John A. Jacquez

Qualitative theory of compartmental systems

Dynamic models of many processes in the biological and physical sciences which depend on local mass balance conditions give rise to systems of ordinary differential equations, many nonlinear, that are called compartmental systems. In this paper, the authors define compartmental systems, specify their relations to other nonnegative systems, and discuss examples of applications.The authors review the qualitative results on linear and nonlinear compartmental systems, including their relation to cooperative systems. They review the results for linear compartmental systems and then integrate and expand the results on nonlinear compartmental systems, providing a framework for unifying them under a few general theorems. In the course of that they complete the solution of a problem posed by Bellman and show that closed nonlinear, autonomous, n-compartment systems can show the full gamut of possible behaviors of systems of ODES.Finally, to provide additional structure to this study, the authors show how to partiti...

Modeling and analyzing HIV transmission: the effect of contact patterns

A compartmental model is presented for the spread of HIV in a homosexual population divided into subgroups by degree of sexual activity. The model includes constant recruitment rates for the susceptibles in the subgroups. It incorporates the long infectious period of HIV-infected individuals and allows one to vary infectiousness over the infectious period. A new pattern of mixing, termed preferred mixing, is defined, in which a fraction of a group’s contacts can be reserved for within-group contacts, the remainder being subject to proportional mixing. The fraction reserved may differ among groups. In addition, the classic definition of reproductive number is generalized to show that for heterogeneous populations in general the endemic threshold is BDc,, where cr is the mean number of contacts per infective. The most important finding is that the pattern of contacts between the different groups has a major effect on the spread of HIV, an effect inadequately recognized or studied heretofore.

Role of the primary infection in epidemics of HIV infection in gay cohorts.

A review of the data on infectivity per contact for transmission of the HIV suggests that the infectivity may be on the order of 0.1-0.3 per anal intercourse in the period of the initial infection, 10(-4) to 10(-3) in the long asymptomatic period, and 10(-3) to 10(-2) in the period leading into AIDS. The pattern of high contagiousness during the primary infection followed by a large drop in infectiousness may explain the pattern of epidemic spread seen in male homosexual cohorts in the early years of the epidemic. Simulations of cohorts of homosexual males, using that range of parameter values, indicate the following: (a) The initial fast rise and then more or less rapid flattening of the incidence curve of seropositives is primarily due to rapid initial spread, yielding a group of infecteds all of whom pass into the low infectivity asymptomatic period at close to the same time. All this occurs only if the basic reproduction number for the primary infection is > 1. (b) The behavioral changes that have been reported all started after the incidence of new infections began to fall, too late to have a major effect on the initial rise. The behavioral changes had a major effect in slowing down the subsequent rise in the number of seropositives. (c) High activity groups play an important role in the early rapid rise of the epidemic. However, it is not likely that the rapid decrease in rate of growth of seropositives is solely due to saturation of these very high activity groups. Although the evidence for this interpretation of the role of the primary infection is not conclusive, its implications for prevention and for vaccine trials are so markedly different from those of other interpretations that we consider it to be an important hypothesis for further testing.

The role of early HIV infection in the spread of HIV through populations

The combination of two factors gives early HIV infection an especially strong influence on transmission dynamics: (a) increased transmission probabilities and (b) increased transmission potential of partners infected during this period. Most attention has been focused on the first factor because it fits the way we usually think about risk factors affecting individuals. The second factor acts not on individuals, but across chains of transmission. It is missed by models with constant partnership formation rates over an individuals life or with random mixing. It cannot be assessed from available data collected from individuals. Its assessment requires data from both individuals in a partnership. We demonstrate that this second effect can be so strong that early infection can dominate transmission dynamics even when transmission probabilities are only modestly increased. This second effect is not directly parameterized in our models but arises from two realistic types of temporal variation in partnership formation: (a) Partnership formation rates vary by age with preferential partnership formation in ones own age group, and (b) individuals of any age can experience transient periods of high-risk partnership formation. In a model with only the age-related effect, early infection is observed to dominate transmission dynamics when 20% of transmissible virus is allocated to the first 6 weeks of infection, 7% to middle infection, and 73% to late infection. This domination occurs both early in the course of an epidemic and later when endemic infection levels have been reached. When the second effect is added, early infection is seen to dominate transmission in a model allocating 10% of transmissible virus to the first 6 months, 40% to middle infection, and 50% to late infection. In this model, transmission probabilities during early infection are only 4.17 times those of middle infection and half those of late-stage infection.

Numerical parameter identifiability and estimability: Integrating identifiability, estimability, and optimal sampling design

We define two levels of parameters. The basic parameters are associated with the model and experiment(s). However, the observations define a set of identifiable observational parameters that are functions of the basic parameters. Starting with this formulation, we show that an implicit function approach provides a common basis for examining local identifiability and estimability and gives a lead-in to the problem of optimal sampling design. A least squares approach based on a large but finite set of observations generated at initial parameter estimates then gives a uniform approach to local identifiability, estimability, and the generation of an optimal sampling schedule.

Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations

A major project in deterministic epidemiological modeling of heterogeneous populations is to find conditions for the local and global stability of the equilibria and to work out the relations among these stability conditions, the thresholds for epidemic take-off and endemicity, and the basic reproduction number(s). Most of the work to date has been on models of homogeneous populations of constant size. Motivated by their analysis of models of the dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS), the authors carry out this project for SI models of diseases that have multiple stages, which can lead to death, and which infect heterogeneous populations with intricate mixing patterns and varying sizes. In such models, it is even difficult to find the analytical expression for the stability threshold and the reproduction number. The authors show how the number of disease-transmitting contacts by infectives can be used as a Lyapunov function to carry out a systematic stabili...

The Stochastic SI Model with Recruitment and Deaths I. Comparison with the Closed SIS Model

We compare the stochastic and deterministic versions of an SI model with recruitment, background deaths, and deaths due to the disease. For the stochastic version, analysis of the mean number of susceptibles, mx, and infecteds, m(y), and of the means conditioned on nonextinction of the infection, m*x and m*y, shows that (1) if R0 < or = 1, the disease dies out monotonically for the deterministic and stochastic models, and (2) if R0 > 1, the disease dies out early with a probability close to (1/R0)a, where a is the number of infecteds introduced, or m(y) rises to a peak and then dies out slowly. For small populations, N, the peak is an obvious maximum. If N > or = 100, the peak in m(y) is hidden in a long, nearly stationary plateau and m*y is close to the deterministic endemic level for a large range of parameter values. The analytical results are illustrated with simulations. The results for the SI model are motivated by and compared with the corresponding results for the closed SIS model.

Nonequilibrium-Facilitated Oxygen Transport in Hemoglobin Solution

We have used the quasi-linearization method to obtain numerical solutions to the equations which describe steady-state diffusion of oxygen through layers of hemoglobin solution. The numerical solutions show how the facilitated flux of oxygen depends upon the layer thickness, reaction-rate coefficients, and other parameters of the system. The results indicate that steady-state oxygen diffusion in layers of hemoglobin solution, similar to those studied by Scholander, should be adequately described by the models which assume chemical equilibrium exists throughout the layer, but for layers of concentrated hemoglobin solution about the thickness of a human erythrocyte, the facilitation of oxygen diffusion should be much less than the equilibrium models predict.

Linear Regression with Non-Constant, Unknown Error Variances: Sampling Experiments with Least Squares, Weighted Least Squares and Maximum Likelihood Estimators

The theory of simple linear regression is extended to the case of non-uniform error variances for the situation in which replicates are available at each sample point in the domain of the independent variable. The performance of least squares (LS), weighted least squares (WLS), and maximum likelihood (ML) estimators of the regression parameters was compared in sampling experiments in which the error variance varied by as much as 16-fold over the domain of the independent variable and for both normal and log-normal distributions for the error. The LS estimators performed significantly better than the others when few replicates were available at each of many sampling points. The WLS estimators improved as the number of replicates increased and were clearly superior when the number of replicates per point exceeded the number of points in the independent variable. The ML estimates were usually close to the WLS estimates but the WLS estimates were more often the better of the two. A first order approximation for the estimators of variances of the WLS estimators is derived as well as an iterative method for finding the ML estimators.

QUASILINEARIZATION AND THE ESTIMATION OF CHEMICAL RATE CONSTANTS FROM RAW KINETIC DATA

Abstract A technique for estimating chemical rate constants from raw kinetic data is suggested. Such problems are viewed as nonlinear multipoint boundary-value problems for systems of nonlinear ordinary differential equations, for which the quasilinearization procedure offers an affective means of numerical solution. The method is illustrated using kinetic data obtained by Bodenstein and Lindler on some gas phase reactions of nitrogen and oxygen.