Abstract
ABSTRACTErlang differential equation models of epidemic processes provide more realistic disease-class transition dynamics from susceptible (S) to exposed (E) to infectious (I) and removed (R) categories than the ubiquitous SEIR model. The latter is itself is at one end of the spectrum of Erlang SEIR models with concatenated E compartments and concatenated I compartments. Discrete-time models, however, are computationally much simpler to simulate and fit to epidemic outbreak data than continuous-time differential equations, and are also much more readily extended to include demographic and other types of stochasticity. Here we formulate discrete-time deterministic analogs of the Erlang models, and their stochastic extension, based on a time-to-go distributional principle. Depending on which distributions are used (e.g. discretized Erlang, Gamma, Beta, or Uniform distributions), we demonstrate that our formulation represents both a discretization of Erlang epidemic models and generalizations thereof. We consider the challenges of fitting SEIR models and our discrete-time analog to data (the recent outbreak of Ebola in Liberia). We demonstrate that the latter performs much better than the former; although confining fits to strict SEIR formulations reduces the numerical challenges, but sacrifices best-fit likelihood scores by at least 7%.
Highlights
An SEIR epidemic modeling framework, where S, E, I and R represent the named susceptible (S), exposed (E: infected but not yet infectious), infectious (I), and removed (R: either dead or recovered and immune) disease classes—and italicized letters S, E, I, and R represent the number of individuals in each of these classes—underpins all infectious disease modeling at the population level [1, 2]
Continuous-time Erlang SEnImR models may be aesthetically appealing because epidemic events can arise at any time, and humped-shaped specific diseaseclass residence times provide better fits to real data than the exponentially distributed residence times associated with the standard single component SEIR model [14,15]
Data is usually aggregated and binned into discrete time intervals, for convenience, and because the precise times of events is not usually known
Summary
An SEIR epidemic modeling framework, where S, E, I and R represent the named susceptible (S), exposed (E: infected but not yet infectious), infectious (I), and removed (R: either dead or recovered and immune) disease classes—and italicized letters S, E, I, and R represent the number of individuals in each of these classes—underpins all infectious disease modeling at the population level [1, 2]. This latter approach can be modeled using a so-called boxcar formulation: the process by which individuals pass through each disease state (i.e., compartment) X is represented by individuals passing through kX concatenated subcompartments at a rate γX in each of the sub-compartments [19,21,22,23] (Figure 1) In this case, the number of individuals n(t) entering at time t still in state X at time t+τ is not given by an exponential function, but rather n(t) multiplied by (1 − FErlang(τ, kX, γX)), where FErlang(τ, kX, γX) is the cumulative Erlang distribution with parameters kX and γX. It remains unclear precisely which circumstances enable the convergence between the mean of an ensemble of stochastic instantiations and the solution of its discrete counterpart This convergence depends, at least, upon the following three aspects: 1) the number of simulations n used to generate the mean; 2) the size N of the population involved; and 3) the value of R0 associated with the value of the parameters used in the model. Due to computational limitations, the most efficient method likely involves fitting a discrete deterministic model first, and exploring the potential spread that an epidemic can exhibit around those resulting parameters using a stochastic formulation of the model
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