On SICA Models for HIV Transmission

Model-free based control of a HIV/AIDS prevention model

Controlling an epidemiological model is often performed using optimal control theory techniques for which the solution depends on the equations of the controlled system, objective functional and possible state and/or control constraints. In this paper, we propose a model-free control approach based on an algorithm that operates in 'real-time' and drives the state solution according to a direct feedback on the state solution that is aimed to be minimized, and without knowing explicitly the equations of the controlled system. We consider a concrete epidemic problem of minimizing the number of HIV infected individuals, through the preventive measure pre-exposure prophylaxis (PrEP) given to susceptible individuals. The solutions must satisfy control and mixed state-control constraints that represent the limitations on PrEP implementation. Our model-free based control algorithm allows to close the loop between the number of infected individuals with HIV and the supply of PrEP medication 'in real time', in such a manner that the number of infected individuals is asymptotically reduced and the number of individuals under PrEP medication remains below a fixed constant value. We prove the efficiency of our approach and compare the model-free control solutions with the ones obtained using a classical optimal control approach via Pontryagin maximum principle. The performed numerical simulations allow us to conclude that the model-free based control strategy highlights new and interesting performances compared with the classical optimal control approach.

A dynamically-consistent nonstandard finite difference scheme for the SICA model

In this work, we derive a nonstandard finite difference scheme for the SICA (Susceptible-Infected-Chronic-AIDS) model and analyze the dynamical properties of the discretized system. We prove that the discretized model is dynamically consistent with the continuous, maintaining the essential properties of the standard SICA model, namely, the positivity and boundedness of the solutions, equilibrium points, and their local and global stability.