precautions and– 20 –he fact that the data might itself be flawed due to improvement in disease-surveillance overtime [25]. But this division is extremely important for understanding whether the lockdownhas had a long-term impact since, as we have explained in great detail above: (a) the effects ofenforced physical distancing due to the lockdown are transient (b) the effects of behaviouralchanges are long-lasting but not necessarily related to the lockdown (c) the effects of increasedtesting-quarantining are long-lasting.Faced with this difficulty, the modellers of INDSCI-SIM appear to have adopted thefollowing procedure, as far as we can discern from v1.2 of the documentation. They simplyestimated a rate of exponential growth by fitting the data until April 3. (The significance ofthis particular date is unclear to us.) Then they extrapolated this rate of exponential growthto May 15, and compared it with the known cases on that date. They declared that thedifference of these two figures was the number of “deaths averted” by the lockdown. Sincethe estimate of the early exponential rate is prone to errors, which can have a significantimpact over this time-period, the modellers obtained the large range of 8,000–32,000 “deathsaverted.”We note two obvious points1. The idea that the calculation above yields the “deaths averted” by the lockdown is ab-surd. In fact, if the slowdown in the rate of growth in Figure 3 arises from a combinationof enforced physical distancing and behavioural changes (rather than increased testing-quarantining), then our analysis shows that the final number of deaths averted by thelockdown itself — even before accounting for its other long-term negative impacts — islikely to be very small.2. The simple analysis of extrapolating the rate of exponential growth beyond April 3can be done without the INDSCI-SIM model, since any model would yield exponentialgrowth in the early stages. Indeed, this procedure is probably the same as the procedureused by other groups mentioned in the Introduction. Therefore, not only is the finalfigure misleading, it is also misleading to suggest that the computation has anything todo with a sophisticated epidemiological model. References [1] V. Chandrashekhar, “1.3 billion people. a 21-day lockdown. can india curb the coronavirus?,”
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