Infectious disease transmission models

We can mathematically describe the transmission of an infection within a population over time with a renewal process based on the Euler-Lotka equation from ecology and demography [5]. This process models communicable pathogen spread from a primary (infected) case to secondary ones at some time s using two key variables: the effective reproduction number, R_s, and the generation time distribution with probabilities {w_u} for all times u. Here R_s defines the number of secondary cases at time s + 1 one primary case at s infects on average, while w_u is the probability that it takes u time units for a primary case to infect a secondary one [5]. As infection events are generally unobserved, we approximate the time of primary and secondary infection with the corresponding times of symptom onset i.e. the serial interval. This amounts to making the common assumption that the serial interval distribution, which can be observed, is a good approximation to the generation time distribution [2, 12].

If I_s counts the newly observed infected cases at s and a Poisson (Poiss) model is used to represent the noise in these observations then the renewal model captures the reproductive dynamics of infectious disease transmission with I_s ∼ Poiss(R_{s − 1} Λ_s) [4]. Here is the total infectiousness of the disease up to time s − 1 and summarises how previous cases contribute to upcoming cases on day s. We use to represent the incidence curve from time 1 to s. A schematic of this approach to epidemic transmission is given in Fig 1. Usually we are interested in estimating the R_s numbers in real time [21, 22] from the progressing , assuming that the serial interval distribution is known (i.e. derived from some other linelist data) [12].

This effective reproduction number is important for forecasting the kinetics of the epidemic. If R_s > 1 then we can expect the number of infections to increase monotonically with time. However, if R_s < 1 is sustained then we can be confident that the epidemic is being controlled and will, eventually, be eliminated [23]. In order to enhance the reliability of these estimates we usually assume that the epidemic transmission properties are stable over a look-back window of size k defined at time s as τ(s) ≔ {s, s − 1, …, s − k + 1} [12, 24]. We let the reproduction number over this window be R_τ(s) and apply a conjugate gamma (Gam) prior distribution assumption: R_τ(s) ∼ Gam(a, 1∏c) with a and c as shape-scale hyperparameters. This formulation, together with the use of gamma prior distributions, is standard in current renewal model frameworks [12, 21, 25].

The posterior distribution of R_τ(s) given the relevant window of the past incidence curve of data i.e. is also gamma distributed as follows [22] (1) with grouped sums i_τ(s) ≔ ∑_u∈τ(s) I_u and λ_τ(s) ≔ ∑_u∈τ(s) Λ_u. If some variable y ∼ Gam(α, β) then and . As a result, Eq (1) yields the posterior mean estimate of with α_τ(s) ≔ a + i_τ(s), β_τ(s) ≔ 1c+ λ_τ(s). Eq (1) allows us to infer the grouped or averaged effective reproduction number over the window τ(s), which is considered an approximation of the unknown R_s.

We can derive the posterior predictive distribution of the next incidence value (at time s + 1) by marginalising over the domain of R_τ(s) as in [22]. If the space of possible predictions at s + 1 is x|I_τ(s) and NB indicates a negative binomial distribution then we obtain (2)

Eq (2) completely describes the uncertainty surrounding one-step-ahead incidence predictions and is causal because all of its terms (including Λ_s+1) only depend on the past observed incidence curve [22].

If a random variable y ∼ NB(α, p) then and . Hence our posterior mean prediction is . The current estimate of R_τ(s) influences our ability to predict upcoming incidence points. Thus, we expect that good estimation of the effective reproduction number is necessary for projecting the future behaviour of an infectious disease epidemic. In Results we rigorously extend and apply this insight to derive an exact method for computing the probability that an epidemic is reliably over at some time s i.e. that no future infections will occur from s + 1 onwards.

Under-reported and imported cases

The above formulation assumes perfect case reporting and that all cases, , are local to the region being monitored. We now relax these assumptions. First, we consider more realistic scenarios where only some fraction of the local cases are reported or observed at any time. We use N_s and U_s for the number of sampled and unreported cases at time s. We consider a general time-varying binomial (Bin) sampling model with 0 ≤ ρ_s ≤ 1 as the probability that a true case is sampled at time s (hence 1 − ρ_s is the under-reporting probability). Then N_s ∼ Bin(I_s, ρ_s). The smaller ρ_s is, the less representative the sampled curve is of the true .

This is a standard model for under-reporting [8, 26] and implies the following statistical relationship (3)

Raikov’s theorem [27] states that if the sum of two independent variables is Poisson then each variable is also Poisson. Consequently, U_s is Poisson with mean (1 − ρ_s)R_s−1Λ_s. Most studies assume that ρ_s = ρ for all s i.e. that constant under-reporting occurs. The persistence of the Poisson relationship in Eq (3) means that we can directly apply the forecasting and estimation results of the previous section to N_s. Practically, if we observe only then unless we have independent knowledge of ρ_s (which can often be difficult to ascertain reliably [18, 26]) we can only construct an approximation to ρ_sΛ_s as with .

Second, we investigate when imported or migrating cases from other regions, denoted by count M_s at time s, are introduced, resulting in the total number of observed cases being C_s. Within this framework we ignore the under-reporting of cases and assume that I_s is observed to avoid confounding factors. We follow the approach of [9] and describe M_s as a Poisson number with some mean at time s of ϵ_s. Using Raikov’s theorem we obtain (4)

Eq (4) models how imported cases combine with existing local ones to propagate future local infections.

While our work does not require assumptions on ϵ_s, for ease of comparison later on we adopt the convention that the sum of imports and local cases drive the epidemic forward with the same reproduction number and serial interval [28]. Consequently, with . Practically, when surveillance is poor (i.e. local and imported cases cannot be distinguished), it is common to assume that all observed cases are local and conform to the approximate model [25]. The forecasting and estimation results of the previous section therefore also apply under these conditions.

In Results we examine the impact of imperfect (our null hypothesis ) and ideal (the alternative ) surveillance within the context of under-reporting and importation in turn. We treat each problem individually to isolate the impact of each bias. Ideal surveillance then represents the ability to know either U_s or M_s (depending on the problem of interest) and hence account for their contributions. Imperfect surveillance refers to only having knowledge of N_s or C_s and basing inferences on these curves under the strong assumption that they approximate the true incidence. This assumption is often made in the literature [2, 12, 21] for the purposes of tractability and means Eqs (1) and (2) are valid. Fig 1C summarises the relationships from Eqs (3) and (4).

An exact method for declaring an outbreak over

We define an epidemic to be eliminated or over [23] at time s if no future, local or indigenous infected cases are observed i.e. I_s+1 = I_s+2 = ⋯ = I_∞ = 0. We can define the estimated probability of elimination, z_s, as (5) with as the incidence curve (data), observed until time s. We refer to z_s as an estimated probability because we do not have perfect knowledge of the epidemic statistics e.g. we cannot know R_s precisely. The importance of this distinction will become clear in the subsequent section (see Eq (10)). However, we observe that if we could have this idealised knowledge then Eq (5) would exactly define the probability of no future cases given .

Declaring the end of an epidemic with confidence μ% translates into solving the optimal stopping time problem (6) with t₉₅, for example, signifying the first time that we are at least 95% sure that the epidemic has ended. Note that z_s is a function of and practically characterises our uncertainty in the outcome of the epidemic (i.e. if it is over or not). This uncertainty derives from the fact that a range of possible epidemics with distinct future incidences can possess the same and values. Some uncertainty exists even if is known perfectly.

Eq (6) presents an event-triggered approach to declaring the end of an epidemic with the μ threshold serving as an informative trigger. Event-triggered formulations have the advantage of being robust to changes in the observed data [13, 14], a point visible from the dependence of z_s and hence t_μ on . While Eq (6) is written in absolute time, we may also clock time relative to the last observed case, t₀. Our waiting time until declaration is then Δt_μ = t_μ − t₀, which is more useful for comparing z_s values from various realisations of and for deriving confidence intervals. Later, we consider differences in the Δt_μ, denoted δt_μ, proposed by comparable methods.

Previous works on end-of-epidemic declarations have either approximated z_s with a simpler, more conservative probability [2] or used simulations to estimate a quantity similar to z_s that is averaged over those simulations [6] [7]. No study has yet (to our knowledge) included real-time estimates of R_s, within its assessment of epidemic elimination, despite the importance of this parameter in foretelling transmission [23]. By taking the renewal process approach to epidemic propagation (see Fig 1), we explicitly embed uncertainty about R_s estimates to obtain an analytic and insightful expression for the probability that the outbreak is over given past observed cases (Eq (5)).

We derive this by inferring R_s within a sequential Bayesian framework from , using a moving window of length k time units. We denote this estimate R_τ(s) with window τ(s) spanning [12, 22]. Our main result is summarised as a theorem below (see Methods for further details and notation). Fig 2 illustrates how our computed z_s probability varies across the lifetime of an example incidence curve, thus providing a real-time, causal and dynamically updating view of our confidence in its end.

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Fig 2. Elimination probabilities across the lifetime of an epidemic.

We simulate a single incidence curve, I_s (blue, case counts on left y-axis), under the serial interval distribution for Ebola virus [29] and a true R_s profile that step changes from 2 to 0.5 at s = 100 days. We compute the true and estimated elimination probabilities, and z_s, conditional on all cases observed up to time s in grey and red respectively (right y-axis). The circle (black) indicates when the outbreak can be declared over with 95% confidence. Observe how z_s and respond to the low I_s at the beginning of the epidemic before remaining 0 until we get to the tail of the outbreak, where a couple fluctuations occur due to some final cases. An estimate of the WHO declaration time, t_WHO [3], which is mostly insensitive to past case profiles is in dark blue. The central question in this study is how few cases need to be observed in the recent past before we can be confident that the epidemic has been eliminated.

https://doi.org/10.1371/journal.pcbi.1008478.g002

Theorem 1. If the posterior distribution of the grouped effective reproduction number, R_τ(s), given the incidence curve has form Gam(α_τ(s), β_τ(s)) then the estimated probability that this epidemic has been eliminated at time s is with and as the mean posterior incidence prediction and effective reproduction number estimate at time j, respectively.

We outline the development of this theorem. First, we decompose Eq (5) into sequentially predictive terms as: (7)

For simplicity, we rewrite Eq (7) as . The factor q_j conditions on , which includes all the epidemic data, and the sequence of assumed zeros beyond that i.e. for j ≥ 1. This sequence is treated as pseudo-data. Observe that q₀ is simply a one-step-ahead prediction of 0 from the available incidence curve.

We solve Eq (7) by making use of known renewal model results derived in [12, 22, 24] and outlined in Methods. The renewal transmission model allows us to estimate the effective reproduction number R_s and hence compute z_s in real time (see Fig 1). This estimate at time s, R_τ(s), uses the look-back window τ(s) of k time units (e.g. days). The posterior over R_τ(s) is shape-scale gamma distributed as Gam(α_τ(s), β_τ(s)) with α_τ(s) ≔ a + i_τ(s) and (see Eq (1)). Here (a, c) are hyperparameters of a gamma prior distribution placed on R_τ(s) and i_τ(s) and λ_τ(s) are grouped sums of the incidence I_u and total infectiousness Λ_u for u ∈ τ(s). The total infectiousness describes the cumulative impact of past cases and is defined in Methods.

Under this formulation, the posterior predictive distribution of the incidence at s + 1 is negative binomially distributed (NB) (see Eq (2)). The probability of I_{s + 1} being zero from this distribution gives by substitution. The next term, q₁, is computed similarly because we condition on I_s+1 = 0 as pseudo-data (i.e. the sequential terms in Eq (7)) and update Λ_s+2, β_τ(s+1) and α_τ(s+1) with this zero. Iterating for all terms yields (8) which is an exact expression for z_s. As a string of zero incidence values accumulate with time Λ_j+1 → 0 and hence q_j → 1. Consequently, only a finite number of terms in Eq (8) need to be computed and the initial ones are the most important for evaluating z_s.

The posterior mean estimate of R_τ(s) is with I_τ(s) as the incidence values in the τ(s) window (the remaining are assumed uninformative [12]). This follows from the Gam distribution and implies a posterior mean incidence prediction from the NB posterior predictive distribution [22]. Substituting these into Eq (8) gives: (9)

This completes the derivation. Theorem 1, when combined with Eq (6), provides a new, analytic and event-triggered approach to adjudging when an outbreak has ended. Eq (9) provides direct and quantifiable insight into what controls the elimination of an epidemic and can be easily computed and updated in real time.

Understanding the probability of elimination

We dissect and verify the implications of Theorem 1, which provides an exact formula for estimating the probability, z_s, that any infectious disease epidemic has been eliminated by time s. Eq (8) formalises the expectation that any decrease in case incidence increases z_s. This results because ∂q_j/∂α_τ(j) < 0 for all α_τ(j), meaning that q_j is monotonically decreasing in α_τ(j) and hence i_τ(j). As z_s is a product of q_j and every q_j is positive then z_s is also monotonically decreasing in all incidence window sums. Consequently, any process that reduces historical incidence surely increases the probability of elimination, provided other variables are relatively fixed.

The main variable controlling z_s is the average predicted incidence (see Eq (9)). Reducing either Λ_j+1 or therefore increases our confidence in a declaration made after a fixed time (the time-triggered approach) or, decreases the time of declaration for a fixed confidence (the event-triggered approach). This highlights the two known ways that sustained interventions, e.g. vaccination, social-distancing or quarantine, can help drive an epidemic to extinction. First, such measures explicitly limit R_j and hence , leading to an expected rise in z_s [23]. Second, they may also implicitly reduce the duration of the serial interval, resulting in smaller Λ_j+1 [30].

Accordingly, under- or over-estimating or using incorrectly smaller or larger Λ_j+1 sums induces spurious fluctuations in z_s and promotes premature or delayed declarations, respectively. This insight underlies later analyses, which investigate how surveillance imperfections can modulate the declaration time. Because we cannot reduce either reproduction numbers or serial intervals to arbitrary values of interest (e.g. certain diseases have intrinsically wider serial interval distributions) some epidemics will be innately harder to control and eliminate [31].

Interestingly, while z_s is controlled by mean estimates and predictions, it appears insensitive to the uncertainty around those means, despite its derivation from the posterior distributions of Eqs (1) and (2). This follows from the inherent data shortage at the tail of an epidemic (there are necessarily many zero incidence points), which likely precludes the inference of higher order statistics [24]. Moreover, when the incidence is small stochastic fluctuations can dominate epidemic dynamics. Consequently, to maximise the reliability of our z_s estimates we recommend using long windows (large k) for . Short windows are more sensitive to recent fluctuations and are more prone to yielding uninformative estimates when many zero incidence points occur [22, 32].

Last, we validate the correctness of our estimated z_s by considering a hypothetical setting in which the true reproduction number, {R_s: s ≥ 0}, is known without error. This allows us to derive the true (but unknowable) probability of elimination at time s, given complete information of the epidemic statistics. Under the renewal model . Repeating this process sequentially for future zero infected cases (akin to describing the likelihood of that observation series) gives: (10)

Clearly depends on the serial interval distribution and past incidence (through Λ_{j + 1}) and the sequence of reproduction numbers R_j, which are the main factors underlying the transmission of the infectious disease.

The true declaration time with confidence μ% is then (see Eq (6)). We can verify our approach to end-of-epidemic declarations if we can prove that t_μ sensibly converges to . At the limit of α_τ(j) → i_τ(j) → ∞, the estimated tends to the true R_j because under those conditions the posterior mean estimate coincides with the grouped maximum likelihood estimate of R_j, which is unbiased. Applying this limit to q_j in Eq (9) we find that as : (11) implying that , and consequently that .

This asymptotic consistency suggests that z_s and t_μ indeed approximate the true but unknowable probability of elimination and declaration time . Other end-of-epidemic metrics in the literature have not presented such theoretical justification. We illustrate z_s and across a simulated and representative incidence curve in Fig 2. There we find a good correspondence between these probabilities and observe their sensitivity to changes in incidence at the beginning and end of this outbreak. Note that z_s and (and hence t_μ and ) depend on and are more precisely written as and . The WHO declaration time, t_WHO, which is included for reference, is mostly independent of the shape of [3], explaining why it provides no confidence guarantee.

Practical comparisons and verification

We have only validated our approach at an asymptotic limit that is not realistic for elimination i.e. the proof that z_s and t_μ converge to their true counterparts requires infinite incidence. While this proof suggests our formulation is mathematically correct, it does not indicate its performance on actual elimination problems. We now verify out method more practically. We first use simulated data to show that Δt_μ = t_μ − t₀ and correspond well over several end-of-epidemic problems, where we are far from this limit, and with t₀ as the time of the last observed case. We characterise this via histograms of the error , which are given in Fig 3A–3C. There we present 95% (μ = 0.95) declaration time errors calculated over 1000 simulated epidemics with serial interval distributions from the COVID-19 pandemic [33], MERS-CoV in Saudi Arabia [25], Marburg virus in Angola [29] and Measles in Germany [12].

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Fig 3. True and estimated declaration times.

We simulate 1000 independent incidence curves under various renewal models and provide normalised histograms of the difference between the estimated and true declaration times i.e. . Panels A–C present these histograms for various infectious diseases under R_s profiles indicating rapidly controlled, recovering and rising and then decaying transmission (boom-bust). The top row of D plots the true R_s curves in absolute time, while the bottom row of D provides the serial interval distributions of the infectious diseases examined. Generally we find that to a reasonable degree. The quality of this approximation depends on the variability of the serial interval distribution (see S1 Fig) and the level of fluctuation in transmission when incidence is small.

https://doi.org/10.1371/journal.pcbi.1008478.g003

We investigate true R_s profiles that describe rapidly controlled (Fig 3A), partially recovering (Fig 3B) and exponentially rising and falling transmission (boom-bust, Fig 3C). For each profile we use the renewal model to simulate conditionally independent curves and compute and using Eqs (9) and (10). The declaration time errors then follow as above and from Eq (6). Fig 3D plots these R_s profiles (top) and the serial interval distributions for each disease (bottom). Generally, we find that t_μ is a good approximation to , with some error naturally emerging from the difficulty of estimating R_s in conditions where data are necessarily scarce [32]. Our prior distribution over R_τ(j) is Gam(1, 5), which is both uninformative and has a large mean of 5.

This error, δt₉₅, is more prominent for diseases featuring wide serial interval distributions, which are fundamentally more difficult to estimate, due to their dependence on much earlier epidemic dynamics. These simulations also demonstrate why time-triggered approaches can be misleading; they do not adapt to the shape of the specific instance of observed. An example of this is given in S1 Fig, where we find that the WHO declaration time Δt_WHO = t_WHO − t₀ is delayed relative to both the true () and estimated (Δt₉₅) event-triggered declaration times, for Ebola virus disease, which has a wide serial interval. Depending on the disease of interest Δt_WHO could also be premature. The large variability among the possible provides a clear visualisation of the non-deterministic nature of epidemic end-points and the need for adaptive metrics with stated confidence.

At present, we have only verified our method under ideal reporting conditions. Practical surveillance is investigated in later sections. We now compare our method to the event-triggered one of [2], which assumes ideal surveillance and models epidemic transmission with a NB branching process that is strictly only valid at the beginning of the outbreak. This notably differs from our renewal model approach and the elimination probabilities derived in [2] are a mathematically conservative approximation to our z_s. We compare both methods on MERS-CoV data from South Korea, examined in [2], by running them on a set of bootstrapped incidence curves generated from fitting the model of [2] to that data and compute 95% confidence intervals on the probability of elimination.

Fig 4 presents our main results with time relative to the last observed case in each bootstrap (Δs) and blue and red curves as the outputs of [2] and our method. While the median 95% relative declaration times (black circles) are close, the approach of [2] yields a delayed declaration. This effect is reduced if we use the lower bound of the z_s curves instead of their median. When z_s is small (which is not practical for defining end-of-epidemic declarations) we find that the methods are less consistent. The WHO declaration time (dark blue) for this epidemic is over one week later than the time proposed by both methods [2]. While our method shows wider uncertainty, the similarity of these intervals suggests that our formulation is robust to moderate model mismatch.

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Fig 4. Empirical method comparison.

We compare 95% confidence intervals on the elimination probability from [2] (blue) and z_s from Eq (9) (red) on bootstrapped epidemics based on the MERS-CoV data from South Korea used in [2]. Black circles define the median relative declaration time (Δt₉₅) when each method deems the epidemic to be over with 95% confidence (the event trigger). Time is relative to the last observed case in each epidemic bootstrap and the WHO (time-triggered) declaration time (Δt_WHO) is in dark blue.

https://doi.org/10.1371/journal.pcbi.1008478.g004

Under-reporting leads to premature declarations

Having verified z_s and hence t_μ as reliable and sensible means of assessing the conclusion of an epidemic, we investigate the effect of model mismatch due to imperfect surveillance. We start with case under-reporting, which affects all infectious disease outbreaks to some degree. While previous works have drawn attention to how constant under-reporting can bias end-of-epidemic declarations [6] [7], no analytic results are available. Moreover, the impact of time-varying under-reporting, which models a wide range of more realistic surveillance scenarios [8, 34], remains unstudied. We provide mathematical background for our under-reporting models in Methods.

Fig 1C illustrates how under-reporting results in only a portion, N_s, of the total local cases, I_s being sampled or observed. We use U_s = I_s − N_s ≥ 0 to denote the unreported cases. We investigate two hypotheses or models about the incidence curve, a null one, , where we assume that the observed cases represent all the infected individuals and an alternative hypothesis , in which the unreported cases (and hence ) are known and distinguished. The estimated elimination probabilities under both surveillance models are: (12)

Here portrays a naive interpretation of the observed (N_s) incidence, while indicates ideal surveillance. Intensive and targeted population testing should interpolate between and . We compute by constructing the sampled total infectiousness and then applying Theorem 1. This follows because N_s can also be described by a Poisson renewal model (see Methods for details). We therefore find that with n_τ(j) and as the sums of N_u and within the τ(j) window and . We get directly from Eq (8) since this is the perfect surveillance case.

This allows us to construct the ratio of to as with ϕ_j = Λ_j+1β_τ(j) and . Here approximates ϕ_j and both are small compared to 1 (for sensible window lengths). This combined with the fact that i_τ(j) is bigger than n_τ(j) means the above ratio is greater than 1 (powers dominate the expression). This result may be violated if becomes large relative to ϕ and the unreported case count in the window, u_τ(j) = i_τ(j) − n_τ(j), is small. However, this is not possible since and ϕ necessarily converge as u_τ(j) tends to 0 (i.e. perfect surveillance). Consequently, we obtain the inflation (13)

At no point have we assumed any form for the under-reporting fraction, denoted ρ_s at time s (see Methods). Our derivation only depends on under-reporting causing smaller (absolute) historical incidence. If we know all R_j (which is unlikely) this result is also easily obtained from (10) since .

Thus any under-reporting, whether constant (i.e. all ρ_s are the same) or time-varying will engender premature or false-positive end-of-epidemic declarations provided N_s is randomly sampled from I_s (so Theorem 1 holds; see Eq (3)). We highlight this principle by examining a random sampling scheme using empirical SARS 2003 data from Hong Kong [12]. We binomially sample the SARS incidence with random probability ρ_s ∼ Beta(a, b). We set b = 40 and compute a so that the mean sampling fraction takes some desired (fixed) value. We investigate various f_ρ and show that premature declarations are guaranteed in Fig 5A and 5B. The impact of ρ_s is especially large when under-reporting leads to early but false sequences of 0 cases, which is additional to the bias from Eq (13). We present results in absolute time to showcase this effect.

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Fig 5. Case under-reporting and importation lead to premature and delayed declarations respectively.

In A and B we binomially sample an empirical SARS 2003 incidence curve from Hong Kong with reporting probabilities drawn from a beta distribution with mean f_ρ. In A we plot the elimination probability z_s when surveillance is ideal i.e. there is no underreporting (red) versus when the under-reporting is unknown (blue). The difference in the 95% declaration times, denoted δt₉₅, from these curves is in B. As f_ρ decreases we are more likely to declare too early. In C and D we consider an empirical MERS-CoV 2014-5 incidence curve from Saudi Arabia with local and imported cases. We increase the mean fraction of imported cases to f_ϵ by adding Poisson imports with mean ϵ and in C compute z_s with (red) and without (blue) accounting for the difference between imports and local cases. The change in t₉₅ is given in D. As ϵ and hence f_ϵ increase later declarations become more likely. We repeat our sampling or importation procedure 1000 times to obtain confidence intervals in A–D. As f_ϵ → 0 or f_ρ → 1 we attain the ideal surveillance scenarios of no unreported or imported cases.

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Importation results in late declarations

The influence of imported cases on end-of-epidemic declarations, to our knowledge, has not been investigated in the literature. Repeated importations or migrations of infected cases are a common means of seeding and re-seeding local infectious epidemics. Failing to ascertain which cases are local or imported can significantly change our perception of transmission [9]. We assume that I_s is the total count of local cases in our region of interest but that at time s there are also M_s imported cases that have migrated from neighbouring regions. The total number of infected cases observed is C_s = I_s + M_s as displayed in Fig 1C. We provide mathematical background on how importations are included within the renewal framework in Methods. We consider two hypotheses about our observed incidence data that reflect real epidemic scenarios.

Under the null hypothesis, , we assume that all cases are local and so we cannot disaggregate the components of C_s. The alternative, , assumes perfect surveillance. Imported cases are distinguished from local ones under and their differing impact considered. The relevant elimination probabilities for each model are (14)

Since deems all cases local, it models C_s as a renewal process with total infectiousness . Thus we use Theorem 1 to obtain the j^th factor of as with . Here c_τ(j) and are sums of C_u and over window τ(j).

Under the imported cases are distinguished but all cases still contribute to ongoing local transmission [9, 28]. Consequently, I_s still adheres to a renewal transmission process and Theorem 1 yields the j^th factor of as . We compare with directly to easily prove that (15)

Not accounting for migrations shrinks the elimination probability leading to false-negative or unnecessarily late declarations. This result makes no assumption on the dynamics for importation other than it possesses Poisson noise (so Theorem 1 is valid for C_s) and so holds quite generally (see Methods for further details).

We illustrate this phenomenon using empirical MERS-CoV data from Saudi Arabia [25] in Fig 5C and 5D. Here repeated importations occur as zoonotic transmissions from camels to humans. We show the increasing effect of importation by adding further (artificial) imports via a Poisson noise variable with mean ϵ (see Eq (4)). The mean fraction of imported to total cases across the incidence curve is then f_ϵ. In Fig 5 we see that larger ϵ promotes increasingly later declaration times. Note that we do not add any imports beyond the time of the last local case. If imports do come after this case, and seed no further local infections, which is likely for epidemics with large heterogeneity, then the t₀ assumed under will be later, and further exacerbate the bias from importation.