Fig. 4. Example of parameter estimation for one individual.
(A) Simulated viral load curves produced using the model (gray lines) with different sets of parameters can be compared against longitudinal data (green points). The model simulation using optimal parameter values (red line) best matches the data (i.e., black bars measuring distance are smallest in total). (B) For any single parameter (here viral infectivity rate ), changing the value of this parameter will change the value of the objective, or scoring, function. Here we show the least-squares objective function (residual sum of squares, RSS) that measures the squared differences between the model solution at time (i.e., ) and the observed data at time (i.e., ) for time points , i.e., the sum of squared black bars in panel (A). The RSS is minimized simultaneously over all parameters to determine the set of parameter values that best fits the observed data. It is possible that a value can admit a low RSS relative to the nearby values (e.g., ), but not achieve a full “global” minimum . Certain numerical algorithms are more robust to escaping such local minima and finding the global value. (C) The parameter values that produce the optimal simulation curve provide estimates of the rates governing the viral dynamics system. Statistics like 95% confidence intervals can also be constructed using the RSS framework.