Methodology of the sensitivity analysis used for modeling an infectious disease
Introduction
For a certain number of infectious diseases, little is known about disease transmission and limited observational data exist; thus, mathematical models may be used to clarify the dynamics of those infectious diseases [1], [2], [3], [4], [5], [6]. Infectious disease models are basically “age-period-cohort” models, to describe the effect of age and time. These models are thus age-structured models and the epidemiological impact of intervention such as vaccine implementation is assessed at specific times after introduction of the vaccination. Disease-specific models may combine various data: demographical, epidemiological, geographical, and spatial data as well as the results of previous clinical trials [7], [8]. The results in terms of incidence, morbidity or mortality depend on the model structure (compartments and flows) as well as on the model parameter values. Imprecise parameters and the extreme sensitivity of many mathematical systems to small changes in parameter values often explain why modeling infectious disease can considerably bias the disease transmission or the assessment of the impact of infection control measures. To avoid wrong conclusions, a sensitivity analysis is necessary; it consists in assessing how model assumptions and parameter uncertainties can alter model results [2], [3], [9].
The lack of robust sensitivity analysis approaches can disturb or mislead in the research process [2], [3]. Well-structured sensitivity analyses applied to epidemiological work are still scarce. On the one hand, serious difficulties arise when a large number of parameters are involved in a model [10]; a high number of computer simulations depending on the number of parameters tested in the model increase the probabilities of wrong conclusions. On the other hand, parameters with important uncertainties are attractive and supposed to influence the outcomes but small changes in parameter values may be associated with large changes in the outcome results; amplification or, on the contrary, attenuation. The outcome results are influenced by the parameters, thus, the parameters to which the results are the most sensitive to are to be objectively assessed.
Hence, a rigorous sensitivity analysis is essential to model the dynamics of a given infectious disease. The factors to involve should be first identified using literature data. The impact of these parameters on the outcome results should then be tested using logic and modeling: impact between individuals of same age, interaction between cohorts during the same period, impact between periods, impact on younger or older subjects (upstream–downstream impact), and immediate-delayed impact. This study consists in proposing a rigorous framework for sensitivity analyses applied to a mathematical model that assesses varicella and herpes zoster incidence as an example. A methodology of sensitivity analysis, based on parameter selection for sensitivity analysis, univariate sensitivity analysis, and multivariate sensitivity analysis, is presented.
Section snippets
Identification of the factors to involve
The first step consists in identifying the values of the assumption and parameters (or factors) which appear to be uncertain and would be likely to influence the outcome results. This typically involves reviewing the existing literature to identify what factors have been previously explored. These parameters may be grouped into different types: demographic, biological, and interventional (e.g., mass vaccination campaign) and each should be assigned a mean value. The chosen mean values may
Parameter selection for sensitivity analysis
The uncertainties concerning most of the VZV model parameters were large: 8/8 biological parameters and 6/7 varicella vaccine-related parameters had a relative deviation varying from 18 to 200%; only a little proportion of parameters had a small relative deviation ranging from 2 to 13% (Table 1, Table 2).
Discussion
Epidemiological models are useful to represent infection processes and assess the spread of a pathogen in a population. The complexity of some models and the high degree of uncertainty in estimating the values of many parameters require a careful selection of the parameters to include in a quantitative analysis and a systematic approach in quantifying the relationships between the factors studied and the outcomes. We propose here a general framework for sensitivity analysis in the context of
Acknowledgments
Conflict of interest statement: Marie-Laure Kürzinger, Hélène Bricout, Tarik Derrough and François Simondon were employed by Sanofi Pasteur MSD. The authors gratefully thank Jean Iwaz for his valuable input and his critical comments on the manuscript.
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