Transmission of and susceptibility to seasonal influenza in Switzerland from 2003–2015

Understanding the seasonal patterns of influenza transmission is critical to help plan public health measures for the management and control of epidemics. Mathematical models of infectious disease transmission have been widely used to quantify the transmissibility of and susceptibility to past influenza seasons in many countries. The objective of this study was to obtain a detailed picture of the transmission dynamics of seasonal influenza in Switzerland from 2003-2015. To this end, we developed a compartmental influenza transmission model taking into account social mixing between different age groups and seasonal forcing. We applied a Bayesian approach using Markov chain Monte Carlo (MCMC) methods to fit the model to the reported incidence of influenza-like-illness (ILI) and virological data from Sentinella, the Swiss Sentinel Surveillance Network. The maximal basic reproduction number, R0, ranged from 1.46 to 1.81 (median). Median estimates of susceptibility to influenza ranged from 29% to 98% for different age groups, and typically decreased with age. We also found a decline in ascertainability of influenza cases with age. Our study illustrates how influenza surveillance data from Switzerland can be integrated into a Bayesian modeling framework in order to assess age-specific transmission of and susceptibility to influenza.

such as malaise, fever, cough, and muscle pain. On the basis of 51 the data collected in this sentinel network, the SFOPH publishes 52 the weekly incidence of ILI-related GP consultations. In addi-53 tion, some of the patients within the network who suffer from 54 ILI are virologically tested through a nasopharyngeal swab that 55 can be used to determine strain-specific positivity for influenza. 56 In this study, we conducted a detailed analysis of the trans-57 mission dynamics of ten influenza seasons in Switzerland from 58 We developed a deterministic, population-based model that describes human influenza transmission across different age groups in Switzerland. Assuming an SEIR (susceptibleexposed-infected-recovered) structure and gamma-distributed latent and infectious periods (Keeling and Rohani, 2008), the model can be described by the following set of ordinary differential equations (ODEs):   In order to embed the incidence of ILI and the virological 133 data into a likelihood model, we followed a similar method to

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While the majority of these ascertainable influenza cases are caused by the national epidemic, there is an additional influx of cases from abroad. We modeled this influx with the constant parameter ζ c . We then described the total incidence of weekly ascertainable influenza cases per 100,000 as follows: where n denotes the corresponding week. We introduced the random variable z a i (n) that describes the sampled ascertainable influenza cases according to a truncated negative binomial distribution with dispersion parameter Φ: The negative binomial distribution can account for variation in the sampling process, e.g., GP consultations that are not uniformly distributed throughout a week, and additionally allows for over-dispersion of cases due to stochastic processes that are not captured by the deterministic model. We used the following parameterization: if X ∼ nBin(µ, Ψ) then E(X) = µ and Var(X) = µ + µ 2 /Ψ. The variable z a i (n) serves as an auxiliary variable, since the ascertained cases cannot be directly derived from the incidence of ILI. Rather, we used z a i to calculate the proportion z a i (n)/z i (n), which describes the probability of detecting influenza using a nasopharyngeal swab within the set of ILI-related GP consultations. We can then describe the total number of influenza-positive cases, v + i (n), among v i (n) individuals that provided a swab test in age group i using a binomial distribution:  vidually. At t 0 , we initialized the ODEs with one exposed indi-168 vidual partitioned across the five age groups according to their 169 size, i.e., E 1i (0) = N i /N. We further introduced the parameter 170 p S i to account for the proportion of susceptible individuals in All other compartments were set to zero. 173 We implemented the MCMC simulations in Stan (Carpenter et al., 2017; Stan Development Team, 2016), a programming language written in C++ using a Hamiltonian Monte Carlo (HMC) procedure with fast convergence. For every season, we sampled two chains of length 1,000 with a burn-in of 500 using UBELIX (http://www.id.unibe.ch/hpc), the HPC cluster at the University of Bern. We visually confirmed convergence using the chain plots together with package ggmcmc of the programming language R (R Core Team, 2015). Since Stan does not support sampling of discrete variables, we discretized the likelihood function resulting in the following equation: where the overall log-likelihood function is the sum of the log-174 arithms of Eq. 5 over all i and n. We ignored data points where between R 0,min and ∆R 0 , between R 0,min and susceptibility, p S i , 199 and between susceptibility, p S i , and ascertainability, p ai (see   Age group 65+ ascertainable influenza incidence per 100,000   (Fig. 3).

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The most pronounced differences compared to the other sea-

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Our modeling approach could be extended in several ways.

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Since we assumed seasonal forcing of influenza transmission,