The Initialization and Sensitivity of Multigroup Models for the Transmission of HIV

doi:10.1006/jtbi.2000.2214

Journal of Theoretical Biology

Volume 208, Issue 2, 21 January 2001, Pages 227-249

https://doi.org/10.1006/jtbi.2000.2214 Get rights and content

Abstract

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.

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Using an agent-based sexual-network model to analyze the impact of mitigation efforts for controlling chlamydia
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Citation Excerpt :
They are distributed as they would be in an emerging epidemic that started sometime in the past. We call these initial conditions balanced because when the simulation starts, the infected and susceptible populations, along with the duration of infection, are in balance with the distributions for an emerging epidemic (Hyman et al., 2001). When the initial conditions are not naturally balanced, a rapid (nonphysical) initial transient of infections quickly settles down as the infected and susceptible populations transition to a realistic infection network.
Chlamydia trachomatis (Ct) is the most reported sexually transmitted infection in the United States, with a major cause of infertility, pelvic inflammatory disease, and ectopic pregnancy among women. Despite decades of screening women for Ct, rates increase among young African Americans (AA). We create and analyze a heterosexual agent-based network model to help understand the spread of Ct. We calibrate the model parameters to agree with survey data showing Ct prevalence of 12% of the women and 10% of the men in the 15–25 year-old AA in New Orleans, Louisiana. Our model accounts for both long-term and casual partnerships. The network captures the assortative mixing of individuals by preserving the joint-degree distributions observed in the data. We compare the effectiveness of intervention strategies based on randomly screening men, notifying partners of infected people, which includes partner treatment, partner screening, and rescreening for infection. We compare the difference between treating partners of an infected person both with and without testing them. We observe that although increased Ct screening, rescreening, and treating most of the partners of infected people will reduce the prevalence, these mitigations alone are not sufficient to control the epidemic. The current practice is to treat the partners of an infected individual without first testing them for infection. The model predicts that if a sufficient number of the partners of all infected people are tested and treated, then there is a threshold condition where the epidemic can be mitigated. This threshold results from the expanded treatment network created by treating an individual’s infected partners’ partners. Although these conclusions can help design future Ct mitigation studies, we caution the reader that these conclusions are for the mathematical model, not the real world, and are contingent on the validity of the model assumptions.
A sexually transmitted infection model with long-term partnerships in homogeneous and heterogenous populations
2019, Infectious Disease Modelling
Population models for sexually transmitted infections frequently use a transmission model that assumes an inherent partnership length of zero. However, in a population with long-term partnerships, the infection status of the partners, the length of the partnership, and the exclusivity of the partnership significantly affect the rate of infection. We develop an autonomous population model that can account for the possibilities of an infection from either a casual sexual partner or a longtime partner who was either infected at the start of the partnership or was newly infected. The impact of the long-term partnerships on the rate of infection is captured by calculating the expected values of the rate of infection from these extended contacts. We present a new method to evaluate partner acquisition rates for casual or long-term partnerships which produces in a more realistic number of lifetime sexual partners. Results include a SI model with different infectiousness levels for the transmission of HIV and HSV-2 with acute and chronic/latent infection stages for homogeneous (MSM) and heterogeneous (WSM-MSW) groups. The accompanying reproduction number and sensitivity studies highlight the impact of both casual and long-term partnerships on infection spread. We construct an autonomous set of equations that handle issues usually ignored by autonomous equations and handled only through simulations or in a non-autonomous form. The autonomous formulation of the model allows for simple numerical computations while incorporating a combination of random instantaneous contacts between individuals and prolonged contacts between specific individuals.
Dynamical behavior of stochastic multigroup S-DI-A epidemic models for the transmission of HIV
2018, Journal of the Franklin Institute
In this paper, we study two stochastic multigroup S-DI-A epidemic models for the transmission of HIV. For the stochastic S-DI-A epidemic model with periodic coefficients, we first obtain sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we show that the model admits a positive T-periodic solution by using Khasminskii theory of periodic solution. Moreover, we establish sufficient conditions for exponential extinction of the infectious disease. For the stochastic S-DI-A epidemic model disturbed by both white and telegraph noises, we first establish sufficient conditions for persistence in the mean of the disease. Then in the case of persistence, we obtain sufficient conditions for the existence of a unique ergodic stationary distribution of the positive solutions by constructing a suitable stochastic Lyapunov function with regime switching and we also obtain sufficient conditions for exponential extinction of the system with regime switching.
A risk-based model for predicting the impact of using condoms on the spread of sexually transmitted infections
2017, Infectious Disease Modelling
Citation Excerpt :
We use the selective mixing model developed by Busenberg & Castillo-Chavez, (1991) to capture the heterogenous mixing among people with different number of partners. Our model is closely related to the STI models for the spread of the HIV/AIDS in heterosexual networks (Hyman & Stanley, 1988, 1989) that distribute the population based on their risk, such as the number of partners (Hyman, Li, & Stanley, 2001, 2003; Hyman & Stanley, 1988, 1989). Chlamydia and gonorrhea are transmitted when infected semen or vaginal fluids contact mucosal surfaces.
We create and analyze a mathematical model to understand the impact of condom-use and sexual behavior on the prevalence and spread of Sexually Transmitted Infections (STIs). STIs remain significant public health challenges globally with a high burden of some Sexually Transmitted Diseases (STDs) in both developed and undeveloped countries. Although condom-use is known to reduce the transmission of STIs, there are a few quantitative population-based studies on the protective role of condom-use in reducing the incidence of STIs. The number of concurrent partners is correlated with their risk of being infectious by an STI such as chlamydia, gonorrhea, or syphilis. We develop a Susceptible-Infectious-Susceptible (SIS) model that stratifies the population based on the number of concurrent partners. The model captures the multi-level heterogeneous mixing through a combination of biased (preferential) and random (proportional) mixing processes between individuals with distinct risk levels, and accounts for differences in condom-use in the low- and high-risk populations. We use sensitivity analysis to assess the relative impact of high-risk people using condom as a prophylactic intervention to reduce their chance of being infectious, or infecting others. The model predicts the STI prevalence as a function of the number of partners of an individual, and quantifies how this distribution of effective partners changes as a function of condom-use. Our results show that when the mixing is random, then increasing the condom-use in the high-risk population is more effective in reducing the prevalence than when many of the partners of high-risk people have high risk. The model quantifies how the risk of being infected increases for people who have more partners, and the need for high-risk people to consistently use condoms to reduce their risk of infection.
Influence of concurrency, partner choice, and viral suppression on racial disparity in the prevalence of HIV infected women
2016, Mathematical Biosciences
Citation Excerpt :
A separate study including age structure would be a natural extension of this research. Differential infectivity and differential susceptibility [33,41,42] is included in our model by the total population subdivided by sexual behavior as well as to two infected classes. In addition, we include in our model the impact of differential infectivity and differential susceptibility as reflected by the difference between the number of contacts per partner and number of partners by sexual behavior.
In 1992, Watts and May introduced a simple dynamical systems model of the spread of HIV based on disease transmission per partnership including the length of partnership duration. This model allowed for the treatment of concurrent partnerships, although it was hampered by the assumption of an important latent phase which generated a non-autonomous system. Subsequent models including concurrency have been based on networks, Monte Carlo, and stochastic simulations which lose a qualitative understanding of the effects of concurrency. We present a new autonomous deterministic model of the effect of concurrent sexual partnerships that allows for an analytical study of disease transmission.
We incorporate the effect of concurrency through the newly derived force of infection term in a mathematical model of the transmission of HIV through sexual contact in a population stratified by sexual behavior and race/ethnicity. The model also includes variations in population mixing (partner choice) and non-uniform Highly Active Anti-Retroviral Treatment (HAART) leading to viral suppression. We use this mathematical model to understand the non-uniform spread of HIV in women who were infected through heterosexual contact. In addition, an analytical study shows the importance of continued condom use in virally suppressed MSM.
Numerical simulations of the reproduction number as a function of concurrency, viral suppression level, and mixing show a reservoir of disease present in both heterosexual and MSM populations. Statistical analysis of parameter values show that viral suppression level, mixing and progression to AIDS without viral suppression have a strong correlation (either positive or negative) with the number of HIV positive women. Concurrency and assortative mixing are shown to be essential to reproduce infection levels in women, as reported by 2010 data from the Center for Disease Control (CDC).
Agent-based and phylogenetic analyses reveal how HIV-1 moves between risk groups: Injecting drug users sustain the heterosexual epidemic in Latvia
2012, Epidemics
Citation Excerpt :
One of the most important factors influencing the spread of an epidemic is the ability of an infected individual to establish a sufficient number of contacts with susceptible individuals during its infectious period. A major focus of previous research was on the importance of super spreaders, individuals with a high number of social contacts, for the spread of the epidemic (Hyman et al., 2001; Metzger et al., 2011). However, the social and geographical structure of the susceptible population also determine these contact rates.
Injecting drug users (IDUs) are a driving force for the spread of HIV-1 in Latvia and other Baltic States, accounting for a majority of cases. However, in recent years, heterosexual cases have increased disproportionately. It is unclear how the changes in incidence patterns in Latvia can be explained, and how important IDUs are for the heterosexual sub-epidemic. We introduce a novel epidemic model and use phylogenetic analyses in parallel to examine the spread of HIV-1 in Latvia between 1987 and 2010. Using a hybrid framework with a mean-field description for the susceptible population and an agent-based model for the infecteds, we track infected individuals and follow transmission histories dynamically formed during the simulation.
The agent-based simulations and the phylogenetic analysis show that more than half of the heterosexual transmissions in Latvia were caused by IDU, which sustain the heterosexual epidemic. Indeed, we find that heterosexual clusters are characterized by short transmission chains with up to 63% of the chains dying out after the first introduction. In the simulations, the distribution of transmission chain sizes follows a power law distribution, which is confirmed by the phylogenetic data. Our models indicate that frequent introductions reduced the extinction probability of an autonomously spreading heterosexual HIV-1 epidemic, which now has the potential to dominate the spread of the overall epidemic in the future. Furthermore, our model shows that social heterogeneity of the susceptible population can explain the shift in HIV-1 incidence in Latvia over the course of the epidemic. Thus, the decrease in IDU incidence may be due to local heterogeneities in transmission, rather than the implementation of control measures. Increases in susceptibles, through social or geographic movement of IDU, could lead to a boost in HIV-1 infections in this risk group. Targeting individuals that bridge social groups would help prevent further spread of the epidemic.

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Regular ArticleThe Initialization and Sensitivity of Multigroup Models for the Transmission of HIV

Abstract

Math. Biosci.

Math. Biosci.

Math. Biosci.

Simulation of Infectious Disease Epidemics

Lessons from people with nonprogressive HIV infection

N. Engl. J. Med.

Risk factors associated with HIV infection in Uganda

J. Infect. Dis.

Sensitivity and uncertainty analysis of complex models of disease transmission: an HIV model, as an example

Int. Stat. Rev.

Virologic and immunologic characterization of long-term survivors of HIV type 1 Infection

N. Engl. J. Med.

AIDS: the epidemiological significance of two different mean rates of partner change

IMA J. Math. Med. Biol.

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

JAIDS

Regular Article
The Initialization and Sensitivity of Multigroup Models for the Transmission of HIV