The study was approved by the Research Ethics Board, University of Toronto. The Incidence Decay and Exponential Adjustment (IDEA) model is based on concept of the basic reproduction number, R₀, defined by Vynnycky and White as “the (average) number of successful transmissions per infected person” [11] when an infected person first enters a completely susceptible population [11], [15]. The rate of growth of an epidemic is a function of both R₀ and the average serial interval, which is defined as the time between symptoms developing in an index case and symptoms developing in a secondary case [11], [15]. We use a symptom-based interval so that the IDEA model is applicable in situations where microbiological or serological diagnosis is not available. Early in an outbreak or epidemic, incident case counts (I) in each serial interval t may be defined as:(1.0)

The basic reproduction number thus describes initial exponential growth of an outbreak or epidemic. As this process continues, the effective reproduction number R is often defined as R₀ x S/N, where S/N is the proportion of the population that remains susceptible to infection (defined as S  =  number of susceptibles divided by N  =  total population size), and the decline in R with time results in ultimate termination of the epidemic. However, many outbreaks rapidly dampen after a short period of time, in a manner that cannot be attributed to a decline in susceptibles [1], [13], [14], [16]. A potential mechanism driving decline in epidemics in the presence of susceptibles is spontaneous or planned reduction in the components of R₀ itself (disease duration, contact rate, and infectiousness of cases) either because of public health interventions, or due to concern about disease among members of the public. As a decline in R for this reason is unlikely to be estimable in real time in the context of an outbreak, we propose that control be modeled empirically in a time-varying manner analogous to a financial discounting function. The impact of this discounting or dampening factor on case counts may be expressed empirically as:(2.0)where d is a discount factor. The model may be fitted to available outbreak data when case counts are aggregated to reflect likely generation times, and can be readily calculated using information on latent and infectious periods, which are available for many pathogens of public health importance [17], or estimated empirically for novel pathogens [1], [18].

In order to more fully understand the model's performance based on varying disease and disease control characteristics, we created a difference equation model with discrete time steps, each representing a single disease generation. The model was specified as follows:(3.0)(3.1)(3.2)Here S is the number of susceptibles in the population, I is the number of infectives, and R is the number of immune individuals. The total population size N  =  S+I+R. Re_t is the time varying effective reproductive number: the number of new infectious cases in a given generation created by each infective individual in the last generation. Re_t is a function of the basic reproductive number, R₀. Typically, Re_t is expressed as R₀S_t/N but such a formulation fails to account for control activities and dynamic changes in population behavior that may reduce transmissibility of infection.

We defined Re_t as: Re_t  =  R₀ κ_t S_t/N where κ is a function of time and represents the proportionate reduction of risk of transmission via control activities. κ_t is defined as the relative risk of disease transmission (RR) raised to some power, such that κ_t  =  RR^x. Here x is some exponential function of t such that x  =  tⁿ and n is an integer > = 0. We refer to n as the “order” of control. For 0^th order control, the impact of control does not change over time, and Re is simply reduced by a constant fraction throughout the epidemic. For first order control, disease risk is reduced in a manner that accelerates with time; second and third order control represent “accelerating acceleration of control”, and so on.

We used this simple difference equation model to evaluate the fit of the IDEA model to simulated epidemics under different assumptions about infectiousness (R₀), varying orders of control, under-reporting of cases, and multiple waves of infection. Models were fit by minimizing root-mean-squared differences (RMSD) between generation-specific case counts by adjustment of the R₀ and d parameters of the IDEA model. When evaluating the performance of the IDEA model as applied to an SIR difference model under different assumptions about the order of κ, we normalized RMSD by dividing by total case counts, as higher order control resulted in smaller epidemics (and consequently smaller RMSD).

In addition to generating empirical estimates of R₀ and d parameters via fitting, the model can be manipulated algebraically to generate estimates of t_max, the generation where the number of new cases is <1, such that the outbreak is effectively over. Multiplication of t_max by serial interval duration in calendar time provides an approximate estimate of outbreak duration. By manipulating [2.0] it can be seen that:(4.0)

Integration of [2.0] over t also provides a complex expression which predicts total outbreak size, such that:(4.1)(4.1.1)Where(4.1.2)Here erf is the so-called “error function”.

To test the ability of the model to describe simple epidemic dynamics in an actual outbreak, we applied the model to an outbreak of pandemic influenza A (H1N1) from the territory of Nunavut, Canada, using an empirically derived serial interval of 5 days [18]. Serial intervals may not be constant throughout outbreaks [19], [20] but are assumed to be nearly constant for the purposes of this model. The model was fitted to empirical case counts by minimizing sum-of-squares differences between model-derived and empirical case counts.

We obtained the daily number of laboratory-confirmed cases of pandemic H1N1 influenza (in which the cases were reported based on the earliest date of symptom onset, initial care, specimen collection, hospital admission, or ICU admission) for each community under study. A laboratory-confirmed case was reported as an individual with influenza-like illness or severe respiratory illness who tested positive for pandemic H1N1 influenza A virus by real-time reverse-transcriptase PCR (RT-PCR) or viral culture as is typical for Canadian influenza surveillance. As such, cases likely represent a subset of total true influenza cases [21], [22]. These data were provided by the Nunavut Department of Health and Social Services (HSS) and their use in this study was been approved by the Nunavut Chief Medical Officer of Health (Dr. Geraldine Osborne) and Michael Ruta (Territorial Epidemiologist) in 2009. No identifying data regarding individual cases was shared with the research team or used for subsequent analyses. All data included in the dataset used for model evaluation were aggregate, daily case counts for de-identified Nunavut communities. As a result, these data were not deemed protected health information by the territory of Nunavut and therefore, no patient consent was deemed necessary.

Cases were normalized to the first day of the outbreak (day 1). The definition of an outbreak was based on the Ontario Ministry of Health and Long Term Care (MOHLTC) guidelines [23]. In this instance, two cases are considered unrelated if they are separated by more than the sum of the incubation period and the period of communicability for the causative agent, which is 6 days for pandemic influenza A (H1N1). Data points which could not be considered part of the same outbreak (more than 6 days apart) and all outbreaks less than three serial intervals (15 days) were excluded from the analysis. It was assumed that short outbreaks (less than 15 days for pandemic influenza A (H1N1)) would essentially be over by the time an effective and intensive public health response was mobilized.

Simulations were performed using the Berkeley Madonna dynamic systems modeling package (University of California, Berkeley; http://www.berkeleymadonna.com), and model fits for Nunavut data were performed using the “Solver” application for Microsoft Excel (Frontline Systems, Incline Village, Nevada; http://www.solver.com).

Normalized sum of squares fits of the IDEA model to simulated data were best with first order control (i.e., κ  =  RR^t), and were better for systems with low or moderate R₀ (i.e., R₀< = 5) than those with higher R₀ (Figure 1). Model projections of final epidemic sizes were extremely accurate for a range of R₀ values; however, as R₀ increased beyond 5.5, model projected end dates for epidemics were later than those seen in simulated data (Figure A in File S1).

For systems with low or moderate R₀, and assuming first order control, stable parameters were identified for the IDEA model within 3–4 generations, and the use of these parameter values accurately projected the full extent of the epidemic curve (Figure 2) in a manner that made IDEA model projections and simulated data indistinguishable. Empiric best-fit values for the “discount parameter” d were approximately 0.054 or 0.055 for all low or moderate R₀ models. Best-fit R₀ values identified for the IDEA model tended to be slightly higher than true R₀ values, and the proportionate degree of over-estimation increased as the true R₀ increased.

In simulated epidemics with high R₀ initial convergence occurred rapidly as the epidemic grew, with best-fit values of d approximately 0.054 or 0.055, and accurate estimation of true R₀ values, in approximately 4 generations. However as the simulated epidemic peak occurred, best-fit R₀ estimates, and d estimates for the IDEA model both increased sharply diverging from initial estimates and, allowing the IDEA model to reproduce epidemic peaks and subsequent declines (Figure 3 and Figures B and C in File S1). For high R₀ systems, R₀ estimates obtained via fitting after the epidemic had peaked were far higher than true R₀ values and than values estimated prior to peaks.

Under-reporting of cases is expected to occur for a variety of diseases of public health importance; we evaluated IDEA fits to SIR model outputs where increasing fractions of cases were unobserved and consequently unavailable for fitting. In fact, we found parameter estimates and final-size-normalized RMSD model fits to be quite stable as long as case reporting fractions exceeded 5% (Figures D, E, and F in File S1); IDEA model fits became unstable only with low absolute numbers of reported cases. We evaluated the impact of multi-wave epidemics on IDEA model fitting, and found that while the structure of the IDEA model made it difficult to fit to multi-wave epidemics, an important indicator of the emergence of a new wave of infection was an increasing gap between sequential best-fit values of the discount factor d as time series used for fitting were extended to include additional waves (Figure G in File S1). We term this indicator Δd, such that(4.2)

Many outbreaks are characterized by sequential “waves” that may either signify the impact of seasonal or behavioural influences on disease transmission [24], signify the movement of epidemics into previously unaffected sub-populations [25], or (as in the case of SARS in Canada) may signify failure of control measures [26].

As the IDEA model appeared to provide a reasonable means of modeling epidemics, especially for R₀< = 5, we evaluated the expected relationship between R0, d, tmax and Itotal mathematically, using formulae 4.0 and 4.1 for a range of possible R0 and d values. The IDEA model generates an estimate of R0 and d at each point in an outbreak, and it is then possible to rapidly project the estimated duration and total cases of the outbreak. These results are presented graphically in Figure 4.

The Nunavut, Canada data illustrate the behaviour of the model in a real outbreak situation (Figure 5). This outbreak took place over 27 serial intervals and included 950 cases. The population of Nunavut in 2011 was 31,906 [27]. Initially, the outbreak is unable to gain momentum, as shown in Figure 5a by the curve predicted when two serial intervals of data are known (SI = 2). Once four serial intervals are known, however, the outbreak grows exponentially and the model (SI = 4) projects a t_max, or outbreak duration, of 74 serial intervals. By SI = 6 (the model fit with 6 serial intervals), the projected t_max is drastically dampened to 15 generations. In these early stages of the outbreak, the IDEA model is able to rapidly determine whether the outbreak is growing or stabilizing, based on the change in t_max and the change in Δd.

In later stages of the outbreak (shown in Figures 5b, c and d), the model continues to provide a rapid analysis of the immediate direction of the outbreak. Figure 5c shows a key inflection point at which the outbreak suddenly began growing again. Such biphasic outbreaks can occur for various reasons, such as the end of school closure periods, the arrival of a newly infected individual into a community or a reduction in a public health intervention such as hand sanitizer provision. The model illustrates the new projection in the outbreak behavior as such events occur.

Estimating the impact of public health interactions and the degree of control over an outbreak is a considerable challenge while an outbreak is ongoing. As a result, the IDEA model was used to compare actual versus projected cases as a means of judging whether the outbreak was under control. Figure 6a shows the Nunavut outbreak with the actual cases of influenza on the × axis and the cases projected by IDEA model on the y axis. Each point on the y axis represents the model fitted to i generations and applied to i+1 generations. On average, projections to the next generation are correct within 20.3% [95% credible interval 11.8, 28.8]. In Figure 6a we propose that when y>x, the model is projecting excess cases, implying that at this snapshot in the outbreak, the current generation had slowed its growth. Similarly, as the outbreak shifts to the y<x side of the line, the model is projecting a total case count lower than the actual outbreak, implying that the outbreak is uncontrolled and that the current generation has exceeded the model's projection. Figure 6b shows the trend in percent error as the outbreak progresses and demonstrates that after the seventh serial interval the percent error remains below 22%.