Seasonal dynamics of recurrent epidemics

Abstract

Seasonality is a driving force that has a major effect on the spatio-temporal dynamics of natural systems and their populations. This is especially true for the transmission of common infectious diseases (such as influenza, measles, chickenpox and pertussis), and is of great relevance for host-parasite relationships in general. Here we gain further insights into the nonlinear dynamics of recurrent diseases through the analysis of the classical seasonally forced SIR (susceptible, infectious or recovered) epidemic model. Our analysis differs from other modelling studies in that the focus is more on post-epidemic dynamics than the outbreak itself. Despite the mathematical intractability of the forced SIR model, we identify a new threshold effect and give clear analytical conditions for predicting the occurrence of either a future epidemic outbreak, or a 'skip'-a year in which an epidemic fails to initiate. The threshold is determined by the population's susceptibility measured after the last outbreak and the rate at which new susceptible individuals are recruited into the population. Moreover, the time of occurrence (that is, the phase) of an outbreak proves to be a useful parameter that carries important epidemiological information. In forced systems, seasonal changes can prevent late-peaking diseases (that is, those having high phase) from spreading widely, thereby increasing population susceptibility, and controlling the triggering and intensity of future epidemics. These principles yield forecasting tools that should have relevance for the study of newly emerging and re-emerging diseases controlled by seasonal vectors.

Outbreak properties of epidemic models: The roles of temporal forcing and stochasticity on pathogen invasion dynamics

“…This is a key advance in the study of infectious disease dynamics, as much of the insight to date on the effects of seasonality on disease dynamics has resulted largely from analytical or numerical analysis of deterministic epidemic (or endemic) models (Bailey, 1975;Bolker and Grenfell, 1993;Dietz, 1976;Moneim, 2007;Stone et al, 2007). While the approach is applied to linearised epidemic models, we demonstrate here that this approach remains robust (and indeed linear in terms of probabilities) even for full non-linear disease systems, particularly when the aim is to gain a better understanding of the invasion dynamics of epidemic infections into vulnerable populations.…”

A century of transitions in New York City's measles dynamics

“…1 Measles exhibits recurrent epidemics, the frequency and amplitude of which change over long timescales. Much research over the last 40 years has attempted to reveal the biological and dynamical processes that give rise to these changing epidemic patterns, especially the transitions evident in NYC measles dynamics [5,11,[14][15][16][17][18]. In this paper, by more than doubling the length of the NYC measles time series, we substantially enhance the opportunity to test hypotheses concerning the mechanisms that drive childhood disease transmission patterns.…”

Effects of additional food in a susceptible-exposed-infected prey–predator model

“…There are many diseases models incorporating seasonality (London and Yorke 1973;Yorke et al 1979;Uziel and Stone 2012;Aron and Schwartz 1984;Stone et al 2007;Buonomo 2011). In order to investigate the qualitative effects of seasonality, we assume the infection-rate cðtÞ ¼ c 0 ½1 þ c 1 sin ð2ptÞ , where c 0 is the average contact rate-constant, c 1 represents the strength of the seasonal forcing between zero and unity and t has units of years (London and Yorke 1973;Yorke et al 1979;Uziel and Stone 2012;Aron and Schwartz 1984;Stone et al 2007;Buonomo 2011). Note that, for the classical epidemic model with contact rate ruled by mass action law, the sinusoidal forcing may strongly influence the long term behaviour (Buonomo 2011).…”