The proposed analytical solution for identifiability in nonlinear ODE models with time-varying parameters can be extended to stochastic ODE models by incorporating stochastic processes into the system characterization. This involves modifying the deterministic framework to account for random perturbations in the state dynamics and the parameters. Specifically, one can introduce stochastic terms, such as Wiener processes or Poisson processes, into the equations governing the state evolution and the output equations.
To achieve this, the system can be represented as follows:
[
\\begin{align*}
dX(t) &= f(X(t), t, U(t), Q, W(t)) dt + \\sigma(X(t), t) dB(t) \\
Y(t) &= h(X(t), t, U(t), Q, W(t)) + \\epsilon(t),
\\end{align*}
]
where (dB(t)) represents a stochastic process (e.g., Brownian motion), and (\\epsilon(t)) is a noise term affecting the output. The identifiability analysis would then need to consider the stochastic nature of the parameters and the state, potentially using tools from stochastic calculus, such as Itô's lemma, to derive conditions for identifiability in the presence of noise.
Moreover, the observability and identifiability conditions would need to be reformulated to account for the stochastic inputs and outputs, possibly leading to a new set of observability matrices that incorporate the stochastic terms. This extension would allow for a more comprehensive understanding of how uncertainty in biological systems, such as those modeled in viral dynamics, affects parameter estimation and system behavior.