Feedback Control Policies and Network Effects in Epidemics Models

The importance of mathematical models able to predict the dynamic behaviour of an infectious disease has been consolidated in the last year. In this thesis, we examine the SIR compartmental model, both in the scalar and network case, to account for endogenous individuals behaviours and population heterogeneities. In the first part, after an initial simulation of the classic behaviour, there is a presentation of several modified versions of the model, in which a mitigation term for social interactions is included. This addition, expressed as a function of the fraction of infected, can express either an individual reaction to the presence of the disease or a governments measures, imposed to limit interactions and contain contagion. Assuming that this function decreases with respect to the number of infected, we demonstrate the existence of a threshold parameter for determining the dynamics, similar to that characteristic of the classic SIR model. The functions considered within the model are different: linear, quadratic, smooth step and smooth. Another analysed case involves piecewise continuous control, in which a lockdown measure is implemented only if the total number of infected reaches a certain threshold level. The long-term behaviour of the system within this strategy consists of the so-called sliding motion, which will be presented and studied. Also within the scalar model, a modification will be proposed to take into account a certain delay between the observation of the epidemic and the subsequent responses of individuals or measures adopted by governments, in order to achieve a better description. By merging the concept of piecewise continuous control and delay, there is a continuous transition between the unrestricted model and the model with measurements and the phenomenon of chattering is outlined. The second part of the thesis focuses on observing the behaviour of the SIR model within a network, in which the population is divided into subpopulations and their interaction is described through a weighted graph. The average behaviour of the model, in which the weights of the average depend on the characteristics of the network, and the behaviour at each component node of the graph are analysed. New results for threshold conditions and stability properties are proposed, focusing in particular on the case of two-nodes networks. In a specific situation of contact between an initially totally healthy node and one with a fraction of infected, we show the occurrence of an atypical phenomenon with respect to the classical SIR theory. We also exhibit how changing the contact between nodes and differentiating the infection rates can affect the dynamics. Then we analysed different versions of lockdown policies within the network model, assuming restrictions between different subpopulations and also within the same subpopulation.