Estimating the parameters of epidemic spread on two-layer random graphs: a classical and a neural network approach

Abstract

In this paper, we propose and analyse various methods for estimating the infection parameter τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} in an SIR (susceptible-infected-recovered) epidemic spread model simulated on two-layer random graphs with a preferential attachment component. The statistics that can be used for the estimates is the number of susceptible and infected vertices. Our classical approach is based on the maximum likelihood method, while the machine learning approach uses a graph neural network (GNN). The underlying graph has a layer consisting of disjoint cliques (representing e.g. households) and a random second layer with a preferential attachment structure. Our simulation study reveals that the estimation is poorer at the beginning of the epidemic, for larger preferential attachment parameter of the graph, and for larger τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}. As for the neural networks, we find that dense networks offer better training datasets. Moreover, GNN perfomance is measured better using the l2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$l_2$$\\end{document} loss function rather than cross-entropy.