Quantitative assessments of when infectious disease outbreaks end are crucial, as resources targeted towards outbreak responses typically remain in place until outbreaks are declared over. Recent improvements and innovations in mathematical approaches for determining when outbreaks end provide public health authorities with more confidence when making endofoutbreak declarations. Although quantitative analyses of outbreaks have a long history, more complex mathematical and statistical methodologies for analysing outbreak data were developed early in the 20th century and continue to be refined. Historically, such methodologies focused primarily on factors affecting the early and middle phases of an outbreak, with less attention given to determining how and when outbreaks end. This review discusses mathematical modelling methods from the last 20 years that have been developed for determining the ends of infectious disease outbreaks, and considers factors that affect the accuracy of such determinations. When disease surveillance systems provide timely and representative data to inform models, the timings of endofoutbreak declarations can be finetuned to allow outbreaks to be declared over quickly and with a low risk of being incorrect. Premature declarations that outbreaks are over can undermine earlier achievements in disease control and may result in a resurgence of cases, but unnecessary delays in declaring outbreaks over can cause significant economic and social harm. Appropriate declarations that balance the benefits of relaxing control measures against the risk of a surge in cases allow public health resources to be conserved (and economic and social pressures to be reduced) while limiting the potential for additional transmission.
Determining when infectious disease outbreaks (used synonymously with epidemics in this article, see Table 1) end is a critical component of outbreak response. In recent or ongoing outbreaks, such as those of coronavirus disease 2019 (COVID19) or pandemic influenza, mathematical modelling has played a central role in policy and public understanding of the course of these outbreaks. Although quantitative analyses of outbreaks have a long history, complex mathematical and statistical methodologies for analysing outbreak data were developed early in the 20th century and continue to be refined. Historically, such methodologies focused primarily on factors affecting the early and middle phases of outbreaks, with less attention given to determining how and when outbreaks end. In this article, we review mathematical modelling methods from the last 20 years that have been developed for determining the ends of infectious disease outbreaks, and consider the factors that affect the accuracy of such determinations. Rather than setting out a history of epidemiological modelling, here we explain the function and methods of quantitative infectious disease modelling, drawing on key practices and case studies to demonstrate the applications, advantages, and limitations of the disciplinary methodologies that have become crucial to understanding outbreaks and how they end.
Development of statistical methods for determining the ends of infectious disease outbreaks has accelerated in recent years, as large crossborder outbreaks attract significant public attention and consume resources that cannot be fully demobilised until the outbreaks are declared over. Largescale outbreaks and associated control interventions can have a profound impact on lives and livelihoods, whether the implicated disease is one of humans, animals, or plants.^{1} As such, it is in the public interest to be able to declare an outbreak over as soon as possible. However, such declarations, if premature, may undermine earlier, effective control measures and result in a resurgence of cases. Declaring the end of an outbreak based on rigorous quantitative methods helps to ensure that a valid risk assessment has been carried out and that there is only a limited risk of a false declaration, while avoiding keeping interventions in place long after transmission has ceased.
To understand how outbreaks end, it is first important to understand how outbreaks are defined.^{2} Outbreaks occur when the number of cases of a disease exceeds the expected range. The upper range of the expected number of cases can form the basis for an “epidemic threshold” that, when exceeded, signals the need for an urgent public health response. If a disease is endemic (consistently found in a region or population under normal conditions), the epidemic threshold may be determined by historical observations. If a disease is nonendemic (no sustained transmission in a region or population) or newly emerged, the epidemic threshold may be simply “more than zero” cases (for outbreakprone diseases) or evidence of ongoing transmission (for diseases not prone to outbreaks). In this article, we use the term “disease” interchangeably with “pathogen” (for example, we use either of the phrases “disease transmission” or “pathogen transmission” to describe spread of the causative agent, and we refer to either the “disease” or “pathogen” being endemic).
As shown in Figure 1, outbreaks can be thought of as occurring in cycles, with the time between cycles—during which the disease is either at endemic levels or eliminated—varying by disease, location, and other related factors. In situations in which a disease is endemic, cyclicality is often seasonal, as is the case for Lassa fever in West Africa, tickborne encephalitis in parts of Siberia, and seasonal influenza.^{3} In nonendemic situations, outbreaks only arise when the disease is introduced or reintroduced, which may occur due to importation or spillover from an animal reservoir.^{4}
Endemic status is often ascribed according to geographic boundaries (that is, country or province). However, it should be noted that incidence is usually unequally distributed within these boundaries, and this can lead to stretches of zero cases being reported in some regions even for endemic diseases due to patterns of localised extinction.^{5} The use of the term “extinction” in mathematical models does not describe a permanent state, but rather is used to express even shortterm decreases in cases to zero (Table 1). In contrast, nonendemicity is often described in terms of epidemiological elimination, which represents a locationspecific longterm or permanent reduction of incidence to zero cases (that is, no pathogen transmission). At a global level, a challenging (and generally improbable) aim may be eradication, which refers to a scenario in which a pathogen no longer exists in hosts anywhere in the world. Following eradication, continued actions to prevent transmission are no longer required. In contrast, for diseases that have been eliminated (but not eradicated), continued actions may be necessary to prevent substantial pathogen transmission from restarting (for example, monitoring of inbound travellers for infection to reduce the risk of pathogen reimportation).^{6} Elimination may also be defined in terms of “elimination as a public health problem,” which typically indicates reaching a low level of prevalence rather than sustaining zero cases. This is important for many pathogens, as it is often improbable to reach zero cases for an extended period of time, with factors such as numerous possible host types, varying modes of transmission, or difficulties in implementing surveillance and control measures decreasing the chance of complete elimination.
Establishing that an outbreak is over is often less clearcut than confirming that an outbreak has started. A dip in daily reported cases below the epidemic threshold (to endemic levels or to zero cases) does not necessarily mean that the outbreak is over, as cases could soon return to epidemic levels. This was evidenced by the 2013–2016 Ebola virus disease (EVD) epidemic in West Africa, when new cases appeared shortly after five occasions on which the outbreak was declared over in three of the countries involved.^{7} Instead, it may be necessary to define an analogous “extinction threshold” that, rather than being defined based on a simple rule about numbers of reported cases, is instead established using quantitative analyses that aim to assess the probability that the outbreak is over (according to a specified definition of an outbreak end—see Table 1). When the extinction threshold is met, an outbreak can be declared over with confidence. For example, an outbreak might be declared over on the first date on which, according to a mathematical model, the probability that cases occur in the future (as a direct result of recent transmission) is estimated to fall below a prespecified threshold.
Predicting whether or not an outbreak has ended, or when an outbreak will end, is particularly complicated in the case of a recently emerged disease such as COVID19, as there is limited historical data to inform predictions. In fact, as of mid2021, more than a year and a half after the first COVID19 cases were detected, it remains to be seen whether the virus that causes COVID19 will be eliminated longterm in any given location or whether it will instead become endemic worldwide. If the disease burden of COVID19 becomes comparable to or lower than that of other existing infectious diseases, such as seasonal influenza, it may become accepted as endemic and local epidemic thresholds established.
Avoiding both false declarations of outbreak ends and unnecessary waiting times prior to endofoutbreak declarations has been a recent focus of some infectious disease modelling efforts.^{8} Improving the timing of endofoutbreak declarations helps conserve public health resources and reduce pressure on healthcare, social, and economic systems. This review summarises quantitative methods used to infer when infectious disease outbreaks are over from a modernday perspective, with some historical context interspersed throughout. In the following sections, we first introduce the concept of establishing freedom from disease. Then, we introduce epidemiological modelling methods used to estimate outbreak end times. Lastly, we consider scenarios in which these methods are applied and the factors affecting endofoutbreak estimates.
In the wake of an infectious disease outbreak, it can be important to establish when a host population is free from infection by the outbreakcausing pathogen. This is not only important for diseases of humans but is also relevant in the context of animal and plant diseases, as trade regulations commonly require certification of disease elimination in locations where trade products originate to protect global biosecurity.^{9} In fact, the concept of freedom from disease emerged in the field of veterinary epidemiology due to the need to establish the absence of infection when trading livestock and meat.
To guarantee that a disease has been eliminated from a host population, it is necessary to assess all members of that population for infection, and for that assessment to be perfectly accurate. Unfortunately, exhaustive assessment is usually rendered unfeasible by financial and logistical constraints. Instead, screening for disease is commonly performed on a random sample of individuals from the population. The results from that sample are then used to infer the prevalence of disease (the proportion of individuals that are currently infected) in the whole population.
Identifying an infected individual in a selected sample confirms that total disease elimination has not been achieved. However, the inverse is not true; finding no infected individuals in the sample does not guarantee that the disease is absent from the population. It is possible that the disease is present, but that by chance only uninfected individuals were included in the sample. Although the absence of evidence of the disease is not proof of the disease's absence, it does give us some information about its plausible maximum prevalence in the population, since observing no infections in the sample is unlikely to be consistent with a very high prevalence.
For example, if a sample of individuals are randomly chosen from the population and all are assessed to be diseasefree, there is a simple rule—the “rule of three”—that provides an estimate of how many individuals in the entire population could be infected.^{10} The rule of three states that, in that scenario, we can say with 95% confidence that the maximum proportion of the whole population that could be infected is . This rule can be derived straightforwardly. Specifically, suppose that the true (unknown) prevalence of disease in the population is . The probability that none of a sample of randomly selected individuals are infected is . If this probability is very small, it is unlikely that we would observe no infections in the sample. We choose a value as a threshold for credibility, so that if , the chance of observing no cases in our sample is “too unlikely” to be believable.
Therefore, if indeed we observe no infections in the sample, a plausible value of is one that gives rise to this observation with probability no less than . That is, must satisfy the equation , which can be rearranged to obtain the upper bound . By convention, the credibility threshold is usually taken to be (the standard threshold for statistical significance in many scientific disciplines). For this value of , the upper bound for may be approximated by . Any prevalence higher than this would make the likelihood of observing no infected cases in our sample less than 5%. Thus, given that our sample returns a positive infection rate of , we can say with 95% confidence that the maximum “true” prevalence that is consistent with this observation is .
The “rule of three” is an example of what is known as a confidence interval: a range of plausible values of a parameter (in this case ), that contains the true value with a given probability, in this case 95%.^{11} Usefully, this rule can also be inverted to obtain the minimum sample size for which observing no cases establishes with 95% confidence that the disease prevalence is at most . This lower bound on is given by , where represents the ceiling function that rounds a value up to the nearest integer. For example, to be 95% sure that the prevalence is no greater than 1%, we must take a sample of size at least (that is, ) and fail to find the disease.
In practice, observing no positive cases amongst sampled individuals is not necessarily a requirement for declaring freedom from disease. For example, the criterion for a memberstate of the European Union to be declared free from bovine tuberculosis is that the prevalence must remain below 0.1% for 6 consecutive years, which allows for a small number of cases to be present in each annual sample. Reasoning analogous to the above derivation may be applied to estimate confidence intervals for the prevalence in scenarios in which cases have been observed in a sample of size . This allows policymakers to determine whether or not freedom from disease has been achieved in the sense of “elimination as a public health problem.” (as defined by a specific criterion about the estimated prevalence).
Although the simplicity of the rule of three (and confidence intervals constructed as described above) is appealing, it relies on several assumptions. For example, it assumes that the screening method applied to detect disease never gives any false positive results (healthy hosts identified as infected) or false negative results (infected hosts identified as healthy), which is unrealistic in practice. It also assumes that the population is large enough that the probability of selecting an infected individual at any point in the sample remains approximately constant, regardless of the disease status of those already chosen. However, the underlying conceptual framework may readily be extended to overcome these assumptions, allowing for imperfect testing methods and populations of any size.^{12}
Similar methods may also be used to evaluate more complex sampling strategies. For example, when attempting to establish whether livestock are diseasefree at a regional or national level, it is often impractical to implement uniform random sampling from the entire population due to the number of individuals involved. Additionally, doing so would not account for the fact that disease tends to cluster within herds rather than being evenly distributed throughout the population. Instead, twostage sampling frameworks are often implemented in practice, in which the first stage is to select herds for testing and the second is to select individuals from within those herds. In that context, probabilistic techniques may be applied to establish the optimal division of sampling resources between and within herds in order to maximise our confidence that the overall prevalence is below an acceptable level.^{13}
Since samples taken at a single point in time do not incorporate information gained from previous sampling rounds, analogous techniques may be applied to construct confidence intervals for the underlying prevalence based on sampling performed at multiple time points. These models may be designed to account for changes in the underlying prevalence over time, to consider the incubation period of the disease (the time during which individuals are infected but presymptomatic), or to consider the risk that the pathogen has been reintroduced into the population from elsewhere.^{14}
As noted above, although establishing the elimination of a disease from a population with 100% certainty is almost always unfeasible, probabilistic techniques may be used to guide statistically sound surveys that allow us to determine with a given level of confidence that disease prevalence is below a given threshold. In addition to diseases of livestock such as bovine tuberculosis, footandmouth disease, and Bluetongue, these methods have been applied in the context of human diseases including Zika, parasitic infections, and COVID19.^{15}
For populations in which samples of randomly chosen individuals are not tested for infection, an alternative approach is required in which the probability that a population is diseasefree is assessed based on the time elapsed since the last observed case, as described in the following two sections.
Guidance developed by the World Health Organization (WHO) for declaring the official end of an outbreak for diseases such as EVD and Lassa fever indicates that an outbreak can be considered over if twice the maximal incubation period has elapsed since the most recent case.^{16} However, the timing of an endofoutbreak declaration cannot be optimized using a fixed length of time. If twice the maximal incubation period has elapsed, the probability that the outbreak is truly over depends on the characteristics of both the disease and the specific outbreak being considered. Infectious disease transmission is dynamic, and given the multitude of possible outbreak trajectories, efficient and accurate endofoutbreak declarations require a method that can dynamically determine the end of an outbreak using information about the outbreak’s characteristics and causal pathogen. One such approach is a statistical method that offers an interpretable estimate of the endofoutbreak probability based on the probability of observing additional cases after the current time. This approach uses the offspring distribution (describing the numbers of secondary cases infected by each infected individual) and the serial interval (describing the time from illness onset in an infected individual to illness onset in a case they infect), and was applied during the outbreak of Middle East respiratory syndrome (MERS) in South Korea in 2015.^{17} In that study, the declaration of the end of a MERS outbreak was recommended when the estimated probability of observing additional cases in future falls below a prespecified threshold value.
More specifically, using the dataset of onset dates for diagnosed cases () for cases , the probability of observing at least one additional case in the future (as estimated at time ) can be approximated by the expression:
Here, is the offspring distribution, which describes the probability that cases are infected by a given primary case, while represents the cumulative distribution function of the serial interval.
This methodology was later extended to consider endofoutbreak probabilities for other diseases, such as EVD and COVID19, as well as to account for the delay between the illness onset of a case and when they are reported in surveillance data. Accounting for reporting delays involves a straightforward extension of the equation above by combining the serial interval and reporting delay distributions.^{18}
Furthermore, the method was adapted to account for multiple modes of transmission by Lee & Nishiura (2019), who considered the potential for both sexual and nonsexual transmission of Ebola virus in the context of the end of the 2013–2016 EVD outbreak in West Africa. Ebola virus can be detected in semen long after illness onset, and sexual transmission increases the risk of reemergence of the virus, as potentially evidenced by the appearance of new cases in West Africa after the virus was believed to have been eliminated.^{19} To account for both modes of transmission, Lee & Nishiura introduced a probability distribution characterising the length of the serial interval using distributions of the serial intervals for each mode of transmission, with the distribution for sexual transmission based on the survival probability of Ebola virus in semen as a function of the time since illness onset.^{20} The authors found that the optimal time at which to declare the outbreak over varies based on the relative frequency of sexual and nonsexual transmission, as well as the level of underreporting of cases. If there is a substantial amount of sexual transmission, a long period is required without observing any new cases for the outbreak to be declared over with confidence.
Transmission of many pathogens can be highly heterogeneous between infectors, with some infectors transmitting the pathogen to many individuals and other infectors transmitting the pathogen to few individuals.^{21} The potential for superspreading has been characterised by the “80/20 rule”, whereby 20% of infected individuals are said to generate around 80% of infections.^{22} Statistical models for assessing the endofoutbreak probability typically account for this superspreading potential. A higher propensity for superspreading acts to increase the time until the end of the outbreak can be declared with confidence.^{23}
A number of recent publications have built on the statistical approach described here.^{24} Specifically, methods have been developed based on renewal equations in which the numbers of cases each day follow a Poisson or negative binomial distribution. Using renewal equations, the probability of cases arising each day in the future can be approximated using model simulations (or calculated analytically), based on the observed diseaseincidence time series up until the current time. For those models, the probability that an outbreak is over based on incidence data up to time t is simply the proportion of forward simulations in which no cases occur after time t. Although these approaches are based on the one by Nishiura, Miyamatsu, & Mizumoto (2016), there are subtle differences in the underlying assumptions, and a quantitative comparison of endofoutbreak probabilities obtained using these different methods remains a target for further research.
The epidemiological models described in the previous section track the number of new cases arising each day. However, the dynamics of infectious disease outbreaks can be complex, with individuals transitioning through a range of infection or symptom states. For that reason, a commonly used mathematical modelling framework is compartmental modelling, in which individuals are categorised over the course of an outbreak according to their infection or symptom status.^{25}
Compartmental epidemiological models have a long history. One of the most basic compartmental models is the SusceptibleInfectedRemoved (SIR) model (Figure 2a), which is a special case of the epidemiological model considered by Kermack and McKendrick (1927). In the SIR model, individuals are classified as (S)usceptible to the outbreak pathogen, (I)nfected and generating new infections, or (R)emoved and no longer generating new infections. As the outbreak progresses, individuals who are susceptible may become infected (and transition from the S compartment to the I compartment), and then subsequently recover or die (and transition from the I compartment to the R compartment). By choosing the parameters of the model (in the SIR model, the infection rate and removal rate parameters) appropriately, output from compartmental models can be tuned to match realworld data from an ongoing outbreak.^{26}
Compartmental models can be categorised into two complementary groups—either deterministic models or stochastic models.^{27} Deterministic models, often represented as systems of ordinary differential equations, generate the same results every time for a specific set of inputs (for example, infection and removal rate parameters, and initial numbers of individuals in each compartment of the model). Stochastic models, on the other hand, reflect the intrinsic and extrinsic randomness inherent in realworld epidemiological systems, with repeated simulations of the model generating different outbreaks, even when the inputs are identical. A simulation of the stochastic SIR model can be thought of as a series of coin tosses, with the result of each coin toss determining whether the next event is an infection event (with an individual in the S compartment transitioning to the I compartment) or a removal event (with an individual in the I compartment transitioning to the R compartment). When considering outbreak extinctions, randomness in when precisely the pathogen goes extinct is important in determining the confidence in an endofoutbreak declaration. For that reason, our focus here is entirely on stochastic, rather than deterministic, epidemiological models.
A key benefit of compartmental models is that they can be extended straightforwardly to include different features that affect transmission. For example, a common extension to the SIR model is to include a time delay between each individual being first infected and becoming infectious. This time delay is termed the latent period and varies between pathogens. For example, the latent period for influenza is typically in the range of 1–3 days, whereas the latent period for measles is around 8–13 days.^{28} A latent period can be included in the SIR model by inserting a new compartment—the (E)xposed compartment—between the S and I compartments.^{29} Individuals in the E compartment are infected but not yet infectious. Other possible features that can be included in compartmental models, and may be appropriate when modelling outbreaks of certain pathogens, are age structure, spatial structure, withinhost dynamics, and transmission via insect vectors.^{30}
In the context of determining whether or not an outbreak has finished, a crucial question is whether the outbreak is over or whether it may still be ongoing in individuals who are not reporting disease (see Section 2, above). For outbreaks in populations of humans, a failure to report disease might arise due to some infected individuals being asymptomatic or showing only limited symptoms, or because individuals are recovering at home without reporting their infection to local health authorities.^{31} In either scenario, some infected individuals may not appear in routinely collected surveillance data. To establish whether or not an outbreak has finished, an important extension to the SIR model described above is therefore to include the possibility that infected individuals do not report disease. This can be done by including separate compartments in the model for infected individuals that do and do not report disease (Figure 2b).
Simulations of stochastic compartmental models of the type shown in Figure 2b can be used to determine the confidence that an outbreak is over, based on the time period since the last case was reported. Specifically, a large number of model simulations can be run, with each simulation generated until the number of infected individuals reporting disease reaches zero. Each simulation can then be continued until a period of X days has passed since an infectionreporting individual was present in the population. The proportion of those simulations in which no unreported infected individuals remain after a time period of X days without reported infections is then a proxy for the probability that the outbreak is over after that period.
The simulation approach described above was applied to study the confidence in endofoutbreak declarations for EVD.^{32} The WHO considers EVD outbreaks to be over at the national level once a time period of 42 days (twice the maximal incubation period) has elapsed since the latest reported case. However, the situation is complicated by several features of EVD outbreaks (see Section 2and Figure 1). First, the virus may be reimported from elsewhere, with local transmission restarting even after the virus has been eliminated locally.^{33} Second, some EVD survivors may be infectious long after they were first infected, again potentially leading to local resurgence after the outbreak appears to have finished.^{34} Third, as described above, reporting is imperfect, and so chains of transmission may persist undetected during the 42 days following the most recent reported case. Thompson et al. (2019) considered this third feature and showed that the confidence in an endofoutbreak declaration is extremely sensitive to the level of underreporting. This led to a target of 79% case detection for EVD outbreaks to be declared over with 95% confidence after a time period of 42 days without reported cases.^{35} The finding that the confidence in an endofoutbreak declaration is sensitive to the level of reporting was echoed in a later analysis by Djaafara et al. (2021) using renewal equations (see Section 2), in which those authors advocated replacing the 42day period with a longer period of 63 days.
A similar approach to the one described above has also been used in the context of polio eradication. Specifically, Eichner and Dietz (1996) considered a synthetic population of 200,000 individuals on the pathway to polio eradication, and demonstrated that the duration of the casefree period must be at least 3 years before a policymaker can be 95% sure that local extinction has occurred. At the global level, polio eradication remains a key challenge, and epidemiological models can be used to assess different public health policies.^{36}
To summarise, compartmental models involve separating individuals according to their infection or symptom status. Extensions to these models (to include, for example, a latent period, age structure, or any number of other epidemiological realisms) involve adding more compartments. The models can be matched to realworld outbreak data by adjusting the values of the parameters governing transmission. Simulations of compartmental models including underreporting can then be used to assess the confidence that an outbreak is over, in a similar fashion to simulations of renewal equations, based on the time period since the last case was observed.
Throughout this review article so far, we have focused on the question of determining whether or not a population is diseasefree at the current time. However, interventions are often planned while an outbreak is ongoing, and so estimating when an outbreak will end during the outbreak is another important area of research.
The timevarying reproduction number, , is often used to indicate whether or not an outbreak is on the path to extinction.^{37} It describes the expected number of secondary cases generated by an infected individual in the population at time . If is (and remains) below one, the disease is unable to sustain itself without repeated introductions and the outbreak will eventually end. However, even once the threshold of is crossed (so that ), although extinction could be inevitable, it is not immediate and control measures may need to be maintained for extinction to occur. Providing stakeholders with the expected remaining duration of an outbreak can help estimate future costs required for controlling the disease. Towards the end of an outbreak, it is therefore important to assess how long the delay will be before extinction occurs.
Stochastic models (see Section 3) can be used to estimate the expected time until extinction given an initial observation of the number of infected individuals in the population. This period depends on the disease dynamics, control measures in place, and the population size. Estimation of the time to extinction was first considered for ecological systems with stochastic models describing deteriorating population dynamics.^{38} Studies with synthetic infectious disease data have demonstrated the capability to approximate the expected time to extinction for both simple epidemiological models (for example, the SIR model) and more complex models with spatial structure, vaccination dynamics, and hostvector transmission.^{39}
The expected time to extinction is typically stated as a function , of the current observed number of infected individuals, , and can be found by solving a set of simultaneous equations. The coefficients of the equations are constructed by considering the possible transitions between compartments. For example, the schematic and equation in Figure 3 describe a stochastic SusceptibleInfectedSusceptible (SIS) model. Figure 3a illustrates how, as the next event, the number of infectious individuals can either increase to (a susceptible individual is infected) or decrease to (an infected individual recovers). For the example in Figure 3, the total number of individuals in the population () is assumed to be constant, and new infections occur at rate , where the parameter governs the rate of transmission between infectious and susceptible individuals. Each infected individual recovers from the infection at a rate , where is the average amount of time that an individual spends infectious before recovering. The infection process will continue until no one in the population is infected and disease extinction is reached (leftmost box in Figure 3a).
The formula shown in Figure 3b indicates that the expected time to extinction from infected individuals depends on the time until the next event (first term), the probability that the next event is a recovery (second term), and the probability that the next event is an infection (third term). These probabilities relate the expected time to extinction, , to the subsequent expected extinction time, which could be or depending on whether the next event is a recovery or an infection. This equation can be simplified by assuming that, when near extinction, very few individuals in the population will be infected and so the number of susceptible individuals is approximately equal to the population size, . Generally, following some rearrangement, the coefficients of this equation can be written as a matrix , where each row of the matrix, , contains the coefficients in the equation for . Finding the extinction times given the current number of infected individuals involves solving .^{40} The matrix is commonly referred to as the transition matrix conditioned on nonextinction, and can be constructed for models with more complex disease dynamics, such as those including latent periods or waning immunity.
As well as the expected time until extinction occurs, understanding the distribution of possible extinction times is important to account for uncertainty in model predictions, and can be used for more indepth analyses, such as finding the date by which there is a 90% chance of extinction.^{41} Stochastic models may appear to fluctuate around an endemic steady state for a long time before extinction is reached. For many epidemiological models, the time to extinction starting from that steadystate value is exponentially distributed.^{42} However, approximating the distribution of possible extinction times can be challenging in some scenarios, such as when the population size is large.^{43} The distribution of extinction times starting from infected individuals can sometimes be estimated using higher moments of .^{44}
An example demonstrating how the remaining duration of an outbreak, after it has been brought under control (), can be estimated using the stochastic SIS model is shown in Figure 4. We ran 10,000 stochastic simulations of the SIS model and recorded the time to extinction in each simulation (the first time that there were zero infected individuals in the population), presented in Figure 4b. The mean extinction time from the equation in Figure 3b is plotted onto the histogram, and we observe that it aligns with the simulation results.
In summary, if but the outbreak has not yet faded out, the mean time to extinction can be estimated, providing an approximation of the remaining duration of the outbreak. The distribution of possible times to extinction can provide more nuanced information about the possible duration of the outbreak remaining. The methods described here are suitable for use with prevalence data (describing the total number of individuals currently infected). However, these data are often unavailable, so future work should consider the development of methods to estimate the time to extinction from incidence data (describing the number of new cases each day). The application of these future methods to predict outbreak extinction dates for realworld pathogens has the potential to inform outbreak responses.
In the previous sections, mathematical approaches for inferring whether or not an outbreak is over and projecting the timing of the outbreak end were reviewed. These quantitative methods provide insights into the uncertainties surrounding outbreak decline and extinction (and, therefore, the confidence levels with which outbreaks can be declared over). However, there are many factors that can affect the accuracy of these estimates. These include data reliability, modes of pathogen transmission, parameter selection and inference, and the precise definition of the host population within which the outbreak is occurring.
As mentioned in Section 3, outbreaks can be declared over most quickly and with highest confidence when disease surveillance systems are highly sensitive and able to detect infectious cases accurately and promptly.^{45} However, the sensitivity and timeliness of surveillance can vary greatly from location to location, as well as by disease, so it is necessary to account for underascertainment of cases and reporting delays. Failing to do so can lead to erroneous endofoutbreak declarations.^{46} Factors that may affect the sensitivity of a human, animal, or plant disease surveillance system include: the level of symptoms associated with infections (for example, asymptomatic or mild infections may be less likely to be detected), failures to seek healthcare, veterinary or agricultural services, inaccurate diagnoses, falsenegative tests, and failures to report cases to public health or other authorities.^{47}
To obtain an accurate picture of when an outbreak will end for a given disease, it is necessary to understand and account for epidemiological differences associated with different modes of transmission. These differences can affect the estimates generated by the methods described here. During the 2013–2016 EVD outbreak in West Africa, the monthslong persistence of the Ebola virus in the semen of some men who recovered from EVD was not originally considered in analyses of when to declare that EVD outbreak over (see Section 2). This led Lee and Nishiura (2019) to explore the use of an epidemiological model that accounts for sexual contacts and survival of the Ebola virus in semen to generate more realistic endofoutbreak assessments.
The values of the parameters of models used to inform endofoutbreak declarations must be considered carefully. For example, the models presented in Section 2 use the serial interval or generation time (which characterises the times between successive infections) to inform estimates. However, the serial interval and generation time can change over the course of an outbreak.^{48} Failing to consider this can result in incorrect assessments of the endofoutbreak probability. Likewise, other parameters governing pathogen transmission, which may be included in the compartmental models presented in Section 3, can vary by host age, pathogen variant, and host immunity (which is affected by vaccination status and past infections), among other factors.^{49} The size of the susceptible population must also be quantified accurately, since sustained pathogen transmission requires the susceptible population size to be sufficiently large.^{50} When deciding which parameters to include in epidemiological models for analysing endofoutbreak dynamics, it is necessary to balance epidemiological realism with tractability and interpretability. The model parameters to include, and the methods used to estimate epidemiological parameter values, require careful consideration.
Most mathematical approaches used for studying epidemiological dynamics at the ends of outbreaks have considered outbreaks as occurring with fixed geographical or administrative boundaries. However, pathogens do not observe these boundaries, and not all outbreaks have a clear geographical scope.^{51} Foodborne disease outbreaks, for example, can cross international borders via food distribution chains, and plant and animal disease outbreaks often follow domestic or international trade networks.^{52} For outbreaks of directly transmitted pathogens, such as the ongoing COVID19 pandemic, outbreaks may likewise cross geographical boundaries, with transmission facilitated by domestic and international travel. Outbreaks can be considered at a number of different spatial scales: for example, a national outbreak or a cluster of cases in a specific setting such as a hospital, prison, or care home.^{53} The optimal scale at which to define the host population when assessing endofoutbreak probabilities depends on factors including the patterns of exposure: for example, for pointsource outbreaks, when exposure is to a single source of infection over a short period of time, it may only be necessary to include attendees at the exposure event in the population under consideration.^{54}
While a suite of approaches exist for analysing the ends of outbreaks, it should be emphasised that the performance of these methods has yet to be assessed rigorously in a range of outbreak response scenarios. A comparison of different approaches, as well as scientific assessments of their relative accuracy and reliability, should be explored systematically through theoretical studies and practical applications during and after outbreaks. While we have focused mainly on approaches relating to local extinction or elimination, quantitative methods for use in scenarios in which the outbreak end represents reaching or returning to endemicity require particular attention. The utility of different approaches from a public health and economic viewpoint must be assessed, and the criteria to use for declaring outbreaks over should be considered carefully. A key component of this decision is the potential consequence of an incorrect endofoutbreak declaration, which must be assessed with a multidisciplinary perspective.
The accurate determination of when outbreaks end allows enhanced surveillance activities and public health interventions to be relaxed safely.^{55} In this article, we have introduced various epidemiological modelling approaches that can be used to infer whether or not, and when, an outbreak has ended. Improving the accuracy of endofoutbreak determinations based on modelling studies is best accomplished by: a) Strengthening surveillance systems, so that cases are found and reported accurately; and b) Supporting research that leads to an improved characterisation of pathogen transmission in the later stages of outbreaks, including identification and estimation of relevant transmission parameters.
However, endofoutbreak declarations must not solely be the jurisdiction of public health advisors or policymakers and epidemiological modellers. It is important that healthcare capacity and socioeconomic considerations are also accounted for when decisions are made. If the risk to human health is low (that is, an incorrect endofoutbreak declaration is only likely to lead to a small number of severe future cases), but the societal costs of outbreak interventions are high, it may be best to declare outbreaks over when the confidence in an endofoutbreak declaration is relatively low. On the other hand, if the risk to human health of outbreak resurgence is particularly high (for example, if healthcare systems may be overwhelmed), it may be preferred to set the threshold confidence for declaring an outbreak over to be much higher. Irrespective of a policymaker's desired level of risk aversion, quantitative approaches for studying the ends of infectious disease outbreaks will remain important for public health decisionmaking whenever pathogens near elimination. It is essential that these methods continue to be developed and applied.

Emerging disease  A disease with incidence that has recently (for example, in the past two decades) increased substantially or threatens to increase in the near future.  
Epidemic  A sudden increase in the occurrence of cases of a disease in excess of normal expectancy for the location or season. This is sometimes also the definition of an outbreak, although the term outbreak is used more often when the geographical range is limited.  
Endemic  When a disease is consistently present in a region or population.  
Elimination (as a public health problem)  Achievement of measurable global targets in reduction of incidence or prevalence set by WHO. When reached, continued actions are required to maintain the targets and/or to advance the interruption of transmission.  
Elimination (of transmission)  Reduction of incidence of a given disease to zero in a defined location and for a defined minimum period of time as a result of deliberate efforts with minimal risk of reintroduction.  
Eradication  Permanent worldwide reduction of incidence of a given disease to zero as a result of deliberate efforts.  
Extinction  Shortterm disappearance of a disease in a defined location or population. This term is frequently used in statistical endofoutbreak analyses. 
^{1} For examples of each, see Ebata, Nisbett, & Gillespie (2021); Sonaiya (2007); Thiermann (2004); R. N. Thompson & BrooksPollock (2019).
^{2} Historiographical article on how outbreaks end: Charters & Heitman (2021). For definitions of different endpoints, see Dowdle (1998) and Table 1.
^{3} For an example for Lassa fever, see Lo Iacono et al. (2016); for tickborne encephalitis, see Korenberg (2000); for seasonal influenza, see Dalziel et al. (2018).
^{4} Examples of importation events: R. N. Thompson (2020); R. N. Thompson, Thompson, Pelerman, Gupta, & Obolski (2018); Wilson (2005). Examples of spillover from an animal reservoir: Borremans, Faust, Manlove, Sokolow, & LloydSmith (2019); LloydSmith et al. (2009); Plowright et al. (2017); Ponce, Kinoshita, & Nishiura (2019).
^{5} Bartlett (1957); Eichner & Dietz (1996); Keeling & Grenfell (2002); Lindholm & Britton (2007); Britton & Neal (2010); Holme (2013).
^{6} SachakPatwa, Byrne, Dyson, & Thompson (2021).
^{7} Lee & Nishiura (2017).
^{8} For example, Djaafara et al. (2021); Nishiura, Miyamatsu, & Mizumoto (2016); Parag, Donnelly, Jha, & Thompson (2020); R. N. Thompson, Morgan, & Jalava (2019).
^{9} See Caporale, Giovannini, & Zepeda (2012); Food and Agriculture Organization (2017); Rüegg et al. (2018); Tratalos et al. (2018).
^{10} Hanley & LippmanHand (2008); Jovanovic & Levy (1997); Louis (1981).
^{11} Hazra (2017).
^{12} Cameron & Baldock (1998a); Cannon (2001); Johnson, Su, Gardner, & Christensen (2004); Nishiura et al. (2016).
^{13} Cameron & Baldock (1998b); Cannon (2001); Rüegg et al. (2018).
^{14} Bourhis, Gottwald, LopezRuiz, Patarapuwadol, & van den Bosch (2019); Bourhis, Gottwald, & van den Bosch (2019); More et al. (2009).
^{15} Bovine tuberculosis: More et al. (2009). Footandmouth disease: Caporale et al. (2012). Bluetongue: Rüegg et al. (2018). Zika: C. N. Thompson et al. (2018). Parasitic infections: Michael et al. (2018). COVID19: Foddai, Lubroth, & EllisIversen (2020); Larsen et al. (2021).
^{16} Hersey et al. (2015).
^{17} Nishiura et al. (2016).
^{18} Linton, Akhmetzhanov, & Nishiura (2021b).
^{19} Deen et al. (2017); Lee & Nishiura (2017).
^{20} For more information about the persistence of Ebola virus in semen, see Eggo et al. (2015).
^{21} LloydSmith, Schreiber, Kopp, & Getz (2005). For COVID19 examples, see Adam et al. (2020); Tariq et al. (2020); Zhao, Zhang, & Li (2020).
^{22} Woolhouse et al. (1997).
^{23} Linton et al. (2021b).
^{24} Djaafara et al. (2021); Parag (2021); Parag, Cowling, & Donnelly (2021); Parag, Donnelly, Jha, & Thompson (2020).
^{25} Brauer (2008); R. N. Thompson (2020).
^{26} Chowell (2017).
^{27} Keeling & Rohani (2008).
^{28} Anderson & May (1991).
^{29} Anderson & May (1991); Bolker & Grenfell (1995); Chowell, Nishiura, & Bettencourt (2007); R. N. Thompson, Gilligan, & Cunniffe (2016).
^{30} Age structure: Davies, Kucharski, et al. (2020); Prem et al. (2020). Spatial structure: Bolker & Grenfell (1995); R. N. Thompson, Thompson, et al. (2018). Withinhost dynamics: Hart, Maini, Yates, & Thompson (2020); Mideo, Alizon, & Day (2008). Transmission via insect vectors: Allen et al. (2019); Kucharski et al. (2016); R. N. Thompson, Gilligan, & Cunniffe (2020).
^{31} Angulo, Finelli, & Swerdlow (2021); Cori et al. (2017); Gignoux et al. (2015).
^{32} R. N. Thompson, Morgan, & Jalava (2019).
^{33} Weah et al. (2017).
^{34} Diallo et al. (2016); Keita et al. (2016); MacDermott & Bausch (2016).
^{35} R. N. Thompson, Morgan, & Jalava (2019).
^{36} See K. M. Thompson & Kalkowska (2020; 2021).
^{37} For more on , see Cori, Ferguson, Fraser, & Cauchemez (2013); Nishiura & Chowell (2009); R. N. Thompson, Stockwin, et al. (2019).
^{38} See Giles Leigh (1981); MacArthur & Wilson (1967).
^{39} SIR model: Barbour (1975). Spatial structure: Britton & Neal (2010); Lindholm & Britton (2007); Swinton (1998). Vaccination dynamics: Andersson & Britton (2000). Hostvector transmission: Aliee, Rock, & Keeling (2020); Britton & Traoré (2017).
^{40} See Keeling & Ross (2008).
^{41} Aliee et al. (2020).
^{42} SIS dynamics: Mangel & Tier (1993). SIR dynamics: Andersson & Britton (2000); Nåsell (1999). Hostvector dynamics: Britton & Traoré (2017).
^{43} Doering, Sargsyan, & Sander (2005); Nåsell (1999).
^{44} Aliee et al. (2020).
^{45} R. N. Thompson, Morgan, & Jalava (2019).
^{46} Akhmetzhanov, Jung, Cheng, & Thompson (2021); Linton et al. (2021b); Parag et al. (2020).
^{47} Gibbons et al. (2014). Poliovirus can persist for years within a population as a “silent” infection, since most of those infected do not show symptoms and are never diagnosed. Eichner & Dietz (1996) estimated how many years of zero case incidence would need to pass before it would be possible to be 95% certain that local extinction of wild poliovirus infection had occurred.
^{48} Noted for SARS by Lipsitch et al. (2003). Later investigated in greater detail by Kenah, Lipsitch, & Robins (2008), as well as Nishiura (2020). Analyses considering COVID19 include Ali et al. (2020) and Linton, Akhmetzhanov, & Nishiura (2021a); Hart, Abbott, et al. (2022); Hart, Miller, et al. (2022).
^{49} Davies, Klepac, et al. (2020); Teunis, Le Guyader, Liu, Ollivier, & Moe (2020).
^{50} Keeling & Grenfell (1997).
^{51} R. N. Thompson, Cobb, Gilligan, & Cunniffe (2016).
^{52} For more on foodborne disease, see Coulombier & Takkinen (2013). For more on plant and animal outbreaks, see Wilkinson et al. (2011).
^{53} For examples of estimating endofoutbreak probabilities using case clusters, see Linton et al. (2021b). For examples of cluster settings, see Furuse et al. (2020).
^{54} Brookmeyer & You (2006).
^{55} The definition of elimination introduced by Dowdle (1998) embraced the need for continued interventions to prevent reemergence and reestablishment of transmission. However, as pointed out by Heymann (2006), all too often a complete cessation of intervention activities follows elimination. Although public health interventions can and should be relaxed once an outbreak ends, some surveillance and continuation of control interventions are necessary to maintain elimination or transmission at endemic levels; these should continue until and unless eradication is achieved.
For historical definitions of eradication and elimination, see Soper (1962) and Dowdle (1998). For a more recent commentary on the definition of eradication, see Arita, Wickett, & Nakane (2004). In practice, elimination of a disease may be defined as elimination of the disease as a public health problem (rather than complete elimination), or in terms of prevalence, rather than incidence. For example, leprosy elimination was defined in 1991 by the World Health Assembly as the reduction of prevalence to a level of < 1 case per 10,000 population. Likewise, tuberculosis elimination as a public health problem has been defined as reduction of prevalence to < 1 case per 1 million population. See also World Health Organization (2015).
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