Modelling the HIV/AIDS epidemic trends in South Africa: insights from a simple mathematical model.

*(English)*Zbl 1215.92055Summary: The HIV/AIDS epidemic is a serious public health challenge in South Africa. In this paper, a simple deterministic HIV/AIDS model incorporating condom use, sexual partner acquisition, behavior change and treatment as HIV/AIDS control strategies is formulated using a system of ordinary differential equations with the object of applying it to the current South African situation. Firstly, a theoretical analysis of the model is presented. The analysis of the model shows that the disease free equilibrium point is globally asymptotically stable for the epidemic threshold R (the model reproduction number) less than unit and unstable when this threshold is greater than unit. A unique endemic equilibrium, whose local stability near the threshold R, is determined using the center manifold theory, is shown to be locally asymptotically stable for \(R > 1\). Global stability of the endemic equilibrium is shown for a special case whenever \(R > 1\). Secondly, the model is fitted to data from UNAIDS/WHO on HIV/AIDS in South Africa. Based on the fit to the prevalence data, projections are made beyond 2007 on the course of the epidemic for various levels of interventions. The results compare very well with other research outcomes on the HIV/AIDS epidemic in South Africa. Projections are made to track the changes in the number of individuals who will be under treatment, an important group as far as public health planning is concerned. The epidemiological implications of such projections on public health planning and management are discussed.

##### MSC:

92D30 | Epidemiology |

34H05 | Control problems involving ordinary differential equations |

34D23 | Global stability of solutions to ordinary differential equations |

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\textit{F. Nyabadza} et al., Nonlinear Anal., Real World Appl. 12, No. 4, 2091--2104 (2011; Zbl 1215.92055)

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