Systems of differential equations have a key role in biological and chemical models. These models come with parameters that show the model’s dependency on the environmental effects and often have unknown values. Model simulations from observation are desired not to be affected by the values of the parameters. In other words, we would like the parameters to be identifiable from the input-output behaviour of the system. The identifiability of parameters is the first step towards estimating parameters from observations, which is a main problem in biological models. If the parameters of a model are unidentifiable, then non-unique values of parameters will give the same fit for the observations.
The parameter identifiability problem has been a popular problem and many scientists have worked on this problem. There are many algorithms proposed for ordinary differential equation (ODE) models. A quite standard method is to look at the Taylor series of the ODE model. Alternatively, many algorithms for this problem are based on computational algebra, as many differential equations of biological models have polynomials on their right-hand side. One common method is to solve for the parameters using Gröbner bases of the ideal of the polynomials in the model. Gröbner bases are special generating sets with many properties that help with solving systems of polynomial equations.
Other than those methods, differential algebra is useful in studying the identifiability problem. In differential algebra, derivatives are treated as new variables and a differential ring is formed which is equipped with a differentiation operator. Then a differential ideal can be formed using the equations of the model and their derivatives so that the ideal is closed under differentiation. Having the differential ideal (whose finite representation is just the mode equation), one can use tools in differential algebra, in particular, the Rosenfeld–Gröbner algorithm, which is roughly speaking the differential version of Gröbner bases and is used for solving the identifiability problem.
Most of the work on the identifiability problem has been on the ODE models and there is little research done on the partial differential equation (PDE) models (and also the PDE models are more complicated). However, there is an increase in the availability of spatio-temporal data that makes PDE models in biology more interesting. In a joint work, myself, together with fellow Oxford Mathematicians Helen Byrne, Heather Harrington, and colleagues Alexey Ovchinnikov, Gleb Pogudin and Pedro Soto, studied parameter identifiability for PDE models. Our goal is to extend the existing algebraic approaches for the ODE models to spatial systems.
We presented a method for testing parameter identifiability in PDE models. Our algorithm is based on differential algebra and uses the Rosenfeld–Gröbner algorithm. For experiments, we considered different types of PDEs (parabolic, elliptic and hyperbolic), with a particular focus on parabolic PDEs that come from mathematical biology. In particular, we applied our method to Fisher’s equation, the coupled reaction-diffusion equations system, and a reaction-diffusion system and showed that they are all identifiable. Our algorithm is based on symbolic computations, which can be too demanding in some cases. However, we demonstrate how numeric computations with random values of parameters give evidence for identifiability in reasonable computing time.
Hamid Rahkooy is a Postdoctoral Research Associate in the Machine Learning and Data Science Research Group in Oxford.