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If a proportion p of newborns is vaccinated then Eqs. 1a and 1d become dS/dt = ν(1 - p) - (βl + μ)S dR/dt = νρ + γl - μR and the other SEIR equations are unchanged. Now consider a simple change of variables: S = S′ (1 - p), E = E′(1 - p), l = l′(1 - p), and R = R′(1 - p) + (ν/μ)p. The dynamical equations for the primed variables are exactly the SEIR equations without vaccination, i.e., Eqs. 1, except that the transmission rate β is replaced everywhere by β(1 - p). Thus, except for an overall reduction in the number of cases, when a proportion of a population is vaccinated the dynamics are identical to those of an unvaccinated SEIR system with a smaller mean transmission rate, 〈β〉 → 〈β〉 szlig;> (1 - p). Because birth rate ν enters the dS/dt equation in exactly the same place as vaccination, a similar argument applies to changes in birth rate. A change in birth rate from ν to ν′ is dynamically equivalent to a change from 〈β〉 to 〈β〉 (ν′/ν). This is why we can predict the character of the population dynamics of measles (or other infections) after vaccination or birth rate changes, and explain a wide variety of data with a single diagram (Fig. 1).
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0342438675
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Good fits to measles incidence times series can be obtained with the SEIR model only if term-time forcing is used. However, in (10) we show that the sequence of bifurcations in the term-time forced SEIR model is reproduced by the sinusoidally forced SEIR model (at much lower forcing amplitude). A relevant bifurcation diagram can therefore be generated using sinusoidal forcing, though only if the appropriate amplitude is determined, which is a somewhat subtle procedure (10). Most previous discussions of measles dynamics have assumed sinusoidal seasonal forcing with an amplitude of 0.28, which yields a chaotic attractor (3); this level of forcing is about three times higher than it should be. A term-time forcing amplitude of 0.25 (used in Fig. 1) corresponds to a sinusoidal forcing amplitude of roughly 0.08 (10).
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36
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0342873241
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Because the seasonal amplitude cannot be measured precisely (and longer time series are not available) we will probably never be certain whether periods of irregularity in measles time series arise from stochastic hopping between coexisting attractors or stochastic interactions with (possibly chaotic) repellors. However, it seems that irregular behavior must be induced to some extent by noise, i.e., the source of irregularity cannot be a chaotic attractor. Seasonal amplitudes above the minimum required for a chaotic attractor to be reached appear to be ruled out by the data (10, 13).
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We thank S. Balshine, O. Bjørnstad, B. F. Finkenstädt, R. A. Johnstone, S. A. Levin, D. A. Price, and the anonymous referees for helpful comments. Supported by a Wellcome Trust postdoctoral research fellowship in mathematical biology (D.J.D.E.), a Natural Environment Research Council postdoctoral research fellowship and a Royal Society University Research Fellowship (P. R.), and the Wellcome Trust (B.T.G.).
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