Qualitative analysis of an HIV transmission model

Modeling the impact of random screening and contact tracing in reducing the spread of HIV

Mathematical models can help predict the effectiveness of control measures on the spread of HIV and other sexually transmitted diseases (STDs) by reducing the uncertainty in assessing the impact of intervention strategies such as random screening and contact tracing. Even though contact tracing is one of the most effective methods used for controlling treatable STDs, it is still a controversial strategy for controlling HIV because of cost and confidentiality issues. To help estimate the effectiveness of these control measures, we formulate two models with random screening and contact tracing based on the differential infectivity (DI) model and the staged-progression (SP) model. We derive formulas for the reproductive numbers and the endemic equilibria and compare the impact that random screening and contact tracing have in slowing the epidemic in the two models. In the DI model the infected population is divided into groups according to their infectiousness, and HIV is largely spread by a small, highly infectious, group of superspreaders. In this model contact tracing is an effective approach to identifying the superspreaders and has a large effect in slowing the epidemic. In the SP model every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. In this model random screening is more effective than for the DI model, and contact tracing is less effective. Thus the effectiveness of the intervention strategy strongly depends on the underlying etiology of the disease transmission.

An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.

The differential infectivity and staged progression models for the transmission of HIV

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transmission studies and suggest that a small subset of infected people may be responsible for a disproportionate number of infections. We use two simple models to study the impact of variations in infectiousness. In the first model, we account for different levels of virus between individuals during the chronic phase of infection, and the increase in the average time from infection to AIDS that goes along with a decreased viral load. The second model follows the more standard hypothesis that infected individuals progress through a series of infection stages, with the infectiousness of a person depending upon his current disease stage. We derive and compare threshold conditions for the two models and find explicit formulas of their endemic equilibria. We show that formulas for both models can be put into a standard form, which allows for a clear interpretation. We define the relative impact of each group as the fraction of infections being caused by that group. We use these formulas and numerical simulations to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease. The acute stage and the most infectious group both appear to have a disproportionate effect, especially on the early epidemic. Contact tracing to identify super-spreaders and alertness to the symptoms of acute HIV infection may both be needed to contain this epidemic.

An SIS epidemic model with variable population size and a delay

The SIS epidemiological model has births, natural deaths, disease-related deaths and a delay corresponding to the infectious period. The thresholds for persistence, equilibria and stability are determined. The persistence of the disease combined with the disease-related deaths can cause the population size to decrease to zero, to remain finite, or to grow exponentially with a smaller growth rate constant. For some parameter values, the endemic infective-fraction equilibrium is asymptotically stable, but for other parameter values, it is unstable and a surrounding periodic solution appears by Hopf bifurcation.

Population size dependent incidence in models for diseases without immunity

Epidemiological models of SIS type are analyzed to determine the thresholds, equilibria, and stability. The incidence term in these models has a contact rate which depends on the total population size. The demographic structures considered are recruitment-death, generalized logistic, decay and growth. The persistence of the disease combined with disease-related deaths and reduced reproduction of infectives can greatly affect the population dynamics. For example, it can cause the population size to decrease to zero or to a new size below its carrying capacity or it can decrease the exponential growth rate constant of the population.

An epidemiological model for HIV/AIDS with proportional recruitment

A model for HIV transmission is formulated for a homosexual population of varying size, with recruitment into the susceptible class proportional to the active population size and with stages of progression to AIDS. Analysis of this model includes identifying the threshold that determines whether the disease dies out or proportions remain endemic and establishing criteria that determine whether the population size decays asymptotically exponentially to zero or grows asymptotically exponentially to infinity. In an analogous heterosexual model, the conservation of heterosexual contacts is shown to imply that this two-sex model reduces to the one-sex model.

AIDS and promiscuity: muddles in the models of HIV prevention

AIDS has been blamed on promiscuity and the promiscuous, and a major goal of many HIV-prevention programs has been to induce people to reduce the number of their sexual partners. Despite the salience of this concept in the AIDS discourse of scientists, policymakers, the media, religious leaders, and the gay community, critical analysis of the role of promiscuity in this epidemic has been lacking. Following a review of promiscuity in various genres of AIDS discourse, this article discusses promiscuity in American society and in HIV-prevention campaigns. The relative risks associated with monogamy, abstinence and promiscuity are examined, and the author concludes that the partner-reduction strategy, instead of contributing to a reduction in HIV transmission has been an impediment to AIDS prevention efforts, exacerbating the problem by undermining the sex-positive approaches to risk reduction that have proven effective. Responsibility for this misguided strategy is attributed to a moralistic approach to AIDS and to the misapplication of epidemiological concepts and inappropriate social science models to the task of promoting healthy forms of sexuality.

On the uniqueness of endemic equilibria of an HIV/AIDS transmission model for a heterogeneous population

In this paper, we examine an HIV/AIDS transmission model which has been widely used for studying the spread of HIV/AIDS through sexual contact. Some sufficient conditions are obtained for the uniqueness of endemic equilibrium. We also present a two-group model whose no-disease equilibrium is unstable but it has at least three positive endemic equilibria.