Approximate endogenous period

FIGURE 1-12 Graph showing the amplitude of oscillations (y axis, peak-trough ratio) as a function of the endogenous oscillation period (x axis) in a stochastic forced S-I-R-S epidemic model for 2,000 sets of randomly chosen parameters. The imposed 1-year seasonal variation in transmission for all instances is set as a sinusoidal curve with only ±4 percent variation. Strong resonance between the endogenous and imposed oscillations occurs when the approximate endogenous period is near 1 year. SOURCE: Dushoff et al. (2004).

Epidemics as Partially Decomposable Systems

Because dynamical systems, such as epidemics, are often composed of two or more semi-independent but partially interacting dynamical subsets (e.g., environmental conditions, immunity, health systems), it is essential to isolate and analyze these component dynamical subsystems so as to be able to understand the effects of any particular forcing factor (e.g., ambient temperature). This field is still in its infancy, but there are a number of techniques available. My colleague Derek Cummings and I have experimented with various time-series decomposition techniques borrowed from physics to tease apart the component subsystems from long-term records of the waxing and waning of epidemic diseases (Cummings et al., 2004). Working with our colleagues at the Thailand Ministry of Public Health, we reviewed epidemiological records over many years, entered them into digital format, and applied decomposition methods. Some of these



FIGURE 1-13 Influenza virus types isolated in the United States between 1997 and 2007. Influenza H3N2 is shown in red, H1N1 in blue, and B in green. Numbers shown are total isolates typed and reported. SOURCE: WHO (2008).

time-series decomposition methods include the well-known Fourier decomposition methods, but we also examined various wavelet decomposition methods, and the Empirical Mode Decomposition. Figure 1-14 is just one example of an analysis of longitudinal epidemic time-series data, showing that this is a partially decomposable system. We applied the Empirical Mode Decomposition (Huang et al., 1998) to analyze 15 consecutive years of the incidence of dengue hemorrhagic fever in Bangkok. A major feature of the Empirical Mode Decomposition is that the method identifies component "modes" of differing frequency, from fast to slow, that together contribute to the full tracing of the epidemic time series. Note that in the Empirical Mode Decomposition, the identified modes are not single standing frequencies, but instead are patterns whose frequencies may vary. For dengue in Bangkok, we identified several major component frequency modes, including a slow 3- to 4-year oscillation that we believe is due to changes in host immunity, a clear 1-year annual oscillation that is probably driven by seasonal changes in weather, and a spiky, irregular, rapid (faster-than-annual) oscillation

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