Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters

Abstract

Summary Influenza A virus (IAV) infection continues to be a global health threat, as evidenced by the outbreak of the novel A/California/7/2009 IAV strain. Previous flu vaccines have proven less effective than hoped for emerging IAV strains, indicating a more thorough understanding of immune responses to primary infection is needed. One issue is the difficulty in directly measuring many key parameters and variables of the immune response. To address these issues, we considered a comprehensive workflow for statistical inference for ordinary differential question (ODE) models with partially observed variables and time-varying parameters, including identifiability analysis, two-stage and NLS estimation, and model selection etc‥ In particular, we proposed a novel one-step method to verify parameter identifiability and formulate estimating equations simultaneously. Thus, the pseudo-LS method can now deal with general ODE models with partially observed state variables for the first time. Using this workflow, we verified the relative significance of various immune factors to virus control, including target epithelial cells, cytotoxic T-lymphocyte (CD8+) cells and IAV specific antibodies (IgG and IgM). Factors other than cytotoxic T-lymphocyte (CTL) killing contributed the most to the loss of infected epithelial cells, though the effects of CTL are still significant. IgM antibody was found to be the major contributor to neutralization of free infectious viral particles. Also, the maximum viral load, which correlates well with mortality, was found to depend more on viral replication rates than infectivity. In contrast to current hypotheses, the results obtained via our methods suggest that IgM antibody and viral replication rates may be worth of further explorations in vaccine development.

Generalized Ordinary Differential Equation Models

“…However, the estimates of ρ E obtained by the ESB method are significantly biased (AREs on the order of 10 7 %), and we think that this parameter cannot be reliably determined without any time-course data for E when the ESB method is used (see the next paragraph for detail). Overall, we conclude that the GODE method has a superior performance over the ESB method in terms of parameter estimation accuracy, which could be due to the approximation error in Ĉ · b ′ ( t ) used in smoothing-based approaches (Liang et al, 2010; Wu et al, 2014); however, we also find that the computing cost of the ESB method is 80% less than that of the GODE method because the ESB method does not use the initial value ODE solver.…”

Robust estimation of constant and time-varying parameters in nonlinear ordinary differential equation models

“…Note that all the state variables are assumed to be observed in this article. In the case that state variables are partially observed, Wu, Miao, Xue, Tppham, and Zand (2014) proposed a novel solution which combines identifiability analysis techniques with smoothing-based approaches. They applied this method to influenza A virus infection data, and eliminated the un-observed state variables by making use of derivatives of observed variables without loss of parameter identifiability.…”