Determination of threshold conditions for a non-linear stochastic partnership model for heterosexually transmitted diseases with stages

• Published in: Mathematical biosciences 2002-05, Vol.177 (1-2), p.287-315
• Main Authors: Gallop, Robert J., Mode, Charles J., Sleeman, Candace K.
• Format: Article
• Language: English
• Subjects: Acquired Immunodeficiency Syndrome - epidemiology; Acquired Immunodeficiency Syndrome - transmission; Algorithms; Computer Simulation; Differential equations; Eigenvalues and eigenfunctions; Embedded deterministic models; Extra-marital sexual contacts; Extramarital Relations; Female; Heterosexual; Heterosexuality; HIV - growth & development; HIV/AIDS; Humans; Male; Marital partnerships; Marriage; Models, Immunological; Monte Carlo methods; Monte Carlo simulation; Parameter estimation; Random processes; Search engines; Stochastic models; Stochastic Processes; System stability; Threshold conditions
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/S0025-5564(01)00093-1

When comparing the performance of a stochastic model of an epidemic at two points in a parameter space, a threshold is said to have been crossed when at one point an epidemic develops with positive probability; while at the other there is a tendency for an epidemic to become extinct. The approach used to find thresholds in this paper was to embed a system of ordinary non-linear differential equations in a stochastic process, accommodating the formation and dissolution of marital partnerships in a heterosexual population, extra-marital sexual contacts, and diseases such as HIV/AIDS with stages. A symbolic representation of the Jacobian matrix of this system was derived. To determine whether this matrix was stable or non-stable at a particular parameter point, the Jacobian was evaluated at a disease-free equilibrium and its eigenvalues were computed. The stability or non-stability of the matrix was then determined by checking if all real parts of the eigenvalues were negative. By writing software to repeat this process for a selected set of points in the parameter space, it was possible to develop search engines for finding points in the parameter space where thresholds were crossed. The results of a set of Monte Carlo simulation experiments were reported which suggest that, by combining the stochastic and deterministic paradigms within a single formulation, it was possible to obtain more informative interpretations of simulation experiments than if attention were confined solely to either paradigm.