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

Abstract

The population dynamics of infectious diseases occasionally undergo rapid qualitative changes, such as transitions from annual to biennial cycles or to irregular dynamics. Previous work, based on the standard seasonally forced 'susceptible-exposed-infectious -removed' (SEIR) model has found that transitions in the dynamics of many childhood diseases result from bifurcations induced by slow changes in birth and vaccination rates. However, the standard SEIR formulation assumes that the stage durations (latent and infectious periods) are exponentially distributed, whereas real distributions are narrower and centred around the mean. Much recent work has indicated that realistically distributed stage durations strongly affect the dynamical structure of seasonally forced epidemic models. We investigate whether inferences drawn from previous analyses of transitions in patterns of measles dynamics are robust to the shapes of the stage duration distributions. As an illustrative example, we analyse measles dynamics in New York City from 1928 to 1972. We find that with a fixed mean infectious period in the susceptible -infectious -removed (SIR) model, the dynamical structure and predicted transitions vary substantially as a function of the shape of the infectious period distribution. By contrast, with fixed mean latent and infectious periods in the SEIR model, the shapes of the stage duration distributions have a less dramatic effect on model dynamical structure and predicted transitions. All these results can be understood more easily by considering the distribution of the disease generation time as opposed to the distributions of individual disease stages. Numerical bifurcation analysis reveals that for a given mean generation time the dynamics of the SIR and SEIR models for measles are nearly equivalent and are insensitive to the shapes of the disease stage distributions.

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

“…Because the mean latent period for measles (T lat ≃ 8 days, [1]) is long relative to the mean infectious period (T inf ≃ 5 days, [1]), it is natural to assume that the SEIR model will represent measles dynamics better than the simpler SIR model, which includes no latent period. However, Krylova & Earn [27] showed that the SIR model displays virtually identical dynamics to the SEIR if the length of the mean generation interval (≃13 days) is used for the mean infectious period in the SIR model. This dynamical correspondence holds also for versions of the SIR and SEIR model with realistically distributed stage durations, rather than the exponential distributions implied by equation (3.1) [27].…”

“…We use the simple sinusoidal forcing function (3.3) rather than more realistic term-time forcing [5,17] because the two yield virtually identical dynamics (for different a values) [5,27]. Introducing time-dependence into the transmission rate generally affects the basic reproduction number R 0 [30,31]; however, for the SIR model (3.1), inserting the mean transmission rate b 0 in place of b in equation (3.2) is correct [32].…”

“…As discussed below, N 0 refers to the population size at a particular 'anchor time' t 0 (as opposed to the current total populations size N ¼ S þ I þ R). We will interpret the 'birth rate' n more generally as the per capita rate of susceptible recruitment (relative to N 0 ) [2,4,5,13,27]. Here m represents the per capita rate of death from natural causes (disease-induced mortality is assumed to be negligible).…”

“…If births balance deaths, then for the SIR model R 0 ¼ bN=(g þ m) [1]. In our situation (3.1), typically n = m and [27] …”

“…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).…”