Stochasticity in staged models of epidemics: quantifying the dynamics of whooping cough

Abstract

Although many stochastic models can accurately capture the qualitative epidemic patterns of many childhood diseases, there is still considerable discussion concerning the basic mechanisms generating these patterns; much of this stems from the use of deterministic models to try to understand stochastic simulations. We argue that a systematic method of analysing models of the spread of childhood diseases is required in order to consistently separate out the effects of demographic stochasticity, external forcing and modelling choices. Such a technique is provided by formulating the models as master equations and using the van Kampen system-size expansion to provide analytical expressions for quantities of interest. We apply this method to the susceptible-exposed -infected -recovered (SEIR) model with distributed exposed and infectious periods and calculate the form that stochastic oscillations take on in terms of the model parameters. With the use of a suitable approximation, we apply the formalism to analyse a model of whooping cough which includes seasonal forcing. This allows us to more accurately interpret the results of simulations and to make a more quantitative assessment of the predictions of the model. We show that the observed dynamics are a result of a macroscopic limit cycle induced by the external forcing and resonant stochastic oscillations about this cycle.

Effects of the infectious period distribution on predicted transitions in childhood disease dynamics

“…(1) Asymptotic analysis (to identify the periods of attractors of the model, which are reached asymptotically) [10][11][12][13][14][15]: construct a bifurcation diagram with R 0 as the control parameter, over a range of R 0 that includes the value estimated for time t 0 and the full range of R 0;eff determined via equation (2.4) (figure 2a). From this diagram, we can easily infer the periods of cyclical attractors of the system.…”

“…Equilibrium stability analyses have been conducted on 'unforced' models that assume constant contact rates [6,7,[29][30][31][32], and bifurcation analyses have been conducted on 'forced' models in which contact rates vary seasonally [6][7][8][9][10][11][12]33]. Lloyd [7] found that the biennial pattern observed in the SI 1 R model is reproduced by the SI n R model but with much weaker seasonality.…”

“…The dynamical effects of exponential versus more realistic distributions of stage durations have been explored extensively in the literature [6][7][8][9][10][11][12], which has revealed that changing the shapes of these distributions while keeping their means fixed can have a large impact on predicted dynamics. Consequently, it is important to re-evaluate any inferences drawn about real data from models that assume exponentially distributed stage durations.…”

Stochastic dynamics and logistic population growth

“…The effects of internal fluctuations have been studied in predator-prey models [15,16], epidemic models [17][18][19][20][21][22][23], cell biology [24], and ecological systems [13]. In particular, extinction of a stochastic population [11,25,26], which is a crucial concern for population biology [27] and epidemiology [28,29], has also attracted scrutiny in cell biochemistry [30] and in physics [31,32].…”

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

“…There has been considerable debate in recent years concerning the most appropriate formulation of the SIR model for childhood infections such as measles [20][21][22][23][24][25][26]. We follow Krylova & Earn [6,27], who showed that for measles, the simplest sinusoidally forced SIR model-when suitably parametrized-makes identical predictions to more complicated versions of the model that include realistically distributed latent and infectious periods (details in §3.1).…”