An extended catalytic model to assess changes in risk for multiple reinfections with SARS-CoV-2 (preprint)

Background: The SARS-CoV-2 pandemic has illustrated that monitoring trends in multiple infections can provide insight into the biological characteristics of new variants. Following several pandemic waves, many people have already been infected and reinfected by SARS-CoV-2 and therefore methods are needed to understand the risk of multiple reinfections. Objectives: In this paper, we extended an existing catalytic model designed to detect increases in the risk of reinfection by SARS-CoV-2 to detect increases in the population-level risk of multiple reinfections. Methods: The catalytic model assumes the risk of reinfection is proportional to observed infections and uses a Bayesian approach to fit model parameters to the number of nth infections among individuals whose nth infection was observed at least 90 days before. Using a posterior draw from the fitted model parameters, a 95% projection interval of daily nth infections is calculated under the assumption of a constant nth infection hazard coefficient. An additional model parameter was introduced to consider the increased risk of reinfection detected during the Omicron wave. Validation was performed to assess the model's ability to detect increases in the risk of third infections. Key Findings: The model parameters converged when applying the model's fitting and projection procedure to the number of observed third SARS-COV-2 infections in South Africa. No additional increase in the risk of third infection was detected after the increase detected during the Omicron wave. The validation of the third infections method showed that the model can successfully detect increases in the risk of third infections under different scenarios. Limitations: Even though the extended model is intended to detect the risk of nth infections, the method was only validated for detecting increases in the risk of third infections and not for four or more infections. The method is very sensitive to low numbers of nth infections, so it might not be usable in settings with small epidemics, low coverage of testing or early in an outbreak. Conclusions: The catalytic model to detect increases in the risk of reinfections was successfully extended to detect increases in the risk of nth infections and could contribute to future detection of increases in the risk of nth infections by SARS-CoV-2 or other similar pathogens.

Incubation-period estimates of Omicron (BA.1) variant from Taiwan, December 2021-January 2022, and its comparison to other SARS-CoV-2 variants: a statistical modeling, systematic search and meta-analysis (preprint)

Background: The ongoing COVID-19 pandemic has seen several variants of concern, including the Omicron (BA.1) variant which emerged in October 2021. Accurately estimating the incubation period of these variants is crucial for predicting disease spread and formulating effective public health strategies. However, existing estimates often conflict because of biases arising from the dynamic nature of epidemic growth and selective inclusion of cases. This study aims to accurately estimate of the Omicron (BA.1) variant incubation period based on data from Taiwan, where disease incidence remained low and contact tracing was comprehensive during the first months of the Omicron outbreak. Methods: We reviewed 100 contact-tracing records for cases of the Omicron BA.1 variant reported between December 2021 and January 2022, and found enough information to analyze 70 of these. The incubation period distribution was estimated by fitting data on exposure and symptom onset within a Bayesian mixture model using gamma, Weibull, and lognormal distributions as candidates. Additionally, a systematic literature search was conducted to accumulate data for estimates of the incubation period for Omicron (BA.1/2, BA.4/5) subvariants, which was then used for meta-analysis and comparison. Results: The mean incubation period was estimated at 3.5 days (95% credible interval: 3.1-4.0 days), with no clear differences when stratified by vaccination status or age. This estimate aligns closely with the pooled mean of 3.4 days (3.0-3.8 days) obtained from a meta-analysis of other published studies on Omicron subvariants. Conclusions: The relatively shorter incubation period of the Omicron variant, as compared to previous SARS-CoV2 variants, implies its potential for rapid spread but also opens the possibility for individuals to voluntarily adopt shorter, more resource-efficient quarantine periods. Continual updates to incubation period estimates, utilizing data from comprehensive contact tracing, are crucial for effectively guiding these voluntary actions and adjusting high socio-economic cost interventions.

Intermediate levels of asymptomatic transmission can lead to the highest levels of epidemic fatalities (preprint)

Asymptomatic infections have hampered the ability to characterize and prevent the transmission of SARS-CoV-2 throughout the ongoing pandemic. Even though asymptomatic infections reduce severity at the individual level, they can make population-level outcomes worse if asymptomatic individuals---unaware they are infected---transmit more than symptomatic individuals. Using an epidemic model, we show that intermediate levels of asymptomatic infection lead to the highest levels of epidemic fatalities when the increase in asymptomatic transmission, due either to individual behavior or mitigation efforts, is strong. We generalize this result to include presymptomatic transmission, showing how intermediate levels of non-symptomatic transmission can lead to the highest levels of fatalities. Finally, we extend our framework to illustrate how the intersection of asymptomatic spread and immunity profiles determine epidemic trajectories, including population-level severity, of future variants.

Inferring the differences in incubation-period and generation-interval distributions of the Delta and Omicron variants of SARS-CoV-2 (preprint)

Estimating the differences in the incubation-period, serial-interval, and generation-interval distributions of SARS-CoV-2 variants is critical to understanding their transmission and control. However, the impact of epidemic dynamics is often neglected in estimating the timing of infection and transmission---for example, when an epidemic is growing exponentially, a cohort of infected individuals who developed symptoms at the same time are more likely to have been infected recently. Here, we re-analyze incubation-period and serial-interval data describing transmissions of the Delta and Omicron variants from the Netherlands at the end of December 2021. Previous analysis of the same data set reported shorter mean observed incubation period (3.2 days vs 4.4 days) and serial interval (3.5 days vs 4.1 days) for the Omicron variant, but the number of infections caused by the Delta variant decreased during this period as the number of Omicron infections increased. When we account for growth-rate differences of two variants during the study period, we estimate similar mean incubation periods (3.8--4.5 days) for both variants but a shorter mean generation interval for the Omicron variant (3.0 days; 95\% CI: 2.7--3.2 days) than for the Delta variant (3.8 days; 95\% CI: 3.7--4.0 days). We further note that the differences in estimated generation intervals may be driven by the "network effect"---higher effective transmissibility of the Omicron variant can cause faster susceptible depletion among contact networks, which in turn prevents late transmission (therefore shortening realized generation intervals). Using up-to-date generation-interval distributions is critical to accurately estimating the reproduction advantage of the Omicron variant.

How time-scale differences in asymptomatic and symptomatic transmission shape SARS-CoV-2 outbreak dynamics (preprint)

Asymptomatic and symptomatic SARS-CoV-2 infections can have different characteristic time scales of transmission. These time-scale differences can shape outbreak dynamics as well as bias population-level estimates of epidemic strength, speed, and controllability. For example, prior work focusing on the initial exponential growth phase of an outbreak foundthat larger time scales for asymptomatic vs. symptomatic transmission can lead to under-estimates of the basic reproduction number as inferred from epidemic case data. Building upon this work, we use a series of nonlinear epidemic models to explore how differences in asymptomatic and symptomatic transmission time scales can lead to changes in the realized proportion of asymptomatic transmission throughout an epidemic. First, we find that when asymptomatic transmission time scales are longer than symptomatic transmission time scales, then the effective proportion of asymptomatic transmission increases as total incidence decreases. Moreover, these time-scale-driven impacts on epidemic dynamics are enhanced when infection status is correlated between infector and infectee pairs (e.g., due to dose-dependent impacts on symptoms). Next we apply these findings to understand the impact of time-scale differences on populations with age-dependent assortative mixing and in which the probability of having a symptomatic infection increases with age. We show that if asymptomatic generation intervals are longer than corresponding symptomatic generation intervals, then correlations between age and symptoms lead to a decrease in the age of infection during periods of epidemic decline (whether due to susceptible depletion or intervention). Altogether, these results demonstrate the need to explore the role of time-scale differences in transmission dynamics alongside behavioural changes to explain outbreak features both at early stages (e.g., in estimating the basic reproduction number) and throughout an epidemic (e.g., in connecting shifts in the age of infection to periods of changing incidence).

From Science to Politics: COVID-19 Information Fatigue on YouTube (preprint)

Objective: The COVID-19 pandemic is the first pandemic where social media platforms relayed information on a large scale, enabling an “infodemic” of conflicting information which undermined the global response to the pandemic. Understanding how the information circulated and evolved on social media platforms is essential for planning future public health campaigns. This study investigated what types of themes about COVID-19 were most viewed on YouTube during the first 8 months of the pandemic, and how COVID-19 themes progressed over this period. Methods: : We analyzed top-viewed YouTube COVID-19-related videos in English from from December 1, 2019 to August 16, 2020 with an open inductive content analysis. We coded 536 videos associated with 1.1 billion views across the study period. East Asian countries were the first to report the virus, while most of the top-viewed videos in English were from the US. Videos from straight news outlets dominated the top-viewed videos throughout the outbreak, and public health authorities contributed the fewest. Although straight news was the dominant COVID-19 video source with various types of themes, its viewership per video was similar to that for entertainment news and YouTubers after March. Results: : We found, first, that collective public attention to the COVID-19 pandemic on YouTube peaked around March 2020, before the outbreak peaked, and flattened afterwards despite a spike in worldwide cases. Second, more videos focused on prevention early on, but videos with political themes increased through time. Third, regarding prevention and control measures, masking received much less attention than lockdown and social distancing in the study period. Conclusion: Our study suggests that a transition of focus from science to politics on social media intensified the COVID-19 infodemic and may have weakened mitigation measures during the first waves of the COVID-19 pandemic. It is recommended that authorities should consider co-operating with reputable social media influencers to promote health campaigns and improve health literacy. In addition, given high levels of globalization of social platforms and polarization of users, tailoring communication towards different digital communities is likely to be essential.

Bounding the levels of transmissibility & immune evasion of the Omicron variant in South Africa (preprint)

A new SARS-CoV-2 variant of concern, Omicron (B.1.1.529), has been identified based on genomic sequencing and epidemiological data in South Africa. Presumptive Omicron cases in South Africa have grown extremely rapidly, despite high prior exposure and moderate vaccination coverage. The available evidence suggests that Omicron spread is at least in part due to evasion of this immune protection, though Omicron may also exhibit higher intrinsic transmissibility. Using detailed laboratory and epidemiological data from South Africa, we estimate the constraints on these two characteristics of the new variant and their relationship. Our estimates and associated uncertainties provide essential information to inform projection and scenario modeling analyses, which are crucial planning tools for governments around the world. One Sentence Summary We report a region of plausibility for the relative transmissibility and immune escape characteristics of the SARS-CoV-2 Omicron variant estimated by integrating laboratory and epidemiological data from South Africa.

SARS-CoV-2 reinfection trends in South Africa: analysis of routine surveillance data (preprint)

Objective: To examine whether SARS-CoV-2 reinfection risk has changed through time in South Africa, in the context of the emergence of the Beta and Delta variants Design: Retrospective analysis of routine epidemiological surveillance data Setting: Line list data on SARS-CoV-2 with specimen receipt dates between 04 March 2020 and 30 June 2021, collected through South Africa's National Notifiable Medical Conditions Surveillance System Participants: 1,551,655 individuals with laboratory-confirmed SARS-CoV-2 who had a positive test result at least 90 days prior to 30 June 2021. Individuals having sequential positive tests at least 90 days apart were considered to have suspected reinfections. Main outcome measures: Incidence of suspected reinfections through time; comparison of reinfection rates to the expectation under a null model (approach 1); empirical estimates of the time-varying hazards of infection and reinfection throughout the epidemic (approach 2) Results: 16,029 suspected reinfections were identified. The number of reinfections observed through the end of June 2021 is consistent with the null model of no change in reinfection risk (approach 1). Although increases in the hazard of primary infection were observed following the introduction of both the Beta and Delta variants, no corresponding increase was observed in the reinfection hazard (approach 2). Contrary to expectation, the estimated hazard ratio for reinfection versus primary infection was lower during waves driven by the Beta and Delta variants than for the first wave (relative hazard ratio for wave 2 versus wave 1: 0.75 (95% CI: 0.59-0.97); for wave 3 versus wave 1: 0.70 (95% CI: 0.55-0.90)). Although this finding may be partially explained by changes in testing availability, it is also consistent with a scenario in which variants have increased transmissibility but little or no evasion of immunity. Conclusion: We conclude there is no population-wide epidemiological evidence of immune escape and recommend ongoing monitoring of these trends.

Testing and Isolation Efficacy: Insights from a Simple Epidemic Model (preprint)

Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases -- surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals' behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control.

Roles of generation-interval distributions in shaping relative epidemic strength, speed, and control of new SARS-CoV-2 variants (preprint)

Inferring the relative strength (i.e., the ratio of reproduction numbers, Rvar/Rwt) and relative speed (i.e., the difference between growth rates, rvar-rwt) of new SARS-CoV-2 variants compared to their wild types is critical to predicting and controlling the course of the current pandemic. Multiple studies have estimated the relative strength of new variants from the observed relative speed, but they typically neglect the possibility that the new variants have different generation intervals (i.e., time between infection and transmission), which determines the relationship between relative strength and speed. Notably, the increasingly predominant B.1.1.7 variant may have a longer infectious period (and therefore, a longer generation interval) than prior dominant lineages. Here, we explore how differences in generation intervals between a new variant and the wild type affect the relationship between relative strength and speed. We use simulations to show how neglecting these differences can lead to biases in estimates of relative strength in practice and to illustrate how such biases can be assessed. Finally, we discuss implications for control: if new variants have longer generation intervals then speed-like interventions such as contact tracing become more effective, whereas strength-like interventions such as social distancing become less effective.

Age-dependence of healthcare interventions for COVID-19 in Ontario, Canada (preprint)

Background: Patient age is the most salient clinical indicator of risk from COVID-19. Age-specific distributions of known SARS-CoV-2 infections and COVID-19-related deaths are available for many regions. Less attention has been given to the age distributions of serious medical interventions administered to COVID-19 patients, which could reveal sources of potential pressure on the healthcare system should SARS-CoV-2 prevalence increase. Methods: We analysed 97,957 known SARS-CoV-2 infection records for Ontario, Canada, from 23 January 2020 to 26 November 2020 and estimated the age distributions of hospitalizations, Intensive Care Unit admissions, intubations, and ventilations. We quantified the probability of hospitalization given known SARS-CoV-2 infection, and of survival given COVID-19-related hospitalization. Results: The distribution of hospitalizations peaks with a wide plateau covering ages 54–90, whereas deaths are concentrated in very old ages. The estimated probability of hospitalization given known infection reaches a maximum of 30.9% at age 80 (95% CI 28.0%–33.9%). The probability of survival given hospitalization is near 100% for adults younger than 40, but declines substantially after this age; for example, a hospitalized 54-year-old patient has a 91.5% chance of surviving COVID-19 (95% CI 87.0%–94.9%). Conclusions: Ontario’s healthcare system has not been overstretched by COVID-19 thanks to wide-spread infection control efforts, yet the probability of survival given hospitalization for COVID-19 is lower than is generally perceived for patients over 40. As pervalence continues to increase during this most recent wave of infection, healthcare capacities are at risk of being exceeded. Survival of individuals in the broad age range requiring acute care could decrease, potentially expanding the distribution of COVID-19-related deaths toward younger ages.

Age-dependence of healthcare interventions for SARS-CoV-2 infection in Ontario, Canada (preprint)

BackgroundPatient age is the most salient clinical indicator of risk from COVID-19. Age-specific distributions of known SARS-CoV-2 infections and COVID-19-related deaths are available for most countries. However, relatively little attention has been given to the age distributions of hospitalizations and serious healthcare interventions administered to COVID-19 patients. We examined these distributions in Ontario, Canada, in order to quantify the age-related impacts of COVID-19, and to identify potential risks should the healthcare system become overwhelmed with COVID-19 patients in the future. MethodsWe analysed known SARS-CoV-2 infection records from the integrated Public Health Information System (iPHIS) and the Toronto Public Health Coronavirus Rapid Entry System (CORES) between 23 January 2020 and 17 June 2020 (N = 30,546), and estimated the age distributions of hospitalizations, ICU admissions, intubations, and ventilations. We quantified the probability of hospitalization given known SARS-CoV-2 infection, and of survival given COVID-19-related hospitalization. ResultsThe distribution of COVID-19-related hospitalizations peaks with a wide plateau covering ages 54-90, whereas deaths are sharply concentrated in very old ages, with a maximum at age 90. The estimated probability of hospitalization given known SARS-CoV-2 infection reaches a maximum of 32.0% at age 75 (95% CI 27.5%-36.7%). The probability of survival given COVID-19-related hospitalization is uncertain for children (due to small sample size), and near 100% for adults younger than 40. After age 40, survival of hospitalized COVID-19 patients declines substantially; for example, a hospitalized 50-year-old patient has a 90.4% chance of surviving COVID-19 (95% CI 81.9%-95.7%). InterpretationConcerted efforts to control the spread of SARS-CoV-2 have kept prevalence of the virus low in the population of Ontario. The healthcare system has not been overstretched, yet the probability of survival given hospitalization for COVID-19 has been lower than is generally recognized for patients over 40. If prevalence of the virus were to increase and healthcare capacities were to be exceeded, survival of individuals in the broad age range requiring acute care would be expected to decrease, potentially expanding the distribution of COVID-19-related deaths toward younger ages.

Cohort-based approach to understanding the roles of generation and serial intervals in shaping epidemiological dynamics (preprint)

G eneration intervals and serial intervals are critical quantities for characterizing outbreak dynamics. Generation intervals characterize the time between infection and transmission, while serial intervals characterize the time between the onset of symptoms in a chain of transmission. They are often used interchangeably, leading to misunderstanding of how these intervals link the epidemic growth rate r and the reproduction number ℛ . Generation intervals provide a mechanistic link between r and ℛ but are harder to measure via contact tracing. While serial intervals are easier to measure from contact tracing, recent studies suggest that the two intervals give different estimates of ℛ from r . We present a general framework for characterizing epidemiological delays based on cohorts (i.e., a group of individuals that share the same event time, such as symptom onset) and show that forward-looking serial intervals, which correctly link ℛ with r , are not the same as “intrinsic” serial intervals, but instead change with r . We provide a heuristic method for addressing potential biases that can arise from not accounting for changes in serial intervals across cohorts and apply the method to estimating ℛ for the COVID-19 outbreak in China using serial-interval data — our analysis shows that using incorrectly defined serial intervals can severely bias estimates. This study demonstrates the importance of early epidemiological investigation through contact tracing and provides a rationale for reassessing generation intervals, serial intervals, and ℛ estimates, for COVID-19. Significance Statement The generation- and serial-interval distributions are key, but different, quantities in outbreak analyses. Recent theoretical studies suggest that two distributions give different estimates of the reproduction number ℛ from the exponential growth rate r ; however, both intervals, by definition, describe disease transmission at the individual level. Here, we show that the serial-interval distribution, defined from the correct reference time and cohort, gives the same estimate of ℛ as the generation-interval distribution. We then apply our framework to serial-interval data from the COVID-19 outbreak in China. While our study supports the use of serial-interval distributions in estimating ℛ , it also reveals necessary changes to the current understanding and applications of serial-interval distribution.

Moving Beyond a Peak Mentality: Plateaus, Shoulders, Oscillations and Other 'Anomalous' Behavior-Driven Shapes in COVID-19 Outbreaks (preprint)

The COVID-19 pandemic has caused more than 300,000 reported deaths globally, of which more than 83,000 have been reported in the United States as of May 16, 2020. Public health interventions have had significant impacts in reducing transmission and in averting even more deaths. Nonetheless, in many jurisdictions (both at national and local levels) the decline of cases and fatalities after apparent epidemic peaks has not been rapid. Instead, the asymmetric decline in cases appears, in some cases, to be consistent with plateau- or shoulder-like phenomena. Here we explore a model of fatality-driven awareness in which individual protective measures increase with death rates. In this model, epidemic dynamics can be characterized by plateaus, shoulders, and lag-driven oscillations after exponential rises at the outset of disease dynamics. We also show that incorporating long-term awareness can avoid peak resurgence and accelerate epidemic decline. We suggest that awareness of epidemic severity is likely to play a critical role in disease dynamics, beyond that imposed by intervention-driven policies.

Intervention Serology and Interaction Substitution: Modeling the Role of 'Shield Immunity' in Reducing COVID-19 Epidemic Spread (preprint)

The COVID-19 pandemic has precipitated a global crisis, with more than 690,000 confirmed cases and more than 33,000 confirmed deaths globally as of March 30, 2020 [1-4]. At present two central public health control strategies have emerged: mitigation and suppression (e.g, [5]). Both strategies focus on reducing new infections by reducing interactions (and both raise questions of sustainability and long-term tactics). Complementary to those approaches, here we develop and analyze an epidemiological intervention model that leverages serological tests [6, 7] to identify and deploy recovered individuals as focal points for sustaining safer interactions via interaction substitution, i.e., to develop what we term 'shield immunity' at the population scale. Recovered individuals, in the present context, represent those who have developed protective, antibodies to SARS-CoV-2 and are no longer shedding virus [8]. The objective of a shield immunity strategy is to help sustain the interactions necessary for the functioning of essential goods and services (including but not limited to tending to the elderly [9], hospital care, schools, and food supply) while decreasing the probability of transmission during such essential interactions. We show that a shield immunity approach may significantly reduce the length and reduce the overall burden of an outbreak, and can work synergistically with social distancing. The present model highlights the value of serological testing as part of intervention strategies, in addition to its well recognized roles in estimating prevalence [10, 11] and in the potential development of plasma-based therapies [12-15].

Potential roles of social distancing in mitigating the spread of coronavirus disease 2019 (COVID-19) in South Korea (preprint)

On January 20, 2020, the first COVID-19 case was confirmed in South Korea. After a rapid outbreak, the number of incident cases has been consistently decreasing since early March; this decrease has been widely attributed to its intensive testing. We report here on the likely role of social distancing in reducing transmission in South Korea. Our analysis suggests that transmission may still be persisting in some regions.

The time scale of asymptomatic transmission affects estimates of epidemic potential in the COVID-19 outbreak (preprint)

We assess the impact of asymptomatic transmission on epidemic potential of novel respiratory pathogens (like COVID-19) -- as measured both by the basic reproduction number (i.e., the expected number of secondary cases generated by an average primary case in a fully susceptible population) and the fraction of new secondary cases attributable to asymptomatic individuals. We show that the impact of asymptomatic transmission depends on generation intervals (i.e., time between when an individual is infected and when that individual infects another person). If the generation-interval distribution of asymptomatic transmission differs from that of symptomatic transmission, then estimates of the basic reproduction number which do not explicitly account for asymptomatic cases may be systematically biased. Specifically, if asymptomatic cases have a shorter generation interval than symptomatic cases, R_0 will be over-estimated, and if they have a longer generation interval, R_0 will be under-estimated. We also show that as the length of asymptomatic generation intervals increase, estimates of the realized proportion of asymptomatic transmission during the exponential phase of the epidemic decrease. Our analysis provides a rationale for assessing the duration of asymptomatic cases of COVID-19 in addition to their prevalence in the population.

Reconciling early-outbreak preliminary estimates of the basic reproductive number and its uncertainty: a new framework and applications to the novel coronavirus (2019-nCoV) outbreak (preprint)

A novel coronavirus (SARS-CoV-2) has recently emerged as a global threat. As the epidemic progresses, many disease modelers have focused on estimating the basic reproductive number Ro -- the average number of secondary cases caused by a primary case in an otherwise susceptible population. The modeling approaches and resulting estimates of Ro vary widely, despite relying on similar data sources. Here, we present a novel statistical framework for comparing and combining different estimates of Ro across a wide range of models by decomposing the basic reproductive number into three key quantities: the exponential growth rate $r$, the mean generation interval $\bar G$, and the generation-interval dispersion $\kappa$. We then apply our framework to early estimates of Ro for the SARS-CoV-2 outbreak. We show that many early Ro estimates are overly confident. Our results emphasize the importance of propagating uncertainties in all components of Ro, including the shape of the generation-interval distribution, in efforts to estimate Ro at the outset of an epidemic.