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Given any epidemic model of an infectious disease, there are various ways of computing a basic reproduction number(\$R_0\$) such as; the next-generation method, survival function, largest eigenvalue of the Jacobian matrix and so on.

My question is,

Do we get the same \$R_0\$ for the same model but using different methods of computing \$R_0\$?

If we don't(if \$R_0\$ is unique) what is the explanation? Are we allowed to use "this" different \$R_0\$ in writing a paper?

It should be noted that many of the methods you talk about aren't necessarily for getting a number but rather a formula for \$R_0\$, all of which should be equivalent.

However, when those methods step into the realm of getting a number by fitting parameters, they may give different results as they handle certain aspects of the data differently.

You are certainly "allowed" to use any formulation of \$R_0\$ you want - just document what you did.

## Hydrobates

These days reproduction numbers for epidemiology are prominent in the popular media. Many people are familiar with the idea that stopping a resurgence of COVID-19 infections in a region has to do with making and keeping something called less than one. They may also be familiar with the informal definition of that it is the number of new infections caused by an infected individual. But how is (or as it is more commonly called by scientists) defined? A mathematician might not expect to find an answer in the media but it might be reasonable to expect one in the scientific literature on epidemiology. In the past I have been frustrated by the extent to which this fails to be the case. What is typically given is a description in words which I never found possible to convert into a precise mathematical account, despite considerable effort. Now, in the context of a project on hepatis C which I have been working on with colleagues from Cameroon, my attention was drawn to a paper of van den Driessche and Watmough (Math. Biosci. 180, 29) which contains some answers of the type I was looking for. I was vaguely aware of this paper before but I had never seriously tried to read it because I did not realise its nature.

The context in which I would have liked to find answers is that of models given by systems of ordinary differential equations where the unknowns are the numbers of individuals in different categories (susceptible, infected, recovered etc.) as functions of time. How the numbers reported in the media are calculated (on the basis of discrete data) is something I have not yet tried to find out. At the moment I would like an answer in the context which bothered me in the past and this is the context treated in the paper mentioned above. A typical situation is that found in the basic model of virus dynamics, a system of three ODE describing the dynamics of a virus within a host, with the unknowns being uninfected cells, infected cells and virions. There is a quantity which can be expressed in terms of the coefficients of the system. If then the only non-negative steady state is virus-free. This is the uninfected state and it is globally asymptotically stable. If there is an uninfected state which is unstable and an infected state which is positive and globally asymptotically stable. This kind of situation is not unique to this example and similar things are seen in many models of infection. There is a reproductive number (or perhaps more than one) which defines a threshold between different types of late-time behaviour.

It is not obvious that the analysis of van den Driessche and Watmough applies to models of in-host dynamics of a pathogen since it is necessary to make a choice of infected and uninfected compartments which is related to the biological interpretation of the variables and not just to the mathematical structure of the model. Their analysis does apply to the basic model of virus dynamics if the infected compartments are chosen to be the infected cells and virions and the reaction fluxes are partitioned in a suitable way. The simple picture of the significance of the reproductive number given above does not always hold. There is also another scenario which can occur and does so in many practical examples and involves the notion of a backward bifurcation. It goes as follows. For sufficiently small the disease-free steady state is globally asymptotically stable but as is increased this property breaks down before is reached. A fold bifurcation occurs which creates a stable and an unstable positive steady state. The unstable steady state moves so as to meet the disease-free state when . For there are exactly two steady states and the positive one is globally asymptotically stable. There is bifurcation for but it has a different structure from that in the classical scenario (which is a transcritical bifurcation). It bears some resemblance to a sub-critical Hopf bifurcation.

## Dog Rabies and Its Control

Darryn L. Knobel , . Katie Hampson , in Rabies (Third Edition) , 2013

### 2.1 Key Epidemiological Parameters

The basic reproductive number (R0) of an infectious agent such as rabies virus is defined as the average number of secondary infections produced by an infected individual in an otherwise susceptible host population ( Anderson & May, 1991 ). R0 determines whether a pathogen can persist in such a population, and it is valuable for assessing control options. When R0 is less than 1, on average each infectious individual infects less than one other individual, and the pathogen will die out in the population. In contrast, when R0 exceeds 1 there is an exponential rise in the number of cases over time, and an epidemic results. R0 is consistently estimated to be between 1 and 2 from rabies outbreaks in dog populations around the world ( Hampson et al., 2009 ), which is relatively close to the extinction threshold of 1.

A closely related concept is that of the effective reproductive number (Re), when transmission occurs in a population that is not entirely susceptible due to implemented control efforts. For dog rabies, Re is determined by the number of susceptible dogs bitten by each infected dog during its infectious period, and the probability that those bitten dogs go on to develop rabies (i.e. become infectious themselves). The number of dogs bitten may depend upon genetic, behavioral, environmental, and anthropogenic factors, whereas the probability of developing rabies once bitten depends upon vaccination status (since there is no natural immunity to rabies), as well as other intrinsic factors including viral dose, location of bite(s), and degree of tissue injury. The aim of control measures is to reduce transmission so that Re is reduced below the threshold of 1. For rabies, these control measures can therefore operate by reducing either the number of dogs bitten or the probability that bitten dogs develop rabies.

Historically, prior to the advent of effective vaccines, domestic dog rabies control measures focused on reducing the number of susceptible dogs bitten, through movement restrictions and culling rabid, bitten and “stray” dogs ( Bögel, 2002 Fooks, Roberts, Lynch, Hersteinsson, & Runolfsson, 2004 Meldrum, 1988 Muir & Roome, 2005 Tierkel, 1959 WHO, 1987 ). The low value of R0 and cultural context of strict confinement and muzzling of dogs probably contributed to the success of these measures in isolated locations such as the UK, but few other successes were reported using only these measures. However, with the advent of effective animal vaccines, mass vaccination has become the mainstay of successful dog rabies control and a more ethically and culturally acceptable measure. The control of infectious diseases through mass vaccination is based on the concept of herd immunity, when the vaccination of a proportion of the population (or “herd”) provides protection for individuals who are not vaccinated. That proportion of the population that needs to be vaccinated to achieve herd immunity and thus control disease depends on R0. For rabies, low values of R0 suggest that the critical vaccination coverage, Pcrit, required to control disease should be roughly between 20 and 40% (i.e., 20–40% of the dog population should be immune at any time in order to prevent sustained outbreaks of rabies, although short chains of transmission can still occur in such partially vaccinated populations Hampson et al., 2009 ). By comparison, some other infectious diseases that have been successfully controlled by mass vaccination (e.g., measles or rinderpest) have considerably higher values of R0, and require coverages well above 90%. These figures thus suggest that dog rabies is amenable to control by mass vaccination.

## The basic reproduction number (R 0) of measles: a systematic review

The basic reproduction number, R nought (R0), is defined as the average number of secondary cases of an infectious disease arising from a typical case in a totally susceptible population, and can be estimated in populations if pre-existing immunity can be accounted for in the calculation. R0 determines the herd immunity threshold and therefore the immunisation coverage required to achieve elimination of an infectious disease. As R0 increases, higher immunisation coverage is required to achieve herd immunity. In July, 2010, a panel of experts convened by WHO concluded that measles can and should be eradicated. Despite the existence of an effective vaccine, regions have had varying success in measles control, in part because measles is one of the most contagious infections. For measles, R0 is often cited to be 12-18, which means that each person with measles would, on average, infect 12-18 other people in a totally susceptible population. We did a systematic review to find studies reporting rigorous estimates and determinants of measles R0. Studies were included if they were a primary source of R0, addressed pre-existing immunity, and accounted for pre-existing immunity in their calculation of R0. A search of key databases was done in January, 2015, and repeated in November, 2016, and yielded 10 883 unique citations. After screening for relevancy and quality, 18 studies met inclusion criteria, providing 58 R0 estimates. We calculated median measles R0 values stratified by key covariates. We found that R0 estimates vary more than the often cited range of 12-18. Our results highlight the importance of countries calculating R0 using locally derived data or, if this is not possible, using parameter estimates from similar settings. Additional data and agreed review methods are needed to strengthen the evidence base for measles elimination modelling.

## Consequences of an infection

Viruses are associated with a variety of human diseases. The diagram below shows some common examples of viral infections that affect different systems of the human body: An overview of human viral diseases and the systems they affect when symptomatic. Image credit: “Prevention and treatment of viral infections: Figure 1, by OpenStax College, Biology, CC BY 4.0. Modification of original work by Mikael Häggström.

## Methods

### Data selection and handling: death data

For mortality due to COVID-19, we used time series provided by the New York Times 12 . We selected the New York Times dataset because it is rigorously curated. We analyzed separately only counties that had records of 100 or more deaths by 23 May, 2020. The threshold of 100 was a balance between including more counties and obtaining reliable estimates of r(t). Preliminary simulations showed that time series with low numbers of deaths would bias r(t) estimates (Supplementary Fig. 2). However, we did not want to use the maximum daily number of deaths as a selection criterion, because this could lead to selection of counties based on data from a single day. It would also involve some circularity, because the information obtained, r(t), would be directly related to the criterion used to select datasets. Therefore, we used the threshold of 100 cumulative deaths. The District of Columbia was treated as a county. Also, because the New York Times dataset aggregated the five boroughs of New York City, we treated them as a single county. For counties with fewer than 100 deaths, we aggregated mortality to the state level to create a single time series. For thirteen states (AK, DE, HI, ID, ME, MT, ND, NH, SD, UT, VM, WV, and WY), the aggregated time series did not contain 100 or more deaths and were therefore not analyzed.

### Data selection and handling: explanatory county-level variables

County-level variables were collected from several public data sources 36,37,38,39,40,41,42 . We selected socio-economic variables a priori in part to represent a broad set of population characteristics.

### Time series analysis: time series model

We used a time-varying autoregressive model 15,56 designed explicitly to estimate the rate of increase of a variable using nonlinear, state-dependent error terms 16 . We assume in our analyses that the susceptible proportion of the population represented by a time series is close to one, and therefore there is no decrease in the infection rate caused by a pool of individuals who were infected, recovered, and were then immune to further infection.

Here, x(t) is the unobserved, log-transformed value of daily deaths at time t, and D(t) is the observed count that depends on the observation uncertainty described by the random variable ϕ(t). Because a few of the datasets that we analyzed had zeros, we replaced zeros with 0.5 before log-transformation. The model assumes that the death count increases exponentially at rate r(t), where the latent state variable r(t) changes through time as a random walk with ωr(t)

N(0, σ 2 r). We assume that the count data follow a quasi-Poisson distribution. Thus, the expectation of counts at time t is exp(x(t)), and the variance is proportional to this expectation.

We fit the model using the extended Kalman filter to compute the maximum likelihood 57,58 . In addition to the parameters σ 2 r and σ 2 ϕ, we estimated the initial value of r(t) at the start of the time series, r0, and the initial value of x(t), x0. The estimation also requires terms for the variances in x0 and r0, which we assumed were zero and σ 2 r, respectively. In the validation using simulated data (Supplementary Methods: Simulation model), we found that the estimation process tended to absorb σ 2 r to zero too often. To eliminate this absorption to zero, we imposed a minimum of 0.02 on σ 2 r.

### Time series analysis: parametric bootstrapping

To generate approximate confidence intervals for the time-varying estimates of r(t) (Eq. 1b), we used a parametric bootstrap designed to simulate datasets with the same characteristics as the real data that are then refit using the autoregressive model. We used bootstrapping to obtain confidence intervals, because an initial simulation study showed that standard methods, such as obtaining the variance of r(t) from the Kalman filter, were too conservative (the confidence intervals too narrow) when the number of counts was small. Furthermore, parametric bootstrapping can reveal bias and other features of a model, such as the lags we found during model fitting (Supplementary Fig. 1a, b).

Changes in r(t) consist of unbiased day-to-day variation and the biased deviations that lead to longer-term changes in r(t). The bootstrap treats the day-to-day variation as a random variable while preserving the biased deviations that generate longer-term changes in r(t). Specifically, the bootstrap was performed by calculating the differences between successive estimates of r(t), Δr(t) = r(t) – r(t-1), and then standardizing to remove the bias, Δrs(t) = Δr(t) – E[Δr(t)], where E[] denotes the expected value. The sequence Δrs(t) was fit using an autoregressive time-series model with time lag 1, AR(1), to preserve any shorter-term autocorrelation in the data. For the bootstrap, a new time series was simulated from this AR(1) model, Δρ(t), and then standardized, Δρs(t) = Δρ(t) – E[Δρ(t)]. The simulated time series for the spread rate was constructed as ρ(t) = r(t) + Δρs(t)/2 1/2 , where dividing by 2 1/2 accounts for the fact that Δρs(t) was calculated from the difference between successive values of r(t). A new time series of count data, ξ(t), was then generated using equation 1 with the parameters from fitting the data. Finally, the statistical model was fit to the reconstructed ξ(t). In this refitting, we fixed the variance in r(t), σ 2 r, to the same value as estimated from the data. Therefore, the bootstrap confidence intervals are conditional of the estimate of σ 2 r.

### Time series analysis: calculating R0

We derived estimates of R(t) directly from r(t) using the Dublin-Lotka equation 21 from demography. This equation is derived from a convolution of the distribution of births under the assumption of exponential population growth. In our case, the “birth” of COVID-19 is the secondary infection of susceptible hosts leading to death, and the assumption of exponential population growth is equivalent to assuming that the initial rate of spread of the disease is exponential, as is the case in equation 1. Thus,

where p(τ) is the distribution of the proportion of secondary infections caused by a primary infection that occurred τ days previously. We used the distribution of p(τ) from Li et al. 59 that had an average serial interval of T0 = 7.5 days smaller or larger values of T0, and greater or lesser variance in p(τ), will decrease or increase R(t) but will not change the pattern in R(t) through time. Note that the uncertainty in the distribution of serial times for COVID-19 is a major reason why we focus on estimating r0, rather than R0: the estimates of r0 are not contingent on time distributions that are poorly known. Computing R(t) from r(t) also does not depend on the mean or variance in time between secondary infection and death. We report values of R(t) at dates that are offset by 18 days, the average length of time between initial infection and death given by Zhou et al. 60 .

### Time series analysis: Initial date of the time series

Many time series consisted of initial periods containing zeros that were uninformative. As the initial date for the time series, we chose the day on which the estimated daily death count exceeded 1. To estimate the daily death count, we fit a Generalized Additive Mixed Model (GAMM) to the death data while accounting for autocorrelation and greater measurement error at low counts using the R package mgcv 61 . We used this procedure, rather than using a threshold of the raw death count, because the raw death count will include variability due to sampling small numbers of deaths. Applying the GAMM to “smooth” over the variation in count data gives a well-justified method for standardizing the initial dates for each time series.

### Time series analysis: validation

We performed extensive simulations to validate the time-series analysis approach (Supplementary Methods: Simulation model).

### Regression analysis for r 0

We applied a Generalized Least Squares (GLS) regression model to explain the variation in estimates of r0 from the 160 county and county-aggregate time series:

where start.date is the Julian date of the start of the time series, log(pop.size) and pop.den 0.25 are the log-transformed population size and 0.25 power-transformed population density of the county or county-aggregate, respectively, and ε is a multivariate Gaussian random variable with covariance matrix σ 2 Σ. We used the transforms log(pop.size) and pop.den 0.25 to account for nonlinear relationships with r0 these transforms give the highest maximum likelihood of the overall regression. The covariance matrix contains a spatial correlation matrix of the form C = uI + (1–u)S(g) where u is the nugget and S(g) contains elements exp(−dij/g), where dij is the distance between spatial locations and g is the range 62 . To incorporate differences in the precision of the estimates of r0 among time series, we weighted by the vector of their standard errors, s, so that Σ = diag(s) * C * diag(s), where * denotes matrix multiplication. With this weighting, the overall scaling term for the variance, σ 2 , will equal 1 if the residual variance of the regression model matches the square of the standard errors of the estimates of r0 from the time series. We fit the regression model with the function gls() in the R package nlme 63 .

To make predictions for new values of r0, we used the relationship

where ει is the GLS residual for data i, (hat e) i is the predicted residual, (ar e) is the mean of the GLS residuals, V is the covariance matrix for data other than i, and vi is a row vector containing the covariances between data i and the other data in the dataset 64 . This equation was used for three purposes. First, we used it to compute R 2 pred for the regression model by removing each data point, recomputing (hat e) i, and using these values to compute the predicted residual variance 23 . Second, we used it to obtain predicted values of r0, and subsequently R0, for the 160 counties and county-aggregates for which r0 was also estimated from time series. Third, we used equation (4) to obtain predicted values of r0, and hence predicted R0, for all other counties. We also calculated the variance of the estimates from 64

Predicted values of R0 were mapped using the R package usmap 65 .

### Regression analysis for SARS-CoV-2 effects on r0

The GISAID metadata 27 for SARS-CoV-2 contains the clade and state-level location for strains in the USA strains G, GH, and GR contain the G614 mutation. For each state, we limited the SARS-CoV-2 genomes to those collected no more than 30 days following the onset of outbreak that we used as the starting point for the time series from which we estimated r0 from these genomes (totaling 5290 from all states), we calculated the proportion that had the G614 mutation. We limited the analyses to the 28 states that had five or more genome samples. For each state, we selected the estimates of r0 from the county or county-aggregate representing the greatest number of deaths. We fit these estimates of r0 with the weighted Least Squares (LS) model as in equation (3) with additional variables for strain. Figure 3 was constructed using the R packages usmap 65 and scatterpie 66 .

### Statistics and reproducibility

The statistics for this study are summarized in the preceding sections of the “Methods”. No experiments were conducted, so experimental reproducibility is not an issue. Nonetheless, we repeated analyses using alternative datasets giving county-level characteristics, and also an alternative dataset on SARS-CoV-2 strains (Supplementary Methods: Analysis of Nextstrain metadata of SARS-CoV-2 strains), and all of the conclusions were the same.

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.

## The Basic Reproduction Number for Complex Disease Systems: Defining R0 for Tick‐Borne Infections

Characterizing the basic reproduction number, \$R_<0>\$ , for many wildlife disease systems can seem a complex problem because several species are involved, because there are different epidemiological reactions to the infectious agent at different life‐history stages, or because there are multiple transmission routes. Tick‐borne diseases are an important example where all these complexities are brought together as a result of the peculiarities of the tick life cycle and the multiple transmission routes that occur. We show here that one can overcome these complexities by separating the host population into epidemiologically different types of individuals and constructing a matrix of reproduction numbers, the so‐called next‐generation matrix. Each matrix element is an expected number of infectious individuals of one type produced by a single infectious individual of a second type. The largest eigenvalue of the matrix characterizes the initial exponential growth or decline in numbers of infected individuals. Values below 1 therefore imply that the infection cannot establish. The biological interpretation closely matches that of \$R_<0>\$ for disease systems with only one type of individual and where infection is directly transmitted. The parameters defining each matrix element have a clear biological meaning. We illustrate the usefulness and power of the approach with a detailed examination of tick‐borne diseases, and we use field and experimental data to parameterize the next‐generation matrix for Lyme disease and tick‐borne encephalitis. Sensitivity and elasticity analyses of the matrices, at the element and individual parameter levels, allow direct comparison of the two etiological agents. This provides further support that transmission between cofeeding ticks is critically important for the establishment of tick‐borne encephalitis.

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## Is the basic reproduction number unique? - Biology

The basic reproduction number (R0), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R0 is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R0 easily misrepresented, misinterpreted, and misapplied. R0 is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R0 cannot be modified through vaccination campaigns. R0 is rarely measured directly, and modeled R0 values are dependent on model structures and assumptions. Some R0 values reported in the scientific literature are likely obsolete. R0 must be estimated, reported, and applied with great caution because this basic metric is far from simple.

The basic reproduction number (R0), pronounced “R naught,” is intended to be an indicator of the contagiousness or transmissibility of infectious and parasitic agents. R0 is often encountered in the epidemiology and public health literature and can also be found in the popular press (16). R0 has been described as being one of the fundamental and most often used metrics for the study of infectious disease dynamics (712). An R0 for an infectious disease event is generally reported as a single numeric value or low–high range, and the interpretation is typically presented as straightforward an outbreak is expected to continue if R0 has a value >1 and to end if R0 is <1 (13). The potential size of an outbreak or epidemic often is based on the magnitude of the R0 value for that event (10), and R0 can be used to estimate the proportion of the population that must be vaccinated to eliminate an infection from that population (14,15). R0 values have been published for measles, polio, influenza, Ebola virus disease, HIV disease, a diversity of vectorborne infectious diseases, and many other communicable diseases (14,1618).

The concept of R0 was first introduced in the field of demography (9), where this metric was used to count offspring. When R0 was adopted for use by epidemiologists, the objects being counted were infective cases (19). Numerous definitions for R0 have been proposed. Although the basic conceptual framework is similar for each, the operational definitions are not always identical. Dietz states that R0 is “the number of secondary cases one case would produce in a completely susceptible population” (19). Fine supplements this definition with the description “average number of secondary cases” (17). Diekmann and colleagues use the description “expected number of secondary cases” and provide additional specificity to the terminology regarding a single case (13).

In the hands of experts, R0 can be a valuable concept. However, the process of defining, calculating, interpreting, and applying R0 is far from straightforward. The simplicity of an R0 value and its corresponding interpretation in relation to infectious disease dynamics masks the complicated nature of this metric. Although R0 is a biological reality, this value is usually estimated with complex mathematical models developed using various sets of assumptions. The interpretation of R0 estimates derived from different models requires an understanding of the models’ structures, inputs, and interactions. Because many researchers using R0 have not been trained in sophisticated mathematical techniques, R0 is easily subject to misrepresentation, misinterpretation, and misapplication. Notable examples include incorrectly defining R0 (1) and misinterpreting the effects of vaccination on R0 (3). Further, many past lessons regarding this metric appear to have been lost or overlooked over time. Therefore, a review of the concept of R0 is needed, given the increased attention this metric receives in the academic literature (20). In this article, we address misconceptions about R0 that have proliferated as this metric has become more frequently used outside of the realm of mathematical biology and theoretic epidemiology, and we recommend that R0 be applied and discussed with caution.

### Variations in R0

For any given infectious agent, the scientific literature might present numerous different R0 values. Estimations of the R0 value are often calculated as a function of 3 primary parameters—the duration of contagiousness after a person becomes infected, the likelihood of infection per contact between a susceptible person and an infectious person or vector, and the contact rate—along with additional parameters that can be added to describe more complex cycles of transmission (19). Further, the epidemiologic triad (agent, host, and environmental factors) sometimes provides inspiration for adding parameters related to the availability of public health resources, the policy environment, various aspects of the built environment, and other factors that influence transmission dynamics and, thus, are relevant for the estimation of R0 values (21). Yet, even if the infectiousness of a pathogen (that is, the likelihood of infection occurring after an effective contact event has occurred) and the duration of contagiousness are biological constants, R0 will fluctuate if the rate of human–human or human–vector interactions varies over time or space. Limited evidence supports the applicability of R0 outside the region where the value was calculated (20). Any factor having the potential to influence the contact rate, including population density (e.g., rural vs. urban), social organization (e.g., integrated vs. segregated), and seasonality (e.g., wet vs. rainy season for vectorborne infections), will ultimately affect R0. Because R0 is a function of the effective contact rate, the value of R0 is a function of human social behavior and organization, as well as the innate biological characteristics of particular pathogens. More than 20 different R0 values (range 5.4–18) were reported for measles in a variety of study areas and periods (22), and a review in 2017 identified feasible measles R0 values of 3.7–203.3 (23). This wide range highlights the potential variability in the value of R0 for an infectious disease event on the basis of local sociobehavioral and environmental circumstances.

### Various Names for R0

Inconsistency in the name and definition of R0 has potentially been a cause for misunderstanding the meaning of R0. R0 was originally called the basic case reproduction rate when George MacDonald introduced the concept into the epidemiology literature in the 1950s (17,19,24,25). Although MacDonald used Z0 to represent the metric, the current symbolic representation (R0) appears to have remained largely consistent since that time. However, multiple variations of the name for the concept expressed by R0 have been used in the scientific literature, including the use of basic and case as the first word in the term, reproduction and reproductive for the second word, and number, ratio, and rate for the final part of the term (13). Although the frequent use of the term basic reproduction rate is in line with MacDonald’s original terminology (9), some users interpret the use of the word rate as suggesting a quantity having a unit with a per-time dimension (7). If R0 were a rate involving time, the metric would provide information about how quickly an epidemic will spread through a population. But R0 does not indicate whether new cases will occur within 24 hours after the initial case or months later, just as R0 does not indicate whether the disease produced by the infection is severe. Instead, R0 is most accurately described in terms of cases per case (7,13). Calling R0 a rate rather than a number or ratio might create some undue confusion about what the value represents.

### R0 and Vaccination Campaigns

Vaccination campaigns reduce the proportion of a population at risk for infection and have proven to be highly effective in mitigating future outbreaks (26). This conclusion is sometimes used to suggest that an aim of vaccination campaigns is to remove susceptible members of the population to reduce the R0 for the event to <1. Although the removal of susceptible members from the population will affect infection transmission by reducing the number of effective contacts between infectious and susceptible persons, this activity will technically not reduce the R0 value because the definition of R0 includes the assumption of a completely susceptible population. When examining the effect of vaccination, the more appropriate metric to use is the effective reproduction number (R), which is similar to R0 but does not assume complete susceptibility of the population and, therefore, can be estimated with populations having immune members (16,20,27). Efforts aimed at reducing the number of susceptible persons within a population through vaccination would result in a reduction of the R value, rather than R0 value. In this scenario, vaccination could potentially end an epidemic, if R can be reduced to a value <1 (16,27,28). The effective reproduction number can also be specified at a particular time t, presented as R(t) or Rt, which can be used to trace changes in R as the number of susceptible members in a population is reduced (29,30). When the goal is to measure the effectiveness of vaccination campaigns or other public health interventions, R0 is not necessarily the best metric (10,20).

### Measuring and Estimating R0

Counting the number of cases of infection during an epidemic can be extremely difficult, even when public health officials use active surveillance and contact tracing to attempt to locate all infected persons. Although measuring the true R0 value is possible during an outbreak of a newly emerging infectious pathogen that is spreading through a wholly susceptible population, rarely are there sufficient data collection systems in place to capture the early stages of an outbreak when R0 might be measured most accurately. As a result, R0 is nearly always estimated retrospectively from seroepidemiologic data or by using theoretical mathematical models (31). Data-driven approaches include the use of the number of susceptible persons at endemic equilibrium, average age at infection, final size equation, and intrinsic growth rate (10). When mathematical models are used, R0 values are often estimated by using ordinary differential equations (810,19,31), but high-quality data are rarely available for all components of the model. The estimated values of R0 generated by mathematical models are dependent on numerous decisions made by the modeler (8,32,33). The population structure of the model, such as the susceptible-infectious-recovered model or susceptible-exposed-infectious-recovered model, which includes compartments for persons who are exposed but not yet infectious, as well as assumptions about demographic dynamics (e.g., births, deaths, and migration over time), are critical model parameters. Population mixing and contact patterns must also be considered for example, for homogeneous mixing, all population members are equally likely to come into contact with one another, and for heterogeneous mixing, variation in contact patterns are present among age subgroups or geographic regions. Other decisions include whether to use a deterministic (yielding the same outcomes each time the model is run) or stochastic (generating a distribution of likely outcomes on the basis of variations in the inputs) approach and which distributions (e.g., Gaussian or uniform distributions) to use to describe the probable values of parameters, such as effective contact rates and duration of contagiousness. Furthermore, many of the parameters included in the models used to estimate R0 are merely educated guesses the true values are often unknown or difficult or impossible to measure directly (31,34,35). This limitation is compounded as models become more complex and, thus, require more input parameters (20,35), such as when using models to estimate the value of R0 for infectious pathogens with more complex transmission pathways, which can include vectorborne infectious agents or those with environmental or wildlife reservoirs. In summary, although only 1 true R0 value exists for an infectious disease event occurring in a particular place at a particular time, models that have minor differences in structure and assumptions might produce different estimates of that value, even when using the same epidemiologic data as inputs (20,31,32,36,37).

### Obsolete R0 Values

New estimates of R0 have been produced for infectious disease events that occurred in recent history, such as the West Africa Ebola outbreak (34,38,39). However, for many vaccine-preventable diseases, the scientific literature reports R0 values calculated much further back in history. For example, the oft-reported measles R0 values of 12–18 are based on data acquired during 1912–1928 in the United States (R0 of 12.5) and 1944–1979 in England and Wales (R0 of 13.7–18.0) (14), even though more recent estimates of the R0 for measles highlight a much greater numeric range and variation across settings (23). For pertussis (R0 of 12–17), the original data sources are 1908–1917 in the United States (R0 of 12.2) and 1944–1979 in England and Wales (R0 of 14.3–17.1) (14). The major changes that have occurred in how humans organize themselves both socially and geographically make these historic values extremely unlikely to match present day epidemiologic realities. Behavioral changes undoubtedly have altered contact rates, which are a key component of R0 calculations. Yet, these R0 values have been repeated so often in the literature that newer R0 values generated by using modern data might be dismissed if they fall outside the range of previous estimates. Given that R0 is often considered when designing and implementing vaccination strategies and other public health interventions, the use of R0 values derived from older data is likely inappropriate (23). Decisions about public health practice should be made with contemporaneous R0 values or R values instead.

### Conclusions

Although R0 might appear to be a simple measure that can be used to determine infectious disease transmission dynamics and the threats that new outbreaks pose to the public health, the definition, calculation, and interpretation of R0 are anything but simple. R0 remains a valuable epidemiologic concept, but the expanded use of R0 in both the scientific literature and the popular press appears to have enabled some misunderstandings to propagate. R0 is an estimate of contagiousness that is a function of human behavior and biological characteristics of pathogens. R0 is not a measure of the severity of an infectious disease or the rapidity of a pathogen’s spread through a population. R0 values are nearly always estimated from mathematical models, and the estimated values are dependent on numerous decisions made in the modeling process. The contagiousness of different historic, emerging, and reemerging infectious agents cannot be fairly compared without recalculating R0 with the same modeling assumptions. Some of the R0 values commonly reported in the literature for past epidemics might not be valid for outbreaks of the same infectious disease today.

R0 can be misrepresented, misinterpreted, and misapplied in a variety of ways that distort the metric’s true meaning and value. Because of these various sources of confusion, R0 must be applied and discussed with caution in research and practice. This epidemiologic construct will only remain valuable and relevant when used and interpreted correctly.

Dr. Delamater is an assistant professor in the Department of Geography and a faculty fellow at the Carolina Population Center at the University of North Carolina at Chapel Hill, Chapel Hill, North Carolina, USA. His research focuses on the geographic aspects of health, disease, and healthcare.

### Acknowledgment

This research was supported through a grant from the George Mason University Provost Multidisciplinary Research Initiative.

## Reproduction: Asexual vs. Sexual

Cell division is how organisms grow and repair themselves. It is also how many organisms produce offspring. For many single-celled organisms, reproduction is a similar process. The parent cell simply divides to form two daughter cells that are identical to the parent. In many other organisms, two parents are involved, and the offspring are not identical to the parents. In fact, each offspring is unique. Look at the family in Figure below. The children resemble their parents, but they are not identical to them. Instead, each has a unique combination of characteristics inherited from both parents.

Family Portrait: Mother, Daughter, Father, and Son. Children resemble their parents, but they are never identical to them. Do you know why this is the case?

Reproduction is the process by which organisms give rise to offspring. It is one of the defining characteristics of living things. There are two basic types of reproduction: asexual reproduction and sexual reproduction.

### Asexual Reproduction

Asexual reproduction involves a single parent. It results in offspring that are genetically identical to each other and to the parent. All prokaryotes and some eukaryotes reproduce this way. There are several different methods of asexual reproduction. They include binary fission, fragmentation, and budding.

• Binary fission occurs when a parent cell splits into two identical daughter cells of the same size.
• Fragmentation occurs when a parent organism breaks into fragments, or pieces, and each fragment develops into a new organism. Starfish, like the one in Figurebelow, reproduce this way. A new starfish can develop from a single ray, or arm. Starfish, however, are also capable of sexual reproduction.
• Budding occurs when a parent cell forms a bubble-like bud. The bud stays attached to the parent cell while it grows and develops. When the bud is fully developed, it breaks away from the parent cell and forms a new organism. Budding in yeast is shown in Figurebelow.

Binary Fission in various single-celled organisms (left). Cell division is a relatively simple process in many single-celled organisms. Eventually the parent cell will pinch apart to form two identical daughter cells. In multiple fission (right), a multinucleated cell can divide to form more than one daughter cell. Multiple fission is more often observed among protists.

Starfish reproduce by fragmentation and yeasts reproduce by budding. Both are types of asexual reproduction.

Asexual reproduction can be very rapid. This is an advantage for many organisms. It allows them to crowd out other organisms that reproduce more slowly. Bacteria, for example, may divide several times per hour. Under ideal conditions, 100 bacteria can divide to produce millions of bacterial cells in just a few hours! However, most bacteria do not live under ideal conditions. If they did, the entire surface of the planet would soon be covered with them. Instead, their reproduction is kept in check by limited resources, predators, and their own wastes. This is true of most other organisms as well.

### Sexual Reproduction

Sexual reproduction involves two parents. As you can see from Figure below, in sexual reproduction, parents produce reproductive cells&mdashcalled gametes&mdashthat unite to form an offspring. Gametes are haploid cells. This means they contain only half the number ofchromosomes found in other cells of the organism. Gametes are produced by a type of cell division called meiosis, which is described in detail in a subsequent concept. The process in which two gametes unite is called fertilization. The fertilized cell that results is referred to as a zygote. A zygote is diploid cell, which means that it has twice the number of chromosomesas a gamete.

Mitosis, Meiosis and Sexual Reproduction is discussed at http://www.youtube.com/watch?v=kaSIjIzAtYA.

Cycle of Sexual Reproduction. Sexual reproduction involves the production of haploid gametes by meiosis. This is followed by fertilization and the formation of a diploid zygote. The number of chromosomes in a gamete is represented by the letter n. Why does the zygote have 2n, or twice as many, chromosomes?