Abstract
Mathematical models play a central role in epidemiology. For example, models unify heterogeneous data into a single framework, suggest experimental designs, and generate hypotheses. Traditional methods based on deterministic assumptions, such as ordinary differential equations (ODE), have been successful in those scenarios. However, noise caused by random variations rather than true differences is an intrinsic feature of the cellular/molecular/social world. Time series data from patients (in the case of clinical science) or number of infections (in the case of epidemics) can vary due to both intrinsic differences or incidental fluctuations. The use of traditional fitting methods for ODEs applied to noisy problems implies that deviation from some trend can only be due to error or parametric heterogeneity, that is noise can be wrongly classified as parametric heterogeneity. This leads to unstable predictions and potentially misguided policies or research programs. In this paper, we quantify the ability of ODEs under different hypotheses (fixed or random effects) to capture individual differences in the underlying data. We explore a simple (exactly solvable) example displaying an initial exponential growth by comparing state-of-the-art stochastic fitting and traditional least squares approximations. We also provide a potential approach for determining the limitations and risks of traditional fitting methodologies. Finally, we discuss the implications of our results for the interpretation of data from the 2014-2015 Ebola epidemic in Africa.
Highlights
Mathematical models play an increasingly central role in the analysis of infectious disease data at both the within-host and epidemiological levels (Perelson et al, 1996; Heesterbeek, 2000; MolinaParís and Lythe, 2011)
We address in this paper two related questions regarding modeling of panel data: (i) can we use a stochastic modeling approach to partition variability into stochastic and parametric components? and (ii) can we quantify the bias induced by modeling the data by a deterministic approach with error? Put in other words, is there a best and a good-enough fitting method for the practitioner? In section 2.1, we consider two simple structural models that will help us emphasize the essence of the problem without having to invoke unnecessary complexities that may cloud our main arguments
The density of the number of incident cases in the kth time period of the jth unit, xj,k, is assumed to be Poisson(xj,k|Ij,k−1α) were Ij,k is the simulated number of extant infected cases in the kth time period of the jth unit and α is the growth rate, which itself may be sampled from a Gamma distribution
Summary
Mathematical models play an increasingly central role in the analysis of infectious disease data at both the within-host and epidemiological levels (Perelson et al, 1996; Heesterbeek, 2000; MolinaParís and Lythe, 2011). The traditional modeling approach involves formulating a set of structural assumptions about the processes involved, such as infection, recovery, death, etc. Often, these structural assumptions are implemented in terms of differential equations, predominantly ordinary (ODE), but sometimes partial (PDE), or delayed (dODE) differential equations. These structural assumptions are implemented in terms of differential equations, predominantly ordinary (ODE), but sometimes partial (PDE), or delayed (dODE) differential equations The advantage of this approach is its amenability for both analytical treatment and powerful numerical and fitting algorithms even for non-linear problems. We will refer to those approaches collectively as deterministic
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