2.1 RV and IAV population dynamics across the US and Canada

We collected time series of weekly tests and detections for RV/EV and IAV from longstanding respiratory virus surveillance systems in the US and Canada at both the national and regional level (a map is shown in Fig A in S1 Appendix). Rapid laboratory diagnostic testing does not differentiate RVs from other EVs, though RVs are expected to account for most detections [56,57]. For brevity, we thus refer to RV/EV simply as RV. We plot longitudinal data at the national level in Fig 1 (see Figs B-C in S1 Appendix for time series at the regional level).

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Fig 1. RV and IAV circulation at the national level (US and Canada).

Top: RV (blue) and IAV (red) surveillance detections in (A) the US and (B) Canada; data have been rescaled to mitigate biases due to circulation of other respiratory pathogens and changes in testing effort (see Materials and Methods §4.1). Mean change in mobility (colored background) during the COVID-19 pandemic were computed from Google COVID-19 Community Mobility Reports [83]. See Figs B-C in S1 Appendix for time series at the regional and provincial level. Bottom: Cross-wavelet transform of IAV and RV detections in (A) the US and (B) Canada. We used the function xwt from the R package biwavelet [93] (see details in Note A in S1 Appendix). Colors indicate standardized cross-wavelet power, with 95% significance region outlined by a contour. Arrows pointing left (resp. right) suggest IAV and RV detections are out-of-phase (resp. in phase) and arrows pointing down (resp. up) suggest IAV detections lead (resp. lag) RV detections (with vertical arrows indicating a phase difference of (quarter cycle)). Here, results confirm the asynchronous circulation of IAV and RV at the national scale: we find that detections are predominantly out-of-phase at subannual periods ( weeks), as well as an anti-phase behavior with RV lagging at more annual periods ( weeks). This pattern is transiently disrupted during the COVID-19 period (2021-2022), but is restored in the subsequent years. See Figs F-G in S1 Appendix for plots at the regional and provincial level.

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Across these locations, RV exhibit biannual outbreaks before the pandemic, with a peak in fall and a (usually smaller) peak in spring. Pre-pandemic IAV dynamics are annual with a peak (of varying magnitude) in winter. We used cross-wavelet transform analyses to confirm and complement the initial visualization and description of the time series. Results in Fig 1 confirm that national IAV and RV detections are predominantly out-of-phase at subannual periods ( weeks), except in 2016 when the IAV outbreak seems delayed; and indicate an anti-phase behavior with RV lagging at more annual periods ( weeks). We recover similar patterns at the regional level, though with some variability across locations (Figs F-G in S1 Appendix).

Importantly, IAV effectively disappeared following NPI implementation at the onset of the COVID-19 pandemic. In the US, this period was followed by a small rebound late 2021/early 2022 and a bigger rebound in the following season, similar to pre-pandemic levels (Fig 1A and Fig B in S1 Appendix). IAV rebounds occurred later in Canada, between mid-2022 and early 2024 across regions of Canada (Fig 1B and Fig C in S1 Appendix). Conversely, while RV detections dropped at the very beginning of the COVID-19 pandemic, they continued to circulate during the pandemic at approximately similar levels (Fig 1 and Figs B-C in S1 Appendix).

In the following, we use a seasonally-forced Susceptible-Infectious-Recovered-Susceptible (SIRS) model to fit RV incidence time series (see Materials and Methods §4.2 and Table 1). Briefly, the transmission rate is decomposed into three terms: (i) a periodic term consisting of weekly transmission rates that can handle any seasonal pattern within a year, similar to [58], (ii) a time-varying term that captures contact changes due to COVID-19 pandemic NPIs and (iii) a time-varying term that phenomenologically captures potential effect of viral interaction due to IAV. We use IAV incidence data as an external input (or covariate, as in [59]), which is assumed to force RV transmission by some factor . Our aim is to detect whether signatures of an interaction of IAV on RV dynamics exist at the population level. These effects could be derived from within-host viral interference mechanisms (e.g., IFNs in the context of innate immunity) or ecological interaction mechanisms (e.g., changes in contact rates after a primary infection) [37,55]. While viral interaction effects can be characterized by their direction, intensity and duration, our approach cannot easily estimate the duration of these effects. We start by using the concurrent weekly incidence of IAV, assuming the duration of interaction is short-term (on the order of days to one week consistent with, e.g., interactions mediated primarily by IFNs).

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Table 1. Main notations. For the sake of simplicity, pathogen-associated state variables and parameters correspond to RV, unless otherwise specified with the subscript IAV.

https://doi.org/10.1371/journal.pcbi.1014784.t001

2.2 Exogenous perturbations as a powerful tool to infer pathogen-pathogen interaction

We first conducted a simulation study to assess our ability to recover model parameter values, and in particular the direction and intensity of the viral interaction of IAV on RV, . Using a two-pathogen transmission model, we first simulated epidemiological trajectories for RV and IAV (Fig H in S1 Appendix), from which we generated synthetic time series; we then fitted RV simulated data while IAV simulated data was only used as an external input. The primary confounder of the population-level effects of viral interactions is seasonality in transmission, so here we generated simulated data under three main scenarios, each consistent with the sub-annual patterns of RV epidemics. In scenario 1, RV transmission corresponds to a bimodal seasonality with no interaction (); in scenario 2, IAV negatively impacts RV () but we adjust the seasonal forcing so that the overall transmission rate matches scenario 1 (“masked interaction”); in scenario 3, RV seasonality corresponds to a classic unimodal, sinusoidal forcing, and the two annual RV outbreaks arise from the interference by IAV. We plot the different seasonal transmission profiles in Fig 2A and 2B. Simulated data include a pre-pandemic period of 6 years and a (post-)pandemic period of 4 years, aligned with our empirical surveillance data. See more details on model structure and fitting in Materials and Methods §4.2-4.3 and simulation experiment in Materials and Methods §4.4.

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Fig 2. Perturbations can be a powerful tool for estimating pathogen-pathogen interactions.

We conducted a simulation study to validate our ability to estimate model parameters. For simplicity, we only display here the results obtained with the main simulations of each scenario. (A) RV seasonal transmission (solid blue lines) and effect of the interference due to IAV when the latter is at its endemic attractor (i.e., no perturbation, dashed blue lines). Importantly, RV and IAV are independent in scenario 1 (no interaction) and this seasonal forcing is mimicked in scenario 2 in the presence of IAV. (B) IAV seasonal transmission (same across scenarios). (C-E) Posterior distributions of the estimated viral interaction term. (F-H) Estimated profiles of RV basic reproduction number (median values (lines) and 95% CrIs (shaded envelopes)), reflecting seasonal forcing terms. Black dashed lines represent true values used in the simulations. See Table A in S1 Appendix for parameter values and see Figs I, J and K in S1 Appendix for the full results.

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Simulation study results showed that including exogenous perturbations can enable us to infer pathogen-pathogen interaction. In scenario 1 (no interaction), the model consistently inferred that RV dynamics is independent of IAV (Fig 2C and top panel of Fig I in S1 Appendix). In scenario 2 (“masked interaction”), the model failed to detect the presence of viral interaction when using the pre-pandemic data alone, but including pandemic perturbations substantially improved the inference, enabling both the viral interaction and the seasonal forcing terms to be estimated correctly (Fig 2D and 2G and top panel of Fig J in S1 Appendix). In scenario 3 (simple seasonality with interaction), model fit using pre-pandemic data correctly detected a negative interaction but did not accurately estimate its magnitude, which was underestimated with a wide CrI (Fig 2E and 2H and top panel of Fig K in S1 Appendix). Here, viral interaction was only partially captured by other parameters (in particular the seasonal forcing), likely because the reduction in the force of infection coincident with the increase in IAV was more abrupt than in scenario 2. However, fitting the model with and without interaction (i.e., estimating or fixing , respectively) yielded similar fits to the pre-pandemic data (top panel of Fig K in S1 Appendix). Including the (post-)pandemic period resulted in much better estimates of both the viral interaction effect and seasonal forcing. Across scenarios, restricting the inference to the pre-pandemic period resulted in wider CrIs for all model parameters, with the exception of initial conditions (Fig 2 and top panels of Figs I, J and K in S1 Appendix); in particular, the mean duration of immune protection and the under-reporting factor were more difficult to estimate precisely and were usually positively correlated. Our results were robust to observation noise (Fig L in S1 Appendix). As a sensitivity analysis, we also varied the chosen values of some key parameters of the viruses, showing that most estimates remained robust across the explored range of values (Fig M in S1 Appendix).

For each scenario, we also examined two extensions (we refer to the previous cases as the main simulations). First, we considered a six-month shift in the timing of NPIs, such that interventions implemented in winter (typically stricter) occurred in the preceding summer, leading to earlier post-pandemic rebounds of IAV. Overall, varying the timing of NPIs resulted in similar results (middle panels of Figs I, J and K in S1 Appendix). Second, we introduced a one-off exogenous perturbation in the dynamics of IAV in the pre-pandemic period (a “kick”, transferring 35% of the susceptibles to IAV to the recovered compartment, e.g., due to a vaccination campaign). Here, in scenarios with viral interference (2 and 3), fitting the model to the pre-pandemic period alone yielded more accurate parameter estimates than for the main simulations, which only featured regular cycles (bottom panels of Figs I, J and K in S1 Appendix). Overall, in the absence of viral interference, the model performed consistently across all cases. In the presence of viral interference, including any form of perturbations enabled accurate estimation of the viral interaction and seasonal forcing terms. Estimation of seasonal forcing was improved in particular for weeks of high IAV circulation, which peaks around week 52, while errors remained consistently smaller during periods of low IAV circulation (trough centered around week 30) (Fig N in S1 Appendix).

In this section, we used simulations to illustrate how perturbations can be a powerful tool for estimating pathogen-pathogen interactions. This analysis indicates that COVID-19 pandemic NPIs can be used as a large-scale natural experiment for examining such interactions, which would not be possible when available data only include endemic limit cycles. In contrast to perturbations solely affecting IAV, NPIs represent a distinct class of perturbations that, in addition to potential indirect effects mediated through IAV dynamics, also directly affects RV transmission. One major limitation of NPIs may be the unknown shape of the perturbation through time. Additional simulations showed that, when the model also had to infer this shape without constraints, it tended here to underestimate the magnitude of the interaction effect (though the estimated direction remained correct here) while overestimating seasonal transmission rates (Fig O in S1 Appendix). Constraining the shape of these perturbations seems thus essential for accurately estimating the different drivers of the force of infection. In practice, the true shape is unknown, and we used mobility data as a proxy.

2.3 Modeling of empirical surveillance data does not support an impact of IAV on RV dynamics at the population level

We then fitted our model to RV detection time series from the US and Canada, including both pre- and (post-)pandemic periods. The model converged for all locations except the US HHS region 1 and the Canadian Atlantic province; examples of MCMC chains and posterior distributions are provided in Fig P in S1 Appendix. Our model captured most of the RV outbreaks (Fig 3 and Figs Q-R in S1 Appendix) – though it underestimated the 2021 fall outbreaks in several locations. Using the current incidence of IAV, the posterior distributions of – and thus of the change in transmission due to viral interaction, with negative values suggesting that IAV epidemics suppress RV activity – are either not significantly different from 0 or relatively weak in magnitude across locations (Fig 4). For example, at the national level, we obtained a median (95% CrI: ) for the US and (95% CrI: ) for Canada. A sensitivity analysis varying the prior constraint on seasonal transmission variability yielded similar results (Fig S in S1 Appendix). Model comparisons of expected log-predictive density against a null model with no viral interaction (fixing ) did not support an effect of IAV on RV dynamics in any location (Fig T in S1 Appendix). We also explored the possibility that interaction might be better captured by considering longer-lasting effects (i.e., if interaction effects would last more than 1 week), so that prior IAV incidence could also affect RV dynamics. Refitting the model using a lagged sum of IAV incidence, rather than current incidence, with a lag of 1, 3, or 5 weeks (see Materials and Methods §4.3) led to similar results (Fig 4 and Fig T in S1 Appendix). Surprisingly, viral interaction estimates tend to be more positive for Canadian locations and more negative for American locations. Whether these small differences are meaningful is unclear and otherwise difficult to explain. The different surveillance systems and the usually earlier timing of IAV rebounds in US locations seem to be the two main differences between the two countries. These results were obtained when fitting the model independently to each location. We explored how fitting the model simultaneously to all regions/provinces impacted the results, constraining to be shared across locations within a given country, and the rate of immune waning across all locations (see Note C in S1 Appendix). In this case, estimates of viral interaction remained centered around 0 for both US and Canada, with no difference in sign, but the Canadian estimate was highly uncertain (Fig U in S1 Appendix). We also considered different extensions of the main analysis, showing that our results remained robust to: (i) post-pandemic behavioral changes (Fig V in S1 Appendix), (ii) the use of total IV incidence, i.e., including influenza B virus (IBV) detections, (Fig W in S1 Appendix) and (iii) the addition of an exposed RV stage, i.e., infected but not yet infectious, (Fig X in S1 Appendix).

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Fig 3. Fitted model results in the US and Canada.

We independently fitted model (3)-(4) with seasonal transmission (8) (see Materials and Methods §4.2-4.3) to rescaled RV detections (black points) in (A) the US and (B) Canada. Detections were rescaled to account for changes in testing patterns. We compare here results of the model where IAV can affect the transmission of RV by estimating (yellow) or not by fixing to 0 (purple). We plot posterior median values (lines) and 95% CrIs (shaded envelopes) for: from top to bottom, (i) the proportion susceptible S/N, (ii) the proportion infected I/N, (iii) the number of detections and (iv) the effective reproduction number (the epidemic grows as soon as (black horizontal line)). Colored backgrounds indicate the pandemic period (mean change in mobility during the COVID-19 pandemic were computed from Google COVID-19 Community Mobility Reports [83]). See Figs Q-R in S1 Appendix for fitted model results at the regional/provincial level.

https://doi.org/10.1371/journal.pcbi.1014784.g003

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Fig 4. Viral interaction estimates for the US and Canada.

Posterior median values (points) and 95% CrIs (segments) of the maximum change in RV transmission due to IAV interaction. We recall that, for the purpose of model fitting, the effect of viral interaction is modeled as , where denotes the observed incidence of IAV, so that represents the maximum change in the force of infection of RV due to viral interaction (i.e., during periods of highest IAV circulation). If is positive, it means IAV facilitates RV transmission, if negative, it suppresses RV transmission, and if zero, there is no interaction. We also considered lagged sums of IAV incidence, substituting by with a maximum lag l of 1, 3 or 5 weeks (l = 0 refers to current incidence only).

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We plot in Fig 5A and 5C the estimated weekly transmission rates of the more parsimonious model (i.e., without viral interaction). Both the magnitude and seasonal profile vary across locations (in particular in the US) but, as expected, a common seasonal pattern includes a drop in transmissibility around the summer (typically between week 20 and 35), followed by an increase in fall. We also plot in Fig 5B and 5D the posterior distributions of the mean duration of immune protection to RV reinfection, . Immune protection was estimated to be quite short (typically around 1–3 months), despite some variability across locations (e.g., a median of 20 weeks (95% CrI: [16.41, 24.43]) in the HHS region 9). Posteriors of the remaining parameters are displayed in Fig Y in S1 Appendix and summarized in Table C in S1 Appendix. Note that estimates of the under-reporting factor cannot be interpreted quantitatively as can also compensate for the arbitrary rescaling of detections. Using our estimates, the median of the basic reproduction number was estimated between for the US (Fig 5A) and for Canada (Fig 5C), which is consistent with existing literature suggesting a range between 1.2 and 2.7 [60–63]. Averaged over the entire seasonal profile, mean values range from 1.16 to 2.04 for the US and from 1.24 to 1.42 for Canada. The effective reproduction number ranges between approximately 0.8 and 1.5 in classic seasons, following the biannual pattern of RV (primary peak in fall and a smaller secondary peak in spring). During the first implementation of NPIs at the onset of the COVID-19 pandemic (March-May 2020), shows its biggest decline, but followed by a rapid return to historical levels (Fig 3 and Figs Z-AA in S1 Appendix).

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Fig 5. Estimated transmissibility and duration of immune protection of RV.

We plot results obtained with the model with no viral interaction (). First column: estimated profiles of RV weekly transmission rate (left y axis) and basic reproduction number (right y axis) in (A) the US and (C) Canada; the lines indicate the median values and the envelopes represent the 95% CrIs of the inferred posterior distributions. Second column: posterior distributions of the mean duration of immune protection in (B) the US and (D) Canada; vertical black lines represent 2.5%, 50% (median) and 97.5% quantiles, respectively. Hat symbols denote estimated values. CA = Canada (national), BC = British Columbia, ON=Ontario, PR = Prairies.

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4.1 Data

We collected time series of weekly detections for RV/EV and IAV in the US (Fig B in S1 Appendix) and Canada (Fig C in S1 Appendix). US data were collected from the Centers for Disease Control and Prevention (CDC), as reported through the National Respiratory and Enteric Virus Surveillance System (NREVSS). Data were downloaded at both the national and regional level (10 Health and Human Services (HHS) regions, Fig A in S1 Appendix) and ranged from January 4, 2014 to June 28, 2025 (regional level) or September 6, 2025 (national level). Canadian time series were reconstructed from publicly available reports provided by the Respiratory Virus Detection Surveillance System (RVDSS, Public Health Agency of Canada [80]). We analyzed data at the national and regional level – 4 provinces/regions: Atlantic, British Columbia, Ontario and Prairies (which includes Alberta, Manitoba and Saskatchewan) (Fig A in S1 Appendix) – from August 31, 2013 to June 8, 2024. NREVSS and RVDSS are decade-long laboratory-based surveillance systems that monitor respiratory virus activity across the US and Canada, respectively. These systems involve a network of participating laboratories across each country testing patients with respiratory symptoms in outpatient and inpatient settings. Testing is conducted by PCR; in recent years the use of multiplex PCR has become more frequent. We note that rapid laboratory diagnostic testing does not differentiate RVs from other EVs (and, a fortiori, does not distinguish RV species and subtypes), and results are typically reported as a combined RV/EV category. While there is no strong evidence for viral interference between IAV and other EVs, RVs are expected to account for most detections, in line with [56,57]. We therefore considered RV/EV patterns to be primarily driven by RV circulation and, for brevity, we refer to RV/EV simply as RV.

Positivity rates (i.e., proportion of detections among tests performed) can yield a biased measure of pathogen activity levels depending on the co-circulation of other respiratory pathogens [58,81]. We thus calculated an incidence proxy to mitigate this bias and to account for changes in testing effort over time. Following an approach similar to [42,55,82], we rescaled incidence data by multiplying raw cases by a weekly testing factor equal to the average number of tests (over the entire studied period) divided by the one-year moving average of the number of tests (see testing patterns in Fig D for RV and Fig E for IAV in S1 Appendix).

We also used Google COVID-19 Community Mobility Reports [83] as a proxy to capture reductions in transmission due to COVID-19 pandemic NPIs. Google mobility reports provide percentage changes in the number of visitors relative to baseline days (median value from the 5‑week period from January 3 to February 6, 2020). As in [55,58], we defined a weekly mobility metric, c, by averaging mobility data across four categories (grocery & pharmacy, retail & recreation, transit stations and workplaces), computing a population-weighted average across states or provinces. Google mobility data were only reported between February 15, 2020 and October 15, 2022 and we assumed c = 0 (i.e., same as baseline) outside of this range (we relaxed this assumption in Fig V in S1 Appendix). We approximated population sizes using 2021 census data for Canada [84] and 2020 census data for the US [85].

4.2 Transmission model

We modeled RV epidemiological dynamics using a seasonally-forced SIRS model (details of notation are provided in Table 1). We assumed a homogeneously mixed population in which susceptible hosts (S) become infected and infectious (I) at a per capita rate (force of infection, where t denotes the current time). Infected hosts then enter the recovered state (R) at a per capita rate . Infection provides full immunity against reinfection but immunity naturally wanes at a per capita rate after recovery (in the following we denote the mean duration of immune protection, i.e., the mean sojourn time in R). Last, we define as the per capita host birth and death rate, such that the population size S(t) + I(t) + R(t) = N remains constant over time. These epidemiological trajectories can be written as the following system of ordinary differential equations (ODEs):

(1)

where the current incidence is computed by tracking the number of new infections . As with NPIs, potential effect of IAV is assumed to affect susceptibility or transmissibility, modeled phenomenologically as a scalar effect on the transmission term. The force of infection is then given by:

(2)

where is the seasonal transmission rate, , the strength of the effect of NPIs (assuming this effect scales linearly with the changes in contacts/visitors, c(t), due to control measures), and where captures the strength and direction of the effect of IAV (assuming this effect scales linearly with IAV prevalence, ).

For the purpose of model fitting, we used in practice the efficient discretization scheme described in [86]:

(3)

, and represent the densities of individuals leaving compartment S, I and R, respectively, between time t and :

(4)

where is the probability of leaving a given compartment when r is the sum of rates out of this compartment. Consequently, the corresponding mean sojourn time is given by , which, taking the limit as , yields the sojourn time in the corresponding ODE, 1/r. Besides, the incidence (i.e., number of new cases per time step) now corresponds to the term:

(5)

Last, we determine the effective reproduction number (i.e., the expected number of secondary infections by a single infectious individual), as well as the basic reproduction number (defined in an otherwise wholly susceptible population), of the discrete model (3). Multiplying the incidence by the mean number of time steps an individual stays in the the infected compartment, , and linearizing around I[t] = 0 yields:

(6)

(we drop the potential effects of NPIs and viral interaction); the case at is a removable singularity. Note that converges towards when .

4.3 Bayesian inference

Using a Bayesian framework, we fitted the model (3)–(4) to our RV detection proxy C[t]. Because C[t] is a positive, continuous variable, we relied on a log-Normal likelihood:

(7)

where is the incidence given in equation (5), accounts for under-reporting and is the residual standard deviation. The log-Normal distribution implies multiplicative observation noise. Poisson and negative binomial likelihoods are also commonly used in similar epidemiological settings, but require discrete case counts.

To estimate the seasonal transmission rate , we used 52 weekly transmission rates (with , the k^th week of the year), modeled as a cyclical random-walk on the log scale:

(8)

with , some independent step increments (centered by their arithmetic mean to enforce cyclicity), whose magnitude is scaled by , and where is a reference transmission rate (by construction, ).

IAV incidence was directly input from surveillance data, similarly to [59]. Specifically, we substituted the unknown IAV prevalence in the viral interaction term by the observed weekly IAV incidence, , which was assumed to be a reasonable proxy as IAV infectious period is also of the order of one week. Values were slightly smoothed using a geometric three-week moving average to reduce the influence of observation noise. We then scaled the time series between 0 and 1 by dividing by its global maximum (across all seasons of the study period) to enforce a lower bound of -1 for , ensuring that the force of infection is always positive while preserving any relative differences across seasons such as epidemic size. When fitting the model to real data, we also considered a lagged sum of IAV incidence instead of current incidence, substituting by with a lag l of 1, 3 or 5 weeks. This approach assumes a more sustained effect, where individuals infected during the past l weeks remain protected, and is intended to approximate varying durations of potential immune protection or reduced susceptibility. Interaction mechanisms mediated by interferon responses (typically resolve within days to a week) are more consistent with short-term effects (lags 0 or 1), whereas ecological mechanisms could potentially last longer.

We fitted models with a weekly time step ( week) using a Hamiltonian Monte Carlo algorithm (no-U-turn sampler) implemented in Stan [87] and its R interface rstan [88]. RV infections are typically associated with an infectious period between 1 or 2 weeks [3,62,89,90], so we assumed infected individuals to recover at rate , that is a mean infectious period of week (around 8 days). We estimated the rate of immune waning because of lack of information, noting that elapsed time between two infections of the same individual is likely to be relatively short due to the wide antigenic diversity of RVs and the low level of cross-immunity [4,69]. We also fixed the mortality rate, assuming . Let be the vector of parameters to estimate and its estimator. Using the seasonal transmission rate (8), we have ; priors can be found in Table B in S1 Appendix. For each estimation process, we independently ran 4 MCMC chains of 4,000 iterations (including a burn-in period of 2,000 iterations). We assessed the convergence of posterior distributions by the absence of warnings from Stan, including no divergent transitions, sufficient maximum tree depth, Gelman-Rubin statistics (R-hat) below 1.05 and effective sample size (ESS) above 400 (number of chains) for each parameter.

4.4 Simulation study

We conducted a simulation study to validate our approach, and in particular our ability to quantify pathogen-pathogen interaction. Simulated data were generated by extending the ODE model (1) to account for two pathogens (see Note B in S1 Appendix). ODEs were solved using the deSolve package [91] in the R programming language [92]. We considered 3 main scenarios:

• Scenario 1 (bimodal seasonality with no interaction): RV seasonal forcing peaks twice a year, and transmission is independent from IAV ();
• Scenario 2 (“masked interaction”): Building upon the previous scenario, IAV now negatively impacts RV (), but we adjust RV seasonal forcing so that its force of infection mimics scenario 1 when IAV is at its endemic attractor (i.e., in the absence of perturbation);
• Scenario 3 (unimodal seasonality with negative interaction): RV seasonality follows a simple sinusoidal forcing (one peak per year), and the two annual RV outbreaks arise from the interference by IAV ().

Seasonal transmission profiles are shown in Fig 2A and 2B and parameter values are listed in Table A in S1 Appendix; virus parameters are based on existing literature and/or to reproduce the observed epidemiological characteristics of RV and IAV (see caption for more details).

In the last two scenarios, the maximum change in transmission due to viral interference lies around %. Starting from initial conditions on the endemic attractor of both pathogens, we simulated the model for 10 years, corresponding to a 6-year pre-pandemic period (endemic limit cycles) followed by a 4-year (post-)pandemic period. We used Google mobility data from Canada to model more realistically the pandemic period. We also considered two extensions: (i) one with a 6-month shift in the timing of NPIs (e.g., NPIs in winter now occurs in the summer), and (ii) one where we introduced a one-off exogenous perturbation in IAV dynamics (a “kick”, moving 35% of the susceptibles to IAV to the recovered compartment) at t = 1.5 year during the pre-pandemic period. All simulations are presented in Fig H in S1 Appendix.

Simulated pathogen detections were generated using a log-Normal distribution with a residual standard deviation of 0.25 on the log scale. We fitted model (3) to RV detections using the seasonal transmission (8) and compared the results with those obtained when the model was only fitted to the pre-pandemic period. Additionally, we also contrasted these fits with a null model with no interaction (fixing ). To account for model discretization, the recovery rate was fixed as , where is the recovery rate used in the continuous ODE model. Likewise, we did not directly compare the true and estimated rates of immune waning , but the mean durations of immune protection : (continuous ODE model) vs. (discretized version).