How generation intervals shape the relationship between growth rates and reproductive numbers

Mathematical models of transmission have become invaluable management tools in planning for the control of emerging infectious diseases. A key variable in such models is the reproductive number R. For new emerging infectious diseases, the value of the reproductive number can only be inferred indirectly from the observed exponential epidemic growth rate r. Such inference is ambiguous as several different equations exist that relate the reproductive number to the growth rate, and it is unclear which of these equations might apply to a new infection. Here, we show that these different equations differ only with respect to their assumed shape of the generation interval distribution. Therefore, the shape of the generation interval distribution determines which equation is appropriate for inferring the reproductive number from the observed growth rate. We show that by assuming all generation intervals to be equal to the mean, we obtain an upper bound to the range of possible values that the reproductive number may attain for a given growth rate. Furthermore, we show that by taking the generation interval distribution equal to the observed distribution, it is possible to obtain an empirical estimate of the reproductive number.     

Estimation of the incubation period of influenza A (H1N1-2009) among imported cases: addressing censoring using outbreak data at the origin of importation

Empirical estimates of the incubation period of influenza A (H1N1-2009) have been limited. We estimated the incubation period among confirmed imported cases who traveled to Japan from Hawaii during the early phase of the 2009 pandemic (n=72). We addressed censoring and employed an infection-age structured argument to explicitly model the daily frequency of illness onset after departure. We assumed uniform and exponential distributions for the frequency of exposure in Hawaii, and the hazard rate of infection for the latter assumption was retrieved, in Hawaii, from local outbreak data. The maximum likelihood estimates of the median incubation period range from 1.43 to 1.64 days according to different modeling assumptions, consistent with a published estimate based on a New York school outbreak. The likelihood values of the different modeling assumptions do not differ greatly from each other, although models with the exponential assumption yield slightly shorter incubation periods than those with the uniform exposure assumption. Differences between our proposed approach and a published method for doubly interval-censored analysis highlight the importance of accounting for the dependence of the frequency of exposure on the survival function of incubating individuals among imported cases. A truncation of the density function of the incubation period due to an absence of illness onset during the exposure period also needs to be considered. When the data generating process is similar to that among imported cases, and when the incubation period is close to or shorter than the length of exposure, accounting for these aspects is critical for long exposure times.     

The transmission dynamics of the aetiological agent of scrapie in a sheep flock

We formulate and investigate the properties of a model framework to mimic the transmission dynamics of the aetiological agent of scrapie in a sheep flock. We derive expressions for summary parameters that characterize transmission scenarios, notably the basic reproduction number R(0) and the mean generation time T(g). The timescale of epidemic outbreaks is expressed in terms of R(0) and cumulants of the generation time distribution. We discuss the relative contributions to the overall rate of transmission of horizontal and vertical routes during invasion and in endemicity. Simplified models are used to obtain analytical insight into the characteristics of the endemic state.     

Contact intervals, survival analysis of epidemic data, and estimation of R(0)

We argue that the time from the onset of infectiousness to infectious contact, which we call the "contact interval," is a better basis for inference in epidemic data than the generation or serial interval. Since contact intervals can be right censored, survival analysis is the natural approach to estimation. Estimates of the contact interval distribution can be used to estimate R(0) in both mass-action and network-based models. We apply these methods to 2 data sets from the 2009 influenza A(H1N1) pandemic.     

The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda

Despite improved control measures, Ebola remains a serious public health risk in African regions where recurrent outbreaks have been observed since the initial epidemic in 1976. Using epidemic modeling and data from two well-documented Ebola outbreaks (Congo 1995 and Uganda 2000), we estimate the number of secondary cases generated by an index case in the absence of control interventions R0. Our estimate of R0 is 1.83 (SD 0.06) for Congo (1995) and 1.34 (SD 0.03) for Uganda (2000). We model the course of the outbreaks via an SEIR (susceptible-exposed-infectious-removed) epidemic model that includes a smooth transition in the transmission rate after control interventions are put in place. We perform an uncertainty analysis of the basic reproductive number R0 to quantify its sensitivity to other disease-related parameters. We also analyse the sensitivity of the final epidemic size to the time interventions begin and provide a distribution for the final epidemic size. The control measures implemented during these two outbreaks (including education and contact tracing followed by quarantine) reduce the final epidemic size by a factor of 2 relative the final size with a 2-week delay in their implementation.     

Generation interval contraction and epidemic data analysis

The generation interval is the time between the infection time of an infected person and the infection time of his or her infector. Probability density functions for generation intervals have been an important input for epidemic models and epidemic data analysis. In this paper, we specify a general stochastic SIR epidemic model and prove that the mean generation interval decreases when susceptible persons are at risk of infectious contact from multiple sources. The intuition behind this is that when a susceptible person has multiple potential infectors, there is a "race" to infect him or her in which only the first infectious contact leads to infection. In an epidemic, the mean generation interval contracts as the prevalence of infection increases. We call this global competition among potential infectors. When there is rapid transmission within clusters of contacts, generation interval contraction can be caused by a high local prevalence of infection even when the global prevalence is low. We call this local competition among potential infectors. Using simulations, we illustrate both types of competition. Finally, we show that hazards of infectious contact can be used instead of generation intervals to estimate the time course of the effective reproductive number in an epidemic. This approach leads naturally to partial likelihoods for epidemic data that are very similar to those that arise in survival analysis, opening a promising avenue of methodological research in infectious disease epidemiology.     

Times from Infection to Disease-Induced Death and their Influence on Final Population Sizes After Epidemic Outbreaks

For epidemic models, it is shown that fatal infectious diseases cannot drive the host population into extinction if the incidence function is upper density-dependent. This finding holds even if a latency period is included and the time from infection to disease-induced death has an arbitrary length distribution. However, if the incidence function is also lower density-dependent, very infectious diseases can lead to a drastic decline of the host population. Further, the final population size after an epidemic outbreak can possibly be substantially affected by the infection-age distribution of the initial infectives if the life expectations of infected individuals are an unbounded function of infection age (time since infection). This is the case for lognormal distributions, which fit data from infection experiments involving tiger salamander larvae and ranavirus better than gamma distributions and Weibull distributions.     

Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions

Recurrent outbreaks of the avian H5N1 influenza virus in Asia represent a constant global pandemic threat. We characterize and evaluate hypothetical public health measures during the 1918 influenza pandemic in the Canton of Geneva, Switzerland. The transmission rate, the recovery rate, the diagnostic rate, the relative infectiousness of asymptomatic cases, and the proportion of clinical cases are estimated through least-squares fitting of the model to epidemic curve data of the cumulative number of hospital notifications. The latent period and the case fatality proportion are taken from published literature. We determine the variance and identifiability of model parameters via a simulation study. Our epidemic model agrees well with the observed epidemic data. We estimate the basic reproductive number for the spring wave R1;=1.49 (95% CI: 1.45-1.53) and the reproductive number for the fall wave R2;=3.75 (95% CI: 3.57-3.93). In addition, we estimate the clinical reporting for these two waves to be 59.7% (95% CI: 55.7-63.7) and 83% (95% CI: 79-87). We surmise that the lower reporting in the first wave can be explained by a lack of initial awareness of the epidemic and the relative higher severity of the symptoms experienced during the fall wave. We found that effective isolation measures in hospital clinics at best would only ensure control with probability 0.87 while reducing the transmission rate by >76.5% guarantees stopping an epidemic.     

Characterizing the epidemiology of the 2009 influenza A/H1N1 pandemic in Mexico

Background

Mexico''s local and national authorities initiated an intense public health response during the early stages of the 2009 A/H1N1 pandemic. In this study we analyzed the epidemiological patterns of the pandemic during April–December 2009 in Mexico and evaluated the impact of nonmedical interventions, school cycles, and demographic factors on influenza transmission.

Methods and Findings

We used influenza surveillance data compiled by the Mexican Institute for Social Security, representing 40% of the population, to study patterns in influenza-like illness (ILIs) hospitalizations, deaths, and case-fatality rate by pandemic wave and geographical region. We also estimated the reproduction number (R) on the basis of the growth rate of daily cases, and used a transmission model to evaluate the effectiveness of mitigation strategies initiated during the spring pandemic wave. A total of 117,626 ILI cases were identified during April–December 2009, of which 30.6% were tested for influenza, and 23.3% were positive for the influenza A/H1N1 pandemic virus. A three-wave pandemic profile was identified, with an initial wave in April–May (Mexico City area), a second wave in June–July (southeastern states), and a geographically widespread third wave in August–December. The median age of laboratory confirmed ILI cases was ∼18 years overall and increased to ∼31 years during autumn (p<0.0001). The case-fatality ratio among ILI cases was 1.2% overall, and highest (5.5%) among people over 60 years. The regional R estimates were 1.8–2.1, 1.6–1.9, and 1.2–1.3 for the spring, summer, and fall waves, respectively. We estimate that the 18-day period of mandatory school closures and other social distancing measures implemented in the greater Mexico City area was associated with a 29%–37% reduction in influenza transmission in spring 2009. In addition, an increase in R was observed in late May and early June in the southeast states, after mandatory school suspension resumed and before summer vacation started. State-specific fall pandemic waves began 2–5 weeks after school reopened for the fall term, coinciding with an age shift in influenza cases.

Conclusions

We documented three spatially heterogeneous waves of the 2009 A/H1N1 pandemic virus in Mexico, which were characterized by a relatively young age distribution of cases. Our study highlights the importance of school cycles on the transmission dynamics of this pandemic influenza strain and suggests that school closure and other mitigation measures could be useful to mitigate future influenza pandemics. Please see later in the article for the Editors'' Summary     

Early real-time estimation of the basic reproduction number of emerging or reemerging infectious diseases in a community with heterogeneous contact pattern: Using data from Hong Kong 2009 H1N1 Pandemic Influenza as an illustrative example

Emerging and re-emerging infections such as SARS (2003) and pandemic H1N1 (2009) have caused concern for public health researchers and policy makers due to the increased burden of these diseases on health care systems. This concern has prompted the use of mathematical models to evaluate strategies to control disease spread, making these models invaluable tools to identify optimal intervention strategies. A particularly important quantity in infectious disease epidemiology is the basic reproduction number, R_0. Estimation of this quantity is crucial for effective control responses in the early phase of an epidemic. In our previous study, an approach for estimating the basic reproduction number in real time was developed. This approach uses case notification data and the structure of potential transmission contacts to accurately estimate R₀ from the limited amount of information available at the early stage of an outbreak. Based on this approach, we extend the existing methodology; the most recent method features intra- and inter-age groups contact heterogeneity. Given the number of newly reported cases at the early stage of the outbreak, with parsimony assumptions on removal distribution and infectivity profile of the diseases, experiments to estimate real time R₀ under different levels of intra- and inter-group contact heterogeneity using two age groups are presented. We show that the new method converges more quickly to the actual value of R₀ than the previous one, in particular when there is high-level intra-group and inter-group contact heterogeneity. With the age specific contact patterns, number of newly reported cases, removal distribution, and information about the natural history of the 2009 pandemic influenza in Hong Kong, we also use the extended model to estimate R₀ and age-specific R₀.     

Breaking the Waves: Modelling the Potential Impact of Public Health Measures to Defer the Epidemic Peak of Novel Influenza A/H1N1

Background

On June 11, 2009, the World Health Organization declared phase 6 of the novel influenza A/H1N1 pandemic. Although by the end of September 2009, the novel virus had been reported from all continents, the impact in most countries of the northern hemisphere has been limited. The return of the virus in a second wave would encounter populations that are still nonimmune and not vaccinated yet. We modelled the effect of control strategies to reduce the spread with the goal to defer the epidemic wave in a country where it is detected in a very early stage.

Methodology/Principal Findings

We constructed a deterministic SEIR model using the age distribution and size of the population of Germany based on the observed number of imported cases and the early findings for the epidemiologic characteristics described by Fraser (Science, 2009). We propose a two-step control strategy with an initial effort to trace, quarantine, and selectively give prophylactic treatment to contacts of the first 100 to 500 cases. In the second step, the same measures are focused on the households of the next 5,000 to 10,000 cases. As a result, the peak of the epidemic could be delayed up to 7.6 weeks if up to 30% of cases are detected. However, the cumulative attack rates would not change. Necessary doses of antivirals would be less than the number of treatment courses for 0.1% of the population. In a sensitivity analysis, both case detection rate and the variation of R0 have major effects on the resulting delay.

Conclusions/Significance

Control strategies that reduce the spread of the disease during the early phase of a pandemic wave may lead to a substantial delay of the epidemic. Since prophylactic treatment is only offered to the contacts of the first 10,000 cases, the amount of antivirals needed is still very limited.     

The feasibility of age-specific travel restrictions during influenza pandemics

Background

Epidemiological studies have shown that imposing travel restrictions to prevent or delay an influenza pandemic may not be feasible. To delay an epidemic substantially, an extremely high proportion of trips (~99%) would have to be restricted in a homogeneously mixing population. Influenza is, however, strongly influenced by age-dependent transmission dynamics, and the effectiveness of age-specific travel restrictions, such as the selective restriction of travel by children, has yet to be examined.

Methods

A simple stochastic model was developed to describe the importation of infectious cases into a population and to model local chains of transmission seeded by imported cases. The probability of a local epidemic, and the time period until a major epidemic takes off, were used as outcome measures, and travel restriction policies in which children or adults were preferentially restricted were compared to age-blind restriction policies using an age-dependent next generation matrix parameterized for influenza H1N1-2009.

Results

Restricting children from travelling would yield greater reductions to the short-term risk of the epidemic being established locally than other policy options considered, and potentially could delay an epidemic for a few weeks. However, given a scenario with a total of 500 imported cases over a period of a few months, a substantial reduction in the probability of an epidemic in this time period is possible only if the transmission potential were low and assortativity (i.e. the proportion of contacts within-group) were unrealistically high. In all other scenarios considered, age-structured travel restrictions would not prevent an epidemic and would not delay the epidemic for longer than a few weeks.

Conclusions

Selectively restricting children from traveling overseas during a pandemic may potentially delay its arrival for a few weeks, depending on the characteristics of the pandemic strain, but could have less of an impact on the economy compared to restricting adult travelers. However, as long as adults have at least a moderate potential to trigger an epidemic, selectively restricting the higher risk group (children) may not be a practical option to delay the arrival of an epidemic substantially.     

Estimation of the basic reproduction number of BSE: the intensity of transmission in British cattle

The basic reproduction number, R0, of an infectious agent is a key factor determining the rate of spread and the proportion of the host population affected. We formulate a general mathematical framework to describe the transmission dynamics of long incubation period diseases with complex pathogenesis. This is used to derive expressions for R0 of bovine spongiform encephalopathy (BSE) in British cattle, and back-calculation methods are used to estimate R0 throughout the time-course of the BSE epidemic. We show that the 1988 meat and bonemeal ban was effective in rapidly reducing R0 below 1, and demonstrate that this indicates that BSE will be unable to become endemic in the UK cattle population even when case clustering is taken into account. The analysis provides some insight into absolute infectiousness for bovine-to-bovine transmission, indicating maximally infectious animals may have infected up to 400 animals each. The relationship between R0 and the early stages of the BSE epidemic and the requirements for additional research are also discussed.     

Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks

Two approximations are commonly used to describe the spread of an infectious disease at its early phase: (i) the branching processes based on the generation concept and (ii) the exponential growth over calendar time. The former is characterized by a mean parameter: the reproduction number R₀. The latter is characterized by a growth rate ρ, also known as the Malthusian number. It is common to use empirically observed ρ to assess R₀ using formulae derived either when both the latent and infectious periods follow exponential distributions or assuming both are fixed non-random quantities. This paper first points out that most of these formulae are special cases when the latent and infectious periods are gamma distributed, given by a closed-form solution in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198]. A more general result will be then established which takes the result in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198] as its special case. Three aspects separately shape the relationship between ρ and R₀. They are: (i) the intensity of infectious contacts as a counting process; (ii) the distribution of the latent period and (iii) the distribution of the infectious period. This article also distinguishes the generation time from the transmission interval. It shows that whereas the distribution of the generation time can be derived by the latent and infectious period distributions, the distribution of the transmission interval is also determined by the intensity of infectious contacts as a counting process and hence by R₀. Some syntheses among R₀, ρ and the average transmission interval are discussed. Numerical examples and simulation results are supplied to support the theoretical arguments.     

The COVID-19 outbreak in Sichuan,China: Epidemiology and impact of interventions

In January 2020, a COVID-19 outbreak was detected in Sichuan Province of China. Six weeks later, the outbreak was successfully contained. The aim of this work is to characterize the epidemiology of the Sichuan outbreak and estimate the impact of interventions in limiting SARS-CoV-2 transmission. We analyzed patient records for all laboratory-confirmed cases reported in the province for the period of January 21 to March 16, 2020. To estimate the basic and daily reproduction numbers, we used a Bayesian framework. In addition, we estimated the number of cases averted by the implemented control strategies. The outbreak resulted in 539 confirmed cases, lasted less than two months, and no further local transmission was detected after February 27. The median age of local cases was 8 years older than that of imported cases. We estimated R₀ at 2.4 (95% CI: 1.6–3.7). The epidemic was self-sustained for about 3 weeks before going below the epidemic threshold 3 days after the declaration of a public health emergency by Sichuan authorities. Our findings indicate that, were the control measures be adopted four weeks later, the epidemic could have lasted 49 days longer (95% CI: 31–68 days), causing 9,216 more cases (95% CI: 1,317–25,545).     

Estimating a Markovian Epidemic Model Using Household Serial Interval Data from the Early Phase of an Epidemic

The clinical serial interval of an infectious disease is the time between date of symptom onset in an index case and the date of symptom onset in one of its secondary cases. It is a quantity which is commonly collected during a pandemic and is of fundamental importance to public health policy and mathematical modelling. In this paper we present a novel method for calculating the serial interval distribution for a Markovian model of household transmission dynamics. This allows the use of Bayesian MCMC methods, with explicit evaluation of the likelihood, to fit to serial interval data and infer parameters of the underlying model. We use simulated and real data to verify the accuracy of our methodology and illustrate the importance of accounting for household size. The output of our approach can be used to produce posterior distributions of population level epidemic characteristics.     

Early outbreak of 2009 influenza A (H1N1) in Mexico prior to identification of pH1N1 virus

Background

In the aftermath of the global spread of 2009 influenza A (pH1N1) virus, still very little is known of the early stages of the outbreak in Mexico during the early months of the year, before the virus was identified.

Methodology/Main Findings

We fit a simple mathematical model, the Richards model, to the number of excess laboratory-confirmed influenza cases in Mexico and Mexico City during the first 15 weeks in 2009 over the average influenza case number of the previous five baseline years of 2004-2008 during the same period to ascertain the turning point (or the peak incidence) of a wave of early influenza infections, and to estimate the transmissibility of the virus during these early months in terms of its basic reproduction number. The results indicate that there may have been an early epidemic in Mexico City as well as in all of Mexico during February/March. Based on excess influenza cases, the estimated basic reproduction number R₀ for the early outbreak was 1.59 (0.55 to 2.62) for Mexico City during weeks 5–9, and 1.25 (0.76, 1.74) for all of Mexico during weeks 5–14.

Conclusions

We established the existence of an early epidemic in Mexico City and in all of Mexico during February/March utilizing the routine influenza surveillance data, although the location of seeding is unknown. Moreover, estimates of R₀ as well as the time of peak incidence (the turning point) for Mexico City and all of Mexico indicate that the early epidemic in Mexico City in February/March had been more transmissible (larger R₀) and peaked earlier than the rest of the country. Our conclusion lends support to the possibility that the virus could have already spread to other continents prior to the identification of the virus and the reporting of lab-confirmed pH1N1 cases in North America in April.     

Climate and the Timing of Imported Cases as Determinants of the Dengue Outbreak in Guangzhou, 2014: Evidence from a Mathematical Model

As the world’s fastest spreading vector-borne disease, dengue was estimated to infect more than 390 million people in 2010, a 30-fold increase in the past half century. Although considered to be a non-endemic country, mainland China had 55,114 reported dengue cases from 2005 to 2014, of which 47,056 occurred in 2014. Furthermore, 94% of the indigenous cases in this time period were reported in Guangdong Province, 83% of which were in Guangzhou City. In order to determine the possible determinants of the unprecedented outbreak in 2014, a population-based deterministic model was developed to describe dengue transmission dynamics in Guangzhou. Regional sensitivity analysis (RSA) was adopted to calibrate the model and entomological surveillance data was used to validate the mosquito submodel. Different scenarios were created to investigate the roles of the timing of an imported case, climate, vertical transmission from mosquitoes to their offspring, and intervention. The results suggested that an early imported case was the most important factor in determining the 2014 outbreak characteristics. Precipitation and temperature can also change the transmission dynamics. Extraordinary high precipitation in May and August, 2014 appears to have increased vector abundance. Considering the relatively small number of cases in 2013, the effect of vertical transmission was less important. The earlier and more frequent intervention in 2014 also appeared to be effective. If the intervention in 2014 was the same as that in 2013, the outbreak size may have been over an order of magnitude higher than the observed number of new cases in 2014.The early date of the first imported and locally transmitted case was largely responsible for the outbreak in 2014, but it was influenced by intervention, climate and vertical transmission. Early detection and response to imported cases in the spring and early summer is crucial to avoid large outbreaks in the future.     

U.S. Human immunodeficiency virus type 1 epidemic: date of origin,population history,and characterization of early strains

Human immunodeficiency virus (HIV) type 1 subtype B sequences (whole envelope and the p17 region of gag) were obtained from peripheral blood mononuclear cell samples collected in 1981 from seven HIV-infected U.S. individuals and in 1982 from one infected Canadian resident. Phylogenetic and nucleotide distance analyses were performed by using database sequences representing North American strains collected from 1978 to 1995. The estimated phylogeny was starlike, with early strains represented on different lineages. When sequences were grouped by years of collection, nucleotide distance comparisons demonstrated an increase in diversity over time and indicated that contemporary strains are more closely related to early epidemic strains than to each other. Using a recently developed likelihood ratio reduction procedure, the date of origin of the U.S. epidemic was estimated to be 1968 +/- 1.4 years. A coalescent approach was also used to estimate the population history of the U.S. subtype B epidemic. Our analyses provide new information that implies an exponential growth rate from the beginning of the U.S. HIV epidemic. The dating results suggest a U.S. introduction date (or date of divergence from the most recent common ancestor) that precedes the date of the earliest known AIDS cases in the late 1970s. Furthermore, the estimated epidemic growth curve shows a period of exponential growth that preceded most of the early documented cases and also indicates a leveling of prevalence rates in the recent past.     

Forecasting the 2001 Foot-and-Mouth Disease Epidemic in the UK

Near real-time epidemic forecasting approaches are needed to respond to the increasing number of infectious disease outbreaks. In this paper, we retrospectively assess the performance of simple phenomenological models that incorporate early sub-exponential growth dynamics to generate short-term forecasts of the 2001 foot-and-mouth disease epidemic in the UK. For this purpose, we employed the generalized-growth model (GGM) for pre-peak predictions and the generalized-Richards model (GRM) for post-peak predictions. The epidemic exhibits a growth-decelerating pattern as the relative growth rate declines inversely with time. The uncertainty of the parameter estimates \( (r{\text{ and }}p) \) narrows down and becomes more precise using an increasing amount of data of the epidemic growth phase. Indeed, using only the first 10–15 days of the epidemic, the scaling of growth parameter (p) displays wide uncertainty with the confidence interval for p ranging from values ~ 0.5 to 1.0, indicating that less than 15 epidemic days of data are not sufficient to discriminate between sub-exponential (i.e., p < 1) and exponential growth dynamics (i.e., p = 1). By contrast, using 20, 25, or 30 days of epidemic data, it is possible to recover estimates of p around 0.6 and the confidence interval is substantially below the exponential growth regime. Local and national bans on the movement of livestock and a nationwide cull of infected and contiguous premises likely contributed to the decelerating trajectory of the epidemic. The GGM and GRM provided useful 10-day forecasts of the epidemic before and after the peak of the epidemic, respectively. Short-term forecasts improved as the model was calibrated with an increasing length of the epidemic growth phase. Phenomenological models incorporating generalized-growth dynamics are useful tools to generate short-term forecasts of epidemic growth in near real time, particularly in the context of limited epidemiological data as well as information about transmission mechanisms and the effects of control interventions.