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Contagion, broadly construed, refers to anything that can spread infectiously from peer to peer. Examples include communicable diseases, rumors, misinformation, ideas, innovations, bank failures, and electrical blackouts. Sometimes, as in the 1918 Spanish flu epidemic, a contagion mutates at some point as it spreads through a network. Here, using a simple susceptible-infected (SI) model of contagion, we explore the downstream impact of a single mutation event. Assuming that this mutation occurs at a random node in the contact network, we calculate the distribution of the number of descendants, $d$, downstream from the initial Patient Zero mutant. We find that the tail of the distribution decays as $d^{-2}$ for complete graphs, random graphs, small-world networks, networks with block-like structure, and other infinite-dimensional networks. This prediction agrees with the observed statistics of memes propagating and mutating on Facebook, and is expected to hold for other effectively infinite-dimensional networks, such as the global human contact network. In a wider context, our approach suggests a possible starting point for a mesoscopic theory of contagion. Such a theory would focus on the paths traced by a spreading contagion, thereby furnishing an intermediate level of description between that of individual nodes and the total infected population. We anticipate that contagion pathways will hold valuable lessons, given their role as the conduits through which single mutations, innovations, or failures can sweep through a network as a whole.
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By incorporating delayed epidemic information and self-restricted travel behavior into the SIS model, we have investigated the coupled effects of timely and accurate epidemic information and peoples sensitivity to the epidemic information on contagion. In the population with only local random movement, whether the epidemic information is delayed or not has no effect on the spread of the epidemic. Peoples high sensitivity to the epidemic information leads to their risk aversion behavior and the spread of the epidemic is suppressed. In the population with only global person-to-person movement, timely and accurate epidemic information helps an individual cut off the connections with the infected in time and the epidemic is brought under control in no time. A delay in the epidemic information leads to an individuals misjudgment of who has been infected and who has not, which in turn leads to rapid progress and a higher peak of the epidemic. In the population with coexistence of local and global movement, timely and accurate epidemic information and peoples high sensitivity to the epidemic information play an important role in curbing the epidemic. A theoretical analysis indicates that peoples misjudgment caused by the delayed epidemic information leads to a higher encounter probability between the susceptible and the infected and peoples self-restricted travel behavior helps reduce such an encounter probability. A functional relation between the ratio of infected individuals and the susceptible-infected encounter probability has been found.
In this paper, we propose a continuous-time stochastic intensity model, namely, two-phase dynamic contagion process(2P-DCP), for modelling the epidemic contagion of COVID-19 and investigating the lockdown effect based on the dynamic contagion model introduced by Dassios and Zhao (2011). It allows randomness to the infectivity of individuals rather than a constant reproduction number as assumed by standard models. Key epidemiological quantities, such as the distribution of final epidemic size and expected epidemic duration, are derived and estimated based on real data for various regions and countries. The associated time lag of the effect of intervention in each country or region is estimated. Our results are consistent with the incubation time of COVID-19 found by recent medical study. We demonstrate that our model could potentially be a valuable tool in the modeling of COVID-19. More importantly, the proposed model of 2P-DCP could also be used as an important tool in epidemiological modelling as this type of contagion models with very simple structures is adequate to describe the evolution of regional epidemic and worldwide pandemic.
Albeit epidemic models have evolved into powerful predictive tools for the spread of diseases and opinions, most assume memoryless agents and independent transmission channels. We develop an infection mechanism that is endowed with memory of past exposures and simultaneously incorporates the joint effect of multiple infectious sources. Analytic equations and simulations of the susceptible-infected-susceptible model in unstructured substrates reveal the emergence of an additional phase that separates the usual healthy and endemic ones. This intermediate phase shows fundamentally distinct characteristics, and the system exhibits either excitability or an exotic variant of bistability. Moreover, the transition to endemicity presents hybrid aspects. These features are the product of an intricate balance between two memory modes and indicate that non-Markovian effects significantly alter the properties of spreading processes.
In this study, we develop the mathematical model to understand the coupling between the spreading dynamics of infectious diseases and the mobility dynamics through urban transportation systems. We first describe the mobility dynamics of the urban population as the process of leaving from home, traveling to and from the activity locations, and engaging in activities. We then embed the susceptible-exposed-infectious-recovered (SEIR) process over the mobility dynamics and develops the spatial SEIR model with travel contagion (Trans-SEIR), which explicitly accounts for contagions both during travel and during daily activities. We investigate the theoretical properties of the proposed model and show how activity contagion and travel contagion contribute to the average number of secondary infections. In the numerical experiments, we explore how the urban transportation system may alter the fundamental dynamics of the infectious disease, change the number of secondary infections, promote the synchronization of the disease across the city, and affect the peak of the disease outbreaks. The Trans-SEIR model is further applied to the understand the disease dynamics during the COVID-19 outbreak in New York City, where we show how the activity and travel contagion may be distributed and how effective travel control can be implemented with only limited resources. The Trans-SEIR model along with the findings in our study may have significant contributions to improving our understanding of the coupling between urban transportation and disease dynamics, the development of quarantine and control measures of disease system, and promoting the idea of disease-resilient urban transportation networks.
The ongoing COVID-19 pandemic has created a global crisis of massive scale. Prior research indicates that human mobility is one of the key factors involved in viral spreading. Indeed, in a connected planet, rapid world-wide spread is enabled by long-distance air-, land- and sea-transportation among countries and continents, and subsequently fostered by commuting trips within densely populated cities. While early travel restrictions contribute to delayed disease spread, their utility is much reduced if the disease has a long incubation period or if there is asymptomatic transmission. Given the lack of vaccines, public health officials have mainly relied on non-pharmaceutical interventions, including social distancing measures, curfews, and stay-at-home orders. Here we study the impact of city organization on its susceptibility to disease spread, and amenability to interventions. Cities can be classified according to their mobility in a spectrum between compact-hierarchical and decentralized-sprawled. Our results show that even though hierarchical cities are more susceptible to the rapid spread of epidemics, their organization makes mobility restrictions quite effective. Conversely, sprawled cities are characterized by a much slower initial spread, but are less responsive to mobility restrictions. These findings hold globally across cities in diverse geographical locations and a broad range of sizes. Our empirical measurements are confirmed by a simulation of COVID-19 spread in urban areas through a compartmental model. These results suggest that investing resources on early monitoring and prompt ad-hoc interventions in more vulnerable cities may prove most helpful in containing and reducing the impact of present and future pandemics.
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