No abstract

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. MSC: 92D30; 34K20

<abstract><p>We present a compartmental model in ordinary differential equations of malaria disease transmission, accommodating the effect of indoor residual spraying on the vector population. The model allows for influx of infected migrants into the host population and for outflow of recovered migrants. The system is shown to have positive solutions. In the special case of no infected immigrants, we prove global stability of the disease-free equilibrium. Existence of a unique endemic equilibrium point is also established for the case of positive influx of infected migrants. As a case study we consider the combined South African malaria region. Using data covering 31 years, we quantify the effect of malaria infected immigrants on the South African malaria region.</p></abstract>

We present a stochastic model of the population dynamics of HIV/AIDS with treatment and inflow of infectives. Starting with a deterministic compartmental model, each of the four ordinary differential equations are stochastically perturbed. An invariant R σ similar to the basic reproduction number of an ordinary differential equation system is introduced. Under conditions which permit the existence of a disease-free equilibrium point, we prove almost sure exponential stability of the disease-free equilibrium for R σ < 1. We also investigate asymptotic behaviour of the solutions to the stochastic model around the endemic equilibrium of the underlying deterministic model. Our theoretical results are illustrated by simulations with parameters applicable to South Africa.