Dynamical models of HIV-AIDS e¤ect on population growth

We review some known dynamical models of epidemics, given by coupled systems of di¤erential equations, and propose a simple SI-type model of HIV-transmision that accounts for population growth. It allows exact analysis of solution and makes speci…c predictions on the spread of HIV and its e¤ect on populations. Then we outline an extended version of the model based on several behavioral/gender groups. We develop numerical code for solving thi system. The extended model corroborates some qualitative predictions of simple 2D system. We compare our predictions with some recent demographic data on the spread of HIV in several African countries. The spread of epidemics could be modeled by coupled systems of di¤erential equations, representing various populations and their interactions (behavioral patterns). Such modeling aims to analyze and predict the spread of epidemics, and also develop control/prevention strategies. 1 SI and SIR models with constant population. The standard models of epidemics involve interacting populations: S susceptible, E exposed, I infectious, R removed (or recovered), etc. (see [3]). When I individual infects S individual a ceratin (latent) period could pass before he becomes infectious (e.g. 7 days for measles). The basic parameters in such models are the recovery/removal rate (1/ mean illness period) and transmission rate . In some cases (short lived epidemics, like ‡u) the total population is considered constant, S + I + ::: = N , in other cases (AIDS) the population (growth) dynamics becomes important. The basic SI model has the form _ S = SI N + I _ I = SI N I (1) where is the transmission rate (number of infections per each I, per unit time) and recovery. Model (1) allows no immunity, so after recovery I group move