Determination of threshold conditions for a non-linear stochastic partnership model for heterosexually transmitted diseases with stages.
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.
Gallop
Robert J
RJ
Department of Mathematics and Applied Statistics, West Chester University, West Chester, PA 19383, USA. rgallop@wcupa.edu
Mode
Charles J
CJ
Sleeman
Candace K
CK
D000163:Acquired Immunodeficiency Syndrome / Q000453:epidemiology* / Q000635:transmission; D000465:Algorithms; D003198:Computer Simulation; D005114:Extramarital Relations; D005260:Female; D006678:HIV / Q000254:growth & development*; D020010:Heterosexuality; D006801:Humans; D008297:Male; D008393:Marriage; D018448:Models, Immunological; D013269:Stochastic Processes