Markovian epidemics in any graph have been mathematically extended to a fractional setting by replacing the ordinary differential operator with a Caputo fractional operator. The strength is that the fractional non-Markovian generalization is analytically solvable. Roughly speaking, exponential time dependencies are generalized to Mittag-Leffler functions in a scaling parameter alpha. If alpha = 1, we retrieve the Markovian regime. Since the Mittag-Leffler distribution is, for small and realistic times, close to a Weibull distribution, measurements from real data may determine the important scaling parameter that quantifies the deviation from Markovian theory. Current mean-field models, used during e.g. the COVID-19 pandemic, are deduced from Markovian epidemics. Generally, the epidemic threshold, which is more precise than the basic reproduction number R0, is very sensitive to perturbations with non-Markovian dynamics. As most of us still recall from the last pandemic, the basic reproduction number R0 was the major control parameter used by governmental agencies. Despites the beautiful mathematical analysis, the physics of fractional process extensions are still insufficiently understood: we cannot precisely simulate the processes to achieve the fractional results. This challenge stand as a main problem for future ViSioN research.
The second pilar concerns the human contact network, that is changing over time and is not a fixed graph. A first result, based on system's theory and periodicity, is able to produce any given graph sequence exactly, but does not lead to good predictions (due to the fact that one finite-in-time realization of the human contact process contains insufficient information to reproduce the entire process). A second result concerns an estimate of the boundary between the time-scale of the epidemic process and the time-scale of the human contact process. In particular, if the human contact process changes much slower than the epidemics, we can regard the graph as fixed and the classical theory applies. If both dynamics are comparable in time, then no obvious simplifications can be made. The boundary estimate determines the highest time-scale of the contact process that still can be regarded as fixed.
A third achievement, dealing with incomplete information, is integrating the epidemics in a network model into a Metropolis algorithm. Many epidemiological data are time series: incidence or prevalence tabulated on a daily, weekly, or monthly basis. Epidemiological data are often incomplete because they cover only a small proportion of the population or because the time resolution is too coarse. The Metropolis algorithm takes these incomplete data as inputs and gives as an output the probable contact graphs and infectivity of the virus. With these outputs, we can construct probabilistic forecasts about the future state of an epidemic in similar language to weather forecasts: e.g. a 90% chance that incidence will decrease this week. With sufficient information, we can even recover the true contact graph.
At last, we have started to explore a generative model for contact graphs, based on the notion of random walkers on a Markov graph. Its properties will be further explored.
