Some state space models of HIV pathogenesis under treatment by anti-viral drugs in HIV-infected individuals.

In this paper we have extended the model of HIV pathogenesis under treatment by anti-viral drugs given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] to a stochastic model. By using this stochastic model as the stochastic system model, we have developed a state space model for the HIV pathogenesis under treatment by anti-viral drugs. In this state space model, the observation model is a statistical model based on the observed numbers of RNA virus copies over different times. For this model we have developed procedures for estimating and predicting the numbers of infectious free HIV and non-infectious free HIV as well as the numbers of different types of T cells through extended Kalman filter method. As an illustration, we have applied the method of this paper to the data of patient Nos. 104, 105 and 107 given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] under treatment by Ritonavir. For these individuals, it is shown that within two weeks since treatment, most of the free HIV are non-infectious, indicating the usefulness of the treatment. Furthermore, the Kalman filter method revealed a much stronger effect of the treatment within the first 10 to 20 h than that predicted by the deterministic model.

A state space model for the HIV epidemic in homosexual populations and some applications.

In this paper we have developed a state space model for the HIV epidemic in homosexual populations which have been divided into subpopulations according to sexual activity levels. In this model, the stochastic dynamic system model is the stochastic model of the HIV epidemic in terms of the chain multinomial model whereas the observation model is a statistical model based on the observed AIDS incidences. This model is applied to the San Francisco homosexual population for estimating the numbers of susceptible people, infective people and AIDS cases and for estimating the probabilities of HIV transmission from infective people to susceptible people given sexual contacts. The results show that the estimated numbers of AIDS incidence trace closely the observed numbers indicating the usefulness of the model. It is observed that the estimated numbers of latent people show multimodal curves and that HIV infection takes place during the primary stage and very late stage. The results have further shown that there are significant differences between the observed AIDS incidences and the estimates by the embedded deterministic model. These results indicate that using the embedded deterministic model to estimate the HIV-infected people and to predict future AIDS cases can be very misleading in some cases.

Stochastic modeling of the dynamics of CD4+ T-cell infection by HIV and some Monte Carlo studies.

In this paper, we develop a stochastic model for the interaction between CD4+ T cells and the human immunodeficiency virus (HIV) virus by taking into account the basic biological mechanism as described in [1-4]. We studied this stochastic model through extensive Monte Carlo simulations. Our results show that, in some cases, there is a positive probability that the virus will be eliminated by the process. We have also shown that, at the earlier stage of the infection, the probability distributions of the CD4+ T cells and free HIV are skewed; however, these distributions will eventually converge to the Gaussian distributions after several years. A real-data example is given to illustrate the application of our model.

Monte Carlo results for the 3-poly test for animal carcinogenicity experiments.

By using a two stage model of carcinogenesis, we generated Monte Carlo studies to assess the efficiency and robustness of the 3-poly test for animal carcinogenicity experiments. The Monte Carlo results indicate that the 3-poly test is quite powerful for detecting the carcinogenic effects of complete carcinogens, moderate promoters, and initiators with moderate or large effect, but, in some cases, it is less powerful for weak initiators or weak promoters. As expected, the 3-poly test is insensitive to the toxicity of many agents.

Characterization of HIV incubation distributions and some comparative studies.

In this paper we use a general stochastic model to characterize the HIV incubation distributions. We generate some Monte Carlo data under different conditions and compare the fitting of HIV incubation distributions by some well known parametric models and some non-parametric methods. The parametric models include most of those that have appeared in the literature. The non-parametric methods include the Kaplan--Meier method, the EMS method, the spline approximation and the Bacchetti method. The comparison criteria are the chi-square statistic, the residual sum of squares, the AIC and the BIC. We show that the non-parametric methods, especially the EMS method, provide excellent fits in almost all cases; for the parametric models, the generalized log-logistic distributions with three and with four parameters fit better than other parametric models.

Characterization of HIV infection and seroconversion by a stochastic model of the HIV epidemic.

In this paper we use a stochastic model for the HIV epidemic in homosexual populations to characterize the HIV infection and seroconversion. Using computer generated data, we compare the fitting of infection distributions and of seroconversion distributions by different parametric models as well as by nonparametric methods. The nonparametric methods include the Kaplan-Meier method, EMS method, Bacchetti's method, and the spline approximation. The parametric models include most of the models which have been used in the literature. The comparison criteria are the chi-square statistic, the AIC (Akaike Information Criterion) and the residual sums of squares. The numerical results suggest that for the proportional mixing pattern, the EMS method, the spline method, Bacchetti's method, and the generalized log-logistic distributions with three and with four parameters provide better fitting for the infection and the seroconversion distributions in most cases. For the restricted mixing patterns, the EMS method, the spline method, Bacchetti's method, and some mixtures of distributions provide close fitting to the infection and the seroconversion distributions.

Estimating the stage-specific numbers of HIV infection using a Markov model and back-calculation.

The back-calculation method has been used to estimate the number of HIV infections from AIDS incidence data in a particular population. We present an extension of back calculation that provides estimates of the numbers of HIV infectives in different stages of infection. We model the staging process with a time-dependent Markov process that partitions the HIV infectious period into the following progressive stages and/or substages: stage 1, infected but antibody negative; substages 2-3; antibody positive but asymptomatic; substages 4-6, pre-AIDS symptoms and/or abnormal haematologic indicator, stage 7, clinical AIDS. We also model an eight stage, decreased due to AIDS. The model allows for time-dependent treatment effects that slow the rate of progression in substages 4-7. We use the estimated AIDS incubation period distribution for the Markov model in back calculation from AIDS incidence data to estimate the total number of HIV infections and the parameters of the infection probability distribution. We then use these estimates in the Markov model to estimate the stage-specific numbers of HIV infections over the course of the epidemic in the population under study. Example calculations employ data for epidemic in San Francisco City, Clinic Cohort.

The Markov process approach to survival analysis.

"Consider a two organs system for which one of the two organs must function for the individual to survive. This paper derives the survival probability distributions under more general situations by adopting [the] Markov process approach, thus providing extensions of results given in Gross, Clark and Liu (1971) and Kodlin (1967). The results obtained are then extended to a k-organs system."

On the absorption probabilities and absorption times of finite homogeneous birth-death processes.

This paper provides some formulas for the absorption probabilities, the mean absorption times and the variances of first absorption times in finite homogeneous absorbing birth-death processes. The results are then applied to a genetic model of Moran [1958] for computing the absorption probability densities, the mean absorption times and the variances of first absorption times. Specifically, it is shown that the probability distribution of the first absorption time is the matrix analog of exponential distribution or a mixture of exponential distributions if the transition matrix is diagonable.