Fig. 1: A graphical illustration of the concept of epidemicity.

Computational results for COVID-19 transmission in two human communities connected by mobility. a System trajectories projected onto the plane spanning the total number of infected people in each of the two communities (Z_i = E_i + P_i + I_i + A_i, i = 1, 2). Trajectories have been initialized with a few exposed individuals in either community (black, yellow, and purple curves), or with a mix of infected individuals in both communities (green, corresponding to the perturbation of the DFE with the fastest growth in the system output²⁵). For this parameter combination (\({\beta }_{1}^{P}={\beta }_{2}^{P}=1.2\times 1{0}^{-1}\) days⁻¹), all trajectories converge to the DFE (\({{\mathcal{R}}}_{0}\,<\, 1\), e₀ < 0). b Temporal dynamics of the total number of infected people in the two communities (Z(t) = ∑_i[E_i(t) + P_i(t) + I_i(t) + A_i(t)]). Transmission chains fueled by the initial seeding of infected people decline rapidly over time. c Temporal dynamics of the system output, defined as the Euclidean norm of the vector whose components correspond to the infection subsystem (w^X = 1, with X ∈ {E, P, I, A}). All trajectories are characterized by a monotonic decline in the system output. d–f As in (a–c), for a parameter combination (\({\beta }_{1}^{P}={\beta }_{2}^{P}=4.2\times 1{0}^{-1}\) days⁻¹) resulting in \({{\mathcal{R}}}_{0}\, <\, 1\) and e₀ > 0. In this case too, all trajectories converge to the DFE (d), but disease prevalence exhibits a peak, later declining slowly over time (e). Also, for suitable initial conditions, a transitory increase of the system output following a pulse perturbation is possible (f). g–i As in (a–c), for a parameter combination (\({\beta }_{1}^{P}={\beta }_{2}^{P}=7.0\times 1{0}^{-1}\) days⁻¹) resulting in \({{\mathcal{R}}}_{0}\,> \, 1\) and e₀ > 0. In this case, trajectories exponentially diverge from the DFE (g), and a large outbreak is observed in both disease prevalence (h) and the system output (i). In these examples, the population size of the first community (N₁ = 10⁶) is twice as large as the size of the second (N₂ = 5 × 10⁵), and the people of the first community are less mobile than those of the second (\({M}_{12}^{S,E,P,A}=1/10\), \({M}_{21}^{S,E,P,A}=2/3\); symptomatic individuals are assumed not to move from either community, \({M}_{12}^{I}={M}_{21}^{I}=0\)). See “Methods” for details and Table 2 for other parameters.