Modeling the variations in pediatric respiratory syncytial virus seasonal epidemics

Primary Children's Medical Center (PCMC) is a 250-bed children's hospital that serves both as a community pediatric hospital for Salt Lake County, Utah (2008 population 1 million [13]), and as a tertiary referral center for five states in the Intermountain West (Utah, Idaho, Wyoming, Nevada, and Montana, total 2008 population 8.36 million [14]). Eighty percent of pediatric hospital admissions occurring in Salt Lake County and 73% occurring in the state of Utah are at PCMC.

During the study period, July 2001 through June 2008, direct respiratory sampling (mainly saline-assisted nasopharyngeal aspiration) for respiratory viral testing was performed for about 70% of children evaluated in the PCMC emergency department for respiratory complaints (unpublished data) and was required for all hospitalized children with respiratory symptoms (eg, upper or lower respiratory tract infection, bronchiolitis, asthma, or bacterial or viral pneumonia). In addition, respiratory viral testing was recommended for all febrile infants one to 90 days of age. Test results were used to inform patient cohorting and isolation procedures and to assist with medical management. All samples were initially tested by direct fluorescent antibody staining (DFA). DFA testing was performed three to five times daily depending on the season, with a mean turnaround time of four hours. For all DFA negative specimens, multiplex polymerase chain reaction (PCR) or viral culture was performed.

The data included in our analyses were all positive test results from the above sampling protocols from any of the testing methods during the study period. The practice of testing and test methods did not change appreciably during the study period (unpublished data on percentage of children tested and methods used). The data were used as daily counts by age group, under two and over two years old.

The RSV epidemic year was defined to be from July 1 of one year through June 30 of the following year. This time period was chosen to place the beginning date close to the middle of the inter-epidemic period, approximately six months from the average historical peak of the seasonal epidemic.

This study was reviewed by the Institutional Review Boards of Intermountain Healthcare and the University of Utah and determined by both organizations to be exempt.

Regression analysis was used to explore the relationship between the initial exponential growth rate and the epidemic season characteristics of size, days to peak, and length using the seven epidemic seasons of RSV data from PCMC. The exponential growth rate, λ_{t0, t1}, for time interval t₀ to t₁ was calculated as , where denotes the cumulative number of cases at time t _i , i = 0,1. The exponential growth rate was calculated at four weeks to assess regression predictions made early in the season. For comparison, exponential growth rate was also calculated at weeks one through six. The total epidemic size was the sum of cases over the epidemic year, including sporadic inter-epidemic cases. An observable seasonal epidemic start date of t₀ was defined as the start of the first week of the epidemic year with at least five confirmed RSV cases. This was the definition used by the hospital epidemiologists at PCMC to declare the start of RSV outbreaks during the study period. The term seasonal epidemic refers to the period from the epidemic start date until the epidemic end date, defined as the end of the last week of the epidemic year with at least five confirmed RSV cases. The number of days until the peak for the epidemic seasons was calculated as the midpoint day of the largest seven-day moving average window minus the epidemic season start day. The length of the epidemic season was calculated as the epidemic season end day minus the epidemic season start day.

Relationships between the initial exponential growth rate and seasonal epidemic characteristics were described using the Pearson correlation coefficient and assessed using standard regression statistics. The fits of the regression models were assessed using the percent error of the model fits from the observed values. To combine across seasons, the absolute values of the percent errors were averaged providing the mean absolute percent error for the model.

We modeled the observed RSV cases using an extension of the SIR model that included individuals (c for children and a for adults) that were susceptible (S^c and S^a), exposed (E^c and E^a), infectious(I^c and I^a), infectious and subsequently detected children (D), and recovered combined across children and adults (R). This SEIDR model was applied to a series of seven epidemic years. The population was split into children less than two years old (children) and those older than two (adults). It has been shown that the initial RSV infection is the most severe and occurs in almost every child in their first two years of life. Transmission is modeled as a function of time using a cosine function to mirror the cyclic nature of epidemics [8]. There is an offset to this cycle (α), which we estimate along with transmission parameter (β). Births and deaths (μ) are accounted for in the susceptible class only. Achievement of age two is accounted for in all age-separated classes (η). Assumptions of simple compartmental models that we made were as presented in Koopman [15].

Our SEIDR transmission model (Figure 2) was defined using the following system of non-linear differential equations:

Here β was the transmission parameter, L the latency period, f the under-two detection fraction, and γ the recovery parameter. All parameters are presented in the next subsection with descriptions, ranges, and reference values from the literature. Solution to the set of differential equations is addressed below.

To fit the SEIDR model to the empiric epidemic data, three parameters-latency period, birth and death rate, and recovery period-were specified based on the literature. Three parameters associated with variation across epidemic years were estimated: 1) the temporal offset of the epidemic cycle (α), 2) detection fraction (f), and 3) transmission parameter (β). Different models were specified to explore the effect of these three parameters. All combinations of these were considered: models with one parameter allowed to vary across seasons, models with two parameters allowed to vary across seasons, and a model with all parameters allowed to vary across seasons.

Each parameter is described below.

The nonlinear equations were solved using the lsoda function from the odesolve library [18] in R statistical software [19]. The parameters were estimated using a grid search. Two fitting statistics were used. The estimates were the values that minimized the square root of the sum of standardized squared errors (RSE) and/or the square root of the sum of squared standardized errors (RMSE). The RSE was calculated as the square root of the sum of the squared errors between the observed daily cases and the fitted model, divided by the fitted value, , where was the fitted value on day i. The RMSE was calculated as the square root of the sum of the squared weighted errors between the observed daily cases and the fitted model; the weight being the fitted value, . The denominator from these measures adjusted for the magnitude of the epidemic curve to avoid fitting the model mainly to the peak, where differences could over-inflate the fitting statistic and under-value differences during the early and late stages of the epidemic. The RMSE reduces the effect of fit to the peak more than does the RSE.

A grid search was used starting with an initial wide range of values for f, β, and α. The search grid was repeated with successively narrowing ranges to minimize the RSE. The grid started with the range of reasonable values, 0 - 1 for β and f and one to 200 days for α. The range was reduced and resolution increased iteratively around minimal RSE and RMSE values. The minimum grid resolution was 0.0001 for β, 0.01 for f, and one day for α. The RSEs and RMSEs from the grid search results were used to select the best parameter estimates within each model type (eg, one model type had only transmission rates that varied by epidemic year).

The model with all three parameters allowed to vary by epidemic year was fit as a saturated model to provide a benchmark for RSE and RMSE, along with the Schwarz Criteria described below, and percent error in estimating epidemic size when evaluating more parsimonious models in which only one of the 3 parameters was allowed to vary by epidemic year. Multiple measures were used to compare the models, in part because the Schwarz criteria assumed the residuals were independent and identically distributed, which was not the case; they are, in fact, autocorrelated.

The Schwarz Information Criterion [20] were calculated based on the weighted least squares method used for parameter estimation. There were n = 2555 data points, 365 days of case data for each of seven years, and k, the number of parameters estimated was 28 in the full model (four parameters for seven years) and 16 in each other model (two parameters for seven years and two parameters overall). The Schwarz Criteria were calculated as: where M represents either the RSE or RMSE fit statistic [21]. The absolute values of the percent error in estimating total epidemic size were summed across seasons for comparison of models.

The number of children with test-positive RSV infection ranged from 682 cases in 2004-5 to 1704 cases in 2007-8 (Table 1). The median size of the annual epidemic was 1113 cases. Overall, 98% of cases were detected between the months of October and April. Larger epidemics alternated with smaller epidemics. The amplitude of this biennial cycle was approximately 600 cases.

The total number of children (under 18 years of age) tested per epidemic year ranged from approximately 3000 to 7000, with numbers of tests increasing over time. Overall, 21% percent of these were positive for RSV, varying according to the biennial cycle. Of children tested, 81% were less than three years old and 95% were less than 11 years old. Of children with positive tests, 92% were less than three years old and 99% were less than 11 years old. Of the children tested, 70% were from Salt Lake County and 77% of children with positive tests were from Salt Lake County.

Exponential growth rates calculated from cases accumulated for four weeks from the observed epidemic season start ranged from 0.034 to 0.081 (Table 1) across the epidemic seasons. The effective reproductive numbers ranged from 1.27 to 1.49 using a serial interval of seven days [16]. In regression analyses (Table 2), the four-week exponential growth rate exhibited a substantial positive correlation with epidemic size (r = 0.69, p = 0.08), and was negatively correlated with start day (r = -0.43, p-value = 0.33), days to peak (r = -0.44, p-value = 0.32), and length of the epidemic (r = -0.58, p-value = 0.17). The regression models provided estimates of epidemic season characteristics that were on average within 16% of observed epidemic season size, 11% of observed days to peak, and 8% of observed epidemic length. Using exponential growth rates calculated from weeks one through six provided, in general, increasing correlation (Table 3).

The saturated SEIDR model was fit to seven epidemic years of observed RSV data with epidemic year-specific RSE values that ranged from 13 to 21, RMSE values that ranged from 0.40 to 0.77 and percent error of total cases that ranged from 1% to 16%. The fit statistics for the models with either transmission parameter or detection fraction estimated as a constant across epidemic year did not differ substantially from those from the saturated model (Table 4). The minimum RSE model with detection fraction held constant across epidemic years had the smallest % error, smallest Schwarz RSE Criterion, and had other fit statistics nearly equal to the saturated model. The minimum RMSE models were, in general, fitting to the tails of the epidemic and resulted in large errors in estimating epidemic size.

The pattern of variation in estimates of offset from all models matched the biennial cycle variation in total epidemic size across epidemic years (Figure 3). The variation in estimates of the transmission parameter and detection fraction did not necessarily match this cycle for all epidemic years. The parameter estimates for the transmission parameter were negatively correlated with total epidemic size.