Model overview
We developed a dynamic model of influenza infection progression and transmission in Texas, California, Connecticut, and Virginia. Our model is a modified susceptible-infected-recovered compartmental framework [27], where by the population is stratified into health-related compartments, and transitions between the compartments occur over time (Fig. 1a). Transitions between the compartments are governed by a series of difference equations (Additional file 1: Model overview, equations 2) [1, 5, 7, 8, 20, 23, 24, 27, 34,35,36,37, 41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76]. To model age-dependent transmission, we stratified the population into five age groups: 0–4 years, 5–19 years, 20–49 years, 50–64 years, and ≥ 65 years. We also distinguished between high-risk and low-risk individuals within each age group based on the ACIP case definition [18]. Altogether, our model includes (5 × 2 = 10) ten subgroups based on age and risk.
At the beginning of each season, susceptible individuals from age group j are in the Sj compartment. In the model, susceptible individuals may interact with infectious individuals and become either asymptomatically or symptomatically infected [72, 73], at which point they can transmit the disease to others until they recover. Upon infection, individuals transition into the iτ = 0, aτ = 0 compartments, representing the first day of their symptomatic or asymptomatic infection, respectively. Consistent with previous models [34, 37,38,39], we explicitly tracked the day of infection τ such that the daily transmissibility was based on the daily viral load [41, 42]. Upon recovery, individuals transition to a Rj compartment, at which point they are fully protected for the remainder of the season. This assumption is supported by prospective studies demonstrating that reinfection within the same season is rare [51, 52] (Fig. 1).
Force of infection
The rate at which infectious individuals transmit the virus depends on (1) age-specific contact rates (Additional file 1: Table S1) between an infected individual and his or her contacts, (2) age-specific susceptibility to infection, and (3) infectiousness of the infected individual based on her/his daily viral loads and the time in the season (Additional file 1: Figure S1; Additional file 1 for details).
We parameterized the age-specific contact rates between an infected individual in age group e and their contact in age group j, Ce, j using data from an extensive survey of daily contacts (Additional file 1: Table S1) [34, 59]. These contact data exhibit frequent mixing between similar age groups, moderate mixing between children and adults in their thirties, and infrequent mixing between other groups. Age-specific susceptibility to infection βj was calibrated using data on the weekly numbers of cases of influenza (see the “Model calibration” section).
The logarithm of the infectious viral load positively correlates with the transmissibility of several respiratory viruses, including influenza [34,35,36,37, 58]. The viral load depends on (1) the day of infection, (2) the risk group of the infected individual (i.e., high-risk/low-risk), (3) the type of infection (symptomatic/asymptomatic), and (4) the day on which the individual received antiviral treatment. We evaluated the viral load, detailed in Additional file 1: Datasets and parameters, by explicitly accounting for these four components based on recent prospective studies (using real-time RT-PCR) that observed young children and adults over the course of their influenza infection with and without treatment [41, 42]. These studies suggest that the viral load in infected individuals peaks on approximately the day of symptom onset. Moreover, untreated infected individuals at high risk exhibited the most viral shedding, while asymptomatic individuals had the least viral shedding. Based on the reduction in the viral shedding, we considered 23.2% (CI 10.4–34.3%) reduction in transmissibility for individuals at high risk who are treated within 0–48 h of symptom onset. Likewise, individuals treated within 48–72 from symptoms onset had 21.1% (CI 9.3–32.2%) [37, 41].
In the USA, influenza incidence is seasonal, with a peak typically occurring in the winter. However, the driver of this seasonality remains uncertain [53]. Thus, we included general seasonal variation in the susceptibility rate of the model as \( T(t)=\left(1+\mathit{\cos}\left[\frac{2\pi \left(t-\phi \right)}{365}\right]\right) \), where ϕ is a seasonal offset. This formulation was previously shown to capture the seasonal variation in the incidence of respiratory diseases in the USA [34, 54, 77]. Altogether, the force of infection for age group j, λj(t), is given by:
$$ {\lambda}_j(t)={\beta}_j\cdot T(t)\cdot \left(\sum \limits_{e=1}^5{C}_{e,j}\left(\sum \limits_{\tau =0}^{\varphi}\sum \limits_{k\in \left\{H,L\right\}}V{L}_{k,S}^{\tau}\cdot {i}_{e,k}^{\tau}\left(t-1\right)+\sum \limits_{\tau =0}^{\varphi}\sum \limits_{k\in \left\{H,L\right\}}V{L}_{k,A}^{\tau}\cdot {a}_{e,k}^{\tau}\left(t-1\right)\right)\right), $$
where βj is the susceptibility rate for an individual in age group j and \( V{L}_{k,S}^{\tau } \) is the logarithm of the viral load in symptomatic individuals on the day of infection τ in risk group k ∈ {H, L} (i.e., high- and low-risk). \( V{L}_{k,A}^{\tau } \) is the logarithm of the viral load in asymptomatic individuals on the day of infection τ in risk group k. The numbers of symptomatic and asymptomatic infected individuals are represented by \( {i}_{j,k}^{\tau },\kern0.5em {a}_{j,k}^{\tau } \). Likewise, we incorporated the effect of antiviral treatment. A detailed description is presented in Additional file 1: Force of Infection.
We chose Texas, California, Connecticut, and Virginia because they reflect the large variability in the USA in terms of population size, climatic factors, geographic location, sociodemographic characteristics, and vaccination coverage. Specifically, Texas and California are the most populated states, with 28,995,881 and 39,512,223 residents, respectively, while Virginia and Connecticut have smaller populations, with 8,535,519 and 3,565,287 residents, respectively [78]. The median ages of the residents of Texas and California are 35.0 and 37.0, respectively. In Virginia and Connecticut, the population is relatively older, with median ages of 38.6 and 41.1, respectively [79, 80]. The vaccination coverage levels in Virginia and Connecticut are above the national average, while those in California and Texas are below the national average [69]. Thus, these four states are arguably adequately representative of the USA.
Hospitalizations
Hospitalization was not modeled explicitly. However, the number of hospitalizations in each age and risk group was computed by multiplying the number of symptomatic infected individuals by the rate of hospitalization given influenza infection (Table 1). These age- and risk-specific rates were obtained from epidemiological studies [60,61,62] (see Additional file 1: Fixed parameters).
High-risk individuals receiving antiviral treatment have a lower rate of hospitalization than nontreated high-risk patients. The age-specific reduction in hospitalization rate among treated high-risk individuals was obtained from previous retrospective and prospective studies (Table 1) [64, 65].
Baseline vaccination and treatment
For each year, we parameterized vaccination coverage from state-specific influenza vaccine coverage data for different age groups, as observed from 2013 to 2018 (Additional file 1: Table S2) [69]. We estimated vaccine efficacy using the Centers for Disease Control and Prevention (CDC) estimates for influenza vaccine efficacy between 2013 and 2018 [7].
The proportion of individuals at high risk receiving antiviral drugs and the timing of antiviral administration after symptom onset were obtained from large-scale studies among US patients [24] (Table 1; Additional file 1: Fixed parameters). These data were used to inform our baseline treatment scenario for each state.
Model calibration
To empirically estimate unknown epidemiological parameters, we calibrated our model to the data on the weekly numbers of cases of influenza (confirmed by viral isolation, antigen detection, or PCR) [75]. These data were collected by the National Respiratory and Enteric Virus Surveillance System of the CDC and state health departments from four different states in the USA from 2014 to 2019.
To obtain the numbers of influenza cases, we multiplied the number of weekly influenza-like illness (ILI) cases by the weekly proportion of specimens positive for influenza. To account for the unreported cases in each age group, we scaled up the number of cases such that the mean attack rate between 2014 and 2019 matched large-scale estimates from a meta-analysis conducted in the USA [5]. Although several studies have attempted to estimate state-level annual influenza attack rates in the USA [1, 5, 76], the state-level rates remain unknown. Therefore, we used the national attack rate to scale up state-specific numbers of influenza cases. Altogether, the yearly attack rate varied considerably between years and states, ranging from 2.8–15.0% in Texas, 4.5–12.0% in California, 5–11.7% in Connecticut, and 4.0–12.3% in Virginia.
Due to the uncertainty related to the actual incidence of influenza, we calibrated our model parameters for each state using different settings reflecting the lowest, average, and highest attack rates across influenza seasons.
Interventions
We evaluated two interventions for increasing the number of high-risk patients seeking care and being treated within the first 2 days. In the first intervention, we increased the number of individuals who were treated within the first two days after symptom onset by assuming that a proportion of those who received treatment after this infection period would receive treatment within the first two days after symptom onset. In the second intervention, we increased the total number of infected high-risk individuals who received treatment while assuming that they all received treatment within the first 2 days after symptom onset. We evaluated the population- and individual-level benefits of these interventions in terms of infections and hospitalizations averted during a single influenza season.
Sensitivity analyses
We conducted sensitivity analyses to examine the robustness of our results. In the first analysis, we investigated the impact of effective vaccination coverage on the effectiveness of antiviral treatment. Effective vaccination coverage is defined as the product of vaccine efficacy times vaccine coverage and represents the level of vaccine-induced immunity in the population. In the second analysis, we conducted a two-way sensitivity analysis to investigate the joint impact of changing both the attack rate and the effective vaccination coverage. There is a high variability in the literature concerning the effectiveness of early antiviral treatment in reducing hospitalizations [20, 64,65,66,67,68]. Thus, we also conducted a sensitivity analysis that considered two main aspects: (1) changes in transmissibility that arise from uncertainty related to the effect of the reduction in the viral load following treatment and (2) uncertainty of the effectiveness of treatment with regard to preventing hospitalizations. Uncertainty regarding the daily reduction in viral load was based on estimates from a clinical trial [42]. We considered the effectiveness of early treatment to range between 11 and 89% in adults and 11 and 81% in children (Table 1).
Model simulations
To determine the population-level and the per person treated benefit of increasing the coverage and timeliness of antiviral treatment, we simulated five influenza seasons with the implementation of the interventions. We evaluated the cases and hospitalizations averted for a range of values of effective vaccination coverage, attack rate, and early treatment coverage. We define early treatment if antiviral treatment is administered within 48 h. We compared the baseline scenario with the reduction in influenza cases and hospitalizations achieved both per person treated and overall across all periods of implementation.