SEAIR Epidemic spreading model of COVID-19

Abstract

We study Susceptible-Exposed-Asymptomatic-Infectious-Recovered (SEAIR) epidemic spreading model of COVID-19. It captures two important characteristics of the infectiousness of COVID-19: delayed start and its appearance before onset of symptoms, or even with total absence of them. The model is theoretically analyzed in continuous-time compartmental version and discrete-time version on random regular graphs and complex networks. We show analytically that there are relationships between the epidemic thresholds and the equations for the susceptible populations at the endemic equilibrium in all three versions, which hold when the epidemic is weak. We provide theoretical arguments that eigenvector centrality of a node approximately determines its risk to become infected.

Keywords: COVID-19; Complex networks; Eigenvector centrality; Epidemic spreading; Jacobian matrix eigenvectors; SEAIR epidemic model.

Fig. 1

Disease spreading on random regular graph and random graph with constant degree distribution.The curves represent the dependence of the number of susceptible individuals at the end of the epidemic on the parameter $\mu$. The meaning of the symbols is the following: orange stars – theoretical values from eq. for (28) for infinite-size random regular graph with node degree 50; blue diamonds – random regular graph with the same degree and 1000 nodes; red circles – random graph with uniform degree distribution in [30,70] and 1000 nodes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2

Discrete-time epidemic model on complex networks at the end of epidemic. In all panels ten different complex networks with 1000 nodes are considered. The seed of generating the BA networks, $m,$ and the link probability for ER networks $p_{\text{ER}}$ is given in the inset. In the top panels are shown the average number of susceptible individuals $〈S〉,$ while at bottom are average correlation coefficients $〈\rho 〉$ between the number of recovered individuals and the principal eigenvector of the respective adjacency matrix. The horizontal axis is given in units of the critical value of the parameter ${\mu }_0$ at the epidemic threshold which is calculated for each network separately.

Fig. 3

Evolution of correlation coefficient between the principal eigenvector of the adjacency matrix and the vector of probability of recovered state in ER (left panel) and BA (right panel) complex networks. The considered networks have 1000 nodes. Each curve is obtained by averaging ten networks with the same parameters and ten randomly chosen initially infected nodes for each network.