Transient sensitivity analysis for nonlinear population models

Summary 1. Performing sensitivity analyses for deterministic multi-stage, multi-parameter population models that include nonlinearity presents a significant computational challenge. 2. We implement a standard forward sensitivity analysis to evaluate partial derivatives of population state variables with respect to parameters at all times for which the solution is known. We present a hybrid Matlab/Maple software package, sensai, which automates the steps required to calculate sensitivities and elasticities. We focus on sensitivities of transient states of populations. Our analysis therefore differs from the more common analysis of linear systems where sensitivities are computed for the asymptotic stable-stage distribution assuming unbounded population growth. The method generalizes the matrix calculus methods for nonlinear, stage-structured matrix models previously developed by Caswell (2009, Journal of Difference Equations and Applications, 15 ,3 49– 369). 3. We present an example of a nonlinear, discrete-time population model in the form of a stagestructured Lefkovitch matrix, and another multi-stage, continuous-time SIR disease model in the form of a system of ordinary differential equations. In addition, we extend the framework for composite nonlinear maps of population demography to include single-locus genetics with natural selection discriminating among genotypes. 4. SENSAI allows for the analysis of a quantity of interest (an arbitrary function of variables and parameters, e.g. the proportion of infected individuals in a disease model), and sensitivity to combinations of model parameters. For example, we can compute sensitivities with respect to parameters and initial conditions at all times during an outbreak in a disease model, before a steady state of disease prevalence is attained. Similarly, in population genetic models, we can compute sensitivity of a response to natural selection (e.g. the change in allele frequencies) in relation to fitness differences among genotypes. 5. We illustrate the capabilities of sensai through a series of canonical models with relatively few variables and parameters. However, sensai has the capability to analyse significantly more complex models.