Epidemic modeling and control of HIV/AIDS dynamics in populations under external interactions: A worldwide challenge

In this chapter the problem of the interaction between groups of subjects singularly characterized by a specific infectious disease is addressed. The dynamical characteristics of an isolated population are preliminary studied, with particular reference to the equilibrium points and their stability. Then, the effects of constant inputs on the dynamics are deeply analyzed also by numerical simulations; this analysis is propaedeutic to the study of the interaction between the groups. The interactions between the different populations are modeled as additional input/output to the single group dynamics introducing total averaged effects including all the external migration effects. This approach focuses on the changes in the dynamics of one population when interactions are present without showing the global migration fluxes, but stressing the influences on each populations. Besides the simplifications of the model, this point of view may be fruitful also with respect of the design of control actions, assuming that each group can adopt the best control strategy for her/his own specific social characteristics. The epidemic case analyzed is HIV-AIDS. This choice has been made since this virus is present all over the world, but with different levels of dangerousness and number of infected patients depending on the economic, social, and cultural habits. The model used is a recently introduced one, which describes this epidemic spread considering two compartments of susceptible people, distinguished by the level of attention with respect to the virus transmission, one of the infected individuals not aware of their status, and two classes of patients, divided according to the level of infection. Additional inputs have been introduced to model fluxes of susceptible individuals and infected but not aware individuals. These effects have been reported in numerous figures showing the results of numerical simulations.