Nonlinear Parameter Optimization Using R Tools

Chapter 20Differential equation models

This chapter follows steps that attempt to illustrate the estimation of an SIWR model from the problem proposed by Marisa Eisenberg in “Lab Session: Intro to Parameter Estimation & Identifiability, 6–17–13.” This serves as a useful overview of dealing with differential equation models and their estimation.

20.1 The model

The description here refers directly to the program code below. The model aims to explain the numbers of susceptible (S), infected (I), and recovered (R) in a cholera outbreak. The particular model includes a variable that measures bacteria concentration in water (). This leads to the SIWR model developed by Tien and Earn 2010, Bi and Bw are parameters that describe the direct (human–human) and indirect (waterborne) cholera transmission. e is the pathogen decay rate in the water.

We do not measure I directly, but a scaled version of it

y = I/k

where k is a combination of the reporting rate, the asymptomatic rate, and the total population size.

Our equations include a recovery rate and a birth–death parameter, but in the present situation, the description of the problem states that the former is fixed at 0.25 and the latter is 0.

Unfortunately, the code suggested fails to do a good job of optimizing the cost function. An e-mail exchange between myself and Nathan Lemoine that prompted this investigation was initiated with ...