The Burr distribution as a model for the delay between key events in an individual’s infection history

Nyall Jamieson; Christiana Charalambous; David M. Schultz; Ian Hall

doi:10.1371/journal.pcbi.1012041

Abstract

Understanding the temporal relationship between key events in an individual’s infection history is crucial for disease control. Delay data between events, such as infection and symptom onset times, is doubly censored because the exact time at which these key events occur is generally unknown. Current mathematical models for delay distributions are derived from heuristic justifications. Here, we derive a new model for delay distributions, specifically for incubation periods, motivated by bacterial-growth dynamics that lead to the Burr family of distributions being a valid modelling choice. We also incorporate methods within these models to account for the doubly censored data. Our approach provides biological justification in the derivation of our delay distribution model, the results of fitting to data highlighting the superiority of the Burr model compared to currently used models when the mode of the distribution is clearly defined or when the distribution tapers off. Under these conditions, our results indicate that the derived Burr distribution is a better-performing model for incubation-period data than currently used methods, with the derived Burr distribution being 13 times more likely to be a better-performing model than the gamma distribution for Legionnaires’ disease based on data from a known outbreak.

Author summary

In public health, it is important to know key temporal properties of diseases (such as how long someone is ill for or infectious for). Mathematical characterisation of properties requires information about patients’ infection histories, such as the number of days between infection and symptom onset. These methods provide useful insights, such as how their infectiousness varies over time since they were infected. However, two key issues arise with these approaches. First, these methods do not have strong arguments for the validity of their usage. Second, the data typically used is provided as a rounded number of days between key events, as opposed to the exact period of time. We address both these issues by developing a new mathematical model to describe the important properties of the infection process of various diseases based on strong biological justification, and further incorporating methods within the mathematical model which consider infection and symptom onset to occur at any point within an interval, as opposed to an exact time. Under certain conditions, our approach provides more preferable results, based on AIC, than existing approaches, enhancing the understanding of properties of diseases such as Legionnaires’ disease.

Citation: Jamieson N, Charalambous C, Schultz DM, Hall I (2024) The Burr distribution as a model for the delay between key events in an individual’s infection history. PLoS Comput Biol 20(12): e1012041. https://doi.org/10.1371/journal.pcbi.1012041

Editor: Eric HY Lau, The University of Hong Kong, CHINA

Received: April 2, 2024; Accepted: November 19, 2024; Published: December 27, 2024

Copyright: © 2024 Jamieson et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the manuscript and its Supporting information files.

Funding: NJ acknowledges support from the Engineering and Physical Sciences Research Council (EPSRC) and Mathematics and Data in Scientific and Industrial Modelling (MADSIM) at the University of Manchester for funding of their studentship. IH was supported by the JUNIPER modelling consortium (grant MR/V038613/1) the National Core Study on Transmission (PROTECT) and by the UKRI Impact Acceleration Account (IAA 386). NJ and IH also acknowledge the UK Health Security Agency (UKHSA) for honorary contracts and funding (for IH). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

In epidemiology, the temporal relationship between key events in an individual’s infection history is important to understand. For example, a disease that has a long delay from infection to onset of infectiousness may be amenable to contact tracing, and the relationship between these two events can be important for disease control [1, 2]. Often these events are a simplification of a continuous process (i.e., infectivity may not start or end at specific times but instead increase and then decrease over time). For diseases such as Legionnaires’ disease, which spread via airborne dispersion from environmental sources (rather than person-to-person contact), characterisation of the incubation period is critical for source identification (or reverse epidemiology).

Here, we consider the time from infection to symptom onset. The relationship between viral or bacterial load in one’s body and onset of symptoms can be difficult to describe. In brief, the presence of a virus or bacteria within an individual results in an inflammatory immune response that leads to an observable response of symptoms. An exact mathematical model accurately describing the infection process is not feasible to develop due to the large number of different cytokines and cell interactions involved in the immune response, as well as a lack of a clear understanding of how the pro-inflammatory cytokines relate to the appearance of symptoms and a lack of data to parameterise each specific process in the immune response. Previous models for the incubation period provide parsimonious simplifications of the infection process and include in-host models (often assuming symptom onset is proportional to bacterial load [3]) through to simpler probability models (justified on model parsimony or computational capacity). In the latter case, popular distributions include the gamma, log-normal and Weibull distributions [4–6].

Application of these common distributions is primarily based on heuristic justification. These common distributions share similarities in that they are right-skewed, defined on a positive support and flexible so that they can model a wide range of incubation-period datasets. More specific arguments for common distributions can be described as follows. The gamma distribution is a generalisation of the Erlang distribution with non-integer shape, n. In which case, the Erlang distribution is the sum of n exponentially distributed random events, and so fitting to data can help inform the structure of compartmental models [7]. The log-normal distribution is a skewed distribution often applied to biological processes in which the process mean time is relatively low, but its variance is large and results from taking the exponential of a series of normally distributed events. Finally, the Weibull distribution is a classic reliability-theory distribution where the hazard of an event occurring is strictly monotonic over time.

To illustrate the heuristic justification of distributions, we consider Legionnaires’ disease and the statistical analysis that has been conducted in the literature for studying the incubation period. In this case, several papers have used a range of days (2–10) prior to symptom onset and considered all days in this period as a potential infection date [8–14]. Alternatively, others have assumed a median incubation period of either five days [15] or seven days [16], with infection dates obtained by subtracting the median from the symptom onset date. Another common approach is to consider a gamma-distributed incubation period [17]. All papers that take this approach have followed the ideas and method proposed in [4] using a gamma distribution to describe an outbreak in Melbourne [18].

One issue arising is that incubation-period data are given as an integer number of days, implying that each case becomes infected at the same moment from the exposure, and that symptoms develop in an integer amount of days. To illustrate this issue, take two cases in which symptom onset occurs the day after infection. The individual could have been infected at 11:59pm and became symptomatic at 00:01am the next day, or alternatively they could have been infected at 00:01am and became symptomatic at 11:59pm the next day. These two scenarios are 2 minutes and 1 day, 23 hours, 58 minutes long, respectively, but they both correspond to one integer day in the dataset. These simplifications give a lower resolution of the time delay between these events due to lack of knowledge of the exact infection and symptom onset times. Essentially, continuous distributions are being fitted to discretized versions of continuous data, and the result is interval data with censored start and end times.

This type of discretized data are commonly used for analysis without consideration for the censoring issue. Using standard probability distributions, as well as censored incubation-period data in statistical analysis, is likely to produce biased inference. In reality, the individual was likely not infected at the beginning of their infection date. Similarly, symptoms likely did not appear at the end of their symptom onset date. Therefore, recorded incubation-periods are likely to be inflated with a positive bias. Using incubation-period data expressed as an integer number of days will likely lead to a false understanding of delays between key events for specific diseases, such as the incubation period, and produce incorrect conclusions. A model describing the incubation period of Legionnaires’ disease has been built with this type of data [4], but the model is flawed and can be improved upon by accounting for the issues mentioned above. There are various ways to handle the censoring issue, which we discuss in the next section.

In this paper, a new model for incubation periods is derived with potentially stronger justification for its validity than methods currently used in the literature. We apply our new model to a variety of diseases and provide statistically significant changes in the mean incubation period, specifically for Legionnaires’ disease, compared to results obtained from using currently accepted and used models. We also apply techniques that remove the bias from fitting models to censored data and allow for reliable model-fitting, providing a new understanding of the incubation periods of various diseases. We apply these methods to anthrax, salmonellosis and campylobacteriosis, as well as taking a specific focus on Legionnaires’ disease to illustrate the typical kind of improvement achievable with these methods. For the successful models, we develop some distribution theory, calculating their moments and quantile functions, which can be found in S1 Appendix in the Supplementary Material.

Materials and methods

In this section, we develop methods for handling both of the problems discussed in the introduction. First, we adapt the methods developed in [19] for use on incubation-period data in order to account for its censored nature. Second, we consider a probabilistic approach to develop a new model for incubation periods of diseases. We assume exponential growth of bacteria early after infection, as well as a further assumption of the probability of symptom onset being proportional to the bacterial load within an individual until saturating once some load has been reached. Third, we discuss the methods for analysing our fitted models and how we determine which model performs better, so that we can conclude whether or not our developed model offers more reliable results than using methods currently developed in the literature. Finally, we introduce the data used for incubation-period analysis and discuss the reasons why this data are considered censored.

Doubly interval-censored modelling

Methods for handling censored data in epidemiological studies have been proposed in the literature to develop discrete analogues of continuous distributions that preserve properties of their continuous counterparts [20]. However, most of these methods either focus on preserving one property, do not result in valid probability mass functions, or assume that infection occurs exactly at midnight. These discrete analogues are not designed to account for the nature in which the continuous-time incubation-period process is recorded as discrete data.

The exact time at which symptoms occur in an individual cannot be determined based on when they reported their illness to authorities. Similarly, the exact time that an individual becomes infected is also difficult to ascertain. We need a method for handling the fact that these times are unknown (i.e., to account for the uncertainty within a model), so that analysis of any subsequent models is reliable. To consider doubly censored data, a natural approach is to forget the assumption that the exact infection and symptom onset times are known and introduce a time period in which these two events may occur, with a probability distribution for the occurrence within this period [19]. The method proposed in [19], which considers doubly interval-censored (DI) data, is described as follows.

Define T and S to be the time of infection and symptom onset respectively (with t and s being realisations of these random variables respectively), and Z = S − T as the incubation period of the infection. Consider two intervals where T and S could lie within because the exact times of T and S are not known. In other words, let T ∈ (T_L, T_R) and S ∈ (S_L, S_R). The incubation period Z is given as a random variable with p.d.f. f(s − t) (Fig 1).

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Fig 1. Diagram visualising the doubly interval-censoring method [19], highlighting the data typically observed, but accounting for the fact that infection and symptom onset times are not observed exactly and intervals of possible times must be considered.

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The p.d.f. of T is defined as f_T(t) and the p.d.f. of S is defined as f_S(s). The time at which a person becomes infected and the time taken from infection to symptom onset are independent, which leads to f_S(s | t) = f(s − t | t) = f(s − t). Finally, define the joint p.d.f. of T and S as From this, the likelihood for a doubly interval-censored observation x is derived. To implement methods found in [19] to incubation-period data, the following approach is taken. Because the data are rounded to the nearest day, a natural assumption is that T_L = 0 and T_R = 1, so infection occurs at any point on the infection date. Defining x to be the number of days from exposure to symptom onset, set S_R = x and S_L = x − 1, so that the symptoms develop at some point on the stated date of symptom onset. There is not much evidence to indicate what distribution f_T(t) might be, so a reasonable assumption would be to let f_T(t) be uniform (i.e., f_T(t) = 1 on t ∈ (0, 1), 0 otherwise). Other options could be to permit a lower chance during nighttime or a higher chance when people are outdoors, but these will depend on specific release scenarios and are not likely particularly identifiable in data. As f(s − t) is the p.d.f. of the incubation period, the log-likelihood is calculated as follows: (1) In the next section, we develop various distributions to describe the incubation period, and later fit the doubly interval-censored model to these distributions to determine which one provides the most optimal fit.

Derivation of the incubation period model

Incubation period data describes the cases who become symptomatic. Given the knowledge that all individuals in the data will become symptomatic, this section discusses different mathematical models for the occurrence of symptoms onset within a population. We explore how the results for these different methods link, and we develop a new model for incubation periods, by starting from a probabilistic approach of symptom-onset occurrence.

A probability-based approach.

A continuous-time mathematical model can be built considering the hazard rate of symptom-onset occurrence. We first consider the option of using an exponential survival model with time-varying hazard. Define N(t) as the population of individuals who are infected, but are not yet symptomatic at time t, and Q(t) as the population of individuals who are symptomatic at time t with N(0) = N₀ and Q(0) = 0 and Q(t) + N(t) = N₀, . Next, assume that a hazard rate function λ(t) describes the risk that a not-yet-symptomatic individual will start to experience symptoms at a point in time t, given that they have not already succumb to symptoms by time t. Then 1 − δtλ(t) will be the probability that the individual will remain asymptomatic within a small interval (t, t + δt), where in this context δt represents a small time increment in time. Hence (1 − δtλ(t))^N(t) is the probability that nobody who is not-yet-symptomatic will start experiencing symptoms within a small increment δt from t, and (1 − δtλ(t))^N(t)δt is the probability that nobody new will experience symptoms in a small increment δt from t. Following this, define δQ(t) = 1 − (1 − δtλ(t))^N(t)δt to be the probability that there is at least one individual who starts to experience new symptoms in a small increment δt from t. By writing μ(t) = −log(1 − δtλ(t)), the probability of any new symptom onset appearance can be written as δQ(t) = 1 − e^{−μ(t)N(t)δt}. Using a Taylor expansion on the exponential term, dividing by δt, and taking the limit δt → 0 changes this probability to a rate as follows: (2) This approach leads to a separable ordinary differential equation analogous to the cumulative distribution of the exponential distribution with a time-varying rate parameter.

It can be deduced that and that is the accumulated hazard. Hence the rate of symptom onset, μ(t), is the hazard function of an individual becoming symptomatic. Therefore, the hazard of an individual becoming symptomatic at a point in time is equal to the rate of symptom onset at that time. The scenario discussed here can be considered from an inhomogeneous Poisson-process perspective, and the results of the hazard are identical to the inhomogeneous exponentially distributed model. It can be noted here that if μ(t) is constant that this would lead to the exponential distribution and if μ(t) ∝ t^a for some constant a this would suggest the incubation period is a Weibull distributed random variable. The Erlang distribution arises by assuming the incubation period is the sum of a number of stages of constant length μ.

However, various studies have shown that the bacterial load within an individual is positively correlated with the probability of symptom onset [3, 21]. The relationship between bacterial load and probability of symptom onset is complex and varies from bacteria-to-bacteria [21]. Additionally, a positive correlation between load and probability of symptom onset has been observed for viral infections [22]. For parsimony, we assume that symptom onset is likely proportional to bacterial (or viral) load at low loads (i.e., the early stages of infection) before saturating at large loads. The bacterial population early after infection will be approximately some exponential function of time [3, 23, 24]. Therefore, the left tail of the c.d.f. of the incubation-period distribution is given by some function , whilst in the later stage, the c.d.f. should tend to 1 exponentially given by a function G₂(t), as is the case of the hazard function above. Mathematically, with a median T, and considering the case where G(t) = G₁(t) = G₂(t), an equation for the c.d.f. that satisfies these conditions is given as follows: (3) where for some function g(s). The ODE that arises in (3) defines the Burr family of distributions and is discussed in further detail in the next section.

Burr distribution.

A Burr distribution is a distribution whose c.d.f., F(t), (4) is the solution of (3). Theoretically, there are no constraints on G(t) in (4). Twelve main distributions within the Burr family have been characterized [25], named as Burr type I, Burr type II, up-to Burr type XII, but we only consider Burr distributions defined over a domain of (0, ∞).

Some delay distributions arising in epidemiology do permit negative values. For example, the time from symptom onset in infector to symptom onset in infectee could be negative. In this paper, we limit consideration to strictly positive cases. A negative incubation period is not possible, nor is a fixed upper-limit constraint expected. The only biologically feasible distributions are types III, X and XII. The type III distribution could be derived from the flexible generalized gamma distribution with the scale parameter following an inverse Weibull distribution [26]. Similarly, the type XII distribution could be derived from the Weibull distribution where the scale parameter follows an inverse generalized gamma distribution [26].

The Burr distributions and the gamma distribution have parameters that share the same symbols for notational simplicity, although they have different interpretations and their fitted estimates cannot be directly compared. To avoid confusion, we provide a subscript for each parameter to clarify which distribution this parameter corresponds to (i.e., α_III for the α parameter in the type III Burr model) in the text but drop this in tables and figures for brevity. Further, the type III, X and XII distributions used in this research are a generalization of types III, X and XII Burr distributions used in the literature [25], where the time variable is scaled by an additional parameter. Type X is defined with two variables that provide models as parsimonious as the three distributions previously trialled: gamma, log-normal, and Weibull. Further, both type III and XII distributions have two shape parameters α_III,XII and β_III,XII. Finally, the type III distribution has a scale parameter γ_III,XII and the type XII distribution has a location parameter, the median T.

General derived Burr distribution.

In (3), g(t) has a physical interpretation; the function tends to the rate of symptom onset μ(t) in individuals at a time t as t increases. Given F(t) = (1 + e^−G(t))⁻¹, in general, then G(t) → t/β_D (or g(t) → 1/β_D) for some constant β_D as t → ∞ on the basis that the hazard rate approaches constant over time for relatively long incubation periods. In principle, F(0) = 0, so G(t) → −∞ for t → 0 (or G(0) is very large if not actually infinite). Taking the above into account, we propose g(t) = 1/β_D + α_D/t, and as such G(t) = t/β_D + α_D log(t) + C, where C is a constant of integration. We define T_D as the median, which satisfies G(T_D) = 0. Hence, C = −T_D/β_D − α_D log(T_D) and thus Equations for the c.d.f. and p.d.f. for the derived Burr distribution, as well as the gamma and other Burr distributions, are given in Table 1. As discussed, T_D is the median of the distribution. The reciprocal of β_D is the eventual constant rate of symptom onset in individuals for t ≫ T_D. Additionally, there are two details worth noting when analysing the physical interpretation of α_D. First, α_D is an exponent of t that controls the increase in probability density, as for t ≪ T_D. Second, the general derived Burr distribution approaches the exponential distribution for t ≫ T_D. The rate at which the derived Burr distribution approaches a constant hazard (as for an exponential distribution) increases for decreasing α_D. Finally, all parameters must be strictly greater than zero.

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Table 1. The Burr distributions valid over (0, ∞) and previously trialled distributions [4] with their corresponding p.d.f and c.d.f.

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Model comparison

We fit each type of Burr distribution to the data, and assess all the models in terms of their goodness of fit in comparison to the more widely used gamma distribution. The most commonly used methods for model selection are the Akaike information criterion (AIC) and Bayesian information criterion (BIC) [27]. Generally, AIC puts more emphasis on good model prediction, whereas BIC favours model parsimony [27]. Because our goal is good model prediction, the AIC will be used in deciding desirable model fits. Additionally, we calculate the Akaike weights ω for each model fit, which can be used for further model comparison [28].

Akaike weights are used to compare the validity of a Burr distributed model over the gamma distributed model once fit to data. The ratio w_i/w_G, where w_i is the weight for the i^th model and w_G is the weight of the gamma distributed model. This ratio may be interpreted as how much more likely model i is a better fitting model than the gamma model. Alternatively, we also derive the normalized probability that the i^th model is preferable to the gamma model, given by w_i/(w_i + w_G).

The final method of comparison considered is the Bayes factor [29]. The maximum likelihood estimates that we obtain can be considered maximum a posteriori estimates with a uniform prior and are used in this context for conducting the Bayes Factor calculations. Larger Bayes factor values indicate stronger evidence to support one model over another.

Incubation-period data

To test these models, we employ incubation-period data from an outbreak of Legionnaires’ disease in Melbourne in April 2000 [18]. The data for the Melbourne outbreak contains the number of days taken for each Legionnaires’ disease case to develop symptoms from their exposure date, and several potential distributions for fitting the data have been compared [4]. The case data only contains individuals that visited the known source once within the weeks before the outbreak and have a known date of symptom onset. Therefore, the timing of infection and symptom onset events are known correctly to a single day. The results indicated that the gamma distribution provided the best fit [4] out of their proposed models.

Further, we gather incubation-period data for anthrax, campylobacteriosis and salmonellosis for analysis. The anthrax outbreak in 1979 contains data for the known incubation-periods of patients [30]. Investigations into the outbreak, climate conditions and human/animal presence highlighted a single exposure on 2 April 1979. The date of symptom onset was provided as known in most cases. However, for the few remaining cases, an estimated symptom onset date was obtained by subtracting 3 days (the mean delay between symptom onset and death) from the date of death. A literature review has been conducted analysing different salmonellosis studies that contain full data of the incubation periods [31]. Awofisayo-Okuyelu et al. [31] noticed that the incubation periods varied between studies. They grouped studies into subsets using a clustering process, in which the grouped studies did not have any statistically significant difference in their incubation-period data. Similarly, Awofisayo-Okuyelu et al. [32] conducted a review for campylobacteriosis in which the incubation periods varied between studies, and they combined datasets which were not statistically significantly different using a clustering process similar to [31]. For both the salmonellosis and campylobacteriosis datasets, a quality assessment was carried out during their literature review [31, 32]. Incubation period data obtained was assessed based on whether cases were linked to a clearly defined exposure and accuracy of the reported symptom onset time, with the lowest level of resolution in the symptom onset being a period of 24 hours [31, 32]. We provide an Excel sheet of the incubation-period data for these other diseases in S1 Data in the Supplementary Material.

The data gathered for these diseases share a similarity with the Legionnaires’ disease data, in that the data contains the integer number of days taken for each case to develop symptoms. The fact the data for all of these diseases contains integer days implies that each case takes an exact multiple of 24 hours from infection to the appearance of symptoms, which is not realistic. If we assume that the dates of infection and symptom onset are accurate, then we know the date of these events, but the specific times on the given days are unknown. We are dealing with doubly censored data.

Results

Now that we have developed the Burr distribution as an incubation-period model based upon biological justifications, the next step is to fit these models to the incubation-period data of various diseases. We begin by fitting the incubation-period models to the Legionnaires’ disease data, to draw comparisons between the models’ performance. Next, we conduct the same analysis on other diseases such as anthrax, campylobacteriosis and salmonellosis. Finally, we conduct two simulations in which incubation-period data are fabricated. First, we compare the results from fitting the incubation-period models to fabricated data, as we compare the parameter estimates obtained from fitting the gamma and derived Burr distributions to this data in an attempt to assess the relationship between these parameters. Second, we fabricate doubly-censored data and fit the derived Burr distribution using the DI likelihood fitting method to this data. We aim to assess bias in the parameter estimates and the appropriate coverage of 95% confidence intervals of the parameter estimates.

Analysis of the Melbourne data

The gamma distribution is currently most frequently used to model Legionnaires’ disease incubation periods [4]. Therefore, we produce models using a gamma-distributed incubation period to allow for comparison between models. Models are fitted using both the standard and doubly interval-censored maximum likelihood fitting methods to offer comparison between the two methods.

We begin this section by providing the results from fitting the incubation-period models to the data (Table 2). We compare the incubation-period models, as well as model-fitting approaches, and the effect that they have on our understanding of Legionnaires’ disease incubation periods. We provide analysis of the moments of these Legionnaires’ disease incubation-period models in S1 Appendix in the Supplementary Material. Further, in this appendix, we provide visual comparison of the accumulated hazard of these models for large time, to examine their ability to accurately display a Markovian property of long incubation periods. The analysis and production of plots was conducted on R, with the code provided in S1 Code in the Supplementary Material.

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Table 2. Results from fitting the gamma and four Burr distribution models to the Melbourne incubation-period data using both the standard and DI likelihood fitting methods.

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When fitting using the standard maximum likelihood method, type III, X, XII distributions and the derived Burr distribution perform better than the gamma distribution regardless of which scoring criterion is used. Because the type X distribution is a two-parameter distribution, the fact that its maximized log-likelihood is higher than gamma’s automatically means that its minimized AIC will be lower. Types III, XII and the derived Burr distributions perform better than the gamma distribution depending on how harshly they are penalized for their extra parameter. Based on AIC, our ideal information criterion for model selection, these perform better than the gamma distribution. On the whole, all Burr distributions perform better than the gamma distribution. From considering the Akaike weights ratio w/w_G, the derived Burr, type III, and type X distributions are at least two times as likely to be a better-performing model than the gamma distributed model. Additionally, each Burr model provides at least a 62% chance of being a better fitting model than the gamma-distributed model, with the derived Burr model being 70% more likely to be better than the gamma model. Looking at the Bayes factor, there is no substantial evidence to favour the type X distribution over the gamma distribution. However, this criterion gives substantial evidence that both type III, XII distributions as well as the derived Burr distribution are all favourable over the gamma distribution.

Next, when fitting using doubly interval-censoring methods, the type X distribution again outperforms the gamma distribution. Types III, XII and the derived Burr distributions perform better than the gamma model, based on AIC, even with one extra parameter. When considering the Akaike weights, all the Burr distributed models perform much better than the gamma distribution, with the derived Burr distribution being over 13 times more likely to be the better-fitting model. Additionally, when considering w/(w + w_G), all Burr models are more likely to be perform better than the gamma distribution, with the derived Burr distribution being 93% likely. Finally, the Bayes factor for the types X and XII distributions both show substantial evidence of a better fit than the gamma distribution. Further, the Bayes factor for type III and the derived Burr distributions both show strong evidence of a better fit than the gamma model.

The same conclusions are drawn regardless of maximum likelihood fitting method; all the distributions provide a better fit than the gamma distribution. Results using the DI method agree with the standard likelihood method in that β_XII in the Burr type XII model has large standard errors. For this distribution, T centers the curve about the median and α scales the curve. The large standard errors for β indicate that β is not as important in the model fitting procedure.

When fitted using standard maximum likelihood methods, all Burr distributions considered offer a similar curve when plotted, as expected, but do vary slightly as to the model value or the value of the p.d.f. at the mode (Fig 2). The Weibull distribution provides a similar modal value for the incubation period, but is more variable than the Burr models. The gamma distribution provides a slightly lower modal value than the Burr models. The log-normal model provides a noticeably different curve to the Burr models and provides a much lower modal incubation period, with a lighter left tail and heavier right tail than all the other distributions.

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Fig 2. A plot of the Melbourne case data with the four fitted Burr distributions included, which offer a visual representation of the incubation-period distributions trialled.

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The maximised log-likelihood decreases when switching to the DI method for some distributions, such as the log-normal and gamma. On the other hand, the maximised log-likelihood increases for other distributions, such as the derived Burr distribution. For distributions fitted using the standard maximum likelihood fitting method that have a lower modal value lower than the modal value of the data, changing to the DI method reduces the modal value further. This larger difference results in a distribution further from the data. Therefore, lower maximised log-likelihood values are typically obtained using the DI method when the mode of the distribution is lower than the mode of the data. However, this reasoning does not hold true for the type III distribution.

The mean of each fitted distribution along with a bootstrapped 95% confidence interval is calculated under both the standard method and the doubly interval-censored method to identify any differences across distributions and across methods, and is provided in S1 Appendix in the Supplementary Material. A common theme exists, which is that, for each distribution, the mean for the doubly interval-censored model is approximately a day less than the standard model (5.3 days compared to 6.3 days), with the confidence intervals for each model having no overlap across all the distributions. We bootstrap from the distributions and apply a two-sample t-test to assess investigate whether, for each distribution, the mean incubation period obtained from the doubly interval-censored method is statistically-significantly lower than the mean incubation period obtained from the standard method. These calculations and the associated p-values are provided in S1 Appendix in the Supplementary Material. These results are statistically significant and provide support for using a doubly interval-censored model to more accurately represent the incubation period of Legionnaires’ disease.

For all of the distributions, the density under the doubly interval-censored approach is shifted more towards the left, indicating that the incubation period is shorter than when just taking the incubation period as exact integer days (Fig 3). Indeed, the doubly interval-censored methods account for a potential delay between the start of the infection day and the time during the infection day that infection occurs as well as a delay between the time during the symptom-onset day that symptoms appear and the end of the symptom-onset day, whereas the standard model does not account for either delay, resulting in longer times for the incubation periods.

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Fig 3. Plots of the Melbourne data with the standard model fits in red and the doubly interval-censored model fit as a step function in yellow.

Each step of the function is a horizontal line from t ∈ (a, b] where a = ⌊t⌋ and b = ⌈t⌉.

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Application to other diseases

To further check the validity of the Burr distribution, we fit the doubly interval-censored models to data of the incubation periods for different diseases: anthrax [30], campylobacteriosis [32] and salmonellosis [31]. Figures of resulting model fits provided in S1 Fig, along with the obtained parameter estimates and standard errors of these estimates contained in S1 Fig in the Supplementary Material. We use both the standard and the doubly interval-censored methods to fit the gamma and the Burr distributions, to compare which model provides a better fit (Table 3).

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Table 3. Comparing Burr, log-normal and Weibull models with the gamma model on anthrax, salmonellosis and campylobacteriosis datasets.

For brevity, we define the datasets as A, S or C to represent anthrax, salmonellosis and campylobacteriosis datasets respectively. The numbers in the dataset column indicate which dataset for a given disease is being referred to, as we have multiple incubation-period datasets for Salmonella and Campylobacter. The value provided is the difference between the recorded AIC between a given model and the gamma distribution. Negative values indicate lower AIC, which is preferable. Similarly, positive values indicate higher AIC, which implies a worse model fit.

https://doi.org/10.1371/journal.pcbi.1012041.t003

For Burr types III, XII and the derived Burr distribution, a difference in AIC between 0 and +2 indicates that the Burr model provides a preferable fit based on maximum likelihood estimation, but the extra parameter results in a higher AIC. Burr type III and X distributions offer mixed results across datasets and do not consistently outperform the gamma distribution. Based on maximum likelihood, the derived Burr distribution outperforms the gamma distribution for every dataset other than the third campylobacteriosis dataset. However, based on AIC, the gamma distribution becomes preferable for the anthrax dataset and the first salmonellosis dataset regardless of maximum likelihood fitting method. Additionally, based on AIC, the gamma distribution becomes preferable to the derived Burr distribution when fitting to the second and fifth campylobacteriosis datasets with the doubly interval-censored and standard maximum likelihood methods respectively. Instances in which the gamma distribution provides preferable results based on AIC is typically due to the penalty from the derived Burr distribution’s extra parameter. Therefore, these datasets result in relatively close model fits between the gamma and derived Burr distributions. These conclusions hold, to a lesser extent, for the type XII distribution.

No clear pattern exists between any of the fitted α_D and β_D parameter estimates and the performance of the derived Burr distribution. Additionally, there is no clear pattern from the anthrax, campylobacteriosis and salmonellosis datasets as to whether the estimate of the median T_D relates to the performance of the derived Burr model. However, the lack of sensitivity for T_D is logical as T_D solely scales the distribution about the median, and the ability of the derived Burr distribution to fit well to incubation period data will depend more on the tails in the curve and around the median, as opposed to the median itself.

We can draw conclusions on which scenarios the derived Burr distribution will outperform the gamma distribution based on plots provided in S1 Fig of the Supplementary Material. The third campylobacteriosis dataset was the only dataset in which the derived Burr distribution did not outperform the gamma distribution based on either maximized likelihood or on AIC. This dataset is unique in that the incubation period ranges from one to five days. As a result, the effect of the censoring bias will be much larger, due to the fact that this incubation period is much shorter. Therefore, this is not an ideal dataset to use to assess model performance.

Next, we consider the datasets in which the derived Burr distribution outperformed the gamma distribution based on maximized likelihood but not on AIC, regardless of model fitting procedure. The anthrax dataset has a high density after the mode and does not tail off, and the probability distribution of the first salmonellosis dataset does not have a clearly defined mode and is negatively skewed. The derived Burr distribution offers close results to the gamma distribution when it comes to modelling incubation periods without a clear mode or tail off in probability of illness, but is a better-performing distribution when this structure is clearer defined.

Finally, fitting to the second and fifth campylobacteriosis datasets resulted in the derived Burr distribution outperforming the gamma distribution on maximized likelihood but not on AIC. The incubation period for these datasets is relatively small, meaning that the bias from the censoring issue is large when fitting models to these datasets. The campylobacteriosis datasets that resulted in the derived Burr distribution outperforming the gamma distribution were the ones in which the modal time was clearly defined and not a wide range of times at the peak of the distribution. These results further supports the hypothesis that the derived Burr distribution becomes more preferable when either the mode is more apparent, or the range of incubation periods in the datasets is not too short that the censoring becomes a larger issue.

Results of model-fitting to simulated data

We now further assess the validity of the Burr distributions by comparing their fits, along with those of the gamma distribution, to fabricated data. Specifically, we aim to analyse how the parameter estimates of the gamma distribution relate to the parameter estimates of the derived Burr distribution for different datasets, to gain a further understanding of how the derived Burr distribution’s parameters can be interpreted.

Initially, we generate a sample of size 1000 from a gamma distribution with given shape α_Γ and mean μ_Γ (scale β_Γ = μ_Γ/α_Γ). Then, the derived Burr distribution parameter estimates are obtained from fitting to this dataset by standard maximum likelihood, so that analysis can be conducted on the effect that varying α_Γ ∈ (0.5, 5) or μ_Γ ∈ (1, 20) has on these estimates. First, this simulation focuses on analysing the relationship between the parameters of the gamma and derived Burr distributions. Therefore, we fit by standard maximum likelihood as opposed to the doubly interval-censored methods. A heatmap is produced to visualise this effect (Fig 4). Additionally, we repeat the same simulation with doubly-censored data to compare the parameter relationships under more realistic conditions. We simulate doubly-censored data by sampling an infection time from the uniform distribution on the infection day and adding this to a sample from a gamma distribution with α_Γ ∈ (0.5, 5) and μ_Γ ∈ (1, 20). We take the ceiling of this sum to produce the doubly-censored incubation period. Repeating this for a sample of size 1000, we fit the derived Burr distribution using the DI method to obtain the corresponding parameter estimates.

Download:

Fig 4. Heatmaps of the results from the first simulation.

The sub-figures (a), (c) and (e) (left column panels) represent the results from the standard likelihood fitting simulation, whereas (b), (d) and (f) (right column panels) represent the results from the DI likelihood fitting simulation. The sub-figures (a) and (b) represent the α_D estimates obtained. Next, (c) and (d) represent the β_D estimates obtained. Finally, (e) and (f) represent the T_D estimates obtained.

https://doi.org/10.1371/journal.pcbi.1012041.g004

The parameter estimates for β_D and T_D are invariant under the maximum likelihood fitting procedure. Fitting the derived Burr distribution using the DI methods to doubly-censored data generated from the gamma distribution results in similar parameter estimates to those obtained from fitting the derived Burr distribution using standard maximum likelihood fitting methods to non-censored data generated from the gamma distribution.

However, a discrepancy exists for estimates of α_D between methods. Fitting to doubly-censored gamma-distributed data with the DI method results in larger estimates of α_D than fitting to gamma-distributed data with the standard method. No clear pattern exists for α_D estimates for small μ_Γ ≈ 1. Because μ_Γ ≈ 1, the gamma-distributed incubation-period data are small in value, which means that the uniform infection time and ceiling function have a larger effect on the data than the distribution that generates the true incubation period. Therefore, in this circumstance the noise introduced in these two windows affects the estimation process for α_D with the DI method. The shape parameter α_D of the derived Burr distribution is more sensitive to small changes in the doubly-censored data when the mean incubation periods are relatively small.

In general, parallels exist between the interpretations of α_Γ and α_D. Increasing α_Γ results in a larger discrepancy between the gamma distribution and the exponential distribution. Thus, larger α_Γ values result in a longer period of time required for the distribution to become Markovian. Therefore, a positive correlation between α_Γ and α_D is expected (Fig 4a and 4b). The results indicate that μ_Γ does not have an effect on the rate at which the gamma distribution becomes Markovian.

Similarly, parallels exist between the interpretations of β_Γ and β_D. The hazard rate for the gamma distribution tends to 1/β_Γ as t → ∞. Hence, 1/β_Γ is the eventual rate of symptom onset for the gamma distribution. Thus, a positive correlation between β_Γ and β_D is logical (Fig 4c and 4d). Therefore, the effect that varying either μ_Γ or α_Γ in μ_Γ = α_Γβ_Γ has on β_Γ is likely to inform the effect that varying either μ_Γ or α_Γ has on β_D.

Finally, a positive correlation between μ_Γ and T_D is expected, as they both represent a form of average. For large α_Γ, the gamma distribution becomes symmetric, hence T_D → μ_Γ. However, the correlation becomes less linear as α_Γ decreases. In this case, μ_Γ − T_D and equivalently the skewness (defined by for the gamma distribution) increases (Fig 4e and 4f).

Following this simulation, we provide a second simulation in which we further assess the performance of the doubly interval-censored maximum likelihood fitting methods with the derived Burr distribution. Doubly-censored data are generated in the same way as the previous simulation, with the derived Burr distribution used instead of the gamma distribution for the true incubation period. We fit the derived Burr distribution to this fabricated data using the DI method to gain parameter estimates for this dataset. We record the bias of the parameter estimates and appropriate coverage for the 95% confidence intervals of the parameter estimates to assess the performance of the doubly interval-censored maximum likelihood fitting procedure.

We opt to vary one parameter and keep the other two fixed for this simulation due to the computational demand of varying two parameters and producing heatmaps as done in the first simulation (Fig 5). The two fixed parameters are fixed at the estimates obtained from fitting to the Legionnaires’ disease dataset.

Download:

Fig 5. Bias and appropriate coverage of 95% confidence intervals obtained from fitting the derived Burr distribution using DI methods to doubly-censored incubation periods generated from the derived Burr distribution.

The sub-figures (a), (b) and (c) represent the bias from the α_D, β_D and T_D-varying simulations respectively. Further, the sub-figures (d), (e) and (f) represent the appropriate coverage from the α_D, β_D and T_D-varying simulations respectively.

https://doi.org/10.1371/journal.pcbi.1012041.g005

For the α_D-varying simulation, the bias for α_D and β_D estimates increase as the true α_D increases. Increasing the true α_D results in a more spiked true incubation-period distribution. Further, taking the convolution of the spiked incubation-period distribution with a uniform infection window distribution and taking the ceiling results in a less-spiked doubly-censored distribution. The DI maximum likelihood fitting method fails to fully capture the original spike and estimates a flatter distribution, which results in positive bias in the estimates of α_D and β_D.

Further, for the β_D-varying simulation, the bias for β_D increases, whereas the bias for α_D decreases as the true β_D increases. Increasing the true β_D results in a less spiked true incubation period and in which case the DI method is able to extract estimates of α_D close to the true value. However, in this case, the relative bias of β_D remains constant as β_D varies.

Finally, for the T_D-varying simulation, the bias in α_D and β_D remains approximately constant, with some variability due to the random sampling when generating datasets. For each parameter-varying simulation, we obtain almost unbiased estimates for T_D, which indicates that regardless of which parameter is varied in the data-generating process, the DI method is successful at locating the median of the distribution.

For the appropriate coverage, we repeat 100 iterations of the simulation to record a proportion of times in which the true parameter is contained within the 95% confidence interval. For the β_D and T_D-varying simulations, the appropriate coverage for each parameter centers roughly around 95%. For the α_D-varying simulation, the appropriate coverage for T_D centers around 95%. However, as the true α_D increases, the coverage for α_D increases and the coverage for β_D decreases.

Discussion

This paper has brought attention to and provides solutions to two distinct issues involved in modelling incubation periods of diseases. First, we derived a new model for delays between key events in an individual’s infection history, specifically the incubation period, that has justifiable mechanistic reasons for its validity. Second, we adapted methods for using incubation-period data, that is given as an integer number of days and has issues with bias, to fit models.

We considered the probability of an individual changing from the not-yet-symptomatic population to symptomatic for deriving our mathematical model. This approach led to obtaining a differential equation equivalent to the equation defining the exponential c.d.f. with a time-varying rate parameter. We then extended the model with further assumptions to further develop the differential equation describing the incubation period. We considered the assumption that the probability of symptom onset after infection is proportional to the bacterial load before saturating at some large load, as well as considering that bacterial population is expected to grow exponentially. Further, we derived a specific distribution within the Burr family that satisfies a Markovian property of long incubation periods. Other trial functions for G(t) may offer results at least as good as this new model, and some in-host dynamics which affect the rate of symptom onset in populations could be considered for specific diseases to provide even more optimal forms of the Burr model.

Further, by considering models that account for unknown infection and symptom onset times (doubly interval-censored models), we have obtained expected incubation periods for Legionnaires’ disease that are statistically significantly less than previously thought (by a whole day) using standard statistical distributions with incubation-period data. The mathematical derivation of the new model and implementation of this model with doubly interval-censored methods address both these problems, as we arrive at a mechanistic model for incubation periods. Our model has few restrictions on which diseases it can be applied to. Additionally, our research highlights the need to account for the censored nature of the data, since we observe a statistically significant difference in the mean incubation period of Legionnaires’ disease when incorporating the DI methods into the model.

Our mathematical derivation leading to the Burr family of distributions provides a valid incubation-period model. This model does not consider factors such as an individual’s age, levels of immune response, susceptibility, doses received or the disease-specific in-host dynamics at play that determine if and when an individual becomes ill with an infection. For example, frailty may mean faster onset of symptoms, as may higher doses. These modelling choices mean that the exact disease-specific in-host dynamics are not considered. To derive a model considering the biological processes at play with a given disease, a different model would have to be derived based on the details of those dynamics. Additionally, our model was derived from the assumption of proportionality between bacterial (or viral) load and probability of symptom onset. Our assumption likely oversimplifies this relationship, and alternate models may be developed to assume different functional forms. However, the exact relationship between bacterial (or viral) load and probability of symptom onset varies between diseases and is not clearly understood [21]. If further research was conducted with consistent observations across diseases for this relationship, one could change the proportionality assumption to derive an alternate incubation-period model with further justification than the Burr distribution that we derived here.

Use of the double interval-censored maximum likelihood fitting methods relies on the assumption that the individual was infected during a single known exposure on the date given in the data. In reality, some outbreak datasets may not have a clear date of infection for individuals. In this case, an individual may have been exposed several times over various days or they may have been subject to a continuous exposure over time. In the former scenario, concern must be placed on whether the low-dose exposure boosts the immune response or has other effects on the individual. In the latter scenario, one may conduct a sensitivity analysis in which different dates within the continuous exposure window are trialled to assess which date provides the best statistical results. One may use the DI framework with an exposure interval wider than one day. However, a preferable approach would be to investigate these two other exposure scenarios and develop a method that accounts specifically for these different assumptions.

In S1 Appendix in the Supplementary Material, we noticed that all Burr distributions valid over (0, ∞), apart from the type X distribution, exhibited a Markovian property for long incubation periods. Consequently, we compared the results of using this model to the other Burr distributions to judge the validity of the Markovian assumption. The type X distribution provides successful results outperforming the gamma distribution in nearly all of the analysis (we obtain mixed results when fitting to other diseases). However, when compared to all of the other Burr distributions, the type X distribution performed the worst when fitting to the original Legionnaires’ disease dataset, the original Legionnaires’ disease dataset with doubly interval-censored methods and the other diseases with doubly interval-censored methods. Further, the type X distribution visually fits the worst to the Legionnaires’ disease data (Fig 2). These consistent results support our Markovian assumption for long incubation periods. Although non-Markovian Burr distributions provide better-performing models to the widely used gamma model under certain circumstances, the Markovian Burr models provide a further improvement in terms of distributional modelling.

Our proposed model can be applied in a number of ways in epidemiology and infectious disease modelling. For example, a common area of research is to study person-to-person transmissible diseases, such as COVID-19. In this case, researchers usually develop compartmental and time-since-infection models where the infectivity of inflicted individuals infecting susceptible individuals in a population is modelled. Typically, an exponential (or Erlang) distribution from the point in time at which they are infected is used for modelling. This use of ‘Gamma’ related distributions remains necessary for ODE based compartmental models and is an appeal for modelling with the gamma distribution. In this work, we have limited to time delay distributions with range of times that are strictly positive, as must be the case with the incubation period. Some epidemiological distributions, such as generation time, are not bound by this constraint and so care would be needed in application.

Furthermore, we may consider diseases that do not have a person-to-person transmissible property such as Legionnaires’ disease, which has been the focus of this research. Researchers typically track backwards from symptom onset date to predict source location of the infection for elimination and public safety. A more reliable model such as the model developed here can provide more accurate results when predicting locations or causes of Legionnaires’ disease cases, which will result in reduction of bacterial hot-spots and consequently cases of this disease.

This paper provides a flexible model that can reliably fit incubation-period data to a level that is not currently in the literature and is valid for a wide range of diseases. The results of fitting the Burr distribution to the diseases considered in this paper indicate that using the Burr family of distributions as a model for incubation periods performs better than currently accepted models [4] when the mode is clearly defined or when the distribution tapers off.

Supporting information

S1 Appendix. Moments calculations for derived Burr and scaled type XII distributions.

Further Legionnaires’ disease modelling analysis of mean incubation period and cumulative hazards.

https://doi.org/10.1371/journal.pcbi.1012041.s001

(PDF)

S1 Fig. Figures and parameter estimates of anthrax, campylobacteriosis and salmonellosis datasets with model fits for gamma, burr types III, X, XII and the derived Burr based on the original and doubly interval-censored methods.

https://doi.org/10.1371/journal.pcbi.1012041.s002

(PDF)

S1 Code. R code for conducting analysis and producing plots in this research.

https://github.com/NyallJamieson/Burr-Incubation-Period.

https://doi.org/10.1371/journal.pcbi.1012041.s003

(R)

S1 Data. Incubation period data for the diseases analysed in this research.

https://doi.org/10.1371/journal.pcbi.1012041.s004

(XLSX)

Acknowledgments

Disclaimer

The views expressed are those of the author(s) and not necessarily those of the Department of Health or UKHSA.

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