Stability analysis of generalized epidemic models over directed networks

In this paper we propose a generalized version of the Susceptible-Exposed-Infected-Vigilant (SEIV) disease spreading model over arbitrary directed graphs. In the standard SEIV model there is only one infectious state. Our model instead allows for the exposed state to also be infectious to healthy individuals. This model captures the fact that infected individuals may act differently when they are aware of their infection. For instance, when the individual is aware of the infection, different actions may be taken, such as staying home from work, causing less chance for spreading the infection. This model generalizes the standard SEIV model which is already known to generalize many other infection spreading models available. We use tools from nonlinear stability analysis to suggest a coordinate transformation that allows us to study the stability of the origin of a relevant linear system. We provide a necessary and sufficient condition for when the disease-free equilibrium is globally exponentially stable. We then extend the results to the case where the infection parameters are not homogeneous among the nodes of the network. Simulations illustrate our results.