Transmission dynamics reveal the impracticality of COVID-19 herd immunity strategies

Results

In the absence of any intervention measures, our modeling suggests SARS-CoV-2 will spread rapidly through the United Kingdom (Fig. 1A), and ultimately infect approximately 77% of the population (Fig. 1B). Using data on the age-specific fatality rate of COVID-19 (22), our results show around 350,000 fatalities among individuals aged over 60 y, and around 60,000 aged below 60 y (Fig. 1C). While we caution that our model makes a number of simplifying assumptions [e.g., no spatial dependence in transmission (11)], the total of fatalities is comparable to predictions made in other UK modeling studies (e.g., within the 95% prediction interval of ref. 20).

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Fig. 1.

Simulated examples of SARS-CoV-2 spread in the United Kingdom using an age-structured Susceptible Exposed Infectious Recovered (SEIR) model under different control scenarios. (A) Daily new cases assuming no intervention measures enacted. (B) Cumulative proportion of the population exposed to SARS-CoV-2 over the course of the epidemic. (C) Cumulative fatalities assuming fixed age-specific case fatality rates (see Materials and Methods). (D–F) Same as A–C, but assuming that specific control measures are introduced when daily cases reached 10,000 (day 57): Older individuals social distance, and symptomatic individuals self-isolate (at 20% effectiveness). (G–I) Same as D–F but, in addition, schools close on day 57. Reopening of schools after 100 d results in a resurgence.

Sustained social distancing by older individuals (assumed to result in a 90% reduction in contacts with individuals under 25 y old, a 70% reduction with 25- to 59-y-olds, and a 50% reduction between one another), and moderately effective self-isolation by symptomatic individuals (at 20% efficacy) results in a shallower epidemic curve (Fig. 1D) and a much smaller outbreak size among individuals aged 60+ y (Fig. 1E). The attendant mortality burden among 60+-y-old individuals is also substantially reduced (to 62,000), with a smaller reduction in fatalities in those aged <60 y (to 43,000; Fig. 1F).

The addition of school (and university) closures, corresponding to a 70% reduction in contacts among school-aged individuals and a 20% reduction with 25- to 59-y-olds, dramatically reduces the rate of epidemic growth (Fig. 1G), although such levels of control are insufficient to suppress the epidemic (i.e., the number of daily cases still rises after implementation). The premature reopening of schools after 100 d (while the virus is still circulating) triggers a second wave of infection, with only a moderately reduced peak in daily new cases, largely eroding any additional gains made (23). The final proportion of the population exposed (Fig. 1H) and the number of fatalities (Fig. 1I) are largely unaltered compared to if schools had not been closed (compare Fig. 1 E and F).

Our modeling indicates that, if sustained, such control measures can lead to the suppression of COVID-19 in the United Kingdom by reducing R₀ to <1 (Fig. 2 A and B). The effectiveness of self-isolation by symptomatic individuals (i.e., its impact on reducing overall transmission) is a product of two factors. Firstly, the proportion of infections generated while the primary case is exhibiting symptoms (including mild symptoms) and, secondly, the engagement of symptomatic individuals with self-isolation policies (see Materials and Methods). As both of these parameters decrease, the self-isolation efficacy drops, and greater social distancing measures are necessary to achieve suppression (Fig. 2A). At present, there is a large uncertainty in the relationship between symptoms and viral shedding (9). For the social distancing strengths considered, suppression is possible if over 14% of infections are caused while the primary case is exhibiting symptoms (ps). The associated self-isolation observance necessary to achieve suppression is inversely proportional to ps, decreasing from 100% if ps=0.14 to 50% if ps=0.28. Given the uncertainty surrounding asymptomatic transmission, the likelihood of successful suppression is greatest if all social distancing measures are enacted (Fig. 2B).

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Fig. 2.

Prospects for disease suppression. Simulations of the age-structured SEIR model performed assuming control measures are initiated when there is a total of 10,000 infectious individuals in the population. The different control measures and strengths are listed in Materials and Methods. (A) Whether suppression is possible depends on both the self-isolation observance rate and the proportion of infections due to symptomatic individuals. More extensive social distancing measures increase the ranges of these two parameters for which suppression is possible. (B) Increases in self-isolation effectiveness drive down the reproductive number, which also depends on the social distancing measures employed. (C) The time taken for COVID-19 to be suppressed (modeled as a 100-fold reduction in infectious individuals) depends on the amount the reproductive number is decreased below one.

The time taken for suppression to be achieved (modeled as a 100-fold reduction in infectious individuals) once control measures are implemented is shown in Fig. 2C. If self-isolation effectiveness is high (>70% reduction in transmission), then suppression can be achieved in 2 mo regardless of any additional social distancing measures. There is little additional decrease in the necessary duration of social distancing unless schools and workplaces are both closed, in which case suppression can be achieved within 2 mo at much lower levels of self-isolation effectiveness (≳45%).

If suppression cannot be achieved (due to unfeasibility or lack of political will to reduce transmission sufficiently), then the objective of control measures is mitigation. Social distancing by 60+-y-old individuals results in a marked reduction of the final fraction of this age group that are exposed; however, unless both schools and workplaces are closed, additional social distancing measures do not lead to much further reduction (Fig. 3A).

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Fig. 3.

Outcomes of disease mitigation attempts, simulated using the age-structured SEIR model. As in Fig. 2, control measures are implemented when there are 10,000 cases in the population. (A) The final fraction of 60+-y-old individuals exposed to COVID-19 for each of the control strategies simulated, assuming that social distancing measures can be maintained at the same strength indefinitely (“without fatigue”). (B) Size and (C) timing of the peak in daily new cases among 60+-y-old individuals. (D) Peak hospital burden, assuming age-specific hospitalization rates (see Materials and Methods) and a mean hospital stay of 7 d (24). (E–H) Simulation results using the same control strategies as in A–D, but assuming that, due to fatigue, schools and workplaces closures last 100 d.

These results are also mirrored in the impacts of social distancing on the daily cases in 60+-y-old individuals (Fig. 3B). Unfortunately, the hospital burden remains high for most intervention strategies, unless self-isolation is very effective (≳50%; Fig. 3D). Taking around 100,000 hospitalized cases to be the upper limit of hospital capacity, we find that there is a relatively small range of parameters where mitigation is successful at preventing hospitals being overwhelmed, but the disease is not also successfully suppressed (Fig. 3D; compare to Fig. 2B). If social distancing is applied to all age groups, this range is 0 to 14% self-isolation effectiveness, whereas, if just the 60+-y age group socially distance, the range is 41 to 54%.

As mentioned previously, if schools and workplaces reopen simultaneously after 100 d [e.g., due to social distancing “fatigue” (10)] and the disease has not been successfully suppressed, then much of the benefit of their closure is lost (Fig. 3 E–H) due to a resurgent second wave. In this scenario, the principle effect of school and workplace closures is in delaying the peak, buying more time for preparations (Fig. 3G).

It has been suggested that children might have reduced susceptibility to infection with SARS-CoV-2 (25). We repeated our analysis of suppression and mitigation assuming 50% reduction in susceptibility of individuals under 20 y old, with transmission rates among adults increased to ensure the same reproductive number, R0=2.3. As might be expected, there is a decreased impact of school closures; however, this is compensated by the increased impact of social distancing by over-20-y-olds, resulting in little net change in our findings if both work and school closures occur (SI Appendix, Figs. S1 and S2).

To summarize, aiming to build herd immunity to SARS-CoV-2 in a population while mitigating the burden on hospitals requires initially reducing the reproductive number to ensure available hospital capacity is not exceeded (Fig. 4A). If social distancing is maintained at a fixed level, hospital capacity needs to be much larger than presently available to achieve herd immunity without exceeding capacity; otherwise, the final outbreak size will be insufficient to achieve herd immunity (Fig. 4B). Relaxing social distancing measures linearly is also insufficient (SI Appendix, Fig. S3).

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Fig. 4.

Summary of prospects for achieving herd immunity. (A) The peak hospital burden (shown on a log scale) is highly sensitive to the reproductive number. There is a narrow window of reproductive number values (shaded) where either 1) the number of COVID-19 cases requiring hospitalization does not overwhelm hospital capacity (modeled at the average hospital burden for April, 17,800 beds) or 2) circulation is suppressed. This window depends subtly on the exact age-specific social distancing configuration; however, all strategies studied fall between the two curves shown. (B) None of the simulated control scenarios shown in Figs. 2 and 3 achieved herd immunity while also keeping cases below hospital capacity. For the parameters considered, a hospital capacity in excess of 300,000 is required for this to be possible—almost 3 times the total UK NHS hospital beds (around 125,000 beds; see Materials and Methods), and around 15 times the average hospital burden of April.

Instead, achieving herd immunity without exceeding hospital capacity requires that social distancing is implemented nonlinearly. An idealized optimal strategy for achieving social distancing, calculated using a two-age group (<60 y old and ≥60 y old) SEIR model, is shown in Fig. 5 A and B. We assumed that individuals ≥60 y old were able to perfectly self-isolate, while herd immunity is built up in the <60-y age group. The epidemic in <60-y-old individuals is initially allowed to grow unimpeded until the hospital burden caused by the number of sick individuals reaches the hospital capacity. At this point, social distancing among those younger than 60 y commences (Fig. 5B), at a level that ensures the epidemic growth stalls (i.e., Reff=1). To achieve herd immunity in minimal time, the rate of new infections must then remain at the maximum afforded by the healthcare capacity. This requires reducing social distancing (i.e., increasing contacts) at the exact rate to balance out reductions in transmission stemming from the decreasing susceptible population (Fig. 5B)—too quickly and the epidemic grows above hospital capacity; too slowly and the epidemic with die out without reaching herd immunity. Over time, social distancing in <60-y-olds drops to zero, after which there are no longer enough susceptible <60-y-old individuals for the epidemic to be sustained. New infections continue but decline in frequency (meanwhile, further decreasing the effective reproductive number), before reaching zero.

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Fig. 5.

Proof-of-principle strategy for achieving herd immunity. This strategy 1) achieves herd immunity with minimal social distancing duration 2) without exceeding hospital capacity, and 3) prevents infection among the most vulnerable. We simulated a reduced two-age group (<60 y old and ≥60 y old) model, assuming that all ≥60-y-old individuals completely isolate. (A) To achieve herd immunity in the minimum time requires control measures that fix the rate of new hospitalized cases to ensure hospital beds are at the maximum acceptable capacity until herd immunity is achieved (results showing taking capacity to be 1× and 2× the average hospital burden of April). (B) The reduction in contacts (via the product of self-isolation and social distancing) among <60-y-olds needs to vary nonlinearly. Until the hospital burden hits the hospital capacity, there is no social distancing; subsequently, it is tuned to ensure Reff=1; that is, the epidemic neither grows nor shrinks. Contact rates have to then gradually increase to exactly balance the reduction in Reff due to susceptible depletion. Eventually, there are no longer enough <60-y-old susceptible individuals to sustain the epidemic, and Reff<1. (C) While this strategy ensures that the final reproductive number drops below 1 if individuals ≥60 y old remain in isolation, if they return to pre-COVID-19 contact rates, then whether the reduction in susceptibles is sufficient to achieve herd immunity depends on the hospital capacity. The greater the proportion of the population that needs protecting, the greater the hospital capacity needed to achieve herd immunity. (D) The duration of social distancing also depends on the available hospital capacity. For the parameters considered, if hospital capacity is less than around 20,000, then it is not possible to achieve herd immunity without individuals aged ≥60 y remaining socially distanced.

While, at this point, the effective reproductive number is below 1 without any social distancing in the <60-y age group, full herd immunity has not necessarily been achieved (Fig. 5C). If ≥60-year-olds cease self-isolating, then the increased contact rates cause the effective reproductive number to increase, and may take it above 1. Achieving herd immunity requires that, while ≥60-y-olds are isolated, the additional new infections after Reff<1 sufficiently further reduce Reff (the “overshoot”) such that, when ≥60-y-olds cease self-isolation, it remains below 1. The amount of overshoot increases with the hospital capacity (Fig. 5C). In addition to increasing the prospects of achieving herd immunity (and also its robustness; see also SI Appendix, Fig. S3C), the necessary duration of social distancing is inversely proportional to hospital capacity (Fig. 5D).

Our modeling required estimates for the case hospitalization probability (22) and mean hospitalization stay (24). We explored sensitivity of the duration of social distancing to these two parameters, and also performed a comparison with a rough calculation of the duration based on estimates of seroprevalance at the end of April (26) (SI Appendix, Fig. S4). Our estimated necessary social distancing duration of 12 mo is close to the estimate from serology of 7 mo to 11 mo, especially considering the uncertainty in the case hospitalization probability (22).

Throughout our analysis, we used R0=2.3, broadly in line with most early estimates; however, some estimates were higher. We repeated our analysis using R0=5.7 (27), finding that, as might be expected, the range of intervention parameters that led to successful suppression was reduced (SI Appendix, Fig. S5). However, our main finding was reinforced: Higher values of R0 narrow the window for successful mitigation (SI Appendix, Fig. S6).