Outcome prediction in mathematical models of immune response to infection

TL;DR

This study investigates the limits of predicting patient outcomes in immune response models, showing that variability among patients fundamentally restricts early prediction accuracy, but accuracy improves with delayed prognosis.

Contribution

It quantifies how patient variability affects outcome prediction accuracy in mathematical immune response models and explores the trade-off between early prediction and accuracy.

Findings

Prediction accuracy approaches 100% when variability is zero.

Increasing elapsed time improves prediction accuracy.

External noise decreases prediction accuracy.

Abstract

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of `virtual patients', each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability $v$ in the ODE models by randomly selecting the model parameters from Gaussian distributions with variance $v$ that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near $100\%$ accuracy for $v=0$, and the accuracy decreases with increasing $v$ for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for $v>0$. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.