Dynamical analysis of a multi-strain model of HIV in the presence of anti-retroviral drugs

One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (R t 0) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where R t 0 > 1, it is shown that the model can have two co-existing endemic equilibria, and competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (R t r) is greater than that of the wild strain (R t w). Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold (R(0)) exceeds unity. For the case when [Formula in text], it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make R t w < R t r.     

Dynamics Analysis of a Multi-strain Cholera Model with an Imperfect Vaccine

A new two-strain model, for assessing the impact of basic control measures, treatment and dose-structured mass vaccination on cholera transmission dynamics in a population, is designed. The model has a globally-asymptotically stable disease-free equilibrium whenever its associated reproduction number is less than unity. The model has a unique, and locally-asymptotically stable, endemic equilibrium when the threshold quantity exceeds unity and another condition holds. Numerical simulations of the model show that, with the expected 50 % minimum efficacy of the first vaccine dose, vaccinating 55 % of the susceptible population with the first vaccine dose will be sufficient to effectively control the spread of cholera in the community. Such effective control can also be achieved if 50 % of the first vaccine dose recipients take the second dose. It is shown that a control strategy that emphasizes the use of antibiotic treatment is more effective than one that emphasizes the use of basic (non-pharmaceutical) anti-cholera control measures only. Numerical simulations show that, while the universal strategy (involving all three control measures) gives the best outcome in minimizing cholera burden in the community, the combined basic anti-cholera control measures and treatment strategy also has very effective community-wide impact.     

Mathematical Study of a Staged-Progression HIV Model with Imperfect Vaccine

A staged-progression HIV model is formulated and used to investigate the potential impact of an imperfect vaccine. The vaccine is assumed to have several desirable characteristics such as protecting against infection, causing bypass of the primary infection stage, and offering a disease-altering therapeutic effect (so that the vaccine induces reversal from the full blown AIDS stage to the asymptomatic stage). The model, which incorporates HIV transmission by individuals in the AIDS stage, is rigorously analyzed to gain insight into its qualitative features. Using a comparison theorem, the model with mass action incidence is shown to have a globally-asymptotically stable disease-free equilibrium whenever a certain threshold, known as the vaccination reproduction number, is less than unity. Furthermore, the model with mass action incidence has a unique endemic equilibrium whenever this threshold exceeds unity. Using the Li-Muldowney techniques for a reduced version of the mass action model, this endemic equilibrium is shown to be globally-asymptotically stable, under certain parameter restrictions. The epidemiological implications of these results are that an imperfect vaccine can eliminate HIV in a given community if it can reduce the reproduction number to a value less than unity, but the disease will persist otherwise. Furthermore, a future HIV vaccine that induces the bypass of primary infection amongst vaccinated individuals (who become infected) would decrease HIV prevalence, whereas a vaccine with therapeutic effect could have a positive or negative effect at the community level.     

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.     

Male circumcision in the general population of Kisumu, Kenya: beliefs about protection, risk behaviors, HIV, and STIs

Using a population-based survey we examined the behaviors, beliefs, and HIV/HSV-2 serostatus of men and women in the traditionally non-circumcising community of Kisumu, Kenya prior to establishment of voluntary medical male circumcision services. A total of 749 men and 906 women participated. Circumcision status was not associated with HIV/HSV-2 infection nor increased high risk sexual behaviors. In males, preference for being or becoming circumcised was associated with inconsistent condom use and increased lifetime number of sexual partners. Preference for circumcision was increased with understanding that circumcised men are less likely to become infected with HIV.     

Mathematical Study of the Role of Gametocytes and an Imperfect Vaccine on Malaria Transmission Dynamics

A mathematical model is developed to assess the role of gametocytes (the infectious sexual stage of the malaria parasite) in malaria transmission dynamics in a community. The model is rigorously analysed to gain insights into its dynamical features. It is shown that, in the absence of disease-induced mortality, the model has a globally-asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold, known as the basic reproduction number (denoted by ℛ₀), is less than unity. Further, it has a unique endemic equilibrium if ℛ₀>1. The model is extended to incorporate an imperfect vaccine with some assumed therapeutic characteristics. Theoretical analyses of the model with vaccination show that an imperfect malaria vaccine could have negative or positive impact (in reducing disease burden) depending on whether or not a certain threshold (denoted by ∇) is less than unity. Numerical simulations of the vaccination model show that such an imperfect anti-malaria vaccine (with a modest efficacy and coverage rate) can lead to effective disease control if the reproduction threshold (denoted by ℛ_vac) of the disease is reasonably small. On the other hand, the disease cannot be effectively controlled using such a vaccine if ℛ_vac is high. Finally, it is shown that the average number of days spent in the class of infectious individuals with higher level of gametocyte is critically important to the malaria burden in the community.     

Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe

We present a sex-structured model for heterosexual transmission of HIV/AIDS in a community. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with a time delay due to incubation period. The sex-structured HIV/AIDS model divides the population into a two sex-structure consisting of females and males. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to focus on the effects of condom use as a single-strategy approach in HIV prevention in the absence of any treatment. Initially we model the use of male condoms and further extend the model to incorporate the use of both female and male condoms. The model includes two primary factors in condom use to control HIV that are condom efficacy and compliance. The exposure risk of infection after each intervention is obtained. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male and female condom use in a community. The models are numerically analysed to assess the effectiveness of condom use on the transmission dynamics of HIV/AIDS using demographic and epidemiological parameters for Zimbabwe. The study demonstrates the use of sex-structured HIV/AIDS models in assessing the effectiveness of female and male condom use as a preventative strategy in a heterosexually active population. Z. Mukandavire would like to acknowledge financial support given by the National University of Science and Technology through a Staff Development Scholarship. The authors are grateful to Eagle Insurance Company of Zimbabwe for financial support.     

“When I Was Circumcised I Was Taught Certain Things”: Risk Compensation and Protective Sexual Behavior among Circumcised Men in Kisumu,Kenya

Background

Male circumcision has been shown to reduce the transmission of HIV from women to men through vaginal sex by approximately 60%. There is concern that men may engage in risk compensation after becoming circumcised, diminishing the benefits of male circumcision.

Methods and Findings

We conducted qualitative interviews with 30 sexually active circumcised men in Kisumu, Kenya from March to November 2008. Most respondents reported no behavior change or increasing protective sexual behaviors including increasing condom use and reducing the number of sexual partners. A minority of men reported engaging in higher risk behaviors either not using condoms or increasing the number of sex partners. Circumcised respondents described being able to perform more rounds of sex, easier condom use, and fewer cuts on the penis during sex.

Conclusions

Results illustrate that information about MC''s protection against HIV has disseminated into the larger community and MC accompanied by counseling and HIV testing can foster positive behavior change and maintain sexual behavior.     

Women's Beliefs about Male Circumcision,HIV Prevention,and Sexual Behaviors in Kisumu,Kenya

It is important to understand how women''s sexual practices may be influenced by male circumcision (MC) as an HIV prevention effort. Women''s beliefs about MC and sexual behaviour will likely influence the scale-up and uptake of medical MC. We conducted qualitative interviews with 30 sexually active women in Kisumu, Kenya. Women discussed MC related to perceived health benefits, condom use, sexual behaviour, knowledge of susceptibility to HIV and sexually transmitted infections (STIs), circumcision preference, and influence on circumcision uptake. Respondents had a good understanding of the partial protection of MC for acquisition of HIV for men. Women perceived circumcised men as cleaner, carrying fewer diseases, and taking more time to reach ejaculation. Male''s circumcision status is a salient factor for women''s sexual decision making, including partner choice, and condom use. It will be important that educational information affirms that MC provides only partial protection against female to male transmission of HIV and some STIs; that other HIV and STI prevention methods such as condoms need to be used in conjunction with MC; that MC does not preclude a man from having HIV; and that couples should develop plans for not having sex while the man is healing.     

Backward bifurcations in dengue transmission dynamics

A deterministic model for the transmission dynamics of a strain of dengue disease, which allows transmission by exposed humans and mosquitoes, is developed and rigorously analysed. The model, consisting of seven mutually-exclusive compartments representing the human and vector dynamics, has a locally-asymptotically stable disease-free equilibrium (DFE) whenever a certain epidemiological threshold, known as the basic reproduction number(R(0)) is less than unity. Further, the model exhibits the phenomenon of backward bifurcation, where the stable DFE coexists with a stable endemic equilibrium. The epidemiological consequence of this phenomenon is that the classical epidemiological requirement of making R(0) less than unity is no longer sufficient, although necessary, for effectively controlling the spread of dengue in a community. The model is extended to incorporate an imperfect vaccine against the strain of dengue. Using the theory of centre manifold, the extended model is also shown to undergo backward bifurcation. In both the original and the extended models, it is shown, using Lyapunov function theory and LaSalle Invariance Principle, that the backward bifurcation phenomenon can be removed by substituting the associated standard incidence function with a mass action incidence. In other words, in addition to establishing the presence of backward bifurcation in models of dengue transmission, this study shows that the use of standard incidence in modelling dengue disease causes the backward bifurcation phenomenon of dengue disease.     

Role of incidence function in vaccine-induced backward bifurcation in some HIV models

The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control. In such a scenario, the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination. This paper addresses the role of the choice of incidence function in a vaccine-induced backward bifurcation in HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with their equivalents that employ mass action incidence. Furthermore, this result is independent of the type of vaccination program adopted. These results emphasize the need for further work on the incidence functions used in HIV models.     

Chancroid transmission dynamics: a mathematical modeling approach

Mathematical models have long been used to better understand disease transmission dynamics and how to effectively control them. Here, a chancroid infection model is presented and analyzed. The disease-free equilibrium is shown to be globally asymptotically stable when the reproduction number is less than unity. High levels of treatment are shown to reduce the reproduction number suggesting that treatment has the potential to control chancroid infections in any given community. This result is also supported by numerical simulations which show a decline in chancroid cases whenever the reproduction number is less than unity.     

Impact of Male Circumcision on the HIV Epidemic in Papua New Guinea: A Country with Extensive Foreskin Cutting Practices

The degree to which adult medical male circumcision (MC) programs can reduce new HIV infections in a moderate HIV prevalence country like Papua New Guinea (PNG) are uncertain especially given the widespread prevalence of longitudinal foreskin cuts among adult males. We estimated the likely impact of a medical MC intervention in PNG using a mathematical model of HIV transmission. The model was age-structured and incorporated separate components for sex, rural/urban, men who have sex with men and female sex workers. Country-specific data of the prevalence of foreskin cuts, sexually transmitted infections, condom usage, and the acceptability of MC were obtained by our group through related studies. If longitudinal foreskin cutting has a protective efficacy of 20% compared to 60% for MC, then providing MC to 20% of uncut males from 2012 would require 376,000 procedures, avert 7,900 HIV infections by 2032, and require 143 MC per averted infection. Targeting uncut urban youths would achieve the most cost effective returns of 54 MC per HIV infection averted. These numbers of MC required to avert an HIV infection change little even with coverage up to 80% of men. The greater the protective efficacy of longitudinal foreskin cuts against HIV acquisition, the less impact MC interventions will have. Dependent on this efficacy, increasing condom use could have a much greater impact with a 10 percentage point increase averting 18,400 infections over this same period. MC programs could be effective in reducing HIV infections in PNG, particularly in high prevalence populations. However the overall impact is highly dependent on the protective efficacy of existing longitudinal foreskin cutting in preventing HIV.     

HIV/AIDS dynamics: Impact of economic classes with transmission from poor clinical settings

We formulate and analyze a nonlinear deterministic HIV/AIDS model with two social classes, namely the poor and the rich including transmission from poor clinical settings with a randomly variable population. Four sub-models are derived from the full model, the disease threshold parameters are computed, and it is shown that the disease will die down if these initial threshold parameters are less than unity and will persist if they exceed unity. The impact of economic classes (along with transmission from poor/inadequate clinical settings) on the disease dynamics is assessed, and we observe that even with a single sexual partner, the reproduction number is slightly greater than unity, implying that the additional transmission can only be from clinical settings. Stability (local and global) of both the disease-free and endemic equilibria are then investigated using various techniques of dynamical systems such as Centre Manifold theory and Lyapunov's second method. Analysis on the bifurcation parameter is carried out to assess the impact of related HIV transmission from poor clinical settings. We estimate some of the model parameter values and numerical simulations of the model are represented graphically. Our results show that the prevalence of HIV in rich communities seems to be higher than that in the poor, but the disease develops faster in impoverished individuals.     

Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment

A two strain HIV/AIDS model with treatment which allows AIDS patients with sensitive HIV-strain to undergo amelioration is presented as a system of non-linear ordinary differential equations. The disease-free equilibrium is shown to be globally asymptotically stable when the associated epidemic threshold known as the basic reproduction number for the model is less than unity. The centre manifold theory is used to show that the sensitive HIV-strain only and resistant HIV-strain only endemic equilibria are locally asymptotically stable when the associated reproduction numbers are greater than unity. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The model is numerically analysed to assess the effects of treatment with amelioration on the dynamics of a two strain HIV/AIDS model. Numerical simulations of the model show that the two strains co-exist whenever the reproduction numbers exceed unity. Further, treatment with amelioration may result in an increase in the total number of infective individuals (asymptomatic) but results in a decrease in the number of AIDS patients. Further, analysis of the reproduction numbers show that antiretroviral resistance increases with increase in antiretroviral use.     

Investigating alcohol consumption as a risk factor for HIV transmission in heterosexual settings in sub-Saharan African communities

Alcohol consumption and abuse is widespread in sub-Saharan Africa where most HIV infections occur and has been associated with risky sexual behaviors. It may therefore be one of the most common, potentially modifiable HIV risk factors in this region. A deterministic system of ordinary differential equations incorporating heterogeneity and biased sexual preferences is formulated to assess the effects of alcohol consumption on the transmission dynamics of the disease in heterosexual settings. Extensive qualitative analysis of the model is carried out and epidemic threshold such as the alcohol-induced reproductive number $({\mathcal{R}}_{A})$ , and equilibria are derived and their stabilities examined. The disease-free equilibrium is found to be globally attracting whenever the reproductive number is less than unity. In the model, heterosexuality is the source of transmissions, and therefore, targeting a reduction of the basic reproductive number $({\mathcal{R}}_{0})$ should be primary objective for any intervention programme. We show that the preference to form partnerships amongst the heterogeneous groups influences the severity of disease and its evolution, and consequently the rate of partnership formation between females and alcohol consumers and their relative infectiousness over nondrinkers has a huge positive correlation with the alcohol-induced reproductive number and hence the epidemic. The proportion or absolute number of drinkers is shown to have minimal influence on the disease dynamics, and in a community with alcohol consumers, it is more prudent to reduce their risk sexual behavior rather than to fight the spread of alcohol consumption. Thus, intervention measures targeted at reducing heterogeneous group interactions and behavior change are the key to disease control in these settings.     

HIV prevention for indigenous people of the Amazon basin.

This study identifies theoretically-based predictors of condom use in a sample of 251 sexually active adults recruited from Sao Gabriel da Cachoeira and six indigneous communities of the Upper Rio Negro region of Amazonas Brazil. The information-motivation-behavioral skills (IMB) model of AIDS-preventive behavior was used to describe the roles of HIV/AIDS knowledge, experiences with and attitudes toward condom use, peer influences, perceived vulnerability, monogamy and behavioral skills. A predictive path analytic model revealed significant predictors of more condom use including male gender, greater sexual HIV knowledge, positive experiences and attitudes about condom use, multiple partners, and greater behavioral skills. Results suggest that attention to behavioral skills for negotiating safer sex and instruction in the correct use of condoms are important elements in reducing high risk behaviors. Increasing the specific knowledge level of indigenous people regarding the complexities of sexual transmission of HIV is crucial and should be addressed. Heightening individuals' understanding of the limited protection of serial monogamy, and the need to conduct gender-specific training for behavior change to reduce transmission of HIV should be an additional goal of Brazilian health professionals. Obstacles to the implementation of the IMB HIV prevention program in Amazonas are noted and an alternative Brazilian HIV/AIDS prevention program is discussed.     

Learning that circumcision is protective against HIV: risk compensation among men and women in Cape Town, South Africa

Objectives

We examined whether knowledge of the HIV-protective benefits of male circumcision (MC) led to risk compensating behavior in a traditionally circumcising population in South Africa. We extend the current literature by examining risk compensation among women, which has hitherto been unexplored.

Methods

We used data on Xhosa men and women from the 2009 Cape Area Panel Study. Respondents were asked if they had heard that MC reduces a man’s risk of contracting HIV, about their perceived risk of contracting HIV, and condom use. For each gender group we assessed whether risk perception and condom use differed by knowledge of the protective benefits of MC using bivariate and then multivariate models controlling for demographic characteristics, HIV knowledge/beliefs, and previous sexual behaviors. In a further check for confounding, we used data from the 2005 wave to assess whether individuals who would eventually become informed about the protective benefits of circumcision were already different in terms of HIV risk perception and condom use.

Results

34% of men (n = 453) and 27% of women (n = 690) had heard that circumcision reduces a man’s risk of HIV infection. Informed men perceived slightly higher risk of contracting HIV and were more likely to use condoms at last sex (p<0.10). Informed women perceived lower HIV risk (p<0.05), were less likely to use condoms both at last sex (p<0.10) and more generally (p<0.01), and more likely to forego condoms with partners of positive or unknown serostatus (p<0.01). The results were robust to covariate adjustment, excluding people living with HIV, and accounting for risk perceptions and condom use in 2005.

Conclusions

We find evidence consistent with risk compensation among women but not men. Further attention should be paid to the role of new information regarding MC, and drivers of HIV risk more broadly, in modulating sexual behavior among women.     

Male circumcision: hope for HIV infection decrease in southern Africa

Given the magnitude of the HIV pandemic, development of new prevention means is necessary. Male circumcision reduces HIV transmission from female to male by 57 % [95 % Confident Interval (CI): 42-68 %]. Its generalization in sub-Saharan Africa could avert, among men and women, from 1 to 4 millions new HIV infections over the next ten years. Acceptability of this new prevention mean is high in countries which could benefit the most from male circumcision, that means located in southern Africa, a region where in majority men are uncircumcised and where HIV prevalence is high. Male circumcision is a cost-effective prevention strategy. Actual prevention means (condoms, sexual abstinence and fidelity) are not used enough to curb the HIV epidemic. Research is ongoing on other prevention means (vaccine, pre- and post-exposition prophylaxis, microbicides, diaphragm) but their efficiency has not been demonstrated yet. Nevertheless, generalization of circumcision in southern Africa is responsible for contestations in part due to the fact that this prevention mean protects only partially from HIV infection. Moreover, for now, only a few countries integrated circumcision in their HIV prevention program in spite of WHO (World Health Organization) recommendations supporting male circumcision acknowledgement as an additional, important strategy for the prevention of heterosexually acquired HIV infection in men. Significant available funding should allow the situation to evolve quickly. At the same time, research goes on in order to know more about the effects and to facilitate the generalization of this prevention mean which is a great hope for southern Africa.     

Analysis of SIR epidemic models with nonlinear incidence rate and treatment

This paper deals with the nonlinear dynamics of a susceptible-infectious-recovered (SIR) epidemic model with nonlinear incidence rate, vertical transmission, vaccination for the newborns of susceptible and recovered individuals, and the capacity of treatment. It is assumed that the treatment rate is proportional to the number of infectives when it is below the capacity and constant when the number of infectives reaches the capacity. Under some conditions, it is shown that there exists a backward bifurcation from an endemic equilibrium, which implies that the disease-free equilibrium coexists with an endemic equilibrium. In such a case, reducing the basic reproduction number less than unity is not enough to control and eradicate the disease, extra measures are needed to ensure that the solutions approach the disease-free equilibrium. When the basic reproduction number is greater than unity, the model can have multiple endemic equilibria due to the effect of treatment, vaccination and other parameters. The existence and stability of the endemic equilibria of the model are analyzed and sufficient conditions on the existence and stability of a limit cycle are obtained. Numerical simulations are presented to illustrate the analytical results.