Modelling the seasonality of respiratory syncytial virus in young children

Respiratory syncytial virus (RSV) is a major cause of acute lower respiratory tract infections in infants and young children. The transmission dynamics of RSV infection among young children are still poorly understood (Hall et al., 2009) and mathematical modelling can be used to better understand the seasonal behaviour of the virus. However, few mathematical models for RSV have been published to date (Moore et al., 2013; Weber et al., 2001; Leecaster et al., 2011) and these are relatively simple, in contrast to studies of other infectious diseases such as measles and influenza. A simple SEIRS (Susceptible, Exposed, Infectious, Recovered, Susceptible) type deterministic ordinary differential equation model for RSV is constructed and then expanded to capture two separate age classes with different transmission parameters, to reflect the age specific dynamics known to exist for RSV. Parameters in the models are based on the available literature. In temperate climates, RSV dynamics are highly seasonal with mid-winter peaks and very low levels of activity during summer months. Often there is an observed biennial seasonal pattern in southern Australia with alternating peak sizes in winter months. To model this seasonality the transmission parameter β(t) is taken to vary sinusoidally with higher transmission during winter months, such as in models presented in Keeling and Rohani (2008) for infections such as measles and pertussis: β(t) = β₀[1 + β₁ sin(²₅₂^πt)]. (1) This seasonal forcing reflects increases in infectivity and susceptibility thought to be due to multiple factors including increased rainfall, variation in humidity, and decreased temperature (Cane, 2001; Weber et al., 1998). Sinusoidally forced SIR-type models are known to support complex multi-periodic and even chaotic solutions. For realistic parameter values, obtained from the literature, and depending on the values selected for β₀ and β₁, the model predicts either annual peaks of the same magnitude, or the observed biennial pattern that can be explained by the interaction of the forcing frequency and the natural frequency of the system. This behaviour is in keeping with what is observed in different climatic zones.