Carl P. Simon

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.

Prevalence of Antibiotic Resistance in Drinking Water Treatment and Distribution Systems

ABSTRACT The occurrence and spread of antibiotic-resistant bacteria (ARB) are pressing public health problems worldwide, and aquatic ecosystems are a recognized reservoir for ARB. We used culture-dependent methods and quantitative molecular techniques to detect and quantify ARB and antibiotic resistance genes (ARGs) in source waters, drinking water treatment plants, and tap water from several cities in Michigan and Ohio. We found ARGs and heterotrophic ARB in all finished water and tap water tested, although the amounts were small. The quantities of most ARGs were greater in tap water than in finished water and source water. In general, the levels of bacteria were higher in source water than in tap water, and the levels of ARB were higher in tap water than in finished water, indicating that there was regrowth of bacteria in drinking water distribution systems. Elevated resistance to some antibiotics was observed during water treatment and in tap water. Water treatment might increase the antibiotic resistance of surviving bacteria, and water distribution systems may serve as an important reservoir for the spread of antibiotic resistance to opportunistic pathogens.

Wastewater treatment contributes to selective increase of antibiotic resistance among Acinetobacter spp.

The occurrence and spread of multi-drug resistant bacteria is a pressing public health problem. The emergence of bacterial resistance to antibiotics is common in areas where antibiotics are heavily used, and antibiotic-resistant bacteria also increasingly occur in aquatic environments. The purpose of the present study was to evaluate the impact of the wastewater treatment process on the prevalence of antibiotic resistance in Acinetobacter spp. in the wastewater and its receiving water. During two different events (high-temperature, high-flow, 31 degrees C; and low-temperature, low-flow, 8 degrees C), 366 strains of Acinetobacter spp. were isolated from five different sites, three in a wastewater treatment plant (raw influent, second effluent, and final effluent) and two in the receiving body (upstream and downstream of the treated wastewater discharge point). The antibiotic susceptibility phenotypes were determined by the disc-diffusion method for 8 antibiotics, amoxicillin/clavulanic acid (AMC), chloramphenicol (CHL), ciprofloxacin (CIP), colistin (CL), gentamicin (GM), rifampin (RA), sulfisoxazole (SU), and trimethoprim (TMP). The prevalence of antibiotic resistance in Acinetobacter isolates to AMC, CHL, RA, and multi-drug (three antibiotics or more) significantly increased (p<0.01) from the raw influent samples (AMC, 8.7%; CHL, 25.2%; RA, 63.1%; multi-drug, 33.0%) to the final effluent samples (AMC, 37.9%; CHL, 69.0%; RA, 84.5%; multi-drug, 72.4%), and was significantly higher (p<0.05) in the downstream samples (AMC, 25.8%; CHL, 48.4%; RA, 85.5%; multi-drug, 56.5%) than in the upstream samples (AMC, 9.5%; CHL, 27.0%; RA, 65.1%; multi-drug, 28.6%). These results suggest that wastewater treatment process contributes to the selective increase of antibiotic resistant bacteria and the occurrence of multi-drug resistant bacteria in aquatic environments.

Percolation on heterogeneous networks as a model for epidemics

We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.

Effective Price Mechanisms

It is known that the price mechanism whereby the rate of change of a price is proportional to the excess demand of the corresponding commodity need not converge to a competitive equilibrium for a pure exchange economy with more than two commodities. On the other hand, there exist convergent price mechanisms, similar to the Newton iterative process, where the rate of change of the prices is determined by the excess demand and the marginal excess demands of all the commodities. This is a considerable informational requirement. It is shown that this requirement cannot be substantially reduced for any convergent price mechanisms, that is for price mechanisms expressed in terms of a difference or differential equation where the solutions converge to a competitive equilibrium.

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population, Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibtium.The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.

Ferrets as a transmission model for influenza: sequence changes in HA1 of type A (H3N2) virus.

Ferrets were used as an animal model to study whether controlled transmission of type A influenza is similar to human transmission when sequence changes in HA1 are used as the outcome. Ferrets were infected initially with A/Sydney/5/97 (H3N2) or A/LA/1/87 (H3N2) intranasally, and transmission chains were established by housing infected ferrets with noninfected ferrets with no influenza antibody titer against the infecting virus. Ferrets infected with A/Sydney were seronegative for A/Sydney and A/LA; ferrets infected with A/LA were seronegative for A/LA but had hemagglutination inhibition titers against A/Sydney. Titers of naturally transmitted influenza were higher than those after direct intranasal infection, but lymphocyte counts from nasal washes diminished with transmission. Ferrets infected with A/LA had 2 amino acid differences in HA1 after transmission through 5 ferret cohorts, but those infected with A/Sydney had none. The results show the value of the ferret model. A/LA resembled the transmission of influenza in humans when under antibody pressure.