Computational aspects of time-lag models of Marchuk type that arise in immunology

In his book published in English translation in 1983, Marchuk proposed a set of evolutionary equations incorporating delay-differential equations, and the corresponding initial conditions as a model ('Marchuk's model') for infectious diseases. The parameters in this model (and its subsequent extensions) represent scientifically meaningful characteristics. For a given infection, the parameters can be estimated using observational data on the course of the infection. Sensitivity analysis is an important tool for understanding a particular model; this can be viewed as an issue of stability with respect to structural perturbations in the model.

Examining the sensitivity of the models based on delay differential equations leads to systems of neutral delay differential equations. Below we formulate a general set of equations for the sensitivity coefficients for models comprising neutral delay differential equations. We discuss computational approaches to the sensitivity of solutions — (i) sensitivity to the choice of model, in particular, to the lag parameter τ > 0 and (ii) sensitivity to the initial function — of dynamical systems with time lag and illustrate them by considering the sensitivity of solutions of time-lag models of Marchuk type.