Immunological characteristics govern the changing severity of COVID-19 during the transition to endemicity

in primary cases being restricted to the lower age groups. Parameters for simulations: ω = 1 and ρ = 0.7. In Panel B, we plot how long it takes for the average CFR (calculated as a 6 month moving average) to fall to 0.001, the CFR associated with seasonal influenza. Grey areas represent simulations where the CFR did not reach 0.001 within ten years. We see that the time taken for the CFR to decline to that of influenza decreases as the transmissibility (R 0) increases and the duration of sterilizing immunity becomes shorter. Results are shown for ρ = 0.7. See SI Sec 2.3 and SI Figs 3-5 for sensitivity analyses and model specifications. 8 . CC-BY-NC-ND 4.0 International licenseIt 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 (which was not certified by peer review) The copyright holder for this preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint COVID (CoV−2)SARS (CoV−1)MERS 2.55.07.510.0 0.000 0.005 0.010 0.015 0.00 0.05 0.10 0.15 0.32 0.34 0.36 Time (yr) Case fatality rate R0R0=2R0=4R0=6 Case Fatality Ratio (CFR) Time (yrsafter introduction)0.0 0.2 0.4 0.6 25 50 75 midpoint of age groups (yr) case fatality rate pathogen CoV−2MERSCoV−1 Case Fatality Ratio (CFR) Age (yrs) A B R0 Figure 4: The age dependence of the CFR determines how the overall CFR changes during the transition from epidemic to endemic dynamics for emerging CoVs. Panel A shows the age depen- dence of the CFRs for the three emerging CoVs. CoV-1 and CoV-2 have a J shaped profile, with a monotonic increase in CFR with age. In contrast, the age dependence of the CFR for MERS is U shaped, with high mortality in the younger as well as older age groups. Details of the statistical smoothing are described in SI Sec 5. Panel B shows how the overall CFR (computed from model- predicted infections in each age group multiplied by the age-specific CFRs for the different CoVs) changes with the transition from epidemic to endemic dynamics. We see the overall CFR declines as CoV-2 transitions to endemicity, and we predict a similar trend should a CoV-1 (SARS)-like viurs spread widely. In contrast, the model predicts the CFR for a MERS-like virus would increase with time; this increase is a consequence of the shape of the age-dependence of the CFR for MERS. 9 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint T able 2: Effects of separated IEs model on control strategies for SARS-CoV-2 Effects Predicted by the Model P ossible Solutions Quaran tine + contact tracing W aning transmission-reducing immunity (IES and IEI) with long-lasting pro- tection against severe disease (IE P ) leads to an increase in asymptomatic re- infections. Identifying and quarantining index cases becomes more difficult. Localized outbreaks are more difficult to control, particularly in places with a large first wave of infections (SI Sec 4). Widespread / universal testing regardless of symptoms Shielding & immunity passports Giv en that prior infection and the presence of CoV-2-specific antibodies may indicate stronger IEP than IES or IEI, this strategy is a double-edged sword. Those with antibodies may be less likely than those without to get infected and transmit, but they may also be more likely to have an asymptomatic infection that can spread without their knowledge of it. The chance of this scenario increases with time since infection. It may not be possible to rely on seropositive individuals as caregivers to shield the elderly and vulnerable as has been proposed [24], but they are a good choice for frontline workers or those caring for COVID patients because they are less likely to have severe pathology following reinfection. • Iden tify immune surrogates of IES, IEP and IEI. • Time stamps on immunity passports. So cial dis- tancing & PPE use Both social distancing and everyday use of personal protective equipment (PPE) reduce the spread of infection (i.e., reduce R0) regardless of symptoms and therefore their efficacy does not depend on detecting symptoms. Increase PPE use to maintain lo- cal R0 < 1 with businesses as open as possible. V accination V accination, like natural infection, may not provide long-lasting transmission- blocking immunity. Consequently herd immunity will not be achieved, and the traditional strategy of vaccinating high-contact rate groups will not provide long-term protection to the rest of the population. Instead, vaccination should target those most vulnerable to severe disease (i.e., the elderly) to reduce the overall CFR during the transition to endemicity and essential workers most frequently exposed (i.e., front-line workers). • T arget vaccination toward the elderly, frontline work- ers, and other vulnerable populations. • Generate a ‘smart vaccine’ that produces better-than- natural, transmission- blocking immunity. 10 . CC-BY-NC-ND 4.0 International licenseIt is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. 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(which was not certified by peer review) The copyright holder for this preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint Immunological characteristics will govern changing severity of COVID-19 during the likely transition to endemicity: Supplement Jennie Lavine, Ottar Bjornstad, Rustom Antia August 21, 2020 1 Serostudy analysis details We used data from a 2013 study [1] on the seroprevalence of IgG and IgM antibodies against the Spike (S) protein of the four endemic HCoV strains in a cross-section of the population in Beijing, none of whom exhibited symptoms. A striking feature of these data is that IgM titers are undetectable in all of the more than 140 subjects ages 15 yr and older. This observation combined with general knowledge of the the IgM response [2] suggests that S-specific IgM is only elicited during primary infection. Additionally, IgM titers decay quickly, which makes IgM seropositivity a useful marker of recent primary infection. We calculate error bars on the seroprevalence for both IgM and IgG and estimate the mean age of primary infection (MAPI) for each of the four strains from the IgM data. We assume the data for both IgM and IgG stem from a binomial process where the probability of seropositivity in age group j is pj, estimated by ˆpj and the sample size for each age group is Nj. The 95% confidence interval around the mean proportion seropositive for each age group is then 1.96 √ ˆpj(1 − ˆpj) Nj (1) We further estimate the MAPI using only the IgM data (we assume that the cases are uniformly dis- tributed within each age group – a more accurate estimate could be obtained from the raw data with smaller age bins; unfortunately we were unable to gain access to it). For each HCoV strain, s, we create a vector, As with lengthL containing the ages ages of IgM seropositivity (using the midpoints of each age range). We assume this is a reasonable reflection of ages of first infection, as IgM titers increase only during primary infection and decay in a matter of weeks [3]. The 95% CI for the MAPI for each strain, s, is therefore estimated by MAPIs = 1 N ∑ As ± 1.96 √ Var(As) N (2) 2 Model derivation We derived the model presented in Fig 1 by combining three key sources of information: (1) classic SIRS dis- ease transmission models, (2) separated functional immune efficacies (IE’s, [4]), and (3) a human reinfection experiment with HCoV 229e in fifteen healthy adult volunteers [5]. In aforementioned reinfection experiment, all subjects had serum specific antibodies at the start, suggest- ing that participants had already experienced a primary infection. The first experimental exposure resulted in viral replication and a boosting of IgG titers in ten of the fifteen participants; the group that supported vi- ral replication had lower serum specific IgG, IgA and nasal IgA levels prior to exposure and shed virus for on average 5.6 days; eight out of ten had cold symptoms. Antibody titers increased significantly approximately a week after infection and then slowly decayed over the course of the next year. Among the five participants who did not get infected following exposure at the beginning of the trial, antibody titers remained relatively constant following exposure; however their IgG titers did drop significantly by the end of the year. 1 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint One year after the initial exposure, fourteen of the fifteen participants were re-exposed. All five who had not sustained an infection the first time became infected (their IES waned) and one developed cold symptoms (IEP was still strong for four of the five). Six of the nine who had supported viral replication a year earlier sustained an infection following exposure the second time (their IES waned within the span of year) but none developed cold symptoms (they retained IEP ); the other three did not become infected following exposure. Among all who got infected at the second time point, the mean duration of viral shedding was only 2.0 days, suggesting that IEI had not waned completely. Based on these observations, the following are the basic equations that correspond to Fig 1. dS dt =µN −βS(I1 +ρI2) −µS (3) dI1 dt =βS(I1 +ρI2) − (γ +µ)I1 (4) dR1 dt =γ(I1 +I2) − (µ +ω)R1 (5) dR2 dt =ωR1 −βR2(I1 +ρI2) −µR2 (6) dI2 dt =βR2(I1 +ρI2) − (γ +µ)I2 (7) 2.1 Steady-state analysis The above equations are used for steady state analysis of the predicted mean ages of primary infection (MAPIs). The model-predicted MAPI for a given set of parameters is calculated as the waiting time from birth to first infection according to the following equation: MAPI = 1 β ˆI1 +ρβ ˆI2 yr (8) where ˆIi is the equilibrium proportion of the population in class Ii. The equilibrium values are calculated by first running a short (100 iterations = 1/10 yr) numerical simulation using a wrapper for the R function lsoda and then using the final values as estimates to start the Newton-Raphson method to find equilibria as implemented in the R package rootSolve, function stode. We additionally calculate the proportion of cases caused by reinfections as follows: ρ ˆI2 ˆI1 +ρ ˆI2 (9) In addition to the results shown in the main text for R0 = 5, we here show figures parallel to Fig 3b for R0 = 2 and R0 = 10 (SI Fig 1). 2.2 Transient to endemic dynamic simulations To incorporate seasonality with a peak in early January (modeling on influenza and seasonal coronaviruses) and the introduction of the virus in early March (as was approximately observed with CoV-2 in the US), cases are introduced at t=0 and we allow β to fluctuate annually according to β =β0[1 +β1 cos(2πt + π 3 )] (10) β0 is the mean value of β, and β1 is the amplitude of the sin wave. All results shown here use β1 = 0.2 (SI Figs 3-6 and Fig 3 in the main text). Additionally, to incorporate the age-specific case fatality rates, death rates ( δ), and the current age distribution of the US population, we separate each immune state, X, into nine age classes Xj with 10-yr widths for the first eight. The aging rates correspond to the width of the age classes; the death rates and age distribution are taken from US data. This yields the following equations: 2 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 0.0 0.5 1.0 1.5 2.0 0.00 0.25 0.50 0.75 1.00 ρ ω age grp (yr) (0,1] (1,2] (2,3.4] (3.4,5.1] (5.1,7] (7,10] NA R0=10 0.0 0.5 1.0 1.5 2.0 0.00 0.25 0.50 0.75 1.00 ρ ω age grp (yr) (0,1] (1,2] (2,3.4] (3.4,5.1] (5.1,7] (7,10] (10,100] NA R0=2 Figure 1: Mean age of primary infection dSj dt =µN +λjSj −βSj J∑ j=1 (I1j +ρI2j) − (δj +λj+1)Sj (11) dI1j dt =λjI1 +βSj J∑ j=1 (I1j +ρI2j) − (γ +δj +λj+1)I1j (12) dR1j dt =λjR1j +γ(I1j +I2j) − (δj +ω +λj+1)R1j (13) dR2j dt =λjR2j +ωR1j −βR2j J∑ j=1 (I1j +ρI2j) − (δj +λj+1)R2 (14) dI2j dt =λjI2j +βR2j J∑ j=1 (I1j +ρI2j) − (γ +δj)I2j (15) whereN(t) = ∑J j=1Xt is the total population size at time t. The birth rate, µ, is a vector of length nine, containing the overall population birthrate (based on demographic data) followed by zeros for all subsequent age classes (i.e., people are only born into the youngest age class). The age-specific death rates, δj, are fixed at values estimated from demographic data. The aging rates, λj, are contained in a vector of length ten: (0, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.1, 0.04, 0). We fix γ at 365 9 corresponding to an infectious period that lasts on average nine days. β0 is calculated according to β0 =R0(γ +µ). We consider a range of values of R0 (2-10), ω (0-2), and ρ (0-1). The age-specific death rates were inferred from CDC data Health Statistics [6], and were calculated as follows: death_rate_age<-as.data.frame(cbind(age=seq(25,95,by=10), number=c(30154,58844,80380,164837,374836,543778,675205,880280), rate=c(70.2,128.8,194.7,395.9,886.7,1783.3,4386.1,13450.7)/100000)) death_rate.glm<-with( death_rate_age,glm(rate ˜ age, 3 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint family=gaussian(link=’log’)) ) pred.ages<-seq(5,85,length=9) pred_death_rate<-exp(predict(death_rate.glm, data.frame(age=seq(0,100,by=0.5)))) mod.pred<-round(exp(predict(death_rate.glm, data.frame(age=pred.ages))),5) print(mod.pred) ## 1 2 3 4 5 6 7 8 9 ## 0.00001 0.00003 0.00008 0.00024 0.00068 0.00197 0.00565 0.01623 0.04662 0 20 40 60 80 100 0.00 0.06 0.12 age death rate Figure 2: Age specific death rate in the US The ages used in the model prediction for death rates were the midpoints of the age classes (e.g. 5, 15, 25yr, etc). We used the death rate for 85 yr olds as the highest death rate. The initial conditions were set so the population was distributed into the nine susceptible classes according to the age distribution of the US population [7]. One infected individuals was seeded into each age group’s I1 class. 2.3 Calculating infections and the case fatality rate from simulations The following steps were used to calculate the number of daily infections and 6-month moving average case fatality rate (CFR): 1. Numerically integrate equations with chosen parameters and initial conditions as described above using the R function lsoda with a time step of one day (1/365 yr). 2. Calculate the probability of staying in I1 for a timestep of one day given that you’re already there. In the simplest version of the model we consider here, this can be calculated using the cumulative distribution function, F: P(stay in I in time step∆t) = 1 −F(γ, ∆t) (16) gamma=365/9 time.step=1/365 prob.stay=1-pexp(rate=gamma, time.step) print(prob.stay) ## [1] 0.8948393 4 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 3. For each age group, calculate the number of new primary infections in each time step, Xt It1 =It0P(stay in I in time step) + Xt (17) Xt =It1 −It0P(stay in I in time step) (18) I_t1 <- tail(out[, ’I1’],-1) I_t0 <- head(out[, ’I1’],-1) X = I_t1 - I_t0 * prob.stay 4. Calculate the projected number of HCoV-induced deaths in each age group based on the age-specific CFR’s. deathsj =Xj ∗ CFRj (19) 5. For every 6-month window, calculate the overall CFR beginning six months into the pandemic ∑J j=1 deathsj ∑J j=1Xj (20) The key result that the overall CFR drops to something akin to seasonal influenza (0.001), is robust across a wide range of values for R0, ω and ρ for COVID-like CFRs. 0.7 1 0.05 0.35 2 4 6 2 4 6 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 R0 ω 0.5 2.0 8.0 yrs to CFR=0.001 Figure 3: Time to CFR=0.001, parallel to main text Fig 3b We additionally show results for simulations with different initial conditions, in which 2000 infections were seeded into each age groups approximating the situation when the pandemic was under control in the U.S. (i.e., there had been already been many infections and R0 was close to 1). The CFR evolves very similarly to the case where only nine cases are introduced. Additionally, we see that keepingR0 below a threshold value (in these simulations approximately 2, e.g., by social distancing and the use of Personal Protective Equipment) allows us to stop the majority of deaths from happening early on, buying time for the development of an effective vaccine and/or treatment (SI Fig 5). 5 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 0.7 1 0.05 0.35 2 4 6 2 4 6 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 R0 ω 0.5 2.0 8.0 yrs to CFR=0.001 Figure 4: Effect of initial outbreak on time to CFR=0.001 0.7 1 0.05 0.35 2 4 6 2 4 6 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 R0 ω 0.25 1.00 4.00 16.00 years to 75% infected Figure 5: The time it takes (in yr) for 75% of the initial population size to become infected. For R0 > 2 the vast majority of infections occur in the first year. By slowing the epidemic down (i.e., decreasing R0, infections are delayed buying time for the development of a vaccine and/or treatment. However, as we slow down the epidemic by decreasing R0, the time scales of disease spread and immune waning meet, and reinfections increasingly drive excess deaths, and the total number of infection-induced deaths within the first decade after emergence is very similar regardless of how fast the infection is spreading (SI Fig 6). 6 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 0.7 1 0.05 0.35 2 4 6 2 4 6 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 R0 ω 1e+06 2e+06 3e+06 4e+06 deaths in yr 1−10 Figure 6: Number of infection-induced deaths in first ten years after emergence. When reinfection oc- curs quickly and is transmissible (high ω and ρ), the total number of infection-induced deaths is close to independent of R0. 3 Loss of immunity kernel We estimate the duration of sterilizing immunity based on Callow et al. [5], in which six out of nine people who got infected at the start of the experiment were susceptible to reinfection one year later. We can calculate the mean of ω given this for exponentially distributed waning times setting the CDF equal to 2/3. 1 ¯ω = 1 log 2/3 = 0.91 (21) lambdas=seq(0,2,by=0.01) plot(lambdas, pexp(1,rate=lambdas), type=’l’) abline(h=0.66667, v=1.099) Given this, the estimate for ω is 1.099 and the mean waiting time is 1/ω = 0.91 yr. We can also find the average duration of immunity for normally distributed waning times. We assume that the variance is the same as for the exponential model, 1 λ2 and the standard deviation is 1 λ. means = seq(0,2,by=0.01) plot(means,pnorm(1, mean=means, sd=1/1.099), type=’l’) abline(h=0.6667) abline(v=0.61) Here, we find the estimate for the mean of ω to be 0.61/yr, so the mean waiting time is 1 0.61 = 1.64yr. 4 Quarantine analysis To better understand how waning of immunity impacts symptom-based quarantining in the first few years after disease emergence, we consider the following three phases. 7 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 0.0 0.5 1.0 1.5 2.0 0.0 0.4 0.8 lambdas pexp(1, rate = lambdas) Figure 7: Estimated duration of immunity given exponentially distributed waning times 4.1 Phase 1 The initial epidemic, before control measures are introduced, burns through a proportion of the population, p1. At some point, let’s call it t0, we introduce highly effective quarantining, which breaks chains of trans- mission. Because the initial epidemic burns through the population quickly, we approximate the time of entry intoR1 as being uniform. Additionally, since chains of transmission are broken upon the introduction of effective quarantining and contact tracing, we assume no new cases arise aftert0 during Phase1. Therefore at t0 (or shortly after): S = 1 −p1 (22) I1 = 0 (23) R1 =p1 (24) 4.2 Phase 2 There is a time period during which the occasional immigrant case enters, but no chains of transmission start. Even if the immigrant with a primary or secondary infection infects a Susceptible person before being quarantined, that I1 is immediately quarantined and does not infect others. If an R2 person is infected, they are not quarantined but neither can they start a chain of transmission because any infection to S

in quarantine, and a transmission chain all among secondary (or later) cases will not happen because RE =R2ρβ0/(γ +µ)< 1. During this time period, people who have recovered from a primary infection are losing their immunity. During this phase: S = 1 −p1 (25) I1 =Q =I2 = 0 (26) R1 <p 1 (27) R2 <R ∗ 2 see phase 3 for definition ofR ∗ 2 (28) 4.3 Phase 3 There is a threshold value of R2, let’s call it R∗ 2, above which an immigrant case can lead to an outbreak of asymptomatic secondary infections, which acts as a reservoir for symptomatic primary infections. This 8 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 0.0 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 means pnorm(1, mean = means, sd = 1/1.099) Figure 8: Estimated duration of immunity given normally distributed waning times Figure 9: Phase 1 transmission occurs even in the presence of a strong quarantining program for symptomatic primary cases. If R∗ 2 > p1, quarantining can continue working indefinitely. However, ifR∗ 2 <p 1, immune waning will lead to a sufficient build up of secondary susceptibles (in R2) to sustain an outbreak. In this scenario, the ‘safe time’ during which strong quarantining of primary cases can prevent outbreaks is influenced by: how fast immunity wanes (ω), how transmissible secondary cases are ( ρβ0) and the proportion of the population that was infected in the initial outbreak (p1). During this phase: S <1 −p1 (29) dI1 dt =ρβ0I2S (30) R1 <p 1 (31) dI2 dt =ρβ0I2R2 (32) R2fluctuates as transmission occurs. (33) 4.4 Window of safety Here, we show results for the window of safety during which strict quarantining of primary cases will prevent an immigrant infection from leading to an outbreak for R0 = 6 and R0 = 1.5 to mimic a high but reason- 9 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint Figure 10: Phase 2 transmission able transmission rate and a rate that may be more commensurate with the current scenario given social distancing. ForR0 = 2 we see quar_thresh_R0 <- expression(1/(rho*propR2)) pars2<-expand.grid(R0=c(2,6), rho=seq(0.1,1,by=0.01), omega=seq(0.5,2,by=0.01), p.pulse=seq(0.3,0.7, by=0.1)) pars2$thresh.R2<-with(pars2, 1/(R0*rho)) pars2<-subset(pars2, thresh.R2<p.pulse) pars2$safe.time.exp<-apply(pars2, 1, function(x){ qexp(x[’thresh.R2’]/x[’p.pulse’], rate=x[’omega’])}) pars2$safe.time.norm <-apply(pars2, 1, function(x){ qnorm(x[’thresh.R2’]/x[’p.pulse’], mean=x[’omega’], sd=sqrt(1/x[’omega’]ˆ2))}) pars2 <- pivot_longer(data=pars2, cols=grep(’safe.time’,colnames(pars2)), names_to = ’model’, values_to=’safe_time’) model.labs<-c(safe.time.norm=’Normal \nwaning times’, safe.time.exp=’Exponential\nwaning times’) p.pulse.labs<-paste(’p1=’,unique(pars2$p.pulse), sep=’’) names(p.pulse.labs)<-unique(pars2$p.pulse) g<-ggplot(data=subset(pars2, R0==2), aes(x=omega, y=rho)) g + geom_tile(aes(fill=safe_time))+ scale_fill_gradient2(mid=’white’, midpoint=2, breaks=c(2,4,6,8)) + facet_grid(cols=vars(p.pulse), rows=vars(model), labeller=labeller(model = model.labs, p.pulse=p.pulse.labs))+ labs(title=’Safe time, R0=2’, x=expression(omega), y=expression(rho), fill=’Time (yr)’) 10 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint Figure 11: Phase 3 transmission p1=0.6 p1=0.7 Exponential waning times Normal waning times 0.5 1.0 1.5 2.00.5 1.0 1.5 2.0 0.8 0.9 1.0 0.8 0.9 1.0 ω ρ 2 4 6 8 Time (yr) Safe time, R0=2 Figure 12: Safe time for R0 = 2 ForR0 = 6, a smaller p1 can support an outbreak (as low as p1=0.3). The amount of time it takes for people’s immunity to wane sufficiently has a broader range. g<-ggplot(data=subset(pars2, R0==6), aes(x=omega, y=rho)) g + geom_tile(aes(fill=safe_time))+ scale_fill_gradient2(mid=’white’, midpoint=2, breaks=c(2,4,6,8)) + facet_grid(cols=vars(p.pulse), rows=vars(model), labeller=labeller(model = model.labs, p.pulse=p.pulse.labs))+ 11 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint labs(title=’Safe time, R0=6’, x=expression(omega), y=expression(rho), fill=’Time (yr)’) p1=0.3 p1=0.4 p1=0.5 p1=0.6 p1=0.7 Exponential waning times Normal waning times 0.5 1.0 1.5 2.00.5 1.0 1.5 2.00.5 1.0 1.5 2.00.5 1.0 1.5 2.00.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 ω ρ 2 4 6 8 Time (yr) Safe time, R0=6 Figure 13: Safe time for R0 = 6 In reality, many primary cases have also been found to be asymptomatic, making symptom-based quar- antine substantially more challenging and highlighting the need for asymptomatic surveillance to protect against transmission to vulnerable individuals [8]. 5 Age-severity curves We estimate the age-severity curves for the three HCoVs to have emerged in the past few decades using published data (CoV-2 [9], SARS CoV-1 [10], and MERS [11]). We then fit a generalized linear models to each data set to estimate a smoothed CFR as a function of age. We use a binomial model in which total cases in an age group is considered the number of trials, and the number of deaths is considered the number of ‘successes’ . For MERS, we allow the function to be a third degree polynomial to account for the non-monotonicity of the data. We consider a second degree polynomial for SARS CoV-1 and -2, since the relationship between age and CFR appears monotonic in the data for these. Our code is included below. require(tidyverse) library(dplyr) library(ggplot2) library(reshape2) ## ## Attaching package: ’reshape2’ ## The following object is masked from ’package:tidyr’: ## ## smiths 12 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint hk.sars 75’)) hk.sars$prev.10000 <- c(0.1,0.8,2,3.8,2.6,2.5,2.4,3.1) hk.sars$cases <- round(1750*(hk.sars$prev.10000/sum(hk.sars$prev.10000))) hk.sars$cfr <- c(0,0.5,1.6, 10,13,25.3,52.5,69.6)/100 hk.sars$deaths<- round(hk.sars$cfr * hk.sars$cases) cfr.age<-data.frame( path=c(rep(c(’mers’, ’covid’),each=9),rep(’sars’,8)), age.mids=c(rep(c(seq(5,75,by=10),92.5),2),c(7.5,20,30,40,50,60,70,85)) ) cfr.age$cases[cfr.age$path==’mers’]<-c(8, 27, 314, 215, 264, 321,333,214, 93) cfr.age$deaths[cfr.age$path==’mers’]=c(5, 8, 52,63, 76, 103, 114, 118, 49) cfr.age$cases[cfr.age$path==’covid’]<-c( 416, 549, 3619, 7600, 8571, 10008, 8583, 3918, 1408) cfr.age$deaths[cfr.age$path==’covid’]<-c( 0,1,7,18,38, 130,309, 312, 208) cfr.age$cases[cfr.age$path==’sars’]<-hk.sars$cases cfr.age$deaths[cfr.age$path==’sars’]=hk.sars$deaths cfr.age$cfr<-cfr.age$deaths/cfr.age$cases #cfr.age$cfr[cfr.age$path==’sars’]<-c(0,0,0.9,3.0, 5.0, 10,17.6, 28, 26.3)/100 cfr.age$cfr.lo<-cfr.age$cfr- 1.96*sqrt(cfr.age$cfr*(1-cfr.age$cfr)/cfr.age$cases) cfr.age$cfr.hi<-cfr.age$cfr+ 1.96*sqrt(cfr.age$cfr*(1-cfr.age$cfr)/cfr.age$cases) cfr.age$surv<-cfr.age$cases-cfr.age$deaths pred.ages<-seq(5,85,length=9) ages.sars<-cfr.age$age.mids[cfr.age$path==’sars’] dat.sars<-as.matrix(cfr.age[cfr.age$path==’sars’,c(’deaths’,’surv’)]) fit.sars<-glm(dat.sars ˜ poly(ages.sars,2), family=’binomial’) sars.logit <- predict(fit.sars, data.frame(ages.sars=pred.ages)) pred.sars<- exp(sars.logit)/(1+exp(sars.logit)) ages.mers<-cfr.age$age.mids[cfr.age$path==’mers’] dat.mers<-as.matrix(cfr.age[cfr.age$path==’mers’,c(’deaths’,’surv’)]) fit.mers<-glm(dat.mers ˜ poly(ages.mers,3), family=’binomial’) mers.logit <- predict(fit.mers, data.frame(ages.mers=pred.ages)) pred.mers<- exp(mers.logit)/(1+exp(mers.logit)) ages.covid<-cfr.age$age.mids[cfr.age$path==’covid’] dat.covid<-as.matrix(cfr.age[cfr.age$path==’covid’,c(’deaths’,’surv’)]) fit.covid<-glm(dat.covid ˜ poly(ages.covid,2), family=’binomial’) covid.logit <- predict(fit.covid, data.frame(ages.covid=pred.ages)) pred.covid<- exp(covid.logit)/(1+exp(covid.logit)) 13 . CC-BY-NC-ND 4.0 International licenseIt 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 preprintthis version posted September 5, 2020. ; https://doi.org/10.1101/2020.09.03.20187856doi: medRxiv preprint 6 Code availability and reproducibility All code will be available as an RMarkdown file.

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