A multi-stage compartmental model for HIV-infected individuals: II--application to insurance functions and health-care costs.

Stochastic population processes have received a lot of attention over the years. One approach focuses on compartmental modeling. Billard and Dayananda (2012) developed one such multi-stage model for epidemic processes in which the possibility that individuals can die at any stage from non-disease related causes was also included. This extra feature is of particular interest to the insurance and health-care industries among others especially when the epidemic is HIV/AIDS. Rather than working with numbers of individuals in each stage, they obtained distributional results dealing with the waiting time any one individual spent in each stage given the initial stage. In this work, the impact of the HIV/AIDS epidemic on several functions relevant to these industries (such as adjustments to premiums) is investigated. Theoretical results are derived, followed by a numerical study.

A multi-stage compartmental model for HIV-infected individuals: I--waiting time approach.

Traditionally, epidemic processes have focused on establishing systems of differential-difference equations governing the number of individuals at each stage of the epidemic. Except for simple situations such as when transition rates are linear, these equations are notoriously intractable mathematically. In this work, the process is described as a compartmental model. The model also allows for individuals to go directly from any prior compartment directly to a final stage corresponding to death. This allows for the possibility that individuals can die earlier due to some non-disease related cause. Then, the model is based on waiting times in each compartment. Survival probabilities of moving from a given compartment to another compartment are established. While our approach can be used for general epidemic processes, our framework is for the HIV/AIDS process. It is then possible to establish the impact of the HIV/AIDS epidemic process on, e.g., insurance premiums and payouts and health-care costs. The effect of changing model parameter values on these entities is investigated.

A stochastic model for PSA levels: behavior of solutions and population statistics.

This paper investigates the partial differential equation for the evolving distribution of prostate-specific antigen (PSA) levels following radiotherapy. We also present results on the behavior of moments for the evolving distribution of PSA levels and estimate the probability of long-term treatment success and failure related to values of treatment and disease parameters. Results apply to a much wider range of parameter values than was considered in earlier studies, including parameter combinations that are patient specific.

A stochastic model for prostate-specific antigen levels.

We introduce a continuous stochastic model for the prostate-specific antigen (PSA) levels following radiotherapy and derive solutions for the associated partial differential (Kolmogorov-Chapman) equation. The solutions describe the evolution of the time-dependent density for PSA levels which take into account an absorbing condition along the boundary and various initial conditions. We include implications for single-dose and multi-dose radiation treatment regimens and discuss parameter estimation and sensitivity issues.

Prostate cancer: progression of prostate-specific antigen after external beam irradiation.

This paper is concerned with the development of a stochastic path of prostate-specific antigen (PSA) level after radiation treatment for prostate cancer. PSA is a biomarker for prostate cancer, higher levels of which indicate the seriousness of the cancer progression. Following the deterministic modeling of the data by the previous authors, Cox et al., this paper is concerned with the theoretical knowledge that could be gained by the stochastic modeling in discrete form of the PSA path over time. The expected value of the PSA level is computed and compared with the deterministic model and it is found that they are the same for about the first year after radiation therapy. The American Society for Therapeutic Radiology has set a consensus panel definition of biochemical failure following radiation therapy: the rise in three consecutive levels of PSA is considered to be a failure of the radiation therapy. Knowledge of the path of PSA presented in this paper would be useful in the management of the radiation treatment and in particular assessing quantitatively any clinically based policy for defining recurrence after radiation therapy. Application of the model is illustrated by fitting it to clinical data available in the University of Michigan cancer center.