A model of HIV/AIDS population dynamics including ARV treatment and pre-exposure prophylaxis

Antiretroviral treatment (ART) and oral pre-exposure prophylaxis (PrEP) have recently been used efficiently in management of HIV infection. Pre-exposure prophylaxis consists in the use of an antiretroviral medication to prevent the acquisition of HIV infection by uninfected individuals. We propose a new model for the transmission of HIV/AIDS including ART and PrEP. Our model can be used to test the effects of ART and of the uptake of PrEP in a given population, as we demonstrate through simulations. The model can also be used to estimate future projections of HIV prevalence. We prove global stability of the disease-free equilibrium. We also prove global stability of the endemic equilibrium for the most general case of the model, i.e., which allows for PrEP individuals to default. We include insightful simulations based on recently published South-African data

A Treatise of Biological Models

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Modelling the epidemiological and economic impact of HIV/AIDS with particular reference to Zimbabwe

HIV/AIDS is a major concern in Zimbabwe, not just in terms of disease spread but also in terms of its impact on society and economic development. The paper presents the epidemiological and economic impact of HIV/AIDS which is analysed using a conventional epidemiological model which captures the dynamics of the epidemic with multi-intervention strategies and a total cost function which analyzes the fiscal economic impact of the pandemic to Zimbabwe. The model assesses the effects of treating using anti-retroviral (ARVs) drugs (based on efficacy and adherence), condom use (based on efficacy and compliance), voluntary testing and counselling and the epidemiological impact on the country. The model shows that reduction in effective sexual contacts by increasing condom use and avoidance of multiple sexual partners has a significant impact in reducing the transmission of the disease. It is shown that without behavioral change, effective treatment of the infected individuals re! suits in more spread of the epidemic. Increasing the incubation by treatment tends to benefit the infected individuals as this increases the number of years lived but the total annual cost maybe too high to be sustained by the government. On the other hand the cost of doing "nothing" as represented by an incubation period of less than 9 years appears to be costly as well. Treatment and caring for those who have developed clinical AIDS constitute a major part of our cost function followed by treatment of those who are infected but have not yet developed clinical AIDS. In Zimbabwe, where the pandemic has reached epidemic proportions, combating the disease and its economic effects successfully will require a large and determined fiscal effort. This study therefore sheds light on the economic consequences of decisions taken in the prevention, treatment and care of HIV/AIDS individuals

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 (RA), 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 (R0) 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

Dynamical properties and thresholds of an HIV model with super-infection.

Super-infection by multiple HIV-1 subtypes, previously thought restricted to high risk groups, has now been reported in the general heterosexual populations at relatively the same incidence rate as in high risk groups. We present a simple deterministic HIV model with super-infection by two HIV-1 subtypes. Mathematical characteristics including the basic reproductive number $(\mathcal{R}_0)$, invasion threshold $(\mathcal{R}_{21},\mathcal{R}_{12})$ and conditions for asymptotic stability are derived. In the absence of super-infection the model exhibits competitive exclusion, and all equilibria are globally attracting if they exist except for the disease free which is a saddle for $\mathcal{R}_0>1.$ The results show that the subtype with the dominant reproductive number exceeding unity dominates the weaker subtype forcing it to extinction regardless of the size of the reproductive number. On the other end, super-infection may promote subtype co-existence whenever the minimum of the subtype specific reproductive numbers $(\mathcal{R}_1,\mathcal{R}_2)$ and the invasion reproductive numbers $(\mathcal{R}_{12},\mathcal{R}_{21})$ exceed unity. Our results demonstrate that if the partial reproductive numbers (\mathcal{R}_1~\mbox{and}~\mathcal{R}_2 ) and the invasion reproductive number for the weaker subtype $(\mathcal{R}_{21})$ satisfy \mathcal{R}_21~\mbox{and}~\mathcal{R}_{21}>1, then primary infection by subtype $1$ may stay the extinction of subtype $2$ despite its relatively low reproductive fitness. For certain parameter ranges, hysteresis (including backward bifurcation) occurs with possible differences in the asymptotic level of disease prevalence. Super-infection may thus facilitate the continued re-generation of reproductively noncompetent subtypes whose subtype specific reproductive numbers will be less than unity while at the same time allowing for the mutual coexistence and persistence of multiple strains. Persistence and co-existence of multiple strains has detrimental effect on vaccine design and development and administration of ART where one or more of the strains are drug resistant

Optimal control of aquatic diseases: a case study of Yemen’s cholera outbreak

We propose a mathematical model for the transmission dynamics of some strains of the bacterium Vibrio cholerae, responsible for the cholera disease in humans. We prove that, when the basic reproduction number is equal to one, a transcritical bifurcation occurs for which the endemic equilibrium emanates from the disease-free point. A control function is introduced into the model, representing the distribution of chlorine water tablets for water purification. An optimal control problem is then proposed and analyzed, where the goal is to determine the fraction of susceptible individuals who should have access to chlorine water tablets in order to minimize the total number of new infections plus the total cost associated with the distribution of chlorine water tablets, over the considered period of time. Finally, we consider real data of the cholera outbreak in Yemen, from April 27, 2017 to April 15, 2018, choosing the values of the parameters of the uncontrolled model that fit the real data. Using our optimal control results, we show, numerically, that the distribution of chlorine water tablets could have stopped, in a fast way, the worst cholera outbreak that ever occurred in human history. Due to the critical situation of Yemen, we also simulate the case where only a small percentage of susceptible individuals has access to chlorine water tablets and obtain an optimal control solution that decreases, substantially, the maximum number of infective individuals affected by the outbreak.publishe