Complete dimensional collapse in the continuum limit of a delayed SEIQR network model with separable distributed infectivity.

Abstract

We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity and consider the continuum limit of the same with a simple separable interaction model for the infectivities beta . Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of beta in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in beta appears through an integral closely related to the moment generating function of u=sqrt{beta }. If the first few moments of u exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single beta . Even otherwise, the new scalar DDE can be solved using either numerical integration over u at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model and will hopefully lead to new analysis of continuum models for epidemics.