key: cord-0885751-g3l5airx
authors: Shaw, Clara L.; Kennedy, David A.
title: What the reproductive number [Formula: see text] can and cannot tell us about COVID-19 dynamics
date: 2021-01-05
journal: Theor Popul Biol
DOI: 10.1016/j.tpb.2020.12.003
sha: 6ad1ffd2c4947f6fa3530b5b8de582efe519ff17
doc_id: 885751
cord_uid: g3l5airx

The reproductive number [Formula: see text] (or [Formula: see text] , the initial reproductive number in an immune-naïve population) has long been successfully used to predict the likelihood of pathogen invasion, to gauge the potential severity of an epidemic, and to set policy around interventions. However, often ignored complexities have generated confusion around use of the metric. This is particularly apparent with the emergent pandemic virus SARS-CoV-2, the causative agent of COVID-19. We address some misconceptions about the predictive ability of the reproductive number, focusing on how it changes over time, varies over space, and relates to epidemic size by referencing the mathematical definition of [Formula: see text] and examples from the current pandemic. We hope that a better appreciation of the uses, nuances, and limitations of [Formula: see text] and [Formula: see text] facilitates a better understanding of epidemic spread, epidemic severity, and the effects of interventions in the context of SARS-CoV-2.

With the emergence of SARS-CoV-2, the novel coronavirus 2 responsible for COVID-19, much attention has been given to the 3 reproductive number, R, and its initial state, R 0 (Viceconte and 4 Petrosillo, 2020) . R 0 is the expected number of infections gener-5 ated by an infected individual in an otherwise fully susceptible 6 population and in the absence of interventions (Anderson and 7 May, 1991; Diekmann et al., 1990) . Since each infection produces 8 an average of R 0 new infections, R 0 describes the exponential 9 growth of infections during the early phase of an epidemic. Under 10 relatively general assumptions, R 0 can be used to determine the 11 probability of an emerging disease will cause an epidemic, the 12 final size of an epidemic, and what level of vaccination would 13 be required to achieve herd immunity (Anderson and May, 1991; 14 Delamater et al., 2019; Heffernan et al., 2005; Roberts, 2007) . In 15 many cases, R 0 has been invaluable such as for predicting the 16 risk of measles resurgence (Béraud et al., 2018; Hens et al., 2015) 17 and for managing emerging infectious diseases like SARS-CoV-1 18 (Lipsitch et al., 2003) and foot and mouth disease (Ferguson 19 et al., 2001) . When interpreted correctly, and in conjunction 20 with additional relevant information, it can yield valuable in-21

sight. However, misinterpretation may lead to faulty conclusions 22

regarding disease dynamics. 23 * Corresponding author. 2020) in addition to 27 severe economic distress. Policy makers have relied on estimates 28 of R 0 to tailor control measures (e.g. Ferguson et al., 2020) , but 29 these estimates vary tremendously within and between popula-30 tions around the globe (Fig. 1) . It is important to understand why 31 these estimates vary. It is also important to understand how the 32 utility of the reproductive number is limited. Here, we derive and 33 explain some of the key nuances of R and R 0 , paying particular 34 attention to insights and limitations with respect to the emerging 35 pathogen SARS-CoV-2. 36 2. Deriving R and R 0 37

How many new infections will be caused by a single infected 39 individual? For a directly transmitted pathogen, the answer to 40 this question can be written as: 41

Above, R is the reproductive number, k τ is the rate of contacts 43 that an infected individual has with susceptible individuals at 44 time τ post infection, b τ is the probability that a contact at time 45 τ results in a new infection, and P τ is the probability of still being YTPBI: 2792 C.L. Shaw and D.A. Kennedy Theoretical Population Biology xxx (xxxx) xxx Chen et al., 2020; Choi and Ki, 2020; Deb and Majumdar, 2020; Giordano et al., 2020; Johndrow et al., 2020; Ke et al., 2020; Korolev, 2020; Lewnard et al., 2020; Li et al., 2020b; Liu et al., 2020; Majumder and Mandl, 2020; Peirlinck et al., 2020; Pitzer et al., 2020; Ranjan, 2020; Read et al., 2020; Riou and Althaus, 2020; Sanche et al., 2020; Senapati et al., 2020; Shim et al., 2020; Singh and Adhikari, 2020; Tang et al., 2020; Wu et al., 2020; Yuan et al., 2020; Zhao et al., 2020) . Each point represents a literature-compiled average R 0 estimate for a different geographic area (sample size noted alongside means, error bars show plus or minus 1 standard error). For individual studies that provided multiple estimates for a single geographic area, the median estimate was used to avoid pseudo-replication. An analysis (not shown) confirmed that R 0 estimation method (transmission model, exponential growth model, or stochastic simulation method) did not drive the pattern of variation in R 0 by location. Recent meta-analyses of R 0 values for SARS-CoV-2 consider the effects of estimation methods in more detail (Alimohamadi et al., 2020; Barber et al., 2020) .

infected at time τ . Notably, we could have combined b τ and P τ 1 into a single parameter since the probability of infection given 2 contact falls to zero after an individual recovers, but we prefer 3 this more explicit formulation. Eq. (1) yields R, the total number 4 of infections one infected individual would generate over the 5 course of their infection. When the population is fully susceptible 6 and when no preventative interventions have been imposed, as 7 would be expected at the beginning of a novel outbreak, Eq. (1) 8 yields R 0 . Note that we have neglected to explicitly incorporate 9

individual variation and temporal variation in contact rates, the 10 probability that a contact results in a new infection, and the time 11

to recovery. However, R and R 0 , are intended to be averages, and 12 so this variation is inherently a part of the reproductive number 13

calculation. This variation could be explicitly included with addi-14 tional subscripts to denote all possible infected hosts, noninfected 15 hosts, times since the epidemic began, and ages of infections. 16

For simplicity, we will not use any subscripts for k, b, and P 17 when we discuss how they relate to the reproductive number of 18 SARS-CoV-2 in the remaining sections of this commentary. 19

Estimating the individual parameters in Eq. (1) requires ex-22

tensive data collection for a specific pathogen, host population, 23 and time. Alternative approaches to estimating R and R 0 therefore 24 frequently rely on epidemiological models and epidemic data. 25

Following the lead of Kermack and McKendrick (1927) , epidemics are often modeled as a set of ordinary differential equations such as for the classical Susceptible-Infected-Recovered (SIR) model. To illustrate the derivation of the reproductive number, we present a model of SARS-CoV-2 transmission that captures some of the key characteristics of the system. Unlike in the classic SIR model, in our model, infected individuals can be asymptomatic or not, and all non-asymptomatic infections progress through two stages, a pre-symptomatic and a symptomatic stage. Transmission is possible from each of these infected classes, but at different rates. Individuals showing symptoms are detected through testing (e.g. fever checks or diagnostic testing), and these individuals are quarantined. We assume no transmission from the quarantined class, and we assume that only symptomatic or quarantined individuals die from infection. Otherwise, all infected individuals eventually recover. In our model, the state variables S, I A , I P , I C , Q , and R are respectively the densities of susceptible, asymptomatically infected, pre-symptomatic, symptomatic (subscript ''C '' used to denote , quarantined, and recovered individuals in a population. Note that ''R'' here is distinct from the reproductive number ''R'', but we use both for historical reasons.

Here, the subscripts A, P, C , and Q denote parameters for asymp-26 tomatic, pre-symptomatic, COVID-19 symptomatic, and quaran-27 

incorporate spatial structure, hospitalization, age structure, host-1 specific exposure risk, superspreading, or time-varying mortality. Numerous methods can be used to derive R 0 from a model (Heffernan et al., 2005) . Perhaps most famously, R 0 is the dominant eigenvalue of the next generation matrix (Diekmann et al., 1990) . In the following text, we use the survival function to calculate R 0 because of its intuitive connection to Eq. (1), but if we had used the next generation matrix, we would have derived the exact same R 0 . The survival function uses three components to determine the number of new infections caused by an initial case:

(1) the rate at which an individual in a particular class causes new infections (from Eq. (1), kb), (2) the probability that an individual is still in the class at time τ (from Eq. (1), P), and (3) the probability that an initial infected individual will enter that class. Note that this third term is implicitly assumed to be one in Eq. (1). The integral of the product of the first two terms multiplied by the third term yields the contribution to R 0 that comes from a focal class. Deriving the overall R 0 requires summing over all possible classes. To illustrate, consider the asymptomatic contribution for an average initial infection. Here, (1) kb is equal to β A S, (2) P is equal to e −γ A τ , and (3) the probability that the initial case enters the asymptomatic class is equal to ϕ. Using these values, and extending the same methods to all infectious classes we derive:

where τ i is the time since entering class i. Since S changes 4 slowly at the beginning of an epidemic when few individuals 5 are infected, we can treat S as a constant with respect to time. 6 This assumption allows us to analytically solve Eq. (3.1), which 7 yields 8

Evaluating Eq. (3.2) at time t rather than time 0 yields R. No-10 tice that the reproductive number has contributions from the 11 asymptomatic, pre-symptomatic, and symptomatic classes, but 12 not from the other classes because the other classes cannot cause 13 new infections. If transmission were possible from these classes, 14 additional terms would need to be included in Eq. (3.2) . Alterna-15 tively, if we simplified our model by assuming that all infections 16

were symptomatic (ϕ = 0) and pre-symptomatic individuals 17

were not infectious (β p = 0), then Eq. (3.2) would simplify to,

2.3. Calculating the reproductive number without an epidemiologi-20 cal model 21 R 0 can also be estimated without an epidemiological model, 22 which can be especially useful if parameter estimates or even 23 an appropriate model structure are not yet known. In principle, 24 one could calculate R 0 by simply counting the cases attributed 25 to infected individuals at or near the beginning of an outbreak. 26 In practice, this method is rarely employed since contact tracing 27 networks are rarely established during the earliest phase of an 28 emerging disease outbreak (but see Pung et al., 2020) and esti-29 mates could be inaccurate due to bias towards observing large 30 chains of transmission. 31 R 0 can also be inferred from the growth rate of cases early in 32 an outbreak. Since the number of susceptible individuals changes 33 slowly during the initial stages of an outbreak, early case growth 34 rates can be approximated by exponential growth: the number 35 (Wallinga and 1 Lipsitch, 2007) . See Zhao et al. (2020) for an example of this 2 method used to calculate the R 0 of SARS-CoV-2. It is important 3

to recognize however that emerging epidemics often grow more 4 slowly than exponential due to stochastic effects, small popu-5 lation sizes, network effects, or preventative measures (Chowell 6 et al., 2016) . In these cases, a generalized growth model may more 7 accurately reflect epidemic growth . 8 After epidemics have started (i.e. when populations are no 9

longer fully susceptible and when interventions may have been 10 imposed), it is still useful to calculate R to understand if cases 11 will continue to grow or decline. A recent review explored various 12 methods for this calculation (Gostic et al., 2020) . Gostic et al. 13 (2020) recommended two methods. The first, developed by Cori 14 et al. (2013) uses time series incidence data and the serial interval 15 to calculate R in real time, which is particularly useful for assess-16

ing the impact of interventions as they are employed. The second, 17 developed by Wallinga and Teunis (2004) uses similar data and 18

is useful for the retrospective calculation of R.

No matter the method used to calculate it, limited data or 20 unreliable data early in an epidemic can make it difficult to 21

constrain R 0 . The World Health Organization originally estimated 22

the R 0 of SARS-CoV-2 to be between 1.4 and 2.5 (WHO, 2020). 23

More recent estimates of R 0 have varied from 2.2 to 6.47 for 24 the beginning of the Wuhan outbreak (Fig. 1 ). This represents 25 tremendous uncertainty when attempting to use R 0 for public 26 health planning. For example, if we were using these estimates 27

to design a vaccine campaign capable of achieving herd immunity 28

for a vaccine with perfect efficacy, our vaccination target (calcu-29 lated as 1 − 1/R 0 under assumptions of Eq. (2.1)-(2.6) would be 30 29% of the population at R 0 = 1.4 or 85% at R 0 = 6.47. Even

with better estimates of R 0 , however, misconceptions around this 32 metric lessen its practical utility. As we have explained, R 0 is calculated during the early stages 36 of an epidemic because of its value in determining future in-37 fection dynamics in the absence of intervention (Anderson and 38 May, 1991; Ma and Earn, 2006) . R is similarly useful for assessing 39 changes in transmission over time and the potential impacts of 40

interventions. However, R 0 and R cannot fully explain future 41 dynamics under particular circumstances. We discuss two impor-42 tant situations in which their value is limited: first, when hosts 43 become aware of infection and alter their behavior, and second, 44 when individual infection risk is heterogeneous. 45 Shifts in behavior that influence contact rates k or the proba-46 bility of infection given contact b can alter R over extremely short 47 timescales (see Eq. (1)). For example, as awareness of the SARS-48

CoV-2 epidemic grew in the United States in March 2020, human 49 mobility ground to a near halt Warren and 50 Skillman, 2020) , presumably reducing contact rates k. Other in-51

dividual behavioral changes such as increased handwashing and 52 mask wearing (Belot et al., 2020; Goldberg et al., 2020) have re-53 duced the probability of transmission given contact b (Eikenberry 54 et al., 2020; Liang et al., 2020) . Such bottom-up forces combined 55

with top-down government-imposed interventions (e.g. school 56

closures, banned gatherings) reduced R to below 1 (the thresh-57 old for epidemic persistence) by late April 2020 in some states 58 (Johndrow et al., 2020; Miller et al., 2020) . Similar reductions to 59 R were documented in China (Li et al., 2020a; Tian et al., 2020) 60 and other countries (Ensser et al., 2020; Giordano et al., 2020; 61 Kupferschmidt, 2020; Yuan et al., 2020) . Indeed, in models of the 62 1918 influenza pandemic, incorporating a behavioral response to 63 death rates improved model fits (Bootsma and Ferguson, 2007; 64 He et al., 2013) . For SARS-CoV-2, behavioral changes have caused 65 R to fluctuate above and below 1 at different times based on the 66 perceived threat of COVID-19 (Santamaría and Hortal, 2020) . 67 While behavioral changes can temporarily reduce R as de-68 scribed above, more sustainable reductions in R are typically 69 achieved when susceptible individuals are removed from popula-70 tions either through naturally acquired immunity or vaccination. 71 When hosts are heterogeneous such that some individuals are 72 more likely to contract infection than others, R declines faster 73 than predicted by Eq. (2.1)-(2.6) or other SIR models lacking 74 host heterogeneity (May and Anderson, 1987) . This is because 75 those most susceptible (for example, due to high exposure or low 76 inherent immunity) will become infected earlier in an epidemic, 77 leaving individuals that are on average more resistant (Langwig 78 et al., 2017; May and Anderson, 1987 (Gomes et al., 2020; Tkachenko et al., 2020) . In some of the hard-90 est hit areas, an appreciable fraction of people have been infected 91 with SARS-CoV-2 (e.g. 22.7% in New York City, USA (Rosenberg 92 et al. 2020) and over 44% in Manaus, Brazil Buss et al., 2020) . 93 However, the impact of these levels of infection on future disease 94 dynamics is unknown, since for SARS-CoV-2, heterogeneity in 95 infection risk is still highly uncertain (Randolph and Barreiro, 96 2020) . Moreover, heterogeneity is likely to change through time 97 due to changes in individual or government-mandated responses 98 (Dolbeault and Turinici, 2020) . Thus, even after herd immunity 99 has been reached, additional waves of infection could occur if 100 heterogeneity changes (Tkachenko et al., 2020) . 101

The R 0 of many pathogens are often referred to as known 104 values. For example, the R 0 of measles is 12-14, polio is 5-7, 105 and pertussis is 12-17 (Doherty et al., 2016) . For SARS-CoV-2, 106 estimates typically range from 2-3 (Liu et al., 2020a) . However, 107 the parameters (k, b, and P in Eq. (1)) that make up R 0 can differ 108 substantially from place to place (Figure 1 , Delamater et al., 2019) . 109 It follows that interventions to reduce R to less than 1 may need 110 to vary in aggressiveness across locations (Stier et al., 2020) . 111 Since R 0 differs between groups of people, combining multiple 112 groups together to estimate a population-wide R 0 can produce 113 misleading notions of disease spread. For example, a high average 114 measles vaccination rate in the United States keeps R below 1 115 nation-wide, but localized communities with high rates of vaccine 116 refusal still experience serious outbreaks (Leslie et al., 2018) . 117 In the current COVID-19 pandemic, disease transmission has so 118 far been much higher in refugee and low income populations 119 compared to non-refugee and high income populations (Chopra 120 and Sobel, 2020; Lau et al., 2020; Ruiz-Euler et al., 2020) and R 0 121 has been documented to vary between US counties (Sly et al., 122 2020) . Since R 0 is an average, combining communities with high 123 and low transmission may yield an estimate of R 0 < 1, yet disease 124 may still readily spread (Li et al., 2011 groups, behavior associated with work, home, and recreation 3 affects contact rates k. As we have discussed above, SIR models 4 often assume that contact rates, and thus R and R 0 , depend on 5 host density. Although, evidence is mixed as to whether larger 6 cities have higher values of R and R 0 for SARS-CoV-2 (Heroy, 7 2020; Stier et al., 2020) , built environments (e.g. hospitals, airport 8 terminals, factories) do often have high values of R (Dietz et al., 9 2020) . This may partially explain patterns of explosive SARS-10

CoV-2 transmission in venues such as cruise ships and meat 11 packing facilities (Althouse et al., Dyal et al., 2020; Mizu-12 moto and Chowell, 2020) . Household contacts have also been 13 particularly important to the virus' transmission dynamics (Bi 14 et al., 2020) . Therefore, differences in R 0 between populations can 15 be partially explained by differences in household sizes between 16 countries and cultures (Singh and Adhikari, 2020) . For example, 17

large, multigenerational households in Italy may have contributed 18 to the large and deadly outbreak there relative to other European 19 countries where multigenerational households are less common 20 (Dowd et al., 2020) . Similarly, the probability of a new infection 21 results from contact b and the probability of remaining infected 22 over time P may vary by population. For SARS-CoV-2, individuals 23

with severe symptoms have 60 times more viral RNA in nasal 24 swabs, which likely increases both their ability to transmit the 25 virus and the amount of time they remain infected (Liu et al., 26 2020b). Since older individuals are more likely to develop se-27

vere infection (Yang et al., 2020) , R 0 is likely to be greater in 28

populations with older individuals, such as in nursing homes 29 (McMichael et al., 2020) or in developed countries (Dowd et al., 30 2020) . 31

us how large an epidemic will be 33

It is tantalizing to imagine that R 0 can be used to predict the 34 extent of an outbreak, since it can be calculated during the early 35 stages of an epidemic. Indeed R 0 is related to final epidemic size 36 (Kermack and McKendrick, 1927) , but this relationship can be 37 substantially affected by the fraction of the population infected 38

initially and by heterogeneity in transmission. 39 If we rescale population sizes such that the initial susceptible 40 population size S 0 = 1, and we assume that population sizes 41

are sufficiently large to neglect demographic stochasticity field and Alizon, 2013; Tildesley and Keeling, 2009) to calculate R 0 so long as the model assumes that the population 47 is well-mixed with homogeneous susceptibility. Note that this 48 equation prominently features I 0 , the fraction of the population 49

infected at the beginning of the outbreak (or at the beginning 50 of an intervention). Efforts to reduce the reproductive number 51 below 1 are understandably a high priority, but when R 0 is close 52 to or less than 1, the final outbreak size is more sensitive to 53 changes in the fraction of infected individuals I 0 than it is to 54 changes in R 0 (Fig. 3) . This is because more infected individuals 55 will fuel the outbreak for longer, infecting a greater proportion 56 of the susceptible population even when each case produces 57 on average less than one new infection. Now that SARS-CoV- 58 2 infection rates are already substantial in populations around 59 the globe, I must be considered in addition to R. For example, 60

Pei et al. (2020) estimated that implementing social distancing 61 policies one week earlier could have reduced the cases in the 62

United States by early May, 2020 by 55% (over 700,000 cases) by 63 keeping the number of infected individuals low at the time such 64 policies were implemented. 65 Fig. 3 . Epidemic size contours and shading show that when R 0 is close to 1, the epidemic is more strongly influenced by a reduction of I 0 than by a reduction of R 0 . For instance, if R 0 = 0.95 (red dashed line), the epidemic could infect from less than 0.1% to greater than 16% of the population as I 0 ranges from just above 0% to 3% of the population. Epidemic size was calculated using the final size equation, Z = S 0 (1 − e −R 0 (Z−I 0 ) ), where S 0 = 1. Shading indicates the cube root of epidemic size with lighter colors corresponding to smaller outbreaks.

While a final epidemic size can be calculated using R 0 and 66 I 0 , the final size equation above does not apply to populations 67 with heterogeneous infection risk (Andreasen, 2011; Ball, 1985; 68 Hébert-Dufresne et al., 2020; Ma and Earn, 2006) . Given the 69 inherent heterogeneity in human social networks, it is surprising 70 that the final size equation is so accurate for many diseases such 71 as childhood diseases (Caudron et al., 2015) . This may be be-72 cause transmission networks among children are close to random 73 (Bjørnstad et al., 2002) or have other features that allow the final 74 size equation to be accurate (Bansal et al., 2007) . For diseases 75 where this is not the case (e.g. many sexually transmitted dis-76 eases), heterogeneity could in principle be incorporated into the 77 final epidemic size equation (Dwyer et al., 2000) , but estimating 78 heterogeneity early in an epidemic can be challenging. Moreover, 79 as we describe in misconception 1, heterogeneity in infection 80 risk can change over time as a result of human behavior or 81 interventions, such as the shutdowns in response to the COVID-19 82 epidemic (Dolbeault and Turinici, 2020; Ruiz-Euler et al., 2020) . 83 Estimates of how future government restrictions and behavioral 84 changes will alter heterogeneity in infection risk are thus critical 85 for assessing the likely impact of the outbreak (Gomes et al., 86 2020) . Such estimates are also key in determining thresholds for 87 herd immunity (Randolph and Barreiro, 2020) and in prioritizing 88 the distribution of interventions such as vaccines when they first 89 become available (Atkinson and Cheyne, 1994; Giambi et al., 90 2019) . 91

As we have discussed, the reproductive number R and its 93 initial value R 0 can be used to assess the potential for disease 94 invasion and persistence, to predict the extent of an epidemic, 95 and to infer the impact of interventions and of relaxing control 96 measures. Determining R in real time is especially helpful for the 97 latter goals (Gostic et al., 2020) . For example, the R 0 of SARS-98 CoV-2 was used to justify implementations of lockdowns in the 99 YTPBI: 2792 C.L. Shaw and D.A. Kennedy Theoretical Population Biology xxx (xxxx) xxx United Kingdom (Ferguson et al., 2020) , and R is one factor that 1 is considered in lockdown relaxation policies (Thompson et al., 2 2020) . 3 However, the utility of R and R 0 can easily be overstated. 4

Though we have focused on three misconceptions that we felt 5 were particularly important for the there 6 are additional considerations to appreciate for accurate interpre-7 tation of R and R 0 (Heffernan et al., 2005; Li et al., 2011; Roberts, 8 2007) . These include complications associated with stochasticity 9 (Keeling and Grenfell, 2000) , superspreading (Lloyd-Smith et al., 10 2005) , metapopulation dynamics (Cross et al., 2007) , seasonal-11

ity (Bjørnstad et al., 2002) , multiple hosts (Roberts, 2007) , and 12 pathogen evolution (Hartfield and Alizon, 2014 

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