key: cord-0872722-dq342gkr
authors: Crellen, T.; Pi, L.; Davis, E.; Pollington, T. M.; Lucas, T. C. D.; Ayabina, D.; Borlase, A.; Toor, J.; Prem, K.; Medley, G. F.; Klepac, P.; Hollingsworth, T. D.
title: Dynamics of SARS-CoV-2 with Waning Immunity in the UK Population
date: 2020-07-25
journal: nan
DOI: 10.1101/2020.07.24.20157982
sha: 801ca064acbd97bc1314718f1efbfd997b5cbb58
doc_id: 872722
cord_uid: dq342gkr

The dynamics of immunity are crucial to understanding the long-term patterns of the SARS-CoV-2 pandemic. While the duration and strength of immunity to SARS-CoV-2 is currently unknown, specific antibody titres to related coronaviruses SARS-CoV and MERS-CoV have been shown to wane in recovered individuals, and immunity to seasonal circulating coronaviruses is estimated to be shorter than one year. Using an age-structured, deterministic model, we explore different potential immunity dynamics using contact data from the UK population. In the scenario where immunity to SARS-CoV-2 lasts an average of three months for non-hospitalised individuals, a year for hospitalised individuals, and the effective reproduction number (Rt) after lockdown is 1.2 (our worst case scenario), we find that the secondary peak occurs in winter 2020 with a daily maximum of 409,000 infectious individuals; almost three-fold greater than in a scenario with permanent immunity. Our models suggests that longitudinal serological surveys to determine if immunity in the population is waning will be most informative when sampling takes place from the end of the lockdown until autumn 2020. After this period, the proportion of the population with antibodies to SARS-CoV-2 is expected to increase due to the secondary peak. Overall, our analysis presents considerations for policy makers on the longer term dynamics of SARS-CoV-2 in the UK and suggests that strategies designed to achieve herd immunity may lead to repeated waves of infection if immunity to re-infection is not permanent.

. Concerns that immunity to SARS-CoV-2 may also wane have therefore motivated the present study [15] .

Dynamic epidemiological models play a major role in shaping the timing and intensity of interventions 23 against SARS-CoV-2 in the UK and elsewhere [16] . Many models or simulations have assumed that infected 24 individuals recover with permanent immunity [16, 17, 18] . In such models the epidemic reaches extinction after 25 running out of infected individuals, although they do not preclude a second wave of infections after lockdown 26 [19]. If immunity wanes over a period of time, or recovered individuals have only partial immunity to re-27 infection, this substantially alters the dynamics of the system [20] . In the absence of stochastic extinction and 28 demography (births and deaths) in a population with equal mixing where; R 0 is the basic reproduction number; 29 γ is the average duration of infection; and ω is the reciprocal of the average duration of immunity; the endemic 30 equilibrium proportion of infected in the population I * , is given by (R 0 − 1) ω/γR 0 and thus, in the absence of 31 interventions, the infection persists indefinitely when R 0 > 1 [21] . 32 In dynamic models which make the assumption of homogeneous mixing in the population, the 'classic' 33 herd immunity threshold is given by 1 − 1/R 0 . As R 0 for SARS-CoV-2 is generally estimated between 2.4-4 34 [22, 23, 24] , this equates to 58-75% of the population requiring immunity to eventually halt the epidemic. 35 Serological studies conducted in affected countries to-date have reported the proportion of the population with 36 antibodies against SARS-CoV-2 to be much lower than this figure [22, 25] . However, when more realistic non- 37 homogeneous mixing is considered, the observed herd immunity threshold is lower than the classical threshold 38 [26]. Recent studies have considered this question for SARS-CoV-2 [27, 28] , with Britton et al. noting that the 39 disease-induced herd immunity threshold could be closer to 40% in an age-structured population, rather than 40 the 60% classic herd immunity threshold when R 0 is 2.5 [28] . This phenomenon is driven by individuals that 41 have more contacts, or greater susceptibility to the virus, getting infected earlier on and leaving the susceptible 42 population; thus decelerating the growth of the epidemic. 43 Kissler et al. considered the dynamics of SARS-CoV-2 in the United States with seasonal forcing, homoge-44 neous mixing and waning immunity that could be boosted by exposure to seasonal circulating betacoronaviruses 45 [13]. Under these assumptions, the incidence of SARS-CoV-2 was predicted to rebound in winter months. Here 46 we do not consider seasonality, but rather the dynamics of transmission in an age-structured population with 47 different periods of waning immunity in the context of the UK emerging from lockdown. 48 We developed a discrete-time gamma delay-distributed (susceptible-exposed-infectious-recovered-susceptible; 49 SEIRS) model, which incorporates current knowledge about the natural history of the virus and the UK popu-50 lation. Our model accounts for symptomatic and asymptomatic transmission, and heterogeneity in both daily 51 contacts and infection susceptibility by age group. We consider different durations of immunity for hospitalised 52 patients (or those with more severe symptoms) compared to non-hospitalised patients (those with less severe 53 symptoms). We use this model to explore a range of scenarios in the UK population in the context of stringent 54 non-pharmaceutical interventions (lockdown) followed by more limited interventions over a two year period from 55 February 2020, and the impact of immunity duration on the longer term disease equilibrium.

Model structure 58 We use current knowledge of the natural history of the virus to construct a plausible epidemiological model 59 ( Figure 1 ). We extend a previously published deterministic compartmental model which has provided general The disease states are susceptible (S), exposed (E), symptomatic infectious (I S ), asymptomatic infectious (I A ), hospitalised recovered (R H ), and non-hospitalised recovered (R N ). Age group specific parameters are indexed by i.

Epidemic transitions for age group i at time t + 1 are given by:

The function f (x, α, B) represents the Erlang delay distribution within classes E, I S , I A , R H and R N ; 89 which is achieved by using α concatenated sub-compartments for each class with rates B between each sub-90 compartment. If n individuals enter state X at time t, by time t + τ there will be remaining n(t)(1 − g(τ, α, B)), 91 where g(τ, α, B) gives the cumulative Erlang distribution with (integer) shape parameter α and rate parameter 92 B:

3 . CC-BY-NC 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 July 25, 2020. . The next generation matrix (K = k ij ) gives the expected number of secondary infections in age group i 94 resulting from contact with an index case in age group j:

The basic reproduction number (R 0 ) is given by the spectral radius ρ(K) which is the largest absolute 96 eigenvalue of K. The force of infection acting on age group i at time t + 1 (λ t+1 ) is given by:

where c i,j is the average number of daily contacts in the population between age groups j and i; N a is the 98 number of discrete age groups (N a = 15); and N j gives the population size of age group j. As we specify the 99 value of R 0 , the transmission parameter β is left as a free parameter which is scaled to the correct value. 

Using data and timing of events from the UK epidemic, we explore four scenarios with varying average durations 102 of immunity to SARS-CoV-2 ( Figure 2 ). 103 S1. Permanent: Where immunity is lifelong for both hospitalised (R H ) and non-hospitalised (R N ) cases. 4 . CC-BY-NC 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 July 25, 2020. [49]. After this time, R t is brought to 0.9, 1, 1.1 or 1.2 until February 2022. We considered the majority of our 116 analysis over a, relatively short, two year period to explore the epidemic up to a secondary peak; beyond this 117 point the dynamics are likely to be altered depending on further interventions or changes to R t . As we simulate 118 disease dynamics over a relatively short period of time, we do not consider demography (births and deaths) or 119 transitions between age classes (ageing). To obtain equilibrium values, we simulated epidemic trajectories for 120 up to five years.

The UK contact matrix (average daily contacts between an individual in age group j with individuals in 

Age structure

The epidemic is driven by the rate of infectious contacts between individuals in different age groups. This is . CC-BY-NC 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 July 25, 2020. . showing the number of secondary cases generated by an index case from age group j (rows) in age group i (columns). The matrices are shown for different time points; at baseline before the implementation of interventions; during the lockdown period; and in the post-lockdown period when the effective reproduction number (R t ) rises from 0.9-1.2. The average number of secondary cases generated by an index case from age group j is the summation of row j. β, which captures the probability of infection given contact, is decreased from 0.13 at baseline to 0.11. In the 144 post-lockdown period daily contacts are increased to a higher proportion of their baseline values (see Methods); 145 in order to keep the reproduction number equal to the dominant eigenvalue of the next generation matrix, β is 146 consequently reduced to 0.05 when R t =0.9 and to 0.07 when R t =1.2. This implies that, to maintain R t below 147 one when more contacts are occurring in the population post-lockdown, the probability that contact results in 148 infection will need to be reduced. 

A secondary peak in infections is expected in spring 2021 where R t = 1.1 or winter 2020 where R t = 1.2 164 (Figure 4 , panels E & G). The height of the secondary peak is determined by the rate at which immunity is 165 lost. In our worst case scenario (S4: short-lived immunity) where immunity lasts an average of three months for 166 non-hospitalised patients, a year for hospitalised patients and R t following lockdown is 1.2, then the secondary 167 peak will exceed the initial peak with a maximum of 409,000 infectious individuals and 133,000 daily new cases 168 in December 2020. This is nearly triple the number of new cases compared with scenario S1 where immunity 169 is permanent; the maximum number of infectious individuals in the secondary peak is 137,000 and there are 170 45,000 daily new cases ( Figure 4G ). We note that the timing of the secondary peak in infection curves across 

When R t following lockdown is 1.1, the differences between the scenarios is even more pronounced with a 175 six-fold difference in the height of the secondary peak of infectious individuals between a scenario of permanent 176 immunity and one of short-lived immunity. When immunity wanes rapidly, a secondary peak is observed in Figure 3 ). Infectious and immune cases as a proportion of the total age group are shown in Figure 5 193 for scenarios S1 & S4 of permanent and short-lived immunity where R t = 1.2 following lockdown. A higher 194 proportion of individuals aged between 20-39 are infected early in the epidemic, and this leads to 10.5-12.6% 195 of individuals in these age groups having antibodies by September 2020 when immunity is life-long ( Figure 5B ).

When immunity wanes, however, by September 2020 this drops to 5.3-6.6% ( Figure 5D ), thus increasing the pool 197 of susceptible individuals to include more of the age groups that drive transmission. This causes the secondary 198 peak of infectious cases to rise more rapidly and to a greater height when immunity wanes ( Figure 5C ), compared 199 with permanent immunity ( Figure 5A ). Our models suggest that the age distribution of cases in the epidemic 200 will not change greatly over time; as seen in Figure 5 the ordering of the proportion of each age group infected 201 remains constant in both scenarios of permanent and short-lived immunity.

Longer term dynamics: extinction or endemic equilibrium 203 We explored the impact of waning immunity and R t on the equilibrium values for the different simulations 204 over a longer, five year, period until February 2025 (Table 2 ). If the post-lockdown R t is suppressed below 205 one following lockdown, then the differences in immunity will have less impact on the longer-term infection 206 dynamics, assuming no imported cases, as transmission of SARS-CoV-2 becomes unsustainable and the virus 207 reaches extinction between April-November 2021 depending on the immunity scenario. In simulations where R t 208 equals one, if immunity is permanent then the epidemic becomes extinct in May 2022. When immunity wanes 209 there is no secondary peak ( Figure 4C ), however the infections persist at a low level of endemicity equivalent 210 to 106, 233 and 1,168 daily cases in immunity scenarios S2-S4, respectively. For larger values of R t , and where 211 immunity wanes, the system oscillates with subsequent peaks of infection over the next five years until a steady 212 state is reached. We find that, if R t = 1.2 post-lockdown and immunity is short-lived, there is the potential for 213 over 76,000 new cases daily; 6,000 hospitalisations; and 1,000 intensive care unit (ICU) admissions (calculated 214 7 . CC-BY-NC 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 July 25, 2020. We predict that surveys to detect waning immunity at the population level would be most effective when 223 carried out in the period between the end of lockdown and autumn 2020, as after this point an upsurge in cases 224 is expected that will increase the proportion of the population with antibodies to SARS-CoV-2. In particular, 225 this will allow evaluation of whether specific antibodies generated against the virus are short-lived if reductions 226 in antibody prevalence are observed at the population level. 227 We find that transmission is driven disproportionately by individuals of working age, and subsequently a 228 higher proportion of individuals aged 20-39 years become infected early in the pandemic and subsequently 229 develop antibodies (Figures 3 & 5) . This prediction is borne out by serological data from Switzerland, which 230 showed that individuals aged 20-49 years were significantly more likely to be seropositive in May 2020 compared 231 with younger and older age groups [54] . We postulate that 'key workers' in the UK population who have con-232 tinued to work during the lockdown are more likely to have antibodies against SARS-CoV-2. Higher immunity 233 among individuals of working age has the effect of slowing the subsequent epidemic when immunity is perma-234 nent. Conversely, when immunity wanes, previously infected individuals of working age re-join the susceptible 235 pool and so contribute again to transmission; leading to a high growth rate and a larger secondary peak of 236 infected cases. In these circumstances, efforts to suppress transmission will be challenging in the absence of a 237 transmission-blocking vaccine [15] . We note that the model structure developed here is capable of simulating 238 the impact of vaccination with a vaccine that provides temporary transmission-blocking immunity, and could 239 be used to predict the optimal timing for booster shots. 8 . CC-BY-NC 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 July 25, 2020. 9 . CC-BY-NC 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 July 25, 2020. 

Our study reinforces the importance of better understanding SARS-CoV-2 immunity among recovered in-249 dividuals of different ages and disease severity. In scenarios where immunity wanes and R t following lockdown 250 is greater than one, the SARS-CoV-2 epidemic never reaches extinction due to herd immunity, but rather the One of the strengths of our study is that the model is calibrated to key features of the UK epidemic. While 261 we did not explicitly fit to data, new cases at the start of the lockdown; cumulative cases between February 262 10 . CC-BY-NC 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 July 25, 2020. . and March; and the proportion of the adult population with antibodies to SARS-CoV-2 are highly comparable between our output and current estimates [25, 50] with this consideration, there are many probability distributions that can be used to capture a median duration 271 of immunity, and our selection of an Erlang distribution with a shape parameter of two is somewhat arbitrary.

Our assumptions on the duration of the latent and infectious periods are more closely informed by estimates from 273 data [29, 46, 47] . We made the decision to capture the expected duration of these states as Erlang distributions 274 rather than the, more conventional, exponential distribution. This has the benefit of closely replicating fitted for asymptomatic and symptomatic infections in subsequent models. 284 We have aimed to capture future infection dynamics at a national level in the UK under a range of scenarios.

Our analysis is limited by not considering regional differences in transmission rates, for instance through a patch CC-BY-NC 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 July 25, 2020. . https://doi.org/10.1101/2020.07.24.20157982 doi: medRxiv preprint

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