Characterising seasonal influenza epidemiology using primary care surveillance data

Understanding the epidemiology of seasonal influenza is critical for healthcare resource allocation and early detection of anomalous seasons. It can be challenging to obtain high-quality data of influenza cases specifically, as clinical presentations with influenza-like symptoms may instead be cases of one of a number of alternate respiratory viruses. We use a new dataset of confirmed influenza virological data from 2011-2016, along with high-quality denominators informing a hierarchical observation process, to model seasonal influenza dynamics in New South Wales, Australia. We use approximate Bayesian computation to estimate parameters in a climate-driven stochastic epidemic model, including the basic reproduction number R0, the proportion of the population susceptible to the circulating strain at the beginning of the season, and the probability an infected individual seeks treatment. We conclude that R0 and initial population susceptibility were strongly related, emphasising the challenges of identifying these parameters. Relatively high R0 values alongside low initial population susceptibility were among the results most consistent with these data. Our results reinforce the importance of distinguishing between R0 and the effective reproduction number (Re) in modelling studies.


Parameter derivations
Model transitions out of the I 1 compartment depend upon two rates: 2γ and λ. An individual remains in this compartment for a time that is distributed Exp(2γ + λ), i.e., the minimum of an Exp(2γ) random variable and an Exp(λ) random variable. The same holds for transitions out of compartment I 2 . So, an individual that was not observed, spends on average 2/(2γ + λ) time units infectious.
We assume that an observed individual should on average spend the same amount of time infectious as an individual who is not observed. To this end, the rates out of O 1 and O 2 must be higher than 2γ, as these individuals undergo an additional Exp(2γ + λ) time while transitioning from I to O. We denoted this rate 2α. We calculated the mean time an observed individual would spend infectious, given that they were observed on the first or the second transition (i.e., progressing through states I 1 → O 1 → O 2 → R, or I 1 → I 2 → O 2 → R, respectively). Using the probability of each of these transitions, we then calculate the overall mean time an observed individual would spend infectious. We equate the mean time that observed and unobserved individuals are infectious, allowing us to solve for α in terms of γ and λ. This results in α = 3γ + λ.
We used the fitted parameters p obs (the probability of observation), and D inf (the mean time infectious) to derive the model parameters γ and λ. This involves solving two simultaneous equations: resulting in:

Full model posterior distributions/outputs
In this section we present: • The H3N2 strain equivalents to Figures 4 and 6 from the main text ( Figure A, B).
• An example of how well model realisations generated from the prior fit the data ( Figure C). • The proportion of susceptible individuals that were infected in a season of a simulated realisation from each accepted parameter set ( Figure H).  Figure B: The relationship betweenR 0 and the population level attack rate of simulated realisations of the process, across 2014 and 2016 H3N2. For each accepted particle, the process was simulated again, and the total number of infected individuals from these new realisations was recorded, and converted into an annual attack rate (by dividing by 2 and the total population). Point colour indicates the probability of an individual seeking treatment; larger attack rates correspond to smaller treatment probabilities given that they all fit the same data. The red line shows denotes where the product ofR 0 and the attack rate would be equal to one; this is the line around which the initial population susceptibility values were situated. This can be interpreted as indicating that, for points under this line, not all susceptible individuals became infected during the season.

Simulated examples
To provide an illustrative example of the challenges of identifying parametersR 0 and initial population susceptibility given the data observed, we simulated H1N1pdm09 epidemics (over both 2011 and 2013, and using the true climate observations and weekly sampling denominators from those years) from the model with parameters based upon accepted particles in different parts of the posterior distribution ( Figure I). Note that the simulations shown are not the realisations that resulted in the particle being accepted, rather an additional realisation independent of the ASPREN data. The 100 particles closest toR 0 = 2, 6, 10, (with associated initial susceptibility of 0.6, 0.18, and 0.1, respectively) were chosen.