Population structure and the spread of disease

A common assumption of many mathematical models for the spread of disease is that there is random mixing among all individuals in the host population. This paper analyzes and develops a model for the spread of disease in a population consisting of several interacting subpopulations. The model considers 2 different types of interactions between individuals: 1) within a subpopulation because of geographic proximity, and 2) of the same or different subpopulations because of attendance at common social functions. A stability analysis performed on the equilibria of the model shows 2 stable states: 1) a population composed solely of susceptible individuals with no disease present, and 2) an interior point where there are susceptible, infective, and recovered individuals present at all times. The analysis shows that the threshold for disease maintenance is more easily exceed in centers that are members of a small local cluster than in randomly mixing centers, but that the spread of the disease throughout the population occurs more rapidly when the initial case attends a randomly mixing center. The conditions under which a disease will become established are dependent upon the transmission rate for the disease, the birth and death rate in each neighborhood, the recovery rate from the disease in each neighborhood, and the movement patterns of the individuals in the population. The study of the spread of disease in a population by means of mathematical models provides a valuable addition to the statistical data analyzed by epidemiologists. This model is relevant any time there is a division of the population into several interacting groups in which the probability of disease spread is a function both of neighborhood contact because of geographic proximity and of social interactions between groups.     

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 equilibrium. 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.     

The effect of structural behavior change on the spread of HIV in a one-sex population.

Within a preferred mixing type of model for the spread of HIV in a one-sex population, the effects of structural behavior change, that is movements of individuals from one activity class to another, with accompanying changes of contact pattern are investigated. It is concluded that such behavior change makes it more difficult for an epidemic to arise if the contact pattern is of the restricted type, whereas the effect is indeterminate in the proportional mixing case. Some of the problems in analyzing sexual activity data from a population within which this behavior change mechanism is at work are also commented upon.     

The initialization and sensitivity of multigroup models for the transmission of HIV

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression 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. We demonstrate the importance of choosing appropriate initial conditions, and define a new approach to distributing the initial population among the subgroups so as to minimize the artificial transients in the solutions due to unbalanced initial conditions. We demonstrate that the rate of removal in and out of a population is an important, yet often neglected, effect. We also illustrate the importance of distinguishing between the number of partners a person has and the number of contacts per partner. By assuming that people with many partners have fewer contacts per partner than people with few partners, we found that the epidemic is less sensitive to the partner acquisition rate than one might expect. However, because the probability of transmission of HIV per contact is low, the epidemic is very sensitive to the number of contacts per partner. Modeling this distinction is particularly important when estimating the impact of programs which encourage people to have fewer sexual partners.     

Modelling the effect of treatment and behavioral change in HIV transmission dynamics

In this paper we analyze a model for the HIV-infection transmission in a male homosexual population. In the model we consider two types of infected individuals. Those that are infected but do not know their serological status and/or are not under any sort of clinical /therapeutical treatment, and those who are. The two groups of infectives differ in their incubation time, contact rate with susceptible individuals, and probability of disease transmission. The aim of this article is to study the roles played by detection and changes in sexual behavior in the incidence and prevalence of HIV. The analytical results show that there exists a unique endemic equilibrium which is globally asymptotically stable under a range of parameter values whenever a detection /treatment rate and an indirect measure of the level of infection risk are sufficiently large. However, any level of detection/ treatment rate coupled with a decrease of the transmission probability lowers the incidence rate and prevalence level in the population. In general, only significant reductions in the transmission probability (achieved through, for example, the adoption of safe sexual practices) can contain effectively the spread of the disease.     

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.     

A stochastic model for the development of an AIDS epidemic in a heterosexual population.

A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part on the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.     

Effects of public health educational campaigns and the role of sex workers on the spread of HIV/AIDS among heterosexuals

This paper presents a sex-structured model for heterosexual transmission of HIV/AIDS in which the population is divided into three subgroups: susceptibles, infectives and AIDS cases. The subgroups are further divided into two classes, consisting of individuals involved in high-risk sexual activities and individuals involved in low-risk sexual activities. The model considers the movement of individuals from high to low sexual activity groups as a result of public health educational campaigns. Thus, in this case public health educational campaigns are resulting in the split of the population into risk groups. The equilibrium and epidemic threshold, which is known as the basic reproductive number (R0), are obtained, and stability (local and global) of the disease-free equilibrium is investigated. The model is extended to incorporate sex workers, and their role in the spread of HIV/AIDS in settings with heterosexual transmission is explored. Comprehensive analytic and numerical techniques are employed in assessing the possible community benefits of public health educational campaigns in controlling HIV/AIDS. From the study, we conclude that the presence of sex workers enlarges the epidemic threshold R0, thus fuels the epidemic among the heterosexuals, and that public health educational campaigns among the high-risk heterosexual population reduces R0, thus can help slow or eradicate the epidemic.     

Serum enhancement of human immunodeficiency virus (HIV) infection correlates with disease in HIV-infected individuals.

The sera from 16 individuals infected with the human immunodeficiency virus (HIV) at different clinical stages were evaluated for antibody-dependent neutralization and/or enhancement of infectivity by HIV. The HIV isolate from each individual (homotypic) and established laboratory strains showing broad cellular host range and cytopathicity were used. All sera could neutralize one of the laboratory-passaged isolates, whereas only two could neutralize the corresponding homotypic strain. Seven homotypic isolates were enhanced by serum from the respective individual. This activity was primarily observed in patients with acquired immune deficiency syndrome. Moreover, the tropism for macrophages of four of these seven viral isolates was found to be enhanced by the homotypic sera. Finally, sequential pairs of HIV and sera obtained from five HIV-infected individuals with different clinical progression were studied over time. The enhancing activity of three of the five sera appeared to increase over time, indicating changes in both the host virus population and the type of antibodies produced. These results suggest that enhancing antibodies contribute to the spread and pathogenesis of HIV in vivo. They emphasize the necessity of studying further the association of enhancing antibodies and disease progression in infected individuals.     

A simulation model of AIDS in San Francisco: I. Model formulation and parameter estimation.

A model is formulated for the spread of the human immunodeficiency virus (HIV) and the subsequent development of acquired immunodeficiency syndrome (AIDS) in the population of homosexual men in San Francisco. The dynamic simulation model includes sexually very active and active subpopulations, migration, and a staged progression of HIV-infected persons to AIDS and death. Numerous data sources are used to estimate parameter values in the model. In a companion paper, simulations using the model and parameter estimates are found that are consistent with HIV and AIDS incidence data.     

Asymptotic worst-case mixing in simple demographic models of HIV/AIDS.

The majority of published modeling work regarding the impact of mixing patterns among subgroups on the spread of HIV infection assumes either that the overall population size remains constant, the aggregate immigration to the population occurs at a constant annual rate, or that no immigration occurs and the population in question declines due to HIV or other causes. In this paper, immigration rates are modeled as simple functions of population size and may be interpreted as aggregate birth rates. This assumption implies asymptotic exponential growth in the disease-free population as long as per capita birth rates exceed per capita mortality rates. The introduction of HIV infection to such a population may change this situation, and the asymptotic population growth rate can be reduced substantially as a result. The specific manner in which this occurs depends in part upon difficult to observe mixing patterns among those with different sexual activity rates. Rather than attempting to explicitly model a variety of mixing patterns, a bound on the impact of worst-case mixing is produced, where "worst case" refers to the mixing pattern that maximizes the asymptotic prevalence of infection, which is equivalent to minimizing the asymptotic population growth rate. These new techniques are illustrated with a numerical example.     

Modelling the spread of HIV immune escape mutants in a vaccinated population

Because cytotoxic T-lymphocytes (CTLs) have been shown to play a role in controlling human immunodeficiency virus (HIV) infection and because CTL-based simian immunodeficiency virus (SIV) vaccines have proved effective in non-human primates, one goal of HIV vaccine design is to elicit effective CTL responses in humans. Such a vaccine could improve viral control in patients who later become infected, thereby reducing onwards transmission and enhancing life expectancy in the absence of treatment. The ability of HIV to evolve mutations that evade CTLs and the ability of these 'escape mutants' to spread amongst the population poses a challenge to the development of an effective and robust vaccine. We present a mathematical model of within-host evolution and between-host transmission of CTL escape mutants amongst a population receiving a vaccine that elicits CTL responses to multiple epitopes. Within-host evolution at each epitope is represented by the outgrowth of escape mutants in hosts who restrict the epitope and their reversion in hosts who do not restrict the epitope. We use this model to investigate how the evolution and spread of escape mutants could affect the impact of a vaccine. We show that in the absence of escape, such a vaccine could markedly reduce the prevalence of both infection and disease in the population. However the impact of such a vaccine could be significantly abated by CTL escape mutants, especially if their selection in hosts who restrict the epitope is rapid and their reversion in hosts who do not restrict the epitope is slow. We also use the model to address whether a vaccine should span a broad or narrow range of CTL epitopes and target epitopes restricted by rare or common HLA types. We discuss the implications and limitations of our findings.     

The penetration of HIV into the population of Moscow homosexuals and its spread

The article deals with the epidemiological investigation of cases of HIV infection among the population of homosexuals. 119 representatives of this population were checked in their sexual contacts; of these, 13 persons proved to be infected with HIV. HIV infection was found to penetrate into the homosexual population of the USSR from the countries of Western Europe and the USA by sexual route. In 1988 the intensive spread of this infection began.     

Cost-effective control of chronic viral diseases: Finding the optimal level of screening and contact tracing

Chronic viral diseases such as human immunodeficiency virus (HIV) and hepatitis B virus (HBV) afflict millions of people worldwide. A key public health challenge in managing such diseases is identifying infected, asymptomatic individuals so that they can receive antiviral treatment. Such treatment can benefit both the treated individual (by improving quality and length of life) and the population as a whole (through reduced transmission). We develop a compartmental model of a chronic, treatable infectious disease and use it to evaluate the cost and effectiveness of different levels of screening and contact tracing. We show that: (1) the optimal strategy is to get infected individuals into treatment at the maximal rate until the incremental health benefits balance the incremental cost of controlling the disease; (2) as one reduces the disease prevalence by moving people into treatment (which decreases the chance that they will infect others), one should increase the level of contact tracing to compensate for the decreased effectiveness of screening; (3) as the disease becomes less prevalent, it is optimal to spend more per case identified; and (4) the relative mix of screening and contact tracing at any level of disease prevalence is such that the marginal efficiency of contact tracing (cost per infected person found) equals that of screening if possible (e.g., when capacity limitations are not binding). We also show how to determine the cost-effective equilibrium level of disease prevalence (among untreated individuals), and we develop an approximation of the path of the optimal prevalence over time. Using this, one can obtain a close approximation of the optimal solution without having to solve an optimal control problem. We apply our methods to an example of hepatitis B virus.     

High prevalence of vitamin D deficiency in HIV-infected individuals in comparison with the general population across Punjab province,Pakistan

The status of vitamin D in individuals infected with human immunodeficiency virus (HIV), particularly in naïve as well as treated patients, has never been reported in the Pakistani population. A cross-sectional study was performed to measure vitamin D in individuals infected with HIV living in various districts of the Punjab, Pakistan. 1000 persons attending various treatment centers of the Punjab were screened for HIV, hepatitis B virus (HBV), hepatitis C virus (HCV), and Syphilis. Total 398 patients met inclusion criteria and vitamin D level was measured in respective cases by using enzyme-linked immunosorbent assay (ELISA) technique. 232 samples from the healthy population were also included in present research. Demographic history and clinical parameters regarding HIV disease were evaluated. Comparison of variables was done to find out the link between vitamin D levels and characteristics of HIV infected persons and comparison to that of healthy individuals was performed. Among 398 HIV patients vitamin D deficiency and insufficiency was found among 15 % and 39 % while majority of the control participants had sufficient levels of vitamin D (78 %). Most of the HIV infected individuals were males (68.6 %) and had age between 24 and 47 years (67.8 %). A significant relationship was found for vitamin D level, lifestyle and CD4 count among HIV + ve non acquired immunodeficiency syndrome (AIDS) subjects (95 % CI; p < 0.001, p = 0.09). For HIV + ve AIDS patients vitamin D had a significant relationship with lifestyle along with HIV viral load and CD4 count. Hypovitaminosis D prevails among the HIV infected population of Punjab, Pakistan.     

Modelling the social dynamics of a sex industry: Its implications for spread of HIV/AIDS

A theoretical model is proposed for a community which has the structure of two classes (direct and indirect) of commercial sex workers (CSW), and two classes of sexually active male customers with different levels of sexual activity. The direct CSW’s work in brothels while the indirect CSW’s are based in commercial establishments such as bars, cafes, and massage parlours where sex can be bought on request and conducted elsewhere. Behaviour change and the resulting change of activity class occurs between the two classes of CSW’s and two classes of males under the setting of the proliferation of human immunodeficiency virus (HIV)/acquired immunodeficiency syndrome epidemic and the subsequent intervention programmes. In recently years, this phenomenon has been observed in several countries in Asia. Given the lower levels of condom use and higher HIV prevalence of the indirect CSW’s, ascertaining the impact of this change in the structure of the sex industry on the spread of HIV is the main focus of this paper. The complete analysis of the disease-free model is given. For the full model, local analysis will be performed for the case of preferred mixing without activity class change, as well as the case with activity class change and restricted mixing. The basic reproduction number for the spread of epidemic will be derived for each case. Results of biological significance include: (i) the change of behaviour by the CSW’s has a more direct effect on the spread of HIV than that of the male customers; (ii) the basic reproduction number is obtained by considering all possible infection cycles of the heterosexual transmission of HIV which indicates the importance of understanding the sexual networking in heterosexual transmission of HIV; (iii) the social dynamics of the sex industry is not just a simple ’supply and demand’ mechanism driven by the demand of the customers, hence highlighting the need for further understanding of the changing structure of the sex industry. The main points of this work will be illustrated with numerical examples using the HIV data of Thailand.     

Mathematical model for Zika virus dynamics with sexual transmission route

Zika virus is a flavivirus transmitted to humans primarily through the bite of infected Aedes mosquitoes. In addition to vector-borne spread, however, the virus can also be transmitted through sexual contact. In this paper, we formulate and analyze a new system of ordinary differential equations which incorporates both vector and sexual transmission routes. Theoretical analysis of this model when there is no disease induced mortality shows that the disease-free equilibrium is locally and globally asymptotically stable whenever the associated reproduction number is less than unity and unstable otherwise. However, when we extend this same model to include Zika induced mortality, which have been documented in Latin America, we find that the model exhibits a backward bifurcation. Specifically, a stable disease-free equilibrium co-exists with a stable endemic equilibrium when the associated reproduction number is less than unity. To further explore model predictions, we use numerical simulations to assess the importance of sexual transmission to disease dynamics. This analysis shows that risky behavior involving multiple sexual partners, particularly among male populations, substantially increases the number of infected individuals in the population, contributing significantly to the disease burden in the community.     

A Multiple siRNA-Based Anti-HIV/SHIV Microbicide Shows Protection in Both In Vitro and In Vivo Models

Human immunodeficiency virus (HIV) types 1 and 2 (HIV-1 and HIV-2) are the etiologic agents of AIDS. Most HIV-1 infected individuals worldwide are women, who acquire HIV infections during sexual contact. Blocking HIV mucosal transmission and local spread in the female lower genital tract is important in preventing infection and ultimately eliminating the pandemic. Microbicides work by destroying the microbes or preventing them from establishing an infection. Thus, a number of different types of microbicides are under investigation, however, the lack of their solubility and bioavailability, and toxicity has been major hurdles. Herein, we report the development of multifunctional chitosan-lipid nanocomplexes that can effectively deliver plasmids encoding siRNA(s) as microbicides without adverse effects and provide significant protection against HIV in both in vitro and in vivo models. Chitosan or chitosan-lipid (chlipid) was complexed with a cocktail of plasmids encoding HIV-1-specific siRNAs (psiRNAs) and evaluated for their efficacy in HEK-293 cells, PBMCs derived from nonhuman primates, 3-dimensional human vaginal ectocervical tissue (3D-VEC) model and also in non-human primate model. Moreover, prophylactic administration of the chlipid to deliver a psiRNA cocktail intravaginally with a cream formulation in a non-human primate model showed substantial reduction of SHIV (simian/human immunodeficiency virus SF162) viral titers. Taken together, these studies demonstrate the potential of chlipid-siRNA nanocomplexes as a potential genetic microbicide against HIV infections.     

Factors influencing the emergence and spread of HIV drug resistance arising from rollout of antiretroviral pre-exposure prophylaxis (PrEP)

Background

The potential for emergence and spread of HIV drug resistance from rollout of antiretroviral (ARV) pre-exposure prophylaxis (PrEP) is an important public health concern. We investigated determinants of HIV drug resistance prevalence after PrEP implementation through mathematical modeling.

Methodology

A model incorporating heterogeneity in age, gender, sexual activity, HIV infection status, stage of disease, PrEP coverage/discontinuation, and HIV drug susceptibility, was designed to simulate the impact of PrEP on HIV prevention and drug resistance in a sub-Saharan epidemic.

Principal Findings

Analyses suggest that the prevalence of HIV drug resistance is influenced most by the extent and duration of inadvertent PrEP use in individuals already infected with HIV. Other key factors affecting drug resistance prevalence include the persistence time of transmitted resistance and the duration of inadvertent PrEP use in individuals who become infected on PrEP. From uncertainty analysis, the median overall prevalence of drug resistance at 10 years was predicted to be 9.2% (interquartile range 6.9%–12.2%). An optimistic scenario of 75% PrEP efficacy, 60% coverage of the susceptible population, and 5% inadvertent PrEP use predicts a rise in HIV drug resistance prevalence to only 2.5% after 10 years. By contrast, in a pessimistic scenario of 25% PrEP efficacy, 15% population coverage, and 25% inadvertent PrEP use, resistance prevalence increased to over 40%.

Conclusions

Inadvertent PrEP use in previously-infected individuals is the major determinant of HIV drug resistance prevalence arising from PrEP. Both the rate and duration of inadvertent PrEP use are key factors. PrEP rollout programs should include routine monitoring of HIV infection status to limit the spread of drug resistance.     

A general model for stochastic SIR epidemics with two levels of mixing

This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R_* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.