(594a) Development of Validated Methods for Initial Value Problems Using A Constraint Satisfaction Approach

ODE systems describing biological systems usually contain parameters that are only approximately known. Rather than describing these parameters as having point values, they are more accurately described as falling within a range of values. There are often orders of magnitude variations in the parameters describing biological systems. Hence, it is of the utmost importance to employ interval ranges rather than point values to represent the system parameters. However, to determine the solution space for a range of parameter values requires an infinite number of computer runs, since every point in the range has to be sampled. Interval methods also referred to as validated methods, for initial value ODE problems (IVPs) are guaranteed to enclose solutions arising from any combination of ranges of parameter values.

To better study biological systems, a new hybrid approach for carrying interval analysis was developed. Unlike other interval analysis techniques, such as those used by VNODE and VSPODE, our approach utilizes a combination of explicit symbolic and numerical strategies. Whereas VNODE is based on the traditional Interval Taylor Series (ITS) method, and VSPODE combines ITS and Taylor model approaches, our strategy utilizes Taylor models together with a branch and prune algorithm. Regardless of the approach used, ultimately each method converges in a limited time interval.

In this presentation we will discuss the use of interval methods for solving a biological system represented by IVPs and compare the results of the different approaches. We applied these IVP approaches to analyze the infection dynamics of the lytic RNA bacteriophage, MS2, and its interaction with its host, Escherichia coli. The model developed focused on intercellular dynamics and accounted for uninfected cells (sensitive and resistant type), infected cells, free phage and substrate (glucose) concentration [1]. The IVP system was discretized into a set of polynomial algebraic equations using Taylor model integration methods. The final sets of equations were solved by an interval solver (RealPaver [2]) that employs constraint satisfaction techniques. The results were useful in identifying the most sensitive parameters and the amount of uncertainty that could be associated with each.

References:

1) R. Jain, A.L. Knorr and R. Srivastava, Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination Analysis and the Implications for Phage Therapy, Biot. Prog. 22(6): 1650-1658.

2) L. Granvilliers and F. Benhamou, RealPaver: An Interval Solver Using Constraint Satisfaction Techniques, ACM Trans. On Math. Software, 32 (2006) 138-156.