More Like this

1. Geometrically Motivated Reparameterization for Identifiability Analysis in Power Systems Models

   https://doi.org/10.1109/NAPS.2018.8600617  

   Transtrum, Mark K.; Francis, Benjamin L.; Youn, Clifford C.; Saric, Andrija T.; Stankovic, Aleksander M. (September 2018, 2018 North American Power Symposium (NAPS))

   This paper describes a geometric approach to parameter identifiability analysis in models of power systems dynamics. When a model of a power system is to be compared with measurements taken at discrete times, it can be interpreted as a mapping from parameter space into a data or prediction space. Generically, model mappings can be interpreted as manifolds with dimensionality equal to the number of structurally identifiable parameters. Empirically it is observed that model mappings often correspond to bounded manifolds. We propose a new definition of practical identifiability based the topological definition of a manifold with boundary. In many ways, our proposed definition extends the properties of structural identifiability. We construct numerical approximations to geodesics on the model manifold and use the results, combined with insights derived from the mathematical form of the equations, to identify combinations of practically identifiable and unidentifiable parameters. We give several examples of application to dynamic power systems models. 

   more » « less
2. The Structural Identifiability of a Humidity-Driven Epidemiological Model of Influenza Transmission

   https://doi.org/10.3390/v14122795  

   Zhang, Chunyang; Zhang, Xiao; Bai, Yuan; Lau, Eric H.; Pei, Sen (December 2022, Viruses)

   Influenza epidemics cause considerable morbidity and mortality every year worldwide. Climate-driven epidemiological models are mainstream tools to understand seasonal transmission dynamics and predict future trends of influenza activity, especially in temperate regions. Testing the structural identifiability of these models is a fundamental prerequisite for the model to be applied in practice, by assessing whether the unknown model parameters can be uniquely determined from epidemic data. In this study, we applied a scaling method to analyse the structural identifiability of four types of commonly used humidity-driven epidemiological models. Specifically, we investigated whether the key epidemiological parameters (i.e., infectious period, the average duration of immunity, the average latency period, and the maximum and minimum daily basic reproductive number) can be uniquely determined simultaneously when prevalence data is observable. We found that each model is identifiable when the prevalence of infection is observable. The structural identifiability of these models will lay the foundation for testing practical identifiability in the future using synthetic prevalence data when considering observation noise. In practice, epidemiological models should be examined with caution before using them to estimate model parameters from epidemic data. 

   more » « less
3. Identifiability and Parameter Estimation of Within-Host Model of HIV with Immune Response

   https://doi.org/10.3390/math12182837  

   Liyanage, Yuganthi R; Mirsaleh_Kohan, Leila; Martcheva, Maia; Tuncer, Necibe (September 2024, Mathematics)

   This study examines the interactions between healthy target cells, infected target cells, virus particles, and immune cells within an HIV model. The model exhibits two equilibrium points: an infection-free equilibrium and an infection equilibrium. Stability analysis shows that the infection-free equilibrium is locally asymptotically stable when R0<1. Further, it is unstable when R0>1. The infection equilibrium is locally asymptotically stable when R0>1. The structural and practical identifiabilities of the within-host model for HIV infection dynamics were investigated using differential algebra techniques and Monte Carlo simulations. The HIV model was structurally identifiable by observing the total uninfected and infected target cells, immune cells, and viral load. Monte Carlo simulations assessed the practical identifiability of parameters. The production rate of target cells (λ), the death rate of healthy target cells (d), the death rate of infected target cells (δ), and the viral production rate by infected cells (π) were practically identifiable. The rate of infection of target cells by the virus (β), the death rate of infected cells by immune cells (Ψ), and antigen-driven proliferation rate of immune cells (b) were not practically identifiable. Practical identifiability was constrained by the noise and sparsity of the data. Analysis shows that increasing the frequency of data collection can significantly improve the identifiability of all parameters. This highlights the importance of optimal data sampling in HIV clinical studies, as it determines the best time points, frequency, and the number of sample points required to accurately capture the dynamics of the HIV infection within a host. 

   more » « less
4. Parameter Estimation in the Age of Degeneracy and Unidentifiability

   https://doi.org/10.3390/math10020170  

   Lederman, Dylan; Patel, Raghav; Itani, Omar; Rotstein, Horacio G. (January 2022, Mathematics)

   Parameter estimation from observable or experimental data is a crucial stage in any modeling study. Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Structural unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern. Therefore, the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments. Classical studies on structural unidentifiability focused on the notion that one can at most identify combinations of unidentifiable model parameters. We have identified a different type of structural degeneracy/unidentifiability present in a family of models, which we refer to as the Lambda-Omega (Λ-Ω) models. These are an extension of the classical lambda-omega (λ-ω) models that have been used to model biological systems, and display a richer dynamic behavior and waveforms that range from sinusoidal to square wave to spike like. We show that the Λ-Ω models feature infinitely many parameter sets that produce identical stable oscillations, except possible for a phase shift (reflecting the initial phase). These degenerate parameters are not identifiable combinations of unidentifiable parameters as is the case in structural degeneracy. In fact, reducing the number of model parameters in the Λ-Ω models is minimal in the sense that each one controls a different aspect of the model dynamics and the dynamic complexity of the system would be reduced by reducing the number of parameters. We argue that the family of Λ-Ω models serves as a framework for the systematic investigation of degeneracy and identifiability in dynamic models and for the investigation of the interplay between structural and other forms of unidentifiability resulting on the lack of information from the experimental/observational data. 

   more » « less
5. Algebraic identifiability of partial differential equation models

   https://doi.org/10.1088/1361-6544/ada510  

   Byrne, Helen M; Harrington, Heather A; Ovchinnikov, Alexey; Pogudin, Gleb; Rahkooy, Hamid; Soto, Pedro (January 2025, Nonlinearity)

   Differential equation models are crucial to scientific processes across many disciplines, and the values of model parameters are important for analyzing the behaviour of solutions. Identifying these values is known as a parameter estimation, a type of inverse problem, which has applications in areas that include industry, finance and biomedicine. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. Checking the global identifiability of model parameters is a useful tool when exploring the well-posedness of a given model. This problem has been intensively studied for ordinary differential equation models, where theory, several efficient algorithms and software packages have been developed. A comprehensive theory for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences. 

   more » « less