key: cord-1051240-054tzr9d
authors: Bonsall, M.; Huntingford, C.; Rawson, T.
title: Optimal time to return to normality: parallel use of COVID-19 vaccines and circuit breakers
date: 2021-02-03
journal: nan
DOI: 10.1101/2021.02.01.21250877
sha: 0c37e71bfb7dd0831497a5376d6a4a27d2a18d27
doc_id: 1051240
cord_uid: 054tzr9d

By January 2020, the COVID-19 illness has caused over two million deaths. Countries have restricted disease spread through non-pharmaceutical interventions (e.g., social distancing). More severe 'lockdowns' have also been required. Although lockdowns keep people safer from the virus, they substantially disrupt economies and individual well-being. Fortunately, vaccines are becoming available. Yet, vaccination programs may take several months to implement, requiring further time for individuals to develop immunity following inoculation. To prevent health services being overwhelmed it may be necessary to implement further lockdowns in conjunction with vaccination. Here, we investigate optimal approaches for vaccination under varying lockdown lengths and/or severities to prevent COVID-19-related deaths exceeding critical thresholds. We find increases in vaccination rate cause a disproportionately larger decrease in lockdowns: with vaccination, severe lockdowns can reduce infections by up to 89%. Notably, we include demographics, modelling three groups: vulnerable, front-line workers, and non-vulnerable. We investigate the sequence of vaccination. One counter-intuitive finding is that even though the vulnerable group is high risk, demographically, this is a small group (per person, vaccination occurs more slowly) so vaccinating this group first achieves limited gains in overall disease control. Better disease control occurs by vaccinating the non-vulnerable group with longer and/or more severe lockdowns

The emergence of the SARS-CoV-2 virus in late 2019 has had, subsequently, devastating consequences across the globe, with all countries reporting levels of virus infection affecting public health 23 responses. Since the emergence of this novel coronavirus, the policies implemented (in part using 24 insights from previous pandemics) have focused on non-pharmaceutical interventions (NPIs) and 25 approaches such as travel bans, limited household mixing, stay-at-home orders ('lockdowns') and 26 quarantines, social distancing and the closure and restriction of large mass gatherings. These NPIs 27 require effective implementation, behavioural shifts and continued political and public support [1, 2] 

The dynamics for infected individuals are such that the number increases as susceptibles pass 78 through an incubation period of length τ become infectious. The number of infected decreases as 79 people either die from COVID-19, die naturally, or recover. Hence I follows:

The dynamics for recovered individuals are such that the number increases as people recover from 81 infection, or decreases due to vaccination, non-COVID-19-related death, or loss of immunity, giving:

Finally the dynamics for the total number of vaccinated individuals is dependent on the vaccination 83 rate, lowered only by a background death rate, and therefore there is an assumption of no immunity 

where I i N i is the frequency-dependent transmission function in the i th group. The expected probabil-117 ity that a susceptible individual (in group j) acquires an infection, from any source, is then the sum 118 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint of the products of transmission rate and proportion of infected individuals in the i th group, for all i.

The use of equation (6) allows us to retain the mathematical structure of the epidemiological 121 dynamics of (1)-(4), now applied to the i th group as:

where vaccination rate (ν i ), disease-induced death rate (α i ) and recovery rate (γ i ) are group-specific 123 parameters. The rate of loss of immunity (σ), background death rate (µ), and virus incubation time 124 (τ ) are population-level parameters are instead independent of group structure. For each group adjoint equations used to identify the optimum vaccination rate are presented in the Appendix.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint Vaccination, different cohorts and lockdowns 138 We also investigate the interplay between vaccination strategies amongst population groups and 139 length of lockdowns, so as to achieve the optimal suppression of the virus (in terms of disease-140 induced mortality) for a variety of considered lockdown lengths and effectiveness, while keeping 141 cumulative mortalities below a critical threshold. To do so we use the structured model (equa-142 tions (7)-(10)). This model framework accounts for the three groups of front-line workers (F R), 143 vulnerable (V ) and non-vulnerable (N V ), and was solved numerically over 150 days. The model 144 was parameterized with parameters estimated to be appropriate to the UK (Tables 1 & 2) . We 145 calculate the optimum vaccination strategies across different population groups.

Optimal lockdown times 147 Importantly, we investigate the hypothesis that lockdowns can be used to support, in parallel, on-148 going vaccination programs in order to prevent infection rates crossing key thresholds. In particu-149 lar, we consider how different strategies concerning the order in which groups vulnerable, front-line 150 workers, and non-vulnerable are inoculated, for the same maximum infection rate thresholds, im-151 pacts on length of lockdown. Hence the optimum we search for is the shortest lockdown strategies 152 for each threshold. Again, we solve the structured epidemiology model (equations (7)-(10)) numer-153 ically for different sequences of vaccination over 150 days. The chosen sequence is to deliver vaccine 154 to first group for 30 days, first and second group from 31-60 days and all groups after 60 days.

Critically, the vaccination rates, expressed as fraction of cohort vaccinated per day, are common to 156 each group. This implies that as the vulnerable group is smaller, the number of people vaccinated 157 per day is smaller, likely reflecting the actual situation. The optimal outcome in terms of group 158 order vaccination strategy to achieve the shortest lockdown times, while maintaining cumulative 159 mortalities below a critical threshold, is also determined for varying lockdown severities (i.e. levels CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint this idea of elasticity in demand has been linked to vaccine uptake and disease prevalence [16] . With 165 inelastic demand in vaccine, as disease prevalence falls and assuming that vaccination levels can be 166 maintained, then the public health benefits of vaccination (and the resulting herd immunity) can 167 be maintained as infection levels fall. However if vaccine uptake is elastic with respect to disease 168 prevalence, such that as disease prevalence falls individuals are less likely to take a vaccine, then 169 the public health control of infections can be more challenging. Elasticity can prevent achieving 170 the vaccination levels needed to reach the critical threshold for herd immunity. severities. Our working hypothesis, that we test, is that lockdowns can be used to offset vaccine 175 elasticities to maintain and/or achieve virus control. We use numerical approaches to solve the full 176 structured model over time (150 days) to find the optimal vaccination level, lockdown duration and 177 severities that keep cumulative mortality below a critical threshold. The code used for all numerical analyses and simulations is available at https://osf.io/xvunt/.

Optimal vaccination 182 Solving the constrained optimisation problem (Appendix) shows that the optimal vaccination strat-183 egy is a function of the ratio between the susceptibles, recovered and vaccinated individuals:

where λ i are adjoint (Lagrange) multiplier variables associated with the state variables (S, R and Solutions for the optimal vaccination rate are shown in Figure 1 . As planned, coupled with the 189 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint threshold mortality condition, an optimal vaccination strategy can suppress the COVID-19 epi-190 demic ( Figure 1A ). However, for increasing lockdown severity (20-80% reduction in transmission The (optimal) vaccination strategy is influenced by the stringency of the mortality threshold (Z) 200 ( Figure 1B ). As might be expected, a more stringent threshold necessitates higher levels of vaccina-201 tion to achieve the expected level of control to ensure disease-induced mortality remains below that 202 critical threshold. As also expected, decreasing potential disease transmission (e.g. through the 203 use of NPIs) offsets the need for high levels of vaccination to achieve the necessary levels of disease 204 control ( Figure 1B) . Notable is the strong non-linearity in Figure 1 and, under high vaccine coverage, the necessity for these lockdowns. While still essential, lockdowns 212 can instead be of short duration when partnered with a vaccination program. This is again to limit 213 cumulative mortalities to different levels. Under lockdowns where the level of transmission is only 214 reduced by 20% or 40%, cumulative mortalities due to the virus are expected to be excessively high 215 unless vaccine coverage is also high. Under more severe lockdowns where transmission is reduced by 216 60% or 80% (Figure 2 ), shorter circuit breakers can be sufficient to limit disease-induced mortalities.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint More specifically, for vaccination to be effective in reducing cumulative mortality (set here at 219 ∼ 50, 000 over 150 days), and in the range of anticipated vaccination rate (0.006 -0.0152), then 220 lockdown severity needs to reduce transmission by at least 60% or lockdowns need to be unac-221 ceptably long and extend for more that 90 days. Furthermore, these sort of lockdowns or circuit 222 breakers can be used to offset weakly efficacious vaccine rates to ensure that disease-related mor- To understand more fully vaccine delivery strategies between different parts of society, we begin by 226 determining how focusing initial vaccine delivery on single groups affects the likelihood of keeping 227 levels of mortality below the (optimal) threshold. If vaccines are delivered singly to vulnerable or 228 key worker groups then lockdowns would still be necessary and they would need to be reasonably 229 severe (> 60% reduction in transmission) to reduce cumulative mortality below key thresholds 230 ( Figure 4 ). Notable is that less severe lockdowns when vaccinating either of these two groups are 231 not sufficient to keep cumulative mortality below a critical threshold. This key finding is critical, and 232 requires consideration in light of the decision by many countries to vaccinate the most vulnerable 233 first. The reason for this finding is due to the demographic differences between these two groups 234 (where population sizes in these groups are relatively small) compared to the non-vulnerable group 235 (which contains the majority of the population). For example, consider a vulnerable (V) group of 236 500,000 people, and a non-vulnerable (NV) group of 10,000,000 people. If we vaccinate roughly 5% 237 of each group each day this would lead to vaccinating either 25,000 vulnerable people, or 500,000 238 non-vulnerable people, daily. Hence, even if the vulnerable people are an order of magnitude more 239 likely to die of COVID-19, more lives will be saved by vaccinating the 500,000 non-vulnerable group.

For some nations this scenario might be likely, if non-vulnerable people can be vaccinated (in units 241 of people per day) at a speed that is an order of magnitude larger -for instance through mass 242 vaccination centres that the vulnerable are unable to reach. In those circumstances, a more robust 243 approach to achieve keeping mortality below the critical threshold would be to vaccinate across the 244 non-vulnerable group and gain from the related shorter and/or less severe lockdown ( Figure 4 ).

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. strategies to be implemented. If a strategy is adopted to target vulnerable and front line workers 271 then it will require moderate to severe lockdowns (60%-80% reductions in transmission) to achieve 272 the optimal outcome for disease control ( Figure 5 ).

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint Vaccine demand and uptake can influence the outcome of disease mitigation and control measures. 275 One concern is that as a substantial number of people are vaccinated, and potentially in tandem 276 with initial declines in infection rates, then there will be an emerging complacency and vaccine 277 adoption will fall. However, exponentially declining vaccine uptake is most likely to disrupt control Here, using mathematical modelling approaches we have investigated how combining vaccinations 287 and lockdowns can be used to control virus levels (in terms of disease induced levels of mortality). 288 We show that for different levels of vaccination (or vaccine efficacy), lockdowns of different dura-289 tion and/or severity can be implemented to mitigate levels of mortality. In particular, we highlight CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. proximation that within and between contacts are similar (β ji = β), the non-random distribution 303 of vaccines can be inefficient at achieving sufficient coverage for the disease to fade out. Similar 304 findings for age-structured models [19] and vaccine sharing strategies (Huntingford et al. in re-305 view) corroborate this finding. Here, this occurs due to population size differences between the 306 groups and the relative differences that this has on mitigating levels of disease-induced mortality.

In particular, we find that vaccinating individuals in the vulnerable group (a relatively small group) 308 must be rolled out together with mass vaccination across the wider larger non-vulnerable group to 309 achieve necessary public health benefits of reducing mortality and herd immunity. These sorts of well-planned vaccination strategies may to lead to spatial and temporal clusterings. Here, we account for these factors, and consider declining vaccination uptake should 318 groups decide to alter behaviours during mass-roll out of the vaccine. As expected, rapid (expo-319 nential) decline of vaccine uptake is most precipitous in terms of optimal disease control outcomes.

Our results confirm that lockdowns and other non-pharmaceutical interventions can be used to 321 mitigate against the effects of social clusterings and loss of vaccine uptake/efficiacies. However, 322 rapidly understanding the way in which these groupings form will be critical to determine how 323 robust in terms of severity and duration lockdowns need to be to achieve optimal disease control 324 outcomes for COVID-19. Our main finding is that in addition, with vaccination, optimal strategies to minimize deaths can, 344 under certain conditions, offset the need for severe or long lasting NPIs. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint As noted, neither of these previous studies consider population structure and the interaction be-360 tween vaccines and NPIs as concomitant approaches to disease control and minimizing disease 361 induced deaths. We argue that these sort of optimal control approaches provide a 'weight of ev-362 idence' for more pluralistic approaches to controlling the infection, and especially as appropriate 363 constraints can be included in solving numerical optimal models of the epidemiological dynamics. a vaccine as disease infection rates fall. We also allow consideration of vaccines that are not fully 381 efficacious as we evaluate optimal outcomes in terms of shortest lockdowns for prescribed maximum 382 levels of mortality. We argue that our use of appropriately developed structured epidemiological 383 models provide a robust way to investigate these epidemiological outcomes. Our optimal control 384 approaches allow the best combinations (here for parameter constraints applicable to the UK) to be 385 determined. However, these outcomes are parameter-dependent, and might change across different 386 locations and/or temporal scales.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint Our headline findings are as follows. As might be expected, to require relatively short lockdowns, 389 NPIs have to be sufficiently severe as to suppress transmission. Less effective vaccines imply longer 390 lockdowns, as does a larger vaccine elasticity. However, to achieve appropriate levels of disease 391 control and contrary to the standard approach of vaccinating the most vulnerable first, we find 392 that the optimal vaccination policy is to inoculate the larger (non-vulnerable) demographic group 393 first, then followed by front-line workers and then the vulnerable. While this finding might ap-394 pear counter-intuitive, given the order-of-magnitude difference in death rate for those encouraged 395 to shield against COVID-19 (i.e. in the vulnerable category). The reason for this finding is our 396 analysis assumes that the time required to vaccinate the vulnerable group is identical to that of the 397 much larger non-vulnerable group. As the non-vulnerable group is much larger, many more people 398 are vaccinated per day under that assumption, causing the disease to decline more quickly and yet 399 still constraining the overall number of deaths. 

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. 

The dynamics for infected individuals follow:

The dynamics for recovered individuals follow:

and the dynamics for the total number of vaccinated individuals are:

where β is the disease transmission rate, σ the loss of immunity, µ is the background death rate, 520 ν(t)is the vaccination rate (on both susceptible and recovered individuals), τ is the incubation win-521 dow, α is the disease induced death rate and γ is the disease recovery rate. N(t)=S(t)+I(t)+R(t)+V(t).

The objective functional is defined in terms of the rate of vaccination, the number of individuals 524 vaccinated and level of disease induced mortality such that during the epidemic of time length T 525 . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint the 'costs' of vaccination increase. The goal is to minimize these 'costs' of vaccination and keep 526 daily disease-induced mortality below a threshold (Z) where the control is vaccine rate (ν(t)):

where T is the length of the disease epidemic wave and Z is the critical level of daily disease induced 528 mortality that can not be exceeded. 

Expressions for the characterization of the control are derived from minimizing the Hamiltonian 537 operator. Each adjoint variable (λ i ) satisfies an equation found by differentiating the Hamiltonian 538 operator with respect to the corresponding state variable, and then negating this derivative:

− λ 4 ν . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint

With appropriate boundary conditions, the optimal control (ν) then minimizes the Hamiltonian 540 operator such that ∂H ∂ν = 0 for ν = ν * :

where ν * is :

The second derivative of H indicates the solution is a minimum as:

This is positive when V > 0, so solutions are determined as the number of vaccinated individuals 544 increases.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted February 3, 2021. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. in β, blue -60% reduction in β, green -40% reduction in β, red -20% reduction in β). From (a) -50K mortality threshold, for a weakly efficacious vaccination rate (0.005), depending on the severity of lockdown, lockdown durations could be 30 days (for 80% reduction in beta) or 70 days (for 60% reduction in beta). At this level of vaccination (0.005) and for a lockdown where β is only reduced by 20% or 40% it is not feasible to keep cumulative mortality below the 50K threshold.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. vaccinated, lockdown duration (in days) and lockdown severities (in terms of reducing transmission -(purple -80% reduction in β, blue -60% reduction in β, green -40% reduction in β, red -20% reduction in β)) on keeping cumulative mortality less than 80K. Vaccination across the nonvulnerable group provides greater opportunities to keep mortality below critical threshold with lockdowns of short duration and/or less restrictive.

. CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint

Length of Lockdown (days) Cumulative Mortality . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted February 3, 2021. ; https://doi.org/10.1101/2021.02.01.21250877 doi: medRxiv preprint

Lockdowns and the COVID-19 pandemic: What is

Understanding COVID-19 vaccine efficacy