Epidemicity conditions in spatial models of infectious diseases

We review the work that has been conducted in the past years to understand when and why significant epidemics may arise even though a disease is not bound to become endemic in the long run i.e. subthreshold outbreaks. In particular, we describe the methods that allow the analysis of such outbreaks not only in time but also in space. To that end, we introduce a broad class of spatially explicit mathematical models of water- and air-borne diseases. These models consist of coupled ordinary differential or difference equations. The spatial coupling is obtained via suitable networks that describe connectivity via waterways or human mobility, or both (multiplex networks). We derive exact epidemicity conditions under which the disease-free equilibrium, although asymptotically stable, may be characterized by bursts of infections possibly coalescing, epitomized by transient increases of the infected compartments given suitable perturbations. The growth of infections is quantified by means of ${\ell }^1$ - or ${\ell }^2$ -norms. We show that epidemicity is guaranteed whenever the dominant eigenvalue of a suitable Hermitian matrix is positive (continuous-time models) or larger than unity (in discrete-time models), while the corresponding reproduction number (quantified as the spectral radius of the next-generation matrix) is smaller than unit. We also apply the methodology to two relevant case studies: the 2010 cholera epidemic in Haiti and the 2020 COVID-19 outbreak in Italy.