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A well-known characteristic of pandemics such as COVID-19 is the high level of transmission heterogeneity in the infection spread: not all infected individuals spread the disease at the same rate and some individuals (superspreaders) are responsible for most of the infections. To quantify this phenomenon requires the analysis of the effect of the variance and higher moments of the infection distribution. Working in the framework of stochastic branching processes, we derive an approximate analytical formula for the probability of an outbreak in the high variance regime of the infection distribution, verify it numerically and analyze its regime of validity in various examples. We show that it is possible for an outbreak not to occur in the high variance regime even when the basic reproduction number $R_0$ is larger than one and discuss the implications of our results for COVID-19 and other pandemics.
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Infectious diseases are caused by pathogenic microorganisms and can spread through different ways. Mathematical models and computational simulation have been used extensively to investigate the transmission and spread of infectious diseases. In other words, mathematical model simulation can be used to analyse the dynamics of infectious diseases, aiming to understand the effects and how to control the spread. In general, these models are based on compartments, where each compartment contains individuals with the same characteristics, such as susceptible, exposed, infected, and recovered. In this paper, we cast further light on some classical epidemic models, reporting possible outcomes from numerical simulation. Furthermore, we provide routines in a repository for simulations.
There are often multiple diseases with cross immunity competing for vaccination resources. Here we investigate the optimal vaccination program in a two-layer Susceptible-Infected-Removed (SIR) model, where two diseases with cross immunity spread in the same population, and vaccines for both diseases are available. We identify three scenarios of the optimal vaccination program, which prevents the outbreaks of both diseases at the minimum cost. We analytically derive a criterion to specify the optimal program based on the costs for different vaccines.
Covid-19 has caused hundred of thousands of deaths and an economic damage amounting to trillions of dollars, creating a desire for the rapid development of vaccine. Once available, vaccine is gradually produced, evoking the question on how to distribute it best. While official vaccination guidelines largely focus on the question to whom vaccines should be provided first (e.g. to risk groups), here we propose a strategy for their distribution in time and space, which sequentially prioritizes regions with a high local infection growth rate. To demonstrate this strategy, we develop a simple statistical model describing the time-evolution of infection patterns and their response to vaccination, for infectious diseases like Covid-19. For inhomogeneous infection patterns, locally well-mixed populations and basic reproduction numbers $R_0sim 1.5-4$ the proposed strategy at least halves the number of deaths in our simulations compared to the standard practice of distributing vaccines proportionally to the population density. For $R_0sim 1$ we still find a significant increase of the survival rate. The proposed vaccine distribution strategy can be further tested in detailed modelling works and could excite discussions on the importance of the spatiotemporal distribution of vaccines for official guidelines.
We extend the classical SIR model of infectious disease spread to account for time dependence in the parameters, which also include diffusivities. The temporal dependence accounts for the changing characteristics of testing, quarantine and treatment protocols, while diffusivity incorporates a mobile population. This model has been applied to data on the evolution of the COVID-19 pandemic in the US state of Michigan. For system inference, we use recent advances; specifically our framework for Variational System Identification (Wang et al., Comp. Meth. App. Mech. Eng., 356, 44-74, 2019; arXiv:2001.04816 [cs.CE]) as well as Bayesian machine learning methods.
We model and calculate the fraction of infected population necessary to reach herd immunity, taking into account the heterogeneity in infectiousness and susceptibility, as well as the correlation between those two parameters. We show that these cause the effective reproduction number to decrease more rapidly, and consequently have a drastic effect on the estimate of the necessary percentage of the population that has to contract the disease for herd immunity to be reached. We quantify the difference between the size of the infected population when the effective reproduction number decreases below 1 vs. the ultimate fraction of population that had contracted the disease. This sheds light on an important distinction between herd immunity and the end of the disease and highlights the importance of limiting the spread of the disease even if we plan to naturally reach herd immunity. We analyze the effect of various lock-down scenarios on the resulting final fraction of infected population. We discuss implications to COVID-19 and other pandemics and compare our theoretical results to population-based simulations. We consider the dependence of the disease spread on the architecture of the infectiousness graph and analyze different graph architectures and the limitations of the graph models.
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