# Introduction

In recent years, the use of differential equations to describe the dynamics of within-host viral infections, most frequently HIV-1 or Hepatitis B or C dynamics, has become quite common. The pioneering work described in [1,2,3,4] provided estimates of both the HIV-1 viral clearance rate, c, and infected cell turnover rate, 5, and revealed that while it often takes years for HIV-1 infection to progress to AIDS, the virus is replicating rapidly and continuously throughout these years of apparent latent infection. In addition, at least two compartments of viral-producing cells that decay at different rates were identified. Estimates of infected cell decay and viral clearance rates dramatically changed the understanding of HIV replication, etiology, and pathogenesis. Since that time, models of this type have been used extensively to describe and predict both in vivo viral and/or immune system dynamics and the transmission of HIV throughout a population. However, there are both mathematical and statistical challenges associated with models of this type, and the goal of this chapter is to describe some of these as well as offer possible solutions or options. In particular statistical aspects associated with parameter estimation, model comparison and study design will be described. Although the models developed by Perelson et al. [3,4] are relatively simple and were developed nearly 20 years ago, these models will be used in this chapter to demonstrate concepts in a relatively simple setting. In the first section, a statistical approach for model comparison is described using the model developed in [4] as the null hypothesis model for formal statistical comparison to an alternative model. In the next section, the concept of the mathematical sensitivity matrix and its relationship to the Fisher information matrix (FIM) will be described, and will be used to demonstrate how to evaluate parameter identifiability in ordinary differential equation (ODE) models. The next section demonstrates how to determine what types of additional data are required to address the problem of nonidentifiable parameters in ODE models. Examples are provided to demonstrate these concepts. The chapter ends with some recommendations.

Throughout the remainder of this chapter, the term "compartments" refers to the time-varying states in a system of ODEs and the term "parameters" refers to the constants (either known or unknown) in the vector field which defines the specific system of ODEs. The following notation will be used. The m states of an ODE with time t as independent variable will be described by X(t) = (X1(t), . . Xm(t)) with a (possibly unknown) parameter vector

© = (01/ • • •, 0p) and vector field, f, which describes the system. The general system of ODEs is described as:

Given the true states X(t), one then observes noisy data from one or more of the states

for j = 1, 2, . . n, where e(tj ) is the measurement error assumed to be independently and identically distributed with zero mean.